**7** MARK GEORG SCHIEBL

Thermodynamically consistent space-time discretization of nonisothermal mechanical systems in the framework of GENERIC

MARK GEORG SCHIEBL

EME consistent space-time discretization of non-isothermal mechanical systems

Mark Georg Schiebel

## **Thermodynamically consistent space-time discretization of non-isothermal mechanical systems in the framework of GENERIC**

## **Schriftenreihe des Instituts für Mechanik Karlsruher Institut für Technologie (KIT)**

**Band 7**

Herausgeber: Prof. Dr.-Ing. habil. Peter Betsch Prof. Dr.-Ing. habil. Thomas Seelig

Eine Übersicht aller bisher in dieser Schriftenreihe erschienenen Bände finden Sie am Ende des Buches.

# **Thermodynamically consistent space-time discretization of non-isothermal mechanical systems in the framework of GENERIC**

by Mark Georg Schiebl

Karlsruher Institut für Technologie Institut für Mechanik

Thermodynamically consistent space-time discretization of non-isothermal mechanical systems in the framework of GENERIC

Zur Erlangung des akademischen Grades eines Doktor-Ingenieurs von der KIT-Fakultät für Bauingenieur-, Geo- und Umweltwissenschaften des Karlsruher Instituts für Technologie (KIT) genehmigte Dissertation

von Mark Georg Schiebl, M.Sc.

Tag der mündlichen Prüfung: 16. April 2021 Referent: Prof. Dr.-Ing. habil. Peter Betsch Korreferent: Prof. Dr.-Ing. habil. Michael Groß

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ISSN 2363-4936 ISBN 978-3-7315-1117-5 DOI 10.5445/KSP/1000136011

This thesis is dedicated to my grandfather Georg Schiebl.

# Abstract

The present thesis addresses the design of structure-preserving numerical methods that emanate from the general equation for non-equilibrium reversible-irreversible coupling (GENERIC) formalism.

For conservative mechanical systems, the GENERIC reduces to the purely Hamiltonian case. Considering this special case rst, novel energy-momentum (EM) consistent timestepping schemes in the realm of molecular dynamics are introduced that take periodic boundary conditions, three-body potentials and interatomic functional potentials into account. This method is thermodynamically consistent because it preserves the energy and the symmetries of the system, independent of the time-step size.

Afterwards, thermodynamical systems are considered in the framework of the GENERIC, and a new variational formulation for large-strain thermoelasticity is proposed. This variational formulation emanates from the GENERIC framework and allows for the free choice of the thermodynamic state variable among three options: (i) the absolute temperature, (ii) the internal energy density, or (iii) the entropy density. In this context the notion "GENERIC-consistent space discretization" is introduced, facilitating the design of novel energy-momentum-entropy (EME) consistent schemes. This method is thermodynamically consistent in the sense that the solution strictly obeys the rst and second law of thermodynamics and the symmetries of such a system, independent of the size of the time-step.

Finally, large-strain thermo-viscoelasticity is considered in the context of the GENERIC framework. A mixed nite element approach for the discretization in space is proposed that incorporates a GENERIC-consistent space discretization. Depending on the choice of the thermodynamic state variable, the plain mid-point rule already yields partially structure-preserving schemes on its own. Thus, this newly developed GENERICbased weak form is particularly well suited for the design of structure-preserving methods. In all cases, numerical investigations are presented that conrm the theoretical ndings and shed light on the numerical stability of the newly developed schemes.

Keywords: structure-preserving space-time integration, GENERIC, molecular dynamics, periodic boundary conditions, interatomic potentials, thermoelasticity, thermo-viscoelasticity, nite element method, nite deformation

# Zusammenfassung

Die vorliegende Dissertation behandelt die Entwicklung von strukturerhaltenden numerischen Methoden, welche aus dem GENERIC (Akkronym für general equation for non-equilibrium reversible-irreversible coupling) Formalismus hervorgehen.

Für konservative mechanische Systeme reduziert sich das GENERIC auf die hamiltonische Beschreibung. Zunächst werden für diesen Spezialfall neuartige energy-momentum (EM) konsistente Zeitschrittverfahren für den Bereich molekulardynamischer Systeme eingeführt, welche periodische Randbedingungen, Dreikörperpotentiale und interatomare Funktionalpotentiale berücksichtigen. Eine derartige Methode ist thermodynamisch konsistent, da sie die Energie und die Symmetrien einer solchen Struktur unabhängig von der Zeitschrittgröße bewahrt.

Anschließend wird der Fall von thermodynamischen Systemen im GENERIC Formalismus betrachtet und eine neue Variationsformulierung für die Thermoelastizität unter Berücksichtigung großer Deformationen vorgeschlagen. Diese Variationsformulierung entstammt dem GENERIC-Gerüst und ermöglicht die freie Wahl der thermodynamischen Zustandsvariable unter folgenden drei Optionen: (i) der absoluten Temperatur, (ii) der inneren Energiedichte, oder (iii) der Entropiedichte. In diesem Zusammenhang wird der Begri der "GENERIC-konsistenten räumlichen Diskretisierung" eingeführt, welcher den Entwurf neuartiger energy-momentum-entropy (EME) konsistenter Schemata ermöglicht. Eine solche vorgeschlagene Methode ist thermodynamisch konsistent, da die Lösung streng dem ersten und zweiten Hauptsatz der Thermodynamik und den Symmetrien eines solchen Systems folgt, unabhängig von der Größe des Zeitschritts.

Schließlich wird die Thermo-Viskoelastizität unter Berücksichtigung großer Deformationen im Kontext des GENERIC-Gerüsts betrachtet. Für die Diskretisierung im Raum wird ein Ansatz mit gemischten niten Elementen vorgeschlagen, welcher sich einer GENERIC-konsistenten räumlichen Diskretisierung bedient. Abhängig von der Wahl der thermodynamischen Variable liefert die einfache Mittelpunktregel bereits teilweise strukturerhaltende Schemata. Damit eignet sich die neuartige GENERIC-basierte schwache Form ideal für die Entwicklung strukturerhaltender Verfahren. In allen Fällen werden numerische Untersuchungen vorgestellt, die die theoretischen Erkenntnisse bestätigen und Aufschluss über die numerische Stabilität der neu entwickelten Verfahren geben.

Schlüsselwörter: Strukturerhaltende Raum-Zeit Integration, GENERIC, Molekulardynamik, periodische Randbedingungen, interatomare Potentiale, Thermoelastizität, Thermo-Viskoelastizität, Finite Elemente Methode, große Deformationen

# Acknowledgements

Over the past six years, I have had the privilege of working as a research associate and doctoral candidate at the Institute for Mechanics (IFM) of the Karlsruhe Institute of Technology (KIT). Financial support for this research was provided by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) within the Project BE 2285/13-1 entitled "Thermodynamisch konsistente Diskretisierungsverfahren für thermomechanisch gekoppelte Probleme" (Projectnumber 388118188) which I gratefully acknowledged.

First and foremost, I want to express my deepest gratitude to my supervisor Prof. Dr.- Ing. habil. Peter Betsch for oering me this great opportunity and for his generous support, which he has given me to the greatest extent. Since my time as a master's degree candidate he has been an exceptional mentor in all respects and his passion for mechanics and numerical methods has inspired and motivated me ever since. Without his support throughout numerous discussions and his invaluable guidance this work could not have been possible and I am very grateful for that.

My special thank also goes to Prof. Dr.-Ing. habil. Michael Groß for acting as the second examiner of this thesis.

Further, I wish to express my gratitude to all my colleagues and friends at the institute, who supported me in both research and teaching over the last years. In particular, I would like to thank Dr.-Ing. Marlon Franke for our successful collaboration and especially for sharing his knowledge in Linux with me. A very special thank you goes to Dr.-Ing. Alexander Janz for his support in research and teaching, the countless tips and advice I received from him, and the many after-work dart sessions we shared. Further, I would like to thank the colleagues and members of the "IFM running club": Jonas Hund, Tobias Laschütza, Robin Pfeerkorn, Vanessa Susana Valdes y Beck and Paul Wasmer not only for their irreplaceable support in research and teaching but also for pushing my limits on the tracks near the University. Furthermore, I would like to thank Timo Ströhle, who accompanied me on almost every walk to the bakery "La Mintbrueck" and for the numerous meaningful conversations we shared. Next, I would like to thank Simeon Schneider for all the intense and insightful discussions, which we could build up starting almost always from simple mechanical and mathematical considerations.

Additionally, I want to thank my other colleagues Julian Bauer and Moritz Hille and former colleagues Michael Strobl and Friedemann Streich for their support and contribution to the pleasant working atmosphere at the institute. Moreover, I would like to thank Gabriele Herrmann and Rosemarie Krikis for their administrative assistance and guidance and

Klaus Neidhardt for his technical support. A special thanks goes to Florian Schrammer for his unconditional support and his good-hearted spirit.

Acknowledgment is also given to Prof. Dr.-Ing. habil. Christian Hesch for his helpful advice and technical support on the ESRA-project. In this regard, I would like to thank Dr.-Ing. Maik Dittmann, Dr.-Ing. Melanie Krüger, Jonathan Schulte, Stefan Schuß and Felix Schmidt for the fruitful collaboration on this project.

Further, I would like to pay my special regards to Prof. Dr. Ignacio Romero. I am very grateful to have had the opportunity to work with him and his group for six months during my research stay in Madrid. From the beginning I was welcomed with open arms and all the discussions and his invaluable insights on mechanics and beyond made my stay a memorable experience. The support of the Karlsruhe House of Young Scientists (KHYS) for the stay abroad is gratefully acknowledged.

Of course, I cannot forget friends who went through hard times but also celebrated each accomplishment together: Marc Galley, Fabian Haas, Geza Hildenbrand, Dr.-Ing. Sven Nagel, Gustav Strauß, Dr.-Ing. David Walter, to name a few of them.

Finally, and most importantly, I would like to express my deepest gratitude towards my entire family. I must thank my parents, Erma Schiebl and Dr.-Ing. Hans Schiebl, and my sister Amira Schiebl for their continuous support, their unconditional love and their endless patience they have given me throughout my life. I am forever thankful to them for giving me the opportunities and experiences that have made me who I am.

Karlsruhe, 2021 Mark Schiebl

# Contents





# List of Figures






# List of Tables


# 1. Introduction<sup>1</sup>

Mathematical models are extensively used by engineers to gain insight into the quantitative and structural behavior of a system. Although simple systems can be understood and analyzed more easily, complexity arises when a more accurate representation of reality is required, necessitating the replacement of pencil and paper by large computer simulations in favor of more powerful design tools.

At the heart of every dynamical computer simulation lies the numerical method, also known as the "integrator", which constructs an approximation of the exact solution by providing discrete snapshots of the system over time for a given problem. Today, driven by the exponential growth of computational power, the qualitative properties of the integrator itself become vital to the success of the simulation, as longer simulation durations require more stable numerical methods. While the equations of motion for a dynamic system may "hide" important properties of the physical system, such as conservation laws, these properties are not inherited by the numerical method in general, which can cause unphysical time evolution and thus introduces a source for numerical instabilities. One feasible approach that counters this physical inconsistency is the use of subtle numerical methods that are numerically accurate and capable of reliably capturing the main characteristics of the underlying physical system. This motivated the development of structure-preserving numerical methods, also known as "geometric integrators", for a wide array of elds in applied and theoretical science. Geometric integrators are designed to respect the fundamental physics of the underlying mathematical model, ranging from time reversibility to rst integrals, by preserving the geometric properties in a discrete setting. Most attention has been focused hereby on Hamiltonian or Lagrangian systems due to their well-understood mathematical (geometric) structure. Based on the common underlying structure, structure-preserving methods can be easily generalized from simple systems such as spring-mass arrays to complex nonlinear elastodynamical structures. Therefore, reports of excellent long-term behavior and numerical stability for a huge variety of systems have laid the foundation for the success of this method.

<sup>1</sup> This chapter is based on the introductions given in [1–4]

## 1.1. Structure-preserving integration

Structure-preserving numerical methods have played a major role in the development of numerical time integration methods for decades, and monographs such as [5–7] beautifully demonstrate the advancement of this eld. While numerical integrators have traditionally been viewed solely as an approximation to a time-continuous problem, geometric integrators consider the integration scheme itself as a discrete dynamical system with its own characteristics, including balance laws, symmetries, symplecticity, reversibility, and many more. Even though it was developed within a dierent context, the well-known mid-point rule can be viewed as a member of this general class because it preserves the symmetry and the symplecticity.

Dierent classes of structure-preserving numerical methods can be distinguished by the features inheritated from the continuous problem and the applied numerical techniques. The earliest developments of structure-preserving numerical methods can be traced back to symplectic methods, rst introduced in [8]. Early schemes of this kind were explicit symplectic integrators [9] and have been applied to various problems of celestial mechanics (see, e.g., [10, 11]), followed by implicit schemes [12, 13] based on [14]. Later, a symplectic family of Runge-Kutta schemes were constructed independently in [15, 16].

For mechanical systems, most researchers have focused their attention on momenta, energy, and symplectic structures. Unfortunately, simultaneously preserving the momenta, energy and symplecticity for a xed time-step method is not possible (for more details, see [17]). Therefore, many geometric integrators in the realm of Lagrangian and Hamiltonian mechanics can be gathered around two classes: variational integrators and energy-momentum (EM) integrators.

## 1.1.1. Variational integrators

A more contemporary approach towards symplectic methods is based on a variational nature and is thus termed "variational integrator". Variational integrators preserve the symplecticity and the momenta of the underlying system and, due to its variational nature, allows the extension to non-conservative systems by utilizing the discrete counterpart of the Lagrange-d'Alembert principle. The origin of this method can be traced back to [18] and [19, 20]. Based on those concepts, a theory of discrete Lagrangian and Hamiltonian mechanics was provided in [21], which has been applied to mechanical problems with multi-symplectic geometry [22, 23], dynamical systems evolving on nonlinear manifolds [24, 25], structural elements [26–28], contact and impact problems [29], multibody dynamics and control [30, 31], stochastic dierential equations [32], constrained and forced problems [33] and dynamic of uids [34] (see, e.g., [35] for more examples).

Sharp phase portraits and long numerical simulations demonstrate the strong performance of this method, especially in explicit calculations. Attempts have been made to express thermoelasticity in the framework of Lagrangian/Hamiltonian mechanics by introducing the concept of thermal displacement [36]. Unfortunately, expressing Fourier's law of heat conduction within this framework remains an unsolved issue [35] 2 . In addition, the loss of symplecticity and thus the loss of one key feature of variational integrators is a consequence of dissipation. For that matter, it seems that structure-preserving numerical methods for thermodynamical systems t better within the energy-momentum integrator approach, which has been preferentially chosen for this work. Nevertheless, besides the preservation of the symplectic structure and the momenta, variational integrators are known for their improved long-term energy behavior<sup>3</sup> compared to classical methods and are excellent numerical time-stepping schemes.

## 1.1.2. Energy-momentum integrators

EM algorithms are an additonal, frequently used class of structure-preserving integration schemes which, in contrast to variational integrators (symplectic-momentum integrators), preserve the momenta and the energy of the system in the discrete setting. This method has its roots in [40–44], whereas the rst energy-momentum scheme for nonlinear elastodynamics was proposed in [45]. Based on these previous reports, a systematic approach using the discrete gradient operator was developed in [46]. This second-order accurate operator ensures the preservation of the energy and the momenta by design and, due to its systemized construction, its remarkable robustness, and its good qualitative accuracy, has made the energy-momentum integrator a popular choice for simulating the governing equations of particle dynamics [40, 41], rigid bodies [43], nonlinear solid mechanics [45, 47, 48], nonlinear shells [49–51] and rods [52–54] multi-body dynamics [55, 56], gradient systems [57], and general PDEs [58]. For a comprehensive overview of energy-momentum integrators in the context of Hamiltonian dynamics, see [59].

## 1.1.3. Dissipation

The conservation of energy is not always a desired feature of Hamiltonian or Lagrangian systems because occasionally there is a need for dissipation in resolving high-frequency modes, which is often related to the spatial discretization of the problem. These "spurious" high frequencies are not present in real systems due to physical damping. However, physical damping is seldom addressed in the mathematical modeling and is usually compensated for by introducing controllable numerical dissipation schemes. The rst well-known representatives of such schemes can be found in the family of the Newmark method [60], the

<sup>2</sup> Eorts have been made in [37] to extend the Hamiltonian to account for the second sound phenomenon by adding a dissipative entropy-ux perturbation term and utilizing the Lagrange-D'Alembert principle. A similar approach has been followed for Fourier-type heat conduction in [38, 39]. It is not clear to the author if and how this adiabatic method can be generalized for arbitrary thermodynamical processes.

<sup>3</sup> For conservative systems, variational integrators capture the evolution of a "perturbated" Hamiltonian exactly. This modied energy-level set is, most likely, closely related to the true energy-level set and explains the characteristic energy oscillation of this method.

HHT method developed in [61], or the Wilson-θ method [62] among other methods that have been successfully implemented in common software packages. Unfortunately, these classical dissipation schemes, being developed in the linear regime, lose their dissipative character in the nonlinear regime [63]. In that sense, several energy-decaying numerical methods have been proposed [64–67].

Nevertheless, the philosophy of the current work is to incorporate physical dissipative processes into the mathematical model instead of considering numerical dissipation schemes. Most real structures can only be idealized as Hamiltonian or Lagrangian systems, yet are of relevance in many areas of science and engineering.

Polymeric solids and rubber-like materials are a perfect example of structures that exhibit viscoelastic material features and are highly temperature sensitive, thus being subjected to viscous dissipation and dissipation due to the conduction of heat. Consequently, a large class of problems involving dissipative mechanisms can not be expressed within the framework of Hamiltonian or Lagrangian mechanics, including, for example, the aforementioned Fourier-type heat conduction, which arises in a wide range of industrial applications, including the following:


Unfortunately, structure-preserving integrators for thermodynamical systems have received little attention, one reason being the lack<sup>4</sup> of a well-understood geometric structure guiding the construction of such numerical method. Numerous attempts have been made to extend structure-preserving time-stepping schemes to the domain of non-conservative mechanical systems, such as port-Hamiltonian systems [68], viscoelasticity [69, 70], elastoplasticity [71], thermoelasticity [72, 73], thermo-viscoelasticity [74–76], isothermal and non-isothermal ber-reinforced continuas [77–80]. Unlike the Lagrangian/Hamiltonian case, these structure-preserving schemes do not emerge from a unifying theory.

However, the metriplectic structure of the general equation for non-equilibrium reversibleirreversible coupling (GENERIC) formalism provides a double-generator framework that expresses thermomechanical models of dissipative materials (or generalized standard materials) in a unifying formalism. Its formulation is based on an additive decomposition of the time-evolution equations into a reversible part and a dissipative part. It can be seen as a natural extension of Hamiltonian mechanics to the dissipative regime (see, e.g., [81] for a relation to port-Hamiltonian systems).

Originally developed in the context of complex uids [82, 83], GENERIC has been applied to a vast range of problems (predominantly in the domain of complex uids), such as the reptation model for entangled linear polymers [84], polymer blends [85], colloidal

<sup>4</sup> At least to the computational mechanics' community

suspensions [86], two-phase systems [87], relativistic hydromechanics [88], discrete formulations of hydromechanics for simulations [89], thermodynamically guided simulations [89], diusion through polymeric membranes [90], rheological model of suspension of red blood cells [91], structure-preserving neural networks [92], and the theory of quantum systems [93], to mention only a few examples. We refer to the book by Öttinger [94] for a comprehensive account of the GENERIC formalism up to the year 2005 and to [95] for a huge list of advanced developments up to the year 2017.

Focusing on thermodynamic models for non-isothermal solids, as one of the main themes of the present work, an early application of the GENERIC formalism to nite-strain thermoelasticity is credited to [96], albeit in an Eulerian setting which is quite uncommon in solid mechanics, and to [97] for continuum damage mechanics. Later, the Lagrangian setting was used in [98, 99] and [100] to develop the GENERIC framework for non-isothermal solid mechanics. Moreover, [98, 99] preferred to use the absolute temperature as the thermodynamic state variable, while in [100] a special form of GENERIC is devised that makes the free choice of the thermodynamic state variable possible.

In the eld of computational solid mechanics, Romero [101, 102] recognized at an early stage the great potential of the GENERIC framework for the design of structure-preserving time-stepping schemes. Since the GENERIC framework automatically ensures the thermodynamic admissibility of the time-evolution equations, it provides an ideal starting point for the development of thermodynamically consistent (TC) integrators. The solutions of this integrator could be classied as thermodynamically guided simulations, in analogy to the thermodynamically guided simulation of consistent coarse-graining schemes (e.g., [103, 104]), as TC integrators comply with the rst and the second law of thermodynamics, independent of the size of the time-step. Therefore, TC integrators may also be termed "energy-entropy" integrators.

If TC integrators also respect the symmetries of the underlying mechanical system, they can be viewed as an extension to the dissipative regime of Energy-Momentum (EM) integrators previously developed for Hamiltonian systems with symmetry. The GENERIC framework also facilitates a concise characterization of momentum maps and associated conservation properties (see [105]).

Hence, the GENERIC formalism provides a common structure for non-conservative systems, similarly to Hamiltonian dynamics for conservative problems, and can therefore be used in a similar manner to guide the design of structure-preserving numerical schemes for thermodynamical problems. Consequently, GENERIC provides a solid theoretical foundation for the design of energy-momentum-entropy (EME) methods, as has been shown in [102] for nite-strain thermo-elasticity and in [106] for nite-strain thermoviscoelasticity.<sup>5</sup>

<sup>5</sup> Of course, the GENERIC framework is not a prerequisite for the development of structure-preserving numerical methods for non-isothermal solid mechanics. In the context of coupled thermomechanical problems, alternative procedures have been proposed, such as in [72], [75], [76] and [73].

## 1.2. Objectives

In this work, we aim to formulate, analyze, implement, and verify second-order structurepreserving numerical methods by employing the GENERIC structure. The present work can be divided into two parts:


## 1.2.1. Energy-momentum conserving integration schemes for molecular dynamics

Given the favorable properties of EM methods, it is remarkable that they have not received more attention in the eld of molecular dynamics, at least when investigating the use of implicit time integration schemes for obtaining numerical solutions. The governing equations of the latter are essentially Hamiltonian and t seamlessly in the framework developed since the 1970's for integrating these types of problems, while preserving the energy and the momenta. It would seem natural that integration schemes designed to preserve the main invariants of the motion would give accurate predictions of the thermodynamic averages, which are of interest in many practical and theoretical situations [107–111], but have rarely been studied [112]. Instead, molecular dynamics codes seem to favor the use of the explicit Verlet method or symplectic methods, mainly due to their smaller computational cost as compared with implicit schemes. Although popular explicit methods have desirable properties in terms of computational cost, accuracy, and geometry preservation, they lack energy conservation, a key invariant that is most important in the simulation of microcanonical ensembles. A thorough investigation of the accuracy of energy and momentum conserving schemes in capturing the statistical behavior of atomistic systems for long periods of simulation is lacking. Preliminary results [112] are promising, but much testing and validation is still required.

The development and implementation of energy and momentum conserving algorithms in the context of molecular dynamics has three specic issues that aect the discretization of the equations and their analysis—issues that do not appear in their application to nonlinear solids, shells, rods, etc., or any of the other systems for which the use of these methods is widespread. The rst critical issue is the treatment of periodic boundary conditions. These are almost invariably required for the study of average properties in particle systems [110, 111, 113], and demand a careful analysis, especially to ascertain whether they spoil the conserving properties of the method or not. Taking this into account requires to consider the geometry and topology of the periodic conguration space, and might also aect the accuracy of the integration scheme.

The second issue that needs to be carefully dealt with is the use of conserving schemes in the context of three-body potentials. These functions are employed in modeling angle interactions in atomic bonding [114, 115], and their impact on the global behavior of some systems is so critical that it must to be accounted for. In fact, atomic systems with potentials of this type allow for large relative motions among the particles, and this is precisely the area where conserving schemes have shown their superiority with regard to other implicit integrators.

The third aspect that requires a detailed analysis is the application of conserving schemes to mechanical systems in which the potential is based on cluster functionals [116–120]. These eective potentials are often required for the correct modeling of complex binding among metallic atoms and again require a careful study when used in combination with conserving schemes. While pair potentials of the Lennard-Jones type [121] have been employed together with EM conserving schemes [41], their formulation for cluster potentials needs to be specically addressed.

Here, we formulate energy and momentum conserving schemes for simulating the dynamics of atomic systems. Some of the methods discussed have already been employed in the literature, and we identify new ones. In all cases, we explain how the three critical issues identied before (i.e., periodic boundary conditions, three-body potentials and functional potentials) aect their formulation. To the author's knownledge, none of these have been previously studied.

## 1.2.2. Structure-preserving integration of large-strain thermo-viscoelasticity in the framework of GENERIC

The aforementioned GENERIC-based numerical methods are subject to two serious limitations: the rst limitation is related to the use of the entropy density as the thermodynamic state variable, and the second limitation is due to the fact that the GENERIC formulation focuses on closed systems and thus does not account for boundary conditions. Of course, both points limit the applicability of numerical methods for the solution of initial boundary value problems in thermo-mechanics.

The rst limitation has been addressed in [122, 123] by using the absolute temperature as the thermodynamic state variable in the underlying GENERIC formulation. In the context of discrete systems, the free choice of the thermodynamic state variable has been recently addressed in [124]. The second limitation has been circumvented in [106] by applying the Lagrange multiplier method to enforce temperature boundary conditions.

We completely resolve the aforementioned limitations of GENERIC-based numerical methods for nite-strain thermo-elasticity by proposing a novel GENERIC-based variational formulation that makes the free choice of the thermodynamic state variable possible. In particular, one may choose (i) the internal energy density, (ii) the entropy density, or (iii) the absolute temperature as the thermodynamic state variable. Moreover, boundary conditions are taken into account by applying a generalized GENERIC formulation for open systems.

This novel, GENERIC-based variational formulation is discretized in time using the wellknown mid-point rule. Depending on the choice of the thermodynamic state variable, partially structure-preserving schemes are obtained. For example, choosing the internal energy density as the thermodynamic state variable leads to an EM scheme. On the other hand, choosing the entropy density as the thermodynamic state variable yields a momentum-entropy (ME) scheme. However, despite their partially structure-preserving properties, all of the mid-point type schemes turn out to be prone to numerical instabilities. These observations led to the conjecture that only fully structure-preserving schemes guarantee numerical stability for dissipative systems in the same way that EM schemes do for Hamiltonian systems.

In this context, the GENERIC-based weak form provides an ideal starting point for the development of EME schemes. First, the GENERIC-based weak form is discretized in space, resulting in a GENERIC-consistent space discretization. Then, the semi-discrete system is discretized in time by applying the partitioned discrete gradient operator in the sense of [125] leading to three dierent EME schemes. These EME schemes satisfy a specic Lyapunov-type stability estimate and thus do not exhibit any numerical instabilities. Hence, we develop novel EME schemes for large-strain thermoelasticity.

Finally, we extend the applicability of this method to the realm of thermo-viscoelasticity. In particular, we aim at a material formulation of isotropic large-strain thermo-viscoelasticity which makes the free choice of the thermodynamic state variable possible. To this end, we build on previous work by [99], who laid the theoretical basis for the GENERIC description of thermo-viscoelasticity.

## 1.3. Overview

The thesis is organized into seven chapters. Each chapter can be read independently—therefore, repetitions are inevitable.<sup>6</sup>

Chapter 1 covers the motivation, the contemporary state of structure-preserving methods, and the objectives of this work.

<sup>6</sup> A consistent notation throughout the present work would result in a tiresome reading experience, but notational shifts between the chapters are kept to a minimum depending on the focus therein.

Chapter 2 introduces the GENERIC framework briey. In Section 2.1 the framework is presented for nite dimensional systems. This includes its relation to Hamilton's equations of motion. Furthermore, we introduce the notation of GENERIC-consistent space discretization. Then, in Section 2.2, the structure of the building blocks of the GENERIC formalism is discussed when transformed to a dierent set of variables.

Chapter 3 addresses the formulation and analysis of EM conserving time integration schemes in the context of particle dynamics, and in particular atomic systems. In Section 3.1, we review the basic topology of periodic systems in order to clearly dene the distance function. Section 3.2 introduces particle dynamics, with special attention to its Hamiltonian structure. Next, in Section 3.3, conserving time integration schemes are presented for particle systems in periodic domains, restricted to those with pair potentials. These are extended to systems with angle potentials in Section 3.4 and to atomic systems with functional potentials in Section 3.5.

Chapter 4 proposes a new GENERIC-based variational formulation for nite-strain thermo-elastodynamics. First, the GENERIC framework is dealt within bracket form for closed systems. Therefore, in Section 4.1, we start with nite-strain elastodynamics and subsequently present in Section 4.2 its extension to thermo-elastodynamics. Then, in Section 4.3, the transition to open systems is performed. In the following Section 4.4, the resulting GENERIC-based weak form of the initial boundary value problem (IBVP) at hand is then discretized in time and space. Representative numerical examples are presented in Section 4.5.

Chapter 5 designs novel EME methods for nite-strain thermo-elastodynamics. Section 5.1 briey recapitulates the variational formulation of large-strain thermo-elasticity. Then, in Section 5.2, the GENERIC-based weak form is discretized in space, resulting in a GENERIC-consistent space discretization. In addition to that, the main balance laws to be preserved under discretization are outlined. In Section 5.3, the semi-discrete system is further discretized in time, leading to three alternative EME schemes. Section 5.4 contains representative numerical examples that conrm both the structure-preserving features and the enhanced numerical stability of the newly developed EME schemes when compared to the standard time integration schemes developed in Chapter 4.

Chapter 6 introduces in Section 6.1 the bracket form of the GENERIC formalism for thermo-viscoelasticity with heat conduction. In particular, in Section 6.2, a new material version of the inelastic dissipative bracket is proposed which allows for the free choice of the thermodynamic state variable. In Section 6.3, a mixed nite element approach is proposed that yields a GENERIC-consistent semi-discrete form of the evolution equations. The mid-point type discretization in time is dealt with in Section 6.4, leading to alternative partially structure-preserving schemes. Numerical investigations are presented in Section 6.5.

Chapter 7 summarizes the ndings. The thesis closes by drawing conclusions and giving an outlook on interesting research topics that follow the direction of this work.

# 2. A brief introduction to the GENERIC framework

The general equation for non-equilibrium reversible-irreversible coupling framework (GENERIC) is a double-generator formalism that expresses thermomechanical models of dissipative materials (or generalized standard materials) in a unifying framework. Its formulation is based on an additive decomposition of the time-evolution equations into a reversible part and a dissipative part. While the reversible part is generated by the total energy of the system, the irreversible part is generated by the total entropy.

The GENERIC framework is typically presented in two alternative forms: an operator version (i) in which the entries of the operator matrices are either generalized functions or dierential operators, and (ii) a bracket version. Moreover, the GENERIC formulation typically focuses on closed systems. That is, neither thermal nor mechanical interactions with the surrounding of the system are considered. This Chapter summarizes the GENERIC framework for nite dimensional systems, which will be repeatedly referred to throughout the following chapters. For a comprehensive account of the GENERIC framework see, e.g., [94].

The fundamental concept of the GENERIC framework for nite dimensional systems also applies (with corresponding modications) to innitesimal dimensional systems, which will be considered in Chapters 4 to 6. In this connection the notion "GENERIC consistent space discretization" is introduced, which facilitates the design of energy-momentumentropy (EME) consistent schemes. For now we will consider the case of isolated systems, for which the GENERIC framework was originally developed for. In the case of isolated (or closed) systems, boundary eects are considered to be irrelevant.

## 2.1. GENERIC framework for nite dimensional systems

The GENERIC framework hinges on an additive decomposition of the evolution equations into reversible and irreversible parts. Assuming that the state of a nite-dimensional system is characterized by the state vector <sup>z</sup>(t) <sup>∈</sup> <sup>R</sup> <sup>n</sup>, the time-evolution of the discrete system is described by

$$\frac{d\mathbf{z}}{dt} = \mathbf{L}\partial\_\mathbf{z}\mathcal{E} + \mathbf{M}\partial\_\mathbf{z}\mathcal{S} \,, \tag{2.1}$$

which corresponds to the operator representation (i). In eq. (2.1) reversible processes are generated by the total energy E(z), while dissipative processes are generated by the total entropy S(z). In this connection, L(z) is the Poisson matrix, which must be skew-symmetric, whereas M(z) is the dissipative matrix, which must be symmetric and positive semi-denite. Two degeneration (or non-interaction) conditions have to hold. Namely,

$$\mathbf{L}\partial\_{\mathbf{z}}\mathcal{S} = \mathbf{0}\,,\tag{2.2}$$

and

$$\mathbf{M}\partial\_{\mathbf{z}}\mathcal{E} = \mathbf{0} \,. \tag{2.3}$$

Using the GENERIC formulation (2.1) guarantees that the following two fundamental properties of a closed thermomechanical system are satised. Firstly, in accordance with the rst law of thermodynamics, the total energy is conserved. That is,

$$\frac{d\mathcal{E}}{dt} = \partial\_{\mathbf{z}} \mathcal{E} \cdot \frac{d\mathbf{z}}{dt} = \partial\_{\mathbf{z}} \mathcal{E} \cdot \mathbf{L} \partial\_{\mathbf{z}} \mathcal{E} + \partial\_{\mathbf{z}} \mathcal{E} \cdot \mathbf{M} \partial\_{\mathbf{z}} \mathcal{S} = 0 \,. \tag{2.4}$$

Note that on the right-hand side of the last equation, the skew-symmetry of the Poisson matrix has been used along with degeneration condition (2.3). Secondly, in compliance with the second law of thermodynamics, the total entropy should be a non-decreasing function of time. In fact, using (2.1) leads to

$$\frac{d\mathcal{S}}{dt} = \partial\_{\mathbf{z}} \mathcal{S} \cdot \frac{d\mathbf{z}}{dt} = \partial\_{\mathbf{z}} \mathcal{S} \cdot \mathbf{L} \partial\_{\mathbf{z}} \mathcal{S} + \partial\_{\mathbf{z}} \mathcal{S} \cdot \mathbf{M} \partial\_{\mathbf{z}} \mathcal{S} \ge 0 \,. \tag{2.5}$$

Here, degeneration condition (2.2) has been used, along with the positive semi-deniteness of the dissipative matrix. The properties of L and M can also be conveniently discussed in terms of two brackets

$$\begin{aligned} \{\mathcal{A}, \mathcal{B}\} &= \partial\_{\mathbf{z}} \mathcal{A} \cdot \mathbf{L} \partial\_{\mathbf{z}} \mathcal{B} \,, \\\ [\mathcal{A}, \mathcal{B}] &= \partial\_{\mathbf{z}} \mathcal{A} \cdot \mathbf{M} \partial\_{\mathbf{z}} \mathcal{B} \,. \end{aligned} \tag{2.6}$$

Hereby are A(z) and B(z) arbitrary functions of the state variables. The brackets are termed Poisson and dissipative bracket, respectively. Using eq. (2.1) these brackets lead to the following evolution equation for an arbitrary function A

$$\frac{\mathrm{d}\mathcal{A}}{\mathrm{d}t} = \left\{ \mathcal{A}, \mathcal{E} \right\} + \left[ \mathcal{A}, \mathcal{S} \right] \,, \tag{2.7}$$

which corresponds to bracket version (ii) of the GENERIC framework. The antisymmetry property of the Poisson matrix can now be stated as

$$\{\mathcal{A}, \mathcal{B}\} = -\{\mathcal{B}, \mathcal{A}\}\,. \tag{2.8}$$

Further, the Poisson bracket has to satisfy the product (or Leibniz) rule and the Jacobi identity. Those properties are well-known from the Poisson bracket of classical mechanics and express the essence of reversible dynamics, see e.g., [126].

The symmetry of the dissipative matrix M can be formulated in terms of the dissipative bracket as

$$\left[\mathcal{A},\mathcal{B}\right] = \left[\mathcal{B},\mathcal{A}\right].\tag{2.9}$$

The positive semi-deniteness can be expressed as

$$\left[\mathcal{A},\mathcal{A}\right] \geq 0 \,. \tag{2.10}$$

Finally, the two degeneration conditions can be restated in terms of the bracket representation

$$\begin{aligned} \{\mathcal{A}, \mathcal{S}\} &= 0 \,, \\ [\mathcal{A}, \mathcal{E}] &= 0 \, . \end{aligned} \tag{2.11}$$

## 2.1.1. Connection to Hamilton's equations of motion

If the system under consideration is purely mechanical and conservative, the choice

z = x 1 , . . . , x <sup>N</sup> , p 1 , . . . , p N ,

where x a is the a-th position, p a the <sup>a</sup>-th momentum and <sup>N</sup> <sup>∈</sup> <sup>Z</sup>+, results in a zero dissipative matrix M and a Poisson matrix L of the form

$$\mathbf{L(z)} = \left[ \cfrac{-\mathbf{I}}{\mathbf{0}} \Bigg| \begin{array}{c} \mathbf{I} \\ \hline \mathbf{0} \\ \hline \end{array} \right], \tag{2.12}$$

where 0 is the zero and I is the identity matrix with appropriate dimension depending on the system. By this means the state vector z has to be viewed as a column vector. The previous matrix is the well-known canonical symplectic matrix. In this case the evolution equation of the GENERIC coincide with Hamilton's equation of motion, which will be considered in Chapter 3.

## 2.1.2. GENERIC-consistent space discretization

If an innite dimensional system at hand is discretized in space in such a way that the GENERIC structure for nite dimensional systems is preserved, then we speak of a GENERIC-consistent space discretization. Such a method is devised in Chapters 5 and 6 of the present work.

## 2.2. Transformation of variables

By reasons of taste or convenience for the choice of the specic variables, it is important to address the relation between the building blocks of the GENERIC for dierent sets of state variables. We therefore consider a one-to-one transformation z 7→ z 0 . As for the scalar energy and entropy generators E and S we assume a simple transformation behavior

$$\begin{aligned} \mathcal{E}'(\mathbf{z}') &= \mathcal{E}(\mathbf{z}(\mathbf{z}')) \; , \\ \mathcal{S}'(\mathbf{z}') &= \mathcal{S}(\mathbf{z}(\mathbf{z}')) \; . \end{aligned} \tag{2.13}$$

Applying a transformation of variables in eq. (2.1) using eq. (2.13) results in the following transformation laws for the Poisson and the dissipative matrix

$$\begin{aligned} \mathbf{L}'(\mathbf{z}') &= \partial\_\mathbf{z} \mathbf{z}' \cdot \mathbf{L}(\mathbf{z}(\mathbf{z}')) \cdot \partial\_\mathbf{z} \mathbf{z}'^\mathsf{T}, \\ \mathbf{M}'(\mathbf{z}') &= \partial\_\mathbf{z} \mathbf{z}' \cdot \mathbf{M}(\mathbf{z}(\mathbf{z}')) \cdot \partial\_\mathbf{z} \mathbf{z}'^\mathsf{T}. \end{aligned} \tag{2.14}$$

Considering that

$$\begin{aligned} \mathcal{A}'(\mathbf{z}') &= \mathcal{A}(\mathbf{z}(\mathbf{z}')) \; , \\ \mathcal{B}'(\mathbf{z}') &= \mathcal{B}(\mathbf{z}(\mathbf{z}')) \; , \end{aligned} \tag{2.15}$$

holds, then the transformation laws eq. (2.14) lead to

$$\{\mathcal{A}', \mathcal{B}'\} = \partial\_{\mathbf{z}'} \mathcal{A}' \cdot \mathbf{L}' \partial\_{\mathbf{z}'} \mathcal{B}' = (\partial\_{\mathbf{z}} \mathcal{A} \cdot \mathbf{L} \partial\_{\mathbf{z}} \mathcal{B}) \vert\_{\mathbf{z} = \mathbf{z}(\mathbf{z}')} = \{\mathcal{A}, \mathcal{B}\} \vert\_{\mathbf{z} = \mathbf{z}(\mathbf{z}')},\tag{2.16}$$

and

$$[\mathcal{A}', \mathcal{B}'] = \partial\_{\mathbf{z}'} \mathcal{A}' \cdot \mathbf{M}' \partial\_{\mathbf{z}'} \mathcal{B}' = (\partial\_{\mathbf{z}} \mathcal{A} \cdot \mathbf{M} \partial\_{\mathbf{z}} \mathcal{B}) \mid\_{\mathbf{z} = \mathbf{z}(\mathbf{z}')} = [\mathcal{A}, \mathcal{B}] \mid\_{\mathbf{z} = \mathbf{z}(\mathbf{z}')} \quad. \tag{2.17}$$

In other terms: the Poisson bracket and the dissipative bracket are independent of the particular choice of state variables.

The time evolution of the system is invariant with respect to the choice of a dierent set of variables. However, the building blocks of the GENERIC formalism, the matrices L, M and the gradients ∂zE, ∂zS may have a simpler form, which can be of advantage for the formulation of a time-stepping integration scheme. Further, as an one-to-one transformation preserves the brackets, the transformations can be very useful when considering the verication of the Jacobi identity (see, e.g., [94]).

# 3. Energy-momentum conserving integration schemes for molecular dynamics<sup>1</sup>

In this chapter we study the dynamic motion of systems of particles. This type of problems is of great importance for the simulation of matter at the atomic scale and a very large body of references study details pertaining to their numerical solution and the information that can be extracted from these simulations.

Such systems can be considered as a prototype of a classical mechanical system: N particles in a periodic domain. The state vector of such a system assumes the form

$$\mathbf{z} = \begin{pmatrix} \mathbf{x}^1, & \dots, & \mathbf{x}^N, \mathbf{p}^1, & \dots, & \mathbf{p}^N \end{pmatrix},$$

to be introduced in the subsequent. Hereby the state vector has to be viewed as a column vector. As only reversible processes are present, the GENERIC evolution eq. (2.1) boils down to Hamilton's equations of motion

$$\frac{d\mathbf{z}}{dt} = \mathbf{L}\partial\_\mathbf{z}\mathcal{E}\,\,,\tag{3.1}$$

where L is the symplectic matrix eq. (2.12), or equivalently the equations of motion in Poisson bracket form

$$\frac{d\mathcal{A}}{dt} = \{\mathcal{A}, \mathcal{E}\}\,. \tag{3.2}$$

Therefore, solely the total energy E (which is the Hamiltonian H in this case) in connection with the Poisson bracket {·, ·} describes the evolution of such systems, where the Poisson bracket is given by

$$\{\mathcal{A}, \mathcal{B}\} = \sum\_{a=1}^{N} \left( \partial\_{\mathbf{x}^{a}} \mathcal{A} \cdot \partial\_{\mathbf{p}^{a}} \mathcal{B} - \partial\_{\mathbf{p}^{a}} \mathcal{A} \cdot \partial\_{\mathbf{x}^{a}} \mathcal{B} \right) \,, \tag{3.3}$$

which is the well known Poisson bracket of classical mechanics. Depending on the specic structure of the total energy, dierent interatomic interactions can be modelled. Further,

<sup>1</sup> This chapter is based on [3]

the GENERIC/Hamiltonian framework guides the construction of structure-preserving numerical methods, as we will show in the following sections.

We address the formulation and analysis of energy and momentum conserving time integration schemes and identify three critical aspects of these models that demand a careful analysis when discretized: (i) the treatment of periodic boundary conditions, (ii) the formulation of approximations of systems with three-body interaction forces, and (iii) their extension to atomic systems with functional potentials. These issues, and in particular their interplay with energy-momentum (EM) integrators, are studied in detail. Novel expressions for these time integration schemes are proposed and numerical examples are given to illustrate their performance.

## 3.1. The topology of periodic domains

Many particle systems of interest are formulated in periodic domains. These allow to study large systems by only discretizing representative volumes, much smaller in size, while hopefully not losing too much information. In this section we gather some topological and geometrical facts of periodic domains that will be necessary to analyze numerical methods.

We start by considering a periodic three-dimensional box B of side length L, noting that all the results are applicable to systems in one and two dimensions, with the corresponding modications. This box is isomorphic to the torus T 3 (see e.g., [108, 110]), which itself can be identied with the product manifold S <sup>1</sup> <sup>×</sup> <sup>S</sup> <sup>1</sup> <sup>×</sup> <sup>S</sup> 1 . Hence, each point <sup>ξ</sup> <sup>∈</sup> <sup>T</sup> 3 can be uniquely characterised by three angles (α, β, γ) and the complete manifold is covered by a single chart.

Numerical methods dened on T <sup>3</sup> pose diculties that can be alleviated by mapping this set into a more convenient one. For that, let us rst dene the following equivalence relation on R 3 : two points <sup>x</sup>, <sup>y</sup> <sup>∈</sup> <sup>R</sup> 3 are dened to be equivalent, and indicated as x ∼ y, if there exists a triplet of integers <sup>z</sup> <sup>∈</sup> <sup>Z</sup> 3 such that x = y + Lz. Using this equivalence relation, we can dene the quotient space P := R <sup>3</sup>/Z 3 that is homeomorphic to the torus. In what follows the equivalent class of a point <sup>x</sup> <sup>∈</sup> <sup>R</sup> <sup>3</sup> will be denoted as [x] <sup>∈</sup> <sup>P</sup>.

When dealing with systems of particles in periodic domains one has to choose one of the two homeomorphic descriptions described above, namely, the torus and P. From the computational point of view, employing the latter has many advantages. The rst one is that given a distance on R 3 , this quotient space naturally inherits a distance, and thus a topology. In terms of the standard Euclidean distance <sup>d</sup>(·, ·) : <sup>R</sup> <sup>3</sup> <sup>×</sup> <sup>R</sup> 3 7→ <sup>R</sup> <sup>+</sup> ∪ {0} we can dene <sup>d</sup><sup>T</sup> (·, ·) : <sup>P</sup> <sup>×</sup> <sup>P</sup> 7→ <sup>R</sup> <sup>+</sup> ∪ {0} by the relation

$$d\_T([\mathbf{x}], [\mathbf{y}]) = \inf\_{\mathbf{x} \in [\mathbf{x}], \mathbf{y} \in [\mathbf{y}]} d(\mathbf{x}, \mathbf{y}) \,. \tag{3.4}$$

Abusing slightly the notation, from this point on we will write d<sup>T</sup> (x, y) instead of d<sup>T</sup> ([x], [y]). The second advantage of such a choice is that, for each equivalent class [x], there exists a unique point <sup>x</sup>¯ <sup>∈</sup> [x] <sup>∩</sup> [−L/2, L/2)<sup>3</sup> that serves as identier of the whole class which, in practical terms, implies that all operations need to be performed as with standard points in a cubic box. This identier can be found using a projection operator

$$
\pi : \mathbb{R}^3 \mapsto \mathcal{B} \,, \tag{3.5}
$$

dened as

$$
\bar{x}\_j = \pi(\mathbf{x})\_j = x\_j - \mathbf{f} \mathbf{1} \mathbf{color} \left( \frac{x\_j + L/2}{L} \right) L \,, \tag{3.6}
$$

where <sup>x</sup><sup>j</sup> , j = 1, <sup>2</sup>, <sup>3</sup> denote the Cartesian coordinates of the point <sup>x</sup> and floor : <sup>R</sup> 7→ <sup>Z</sup> is the function that gives the largest integer smaller than or equal to a given real number. See Fig. 3.1 for an illustration of the projection operator.

Figure 3.1.: Graph of the projection operator π restricted to one of the three coordinates of Euclidean space.

The projection map π determines some of the properties and limitations of the numerical methods employed on systems with periodic boundary conditions, and it is helpful to summarize some of its properties:


$$\pi(\mathbf{x}) = \arg\inf\_{\mathbf{x} \in [\mathbf{x}]} d(\mathbf{x}, \mathbf{0})\,. \tag{3.7}$$

3. In general, for arbitrary <sup>x</sup>, <sup>y</sup> <sup>∈</sup> <sup>R</sup> 3 ,

$$d\_T(\mathbf{x}, \mathbf{y}) = |\pi(\mathbf{x} - \mathbf{y})| \neq |\pi(\mathbf{x}) - \pi(\mathbf{y})|\ . \tag{3.8}$$

4. The map <sup>π</sup> is <sup>C</sup><sup>∞</sup> except on the planes <sup>x</sup><sup>i</sup> <sup>=</sup> L/2+kiL, with <sup>k</sup><sup>i</sup> <sup>∈</sup> <sup>Z</sup> and <sup>i</sup> = 1, <sup>2</sup>, <sup>3</sup>, where it is discontinuous. Away from these planes, the gradient ∂xπ is the identity I : R 3 7→ <sup>R</sup> 3 . A one-dimensional illustration of the gradient ∂xπ is given in Fig. 3.2.

Figure 3.2.: Graph of the gradient of the projection operator ∂xπ restricted to one of the three coordinates of Euclidean space.

5. For any <sup>x</sup> <sup>∈</sup> <sup>R</sup> 3

$$
\pi(\mathbf{x}) = -\pi(-\mathbf{x})\,.\tag{3.9}
$$

6. For any <sup>x</sup>, <sup>y</sup> <sup>∈</sup> <sup>R</sup> 3 and <sup>a</sup> <sup>∈</sup> <sup>R</sup> 3

$$d\_T(\mathbf{x} + \mathbf{a}, \mathbf{y} + \mathbf{a}) = d\_T(\mathbf{x}, \mathbf{y})\,. \tag{3.10}$$

However, in general, if Q ∈ SO(3),

$$d\_T(\mathbf{x}, \mathbf{y}) \neq d\_T(\mathbf{Q} \mathbf{x}, \mathbf{Q} \mathbf{y}) \,. \tag{3.11}$$

Hence, in contrast with the Euclidean distance, the function d<sup>T</sup> (·, ·) is not invariant under the action of the special orthogonal group.

In this section we provide the basic ingredients that describe the dynamics of particulate systems. This dynamical system is governed by Hamilton's equations of motion, and most of its complexity comes from the particle interactions, as given by the potential energy. Here we present a fairly general class of potentials that will be examined more carefully in Sections 3.3, 3.4, and 3.5.

## 3.2. Particle dynamics: basic description

## 3.2.1. System description

Let us consider a system of N particles labeled a = 1, 2, . . . , N moving inside a periodic box <sup>B</sup>. Let the mass of the <sup>a</sup>-th particle be denoted as <sup>m</sup><sup>a</sup> , its position as x a , and its velocity as v <sup>a</sup> = x˙ a , where the dot indicates the derivative with respect to time.

A system of particles such as the one introduced possesses a kinetic energy dened by

$$T := \sum\_{a=1}^{N} \frac{1}{2} m^a |\mathbf{v}^a|^2 \,. \tag{3.12}$$

Introducing the momentum p <sup>a</sup> = m<sup>a</sup>v <sup>a</sup> of particle a the kinetic energy reads

$$T = \sum\_{a=1}^{N} \frac{1}{2} \frac{1}{m^a} |\mathbf{p}^a|^2,\tag{3.13}$$

and the potential energy is given by

$$V = \hat{V}(\{\mathbf{x}^a\}\_{a=1}^N)\,,\tag{3.14}$$

modeling the energetic interactions among all the particles. This potential energy is always a model of the true interatomic interaction and as such there exists a large number of simple eective potentials that have proven their value in dierent contexts (gases, uids, organic molecules, metals, etc.).

## 3.2.2. Equations of motion

The dynamics of systems of particles may be expressed in the GENERIC format which is governed by

$$\frac{d\mathbf{z}}{dt} = \mathbf{L}\partial\_\mathbf{z}\mathcal{E}\,\,,\tag{3.15}$$

where the total energy E = T + V is given as the sum of kinetic and potential energy. As no irreversible eects are present the GENERIC framework coincide with Hamilton's equation of motion (see Section 2.1.1). Eq. (3.15) gives rise to the following evolution equation for the state variables

$$\begin{split} \dot{\mathbf{x}}^a &= \frac{1}{m^a} \mathbf{p}^a \,, \\ \dot{\mathbf{p}}^a &= -\partial\_{\mathbf{x}^a} \hat{V} (\{\mathbf{x}^b\}\_{b=1}^N) \, . \end{split} \tag{3.16}$$

19

The gradient of the potential energy is related to the forces acting on the particles, and we dene

$$\mathbf{f}^a := -\partial\_{\mathbf{x}^a} \hat{V}(\{\mathbf{x}^b\}\_{b=1}^N)\,,\tag{3.17}$$

to be the resultant of all forces applied on particle a.

These standard equations need to be carefully studied since the topology of B is not identical to that of Euclidean space and the notion of derivative has to be re-examined. For the moment being, let us assume that this object is well-dened, deferring until Section 3.3 a more detailed inspection.

## 3.2.3. Conserved quantities

Eqs. (3.16) describe the motion of systems of particles that often possess rst integrals, that is, conserved quantities along their trajectories. These quantities are of great relevance to understand the qualitative dynamics of the system, to develop controls, etc. These momentum maps are related to the symmetries of the equations, according to Noether's theorem (see, e.g., [126]). We review them very briey, since they have a direct impact in the formulation of conserving schemes.

We consider only potential energies with translational invariant, that is, functions Vˆ such that for every vector <sup>c</sup> <sup>∈</sup> <sup>R</sup> 3 satisfy

$$
\hat{V}(\{\mathbf{x}^a\}\_{a=1}^N) = \hat{V}(\{\mathbf{x}^a + \mathbf{c}\}\_{a=1}^N) \,. \tag{3.18}
$$

Dierentiating both sides of this equation with respect to c and setting later c = 0 we obtain the relation

$$\mathbf{0} = \sum\_{a=1}^{N} \partial\_{\mathbf{x}^{a}} \hat{V}(\{\mathbf{x}^{b}\}\_{b=1}^{N}) = -\sum\_{a=1}^{N} \mathbf{f}^{a} \,. \tag{3.19}$$

This invariance condition is related to the conservation of the linear momentum of the system, dened as

$$\mathbf{L} = \sum\_{a=1}^{N} \mathbf{p}^{a} \,. \tag{3.20}$$

The time derivative of this quantity follows from the denition of particle momentum and eq. (3.19):

$$\dot{\mathbf{L}} = \sum\_{a=1}^{N} \dot{\mathbf{p}}^a = \sum\_{a=1}^{N} \mathbf{f}^a = \mathbf{0}\,. \tag{3.21}$$

Systems of particles moving in the Euclidean space R <sup>n</sup>, with n = 2 or 3, often conserve angular momentum. This is a consequence of the rotational invariance of the potential energy. However, systems dened on a periodic domain do not preserve it, in general (see, e.g., [127, 128]). One way to explain this loss of symmetry is by noting that the projection (3.6) is not rotationally invariant (see eq. (3.11)) and thus when used in the denition of the potential energy, it spoils the invariance of the whole system.

The total energy of the system is given as the sum of kinetic energy T and potential energy V . The time derivative of the energy can be evaluated using the equations of motion (3.16), giving

$$\dot{\mathcal{E}} = \sum\_{a=1}^{N} m^a \mathbf{v}^a \cdot \dot{\mathbf{v}}^a - \sum\_{a=1}^{N} \mathbf{f}^a \cdot \dot{\mathbf{x}}^a = \sum\_{a=1}^{N} \mathbf{f}^a \cdot \mathbf{v}^a - \sum\_{a=1}^{N} \mathbf{f}^a \cdot \dot{\mathbf{x}}^a = 0 \,, \tag{3.22}$$

proving that the total energy must be a rst integral of the motion. Given the relevance of the aforementioned conservation laws, numerical schemes have been proposed that attempt to preserve them. In particular, energy-momentum schemes, the ones under study in this Chapter, have been designed to integrate the equations of Hamiltonian systems while preserving both linear and angular momentum, in addition to the total energy. From the previous discussion, however, it follows that when dealing with periodic systems, one might focus on the preservation of linear momentum and energy, only.

## 3.2.4. The hierarchical denition of the potential energy

The potential energy of a system of particles is a function with the general form given in eq. (3.14) satisfying the invariance condition (3.18). A hierarchy of functions of growing complexity can be dened considering interactions involving an increasing number of particles. This is abstractly expressed as

$$V = V\_0 + \sum\_{\substack{a,b=1\\a \neq b}}^N V\_2(\mathbf{x}\_a, \mathbf{x}\_b) + \sum\_{\substack{a,b,c=1\\a \neq b \neq c}}^N V\_3(\mathbf{x}\_a, \mathbf{x}\_b, \mathbf{x}\_c) + \dots \, , \tag{3.23}$$

where V<sup>k</sup> is a function involving k−tuples of atoms and satisfying eq. (3.18). The formulation of accurate potentials is an active eld of research and we limit our exposition to the most common types. The reader may consult standard references for a detailed motivation and derivation of other types (e.g., [120]).

A convenient way of formulating potential functions that are translationally invariant is to include atomic interactions only via the distance between pairs of particles. In general, this would mean that the functions V<sup>k</sup> employed in eq. (3.23) must be of the form

$$\begin{aligned} V\_2(\mathbf{x}\_a, \mathbf{x}\_b) &= \tilde{V}\_2(d(\mathbf{x}\_a, \mathbf{x}\_b)) \;, \\ V\_3(\mathbf{x}\_a, \mathbf{x}\_b, \mathbf{x}\_c) &= \tilde{V}\_3(d(\mathbf{x}\_a, \mathbf{x}\_b), d(\mathbf{x}\_b, \mathbf{x}\_c), d(\mathbf{x}\_c, \mathbf{x}\_a)) \;, \end{aligned} \tag{3.24}$$

and similarly for higher order terms.

## 3.2.5. Dynamics in periodic domains

Eqs. (3.16) dene the motion of particles in periodic domains, but special care has to be taken with the denition of the potential energy and its derivative.

With respect to the potential, we note that, when formulating the dynamics of particles in periodic domains, the distance function d(·, ·) in eq. (3.24) should be replaced with the distance d<sup>T</sup> (·, ·) dened in eq. (3.4).

An aspect with important practical implications is that hierarchical potential functions of the form (3.23) are invariably dened employing a cut-o radius that eectively limits the number of particles that interact with those within that distance, in the sense of d<sup>T</sup> (·, ·). Moreover, in order to avoid the singularities in the denition of the gradient of this distance, the cut-o radius is always chosen to be strictly smaller than L/2 (see Fig. 3.2). Equivalently, the dimension L of the periodic box must be selected larger than twice the cut-o radius. Under this condition, we observe that a collection of N particles in a periodic box B is a mathematical representation of an innite domain consisting of boxes of dimension L × L × L that repeat themselves in the three directions of space.

The formulation of energy and momentum conserving schemes in this kind of domains must take these two remarks into consideration, and we explore them in the following sections, starting from the simplest potential function possible.

In this section we study the formulation of energy and momentum conserving algorithms for systems of particles in periodic domains where the potential energy includes only pairwise interactions. In terms of practical applications, only the simplest potentials belong to this class (for example, Lennard-Jones'). They are only accurate for modeling noble gases, but are very often employed for benchmarking and the study of numerical methods.

## 3.3. Energy-momentum methods for periodic systems with pairwise interactions

## 3.3.1. Equations of motion

We consider again a system of N particles in a periodic box B of side L with equations of motion (3.16) and an eective potential

$$V = \frac{1}{2} \sum\_{\substack{a,b=1\\b \neq a}}^{N} \tilde{V} \left( d\_T(\mathbf{x}^a, \mathbf{x}^b) \right) \,, \tag{3.25}$$

where V˜ : R <sup>+</sup> ∪ {0} 7→ <sup>R</sup>. Using the denitions

$$
\bar{\mathbf{r}}^{ab} := \pi(\mathbf{x}^b - \mathbf{x}^a) \qquad \text{and} \qquad \bar{r}^{ab} = d\_T(\mathbf{x}^a, \mathbf{x}^b) := |\bar{\mathbf{r}}^{ab}| \, , \tag{3.26}
$$

the forces (3.17) deriving from a pairwise potential can be written as

$$\mathbf{f}^{a} = \sum\_{\substack{b=1\\a \neq b}}^{N} \mathbf{f}^{ab}, \qquad \text{with} \qquad \mathbf{f}^{ab} = \bar{V}'(\bar{r}^{ab}) \frac{\bar{\mathbf{r}}^{ab}}{\bar{r}^{ab}} \,. \tag{3.27}$$

For the following sections we further dene

$$\mathbf{r}^{ab} := \mathbf{x}^b - \mathbf{x}^a \qquad \text{and} \qquad r^{ab} = d(\mathbf{x}^a, \mathbf{x}^b) = |\mathbf{r}^{ab}| \,. \tag{3.28}$$

## 3.3.2. Time discretization

We consider now the integration in time of the equations of motion (3.16) of a system in the periodic box B and eective potential (3.25). To approximate their solution we will employ implicit time stepping schemes that partition the integration interval [0, T] into disjoint subintervals [tn, tn+1] with t<sup>n</sup> = n∆t, and ∆t being the time step size, assumed to be constant to simplify the notation. In the algorithms dened below, we will use the notation x<sup>n</sup> to denote the approximation to x(tn), and similarly for the velocity. Moreover, the symbol fn+<sup>α</sup> will denote the convex combination (1 − α)f<sup>n</sup> + αfn+1 for any variable f and 0 ≤ α ≤ 1.

#### 3.3.2.1. Mid-point scheme

The canonical mid-point rule approximates the equations of motion (3.16) by the implicit formula

$$\frac{\mathbf{x}\_{n+1}^a - \mathbf{x}\_n^a}{\Delta t} = \mathbf{v}\_{n+1/2}^a \,,$$

$$m^a \frac{\mathbf{v}\_{n+1}^a - \mathbf{v}\_n^a}{\Delta t} = \sum\_{\substack{b=1 \\ a \neq b}}^N \mathbf{f}\_{\text{MP}}^{ab} \,. \tag{3.29}$$

Here we introduced f ab MP, the mid-point approximation of the force acting on particle a due to the presence of particle b, that is,

$$\mathbf{f}\_{\rm MP}^{ab} := \tilde{V}'(\bar{r}\_{n+1/2}^{ab}) \frac{\bar{\mathbf{r}}\_{n+1/2}^{ab}}{|\bar{\mathbf{r}}\_{n+1/2}^{ab}|}\,, \tag{3.30}$$

with

$$\begin{aligned} \bar{\mathbf{r}}\_{n+1/2}^{ab} &= \pi(\mathbf{x}\_{n+1/2}^{b} - \mathbf{x}\_{n+1/2}^{a}) \; , \\ \bar{r}\_{n+1/2}^{ab} &= d\_T(\mathbf{x}\_{n+1/2}^{a}, \mathbf{x}\_{n+1/2}^{b}) \; . \end{aligned} \tag{3.31}$$

As in the continuous case, the condition

$$\mathbf{f}\_{\rm MP}^{ab} = -\mathbf{f}\_{\rm MP}^{ba}\,,\tag{3.32}$$

holds due to eq. (3.9). The properties of the mid-point rule are well known. For example, this method preserves the total liner momentum of the system, dened at an instant t<sup>n</sup> as

$$\mathbf{L}\_n = \sum\_{a=1}^N m^a \mathbf{v}\_n^a \,. \tag{3.33}$$

To prove this property it suces to verify

$$\frac{\mathbf{L}\_{n+1} - \mathbf{L}\_n}{\Delta t} = \sum\_{a=1}^{N} m^a \frac{\mathbf{v}\_{n+1}^a - \mathbf{v}\_n^a}{\Delta t} = \sum\_{\substack{a,b=1\\a \neq b}}^{N} \mathbf{f}\_{\text{MP}}^{ab} = \mathbf{0} \,, \tag{3.34}$$

where we have employed eq. (3.32).

#### 3.3.2.2. Energy and momentum conserving discretization

It is possible to construct a perturbation of the mid-point rule that, in addition to preserving the linear momentum of the system, preserves its total energy. The key ingredient of such methods is the so called discrete gradient operator, an approximation to the gradient that guarantees the strict conservation of energy and momentum along the discrete trajectories generated by the integrator.

Conserving integrators for problems in molecular dynamics have been studied since the 1970's [40, 41, 46], although never for systems with periodic boundary conditions, with the exception of [112]. In all of these works, the conserving schemes are variations of the mid-point rule (3.29) of the form

$$\begin{aligned} \frac{\mathbf{x}\_{n+1}^a - \mathbf{x}\_n^a}{\Delta t} &= \mathbf{v}\_{n+1/2}^a, \\ m^a \frac{\mathbf{v}\_{n+1}^a - \mathbf{v}\_n^a}{\Delta t} &= -\mathbf{D}\_{\mathbf{x}^a} V, \end{aligned} \tag{3.35}$$

where Dx<sup>a</sup> is precisely the discrete gradient operator, an algorithmic approximation to the derivative ∂<sup>x</sup> <sup>a</sup> . In analogy to expression (3.30), the discrete gradient denes a force contribution, to be specied later, such that

$$\mathbf{D}\_{\mathbf{x}^{a}}V = -\sum\_{\substack{b=1\\a\neq b}}^{N} \mathbf{f}\_{\text{algo}}^{ab} \,. \tag{3.36}$$

If the following condition holds

$$\mathbf{f}\_{\text{algo}}^{ab} = \mathbf{f}\_{\text{MP}}^{ab} + \mathcal{O}(\Delta t^2) \,, \tag{3.37}$$

the second order accuracy of the mid-point rule will be preserved. If we want the new method to preserve linear momentum we note from the previous section that it suces that the pairwise forces f ab algo mimic the symmetry condition (3.32), that is,

$$\mathbf{f}\_{\text{algo}}^{ab} = -\mathbf{f}\_{\text{algo}}^{ba} \,. \tag{3.38}$$

Any second order perturbation of f ab MP with this property will result in a second order accurate integrator that preserves linear momentum. The "classical" EM method is constructed in such a way, and preserves, in addition to the total energy, the linear and angular momenta of the system, the latter being important in domains without periodic boundary conditions [46]. For problems in molecular dynamics interacting through pair potentials and posed on periodic domains, the "classical" EM method is based on the discrete gradient (3.36) with

$$\mathbf{f}\_{\text{algo}}^{ab} = \mathbf{f}\_{\text{algo}}^{ab}(\mathbf{r}\_n^{ab}, \mathbf{r}\_{n+1}^{ab}) := \frac{\tilde{V}\_{n+1}^{ab} - \tilde{V}\_n^{ab}}{\bar{r}\_{n+1}^{ab} - \bar{r}\_n^{ab}} \frac{\bar{\mathbf{r}}\_{n+1}^{ab} + \bar{\mathbf{r}}\_n^{ab}}{\bar{r}\_{n+1}^{ab} + \bar{r}\_n^{ab}} \,, \tag{3.39}$$

where we have introduced the notation

$$
\tilde{V}\_n^{ab} = \tilde{V}(\bar{r}\_n^{ab})\,, \tag{3.40}
$$

and where the appropriate limit must be taken in eq. (3.39) when |r¯ ab <sup>n</sup>+1 − r¯ ab n | 7→ 0. This form of the discrete gradient is frequently cited in the literature of integration algorithms (see, e.g., [112]) and is responsible for the conservation properties of the method, also in periodic domains. However, it is not the only possible form and, in fact, it can be shown that there are an innite number of discrete gradients [129], some of which can be more easily extended to the periodic case. More precisely, a second class of EM schemes follows from a new denition of the algorithmic approximation of the pairwise forces:

$$\mathbf{f}\_{\text{algo}}^{ab}(\mathbf{r}\_{n}^{ab}, \mathbf{r}\_{n+1}^{ab}) := \mathbf{f}\_{\text{MP}}^{ab} + \frac{\tilde{V}\_{n+1}^{ab} - \tilde{V}\_{n}^{ab} + \mathbf{f}\_{\text{MP}}^{ab} \cdot (\mathbf{r}\_{n+1}^{ab} - \mathbf{r}\_{n}^{ab})}{|\mathbf{r}\_{n+1}^{ab} - \mathbf{r}\_{n}^{ab}|} \text{ n} \,, \tag{3.41}$$

where n is the normalized direction given by

$$\mathbf{n} = \frac{\mathbf{r}\_{n+1}^{ab} - \mathbf{r}\_n^{ab}}{|\mathbf{r}\_{n+1}^{ab} - \mathbf{r}\_n^{ab}|} \,. \tag{3.42}$$

The expression (3.41) results from perturbing the force f ab MP in the direction that produces work and then imposing precisely that this work should coincide with the dierence V˜ ab <sup>n</sup>+1 <sup>−</sup> <sup>V</sup>˜ ab n , an idea that has been previously employed in the literature in order to formulate EM methods [129]. The proposed denition corrects the pairwise force between the particles a and b with a "small" term in the direction n depending on unprojected relative positions. This is the result of the fact that the rst equation in (3.35) is not posed in the quotient space but rather in the full R 3 . It might be argued that such a correction is nonphysical because it is not dened on the quotient space, where the problem is posed. While this is true, the velocity equation is not posed on this space from the outset, and the proposed correction results from this mismatch.

The EM force given in eq. (3.41) is symmetric in a and b. Hence, the method dened by (3.35) and (3.41) preserves linear momentum. To show that the method indeed preserves exactly the total energy exactly, it suces to take the dot product of the (3.35)<sup>1</sup> with the left hand side of (3.35)2, and vice versa, and then add the result over all particles, that is,

$$\begin{split} \sum\_{a=1}^{N} m^{a} \frac{\mathbf{x}\_{n+1}^{a} - \mathbf{x}\_{n}^{a}}{\Delta t} \cdot \frac{\mathbf{v}\_{n+1}^{a} - \mathbf{v}\_{n}^{a}}{\Delta t} &= \sum\_{a=1}^{N} m^{a} \mathbf{v}\_{n+1/2}^{a} \cdot \frac{\mathbf{v}\_{n+1}^{a} - \mathbf{v}\_{n}^{a}}{\Delta t} \\ \sum\_{a=1}^{N} m^{a} \frac{\mathbf{v}\_{n+1}^{a} - \mathbf{v}\_{n}^{a}}{\Delta t} \cdot \frac{\mathbf{x}\_{n+1}^{a} - \mathbf{x}\_{n}^{a}}{\Delta t} &= \sum\_{\substack{a,b=1 \\ a \neq b}}^{N} \mathbf{f}\_{\text{algo}}^{ab} \cdot \frac{\mathbf{x}\_{n+1}^{a} - \mathbf{x}\_{n}^{a}}{\Delta t} . \end{split} \tag{3.43}$$

Subtracting (3.43)<sup>2</sup> from (3.43)<sup>1</sup> gives

$$\frac{T\_{n+1} - T\_n}{\Delta t} - \sum\_{\substack{a,b=1 \\ a \neq b}}^N \mathbf{f}\_{\text{algo}}^{ab} \cdot \frac{\mathbf{x}\_{n+1}^a - \mathbf{x}\_n^a}{\Delta t} = 0 \,, \tag{3.44}$$

where the total discrete kinetic energy at time t<sup>n</sup> is given by

$$T\_n = \sum\_{a=1}^{N} \frac{1}{2} m^a \mathbf{v}\_n^a \cdot \mathbf{v}\_n^a \,. \tag{3.45}$$

In the spirit of eq. (3.41), further rewriting results in

$$\begin{split} &\sum\_{\begin{subarray}{c}a,b=1\\a\neq b\end{subarray}}^{N} \mathbf{f}\_{\text{algo}}^{ab} \cdot \frac{\mathbf{x}\_{n+1}^{a} - \mathbf{x}\_{n}^{a}}{\Delta t} = \frac{1}{\Delta t} \sum\_{\begin{subarray}{c}a,b=1\\a$$

Therefore, a necessary and sucient condition for energy conservation is that the following directionality condition is satised

$$\sum\_{\substack{a,b=1\\a$$

It is important to emphasize that the proof is based on the inner product between the algorithmic EM force and the unprojected relative position vectors. Finally, to show that the EM scheme (3.41) is indeed a second-order accurate method, it suces to prove that the correction term in the denition (3.41) is of size O(∆t 2 ). Making use of the relation

$$\mathbf{f}\_{\rm MP}^{ab} = -\partial\_{\mathbf{x}^a} \tilde{V}(|\bar{\mathbf{r}}\_{n+1/2}^{ab}|) = \partial\_{\mathbf{r}^{ab}} \tilde{V}(|\bar{\mathbf{r}}\_{n+1/2}^{ab}|) \,, \tag{3.48}$$

a Taylor series expansion around the point r ab n+1/2 gives

$$
\tilde{V}\_{n+1}^{ab} - \tilde{V}\_n^{ab} = \mathbf{f}\_{\rm MP}^{ab} \cdot \left(\mathbf{r}\_{n+1}^{ab} - \mathbf{r}\_n^{ab}\right) + \mathcal{O}(\Delta t^3) \,. \tag{3.49}
$$

Then, since the direction vector of the correction has size

$$\frac{\mathbf{r}\_{n+1}^{ab} - \mathbf{r}\_{n}^{ab}}{|\mathbf{r}\_{n+1}^{ab} - \mathbf{r}\_{n}^{ab}|} = \mathcal{O}(1) \,, \tag{3.50}$$

and |r ab <sup>n</sup>+1 − r ab n | is O(∆t), we conclude that the correction term is indeed O(∆t 2 ).

#### 3.3.2.3. Time reversibility

Time-reversible (or symmetric) integration schemes are often favored for the approximation of Hamiltonian systems for two main reasons. First, the Hamiltonian ow itself is

symmetric, so it is desirable that its numerical approximation also possesses this property. Second, symmetric numerical schemes are known to have several favorable properties [7], especially in long-term simulations. The class of EM integration schemes dened in this section have also this property. This is a direct consequence of the time reversibility of the algorithmic approximation of the pairwise forces, namely,

$$\mathbf{f}\_{\text{algo}}^{ab}(\mathbf{r}\_n^{ab}, \mathbf{r}\_{n+1}^{ab}) = \mathbf{f}\_{\text{algo}}^{ab}(\mathbf{r}\_{n+1}^{ab}, \mathbf{r}\_n^{ab})\,,\tag{3.51}$$

that is trivially satised by both (3.39) and (3.41).

## 3.3.3. Interatomic potential

For the following numerical examples we consider the well-known Lennard-Jones potential [121] with r = ¯r ab, that is,

$$\tilde{V}(r) = 4\varepsilon \left[ \left( \frac{\sigma}{r} \right)^{12} - \left( \frac{\sigma}{r} \right)^{6} \right],\tag{3.52}$$

where ε and σ are material constants.

#### 3.3.3.1. Cut-o distance considerations

In the Lennard-Jones potential, atomic interactions between distant particles are negligible. For this reason, a cut-o distance r<sup>c</sup> is often introduced beyond which the interaction is completely ignored (see, e.g., [108, 110]). However, simply trimming the Lennard-Jones potential beyond the cut-o distance leads to a discontinuity in this function at r = r<sup>c</sup> that might aect the properties of the integration scheme. Since the derivative of the potential enters the equations of motion (3.16), this discontinuity precludes the computation of the interatomic force at r = rc. The discontinuity can be avoided, rst, by shifting the potential function by the amount V (rc), leading to the shifted potential (SP)

$$\tilde{V}\_{\rm SP}(r) = \begin{cases} V(r) - V(r\_c) & \text{if } r < r\_c \text{ ,} \\ 0 & \text{if } r \ge r\_c \text{ .} \end{cases} \tag{3.53}$$

The derivative of this function at r = r<sup>c</sup> is still not dened and neither is the force. To resolve this physical inconsistency one can introduce a shifted and linearly truncated potential (SF), which is equivalent to a shift in the force (see, e.g., [110, 128]), and given by

$$\tilde{V}\_{\text{SF}}(r) = \begin{cases} V(r) - V(r\_c) - (r - r\_c)V'(r\_c) & \text{if } r < r\_c \\ 0 & \text{if } r \ge r\_c \end{cases} \tag{3.54}$$

One can further introduce a quadratic correction term that yields the shifted and quadratically truncated potential (STF), which is equivalent to a shift and a linear truncation in the force,

$$\tilde{V}\_{\rm STF}(r) = \begin{cases} V(r) - V(r\_c) - (r - r\_c)V'(r\_c) - \frac{1}{2}(r - r\_c)^2 V''(r\_c) & \text{if } r < r\_c \\ 0 & \text{if } r \ge r\_c \end{cases} \tag{3.55}$$

This potential is twice dierentiable. Due to its higher smoothness, it is better suited for structure-preserving schemes than the standard potential since it eliminates numerical oscillations in the energy evolution that sometimes appear when employing non-smooth potentials. As mentioned in Section 3.2.5, it is important to stress out that the cuto distance must not be greater than L/2 for consistency with the minimum image convention. Due to the quadratic term in the corrected potential, a cut-o radius of r<sup>c</sup> = 5σ is suggested.

## 3.3.4. Numerical evaluation

All numerical examples are based on a set of dimensionless units. The implementation of periodic boundary conditions follows the details of Section 3.1 (see also [108, 110, 111, 113]), and the order of magnitude of the time step sizes has been selected using standard criteria from molecular dynamics simulations (see, e.g., [130] and references therein.)

#### 3.3.4.1. Accuracy study

In the rst numerical example, we consider a two-dimensional box [−L/2, L/2]<sup>2</sup> with two particles. The initial positions and velocities are given, respectively, by

$$\begin{aligned} \mathbf{x}^1 &= \begin{pmatrix} 0,0 \end{pmatrix}^\mathrm{T}, & \mathbf{v}^1 &= \begin{pmatrix} 0,0 \end{pmatrix}^\mathrm{T}, \\ \mathbf{x}^2 &= \begin{pmatrix} 1.9,1 \end{pmatrix}^\mathrm{T}, & \mathbf{v}^2 &= \begin{pmatrix} 5,0 \end{pmatrix}^\mathrm{T}, \end{aligned}$$

where the rst particle is constrained to remain on the center of the box by holding xed its degrees of freedom and only the second particle is allowed to move freely. For this simulation we used the Lennard-Jones potential with a simple spherical cut-o distance of r<sup>c</sup> = 2.5σ. The numerical values of the remaining parameters of the simulation can be found in Table 3.1.


Table 3.1.: Accuracy study: Data used in the simulation

Next we compare the "classical" EM (3.39), the newly dened EM (3.41), and the mid-point rule. For that, we compare the errors in the position and the linear momentum made by these three methods when using increasingly smaller time steps. In the two analyses, we use as reference the mid-point rule solution and dene the errors, respectively, as

$$\mathbf{e}\_x^{\text{(MP)}} = \frac{||\mathbf{x}^a - \mathbf{x}\_r^a||\_2}{||\mathbf{x}\_r^a||\_2}, \quad \mathbf{e}\_p^{\text{(MP)}} = \frac{||\mathbf{p}^a - \mathbf{p}\_r^a||\_2}{||\mathbf{p}\_r^a||\_2},\tag{3.56}$$

where <sup>a</sup> r , with ∈ {x, <sup>p</sup>}, is the solution at time <sup>T</sup> calculated with the mid-point rule using the reference time step size ∆tref and where <sup>a</sup> is the solution of the considered scheme at time T for each time step size ∆t. Fig. 3.3 conrm that all the schemes under consideration are second order accurate.

Figure 3.3.: Accuracy study: Relative error in the position w.r.t. mid-point rule (left), Relative error in the linear momentum w.r.t. mid-point rule (right)

#### 3.3.4.2. Energy consistency study

The second numerical example investigates the energy conservation properties of the integrators described in Section 3.3. We consider 150 arbitrarily positioned particles inside a three-dimensional periodic box [−L/2, L/2]<sup>3</sup> such that the initial distance between the particles is greater than 2 <sup>1</sup>/6σ. Starting from rest, the kinetic energy of the system will rise until the system is in equilibrium due to the random order of the particles. For this simulation, we consider the shifted and quadratically truncated Lennard-Jones potential (STF) with a spherical cut-o distance of r<sup>c</sup> = 5σ. Numerical values for the parameters of the simulation can be found in Table 3.2.


Table 3.2.: Energy consistency study: Data used in the simulation

For this relatively large time step size, the mid-point rule introduces energy into the system leading to an energy blow-up and eventually to a termination of the simulation, indicated with a vertical line in Fig. 3.4 (left).

Figure 3.4.: Energy consistency study: Total energy (left), Total energy dierence (right)

Figure 3.5.: Energy consistency study: Kinetic energy

Both EM methods described, on the other hand, preserve the total energy up to machine precision for the whole duration of the simulation, see Fig. 3.4 (right). Hence, as long as the nonlinear iterative solver nds the step update, the solution of the EM will never blow up. Of course, if the time step size is xed to value that is too large, the iterative solver will fail to solve the update equations and the method will not be able to proceed. Fig. 3.5 shows the evolution of the kinetic energy throughout the simulation. Since this energy is proportional to the temperature in the system, the gure reveals that only the EM methods can compute the evolution of the system until it reaches equilibrium, for the chosen time step size.

We compare the newly proposed EM method with an explicit integrator, namely, the symplectic velocity-Verlet scheme provided by the molecular dynamics code LAMMPS<sup>2</sup> [131]. For this example we use the shifted and linearly truncated potential (SF) instead of the shifted and quadratically truncated potential (STF), since the former is available in the software package. Moreover, we extended the simulation duration to T = 80. Keeping the time step size of the EM method ∆tEM = 0.08 constant, the time step size of the velocity-Verlet integrator ∆tvV is reduced until the energy uctuations become suciently small. As Fig. 3.6 reveals, the time step size employed for the explicit method is ∆tvV = 0.0008, 100 times smaller than the time step size of the EM method. This is the time step size required to keep the relative energy uctuations in the explicit solution approximately below 0.1‰.

<sup>2</sup> https://lammps.sandia.gov

Figure 3.6.: Energy consistency study: Relative energy drift for ∆tEM = 100∆tvV

Regarding the computational cost per step, the proposed implicit integrator is naturally more expensive than the explicit scheme, but comparable to that of other second-order implicit methods like the mid-point rule.

Figure 3.7.: Energy consistency study: Relative error in the position w.r.t. velocity-Verlet scheme (left), Relative error in the momentum w.r.t. velocity-Verlet scheme (right)

In addition to the previous investigation, we perform an accuracy comparison between the EM and the velocity-Verlet methods. We use again the system described in Table 3.2 and obtain a reference solution with the explicit method and a small time step ∆tref = 0.000001. The displacement and momentum error measures of eq. (3.56) will be used to compare the accuracy of the two integrators for the time step sizes ∆t = {0.0001, 0.0002, 0.00025, 0.0004, 0.0005, 0.00625, 0.001, 0.0025, 0.004, 0.005, 0.01} and a nal simulation time T = 4. As shown in Fig. 3.7, both schemes are second order accurate in the position and the linear momentum.

## 3.4. Energy-momentum methods for periodic systems with three-body interactions

In Section 3.3 we derived the equation of motion of a system of N particles moving inside a period box B where the interactions were based on pair potentials. For materials with strong covalent-bonding character, however, we further need to incorporate a bond-angle dependency to the eective potential. This can be archived by including three-body terms in the expression of the potential.

## 3.4.1. Equation of motion

We consider now a system of N particles in a periodic box B of side L with equations of motion (3.16) and an eective potential that depends only on the interactions among all triplets of particles. Such a potential must be of the form

$$V = \frac{1}{3!} \sum\_{\substack{a,b,c = 1 \\ a \neq b \neq c}}^{N} \tilde{V} \left( d\_T(\mathbf{x}^a, \mathbf{x}^b), d\_T(\mathbf{x}^a, \mathbf{x}^c), d\_T(\mathbf{x}^b, \mathbf{x}^c) \right) \,, \tag{3.57}$$

where V˜ : R <sup>+</sup> ∪ {0} × <sup>R</sup> <sup>+</sup> ∪ {0} × <sup>R</sup> <sup>+</sup> ∪ {0} 7→ <sup>R</sup> is a three-body potential between the a-th, b-th, and the c-th particle. The total kinetic energy of the system is given by eq. (3.13). The forces acting on the particles dened by eq. (3.17) can be written using the denitions in eq. (3.26) as

$$\mathbf{f}^{a} = \sum\_{\substack{b=1\\a \neq b}}^{N} \mathbf{f}^{ab} \qquad \text{with} \qquad \mathbf{f}^{ab} := \varphi^{ab} \frac{\bar{\mathbf{r}}^{ab}}{\bar{r}^{ab}} \,. \tag{3.58}$$

The strength of the force is now obtained by

$$\varphi^{ab} = \sum\_{\substack{c = 1 \\ a \neq b \neq c}}^{N} \partial\_{\bar{r}^{ab}} \tilde{V}(\bar{r}^{ab}, \bar{r}^{ac}, \bar{r}^{bc}) \,. \tag{3.59}$$

## 3.4.2. Time discretization

Next, we consider the integration in time of the equations of motion (3.16) of a system in the periodic box B and eective potential (3.57) and employ the time integration strategy outlined in Section 3.3.2.

#### 3.4.2.1. Mid-point scheme

The canonical mid-point rule approximates the equation of motion by the implicit formula (3.29), where the mid-point approximation of the force acting on the a-th particle in the direction of the b-th particle is given by

$$\begin{aligned} \mathbf{f}\_{\text{MP}}^{ab} &= \varphi\_{\text{MP}}^{ab} \frac{\bar{\mathbf{r}}\_{n+1/2}^{ab}}{|\bar{\mathbf{r}}\_{n+1/2}^{ab}|} \; , \\ \varphi\_{\text{MP}}^{ab} &= \sum\_{\substack{c=1\\a \neq b \neq c}}^{N} \partial\_{\bar{r}\_{n+1/2}^{ab}} \tilde{V} \left( \bar{r}\_{n+1/2}^{ab}, \bar{r}\_{n+1/2}^{ac}, \bar{r}\_{n+1/2}^{bc} \right) \; . \end{aligned} \tag{3.60}$$

As in the continuous case, the weak law of action and reaction is satised and therefore the approximation preserves the total linear momentum of the system, see Section 3.3.2.1.

#### 3.4.2.2. Energy and momentum conserving discretization

As in Sections 3.3.2.2, it is possible to construct a perturbation of the mid-point rule which, in addition to preserving the total linear momentum, preserves the total energy. Instead of the discrete gradient operator the partitioned discrete gradient operator [46] will now be employed, which is an approximation similar to the discrete gradient but is applicable to functions with more than one independent variable. To this end, let us rewrite the potential energy function (3.57) in terms of N(N − 1)/2 independent variables, e.g.,

$$\tilde{V} = \tilde{V}(\bar{r}^{12}, \bar{r}^{13}, \dots, \bar{r}^{1N}, \bar{r}^{23}, \dots, \bar{r}^{N-1, N}) = \tilde{V}(\{\bar{r}^{ab}\})\,,\tag{3.61}$$

where the double indexed set is given by

$$\{\bar{r}^{ab}\} = \{\bar{r}^{ab} \mid a, b \in \langle 1, \dots, N \rangle, \ a < b\}\ . \tag{3.62}$$

To simplify the denition of the partitioned discrete gradient, it proves useful to re-label the interatomic distances r¯ ab using only their position in the array {r¯ ab}. Therefore, a single indexed set is dened by

$$\{\bar{r}^{\alpha}\} = \{\bar{r}^{\alpha} \mid \alpha \in \{1, \ldots, N(N-1)/2\}\}\,. \tag{3.63}$$

Note that here an ordering of the <sup>N</sup> 2 pairs (a, b) has been established. For example, the map (a, b) 7→ α could be chosen to be lexicographic (e.g., [132, pg. 43]). Then, the potential energy can be expressed, abusing slightly the notation, as

$$V = \tilde{V}(\{\bar{r}^{\alpha}\})\,. \tag{3.64}$$

For a potential function like this one, the discrete gradient operator is dened as

$$\mathbf{D}\_{\mathbf{x}^{a}}V = -\sum\_{\substack{b=1\\a\neq b}}^{N} \mathbf{f}\_{\text{algo}}^{ab} = -\sum\_{\substack{b=1\\a\neq b}}^{N} \frac{1}{2} \left(\mathbf{f}\_{n,n+1}^{ab} + \mathbf{f}\_{n+1,n}^{ab}\right) \,,\tag{3.65}$$

with the contributions

$$\begin{split} \mathbf{f}\_{n,n+1}^{ab} &= \mathbf{f}\_{\text{MP}}^{ab} + \frac{\tilde{V}\_{n,n+1}^{\alpha}(\bar{r}\_{n+1}^{\alpha}) - \tilde{V}\_{n,n+1}^{\alpha}(\bar{r}\_{n}^{\alpha}) - \mathbf{f}\_{\text{MP}}^{ab} \cdot (\mathbf{r}\_{n+1}^{ab} - \mathbf{r}\_{n}^{ab})}{|\mathbf{r}\_{n+1}^{ab} - \mathbf{r}\_{n}^{ab}|} \text{ n} \,, \\ \mathbf{f}\_{n+1,n}^{ab} &= \mathbf{f}\_{\text{MP}}^{ab} + \frac{\tilde{V}\_{n+1,n}^{\alpha}(\bar{r}\_{n+1}^{\alpha}) - \tilde{V}\_{n+1,n}^{\alpha}(\bar{r}\_{n}^{\alpha}) - \mathbf{f}\_{\text{MP}}^{ab} \cdot (\mathbf{r}\_{n+1}^{ab} - \mathbf{r}\_{n}^{ab})}{|\mathbf{r}\_{n+1}^{ab} - \mathbf{r}\_{n}^{ab}|} \text{ n} \,, \end{split} \tag{3.66}$$

for which we introduced the compact notation

$$\begin{split} \tilde{V}\_{n,n+1}^{\alpha}(\bar{r}^{\alpha}) &= \tilde{V}(\bar{r}\_{n}^{1}, \dots, \bar{r}\_{n}^{\alpha-1}, \bar{r}^{\alpha}, \bar{r}\_{n+1}^{\alpha+1}, \dots, \bar{r}\_{n+1}^{N(N-1)/2}) \\ \tilde{V}\_{n+1,n}^{\alpha}(\bar{r}^{\alpha}) &= \tilde{V}(\bar{r}\_{n+1}^{1}, \dots, \bar{r}\_{n+1}^{\alpha-1}, \bar{r}^{\alpha}, \bar{r}\_{n}^{\alpha+1}, \dots, \bar{r}\_{n}^{N(N-1)/2}) \end{split} \tag{3.67}$$

To show that the proposed integrator exactly preserves the total linear momentum, it suces to follow the proof outlined in Section 3.3.2.2 and details are omitted. Similarly, to prove the energy conservation property, it is sucient to show that

$$\sum\_{\substack{a,b=1\\a$$

A straightforward manipulation shows that the proposed method with algorithmic forces given by eq. (3.65) satises this condition.

Remark 1. Many three-body potentials are expressed in terms of the bond angles ¯θ bac at particle a, between the bonds ab and ac. Since the angle itself can be written in terms of the distances, that is,

$$\bar{\theta}^{bac} = g(\bar{r}^{ab}, \bar{r}^{ac}, \bar{r}^{bc}) = \arccos\left(\frac{(\bar{r}^{ab})^2 + (\bar{r}^{ac})^2 - (\bar{r}^{bc})^2}{2(\bar{r}^{ab})(\bar{r}^{ac})}\right),\tag{3.69}$$

the composition <sup>V</sup>˜ ◦<sup>g</sup> has again the structure of the potential (3.61) and thus the partitioned discrete gradient operator dened before can be employed without modications.

Remark 2. In problems without periodic boundary conditions, using eq. (3.28), the EM method reads

$$\mathbf{D}\_{\mathbf{x}^{a}}V = -\sum\_{\substack{b=1\\a\neq b}}^{N} \mathbf{f}\_{\text{algo}}^{ab} = -\sum\_{\substack{b=1\\a\neq b}}^{N} \frac{1}{2} \left(\mathbf{f}\_{n,n+1}^{ab} + \mathbf{f}\_{n+1,n}^{ab}\right) \,,\tag{3.70}$$

with the contributions

$$\begin{split} \mathbf{f}\_{n,n+1}^{ab} &= \frac{\tilde{V}\_{n,n+1}^{\alpha}(r\_{n+1}^{\alpha}) - \tilde{V}\_{n,n+1}^{\alpha}(r\_n^{\alpha})}{r\_{n+1}^{\alpha} - r\_n^{\alpha}} \frac{\mathbf{r}\_{n+1}^{ab} + \mathbf{r}\_n^{ab}}{r\_{n+1}^{ab} + r\_n^{ab}}, \\ \mathbf{f}\_{n+1,n}^{ab} &= \frac{\tilde{V}\_{n+1,n}^{\alpha}(r\_{n+1}^{\alpha}) - \tilde{V}\_{n+1,n}^{\alpha}(r\_n^{\alpha})}{r\_{n+1}^{\alpha} - r\_n^{\alpha}} \frac{\mathbf{r}\_{n+1}^{ab} + \mathbf{r}\_n^{ab}}{r\_{n+1}^{ab} + r\_n^{ab}}, \end{split} \tag{3.71}$$

where we used the compact notation

$$\begin{aligned} \check{V}\_{n,n+1}^{\alpha}(r^{\alpha}) &= \tilde{V}(r\_n^1, \dots, r\_n^{\alpha-1}, r^{\alpha}, r\_{n+1}^{\alpha+1}, \dots, r\_{n+1}^{N(N-1)/2}) \; , \\ \check{V}\_{n+1,n}^{\alpha}(r^{\alpha}) &= \tilde{V}(r\_{n+1}^1, \dots, r\_{n+1}^{\alpha-1}, r^{\alpha}, r\_n^{\alpha+1}, \dots, r\_n^{N(N-1)/2}) \; . \end{aligned} \tag{3.72}$$

This EM method preserves the total angular momentum in addition to the total energy and the total linear momentum. It can be used, e.g., for the simulation of bonded three-body interactions between macromolecules.

## 3.4.3. Interatomic potential

For our numerical simulation we consider the Stillinger-Weber potential [115], which includes two- and three-body contributions

$$\tilde{V} = \frac{1}{2!} \sum\_{\substack{a,b=1\\a \neq b}}^N \varepsilon \tilde{f}\_2(\bar{r}^{ab}/\sigma) + \frac{1}{3!} \sum\_{\substack{a,b,c=1\\a \neq b \neq c}}^N \varepsilon \tilde{f}\_3(\bar{r}^{ab}/\sigma, \bar{r}^{ac}/\sigma, \bar{r}^{bc}/\sigma) \,. \tag{3.73}$$

The pair contribution is given by

$$\vec{f}\_2(\hat{r}) = \begin{cases} A\left(B\hat{r}^{-q} - \hat{r}^{-p}\right)g\_2(\hat{r}) & \text{if } \hat{r} < a\\ 0 & \text{if } r \ge a \end{cases},\tag{3.74}$$

where the hats indicate the normalization of the distances by σ. After composing with the law of cosines (3.69), the three-body contribution takes the form

$$\tilde{f}\_3(\hat{r}^{ab}, \hat{r}^{ac}, \hat{r}^{bc}) = h(\hat{r}^{ab}, \hat{r}^{ac}, \hat{r}^{bc}) + h(\hat{r}^{ab}, \hat{r}^{bc}, \hat{r}^{ac}) + h(\hat{r}^{ac}, \hat{r}^{bc}, \hat{r}^{ab}) \,, \tag{3.75}$$

with

$$h(\hat{r}^1, \hat{r}^2, \hat{r}^3) = \begin{cases} \lambda \left( \frac{\left(\hat{r}^1\right)^2 + \left(\hat{r}^2\right)^2 - \left(\hat{r}^3\right)^2}{2\hat{r}^1 \hat{r}^2} + \frac{1}{3}\right)^2 g\_3(\hat{r}^1, \hat{r}^2) & \text{if } \hat{r}^1 < a \text{ and } \hat{r}^2 < a\\ 0 & \text{otherwise} \end{cases} \tag{3.76}$$

Additionally, we introduce the functions

$$\begin{aligned} g\_2(\hat{r}) &= \exp\left( [\hat{r} - a]^{-1} \right), \\ g\_3(\hat{r}^1, \hat{r}^2) &= \exp\left( \gamma [\hat{r}^1 - a]^{-1} + \gamma [\hat{r}^2 - a]^{-1} \right) \ . \end{aligned} \tag{3.77}$$

Here ˜θ bac = g(¯r ab , r¯ ac , r¯ bc) = arccos(−1/3) <sup>≈</sup> <sup>109</sup>.47◦ minimizes the function <sup>h</sup> given in eq. (3.76), that corresponds to the underlying diamond structure of silicon. From this reasoning it follows that the function h penalizes bond-angles which dier from the ones in this crystal structure. The parameters in the potential are A, B, a, ε, σ, q, p, λ, and γ.

## 3.4.4. Numerical evaluation

We consider now the numerical solution of systems of particles with the Stillinger-Weber eective potential. The pairwise contributions to the potential are discretized according to Section 3.3; the remaining three-body interactions are dened as in Section 3.4.2.2.

#### 3.4.4.1. Accuracy study

We consider in this example a three-dimensional box [−L/2, L/2]<sup>3</sup> lled with 5 particles. The initial conguration of the system has a particle at the center of the box and the rest of the particles form bonds with the rst one with angle arccos(−1/3). In addition, these four particles are at distance 1 from the center. Starting from rest, the motion of the system results from the non-vanishing interacting forces among particles away from the center. Further parameters of the simulation can be found in Table 3.3.


Table 3.3.: Accuracy study: Data used in the simulation

As in Section 3.3.4.1, we perform rst an accuracy analysis of the EM integrator using the mid-point rule as a reference. This study is carried out using ten dierent time step sizes for both integrators and employing the error measures eq. (3.56). Fig. 3.8 reveals that both schemes are second order accurate.

Figure 3.8.: Accuracy study: Relative error in the position w.r.t. mid-point rule (left), Relative error in the linear momentum w.r.t. mid-point rule (right)

## 3.4.5. Energy consistency study



Table 3.4.: Energy consistency study: Data used in the simulation

This lattice will be disrupted during the simulation as we consider an initial velocity associated to each atom such that the total initial kinetic energy is approximately 768.22. For the chosen time step size, the mid-point rule clearly violates the conservation of the total energy, see Fig. 3.9, leading to an energy blow-up and nally to a termination of the simulation, indicated by the black line on the same Figure.

Figure 3.9.: Energy consistency study: Total energy

Figure 3.10.: Energy consistency study: Potential energies using the MP rule (left), Potential energies using the EM method (right). V<sup>i</sup> refers to the i−body contribution to the potential

The largest contribution to the algorithmic energy error is due to the mid-point approximation of the forces generated from the three-body contribution of the Stillinger-Weber potential. It can be observed in Fig. 3.10 (left) that the potential energy of the two-body terms remains bounded, while the potential energy of the three-body terms increases unphysically causing the energy blow-up and, ultimately, the termination of the simulation.

Figure 3.11.: Energy consistency study: Total energy dierence (left), kinetic energy (right)

In contrast, using the EM method both contributions to the potential energy of the system remain bounded, see Fig. 3.10 (right). As expected, the EM method preserves the total energy up to round-o errors, see Fig. 3.11 (left). Eventually, as illustrated in Fig. 3.11 (right), the evolution of the kinetic energy reveals that the EM solution reaches thermodynamic equilibrium for the chosen time step size, in contrast with the mid-point rule.

## 3.5. Energy-momentum methods for periodic systems described by the embedded-atom method

In addition to three-body potentials of the type described in Section 3.4, more elaborate and accurate potentials include multi-body eects through an environment-dependent variable, resulting in eective potential that are extremely common, for example, in the simulation of metals. One of the most popular interatomic potentials of this class is the one employed in the embedded-atom method (EAM) [116–118], whose use in the context of conserving schemes is analyzed in this section. The interatomic potential in this case is of the form

$$V = \frac{1}{2} \sum\_{\substack{a,b=1\\b \neq a}}^{N} \tilde{V} \left( d\_T(\mathbf{x}^a, \mathbf{x}^b) \right) + \sum\_{a=1}^{N} \tilde{F} \left( \overline{\rho}^a \right) \,. \tag{3.78}$$

The EAM potential consists of a pair potential contribution and electronic energies F˜(¯ρ a ). The latter is due to the embedding of a-th atom in a homogeneous electron gas of density

ρ¯ a [120]. The background electron density function ρ¯ a is a linear superposition of contributions from each neighbor atom such that the electronic energy F˜(¯ρ a ) of the a-th atom depends on the interatomic distance d<sup>T</sup> (x a , x b ) to each neighbor.

In the case of metals, the environment of each atom is a nearly uniform electron gas and therefore the embedded-atom approximation is reasonable. Since we already investigated pair potentials in Section 3.3, we now investigate the discretization of the forces due to the electronic energy, noting that the resultant forces in an EAM potential must include the forces due to the pair potential, as well.

## 3.5.1. Equation of motion

We consider a system of N particles in a periodic box B of side L with equations of motion (3.16) and an eective potential that depends only on the electronic energy of each particle, that is

$$V = \sum\_{a=1}^{N} \ddot{F} \left( \sum\_{\substack{b=1\\b \neq a}}^{N} g\_b \left( d\_T(\mathbf{x}^a, \mathbf{x}^b) \right) \right), \tag{3.79}$$

where V˜ : R <sup>+</sup> ∪ {0} 7→ <sup>R</sup>. Here, <sup>g</sup><sup>a</sup> : <sup>R</sup> <sup>+</sup> ∪ {0} 7→ <sup>R</sup> is a function of the relative interatomic distance which represents a spherical electron density eld around the isolated a-th particle [120]. Using the denitions in eq. (3.26), the background energy density is then given by

$$\bar{\rho}^a = \sum\_{b \equiv 1 \atop b \neq a}^N g\_b \left( \bar{r}^{ab} \right) \,. \tag{3.80}$$

The forces acting on the particles are dened by eq. (3.17) and have the standard form

$$\mathbf{f}^{a} = \sum\_{\substack{b=1 \\ a \neq b}}^{N} \mathbf{f}^{ab}, \qquad \text{with} \qquad \mathbf{f}^{ab} := \varphi^{ab} \frac{\bar{\mathbf{r}}^{ab}}{\bar{r}^{ab}}, \tag{3.81}$$

where the strength of the force has now the structure

$$
\varphi^{ab} = \tilde{F}'(\bar{\rho}^a) \, g\_b' \left( \bar{r}^{ab} \right) + \tilde{F}'(\bar{\rho}^b) g\_a' \left( \bar{r}^{ab} \right) \, . \tag{3.82}
$$

We observe that, for this potential contribution, the direction of the interatomic force depends only on the distance between the a-th and the b-th particle, while its strength is further determined by the background electron density at the a-th and b-th particle.

## 3.5.2. Time discretization

We consider now the integration in time of the equations of motion (3.16) of a system in the periodic box B and eective potential (3.78) and employ the same time integration strategy as outlined in Section 3.3.2.

#### 3.5.2.1. Mid-point scheme

The canonical mid-point rule approximates the equation of motion by the implicit formula (3.29), where the mid-point approximation of the force acting on the a-th particle in the direction of the b-th particle is given by

$$\begin{split} \mathbf{f}\_{\rm MP}^{ab} &= \varphi\_{\rm MP}^{ab} \frac{\bar{\mathbf{r}}\_{n+1/2}^{ab}}{|\bar{\mathbf{r}}\_{n+1/2}^{ab}|}, \\ \varphi\_{\rm MP}^{ab} &= \tilde{F}' \left( \bar{\rho}\_{\rm MP}^{a} \right) g\_b' \left( \bar{r}\_{n+1/2}^{ab} \right) + \tilde{F}' (\bar{\rho}\_{\rm MP}^{b}) g\_a' \left( \bar{r}\_{n+1/2}^{ab} \right) \,, \end{split} \tag{3.83}$$

and the mid-point approximation of the background energy density reads

$$\bar{\rho}\_{\rm MP}^a = \sum\_{\substack{b=1\\b \neq a}}^N g\_b \left( \bar{r}\_{n+1/2}^{ab} \right) \,. \tag{3.84}$$

As in the continuous case, the weak law of action and reaction is satised and responsible for the conservation of total linear momentum of the system, see Section 3.3.2.1.

#### 3.5.2.2. Energy and momentum conserving discretization

As illustrated in previous sections, it is possible to construct a perturbation of the midpoint rule which, in addition to preserving the total linear momentum, preserves the total energy. For that, we follow once more the steps presented in Section 3.3.2.2. The partitioned discrete gradient will be used again with a slightly dierent structure due to the nature of the embedded function. The partitioned discrete gradient assumes the form

$$\mathbf{D}\_{\mathbf{x}^{a}}V = -\sum\_{\substack{a,b=1\\a\neq b}}^{N} \mathbf{f}\_{\text{alg}}^{ab} = -\sum\_{\substack{a,b=1\\a\neq b}}^{N} \frac{1}{2} \left( \mathbf{f}\_{n,n+1}^{ab} + \mathbf{f}\_{n+1,n}^{ab} \right),\tag{3.85}$$

with the contributions

$$\begin{split} \mathbf{f}\_{n,n+1}^{ab} &= \mathbf{f}\_{\text{MP}}^{ab} + \frac{\Delta \tilde{F}\_{n,n+1}^{ab} - \mathbf{f}\_{\text{MP}}^{ab} \cdot (\mathbf{r}\_{n+1}^{ab} - \mathbf{r}\_{n}^{ab})}{|\mathbf{r}\_{n+1}^{ab} - \mathbf{r}\_{n}^{ab}|} \text{ n} \,, \\ \mathbf{f}\_{n+1,n}^{ab} &= \mathbf{f}\_{\text{MP}}^{ab} + \frac{\Delta \tilde{F}\_{n+1,n}^{ab} - \mathbf{f}\_{\text{MP}}^{ab} \cdot (\mathbf{r}\_{n+1}^{ab} - \mathbf{r}\_{n}^{ab})}{|\mathbf{r}\_{n+1}^{ab} - \mathbf{r}\_{n}^{ab}|} \text{ n} \,, \end{split} \tag{3.86}$$

for which we introduced the compact notation

$$\begin{split} \Delta \tilde{F}\_{n,n+1}^{ab} &= \tilde{F}(\bar{\rho}\_{n,n+1}^{a}(\bar{r}\_{n+1}^{ab})) - \tilde{F}(\bar{\rho}\_{n,n+1}^{a}(\bar{r}\_{n}^{ab})) \\ &+ \tilde{F}(\bar{\rho}\_{n,n+1}^{b}(\bar{r}\_{n+1}^{ab})) - \tilde{F}(\bar{\rho}\_{n,n+1}^{b}(\bar{r}\_{n}^{ab})) \,, \\ \Delta \tilde{F}\_{n+1,n}^{ab} &= \tilde{F}(\bar{\rho}\_{n+1,n}^{a}(\bar{r}\_{n+1}^{ab})) - \tilde{F}(\bar{\rho}\_{n+1,n}^{a}(\bar{r}\_{n}^{ab})) \\ &+ \tilde{F}(\bar{\rho}\_{n+1,n}^{b}(\bar{r}\_{n+1}^{ab})) - \tilde{F}(\bar{\rho}\_{n+1,n}^{b}(\bar{r}\_{n}^{ab})) \,, \end{split} \tag{3.87}$$

and

$$\begin{split} \bar{\rho}\_{n,n+1}^{a}(\bar{r}^{ab}) &= \sum\_{d=1}^{b-1} g\_a(\bar{r}\_n^{ad}) + g\_a(\bar{r}^{ab}) + \sum\_{e=b+1}^{N} g\_a(\bar{r}\_{n+1}^{ae}) \,, \\ \bar{\rho}\_{n+1,n}^{a}(\bar{r}^{ab}) &= \sum\_{d=1}^{b-1} g\_a(\bar{r}\_{n+1}^{ad}) + g\_a(\bar{r}^{ab}) + \sum\_{e=b+1}^{N} g\_a(\bar{r}\_n^{ae}) \, . \end{split} \tag{3.88}$$

The densities ρ¯ b n,n+1(¯r ab) and ρ¯ b <sup>n</sup>+1,n(¯r ab) are dened similarly.

To show that the integrator exactly preserves the total momentum, it suces to follow the proof in Section 3.3.2.2. The critical condition that a conserving scheme must satisfy reads now

$$\sum\_{\substack{b=1\\a$$

which is indeed satised by the proposed method.

## 3.5.3. Interatomic potential

We consider for our subsequent analysis the Lennard-Jones-Baskes (LJB) EAM model [118, 133], which is the extension of the Lennard-Jones material model into the many-body regime of the EAM formalism [134]. The total potential energy of the LJB model is given by

$$\tilde{V} = \frac{1}{2} \sum\_{\substack{a,b=1\\a \neq b}}^{N} V(\bar{r}^{ab}) + \sum\_{a=1}^{N} \tilde{F}(\bar{\rho}^a) \,. \tag{3.90}$$

The two body part has been introduced in Section 3.3.3. For the EAM contribution we further dene

$$\begin{aligned} \tilde{F}(\bar{\rho}^a) &= \frac{1}{2} \varepsilon A Z\_1 \bar{\rho}^a \left( \ln \left( \bar{\rho}^a \right) - 1 \right), \\ \bar{\rho}^a &= \frac{1}{Z\_1} \sum\_{\substack{b=1 \\ b \neq a}}^N g\_b \left( |\bar{\mathbf{r}}^{ab}| \right), \\ g\_b \left( |\bar{\mathbf{r}}^{ab}| \right) &= \begin{cases} \exp \left( -\beta \left( \sigma^{-1} |\bar{\mathbf{r}}^{ab}| - 1 \right) \right) & \text{if } |\bar{\mathbf{r}}^{ab}| < r\_c \\ 0 & \text{otherwise} \end{cases}, \end{aligned} \tag{3.91}$$

where ε, σ, β, A, r<sup>c</sup> and Z<sup>1</sup> are material parameters.

## 3.5.4. Numerical evaluation

Since we already investigated pair potentials in Section 3.3, we focus on the EAM part of the LJB material model for the following numerical evaluations.

#### 3.5.4.1. Accuracy study

A three-dimensional box [−L/2, L/2]<sup>3</sup> is now considered with 5 particles inside it. The particles form a unit body-centered cubic (BCC) cell of side 2, centered within the box. Starting from rest, the system builds up kinetic energy due to the fact that the particles in the exterior of the BCC crystal are not in equilibrium. Further parameters of the simulation can be found in Table 3.5.


Table 3.5.: Accuracy study: Data used in the simulation

Proceeding as in Section 3.3.4.1, we perform an accuracy analysis of the EM integrator using the mid-point rule as the reference solution. To study the convergence of the numerical solutions, we employ ten time steps of decreasing size and the error measures dened in (3.56). Fig. 3.12 conrms again that both the mid-point rule and the EM method are second order accurate approximations.

Figure 3.12.: Accuracy study: Relative error in the position w.r.t. mid-point rule (left), Relative error in the momentum w.r.t. mid-point rule (right)

## 3.5.5. Energy consistency study

In this example we consider 108 atoms inside a three-dimensional box [−L/2, L/2]<sup>3</sup> , where the atoms are arranged in a perfect face-centered cubic (FCC) lattice structure. This FCC lattice is perturbed by the initial velocities of the atoms which are imparted in such a way that the total initial kinetic energy is approximately 3.251.


Table 3.6.: Energy consistency study: Data used in the simulation

Further parameters of the simulation can be found in Table 3.6.

Figure 3.13.: Energy consistency study: Total energy (left), Total energy dierence (right)

For the time step selected, the mid-point rule clearly violates the conservation of total energy, see Fig. 3.13 (left). As in previous examples, this leads to an energy blow-up and nally to a termination of the simulation, indicated by the black line on the same gure. In contrast, the EM method preserves the total energy up to round o errors, see Fig. 3.13 (right). Finally, and as in the previous examples, Fig. 3.14 shows that the EM, but not the mid-point rule, is able to evolve the system of particles up to thermodynamic equilibrium.

Figure 3.14.: Energy consistency study: Kinetic energy

# 4. The GENERIC framework for large-strain thermo-elasticity<sup>1</sup>

In this chapter we study the motion of large-strain thermo-elastic solids. Due the presence of heat conduction irreversible eects have to be considered. The continuum based description will resort to the GENERIC framework for innitesmial dimensional systems, where the time-evolution of an arbitrary functional A can be expressed as

$$\frac{\mathrm{d}\mathcal{A}}{\mathrm{d}t} = \left\{ \mathcal{A}, \mathcal{E} \right\} + \left[ \mathcal{A}, \mathcal{S} \right] \,. \tag{4.1}$$

This equation represents a 2-generator formalism in which the reversible part is generated by the total energy E of the system via the Poisson bracket {·, ·}, while the irreversible part is generated by the total entropy S of the system via the dissipative bracket [·, ·]. In contrast to Chapters 2 and 3 the brackets now imply an integration over continues labels rather then summation over discrete indices, and functional derivatives are considered rather then partial derivatives. A concise denition of these quantities will be presented in the sequel within the context of a Lagrangian description of large-strain thermo-elasticity.

We propose a new variational formulation for nite-strain thermo-elastodynamics, which emanantes from the GENERIC formalism and makes the free choice of the thermodynamic state variable possible. In particular, one may choose the absolute temperature, the internal energy density or the entropy density as thermodynamic state variable. To solve initial boundary value problems, the GENERIC formalism is extended to open systems. The discretization in time makes use of the standard mid-point rule. Depending on the choice of the thermodynamic state variable, partially structure-preserving schemes are obtained. For example, choosing the internal energy as state variable yields a new energymomentum consistent scheme. Thus the newly developed GENERIC-based weak form is particularly well suited for the design of (fully) structure-preserving methods as will be shown in Chapter 5. Furthermore, numerical investigations are presented which conrm the theoretical ndings and shed light on the numerical stability of the newly developed schemes.

The rst part of the present chapter deals with the description of the GENERIC framework which was originally developed for isolated systems. In the case of isolated (or closed) systems boundary eects are considered to be irrelevant.

<sup>1</sup> This Chapter is based on [1]

## 4.1. The reversible part: Isothermal elastodynamics

We start with the formulation of large-strain elastodynamics. Since elastic deformations are reversible the GENERIC evolution equation (4.1) reduces to the Hamiltonian form which is solely based on the Poisson bracket.

Figure 4.1.: Reference conguration B and deformed conguration ϕ(B, t) at time t. In Section 4.1 the focus is on the motion of isolated elastic solids. That is, external tractions acting on the boundary are disregarded.

Consider a continuum body with material points X = X<sup>A</sup>e<sup>A</sup> in the reference conguration B ⊂ <sup>R</sup> 3 (Fig. 4.1). Here and in the sequel the summation convention applies to repeated indices. Moreover, e<sup>A</sup> denote the canonical base vectors in R 3 . Within the Lagrangian description of continuum mechanics the deformed conguration of the body at time <sup>t</sup> is characterized by the deformation map <sup>ϕ</sup> : B × I 7→ <sup>R</sup> 3 , where I = [0, T] is the time interval of interest. Accordingly, the placement at time t of the material point X ∈ B is given by x = ϕ(X, t). Its velocity is dened by v = ϕ˙ , where a superposed dot denotes the material time derivative.

The deformation gradient corresponds to the Jacobian of the deformation map

$$\mathbf{F} = \partial\_{\mathbf{X}} \varphi \,. \tag{4.2}$$

The Hamiltonian formulation of nonlinear elastodynamics is based on the introduction of the momentum conjugate to ϕ. To this end we consider the Lagrangian functional

$$\mathcal{L}(\varphi, \dot{\varphi}) = \int\_{\mathcal{B}} l(\varphi, \mathbf{F}, \dot{\varphi}) \, \mathrm{d}V \,\,, \tag{4.3}$$

which corresponds to the dierence between the kinetic energy and the potential energy. The associated Lagrangian density function is given by

$$l(\varphi, \mathbf{F}, \dot{\varphi}) = \frac{1}{2} \rho \dot{\varphi} \cdot \dot{\varphi} - \left(W(\mathbf{F}) - \mathbf{b} \cdot \varphi\right)\,,\tag{4.4}$$

where <sup>ρ</sup> : B 7→ <sup>R</sup><sup>+</sup> is the mass density in the reference conguration, <sup>W</sup> : <sup>R</sup> 3×3 7→ <sup>R</sup> is a stored energy function which characterizes hyperelastic material behavior, and b : B 7→ R 3 represents prescribed body forces, which are assumed to be dead loads. The conjugate momentum density is now dened by

$$\mathbf{p} = \partial \dot{\varphi} l(\varphi, \mathbf{F}, \dot{\varphi}) = \rho \dot{\varphi} \,. \tag{4.5}$$

Moreover, the Hamiltonian density function results from a Legendre tranformation of the Lagrangian density function (4.4) with respect to the velocity leading to

$$h(\varphi, \mathbf{F}, \mathbf{p}) = \mathbf{p} \cdot \dot{\varphi} - l(\varphi, \mathbf{F}, \dot{\varphi}) \,, \tag{4.6}$$

subject to relation (4.5). With regard to (4.4), the Hamiltonian density function reads

$$h(\varphi, \mathbf{F}, \mathbf{p}) = \frac{1}{2} \rho^{-1} \mathbf{p} \cdot \mathbf{p} + W(\mathbf{F}) - \mathbf{b} \cdot \boldsymbol{\varphi} \,. \tag{4.7}$$

The equations of motion can now be obtained by applying Hamilton's phase space principle. Accordingly, we consider the stationarity of the functional

$$A(\varphi, \mathbf{p}) = \int\_0^T \int\_{\mathcal{B}} \left( \mathbf{p} \cdot \dot{\varphi} - h(\varphi, \mathbf{F}, \mathbf{p}) \right) \, \mathrm{d}V \mathrm{d}t \,, \tag{4.8}$$

with xed endpoints in time (t = 0 and t = T > 0). The rst stationary condition is given by

$$\begin{split} DA(\boldsymbol{\varphi}, \mathbf{p}) \cdot \delta \boldsymbol{\varphi} &= \frac{d}{d\varepsilon} \left. \int\_{0}^{T} \int\_{\mathcal{B}} \left( \mathbf{p} \cdot (\dot{\boldsymbol{\varphi}} + \varepsilon \delta \dot{\boldsymbol{\varphi}}) - h(\boldsymbol{\varphi} + \varepsilon \delta \boldsymbol{\varphi}, \mathbf{F} + \varepsilon \nabla \delta \boldsymbol{\varphi}, \mathbf{p}) \right) \, \mathrm{d}V \mathrm{d}t \right|\_{\boldsymbol{\varepsilon}=0} \\ &= \int\_{0}^{T} \left( \int\_{\mathcal{B}} \mathbf{p} \cdot \delta \dot{\boldsymbol{\varphi}} \, \mathrm{d}V - \frac{d}{d\varepsilon} \int\_{\mathcal{B}} h(\boldsymbol{\varphi} + \varepsilon \delta \boldsymbol{\varphi}, \mathbf{F} + \varepsilon \nabla \delta \boldsymbol{\varphi}, \mathbf{p}) \, \mathrm{d}V \bigg|\_{\boldsymbol{\varepsilon}=0} \right) \mathrm{d}t \\ &= 0 \end{split} \tag{4.9}$$

where the variations <sup>δ</sup><sup>ϕ</sup> : B 7→ <sup>R</sup> 3 are subject to the endpoint conditions in time, δϕ<sup>0</sup> = 0 and δϕ<sup>T</sup> = 0. The second integral on the right-hand side of (4.9) can be recast in the form

$$\begin{split} \frac{d}{d\varepsilon} \int\_{\mathcal{B}} h(\boldsymbol{\varphi} + \varepsilon \delta \boldsymbol{\varphi}, \mathbf{F} + \varepsilon \nabla \delta \boldsymbol{\varphi}, \mathbf{p}) \, \mathrm{d}V \Big|\_{\varepsilon=0} &= \int\_{\mathcal{B}} (\partial\_{\boldsymbol{\varphi}} h \cdot \delta \boldsymbol{\varphi} + \partial\_{\mathbf{F}} h : \nabla \delta \boldsymbol{\varphi}) \, \mathrm{d}V \\ &= \int\_{\mathcal{B}} \delta \boldsymbol{\varphi} \cdot (\partial\_{\boldsymbol{\varphi}} h - \mathrm{Div}(\partial\_{\mathbf{F}} h)) \, \mathrm{d}V \\ &= \int\_{\mathcal{B}} \delta \boldsymbol{\varphi} \cdot \delta\_{\boldsymbol{\varphi}} h \, \mathrm{d}V \,, \end{split}$$

where δϕh denotes the functional derivative (sometimes also called the Volterra derivative [135]) of the Hamiltonian density with respect to ϕ. Note that in the above equation integration by parts has been applied along with the neglect of the boundary terms. As has been mentioned above, the neglect of boundary integrals is related to the fact that the focus in this section is on closed systems in which boundary eects are irrelevant. Performing integration by parts with respect to time on the rst integral on the right-hand side of (4.9), the rst stationary condition eventually yields

$$\int\_{0}^{T} \int\_{\mathcal{B}} \delta \boldsymbol{\varphi} \cdot (\dot{\mathbf{p}} + \delta\_{\mathbf{p}} h) \, \mathrm{d}V \mathrm{d}t = 0 \, . \tag{4.10}$$

The second stationary condition is given by

$$\begin{split} DA(\boldsymbol{\varphi}, \mathbf{p}) \cdot \delta \mathbf{p} &= \frac{d}{d\varepsilon} \int\_{0}^{T} \int\_{\mathcal{B}} \left( (\mathbf{p} + \varepsilon \delta \mathbf{p}) \cdot \dot{\boldsymbol{\varphi}} - h(\boldsymbol{\varphi}, \mathbf{F}, \mathbf{p} + \varepsilon \delta \mathbf{p}) \right) \, \mathrm{d}V \, \mathrm{d}t \bigg|\_{\varepsilon=0} \\ &= \int\_{0}^{T} \int\_{\mathcal{B}} \delta \mathbf{p} \cdot (\dot{\boldsymbol{\varphi}} - \delta\_{\mathbf{p}} h) \, \mathrm{d}V \, \mathrm{d}t \\ &= 0 \end{split} \tag{4.11}$$

Since the variations δp and δϕ are arbitrary (apart from the endpoint conditions on δϕ), the stationary conditions (4.10) and (4.11) give rise to the local form of the equations of motion

$$\begin{aligned} \dot{\varphi} &= \delta\_{\mathbf{p}} h \ & \text{,} \\ \dot{\mathbf{p}} &= -\delta\_{\mathbf{p}} h \ & \text{,} \end{aligned} \tag{4.12}$$

for all X ∈ B and t ∈ [0, T]. Taking into account the specic form of the Hamiltonian density function (4.7), the functional derivatives on the right-hand side of (4.12) are given by

$$
\delta\_{\varphi}h = \partial\_{\varphi}h - \text{Div}(\partial\_{\mathbf{F}}h) = -\mathbf{b} - \text{Div}\left(\partial\_{\mathbf{F}}W\right)\,,
$$

and

$$
\delta\_\mathbf{p} h = \partial\_\mathbf{p} h = \rho^{-1} \mathbf{p} \cdot \mathbf{n}
$$

Upon introduction of the rst Piola-Kirchho stress tensor P = ∂FW, the local form of the equations of motion (4.12) can be recast as

$$
\rho \ddot{\varphi} = \text{Div}\left(\mathbf{P}\right) + \mathbf{b} \dots
$$

This is the familiar Lagrangian form of the balance law for linear momentum (see, e.g., [136]). To translate the equations of motion (4.12) into the GENERIC bracket form (4.1), we now consider functionals of the form

$$\mathcal{A}(\varphi, \mathbf{p}) = \int\_{\mathcal{B}} a(\varphi, \mathbf{F}, \mathbf{p}) \, \mathrm{d}V \, . \tag{4.13}$$

Dierentiation with respect to time yields

$$\begin{split} \frac{d\mathcal{A}}{dt} &= \int\_{\mathcal{B}} \frac{d}{dt} a(\boldsymbol{\varphi}, \mathbf{F}, \mathbf{p}) \, \mathrm{d}V \\ &= \int\_{\mathcal{B}} \left( \partial\_{\boldsymbol{\varphi}} a \cdot \dot{\boldsymbol{\varphi}} + \partial\_{\mathbf{F}} a : \dot{\mathbf{F}} + \partial\_{\mathbf{p}} a \cdot \dot{\mathbf{p}} \right) \, \mathrm{d}V \\ &= \int\_{\mathcal{B}} \left( \delta\_{\boldsymbol{\varphi}} a \cdot \dot{\boldsymbol{\varphi}} + \delta\_{\mathbf{p}} a \cdot \dot{\mathbf{p}} \right) \, \mathrm{d}V \\ &= \int\_{\mathcal{B}} \left( \delta\_{\boldsymbol{\varphi}} a \cdot \delta\_{\mathbf{p}} h - \delta\_{\mathbf{p}} a \cdot \delta\_{\boldsymbol{\varphi}} h \right) \, \mathrm{d}V =: \left\{ \mathcal{A}, \mathcal{H} \right\} \end{split}$$

Note that integration by parts has been applied. In this connection the boundary terms have been neglected again. The resulting equation

$$\frac{\mathrm{d}\mathcal{A}}{\mathrm{d}t} = \left\{\mathcal{A}, \mathcal{H}\right\},\tag{4.14}$$

.

holds for arbitrary functionals of the form (4.13) and is sometimes referred to as the equation of motion in Poisson bracket form (see, e.g., [24]). Note that this equation represents the reversible part of the GENERIC evolution equation (4.1). The Poisson bracket introduced above assumes the canonical form

$$\left\{\mathcal{A}, \mathcal{B}\right\} = \int\_{\mathcal{B}} \left(\delta\_{\varphi} a \cdot \delta\_{\mathbf{p}} b - \delta\_{\mathbf{p}} a \cdot \delta\_{\varphi} b\right) \,\mathrm{d}V\,,\tag{4.15}$$

being the innitesimal version of the classical discrete Poisson bracket eq. (3.3). Hereby are A and B two arbitrary functionals of the form (4.13). It can be easily observed that the canonical Poisson bracket (4.15) is skew-symmetric, that is

$$\left\{\mathcal{A},\mathcal{B}\right\} = -\left\{\mathcal{B},\mathcal{A}\right\}.\tag{4.16}$$

Note that conservation of the Hamiltonian is an immediate consequence of property (4.16) implying {H, H} = 0, so that (4.14) yields dH/dt = 0. Moreover, it is well-known that the Poisson bracket satises the Jacobi identity which is of paramount importance for the time-structure invariance of reversible dynamics (see, e.g., [94]).

## 4.2. Extension to dissipative systems: Thermo-elastodynamics

Departing from the isothermal problem outlined above, we now aim at the inclusion of thermal eects. In particular, we focus on the dynamics of thermoelastic solids with heat conduction (see [136, Sec. 9.1] for a concise denition of a thermoelastic solid). We stress that we still focus on closed (or isolated) systems.

In thermal solid mechanics the selection of the independent thermodynamic variable is a crucial point. From the viewpoint of constitutive modeling the temperature is a natural choice. The temperature eld <sup>θ</sup> : B × I 7→ <sup>R</sup><sup>+</sup> typically enters the denition of the Helmholtz free energy density

$$
\psi = \overline{\psi}(\mathbf{F}, \theta) \,. \tag{4.17}
$$

Alternatively, the entropy density eld <sup>η</sup> : B × I 7→ <sup>R</sup> dened by

$$\eta = \overline{\eta}(\mathbf{F}, \theta) = -\partial\_{\theta} \overline{\psi}(\mathbf{F}, \theta) \ , \tag{4.18}$$

could be used as thermodynamic variable. Then, for example, the temperature can be expressed in terms of the entropy density by inverting the last equation leading to

$$
\theta = \ddot{\Theta}(\mathbf{F}, \eta) \,. \tag{4.19}
$$

Yet another viable alternative is to use the internal energy density eld <sup>u</sup> : B × I 7→ <sup>R</sup>, which follows from a Legendre transformation of the free energy density with respect to the temperature. Accordingly,

$$u = \overline{u}(\mathbf{F}, \theta) = \theta \,\overline{\eta}(\mathbf{F}, \theta) + \overline{\psi}(\mathbf{F}, \theta) \,. \tag{4.20}$$

As usual, we assume that ∂θu > 0 and ∂θη > 0 for all (F, θ) with det(F) > 0 and θ > 0. Using the internal energy density as independent thermodynamic variable, the temperature can be obtained by inverting the last equation leading to <sup>θ</sup> <sup>=</sup> Θ( <sup>b</sup> <sup>F</sup>, u). Making use of this relation, the entropy density (4.18) can be expressed in terms of the internal energy density via

$$
\eta = \overline{\eta}(\mathbf{F}, \hat{\Theta}(\mathbf{F}, u)) = \hat{\eta}(\mathbf{F}, u) \,. \tag{4.21}
$$

Furthermore, inserting (4.19) into (4.20), the internal energy density can be expressed in terms of the entropy density, i.e.

$$u = \widetilde{u}(\mathbf{F}, \eta) \,. \tag{4.22}$$

As a further consequence of the Legendre transform (4.20) we get the following formula for the temperature in terms of the entropy density:

$$
\theta = \Theta(\mathbf{F}, \eta) = \partial\_{\eta} \tilde{u}(\mathbf{F}, \eta) \,. \tag{4.23}
$$

A clear exposition of further interrelationships between the alternative thermodynamic variables can be found in [100, Sec. 2.3].

## 4.2.1. Free choice of the thermodynamic variable

Guided by [100], in what follows, we allow for the free choice of the independent thermodynamic state variable by introducing the variable <sup>τ</sup> : B × I 7→ <sup>R</sup>, which can be selected among the three aforementioned alternatives, i.e. τ ∈ {θ, η, u}. In particular, in view of (4.19) and (4.22), the thermodynamic state variable τ can be expressed in terms of the entropy density via

$$
\tau = \widetilde{\tau}(\mathbf{F}, \eta) \,. \tag{4.24}
$$

On the other hand, the entropy density can also be expressed in tems of τ , as can be observed from formulas (4.18) and (4.21). Accordingly, we introduce the map η 0 such that

$$
\eta = \eta'(\mathbf{F}, \tau) \,, \tag{4.25}
$$

for τ ∈ {θ, η, u}. The following relations will be needed in the sequel:

$$\begin{aligned} \partial\_{\eta} \widetilde{\tau} &= \left( \partial\_{\tau} \eta' \right)^{-1}, \\ \partial\_{\mathbf{F}} \widetilde{\tau} &= - \left( \partial\_{\tau} \eta' \right)^{-1} \partial\_{\mathbf{F}} \eta'. \end{aligned} \tag{4.26}$$

To proof these relations we consider the identity η = η 0 (F, <sup>τ</sup>e(F, η)) which follows from (4.24) and (4.25). Since ∂ηη = 1, we get 1 = ∂<sup>τ</sup> η <sup>0</sup>∂ητe, from which follows(4.26)1. Similarly, since ∂Fη = 0, we have 0 = ∂Fη <sup>0</sup> + ∂<sup>τ</sup> η <sup>0</sup>∂Fτe, from which follows (4.26)2. Eventually, we recall a very important formula for the temperature that is given by (see also [94] or [100])

$$
\Theta'(\mathbf{F}, \tau) = \frac{\partial\_{\tau} u'(\mathbf{F}, \tau)}{\partial\_{\tau} \eta'(\mathbf{F}, \tau)} \,. \tag{4.27}
$$

Note that if the entropy density is chosen as thermodynamic variable (i.e. τ = η), formula (4.23) is recovered. In the case τ = θ, formula (4.27) yields the relationship θ = ∂θu/∂θη which is an immediate consequence of the Legendre transform (4.20).

## 4.2.2. Entropy-based brackets

It is well-known that both the Poisson and the dissipative brackets are invariant under changes of the independent variables (see, e.g., [94] or Section 2.2, where this property has been veried for discrete systems). The Poisson bracket assumes a particularly simple form when the entropy density is chosen as the thermodynamic state variable. This is quite obvious since the reversible dynamics does not aect the entropy. Similarly, the dissipative bracket assumes its simplest form when the internal energy density is chosen as the thermodynamic state variable. This has been observed in [137] in the context of

a compressible, non-isothermal uid. Moreover, this observation lies at the heart of the special form of GENERIC proposed in [100].

In the present work we start with the entropy-based formulation of thermo-elastodynamics and subsequently apply a transformation of variables to obtain the expressions for the brackets depending on the choice of the thermodynamic state variable τ ∈ {θ, η, u}. In the entropy-based formulation we consider functionals of the form

$$\mathcal{A} = \tilde{\mathcal{A}}(\varphi, \mathbf{p}, \eta) = \int\_{\mathcal{B}} \tilde{a}(\varphi, \mathbf{F}, \mathbf{p}, \eta) \, \mathrm{d}V \, . \tag{4.28}$$

Remarkably, in the entropy-based formulation the Poisson bracket retains the canonical form (4.15), i.e.

$$\left\{\tilde{\mathcal{A}}, \tilde{\mathcal{B}}\right\} = \int\_{\mathcal{B}} \left(\delta\_{\varphi}\tilde{a} \cdot \delta\_{\mathbf{p}}\tilde{b} - \delta\_{\mathbf{p}}\tilde{a} \cdot \delta\_{\varphi}\tilde{b}\right) \,\mathrm{d}V\,,\tag{4.29}$$

while the dissipative bracket featuring in the GENERIC evolution equation (4.1) is given by (see, e.g., [105])

$$\mathbb{E}\left[\widetilde{\mathcal{A}},\widetilde{\mathcal{B}}\right] = \int\_{\mathcal{B}} \nabla \left(\widetilde{\Theta}^{-1} \delta\_{\eta} \widetilde{a}\right) \cdot \widetilde{\Theta}^{2} \mathbf{K} \nabla \left(\widetilde{\Theta}^{-1} \delta\_{\eta} \widetilde{b}\right) \,\mathrm{d}V\,\,. \tag{4.30}$$

Here <sup>K</sup> <sup>=</sup> <sup>K</sup><sup>e</sup> (F, η) is a positive semi-denite and symmetric second-order material conductivity tensor. Note that in the entropy-based formulation the temperature is given by (4.23). That is, in (4.30), Θ( <sup>e</sup> <sup>F</sup>, η) = <sup>∂</sup>ηue(F, η). It can be easily observed that the dissipative bracket (4.30) is symmetric, i.e.

$$\left[\widetilde{\mathcal{A}}, \widetilde{\mathcal{B}}\right] = \left[\widetilde{\mathcal{B}}, \widetilde{\mathcal{A}}\right], \tag{4.31}$$

and positive semi-denite, i.e. [Ae, <sup>A</sup>e] <sup>≥</sup> <sup>0</sup>.

## 4.2.3. Change of the thermodynamic variable

We next aim at a uniform description of the Poisson bracket and the dissipative bracket in terms of the thermodynamic state variable τ ∈ {θ, η, u}. To this end we consider functionals of the form

$$\mathcal{A} = \mathcal{A}'(\varphi, \mathbf{p}, \tau) = \int\_{\mathcal{B}} a'(\varphi, \mathbf{F}, \mathbf{p}, \tau) \, \mathrm{d}V \, \,, \tag{4.32}$$

and perform a change of the thermodynamic state variable. As has been outlined above the thermodynamic state variable τ can be expressed in terms of the entropy density through relation (4.24), i.e. <sup>τ</sup> <sup>=</sup> <sup>τ</sup>e(F, η). Consequently, the two alternative expressions for the functional A given by (4.28) and (4.32) can be linked to obtain

$$\widetilde{\mathcal{A}}(\varphi, \mathbf{p}, \eta) = \int\_{\mathcal{B}} \widetilde{a}(\varphi, \mathbf{F}, \mathbf{p}, \eta) \, \mathrm{d}V = \int\_{\mathcal{B}} a'(\varphi, \mathbf{F}, \mathbf{p}, \widetilde{\tau}(\mathbf{F}, \eta)) \, \mathrm{d}V \, \mathrm{d}V$$

To get the functional derivative of <sup>A</sup><sup>e</sup> with respect to <sup>ϕ</sup> we consider

$$\begin{split} & \frac{d}{d\varepsilon} \int\_{\mathcal{B}} a'(\boldsymbol{\varphi} + \varepsilon \delta \boldsymbol{\varphi}, \mathbf{F} + \varepsilon \nabla \delta \boldsymbol{\varphi}, \mathbf{p}, \tilde{\tau} (\mathbf{F} + \varepsilon \nabla \delta \boldsymbol{\varphi}, \boldsymbol{\eta})) \, \mathrm{d}V \bigg|\_{\varepsilon=0} \\ & \qquad = \int\_{\mathcal{B}} \left( \partial\_{\boldsymbol{\varphi}} a' \cdot \delta \boldsymbol{\varphi} + \left( \partial\_{\mathbf{F}} a' + \partial\_{\boldsymbol{\tau}} a' \partial\_{\mathbf{F}} \tilde{\tau} \right) : \nabla \delta \boldsymbol{\varphi} \right) \, \mathrm{d}V \\ & \qquad = \int\_{\mathcal{B}} \delta \boldsymbol{\varphi} \cdot \left( \partial\_{\boldsymbol{\varphi}} a' - \mathrm{Div} (\partial\_{\mathbf{F}} a' + \partial\_{\boldsymbol{\tau}} a' \partial\_{\mathbf{F}} \tilde{\tau}) \right) \, \mathrm{d}V \\ & \qquad = \int\_{\mathcal{B}} \delta \boldsymbol{\varphi} \cdot \delta\_{\boldsymbol{\varphi}} \tilde{a} \, \mathrm{d}V \, . \end{split}$$

Here, again integration by parts and neglect of the boundary terms has been applied. Similarly, we obtain

$$\begin{split} & \left. \frac{d}{d\varepsilon} \int\_{\mathcal{B}} a'(\boldsymbol{\varphi}, \mathbf{F}, \mathbf{p}, \widetilde{\tau}(\mathbf{F}, \boldsymbol{\eta} + \varepsilon \delta \boldsymbol{\eta})) \, \mathrm{d}V \right|\_{\boldsymbol{\varepsilon} = 0} \\ & \qquad = \int\_{\mathcal{B}} \delta \boldsymbol{\eta} \partial\_{\tau} a' \partial\_{\boldsymbol{\eta}} \widetilde{\tau} \, \mathrm{d}V \\ & \qquad = \int\_{\mathcal{B}} \delta \boldsymbol{\eta} \delta\_{\boldsymbol{\eta}} \widetilde{a} \, \mathrm{d}V \, \mathrm{d}. \end{split}$$

To summarize, we have the following relationships

$$\begin{aligned} \delta\_{\varphi}\tilde{a} &= \delta\_{\varphi}a' - \text{Div}(\delta\_{\tau}a'\partial\_{\mathbf{F}}\tilde{\tau})\ , \\ \delta\_{\mathbf{p}}\tilde{a} &= \delta\_{\mathbf{p}}a' \ , \\ \delta\_{\eta}\tilde{a} &= \delta\_{\tau}a'\partial\_{\eta}\tilde{\tau}\ , \end{aligned}$$

where, in view of (4.32), δτa <sup>0</sup> = ∂τa 0 . The derivatives <sup>∂</sup>ητ<sup>e</sup> and <sup>∂</sup>Fτ<sup>e</sup> in the above equation can be substituted from (4.26) to obtain

$$\begin{split} \delta\_{\mathbf{p}}\widetilde{a} &= \delta\_{\mathbf{p}}a' + \mathrm{Div}\left(\frac{\delta\_{\tau}a'}{\partial\_{\tau}\eta'}\partial\_{\mathbf{F}}\eta'\right), \\ \delta\_{\mathbf{p}}\widetilde{a} &= \delta\_{\mathbf{p}}a', \\ \delta\_{\eta}\widetilde{a} &= \frac{\delta\_{\tau}a'}{\partial\_{\tau}\eta'}. \end{split} \tag{4.33}$$

These relationships can now be applied in the transformation of the brackets. Inserting (4.33)1,<sup>2</sup> into the entropy-based expression for the Poisson bracket (4.29), we arrive at the Poisson bracket formulated in terms of τ ∈ {θ, η, u}:

$$\begin{split} \left\{ \mathcal{A}', \mathcal{B}' \right\} &= \int\_{\mathcal{B}} \left( \delta\_{\varphi} a' \cdot \delta\_{\mathbf{p}} b' - \delta\_{\mathbf{p}} a' \cdot \delta\_{\mathbf{p}} b' \right) \mathrm{d}V \\ &+ \int\_{\mathcal{B}} \left( \mathrm{Div} \left( \frac{\delta\_{\tau} a'}{\partial\_{\tau} \eta'} \partial\_{\mathbf{F}} \eta' \right) \cdot \delta\_{\mathbf{p}} b' - \delta\_{\mathbf{p}} a' \cdot \mathrm{Div} \left( \frac{\delta\_{\tau} b'}{\partial\_{\tau} \eta'} \partial\_{\mathbf{F}} \eta' \right) \right) \mathrm{d}V \,. \end{split} \tag{4.34}$$

59


Table 4.1.: Specic relationships needed for the evaluation of the uniform expressions for the Poisson bracket (4.34) and the dissipative bracket (4.35), depending on the choice for the thermodynamic state variable τ ∈ {θ, η, u}.

Obviously, symmetry property (4.16) still holds. That is, {A<sup>0</sup> , B <sup>0</sup>} = −{B<sup>0</sup> , A<sup>0</sup>}. Substituting (4.33)<sup>3</sup> into expression (4.30) for the entropy-based dissipative bracket we get the dissipative bracket in terms of τ ∈ {θ, η, u}:

$$\left[\mathcal{A}',\mathcal{B}'\right] = \int\_{\mathcal{B}} \nabla \left(\frac{\delta\_{\tau}a'}{\partial\_{\tau}u'}\right) \cdot (\Theta')^2 \mathbf{K} \nabla \left(\frac{\delta\_{\tau}b'}{\partial\_{\tau}u'}\right) \mathrm{d}V \,. \tag{4.35}$$

Note that in the last equation Θ<sup>0</sup> is given by formula (4.27) for the temperature and the material conductivity tensor is given by K = K(F, Θ<sup>0</sup> ) = K<sup>0</sup> (F, τ ). Moreover, symmetry property (4.31) is still satised being [A<sup>0</sup> , B 0 ] = [B 0 , A<sup>0</sup> ].

Depending on the choice for τ ∈ {θ, η, u}, specic expressions for the respective brackets can be obtained from (4.34) and (4.35). In this connection Table 4.1 provides a summary of specic formulas. With regard to Table 4.1 it can be easily concluded that the Poisson bracket (4.34) assumes its simplest form in the entropy-based formulation, while the dissipative bracket (4.35) assumes its simplest form in the formulation based on the internal energy density.

Example: The Helmholtz free energy density (4.17) that will be used in the numerical investigations (see Section 4.5 and, for the following Chapters, Sections 5.4 and 6.5) is given by

$$
\psi(\mathbf{F}, \theta) = \psi\_1(\mathbf{F}) + \psi\_2(\theta) - (\theta - \theta\_0)\psi\_3(J) \, , \tag{4.36}
$$

where

$$\begin{aligned} \psi\_1(\mathbf{F}) &= \frac{\mu}{2} \left( \mathbf{F} : \mathbf{F} - 3 - 2 \log J - \frac{2}{3} \left( J - 1 \right)^2 \right) + W\_{\text{vol}}(J) \,, \\ \psi\_2(\theta) &= c \left( \theta - \theta\_0 - \theta \log \left( \frac{\theta}{\theta\_0} \right) \right) \,, \\ \psi\_3(J) &= 3 \beta W\_{\text{vol}}'(J) \,, \\ W\_{\text{vol}}(J) &= \frac{\lambda + \frac{2}{3} \mu}{4} \left( (\log J)^2 + (J - 1)^2 \right) \, . \end{aligned} \tag{4.37}$$

Here, J = det(F) is the determinant of the deformation gradient and µ, λ are prescribed parameters, c > 0 is the specic heat capacity at constant deformation, β is the coecient of thermal expansion, and θ<sup>0</sup> is the reference temperature. We refer to [74, App. C] for a detailed investigation of the specic Helmholtz free energy density (4.36). It is worth noting that the Helmholtz free energy density (4.36) is consistent with the special case of linear thermo-elasticity (see Appendix A.1). It is now a straightforward exercise to calculate the quantities needed in Table 4.1. In particular, the temperature-based formulation (cf. θ-row in Table 4.1) yields

$$\begin{aligned} \overline{\eta}(\mathbf{F},\theta) &= c \log \left(\frac{\theta}{\theta\_0}\right) + \psi\_3(J) \,, \\ \overline{u}(\mathbf{F},\theta) &= \psi\_1(\mathbf{F}) + c(\theta - \theta\_0) + \theta\_0 \psi\_3(J) \,, \\ \partial\_\theta \overline{\eta}(\mathbf{F},\theta) &= \frac{c}{\theta} \,, \\ \partial\_\theta \overline{u}(\mathbf{F},\theta) &= c \,, \\ \partial\_\mathbf{F} \overline{\eta}(\mathbf{F},\theta) &= \psi\_3'(J) \text{cof}(\mathbf{F}) \,, \\ \partial\_\mathbf{F} \overline{u}(\mathbf{F},\theta) &= \partial\_\mathbf{F} \psi\_1(\mathbf{F}) + \theta\_0 \psi\_3'(J) \text{cof}(\mathbf{F}) \, . \end{aligned}$$

Here, cof(F) = J(F) <sup>−</sup><sup>T</sup> denotes the cofactor of the deformation gradient. The formulation based on the entropy density (cf. η-row in Table 4.1) gives

$$\begin{aligned} \widetilde{u}(\mathbf{F}, \eta) &= \psi\_1(\mathbf{F}) + c \left( \theta\_0 e^{\frac{\eta - \psi\_3(J)}{c}} - \theta\_0 \right) + \theta\_0 \psi\_3(J) \,, \\\ \partial\_{\eta} \widetilde{u}(\mathbf{F}, \eta) &= \theta\_0 e^{\frac{\eta - \psi\_3(J)}{c}} = \widetilde{\Theta}(\mathbf{F}, \eta) \\\ \partial\_{\mathbf{F}} \widetilde{u}(\mathbf{F}, \eta) &= \partial\_{\mathbf{F}} \psi\_1(\mathbf{F}) - \left( \widetilde{\Theta}(\mathbf{F}, \eta) - \theta\_0 \right) \psi\_3'(J) \text{cof}(\mathbf{F}) \, . \end{aligned}$$

Moreover, the formulation based on the internal energy density (cf. u-row in Table 4.1) leads to

$$
\begin{split}
\widehat{\eta}(\mathbf{F},u) &= c \log \left(\frac{u + c\theta\_0 - \psi\_1(\mathbf{F}) - \theta\_0\psi\_3(J)}{c}\right) + \psi\_3(J) \,, \\
\partial\_u \widehat{\eta}(\mathbf{F},u) &= \left(\frac{u + c\theta\_0 - \psi\_1(\mathbf{F}) - \theta\_0\psi\_3(J)}{c}\right)^{-1} = \left[\widehat{\Theta}(\mathbf{F},u)\right]^{-1} \,, \\
\partial\_\mathbf{F} \widehat{\eta}(\mathbf{F},u) &= -\left[\widehat{\Theta}(\mathbf{F},u)\right]^{-1} \left(\partial\_\mathbf{F} \psi\_1(\mathbf{F}) + \theta\_0 \psi\_3'(J) \text{cof}(\mathbf{F})\right) + \psi\_3'(J) \text{cof}(\mathbf{F}) \; .
\end{split}
$$

## 4.2.4. GENERIC evolution equation

The GENERIC evolution equation (4.1) can now be considered in more detail by using the previously derived Poisson bracket (4.34) along with the dissipative bracket (4.35). Accordingly, we obtain

$$\frac{\mathrm{d}\mathcal{A}'}{\mathrm{d}t} = \left\{ \mathcal{A}', \mathcal{E}' \right\} + \left[ \mathcal{A}', \mathcal{S}' \right] \,, \tag{4.38}$$

for functionals A<sup>0</sup> of the form (4.32). Note that the evolution equation (4.38) represents a uniform description of thermo-elastodynamics in terms of the thermodynamic state variable τ ∈ {θ, η, u}. In (4.38), the total energy of the system is given by the functional

$$\mathcal{E}'(\varphi, \mathbf{p}, \tau) = \int\_{\mathcal{B}} e'(\varphi, \mathbf{F}, \mathbf{p}, \tau) \, \mathrm{d}V \, \,, \tag{4.39}$$

with associated density function

$$e'(\varphi, \mathbf{F}, \mathbf{p}, \tau) = \frac{1}{2} \rho^{-1} \mathbf{p} \cdot \mathbf{p} + u'(\mathbf{F}, \tau) - \mathbf{b} \cdot \boldsymbol{\varphi} \,. \tag{4.40}$$

Note that in the isothermal case the energy density (4.40) reduces to the Hamiltonian density (4.7). In addition to the total energy, the total entropy of the system acts as second generator in the GENERIC evolution equation (4.38) and is given by the functional

$$\mathcal{S}'(\varphi,\tau) = \int\_{\mathcal{B}} \eta'(\mathbf{F},\tau) \, \mathrm{d}V \, \mathrm{d}V \,. \tag{4.41}$$

In the last equation, <sup>η</sup> : B × I 7→ <sup>R</sup> is the entropy density which has been introduced in (4.25). For later use we mention that the following functional derivatives follow from the denition of the total energy (4.39) and the total entropy (4.41), respectively:

$$\begin{aligned} \delta\_{\mathbf{p}}e' &= \partial\_{\mathbf{p}}e' - \text{Div}\partial\_{\mathbf{F}}e' = -\mathbf{b} - \text{Div}\partial\_{\mathbf{F}}u', \\ \delta\_{\mathbf{p}}e' &= \partial\_{\mathbf{p}}e' = \rho^{-1}\mathbf{p} \ , \\ \delta\_{\tau}e' &= \partial\_{\tau}e' = \partial\_{\tau}u' \ , \end{aligned} \tag{4.42}$$

and

$$\begin{aligned} \delta\_{\mathbf{p}} \eta' &= -\text{Div}\partial\_{\mathbf{F}} \eta', \\ \delta\_{\mathbf{p}} \eta' &= \mathbf{0} \end{aligned} \tag{4.43}$$
 
$$\begin{aligned} \delta\_{\tau} \eta' &= \partial\_{\tau} \eta' \end{aligned} \tag{4.43}$$

Now the left-hand side of (4.38) can be written as

$$\begin{split} \frac{d}{dt} \mathcal{A}' &= \int\_{\mathcal{B}} \frac{d}{dt} a'(\boldsymbol{\varphi}, \mathbf{F}, \mathbf{p}, \tau) \, \mathrm{d}V \\ &= \int\_{\mathcal{B}} \left( \partial\_{\boldsymbol{\varphi}} a' \cdot \dot{\boldsymbol{\varphi}} + \partial\_{\mathbf{F}} a' : \dot{\mathbf{F}} + \partial\_{\mathbf{p}} a' \cdot \dot{\mathbf{p}} + \partial\_{\tau} a' \dot{\tau} \right) \, \mathrm{d}V \\ &= \int\_{\mathcal{B}} \left( \delta\_{\boldsymbol{\varphi}} a' \cdot \dot{\boldsymbol{\varphi}} + \delta\_{\mathbf{p}} a' \cdot \dot{\mathbf{p}} + \delta\_{\tau} a' \dot{\tau} \right) \, \mathrm{d}V \, . \end{split} \tag{4.44}$$

Note that again integration by parts along with the neglect of boundary terms has been applied. The Poisson bracket on the right-hand side of (4.38) follows from (4.34) together with (4.40). Thus we obtain

$$\begin{split} \left\{ \mathcal{A}', \mathcal{E}' \right\} &= \int\_{\mathcal{B}} \left( \delta\_{\varphi} a' \cdot \partial\_{\mathbf{p}} e' - \delta\_{\mathbf{p}} a' \cdot \delta\_{\mathbf{p}} e' \right) \mathrm{d}V \\ &+ \int\_{\mathcal{B}} \left( \mathrm{Div} \left( \frac{\delta\_{\tau} a'}{\partial\_{\tau} \eta'} \partial\_{\mathbf{F}} \eta' \right) \cdot \partial\_{\mathbf{p}} e' - \delta\_{\mathbf{p}} a' \cdot \mathrm{Div} \left( \frac{\partial\_{\tau} u'}{\partial\_{\tau} \eta'} \partial\_{\mathbf{F}} \eta' \right) \right) \mathrm{d}V \,. \end{split} \tag{4.45}$$

Similarly, the dissipative bracket on the right-hand side of (4.38) follows from inserting (4.41) into (4.35). Accordingly,

$$\left[\mathcal{A}',\mathcal{S}'\right] = \int\_{\mathcal{B}} \nabla \left(\frac{\delta\_{\tau}a'}{\partial\_{\tau}u'}\right) \cdot (\Theta')^2 \mathbf{K} \nabla \left(\frac{\partial\_{\tau}\eta'}{\partial\_{\tau}u'}\right) \mathrm{d}V \,. \tag{4.46}$$

Next, we introduce the material heat ux vector <sup>Q</sup> : B × I 7→ <sup>R</sup> 3 through Q = Q 0 (F, τ ), where

$$\begin{split} \mathbf{Q}' &= (\Theta')^2 \mathbf{K} \nabla \left( \frac{\partial\_{\tau} \eta'}{\partial\_{\tau} u'} \right) \\ &= (\Theta')^2 \mathbf{K} \nabla \left( \frac{1}{\Theta'} \right) \\ &= -\mathbf{K} \nabla \Theta' \end{split} \tag{4.47}$$

Note that in the above equation use has been made of relationship (4.27) for the temperature eld Θ<sup>0</sup> (F, τ ). Now, the bracket (4.46) can be recast in the form

$$\left[\mathcal{A}',\mathcal{S}'\right] = \int\_{\mathcal{B}} \nabla \left(\frac{\delta\_{\tau}a'}{\partial\_{\tau}u'}\right) \cdot \mathbf{Q}' \,\mathrm{d}V \,. \tag{4.48}$$

It is important to emphasize that the present brackets need to satisfy the fundamental degeneracy requirements of the GENERIC framework

$$\begin{aligned} \left\{ \mathcal{A}', \mathcal{S}' \right\} &= 0 \\ \left[ \mathcal{A}', \mathcal{E}' \right] &= 0 \end{aligned} \tag{4.49}$$

The validity of these equations for arbitrary functionals A<sup>0</sup> can be easily veried. The degeneracy requirements (4.49) ensure the thermodynamical consistency of the GENERIC evolution equation (4.38) in the sense that dE <sup>0</sup>/dt = 0 and dS <sup>0</sup>/dt = - S 0 , S 0 ≥ 0. Note that the last inequality follows from the above expression for the dissipative bracket together with the positive semi-deniteness of the material conductivity tensor K. Accordingly, the total entropy is non-decreasing in the present case of isolated systems in which the net heating vanishes.

Taking into account relations (4.44), (4.45) and (4.48), the GENERIC evolution equation (4.38) can be recast in the form

$$\begin{split} 0 &= \int\_{\mathcal{B}} \delta\_{\mathbf{p}} a' \cdot (\dot{\varphi} - \partial\_{\mathbf{p}} e') \, \mathrm{d}V \\ &+ \int\_{\mathcal{B}} \delta\_{\mathbf{p}} a' \cdot \left( \dot{\mathbf{p}} + \delta\_{\mathbf{p}} e' + \mathrm{Div} \left( \frac{\partial\_{\tau} u'}{\partial\_{\tau} \eta'} \partial\_{\mathbf{F}} \eta' \right) \right) \, \mathrm{d}V \\ &+ \int\_{\mathcal{B}} \left( \delta\_{\tau} a' \dot{\tau} - \mathrm{Div} \left( \frac{\delta\_{\tau} a'}{\partial\_{\tau} \eta'} \partial\_{\mathbf{F}} \eta' \right) \cdot \partial\_{\mathbf{p}} e' - \nabla \left( \frac{\delta\_{\tau} a'}{\partial\_{\tau} \eta'} \right) \cdot \mathbf{Q}' \right) \, \mathrm{d}V \, . \end{split} \tag{4.50}$$

This equation has to hold for arbitrary functionals A<sup>0</sup> of the form (4.32). Consequently, the rst two integrals in the last equation yield the local equations

$$\begin{aligned} \dot{\varphi} &= \rho^{-1} \mathbf{p} \ , \\ \dot{\mathbf{p}} &= \mathbf{b} + \text{Div} \mathbf{P} \ , \end{aligned} \tag{4.51}$$

that must be satised for all X ∈ B and t > 0. In (4.51)2, the rst Piola-Kirchho stress tensor P = P 0 (F, τ ) can be identied as

$$\mathbf{P}' = \partial\_{\mathbf{F}} u' - \Theta' \partial\_{\mathbf{F}} \eta' \,, \tag{4.52}$$

where Θ<sup>0</sup> is given by eq. (4.27). Concerning the third integral in (4.50), integration by parts and neglect of the boundary terms yields

$$\int\_{\mathcal{B}} \delta\_{\tau} a' \left( \dot{\tau} + \frac{1}{\partial\_{\tau} \eta'} \partial\_{\mathbf{F}} \eta' : \nabla (\rho^{-1} \mathbf{p}) + \frac{1}{\partial\_{\tau} u'} \text{Div} \mathbf{Q}' \right) \, \mathrm{d}V = 0 \,\dots$$

Correspondingly, we arrive at the local form of the evolution of the thermodynamic variable τ ∈ {θ, η, u}, given by

$$\dot{\tau} = -\frac{1}{\partial\_{\tau} \eta'} \partial\_{\mathbf{F}} \eta' : \nabla(\rho^{-1} \mathbf{p}) - \frac{1}{\partial\_{\tau} u'} \text{Div} \mathbf{Q}' \,, \tag{4.53}$$

which has to hold for all X ∈ B and t > 0.

Next, we summarize main ingredients of the present GENERIC-based formulation resulting from the specic choice of the thermodynamic state variable. Choosing τ = θ leads to the temperature-based formulation. With regard to (4.52), the rst Piola-Kirchho stress tensor assumes the form

$$
\overline{\mathbf{P}} = \partial\_{\mathbf{F}} \overline{u} - \theta \partial\_{\mathbf{F}} \overline{\eta} \ .
$$

Furthermore, (4.53) yields the temperature evolution equation given by

$$\theta = -\frac{1}{\partial\_{\theta}\overline{\eta}}\partial\_{\mathbf{F}}\overline{\eta} : \nabla(\rho^{-1}\mathbf{p}) - \frac{1}{\partial\_{\theta}\overline{u}}\text{Div}\overline{\mathbf{Q}} \ .$$

The formulation in terms of the entropy density results from the choice τ = η leading to the rst Piola-Kirchho stress tensor

$$\mathbf{P} = \partial\_{\mathbf{F}} \widetilde{u} \, .$$

The evolution of the specic entropy is governed by

$$
\dot{\eta} = -\frac{1}{\tilde{\Theta}} \text{Div} \tilde{\mathbf{Q}} \, .
$$

Choosing τ = u yields the formulation based on the internal energy density with associated rst Piola-Kirchho stress tensor

$$
\widehat{\mathbf{P}} = -\widehat{\Theta}\partial\_{\mathbf{F}}\widehat{\eta}\ .
$$

Furthermore, the evolution of the internal energy density follows from

$$\dot{u} = -\hat{\Theta}\partial\_{\mathbf{F}}\hat{\eta} : \nabla(\rho^{-1}\mathbf{p}) - \text{Div}\hat{\mathbf{Q}} \ .$$

Remark 3. Apart from the dead loads b = −∂ϕe 0 , non-potential material body forces per unit of reference volume, <sup>f</sup> : B × I 7→ <sup>R</sup> 3 , and a heat supply eld per unit of reference volume, <sup>R</sup> : B × I 7→ <sup>R</sup>, can also be included into the present framework. To this end the supply term

$$\left(\mathcal{A}',\mathcal{F}'\right) = \int\_{\mathcal{B}} \left(\delta\_{\mathbf{p}} a' \cdot \mathbf{f} + \frac{\delta\_{\tau} a'}{\partial\_{\tau} u'} R\right) \,\mathrm{d}V\,,\tag{4.54}$$

needs be appended to the right-hand side of the GENERIC evolution equation (4.38).

## 4.3. GENERIC for open systems

As pointed out in [94], there is no need to pay special attention to the boundary conditions if one is interested in the local eld equations within an isolated system. Indeed, the majority of works dealing with the GENERIC formalism have been conducted in the context of closed systems. However, when it comes to numerical simulation, the boundary conditions are obviously of paramount importance. Consequently, we now aim at the inclusion of boundary eects (Fig. 4.2) into the GENERIC-based framework for nite-strain thermo-elastodynamics developed above. To this end, we build on ideas presented in [138] and apply them to the present framework for thermoelastic solids. Accordingly, the Poisson bracket (4.34) and the dissipative bracket (4.35) are now viewed as full brackets that can be decomposed according to

$$\begin{aligned} \left\{ \mathcal{A}', \mathcal{B}' \right\} &= \left\{ \mathcal{A}', \mathcal{B}' \right\}\_{\text{bulk}} + \left\{ \mathcal{A}', \mathcal{B}' \right\}\_{\text{bound}}, \\\left[ \mathcal{A}', \mathcal{B}' \right] &= \left[ \mathcal{A}', \mathcal{B}' \right]\_{\text{bulk}} + \left[ \mathcal{A}', \mathcal{B}' \right]\_{\text{bound}}. \end{aligned} \tag{4.55}$$

That is, the full brackets are split additively into bulk and boundary parts, respectively.

Figure 4.2.: Reference conguration B with boundary ∂B and current conguration ϕ(B, t) at time t. External tractions t = PN act on the boundary of the current conguration. In addition to that, the heat ux across the current boundary is denoted by q = Q · N.

This split can be accomplished by applying integration by parts to the full brackets. Following [138], the proper interpretation of the GENERIC evolution equation (4.38) is now given by

$$\frac{\mathrm{d}\mathcal{A}'}{\mathrm{d}t} = \left\{ \mathcal{A}', \mathcal{E}' \right\}\_{\mathrm{bulk}} + \left[ \mathcal{A}', \mathcal{S}' \right]\_{\mathrm{bulk}}.\tag{4.56}$$

The degeneracy requirements (4.49) for the full brackets transfer to the corresponding degeneracy requirements for the bulk brackets

$$\begin{aligned} \left\{ \mathcal{A}', \mathcal{S}' \right\}\_{\text{bulk}} &= 0 \end{aligned} , \tag{4.57}$$
  $\left[ \mathcal{A}', \mathcal{E}' \right]\_{\text{bulk}} = 0 \,, \tag{4.57}$ 

for arbitrary functionals A<sup>0</sup> .

Invoking the symmetry properties of the full brackets, i.e. {A<sup>0</sup> , B <sup>0</sup>} = −{B<sup>0</sup> , A<sup>0</sup>} and [A<sup>0</sup> , B 0 ] = [B 0 , A<sup>0</sup> ], (4.55) leads to

$$\begin{aligned} \left\{ \mathcal{A}', \mathcal{B}' \right\}\_{\text{bulk}} &= -\left\{ \mathcal{B}', \mathcal{A}' \right\}\_{\text{bulk}} - \left( \left\{ \mathcal{A}', \mathcal{B}' \right\}\_{\text{bound}} + \left\{ \mathcal{B}', \mathcal{A}' \right\}\_{\text{bound}} \right) \,, \\ \left[ \mathcal{A}', \mathcal{B}' \right]\_{\text{bulk}} &= \left[ \mathcal{B}', \mathcal{A}' \right]\_{\text{bulk}} - \left( \left[ \mathcal{A}', \mathcal{B}' \right]\_{\text{bound}} - \left[ \mathcal{B}', \mathcal{A}' \right]\_{\text{bound}} \right) \,, \end{aligned} \tag{4.58}$$

Substitution from (4.58) into (4.56) yields the GENERIC evolution equation for open systems

$$\begin{split} \frac{\mathrm{d}\mathcal{A}'}{\mathrm{d}t} &= -\left\{ \mathcal{E}', \mathcal{A}' \right\}\_{\mathrm{bulk}} + \left[ \mathcal{S}', \mathcal{A}' \right]\_{\mathrm{bulk}} \\ &- \left( \left\{ \mathcal{A}', \mathcal{E}' \right\}\_{\mathrm{bound}} + \left\{ \mathcal{E}', \mathcal{A}' \right\}\_{\mathrm{bound}} + \left[ \mathcal{A}', \mathcal{S}' \right]\_{\mathrm{bound}} - \left[ \mathcal{S}', \mathcal{A}' \right]\_{\mathrm{bound}} \right) \ . \end{split} \tag{4.59}$$

In the sequel, the above equation will be used to formulate important balance laws and to derive the variational formulation of large-strain thermo-elastodynamics. It is worth mentioning that Remark 3 also applies to evolution equation (4.59).

## 4.3.1. Specic brackets

Next, we provide the specic brackets featuring in the GENERIC evolution equation (4.59) pertaining to thermoelastic solids. Proceeding along the lines of [138], we start from the full brackets derived above and apply integration by parts. In this connection, the goal is to get the derivatives (δϕa 0 , δpa 0 , δτa 0 ) free of any spatial derivatives. To this end, we rewrite the full Poisson bracket (4.34) as

$$\begin{split} \left\{ \mathcal{A}', \mathcal{B}' \right\} &= \int\_{\mathcal{B}} \left( \partial\_{\mathbf{p}} a' \cdot \partial\_{\mathbf{p}} b' - \partial\_{\mathbf{p}} a' \cdot \left( \partial\_{\mathbf{p}} b' - \text{Div} \left( \partial\_{\mathbf{F}} b' - \frac{\partial\_{\tau} b'}{\partial\_{\tau} \eta} \partial\_{\mathbf{F}} \eta' \right) \right) \right) \text{d}V \\ &- \int\_{\mathcal{B}} \text{Div} \left( \partial\_{\mathbf{F}} a' - \frac{\partial\_{\tau} a'}{\partial\_{\tau} \eta'} \partial\_{\mathbf{F}} \eta' \right) \cdot \partial\_{\mathbf{p}} b' \,\text{d}V \,. \end{split} \tag{4.60}$$

Applying integration by parts to the second integral on the right-hand side of (4.60), and taking into account the additive decomposition (4.55)1, we obtain

$$\begin{split} \left\{ \mathcal{A}', \mathcal{B}' \right\}\_{\text{bulk}} &= \int\_{\mathcal{B}} \left( \partial\_{\varphi} a' \cdot \partial\_{\mathbf{p}} b' - \partial\_{\mathbf{p}} a' \cdot \left( \partial\_{\varphi} b' - \text{Div} \left( \partial\_{\mathbf{F}} b' - \frac{\partial\_{\tau} b'}{\partial\_{\tau} \eta'} \partial\_{\mathbf{F}} \eta' \right) \right) \right) \text{d}V \\ &+ \int\_{\mathcal{B}} \left( \partial\_{\mathbf{F}} a' - \frac{\partial\_{\tau} a'}{\partial\_{\tau} \eta'} \partial\_{\mathbf{F}} \eta' \right) : \nabla \partial\_{\mathbf{p}} b' \,\mathrm{d}V \,, \tag{4.61} \end{split} \tag{4.61}$$

and

$$\left\{ \mathcal{A}', \mathcal{B}' \right\}\_{\text{bound}} = - \int\_{\partial \mathcal{B}} \partial\_{\mathbf{p}} b' \cdot \left( \partial\_{\mathbf{F}} a' - \frac{\partial\_{\tau} a'}{\partial\_{\tau} \eta'} \partial\_{\mathbf{F}} \eta' \right) \mathbf{N} \, \mathrm{d}A \, . \tag{4.62}$$

Here, the vector N denotes the unit outward normal eld on the boundary ∂B of the reference conguration (Fig. 4.2). Similarly, applying integration by parts to the full dissipative bracket (4.35), yields the corresponding bulk bracket

$$\left[\mathcal{A}',\mathcal{B}'\right]\_{\text{bulk}} = -\int\_{\mathcal{B}} \frac{\partial\_{\tau}a'}{\partial\_{\tau}u'} \text{Div}\left((\Theta')^2 \mathbf{K} \nabla \left(\frac{\partial\_{\tau}b'}{\partial\_{\tau}u'}\right)\right) \text{d}V \,, \tag{4.63}$$

along with the boundary bracket

$$\left[\mathcal{A}',\mathcal{B}'\right]\_{\text{boun}} = \int\_{\partial\mathcal{B}} \frac{\partial\_{\tau}a'}{\partial\_{\tau}u'} \mathbf{N} \cdot (\Theta')^2 \mathbf{K} \nabla \left(\frac{\partial\_{\tau}b'}{\partial\_{\tau}u'}\right) \mathrm{d}A \,. \tag{4.64}$$

It can be easily veried that the degeneracy conditions (4.57) are satised by the bulk brackets (4.61) and (4.63), respectively. Table 4.2 contains a summary of the specic brackets appearing in the GENERIC evolution equations (4.59) for open systems. The brackets in Table 4.2 can be easily obtained from expressions (4.61) through (4.64) by taking into account the generators in (4.39) and (4.41).

Table 4.2.: Summary of the brackets featuring in the GENERIC evolution equations (4.59) for open systems.

 E 0 , A 0 bulk = − Z B b · ∂pa <sup>0</sup> + ρ <sup>−</sup><sup>1</sup><sup>p</sup> · ∂ϕa <sup>0</sup> <sup>−</sup> Div ∂Fa 0 − ∂τa 0 ∂<sup>τ</sup> η 0 ∂Fη 0 <sup>d</sup><sup>V</sup> + Z B P 0 : ∇∂pa <sup>0</sup> dV , - S 0 , A 0 bulk = − Z B 1 Θ<sup>0</sup> Div (Θ<sup>0</sup> ) <sup>2</sup>K<sup>∇</sup> ∂τa 0 ∂τu 0 <sup>d</sup>V , A 0 , E 0 boun = − Z ∂B ρ <sup>−</sup><sup>1</sup><sup>p</sup> · ∂Fa 0 − ∂τa 0 ∂<sup>τ</sup> η 0 ∂Fη 0 N dA , E 0 , A 0 boun = − Z ∂B ∂pa 0 · P <sup>0</sup>N dA , - A 0 , S 0 boun = Z ∂B ∂τa 0 ∂τu <sup>0</sup> <sup>N</sup> · <sup>Q</sup> 0 dA , - S 0 , A 0 boun = Z ∂B 1 <sup>Θ</sup><sup>0</sup> <sup>N</sup> · (Θ<sup>0</sup> ) <sup>2</sup>K<sup>∇</sup> ∂τa 0 ∂τu 0 dA .

## 4.3.2. Balance laws

Next, we deduce the balance laws in material (or Lagrangian) form from the GENERIC evolution equation (4.59). We start with the total linear momentum of the continuum body dened by L = R B p dV . In particular, we choose the density a <sup>0</sup> = l 0 ξ , with l 0 <sup>ξ</sup> = ξ · p, where <sup>ξ</sup> <sup>∈</sup> <sup>R</sup> 3 is arbitrary and constant. This corresponds to the functional L 0 <sup>ξ</sup> = ξ · L. Substituting L 0 ξ for A<sup>0</sup> in (4.59), the non-zero contributions on the right-hand side of (4.59) are given by E 0 ,L 0 ξ bulk = − R B b · ξ dV and E 0 ,L 0 ξ boun = − R ∂B ξ · PN dA. Consequently, we obtain

$$\boldsymbol{\xi} \cdot \frac{\mathrm{d}\mathbf{L}}{\mathrm{d}t} = \boldsymbol{\xi} \cdot \left( \int\_{\mathcal{B}} \mathbf{b} \, \mathrm{d}V + \int\_{\partial \mathcal{B}} \mathbf{PN} \, \mathrm{d}A \right) \, . \tag{4.65}$$

Due to the arbitrariness of <sup>ξ</sup> <sup>∈</sup> <sup>R</sup> 3 , (4.65) coincides with the balance law for linear momentum. Note that the parentheses on the right-hand side of (4.65) contain the resultant external loads applied to the continuum body (see also Fig. 4.2).

The total angular momentum relative to the origin of the inertial frame is dened by J = R <sup>B</sup> <sup>ϕ</sup> <sup>×</sup> <sup>p</sup> <sup>d</sup><sup>V</sup> . We choose <sup>a</sup> 0 = j 0 ξ , where j 0 <sup>ξ</sup> = ξ · (ϕ × p). Correspondingly, we substitute the functional J 0 <sup>ξ</sup> = ξ · J for A<sup>0</sup> in (4.59). The non-zero terms emanating from the right-hand side of (4.59) are given by E 0 ,J 0 ξ bulk = − R B b · (ξ × ϕ) dV and E 0 ,J 0 ξ boun = − R ∂B (ξ × ϕ) · PN dA. Note that in the bulk bracket use has been made of the symmetry condition FP<sup>T</sup> = PF<sup>T</sup> . Consequently, the GENERIC evolution equation (4.59) yields

$$\boldsymbol{\xi} \cdot \frac{\mathrm{d} \mathbf{J}}{\mathrm{d}t} = \boldsymbol{\xi} \cdot \left( \int\_{\mathcal{B}} \boldsymbol{\varphi} \times \mathbf{b} \, \mathrm{d}V + \int\_{\partial \mathcal{B}} \boldsymbol{\varphi} \times \mathbf{PN} \, \mathrm{d}A \right) \,. \tag{4.66}$$

Since <sup>ξ</sup> <sup>∈</sup> <sup>R</sup> 3 is arbitrary, the last equation corresponds to the balance of angular momentum. Note that the parentheses on the right-hand side of (4.66) contain the resultant external torque about the origin (see also Fig. 4.2).

Employing expression (4.39) for the total energy in the GENERIC evolution equation (4.59), we obtain

$$
\begin{split} \frac{d\mathcal{E}'}{dt} &= -\left\{\mathcal{E}',\mathcal{E}'\right\}\_{\text{bulk}} + \left[\mathcal{S}',\mathcal{E}'\right]\_{\text{bulk}} \\ &- \left(\left\{\mathcal{E}',\mathcal{E}'\right\}\_{\text{bound}} + \left\{\mathcal{E}',\mathcal{E}'\right\}\_{\text{bound}} + \left[\mathcal{E}',\mathcal{S}'\right]\_{\text{bound}} - \left[\mathcal{S}',\mathcal{E}'\right]\_{\text{bound}}\right) \\ &= -\left\{\mathcal{E}',\mathcal{E}'\right\} + \left[\mathcal{S}',\mathcal{E}'\right] - \left\{\mathcal{E}',\mathcal{E}'\right\}\_{\text{bound}} - \left[\mathcal{E}',\mathcal{S}'\right]\_{\text{bound}} \\ &= -\left(\left\{\mathcal{E}',\mathcal{E}'\right\}\_{\text{bound}} + \left[\mathcal{E}',\mathcal{S}'\right]\_{\text{bound}}\right) .\end{split}
$$

In the above equation use has been made of the decomposition of the full brackets in (4.55) along with the skew-symmetry of the full Poisson bracket and the degeneracy condition (4.49)2. Taking into account the specic brackets E 0 , E 0 boun = − R ∂B ρ <sup>−</sup><sup>1</sup><sup>p</sup> · PN <sup>d</sup><sup>A</sup> and - E 0 , S 0 boun = R <sup>∂</sup><sup>B</sup> <sup>N</sup> · <sup>Q</sup> <sup>d</sup>A, the above rate of change of the total energy can be recast in the form

$$\frac{d}{dt} \int\_{\mathcal{B}} \left(\frac{1}{2} \rho^{-1} \mathbf{p} \cdot \mathbf{p} + u'\right) \mathrm{d}V = \int\_{\mathcal{B}} \mathbf{b} \cdot \dot{\varphi} \, \mathrm{d}V + \int\_{\partial \mathcal{B}} \left(\rho^{-1} \mathbf{p} \cdot \mathbf{P} \mathbf{N} - \mathbf{N} \cdot \mathbf{Q}\right) \mathrm{d}A. \tag{4.67}$$

This is the balance of energy in material form which is in line with the rst law of thermodynamics (see, for example, [139, Sec. 2.3]).

Inserting expression (4.41) for the total entropy into the GENERIC evolution equation (4.59) yields

$$\begin{split} \frac{\mathrm{d}\mathcal{S}'}{\mathrm{d}t} &= -\left\{ \mathcal{E}', \mathcal{S}' \right\} + \left[ \mathcal{S}', \mathcal{S}' \right] - \left\{ \mathcal{S}', \mathcal{E}' \right\}\_{\mathrm{boun}} - \left[ \mathcal{S}', \mathcal{S}' \right]\_{\mathrm{boun}}, \\ &= \left[ \mathcal{S}', \mathcal{S}' \right] - \left[ \mathcal{S}', \mathcal{S}' \right]\_{\mathrm{boun}}, \end{split}$$

where again use has been made of the decomposition of the full brackets in (4.55), degeneracy condition (4.49)1, and S 0 , E 0 boun = 0. Since, as has been shown above, - S 0 , S 0 ≥ 0, and - S 0 , S 0 boun = R ∂B (Θ0 ) <sup>−</sup>1<sup>N</sup> · <sup>Q</sup> <sup>d</sup><sup>A</sup> the above equation implies

$$\frac{d\mathcal{S}'}{dt} \ge -\int\_{\partial\mathcal{B}} \frac{1}{\Theta'} \mathbf{N} \cdot \mathbf{Q} \,\mathrm{d}A\,\mathrm{.}\tag{4.68}$$

This corresponds to the Clausius-Duhem form of the second law of thermodynamics (see, for example, [136, Sec. 5]).

## 4.3.3. Initial boundary value problem

We next deal with the initial boundary value problem (IBVP) pertaining to large-strain thermo-elastodynamics. To this end, we decompose the boundary ∂B of the continuum body into a displacement boundary ∂ϕB, on which ϕ = ϕ, and a traction boundary ∂σB, on which PN = t, where ϕ and t are prescribed functions for t ≥ 0 (Fig. 4.3). Moreover, ∂ϕB ∪ ∂σB = ∂B and ∂ϕB ∩ ∂σB = ∅. Similarly, for the thermal part we consider the subsets ∂τB and ∂qB, with the properties ∂τB ∪ ∂qB = ∂B and ∂τB ∩ ∂qB = ∅ (Fig. 4.4). Here, the thermodynamic state variable is prescribed on ∂τB, i.e. τ = τ , whereas the heat ux is prescribed on ∂qB, i.e. Q · N = q.

The goal is now to determine the motion <sup>ϕ</sup> : B × I 7→ <sup>R</sup> 3 , the linear momentum <sup>p</sup> : B×I 7→ <sup>R</sup> 3 , and the thermodynamic state variable <sup>τ</sup> : B×I 7→ <sup>R</sup>. The unknown elds are subject to initial conditions of the form ϕ(·, 0) = X, p(·, 0) = ρV0, and τ (·, 0) = τ ini in B. Here, V<sup>0</sup> is a prescribed material velocity eld and τ ini is a prescribed eld of the thermodynamic state variable τ ∈ {θ, η, u}. The unknown elds are determined by the variational problem to be dealt with in the next section.

Figure 4.3.: Mechanical part of the IBVP. Note that t = PN denotes prescribed external Piola tractions acting on the current boundary expressed per unit area of the reference boundary ∂σB.

Figure 4.4.: Thermal part of the IBVP. Note that q = Q · N is the prescribed rate of heat transfer accross the current boundary expressed per unit area of the reference boundary ∂qB.

#### 4.3.3.1. Variational formulation

To deduce the variational formulation of the present IBVP from the GENERIC evolution equation (4.59), we choose the specic density function a <sup>0</sup> = w 0 , where

$$\mathbf{w}' = \mathbf{w}\_{\varphi} \cdot \varphi + \mathbf{w}\_{\mathbf{p}} \cdot \mathbf{p} + \mathbf{w}\_{\tau} \tau \;. \tag{4.69}$$

Here, <sup>w</sup>ϕ, <sup>w</sup><sup>p</sup> : B 7→ <sup>R</sup> 3 and <sup>w</sup><sup>τ</sup> : B 7→ <sup>R</sup> are test functions that have to satisfy the boundary conditions w<sup>ϕ</sup> = 0 and w<sup>p</sup> = 0 on ∂ϕB, and w<sup>τ</sup> = 0 on ∂τB. With the choice (4.69), the left-hand side of the GENERIC evolution equation (4.59) yields

$$\frac{d}{dt}\mathcal{W}' = \int\_{\mathcal{B}} \left( \mathbf{w}\_{\varphi} \cdot \dot{\varphi} + \mathbf{w}\_{\mathbf{p}} \cdot \dot{\mathbf{p}} + \mathbf{w}\_{\tau} \dot{\tau} \right) \, \mathrm{d}V \, . \tag{4.70}$$

On the other hand, the specic brackets on the right-hand side of (4.59) assume the form (cf. Table 4.2)

$$\begin{split} & \left\{ \mathcal{E}', \mathcal{W}' \right\}\_{\text{bulk}} = -\int\_{\mathcal{B}} \left( \mathbf{b} \cdot \mathbf{w}\_{\text{p}} + \boldsymbol{\rho}^{-1} \mathbf{p} \cdot \left( \mathbf{w}\_{\text{q}} + \text{Div} \left( \frac{\mathbf{w}\_{\tau}}{\partial \tau, \eta} \partial \mathbf{p} \boldsymbol{\eta}^{\prime} \right) \right) - \mathbf{P}' : \nabla \mathbf{w}\_{\text{p}} \right) \mathrm{d}V \,, \\ & \left\{ \mathcal{S}', \mathcal{W}' \right\}\_{\text{bulk}} = -\int\_{\mathcal{B}} \frac{1}{\mathcal{B}\mathcal{V}} \text{Div} \left( (\Theta')^{2} \mathbf{K} \nabla \left( \frac{\mathbf{w}\_{\tau}}{\partial \tau, \boldsymbol{u}^{\prime}} \right) \right) \mathrm{d}V \,, \\ & \left\{ \mathcal{W}', \mathcal{E}' \right\}\_{\text{bounce}} = -\int\_{\partial \mathcal{B}} \rho^{-1} \mathbf{p} \cdot \left( -\frac{\mathbf{w}\_{\tau}}{\partial \tau, \eta} \partial \mathbf{p} \, \eta^{\prime} \right) \mathbf{N} \, \mathrm{d}A \,, \\ & \left\{ \mathcal{E}', \mathcal{W}' \right\}\_{\text{bour}} = -\int\_{\partial \mathcal{B}} \mathbf{w}\_{\text{p}} \cdot \mathbf{P}' \mathbf{N} \, \mathrm{d}A \,, \\ & \left[ \mathcal{W}', \mathcal{S}' \right]\_{\text{bour}} = \int\_{\partial \mathcal{B}} \frac{\mathbf{w}\_{\tau}}{\partial \tau, \boldsymbol{u}^{\prime}} \mathbf{N} \cdot \mathbf{Q}' \, \mathrm{d}A \,, \\ & \left[ \mathcal{S}', \mathcal{W$$

Accordingly, the GENERIC evolution equation (4.59) gives rise to the equations

$$\begin{split} 0 &= \int\_{\mathcal{B}} \mathbf{w}\_{\varphi} \cdot \left( \dot{\varphi} - \rho^{-1} \mathbf{p} \right) \, \mathrm{d}V \,, \\ 0 &= \int\_{\mathcal{B}} \left( \mathbf{w}\_{\mathbf{p}} \cdot \left( \dot{\mathbf{p}} - \mathbf{b} \right) + \mathbf{P}' : \nabla \mathbf{w}\_{\mathbf{p}} \right) \, \mathrm{d}V - \int\_{\partial \mathcal{B}} \mathbf{w}\_{\mathbf{p}} \cdot \mathbf{P}' \mathbf{N} \, \mathrm{d}A \,, \end{split} \tag{4.71}$$

along with

$$\begin{split} 0 &= \int\_{\mathcal{B}} \left( \mathbf{w}\_{\tau} \dot{\boldsymbol{\tau}} - \text{Div} \left( \frac{\mathbf{w}\_{\tau}}{\partial\_{\tau} \eta'} \partial\_{\mathbf{F}} \eta' \right) \cdot \boldsymbol{\rho}^{-1} \mathbf{p} + \frac{1}{\Theta'} \text{Div} \left( (\Theta')^{2} \mathbf{K} \nabla \left( \frac{\mathbf{w}\_{\tau}}{\partial\_{\tau} u'} \right) \right) \right) \text{d}V \\ &+ \int\_{\partial \mathcal{B}} \left( \boldsymbol{\rho}^{-1} \mathbf{p} \cdot \left( \frac{\mathbf{w}\_{\tau}}{\partial\_{\tau} \eta'} \partial\_{\mathbf{F}} \eta' \right) \mathbf{N} - \frac{1}{\Theta'} \mathbf{N} \cdot (\Theta')^{2} \mathbf{K} \nabla \left( \frac{\mathbf{w}\_{\tau}}{\partial\_{\tau} u'} \right) + \frac{\mathbf{w}\_{\tau}}{\partial\_{\tau} u'} \mathbf{N} \cdot \mathbf{Q}' \right) \text{d}A \ . \end{split}$$

Integrating by parts twice, we arrive at an alternative representation of the last equation given by

$$\begin{split} 0 &= \int\_{\mathcal{B}} \left( \mathbf{w}\_{\tau} \dot{\boldsymbol{\tau}} + \nabla (\rho^{-1} \mathbf{p}) : \left( \frac{\mathbf{w}\_{\tau}}{\partial\_{\tau} \eta'} \partial\_{\mathbf{F}} \eta' \right) - \nabla \left( \frac{1}{\Theta'} \right) \cdot (\Theta')^2 \mathbf{K} \nabla \left( \frac{\mathbf{w}\_{\tau}}{\partial\_{\tau} u'} \right) \right) \mathrm{d}V \\ &+ \int\_{\partial \mathcal{B}} \frac{\mathbf{w}\_{\tau}}{\partial\_{\tau} u'} \mathbf{N} \cdot \mathbf{Q}' \mathrm{d}A \, . \end{split}$$

Taking into account expression (4.47) for the material heat ux, and the symmetry property K = K<sup>T</sup> , the above equation can be recast in the form

$$0 = \int\_{\mathcal{B}} \left( \mathbf{w}\_{\tau} \dot{\boldsymbol{\tau}} + \nabla(\rho^{-1} \mathbf{p}) : \left( \frac{\mathbf{w}\_{\tau}}{\partial\_{\tau} \eta'} \partial\_{\mathbf{F}} \eta' \right) - \nabla \left( \frac{\mathbf{w}\_{\tau}}{\partial\_{\tau} u'} \right) \cdot \mathbf{Q}' \right) \mathrm{d}V \\ + \int\_{\partial \mathcal{B}} \frac{\mathbf{w}\_{\tau}}{\partial\_{\tau} u'} \mathbf{N} \cdot \mathbf{Q}' \mathrm{d}A \,, \tag{4.72}$$

Taking into account the above stated boundary conditions, (4.71) and (4.72) give rise to the following variational formulation of the present IBVP:

$$\begin{split} 0 &= \int\_{\mathcal{B}} \mathbf{w}\_{\mathcal{P}} \cdot \left( \dot{\boldsymbol{\varphi}} - \boldsymbol{\rho}^{-1} \mathbf{p} \right) \, \mathrm{d}V \,, \\ 0 &= \int\_{\mathcal{B}} \left( \mathbf{w}\_{\mathbf{p}} \cdot \left( \dot{\mathbf{p}} - \mathbf{b} \right) + \mathbf{P}' : \nabla \mathbf{w}\_{\mathbf{p}} \right) \, \mathrm{d}V - \int\_{\partial\_{\sigma} \mathcal{B}} \mathbf{w}\_{\mathbf{p}} \cdot \overline{\mathbf{f}} \, \mathrm{d}A \,, \\ 0 &= \int\_{\mathcal{B}} \left( \mathbf{w}\_{\tau} \dot{\tau} + \nabla (\boldsymbol{\rho}^{-1} \mathbf{p}) : \left( \frac{\mathbf{w}\_{\tau}}{\partial\_{\tau} \eta'} \partial\_{\mathbf{F}} \eta' \right) - \nabla \left( \frac{\mathbf{w}\_{\tau}}{\partial\_{\tau} u'} \right) \cdot \mathbf{Q}' \right) \, \mathrm{d}V + \int\_{\partial\_{\partial\_{\theta} \mathcal{B}}} \frac{\mathbf{w}\_{\tau}}{\partial\_{\tau} u'} \overline{\mathbf{g}} \, \mathrm{d}A . \end{split} \tag{4.73}$$

These equations have to hold for all times t ≥ 0 and for arbitrary test functions subject to the above mentioned boundary conditions.

Remark 4. The variational formulation (4.73) can be easily extended to account for nonpotential body forces and a heat supply eld. To this end, the supply term introduced in Remark 3 just needs be included in the above derivation of the variational formulation.

Remark 5. The majority of previously developed numerical methods for nite-strain thermomechanics rely on the use of the temperature as the thermodynamic state variable. Choosing τ = θ in the GENERIC-based weak form (4.73)3, leads to the variational equation for the temperature evolution given by

$$\int\_{\mathcal{B}} \left( \mathbf{w}\_{\theta} \dot{\boldsymbol{\theta}} + \nabla (\boldsymbol{\rho}^{-1} \mathbf{p}) : \left( \frac{\mathbf{w}\_{\theta}}{\partial\_{\theta} \overline{\boldsymbol{\eta}}} \partial\_{\overline{\mathbf{v}}} \overline{\boldsymbol{\eta}} \right) - \nabla \left( \frac{\mathbf{w}\_{\theta}}{\partial\_{\theta} \overline{\boldsymbol{u}}} \right) \cdot \overline{\mathbf{Q}} \right) \mathrm{d}V + \int\_{\partial\_{\mathbf{q}} \mathbf{g}} \frac{\mathbf{w}\_{\theta}}{\partial\_{\theta} \overline{\boldsymbol{u}}} \overline{\boldsymbol{q}} \, \mathrm{d}A = \boldsymbol{0} \, \mathrm{d}V \tag{4.74}$$

The last equation can be recast in the form

$$\int\_{\mathcal{B}} \left\{ \frac{\mathbf{w}\_{\theta}}{\partial\_{\theta}\overline{u}} \left( \partial\_{\theta}\overline{u}\dot{\theta} + \frac{\partial\_{\theta}\overline{u}}{\partial\_{\theta}\overline{\eta}} \nabla(\rho^{-1}\mathbf{p}) : \partial\_{\mathbf{F}}\overline{\eta} \right) - \nabla \left( \frac{\mathbf{w}\_{\theta}}{\partial\_{\theta}\overline{u}} \right) \cdot \overline{\mathbf{Q}} \right\} \, \mathrm{d}V \,, + \int\_{\partial\_{\mathbf{q}}\mathbf{g}} \frac{\mathbf{w}\_{\theta}}{\partial\_{\theta}\overline{u}} \overline{q} \, \mathrm{d}A = 0 \,, \quad \frac{\mathbf{w}\_{\theta}}{\partial\_{\theta}\overline{u}} \frac{\mathbf{w}\_{\theta}}{\partial\_{\theta}\overline{u}} = 0 \,, \quad \frac{\mathbf{w}\_{\theta}}{\partial\_{\theta}\overline{u}} = 0$$

or

$$\int\_{\mathcal{B}} \left\{ v\_{\theta} \left( \partial\_{\theta} \overline{u} \dot{\theta} + \theta \nabla (\rho^{-1} \mathbf{p}) : \partial\_{\mathbf{F}} \eta \right) + \nabla v\_{\theta} \cdot \mathbf{K} \nabla \theta \right\} \, \mathrm{d}V + \int\_{\partial\_{\mathbf{q}} \mathcal{B}} v\_{\theta} \overline{q} \, \mathrm{d}A = 0 \, . \tag{4.75}$$

In the last equation the modied test function v<sup>θ</sup> = w<sup>θ</sup> ∂θu has been introduced. In addition to that, temperature formula (4.27) has been used along with constitutive equation (4.47) for the material heat ux vector. In essence, weak form (4.75) of the temperature evolution equation lies at the heart of numerous works on nite-strain thermomechanics such as, for example, [140–142]. Alternatively, dierentiating (4.20) with respect to time, one arrives at the relation

$$
\partial\_{\theta}\overline{u}\dot{\theta} = \theta\dot{\overline{\eta}} - \theta\partial\_{\mathbf{F}}\overline{\eta} : \dot{\mathbf{F}} \ .
$$

Inserting the last equation into (4.75), and making use of the identity ϕ˙ = ρ <sup>−</sup>1p, we obtain the entropy form of (4.75) given by

$$\int\_{\mathcal{B}} \left\{ v\_{\theta} \theta \dot{\overline{\eta}} + \nabla v\_{\theta} \cdot \mathbf{K} \nabla \theta \right\} \, \mathrm{d}V + \int\_{\partial\_{\theta} \mathfrak{B}} v\_{\theta} \overline{q} \, \mathrm{d}A = 0 \,. \tag{4.76}$$

In essence, variational formulation (4.76) lies at the origin of alternative temperature-based numerical methods such as, for example, [143, 144]. It is worth mentioning that in (4.76), η˙ = d dt η(F, θ) is the time derivative of the "constitutive entropy" η(F, θ). This is to be contrasted with the entropy-based formulation emanating from the GENERIC-based weak form (4.73)3. Specically, choosing τ = η, (4.73)<sup>3</sup> yields

$$\int\_{\mathcal{B}} \left\{ \mathbf{w}\_{\eta} \dot{\eta} - \nabla \left( \mathbf{w}\_{\eta} \widetilde{\Theta}^{-1} \right) \cdot \widetilde{\Theta}^{2} \mathbf{K} \nabla \left( \widetilde{\Theta}^{-1} \right) \right\} \, \mathrm{d}V + \int\_{\partial\_{\eta} \mathfrak{B}} \frac{\mathbf{w}\_{\eta}}{\widetilde{\Theta}} \overline{\mathbf{q}} \, \mathrm{d}A = \mathbf{0} \,,\tag{4.77}$$

where Θ = <sup>e</sup> <sup>∂</sup>ηue(F, η).

#### 4.3.3.2. Balance laws revisited

The balance laws dealt with in Section 4.3.2 can also be recovered from the variational formulation (4.73), by choosing specic test functions. For that purpose we conne our attention in this section to the pure Neumann problem (i.e. ∂σB = ∂qB = ∂B).

Consider w<sup>p</sup> = ∂pl 0 ξ , where l 0 <sup>ξ</sup> = ξ · p has been introduced in Section 4.3.2. Correspondingly, inserting w<sup>p</sup> = ξ into (4.73)2, we directly recover the balance law (4.65) for linear momentum.

Next, we choose w<sup>ϕ</sup> = ∂ϕj 0 ξ and w<sup>p</sup> = ∂pj 0 ξ , where j 0 <sup>ξ</sup> = ξ ·(ϕ×p) has been introduced in Section 4.3.2. Accordingly, inserting w<sup>ϕ</sup> = p × ξ and w<sup>p</sup> = ξ × ϕ into (4.73)1,2, and subsequently adding both equations, we recover the balance law (4.66) for angular momentum. To verify the balance of energy we choose for the test functions in (4.73)

$$\begin{aligned} \mathbf{w}\_{\varphi} &= \partial\_{\varphi} e' \,, & \mathbf{w}\_{\varphi} &= -\mathbf{b} \,, \\ \mathbf{w}\_{\mathbf{p}} &= \partial\_{\mathbf{p}} e' \, & \text{or} & \mathbf{w}\_{\mathbf{p}} &= \rho^{-1} \mathbf{p} \,, \\ \mathbf{w}\_{\tau} &= \partial\_{\tau} e' \, & \text{or} & \mathbf{w}\_{\tau} &= \partial\_{\tau} u' \, . \end{aligned}$$

Substituting these quantities into (4.73) and subsequently adding the three resulting equations, a straightforward calculation recovers the balance law (4.67) for energy. In this connection, the following relation is of importance:

$$\begin{split} &\int\_{\mathcal{B}} \left( \mathbf{P'} : \nabla(\rho^{-1}\mathbf{p}) + \partial\_{\tau}u'\dot{\tau} + \nabla(\rho^{-1}\mathbf{p}) : \left( \frac{\partial\_{\tau}u'}{\partial\_{\tau}\eta'} \partial\_{\mathbf{F}}\eta' \right) \right) \mathrm{d}V \\ &= \int\_{\mathcal{B}} \left( \partial\_{\tau}u'\dot{\tau} + \left( \underbrace{\partial\_{\mathbf{F}}u' - \frac{\partial\_{\tau}u'}{\partial\_{\tau}\eta'} \partial\_{\mathbf{F}}\eta'}\_{=\mathbf{P'}} + \frac{\partial\_{\tau}u'}{\partial\_{\tau}\eta'} \partial\_{\mathbf{F}}\eta' \right) : \nabla(\rho^{-1}\mathbf{p}) \right) \mathrm{d}V \\ &= \int\_{\mathcal{B}} \left( \partial\_{\tau}u'\dot{\tau} + \partial\_{\mathbf{F}}u' : \dot{\mathbf{F}} \right) \mathrm{d}V \\ &= \frac{\mathrm{d}}{\mathrm{d}t} \int\_{\mathcal{B}} u' \, \mathrm{d}V \, . \end{split} \tag{4.78}$$

Here, formula (4.52) for the rst Piola-Kirchho stress tensor has been used, along with the identity ϕ˙ = ρ <sup>−</sup><sup>1</sup>p.

Concerning the balance of entropy, we insert w<sup>τ</sup> = ∂<sup>τ</sup> η 0 into (4.73)3, to obtain

$$\begin{split} 0 &= \int\_{\mathcal{B}} \left( \partial\_{\tau} \eta' \dot{\tau} + \nabla (\rho^{-1} \mathbf{p}) : (\partial\_{\mathbf{F}} \eta') - \nabla \left( \frac{\partial\_{\tau} \eta'}{\partial\_{\tau} u'} \right) \cdot \mathbf{Q}' \right) \mathrm{d}V + \int\_{\partial \mathcal{B}} \frac{\partial\_{\tau} \eta'}{\partial\_{\tau} u'} \mathbf{Q}' \cdot \mathbf{N} \, \mathrm{d}A \\ &= \int\_{\mathcal{B}} \left( \frac{\mathrm{d} \eta'}{\mathrm{d}t} - \nabla \left( \frac{1}{\Theta'} \right) \cdot (\Theta')^2 \mathbf{K} \nabla \left( \frac{1}{\Theta'} \right) \right) \mathrm{d}V + \int\_{\partial \mathcal{B}} \frac{1}{\Theta'} \mathbf{Q}' \cdot \mathbf{N} \, \mathrm{d}A \, . \end{split}$$

Here, use has been made of formula (4.27) for the temperature along with expression (4.47) for the material heat ux vector. Moreover, the identity ϕ˙ = ρ <sup>−</sup><sup>1</sup>p has again been taken into account. The above equation can be rewritten as

$$\frac{d\mathcal{S}'}{dt} = \underbrace{\int\_{\mathcal{B}} \nabla \left(\frac{1}{\Theta'}\right) \cdot (\Theta')^2 \mathbf{K} \nabla \left(\frac{1}{\Theta'}\right) \,\mathrm{d}V}\_{\geq 0} - \int\_{\partial\mathcal{B}} \frac{1}{\Theta'} \mathbf{Q}' \cdot \mathbf{N} \,\mathrm{d}A \,,\tag{4.79}$$

which recovers the second law of thermodynamics in the form (4.68).

## 4.4. Discretization in time and space

## 4.4.1. Discretization in time

We rst perform the discretization in time of the variational formulation (4.73). To this end we focus on a representative time interval [tn, tn+1] with corresponding time-step size ∆t = tn+1 − tn. The discrete approximations at times t<sup>n</sup> and tn+1 of the continuous variable (•)<sup>t</sup> will be denoted by (•)<sup>n</sup> and (•)n+1, respectively. Moreover, the approximation of any state variable (•)<sup>t</sup> at mid-point time tn+ <sup>1</sup> 2 = 1 2 (t<sup>n</sup> + tn+1) is given by

$$(\bullet)\_{\mathfrak{n}+\frac{1}{2}} = \frac{1}{2} \left( (\bullet)\_{\mathfrak{n}} + (\bullet)\_{\mathfrak{n}+1} \right) \dots$$

Assume that the state variables ϕ<sup>n</sup> , p<sup>n</sup> : B 7→ <sup>R</sup> 3 and <sup>τ</sup><sup>n</sup> : B 7→ <sup>R</sup>, <sup>τ</sup><sup>n</sup> ∈ {θn, ηn, un} are given. We aim at the determination of the corresponding state variables at time tn+1 (ϕn+1, pn+1, τn+1) through the mid-point type discretization of the variational formulation (4.73) given by

0 =<sup>Z</sup> B w<sup>ϕ</sup> · ϕn+1 − ϕ<sup>n</sup> ∆t − ρ <sup>−</sup><sup>1</sup>pn+ <sup>1</sup> 2 dV , 0 =<sup>Z</sup> B w<sup>p</sup> · pn+1 − p<sup>n</sup> ∆t − w<sup>p</sup> · b + ∇w<sup>p</sup> : P 0 tn+ <sup>1</sup> 2 dV − Z ∂<sup>σ</sup> B w<sup>p</sup> · tn+ <sup>1</sup> 2 dA , 0 =<sup>Z</sup> B w<sup>τ</sup> τn+1 − τ<sup>n</sup> ∆t + ∇(ρ <sup>−</sup><sup>1</sup>pn+ <sup>1</sup> 2 ) : ∂Fη 0 ∂<sup>τ</sup> η 0 tn+ <sup>1</sup> 2 <sup>d</sup><sup>V</sup> − Z B ∇ w<sup>τ</sup> ∂τu 0 | tn+ <sup>1</sup> 2 · <sup>Q</sup> 0 tn+ <sup>1</sup> 2 dV + Z ∂qB w<sup>τ</sup> ∂τu 0 | tn+ <sup>1</sup> 2 qn+ <sup>1</sup> 2 dA . (4.80)

Note that depending on the choice for the thermodynamic state variable τ ∈ {θ, η, u}, three alternative semi-discrete formulations result from (4.80). With regard to (4.52), the time-discrete version of the rst Piola-Kirchho stress tensor in (4.80)<sup>2</sup> assumes the form

$$\begin{split} \left. \mathbf{P}' \right|\_{t\_{n+\frac{1}{2}}} &= \mathbf{P}'(\mathbf{F}\_{n+\frac{1}{2}}, \tau\_{n+\frac{1}{2}}) \\ &= \left. \partial\_{\mathbf{F}} u' \right|\_{t\_{n+\frac{1}{2}}} - \left. \left( \frac{\partial\_{\tau} u'}{\partial\_{\tau} \eta'} \partial\_{\mathbf{F}} \eta' \right) \right|\_{t\_{n+\frac{1}{2}}} . \end{split} \tag{4.81}$$

A frame-indierent thermoelastic formulation requires that the functions u 0 and η 0 can be expressed as

$$\begin{aligned} u'(\mathbf{F}, \tau) &= u''(\mathbf{C}, \tau) \; , \\ \eta'(\mathbf{F}, \tau) &= \eta''(\mathbf{C}, \tau) \; , \end{aligned} \tag{4.82}$$

where C = F <sup>T</sup>F is the right Cauchy-Green tensor written in terms of the deformation gradient F. Taking into account (4.82), we obtain

$$\begin{split} \left. \partial\_{\mathbf{F}} u' \right|\_{t\_{\mathbf{n}+\frac{1}{2}}} &= 2 \mathbf{F}\_{\mathbf{n}+\frac{1}{2}} \partial\_{\mathbf{C}} u'' (\left. \mathbf{C} \right|\_{t\_{\mathbf{n}+\frac{1}{2}}}, \tau\_{\mathbf{n}+\frac{1}{2}}) \,, \\ \left. \partial\_{\mathbf{F}} \eta' \right|\_{t\_{\mathbf{n}+\frac{1}{2}}} &= 2 \mathbf{F}\_{\mathbf{n}+\frac{1}{2}} \partial\_{\mathbf{C}} \eta'' (\left. \mathbf{C} \right|\_{t\_{\mathbf{n}+\frac{1}{2}}}, \tau\_{\mathbf{n}+\frac{1}{2}}) \,, \end{split} \tag{4.83}$$

together with

$$\begin{split} \left. \partial\_{\tau} u' \right|\_{t\_{\mathbf{n}+\frac{1}{2}}} &= \partial\_{\tau} u''(\left. \mathbf{C} \right|\_{t\_{\mathbf{n}+\frac{1}{2}}}, \tau\_{\mathbf{n}+\frac{1}{2}}) \,, \\ \left. \partial\_{\tau} \eta' \right|\_{t\_{\mathbf{n}+\frac{1}{2}}} &= \partial\_{\tau} \eta''(\left. \mathbf{C} \right|\_{t\_{\mathbf{n}+\frac{1}{2}}}, \tau\_{\mathbf{n}+\frac{1}{2}}) \,, \end{split} \tag{4.84}$$

where C| tn+ <sup>1</sup> 2 denotes the right Cauchy-Green tensor evaluated in the mid-point conguration. That is,

$$\mathbf{C}|\_{t\_{\mathbf{n}+\frac{1}{2}}} = \mathbf{F}\_{\mathbf{n}+\frac{1}{2}}^{\mathbf{T}} \mathbf{F}\_{\mathbf{n}+\frac{1}{2}} \ . \tag{4.85}$$

Moreover, in view of (4.47), the time-discrete version of the material heat ux vector Q 0 tn+ <sup>1</sup> 2 featuring in (4.80)<sup>3</sup> is given by

$$\begin{split} \left. \mathbf{Q}' \right|\_{t\_{n+\frac{1}{2}}} &= \mathbf{Q}'(\mathbf{F}\_{n+\frac{1}{2}}, \tau\_{n+\frac{1}{2}}) \\ &= -\left. \mathbf{K}' \right|\_{t\_{n+\frac{1}{2}}} \nabla \left. \left( \frac{\partial\_{\tau} u'}{\partial\_{\tau} \eta'} \right) \right|\_{t\_{n+\frac{1}{2}}}, \end{split} \tag{4.86}$$

where the time-discrete version of the material conductivity tensor K<sup>0</sup> tn+ <sup>1</sup> 2 follows from the relation

$$\left. \mathbf{K}' \right|\_{t\_{\mathrm{n}+\frac{1}{2}}} = \mathbf{K}'(\mathbf{F}\_{\mathrm{n}+\frac{1}{2}}, \tau\_{\mathrm{n}+\frac{1}{2}}) = \mathbf{K}''(\mathbf{C}|\_{t\_{\mathrm{n}+\frac{1}{2}}}, \tau\_{\mathrm{n}+\frac{1}{2}}) \,. \tag{4.87}$$

#### 4.4.1.1. Semi-discrete balance laws

We next investigate to what extent the balance laws outlined in Sections 4.3.2 and 4.3.3.2 are inherited by the three alternative semi-discrete formulations developed in the last section. To this end we proceed along the lines of Section 4.3.3.2. We again conne our attention to the pure Neumann problem (i.e. ∂σB = ∂qB = ∂B). Inserting w<sup>p</sup> = ξ into (4.80)2, we directly obtain

$$
\xi \cdot \frac{\mathbf{L}\_{\mathrm{n}+1} - \mathbf{L}\_{\mathrm{n}}}{\Delta t} = \xi \cdot \left( \int\_{\mathcal{B}} \mathbf{b} \, \mathrm{d}V + \int\_{\partial \mathcal{B}} \overline{\mathbf{t}}\_{\mathrm{n}+\frac{1}{2}} \, \mathrm{d}A \right) \,, \, \xi
$$

where tn+ <sup>1</sup> 2 = P 0 tn+ <sup>1</sup> 2 N. The last equation corresponds to the semi-discrete version of the balance of linear momentum (4.65). Inserting w<sup>ϕ</sup> = pn+ <sup>1</sup> 2 × ξ and w<sup>p</sup> = ξ × ϕn+ <sup>1</sup> 2 into (4.80)1,2, and subsequently adding both equations, a straightforward calculation yields

$$\xi \cdot \frac{\mathbf{J}\_{\mathrm{n}+1} - \mathbf{J}\_{\mathrm{n}}}{\Delta t} = \xi \cdot \left( \int\_{\mathcal{B}} \varphi\_{\mathrm{n}+\frac{1}{2}} \times \mathbf{b} \, \mathrm{d}V + \int\_{\partial \mathcal{B}} \varphi\_{\mathrm{n}+\frac{1}{2}} \times \overline{\mathbf{t}}\_{\mathrm{n}+\frac{1}{2}} \, \mathrm{d}A \right) \,,$$

which corresponds to the semi-discrete version of the balance of angular momentum (4.66). Accordingly, independent of the choice for τ ∈ {θ, η, u}, the three alternative formulations at hand consistently reproduce the balance laws for linear and angular momentum for arbitrary time-steps ∆t.

Inserting w<sup>ϕ</sup> = −b, w<sup>p</sup> = ρ <sup>−</sup><sup>1</sup>pn+ <sup>1</sup> 2 , and w<sup>τ</sup> = ∂τu 0 | tn+ <sup>1</sup> 2 into the semi-discrete formulation (4.80), and subsequently adding the three equations, we arrive at

$$\begin{split} &\frac{1}{2}\int\_{\mathcal{B}}\rho^{-1}\frac{\mathbf{p}\_{\mathrm{n}+1}\cdot\mathbf{p}\_{\mathrm{n}+1}-\mathbf{p}\_{\mathrm{n}}\cdot\mathbf{p}\_{\mathrm{n}}}{\Delta t}+\left(\left.\partial\_{\tau}u'\right|\_{t\_{\mathrm{n}+\frac{1}{2}}}\frac{\tau\_{\mathrm{n}+1}-\tau\_{\mathrm{n}}}{\Delta t}+\left.\partial\_{\mathsf{F}}u'\right|\_{t\_{\mathrm{n}+\frac{1}{2}}}:\nabla(\rho^{-1}\mathbf{p}\_{\mathrm{n}+\frac{1}{2}})\right) \mathrm{d}V \\ &=\int\_{\mathcal{B}}\mathbf{b}\cdot\frac{\varphi\_{\mathrm{n}+1}-\varphi\_{\mathrm{n}}}{\Delta t}\mathrm{d}V+\int\_{\partial\mathcal{B}}\left(\rho^{-1}\mathbf{p}\_{\mathrm{n}+\frac{1}{2}}\cdot\overline{\mathbf{t}}\_{\mathrm{n}+\frac{1}{2}}-\overline{\eta}\_{\mathrm{n}+\frac{1}{2}}\right)\mathrm{d}A\,,\end{split}$$

where qn+ <sup>1</sup> 2 = Q 0 tn+ <sup>1</sup> 2 · N. The last equation can be viewed as semi-discrete version of the balance of energy (4.67) provided that the second integral on the left-hand side of the last equation is equal to <sup>1</sup> ∆t R B u 0 (Fn+1, τn+1) − u 0 (Fn, τn) dV . However, for τ ∈ {θ, η} this is generally not the case. In contrast to that, for τ = u, the above equation reads

$$\begin{split} &\frac{1}{2} \int\_{\mathcal{B}} \rho^{-1} \frac{\mathbf{p}\_{\mathrm{n}+1} \cdot \mathbf{p}\_{\mathrm{n}+1} - \mathbf{p}\_{\mathrm{n}} \cdot \mathbf{p}\_{\mathrm{n}}}{\Delta t} \mathrm{d}V + \int\_{\mathcal{B}} \frac{u\_{\mathrm{n}+1} - u\_{\mathrm{n}}}{\Delta t} \, \mathrm{d}V \\ & \qquad = \int\_{\mathcal{B}} \mathbf{b} \cdot \frac{\varphi\_{\mathrm{n}+1} - \varphi\_{\mathrm{n}}}{\Delta t} \, \mathrm{d}V + \int\_{\partial \mathcal{B}} \left( \rho^{-1} \mathbf{p}\_{\mathrm{n}+\frac{1}{2}} \cdot \overline{\mathbf{t}} \big|\_{t\_{\mathrm{n}+\frac{1}{2}}} - \overline{q}\_{\mathrm{n}+\frac{1}{2}} \right) \, \mathrm{d}A \,, \end{split} \tag{4.88}$$

which indeed corresponds to the semi-discrete counterpart of the balance of energy (4.67). Accordingly, the formulation in terms of the internal energy density leads to an energy-momentum scheme.

Eventually, inserting w<sup>τ</sup> = ∂<sup>τ</sup> η 0 | tn+ <sup>1</sup> 2 into (4.80)3, we obtain

$$\begin{split} &\int\_{\mathcal{B}} \left( \left. \partial\_{\tau} \eta' \right|\_{t\_{n+\frac{1}{2}}} \frac{\tau\_{n+1} - \tau\_{n}}{\Delta t} + \left. \partial\_{\mathbf{F}} \eta' \right|\_{t\_{n+\frac{1}{2}}} : \nabla (\rho^{-1} \mathbf{p}\_{n+\frac{1}{2}}) \right) \mathrm{d}V \\ &= \underbrace{\int\_{\mathcal{B}} \nabla \left( \left. \Theta' \right|\_{t\_{n+\frac{1}{2}}}^{-1} \right) \cdot \left. \Theta' \right|\_{t\_{n+\frac{1}{2}}}^{2} \left. \mathbf{K}' \right|\_{t\_{n+\frac{1}{2}}} \nabla \left( \left. \Theta' \right|\_{t\_{n+\frac{1}{2}}}^{-1} \right) \mathrm{d}V - \int\_{\partial \mathcal{B}} \Theta' \big|\_{t\_{n+\frac{1}{2}}}^{-1} \overline{q}\_{n+\frac{1}{2}} \, \mathrm{d}A \right. \end{split}$$

The last equation does conform with the balance of entropy provided that the integral on the left-hand side of the last equation equals

$$\frac{1}{\Delta t} \int\_{\mathcal{B}} \eta'(\mathbf{F}\_{\mathbf{n}+1}, \tau\_{\mathbf{n}+1}) - \eta'(\mathbf{F}\_{\mathbf{n}}, \tau\_{\mathbf{n}}) \, \mathrm{d}V \, . $$

However, in general this is not true for τ ∈ {θ, u}. In contrast to that, for τ = η, the last equation yields

$$\begin{split} \displaystyle \int\_{\mathcal{B}} & \frac{\eta\_{n+1} - \eta\_{n}}{\Delta t} \, \mathrm{d}V = \\ & \underbrace{\int\_{\mathcal{B}} \nabla \left( \left. \widetilde{\Theta} \right|\_{t\_{n+\frac{1}{2}}}^{-1} \right) \cdot \left. \widetilde{\Theta} \right|\_{t\_{n+\frac{1}{2}}}^{2} \, \widetilde{\mathbf{K}} \Big|\_{t\_{n+\frac{1}{2}}} \, \nabla \left( \left. \widetilde{\Theta} \right|\_{t\_{n+\frac{1}{2}}}^{-1} \right) \, \mathrm{d}V - \int\_{\partial \mathcal{B}} \left. \widetilde{\Theta} \right|\_{t\_{n+\frac{1}{2}}}^{-1} \, \overline{q}\_{n+\frac{1}{2}} \, \mathrm{d}A, \end{split}$$

which corresponds to the semi-discrete counterpart of the second law of thermodynamics in the form (4.68). Accordingly, the formulation in terms of the entropy density leads to an momentum-entropy scheme.

## 4.4.2. Discretization in space

We apply standard isoparametric nite elements (see, for example, [145]) based on nitedimensional approximations of the state variables at time t, given by

$$\varphi\_{\mathbf{t}}^{\mathbf{h}}(\mathbf{X}) = \sum\_{a=1}^{n\_{node}} N^a(\mathbf{X}) \,\varphi\_a(t) \,, \qquad \mathbf{p}\_{\mathbf{t}}^{\mathbf{h}}(\mathbf{X}) = \sum\_{a=1}^{n\_{node}} N^a(\mathbf{X}) \, \mathbf{p}\_a(t) \,, \tag{4.89}$$

and

$$\tau\_{\mathbf{t}}^{\mathbf{h}}(\mathbf{X}) = \sum\_{a=1}^{n\_{node}} N^a(\mathbf{X}) \,\tau\_a(t) \,. \tag{4.90}$$

Here, N<sup>a</sup> : B 7→ <sup>R</sup> denote the nodal shape functions and <sup>ϕ</sup><sup>a</sup> (t), p<sup>a</sup> (t) <sup>∈</sup> <sup>R</sup> 3 , <sup>τ</sup>a(t) <sup>∈</sup> <sup>R</sup> are the respective nodal values at time t. Moreover, nnode denotes the total number of nodes in the nite element mesh. The standard (Bubnov) Galerkin approach relies on analogous approximations for the test functions in the variational equations (4.80), wϕ, w<sup>p</sup> and w<sup>τ</sup> , denoted by w<sup>ϕ</sup> h , w<sup>p</sup> h and w h τ .

It is worth noting that the semi-discrete balance laws investigated in the last section still hold for the fully discrete nite element formulation as can be easily veried by proceeding along the lines of the last section.

## 4.5. Numerical investigations

In this section, the three alternative mid-point type schemes newly developed in the present work are applied to representative numerical examples dealing with nite-strain thermo-elastodynamics. Depending on the specic choice of the thermodynamic state variable τ ∈ {θ, η, u}, the following methods are applied:


Concerning the constitutive equation (4.47) for the material heat ux vector, we assume thermally isotropic material, with material conductivity tensor given by

$$\mathbf{K} = kJ\mathbf{C}^{-1}.\tag{4.91}$$

Here, k is a prescribed coecient of thermal conductivity and, as before, J = p det(C). Moreover, the Helmholtz free energy assumes the form (4.36).

In the numerical investigations we shall focus on momentum maps associated with symmetries of the mechanical system at hand, and the balance laws associated with the two fundamental laws of thermodynamics. In this connection we also consider the functional

$$
\mathcal{L} = \mathcal{E} - \theta\_0 \mathcal{S} \,. \tag{4.92}
$$

According to [146] (cf. Appendix A.2), for certain types of environments, L is a natural Lyapunov function and thus qualies as estimate for the numerical stability of the schemes under consideration. Concerning the numerical solution of the resulting algebraic system of nonlinear equations we apply Newton's method together with the convergence criteria summarized in Appendix A.3.

## 4.5.1. Flying L-shaped block

Figure 4.5.: L-shaped block: Discretised block with initial temperature distribution and mechanical boundary conditions (left), load function over time (right)

The rst numerical example deals with the L-shaped block depicted in Fig. 4.5. The spatial discretization of the block relies on 117 tri-linear nite elements leading to 224 nodes. The initial temperature eld is varying linearly over the height (x<sup>3</sup> direction) of the block. In particular, at x<sup>3</sup> = 0, the initial temperature is prescribed as θa, while at x<sup>3</sup> = h, the temperature is prescribed as θb. The whole block is assumed to be thermally insulated (<sup>q</sup> = 0 on <sup>∂</sup>q<sup>B</sup> <sup>=</sup> <sup>∂</sup>B). Starting at rest, Piola traction vectors ¯t<sup>a</sup> and ¯t<sup>b</sup> are acting on two parts of the boundary surface of the block (Fig. 4.5). The external loads are applied in the form of a hat function over time. In particular, the traction vectors are given by

$$\begin{array}{ll} \mathbf{t}\_a = -\mathbf{t}\_b = f(t) \begin{pmatrix} 256/9 \\ 512/9 \\ 768/9 \end{pmatrix} \text{Pa} \,, \quad f(t) = \begin{cases} t & \text{for} \quad 0 \le t \le 2.5 \text{s} \\ 5-t & \text{for} \quad 2.5 \text{s} \le t \le 5 \text{s} \\ 0 & \text{for} \quad t > 5 \text{s} \end{cases} , \end{array} \tag{4.93}$$

Table 4.3 provides a summary of the data used in the simulations. No Dirichlet boundary conditions are applied.


Table 4.3.: L-shaped block: Data used in the simulations

Since in the initial loading phase the external forces are equilibrated, the total linear momentum of the block is a conserved quantity. In addition to that, after the loading phase (t > 5s) no external torque is acting on the block.

Figure 4.6.: L-shaped block: Total discrete linear momentum (EM)<sup>u</sup> scheme (left), Total discrete angular momentum (EM)<sup>u</sup> scheme (right)

Consequently, the total angular momentum is conserved as well. All of the integrators under consideration are capable to exactly conserve both momentum maps (up to numerical round-o), independent of the chosen time step. This can be observed from Fig. 4.6, where representative numerical results of the energy-momentum integrator (EM)<sup>u</sup> are shown. In the simulations three alternative constant time steps ∆t ∈ {0.08, 0.2, 0.4}s have been used. After the loading phase the total energy must be a conserved quantity. As expected, the (EM)<sup>u</sup> scheme does exactly reproduce this conservation law (up to numerical round-o), see Fig. 4.7. In contrast to that, the schemes (M)<sup>θ</sup> and (ME)<sup>η</sup> are not capable of conserving the total energy.

Figure 4.7.: L-shaped block: Total energy (EM)<sup>u</sup> scheme (left), Incremental change of total energy (EM)<sup>u</sup> scheme (right)

It can be observed from Fig. 4.8 that both schemes have a tendency to increase the energy, depending on the time step. Typically, such energy blow-ups eventually lead to a failure of the iterative (Newton-Raphson) solution procedure. In the diagrams the break down of the simulation is indicated by vertical lines. Despite the capability of the (EM)<sup>u</sup> scheme to conserve the total energy, the simulation typically breaks down at about the same point in time as the break down of the other two schemes occurs. The numerical instability of the (EM)<sup>u</sup> is accompanied by a nonphysical decrease of the total entropy as can be observed from Fig. 4.9. In fact, the total entropy ought to be a non-decreasing function of time. Only the (ME)<sup>η</sup> scheme is capable to correctly adhere to the second law of thermodynamics, independent of the time step (Fig. 4.10). Indeed, in each time step the total entropy does increase, as can be observed from Fig. 4.10 (right). The above investigations indicate that all of the three schemes under consideration are prone to numerical instability.

Figure 4.8.: L-shaped block: Total energy (M)<sup>θ</sup> scheme (left), Total energy (ME)<sup>η</sup> scheme (right)

Figure 4.9.: L-shaped block: Total entropy (EM)<sup>u</sup> scheme (left), Total entropy (M)<sup>θ</sup> scheme (right)

Figure 4.10.: L-shaped block: Total entropy (ME)<sup>η</sup> scheme (left), Incremental change of total entropy (ME)<sup>η</sup> scheme (right)

After the loading phase, the environment of the present example can be characterized as thermally perfect in the sense of [146] (cf. Appendix A.2). Thus L dened in (4.92) plays the role of a Lyapunov function that has to decrease with time. However, none of the schemes under investigation does correctly reproduce this behavior, as can be seen from Figs. 4.11 and 4.12. That is, depending on the time step and the duration of the simulation, all of the schemes inevitably exhibit numerical instabilities characterized by increasing values of L.

Figure 4.11.: L-shaped block: Lyapunov function (EM)<sup>u</sup> scheme (left), Lyapunov function (M)<sup>θ</sup> scheme (right)

Figure 4.12.: L-shaped block: Lyapunov function (ME)<sup>η</sup> scheme

Eventually, the motion of the L-shaped block is illustrated in Figs. 4.13 and 4.14 with snapshots at successive points in time. In addition to that, the distribution of the temperature and det(F) over the block is shown.

Figure 4.13.: L-shaped block: Snapshots of the motion along with the temperature distribution over the block at t ∈ {0, 32, 64, 96, 128, 160, 192, 224}s, obtained with the (M)<sup>θ</sup> scheme and time step ∆t = 0.08s

Figure 4.14.: L-shaped block: Snapshots of the motion along with the distribution of the determinant of the deformation gradient over the block at t ∈ {0, 32, 64, 96, 128, 160, 192, 224}s, obtained with the (M)<sup>θ</sup> scheme and time step ∆t = 0.08s

## 4.5.2. Rotating disc

The second example deals with a rotating disc subjected to prescribed heat ow over part of the boundary surface (Fig. 4.15). The spatial discretization of the disc is based on 200 tri-linear nite elements leading to a total of 360 nodes.

Figure 4.15.: Rotating disc: Initial conguration and thermal boundary conditions (left), function f(t) for the prescribed heat ow over part of the boundary surface (right)

The initial velocity distribution over the disc results from a prescribed angular velocity <sup>ω</sup><sup>0</sup> <sup>∈</sup> <sup>R</sup> 3 and is given by

$$\mathbf{v}\_0(\mathbf{X}) = \omega\_0 \times \mathbf{X}, \qquad \omega\_0 = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \frac{1}{\mathbf{s}} \cdot \mathbf{s}$$

The initial temperature of the disc is homogeneously distributed and equal to the reference temperature θ0. In an initial period of time, t ∈ [0, 4]s, heat ow is prescribed over one quarter of the lateral boundary surface (Fig. 4.15). In particular, the heat ow into the disc is described by

$$\overline{q} = -\frac{2000W}{\pi \text{m}^2} f(t) \,, \qquad f(t) = \begin{cases} \sin(\frac{\pi}{4s}t) & \text{for} \quad 0 \le t \le 4s \text{ }, \\ 0 & \text{for} \quad t > 4s \text{ }. \end{cases}$$

A plot of function f(t) can be found in Fig. 4.15. The rest of the boundary surface of the disc is assumed to be thermally insulated (q = 0). Note that the prescribed heat ow vanishes after t = 4s. For t > 4s, the complete boundary surface of the disc is assumed to be thermally insulated (q = 0 on ∂qB = ∂B). Then, the environment of the disc is thermally perfect in the sense of [146] (cf. Appendix A.2). A summary of the data used in the simulation of the rotating disc can be found in Table 4.4.

Table 4.4.: Rotating disc: Data used in the simulations

In the simulations constant time steps of size ∆t ∈ {0.04, 0.08, 0.1}s are applied. Since neither external loads act on the disc, nor displacement boundary conditions are imposed, the mechanical system at hand has translational and rotational symmetry. Consequently, the corresponding momentum maps are rst integrals of the motion.

Figure 4.16.: Rotating disc: Total linear momentum (ME)<sup>η</sup> scheme (left), Total angular momentum (ME)<sup>η</sup> scheme (right)

All three integrators at hand are capable to conserve the respective momentum map. Representative numerical results are shown in Fig. 4.16.

Figure 4.17.: Rotating disc: Total energy (EM)<sup>u</sup> scheme (left), Discrete change of total energy (EM)<sup>u</sup> scheme (right)

Figure 4.18.: Rotating disc: Total energy (M)<sup>θ</sup> scheme (left), Total energy (ME)<sup>η</sup> scheme (right)

Since heat ow into the system is prescribed in the initial time period [0, 4]s, the total energy is expected to increase. For t > 4s, the system is closed and the total energy should stay constant. Again the (EM)<sup>u</sup> scheme is capable to correctly reproduce the rst law (Fig. 4.17). However, despite the algorithmic energy conservation (for t > 4s), the (EM)<sup>u</sup> scheme gets unstable, depending on the time step. The corresponding point in time of the break down of the simulation is indicated with a vertical line in the diagrams. At about the same points in time the other two schemes break down as well (Fig. 4.18). For these schemes the break down is accompanied by a sudden increase of the total energy leading to the divergence of the Newton-Raphson iterations. For the considered duration of the simulation (t ∈ [0, 30]s), a time step of ∆t = 0.04s is small enough to retain numerical stability of the three schemes at hand.

Figure 4.19.: Rotating disc: Total entropy (EM)<sup>u</sup> scheme (left), Total entropy (M)<sup>θ</sup> scheme (right)

Figure 4.20.: Rotating disc: Total entropy (ME)<sup>η</sup> scheme (left), Discrete change of total entropy (ME)<sup>η</sup> scheme (right)

Due to the prescribed heat ow into the disc, the total entropy of the disc is expected to increase in the initial time period [0, 4]s. For t > 4s, the system is closed and the total entropy should be a non-decreasing function of time. As expected, the (ME)<sup>η</sup> scheme is capable to exactly satisfy the second law, independent of the time step. This can be observed from Fig. 4.20. In particular, Fig. 4.20 (right) conrms that the change per time step of the total entropy is always positive. The (EM)<sup>u</sup> scheme does not correctly reproduce

the second law as can be seen from Fig. 4.19. Accordingly, the divergence of the iterative solution procedure is accompanied by an nonphysical decrease of the total entropy. The (M)<sup>θ</sup> closely adheres to the second law as can be observed from Fig. 4.19.

To shed further light on the numerical stability of the present schemes, we consider the Lyapunov function L dened in (4.92). For t > 4s the system is closed and the function L should decrease with time (see also Appendix A.2). All three schemes fail to correctly reproduce this behavior, depending on the size of the time step and the duration of the simulation (Figs. 4.21 and 4.22). In particular, it can be seen that the smallest time step ∆t = 0.04s yields a stable numerical simulation, at least in the considered time interval [0, 30]s. However, for larger time steps ∆t = 0.08s and ∆t = 0.1s, the numerical instability of each scheme becomes visible through the increase of L.

Figure 4.21.: Rotating disc: Lyapunov function (EM)<sup>u</sup> scheme (left), Lyapunov function (M)<sup>θ</sup> scheme (right)

Figure 4.22.: Rotating disc: Lyapunov function (ME)<sup>η</sup> scheme

In addition we investigate the order of accuracy of the present schemes. We therefore introduce the relative error in the position, linear momentum density and absolute temperature dened by

$$\mathbf{e}\_{\varphi} = \frac{||\varphi - \varphi\_r||\_{L\_2}}{||\mathfrak{P}\_r||\_{L\_2}}, \quad \mathbf{e}\_{\mathfrak{p}} = \frac{||\mathbf{p} - \mathbf{p}\_r||\_{L\_2}}{||\mathbf{p}\_r||\_{L\_2}}, \quad \mathbf{e}\_{\theta} = \frac{||\theta - \theta\_r||\_{L\_2}}{||\theta\_r||\_{L\_2}}, \quad ||\bullet||\_{L\_2} = \left[\int\_{\mathcal{B}} \langle \bullet, \bullet \rangle \, \mathrm{d}V\right]^{\frac{1}{2}},$$

where index r refers to the reference solution obtained with a very small time step. Moreover, h•, •i denotes the inner product. As can be observed from Figs. 4.23 and 4.24, all schemes are second order accurate in the position, linear momentum density and absolute temperature.

Figure 4.23.: Rotating disc: Error in the position (left), Error in the linear momentum density (right)

Figure 4.24.: Rotating disc: Error in the absolute temperature

Eventually, the motion of the disc is illustrated in Fig. 4.25 with snapshots at successive points in time. In addition to that, the distribution of the temperature over the disc is shown.

Figure 4.25.: Rotating disc: Snapshots of the motion at successive points in time t ∈ {0, 4, 8, 12, 16 ,18 ,24, 28}s, and corresponding temperature distribution, calculated with the (M)<sup>θ</sup> scheme and ∆t = 0.04s

## 4.5.3. Rotating disc in a thermally perfect environment

The last example of this Chapter deals with a slight modication of the previous example which renders a thermally perfect environment for the disc throughout the whole duration of the simulation (Fig. 4.26).

Figure 4.26.: Rotating disc in a thermally perfect environment: Initial conguration, temperature boundary conditions imposed on a quarter of the lateral surface of the disc and initial distribution of the temperature

Instead of prescribing the heat ow over a quarter of the lateral surface of the disc, we now impose temperature boundary conditions on the same portion of the lateral surface of the disc. In particular, the temperature at the nite element nodes located on the boundary in question is constrained to be equal to the reference temperature θ0. The

initial temperature at the remaining nodes of the nite element mesh is chosen to assume the value θ<sup>0</sup> + 80K. The remaining simulation data are identical to those used in the previous example (see also Table 4.4). A summary of the data used in the simulation of the rotating disc in a thermally perfect environment can be found in Table 4.5. In this example we focus on the temperature-based (M)<sup>θ</sup> scheme, which makes possible to impose the temperature boundary conditions in a straightforward way through standard Dirichlet boundary conditions. Constant time steps of size ∆t ∈ {0.04, 0.075, 0.0875}s are applied. Again the mechanical system at hand has both translational and rotational symmetry.


Table 4.5.: Rotating disc in a thermally perfect environment: Data used in the simulations

The corresponding momentum maps are exactly conserved by the (M)<sup>θ</sup> scheme as can be observed from Fig. 4.27.

Figure 4.27.: Rotating disc in a thermally perfect environment: Total linear momentum (M)<sup>θ</sup> scheme (left), Total angular momentum (M)<sup>θ</sup> scheme (right)

Due to the temperature boundary condition heat is owing out of the disc. The corresponding evolution of the total energy is depicted in Fig. 4.28 (left). In addition to that, the evolution of the total entropy is shown in Fig. 4.28 (right).

Figure 4.28.: Rotating disc in a thermally perfect environment: Total energy (left), Total entropy (right)

Figure 4.29.: Rotating disc in a thermally perfect environment: Lyapunov function

As before, the break down of a simulation due to numerical instability is indicated by vertical lines. Again the onset of numerical instability can be detected from the plot of the Lyapunov function in Fig. 4.29, although this time the nonphysical increase is not as pronounced as in the two previous examples. Eventually, both the motion and the evolution of the temperature distribution are shown in Fig. 4.30 with snapshots at successive points in time.

Figure 4.30.: Rotating disc in a thermally perfect environment: Snapshots of the motion at t ∈ {0, 10, 20, 30, 40, 50, 60, 70}s and corresponding temperature distribution, calculated with the (M)<sup>θ</sup> scheme and ∆t = 0.04s

# 5. Energy-momentum-entropy consistent numerical methods for large-strain thermo-elasticity based on the GENERIC formalism<sup>1</sup>

The results of Chapter 4 lead to the conclusion that all major balance laws should be maintained in the discrete setting in order to enhance the numerical stability of numerical methods for nite-strain thermo-elastodynamics. Therefore fully structure-preserving numerical methods are developed in the present chapter. We will resort to the GENERICbased description of large-strain thermo-elasticity, which makes the free choice of the thermodynamic state variable possible. In particular, one may choose (i) the internal energy density, (ii) the entropy density, or (iii) the absolute temperature as the thermodynamic state variable. Three alternative energy-momentum-entropy (EME) schemes result from the present approach that make use of a GENERIC-consistent space discretization. These schemes are directly linked to the respective choice of the thermodynamic state variable. Numerical examples conrm the structure-preserving properties of the newly developed (EME) schemes, which exhibit superior numerical stability.

## 5.1. Large-strain thermoelasticty

In this section we briey recapitulate the variational formulation of large-strain thermoelasticity with heat conduction of Chapter 4 which lies at the heart of the proposed discretization in space and time. In contrast to Chapter 4 we introduce a frame-invariant formulation from the outset.

## 5.1.1. Underlying variational formulation

We consider a continuum body with material points X = X<sup>A</sup>e<sup>A</sup> in the reference conguration B ⊂ <sup>R</sup> 3 , see Fig. 5.1. Here and in the sequel the summation convention applies to repeated indices. Moreover, e<sup>A</sup> denote the canonical base vectors in R 3 .

<sup>1</sup> This Chapter is based on [2]

Figure 5.1.: Reference conguration B with boundary ∂B and current conguration ϕ(B, t) at time t. External tractions t = FSN act on the boundary of the current conguration. In addition to that, the heat ux across the current boundary is denoted by q = Q · N.

Within the Lagrangian description of continuum mechanics the deformed conguration of the body at time <sup>t</sup> is characterised by the deformation map <sup>ϕ</sup> : B × I 7→ <sup>R</sup> 3 , where I = [0, T] is the time interal of interest. The velocity of the material point X ∈ B located at x = ϕ(X, t) is given by v = ϕ˙ , where a superposed dotes denotes the material time derivate. The deformation gradient is given by

$$\mathbf{F} = \partial \mathbf{x} \boldsymbol{\varphi} \,. \tag{5.1}$$

Main ingredients of the GENERIC framework are the internal energy and the entropy. In addition to that, the choice of the thermodynamic state variable plays an important role. Following Chapter 4 we allow for the free choice of the thermodynamic state variable <sup>τ</sup> : B × I 7→ <sup>R</sup> from among three options <sup>τ</sup> ∈ {θ, η, u}. These options are: (i) the absolute temperature θ, (ii) the entropy density η, and (iii) the internal energy density u. Depending on the choice of the thermodynamic state variable, the absolute temperature can be written in the form (see also [94] and [100])

$$\Theta'(\mathbf{C},\tau) = \frac{\partial\_{\tau} u'(\mathbf{C},\tau)}{\partial\_{\tau} \eta'(\mathbf{C},\tau)}\,. \tag{5.2}$$

Now the internal energy density and the entropy density, respectively, are given by

$$u = u'(\mathbf{C}, \tau) \quad \text{and} \quad \eta = \eta'(\mathbf{C}, \tau) \,. \tag{5.3}$$

In this connection, a frame-indierent constitutive formulation for thermoelastic materials is based on the right Cauchy-Green tensor C = F <sup>T</sup>F. The GENERIC-based weak form pertaining to large-strain thermo-elasticity can be written in the form (see Chapter 4)

$$\begin{split} 0 &= \int\_{\mathcal{B}} \mathbf{w}\_{\mathcal{P}} \cdot \left( \dot{\boldsymbol{\varphi}} - \rho^{-1} \mathbf{p} \right) \, \mathrm{d}V \,, \\ 0 &= \int\_{\mathcal{B}} \left( \mathbf{w}\_{\mathbf{p}} \cdot \left( \dot{\mathbf{p}} - \mathbf{b} \right) + \mathbf{F} \mathbf{S} : \nabla \mathbf{w}\_{\mathbf{p}} \right) \, \mathrm{d}V - \int\_{\partial\_{\sigma} \mathcal{B}} \mathbf{w}\_{\mathbf{p}} \cdot \overline{\mathbf{t}} \, \mathrm{d}A \,, \\ 0 &= \int\_{\mathcal{B}} \left( \mathbf{w}\_{\tau} \dot{\tau} + \nabla (\rho^{-1} \mathbf{p}) : \left( \frac{\mathbf{w}\_{\tau}}{\partial\_{\tau} \eta'} 2 \mathbf{F} \partial\_{\mathbf{C}} \eta' \right) - \nabla \left( \frac{\mathbf{w}\_{\tau}}{\partial\_{\tau} u'} \right) \cdot \mathbf{Q} \right) \, \mathrm{d}V + \int\_{\partial\_{\mathbf{q}} \mathcal{B}} \frac{\mathbf{w}\_{\tau}}{\partial\_{\tau} u'} \overline{q} \, \mathrm{d}A \,, \end{split} \tag{5.4}$$

where <sup>ρ</sup> : B 7→ <sup>R</sup><sup>+</sup> is the mass density in the reference conguration. Moreover, <sup>p</sup> : B × I 7→ <sup>R</sup> 3 is the linear momentum density and <sup>b</sup> : B 7→ <sup>R</sup> 3 represent prescribed body forces which, for simplicity, are assumed to be dead loads. The second Piola-Kirchho stress tensor is given by

$$\mathbf{S} = \mathbf{S}'(\mathbf{C}, \tau) = 2 \left( \partial\_{\mathbf{C}} u' - \Theta' \partial\_{\mathbf{C}} \eta' \right) \,. \tag{5.5}$$

Furthermore, the material heat ux vector assumes the form

$$\mathbf{Q} = \mathbf{Q}'(\mathbf{C}, \tau) = (\Theta')^2 \mathbf{K}' \nabla \left(\frac{1}{\Theta'}\right),\tag{5.6}$$

where K = K<sup>0</sup> (C, τ ) is the positive semi-denite material conductivity tensor. The weak form needs be supplemented with initial and boundary conditions, respectively. For that purpose, the boundary ∂B of the continuum body is decomposed into a displacement boundary ∂ϕB, on which ϕ = ϕ is prescribed, and a traction boundary ∂σB, on which the external traction t is prescribed such that FSN = t (Fig. 5.2). In this connection, the standard relations ∂ϕB ∪ ∂σB = ∂B and ∂ϕB ∩ ∂σB = ∅ hold. Similarly, for the thermal part we consider the subsets ∂τB and ∂qB, with the properties ∂τB ∪ ∂qB = ∂B and ∂τB ∩ ∂qB = ∅ (Fig. 5.3). The thermodynamic state variable is prescribed on ∂τB, i.e. τ = τ , whereas the heat ux is prescribed on ∂qB, i.e. Q · N = q.

The unknown elds are subject to initial conditions of the form ϕ(·, 0) = X, p(·, 0) = ρv0, and τ (·, 0) = τ<sup>0</sup> in B. Here, v<sup>0</sup> is a prescribed velocity eld and τ<sup>0</sup> is a prescribed eld of the thermodynamic state variable τ ∈ {θ, η, u}.

Figure 5.2.: Mechanical part of the IBVP. Note that t = FSN denotes prescribed external Piola tractions acting on the current boundary expressed per unit area of the reference boundary ∂σB.

Figure 5.3.: Thermal part of the IBVP. Note that q = Q · N is the prescribed rate of heat transfer accross the current boundary expressed per unit area of the reference boundary ∂qB.

## 5.1.2. Frame-invariant local form of the eld equations

We next deal with the frame-invariant local form of the eld equations corresponding to the GENERIC-based weak form (5.4). In this context, we consider alternative frameinvariant formulations related to the specic choice of the thermodynamic state variable

τ ∈ {θ, η, u}. Using standard arguments from the calculus of variations, weak form (5.4) gives rise to the corresponding local form of the eld equations

$$\begin{aligned} \dot{\varphi} &= \rho^{-1} \mathbf{p} \ , \\ \dot{\mathbf{p}} &= \mathbf{b} + \text{Div} \left( \mathbf{F} \mathbf{S} \right) \ , \end{aligned} \tag{5.7}$$

together with

$$\dot{\tau} = -\frac{1}{\partial\_{\tau} \eta'} 2\mathbf{F} \partial\_{\mathbf{C}} \eta' : \nabla(\rho^{-1} \mathbf{p}) - \frac{1}{\partial\_{\tau} u'} \text{Div} \mathbf{Q} \,. \tag{5.8}$$

These equations have to hold for all X ∈ B and t > 0. The last equation governs the time-evolution of the thermodynamic state variable τ ∈ {θ, η, u}. Note that the second Piola-Kirchho stress tensor S in (5.7)<sup>2</sup> assumes the form (5.5), and the material heat ux vector Q in (5.8) is given by (5.6).

Choosing τ = θ leads to the temperature-based formulation, where the second Piola-Kirchho stress tensor follows from (5.5) and assumes the form

$$\overline{\mathbf{S}} = 2 \left[ \partial\_{\mathbf{C}} \overline{u}(\mathbf{C}, \theta) - \theta \partial\_{\mathbf{C}} \overline{\eta}(\mathbf{C}, \theta) \right] \,. \tag{5.9}$$

The temperature typically enters the denition of the Helmholtz free energy density

$$
\psi = \overline{\psi}(\mathbf{C}, \theta) \,. \tag{5.10}
$$

Then the entropy density is dened by

$$\eta = \overline{\eta}(\mathbf{C}, \theta) = -\partial\_{\theta} \overline{\psi}(\mathbf{C}, \theta) \,. \tag{5.11}$$

The internal energy density function then follows from a Legende transformation of the free energy density. Accordingly, the internal energy density is given by

$$u = \overline{u}(\mathbf{C}, \theta) = \psi(\mathbf{C}, \theta) + \theta \,\overline{\eta}(\mathbf{C}, \theta) \,. \tag{5.12}$$

Taking into account the temperature formula (5.2) along with ψ = u − θη, it can be seen that expression (5.9) for the second Piola-Kirchho stress tensor is equivalent to

$$\mathbf{S} = \overline{\mathbf{S}}(\mathbf{C}, \theta) = 2\partial\_{\mathbf{C}}\overline{\psi}(\mathbf{C}, \theta) \,. \tag{5.13}$$

Equations (5.11) and (5.13) are sometimes called thermal equations of state (see, for example, [147]).

Alternatively, we may use the entropy density <sup>η</sup> : B × I 7→ <sup>R</sup> as thermodynamic state variable leading to the entropy-based formulation. Accordingly, choosing τ = η, the second Piola-Kirchho stress tensor follows from (5.5) and assumes the form

$$\mathbf{S} = \mathbf{S}(\mathbf{C}, \eta) = 2\partial\_{\mathbf{C}} \tilde{u}(\mathbf{C}, \eta) \,. \tag{5.14}$$


Table 5.1.: Notation used in the present Chapter, depending on the choice of the thermodynamic state variable τ ∈ {θ, η, u}.

Temperature formula (5.2) leads to

$$
\Theta(\mathbf{C}, \eta) = \partial\_{\eta} \widetilde{u}(\mathbf{C}, \eta) \,. \tag{5.15}
$$

The last two equations belong to the caloric equations of state (see, for example, [136]).

Yet another option is to use the internal energy density <sup>u</sup> : B × I 7→ <sup>R</sup> as thermodynamic state variable leading to the internal-energy-based formulation. Thus, choosing τ = u, the second Piola-Kirchho stress tensor follows from (5.5) and assumes the form

$$\mathbf{S} = \mathbf{S}(\mathbf{C}, u) = -2\Theta \partial\_{\mathbf{C}} \hat{\eta}(\mathbf{C}, u) \,. \tag{5.16}$$

Furthermore, temperature formula (5.2) yields

$$
\widehat{\Theta}(\mathbf{C}, u) = \frac{1}{\partial\_u \widehat{\eta}(\mathbf{C}, u)}\,. \tag{5.17}
$$

The notation used in the present work is summarized in Table 5.1. A more detailed treatment of the alternative choice for the thermodynamic state variables can be found in [100, Sec. 2.3] (see also Section 4.2).

## 5.1.3. Balance laws

We next consider important balance laws of the underlying continuous formulation of nite-strain thermo-elasticity. The main goal of the present work is to preserve these balance laws under discretization. In this section we focus on the pure Neumann problem (i.e. ∂σB = ∂qB = ∂B).

The linear momentum of the continuum body is dened by L = R B p dV . Inserting <sup>w</sup><sup>p</sup> <sup>=</sup> <sup>ξ</sup> into (5.4)2, where <sup>ξ</sup> <sup>∈</sup> <sup>R</sup> 3 is arbitrary but constant, leads to

$$\boldsymbol{\xi} \cdot \frac{\mathrm{d}\mathbf{L}}{\mathrm{d}t} = \boldsymbol{\xi} \cdot \left( \int\_{\mathcal{B}} \mathbf{b} \, \mathrm{d}V + \int\_{\partial \mathcal{B}} \mathbf{F} \mathbf{SN} \, \mathrm{d}A \right) \,. \tag{5.18}$$

Due to the arbitrariness of <sup>ξ</sup> <sup>∈</sup> <sup>R</sup> 3 , the last equation recovers the balance law for linear momentum. The parentheses on the right-hand side of (5.18) contain the resultant external load applied on the continuum body (see also Fig. 5.1).

The total angular momentum relative to the origin of the inertial frame is dened by J = R <sup>B</sup> <sup>ϕ</sup> <sup>×</sup> <sup>p</sup> <sup>d</sup><sup>V</sup> . Substituting <sup>w</sup><sup>ϕ</sup> <sup>=</sup> <sup>p</sup> <sup>×</sup> <sup>ξ</sup>, <sup>w</sup><sup>p</sup> <sup>=</sup> <sup>ξ</sup> <sup>×</sup> <sup>ϕ</sup>, into (5.4) and subsequently adding the equations yields

$$\boldsymbol{\xi} \cdot \frac{\mathrm{d} \mathbf{J}}{\mathrm{d}t} = \boldsymbol{\xi} \cdot \left( \int\_{\mathcal{B}} \boldsymbol{\varphi} \times \mathbf{b} \, \mathrm{d}V + \int\_{\partial \mathcal{B}} \boldsymbol{\varphi} \times \mathbf{F} \mathbf{SN} \, \mathrm{d}A \right) \, . \tag{5.19}$$

Due to the arbitrariness of <sup>ξ</sup> <sup>∈</sup> <sup>R</sup> 3 , the last equation recovers the balance of angular momentum. Note that the parentheses on the right-hand side of (5.19) contain the resultant external torque about the origin (see also Fig. 5.1).

For verication of the balance of energy we choose the following test functions in (5.4)

$$\begin{aligned} \mathbf{w}\_{\varphi} &= \partial\_{\varphi} e' \,, \qquad \qquad \mathbf{w}\_{\varphi} = -\mathbf{b} \,, \\ \mathbf{w}\_{\mathbf{p}} &= \partial\_{\mathbf{p}} e' \,, \qquad \qquad \text{or} \qquad \qquad \mathbf{w}\_{\mathbf{p}} = \rho^{-1} \mathbf{p} \,, \\ \mathbf{w}\_{\tau} &= \partial\_{\tau} e' \, \,, \qquad \qquad \qquad \mathbf{w}\_{\tau} = \partial\_{\tau} u' \, \,, \end{aligned}$$

where the total energy density e 0 is given by

$$e'(\varphi, \mathbf{F}, \mathbf{p}, \tau) = \frac{1}{2} \rho^{-1} \mathbf{p} \cdot \mathbf{p} + u'(\mathbf{C}, \tau) - \mathbf{b} \cdot \varphi \,. \tag{5.20}$$

Substituting these quantities into (5.4) and subsequently adding the three resulting equations, a straightforward calculation recovers the balance law for energy

$$\frac{d\mathcal{E}'}{dt} = \int\_{\partial\mathcal{B}} \left( \rho^{-1} \mathbf{p} \cdot \mathbf{F} \mathbf{S} \mathbf{N} - \mathbf{N} \cdot \mathbf{Q} \right) \, \mathrm{d}A \, . \tag{5.21}$$

Concerning the balance of entropy, we insert w<sup>τ</sup> = ∂<sup>τ</sup> η 0 into (5.4)3, to obtain

$$\begin{split} 0 &= \int\_{\mathcal{B}} \left( \partial\_{\tau} \eta' \dot{\tau} + \nabla (\rho^{-1} \mathbf{p}) : (2 \mathbf{F} \partial\_{\mathbf{C}} \eta') - \nabla \left( \frac{\partial\_{\tau} \eta'}{\partial\_{\tau} u'} \right) \cdot \mathbf{Q} \right) \mathrm{d}V + \int\_{\partial \mathcal{B}} \frac{\partial\_{\tau} \eta'}{\partial\_{\tau} u'} \mathbf{Q} \cdot \mathbf{N} \, \mathrm{d}A \\ &= \int\_{\mathcal{B}} \left( \frac{\mathrm{d} \eta'}{\mathrm{d}t} - \nabla \left( \frac{1}{\Theta'} \right) \cdot (\Theta')^2 \mathbf{K} \nabla \left( \frac{1}{\Theta'} \right) \right) \mathrm{d}V + \int\_{\partial \mathcal{B}} \frac{1}{\Theta'} \mathbf{Q} \cdot \mathbf{N} \, \mathrm{d}A \, . \end{split}$$

Here, use has been made of formula (5.2) for the temperature along with expression (5.6) for the material heat ux vector. Moreover, the identity ϕ˙ = ρ <sup>−</sup><sup>1</sup>p has again been taken into account. The above equation can be rewritten as

$$\frac{\mathrm{d}\mathcal{S}'}{\mathrm{d}t} = \underbrace{\int\_{\mathcal{B}} \nabla \left(\frac{1}{\Theta'}\right) \cdot \Theta'^2 \mathbf{K} \nabla \left(\frac{1}{\Theta'}\right) \mathrm{d}V}\_{\geq 0} - \int\_{\partial\mathcal{B}} \frac{1}{\Theta} \mathbf{Q} \cdot \mathbf{N} \, \mathrm{d}A \, . \tag{5.22}$$

The last equation corresponds to the Clausius-Duhem form of the second law of thermodynamics (see, for example, [136, Sec. 5]).

## 5.2. Discretization in space

Next, we perform the discretization in space of the GENERIC-based weak form (5.4). To this end we apply the isoparametric nite element approach (see, for example, [145]), based on nite-dimensional approximations of the following quantities

$$\varphi^{\mathbf{h}}(\mathbf{X},t) = N^a(\mathbf{X})\,\mathbf{q}\_a(t), \qquad \mathbf{v}^{\mathbf{h}}(\mathbf{X},t) = N^a(\mathbf{X})\,\mathbf{v}\_a(t) \,, \tag{5.23}$$

and

$$
\tau^{\mathbf{h}}(\mathbf{X},t) = N^a(\mathbf{X})\,\tau\_a(t)\,. \tag{5.24}
$$

Here, the summation convention applies, where a = 1, . . . , N, and N denotes the total number of nodes in the nite element mesh. Moreover, N<sup>a</sup> : B → <sup>R</sup> are the nodal shape functions and q<sup>a</sup> (t), <sup>v</sup>a(t) <sup>∈</sup> <sup>R</sup> 3 , <sup>τ</sup>a(t) <sup>∈</sup> <sup>R</sup> are the respective nodal values at time <sup>t</sup>. Analogous approximations are used for the test functions wϕ, w<sup>p</sup> and w<sup>τ</sup> , denoted by w<sup>h</sup> <sup>ϕ</sup>, w<sup>h</sup> <sup>p</sup> and w h τ . Then weak form (5.4) leads to the following semi-discrete equations:

$$\begin{split} 0 &= \int\_{\mathcal{B}} \mathbf{w}\_{\mathcal{P}}^{\mathbf{h}} \cdot \left( \dot{\varphi}^{\mathbf{h}} - \mathbf{v}^{\mathbf{h}} \right) \, \mathrm{d}V \,, \\ 0 &= \int\_{\mathcal{B}} \left( \mathbf{w}\_{\mathbf{p}}^{\mathbf{h}} \cdot \left( \rho \dot{\mathbf{v}}^{\mathbf{h}} - \mathbf{b} \right) + \nabla \mathbf{w}\_{\mathbf{p}}^{\mathbf{h}} : \mathbf{F}^{\mathbf{h}} \mathbf{S}^{\mathbf{h}} \right) \, \mathrm{d}V - \int\_{\partial\_{\sigma} \mathcal{B}} \mathbf{w}\_{\mathbf{p}}^{\mathbf{h}} \cdot \overline{\mathbf{t}} \, \mathrm{d}A \,, \\ 0 &= \int\_{\mathcal{B}} \mathbf{w}\_{\tau}^{\mathbf{h}} \left( \dot{\tau}^{\mathbf{h}} + \nabla \mathbf{v}^{\mathbf{h}} : \left( \frac{2}{\Pi\_{h}(\partial\_{\tau} \eta^{\mathbf{h}})} \mathbf{F}^{\mathbf{h}} \partial\_{\mathbf{C}} \eta^{\mathbf{h}} \right) \right) \, \mathrm{d}V \\ &- \int\_{\mathcal{B}} \nabla \left( \frac{\mathbf{w}\_{\tau}^{\mathbf{h}}}{\Pi\_{h}(\partial\_{\tau} \iota^{\mathbf{h}})} \right) \cdot \mathbf{Q}^{\mathbf{h}} \, \mathrm{d}V + \int\_{\partial\_{\eta} \mathcal{B}} \frac{\mathbf{w}\_{\tau}^{\mathbf{h}}}{\Pi\_{h}(\partial\_{\tau} \iota^{\mathbf{h}})} \overline{\mathbf{q}} \, \mathrm{d}A \,, \end{split} \tag{5.25}$$

where

$$\begin{aligned} \mathbf{S}^{\mathrm{h}} &= 2 \left( \partial\_{\mathbf{C}} \boldsymbol{u}^{\mathrm{h}} - \Theta^{\mathrm{h}} \partial\_{\mathbf{C}} \eta^{\mathrm{h}} \right) \,, \\ \mathbf{Q}^{\mathrm{h}} &= (\Theta^{\mathrm{h}})^{2} \mathbf{K}^{\mathrm{h}} \nabla \left( \frac{1}{\Theta^{\mathrm{h}}} \right) \,, \\ \Theta^{\mathrm{h}} &= \frac{\Pi\_{\mathrm{h}} (\partial\_{\boldsymbol{\tau}} \boldsymbol{u}^{\mathrm{h}})}{\Pi\_{\boldsymbol{h}} (\partial\_{\boldsymbol{\tau}} \eta^{\mathrm{h}})} \, . \end{aligned} \tag{5.26}$$

In this connection, u <sup>h</sup> = u 0 (C h , τ <sup>h</sup> ), η <sup>h</sup> = η 0 (C h , τ <sup>h</sup> ), and K<sup>h</sup> = K<sup>0</sup> (C h , τ <sup>h</sup> ). The interpolation formulas in (5.23) give rise to

$$\mathbf{F}^{\mathbf{h}} = \mathbf{q}\_a \otimes \nabla N^a \quad \text{and} \quad \mathbf{C}^{\mathbf{h}} = \mathbf{q}\_a \cdot \mathbf{q}\_b \nabla N^a \otimes \nabla N^b. \tag{5.27}$$

Moreover, Πh(∂τu h ) denotes the L<sup>2</sup> projection of function ∂τu h into the nite-dimensional space spanned by the shape functions N<sup>a</sup> , a = 1, . . . , N. That is,

$$
\Pi\_h(\partial\_\tau u^\mathbf{h}) = N^a (\partial\_\tau u)\_a \,, \tag{5.28}
$$

where the nodal values (∂τu)<sup>a</sup> are determined by

$$\int\_{\mathcal{B}} N^a \left( \partial\_\tau u^\mathbf{h} - \Pi\_h (\partial\_\tau u^\mathbf{h}) \right) \, \mathrm{d}V = 0 \,,\tag{5.29}$$

for a = 1, . . . , N. In particular, (5.29) together with (5.28) constitute a linear system of algebraic equations given by

$$H^{ab}(\partial\_\tau u)\_b = \int\_{\mathcal{B}} N^a \partial\_\tau u^\mathbf{h} \, \mathrm{d}V \,, \tag{5.30}$$

where Hab denote the components of the positive denite Gram matrix [Hab] dened by

$$H^{ab} = \int\_{\mathcal{B}} N^a N^b \,\mathrm{d}V \,. \tag{5.31}$$

We note for later use that (5.28) in conjunction with (5.30) lead to the relationship

$$
\Pi\_h(\partial\_\tau u^\mathbf{h}) = N^a H\_{ab} \int\_{\mathcal{B}} N^b \partial\_\tau u^\mathbf{h} \,\mathrm{d}V \,\,, \tag{5.32}
$$

where Hab denote the components of the inverse of the Gram matrix, i.e. [Hab] = [Hab] −1 .

Analogous relationships hold for the projection of function ∂<sup>τ</sup> η h , Πh(∂<sup>τ</sup> η h ). It is worth mentioning that the projections Πh(∂τu h ) and Πh(∂<sup>τ</sup> η h ) are only required if the functions ∂τu h and ∂<sup>τ</sup> η <sup>h</sup> do not belong to the nite element space spanned by the shape functions N<sup>a</sup> . For example, in the temperature-based formulation (i.e. for τ = θ), ∂θu <sup>h</sup> = ∂θu(C h , θ<sup>h</sup> ) corresponds to the specic heat at constant deformation. Thus, if this quantity is prescribed to be constant, the projection Πh(∂θu) need not be performed. However, in general the temperature-based formulation requires both projections (i.e. Πh(∂θu) and Πh(∂θη)).

In contrast to that, the two alternative formulations based on the choice τ ∈ {η, u} in general require only one projection. In particular, the formulation in terms of the internal energy density relies on <sup>Π</sup>h(∂uηb), whereas the entropy-based formulation relies on <sup>Π</sup>h(∂ηue). Originally, the projection has been introduced in the framework of the entropy-based formulation in [102] (see also [106]).

In the next Section we will show that the proposed discretization in space of the GENERICbased weak form (5.4) yields semi-discrete equations which can be brought into GENERIC form for nite dimensional systems (2.1). Thus, we speak of a GENERIC-consistent space discretization, see Section 2.1.2. In this connection the introduction of the above projections is essential for retaining consistency of the semi-discrete formulation.

## 5.2.1. GENERIC-consistent space discretization

In the following we will show that the proposed discretization in space of the GENERICbased weak form (5.4) yields semi-discrete equations which can be brought into GENERIC form (2.1), thus we show that the above proposed space discretization is GENERICconsistent in the sense of Section 2.1.2.

The present interpolation of the velocity eld gives rise to nodal velocity vectors va, cf. (5.23). We rst introduce the conjugate nodal momentum vectors via a standard Legendre transformation. Accordingly, we express the kinetic energy Ekin = E 0 kin(p) = R B 1 2 ρ <sup>−</sup><sup>1</sup><sup>p</sup> · <sup>p</sup> <sup>d</sup><sup>V</sup> in terms of the velocity eld by using the identity <sup>p</sup> <sup>=</sup> <sup>ρ</sup>v, leading to Ekin = T (v) = R B 1 2 ρv · v dV . Using the discrete velocity eld v <sup>h</sup> yields the kinetic energy of the discrete system given by

$$T = \mathcal{T}(\mathbf{v}^{\mathrm{h}}) = \frac{1}{2}M^{ab}\mathbf{v}\_a \cdot \mathbf{v}\_b\,,\tag{5.33}$$

where the components of the standard consistent mass matrix [Mab] are given by

$$M^{ab} = \int\_{\mathcal{B}} \rho N^a N^b \,\mathrm{d}V \,. \tag{5.34}$$

Note that the mass matrix [Mab] is symmetric and positive denite. Now the conjugate nodal momentum vectors are dened by

$$\mathbf{p}^a = \partial\_{\mathbf{v}\_a} T = M^{ab} \mathbf{v}\_b \,. \tag{5.35}$$

The last equation implies that the nodal velocity vectors can be written in terms of the nodal momentum vectors via

$$\mathbf{v}\_a = M\_{ab}\mathbf{p}^b,\tag{5.36}$$

where Mab are the components of the inverse mass matrix, i.e. [Mab] = [Mab] −1 . Now we introduce the state vector of the discrete system at hand which is comprised of the nodal quantities q<sup>a</sup> , p a , τa, a = 1, . . . , N. We arrange these quantities in the nodal state vector of the system given by

$$\mathbf{z} = \begin{pmatrix} \mathbf{q}\_1, & \dots, & \mathbf{q}\_N, & \mathbf{p}^1, & \dots, & \mathbf{p}^N, & \tau\_1, & \dots, & \tau\_N \end{pmatrix} \,. \tag{5.37}$$

Hereby the state vector has to be viewed as a column vector. Next we aim at the total energy and the total entropy of the discrete system which play the role of generators in the GENERIC formulation (2.1). The two required functions assume the form

$$\begin{split} \mathcal{E}(\mathbf{z}) &= \int\_{\mathcal{B}} u^{\mathrm{h}} \mathrm{d}V + \frac{1}{2} M\_{ab} \mathbf{p}^{a} \cdot \mathbf{p}^{b} - \mathbf{q}\_{a} \cdot \int\_{\mathcal{B}} N^{a} \mathbf{b} \, \mathrm{d}V \,, \\ \mathcal{S}(\mathbf{z}) &= \int\_{\mathcal{B}} \eta^{\mathrm{h}} \, \mathrm{d}V \, . \end{split} \tag{5.38}$$

Concerning the derivatives ∂zE and ∂zS, we obtain the individual contributions

$$\begin{split} \partial\_{\mathbf{q}\_a} \mathcal{E} &= \int\_{\mathcal{B}} \partial\_{\mathbf{F}} u^{\mathbf{h}} \nabla N^a \, \mathrm{d}V - \int\_{\mathcal{B}} N^a \mathbf{b} \, \mathrm{d}V \,, \\ \partial\_{\mathbf{p}^a} \mathcal{E} &= M\_{ab} \mathbf{p}^b \,, \\ \partial\_{\tau\_a} \mathcal{E} &= \int\_{\mathcal{B}} N^a \partial\_{\tau} u^{\mathbf{h}} \, \mathrm{d}V \,, \end{split} \tag{5.39}$$

and

$$\begin{split} \partial\_{\mathbf{q}\_{a}} \mathcal{S} &= \int\_{\mathcal{B}} \partial\_{\mathbf{F}} \eta^{\mathbf{h}} \nabla N^{a} \, \mathrm{d}V \,, \\ \partial\_{\mathbf{p}^{a}} \mathcal{S} &= \mathbf{0} \,, \\ \partial\_{\tau\_{a}} \mathcal{S} &= \int\_{\mathcal{B}} N^{a} \partial\_{\tau} \eta^{\mathbf{h}} \, \mathrm{d}V \, . \end{split} \tag{5.40}$$

Next, we take a closer look at the semi-discrete formulation emanating from the spacediscrete weak form (5.25). To this end we focus on the pure Neumann problem and neglect the boundary integrals in (5.25). This is in accordance with the fact that the basic form (2.1) of the GENERIC formulation is valid for closed systems. The kinematic equation (5.25)<sup>1</sup> directly leads to <sup>H</sup>ab(q˙ <sup>a</sup> <sup>−</sup> <sup>v</sup>a) = <sup>0</sup>, where the components of the Gram matrix [Hab] have been introduced in (5.31). Due to the positive deniteness of the Gram matrix one obtains

$$
\dot{\mathbf{q}}\_a = M\_{ab} \mathbf{p}^b = \partial\_{\mathbf{p}^a} \mathcal{E} \,. \tag{5.41}
$$

The space-discrete weak form of the balance law for linear momentum, (5.25)2, yields

$$M^{ab}\dot{\mathbf{v}}\_b = \int\_{\mathcal{B}} N^a \mathbf{b} \, \mathrm{d}V - \int\_{\mathcal{B}} \mathbf{F}^{\mathrm{h}} \mathbf{S}^{\mathrm{h}} \nabla N^a \, \mathrm{d}V \, \mathrm{.}\tag{5.42}$$

Taking into account (5.35), (5.26)1,<sup>3</sup> and

$$\begin{aligned} \partial\_{\mathbf{F}} u^{\mathbf{h}} &= 2 \mathbf{F}^{\mathbf{h}} \partial\_{\mathbf{C}} u^{\mathbf{h}}, \\ \partial\_{\mathbf{F}} \eta^{\mathbf{h}} &= 2 \mathbf{F}^{\mathbf{h}} \partial\_{\mathbf{C}} \eta^{\mathbf{h}}, \end{aligned} \tag{5.43}$$

into account eq. (5.42) can be recast in the form

$$\dot{\mathbf{p}}^a = \int\_{\mathcal{B}} N^a \mathbf{b} \, \mathrm{d}V - \int\_{\mathcal{B}} \partial\_{\mathbf{F}} u^{\mathbf{h}} \nabla N^a \, \mathrm{d}V + \int\_{\mathcal{B}} \frac{\Pi\_h(\partial\_{\tau} u^{\mathbf{h}})}{\Pi\_h(\partial\_{\tau} \eta^{\mathbf{h}})} \partial\_{\mathbf{F}} \eta^{\mathbf{h}} \nabla N^a \, \mathrm{d}V \,. \tag{5.44}$$

With regard to (5.32) and (5.39)3, we get

$$
\Pi\_h(\partial\_\tau u^\mathbf{h}) = N^a H\_{ab} \partial\_{\tau\_b} \mathcal{E} \ . \tag{5.45}
$$

109

Accordingly, taking into account the last equation and (5.39)1, (5.44) can be rewritten as

$$\dot{\mathbf{p}}^a = -\partial\_{\mathbf{q}\_a} \mathcal{E} + \int\_{\mathcal{B}} \frac{N^d}{\Pi\_h(\partial\_\tau \eta^\mathbf{h})} \partial\_\mathbf{F} \eta^\mathbf{h} \nabla N^a \, \mathrm{d}V H\_{db} \partial\_{\tau\_b} \mathcal{E} \,. \tag{5.46}$$

The space-discrete weak form of the energy equation, (5.25)3, yields

$$\begin{split} H^{ab}\dot{\tau}\_{b} &= -\mathbf{v}\_{d} \cdot \int\_{\mathcal{B}} \frac{N^{a}}{\Pi\_{h}(\partial\_{\tau}\eta^{\mathbf{h}})} \partial\_{\mathbf{F}}\eta^{\mathbf{h}} \nabla N^{d} \, \mathrm{d}V \\ &+ \nabla \left( \frac{N^{a}}{\Pi\_{h}(\partial\_{\tau}u^{\mathbf{h}})} \right) \cdot (\Theta^{\mathbf{h}})^{2} \mathbf{K}^{\mathbf{h}} \nabla \left( \frac{\Pi\_{h}(\partial\_{\tau}\eta^{\mathbf{h}})}{\Pi\_{h}(\partial\_{\tau}u^{\mathbf{h}})} \right) \, \mathrm{d}V \, . \end{split} \tag{5.47}$$

In analogy to (5.45), we have

$$
\Pi\_h(\partial\_\tau \eta^\mathbf{h}) = N^a H\_{ab} \partial\_{\tau\_b} \mathcal{S} \ .
$$

Using the last equation together with (5.36) and (5.39)2, (5.47) can be recast in the form

$$\begin{split} \dot{\tau}\_{a} &= -H\_{ab}\partial\_{\mathbf{p}^{d}}\mathcal{E} \cdot \int\_{\mathcal{B}} \frac{N^{b}}{\Pi\_{h}(\partial\_{\tau}\eta^{\mathbf{h}})} \partial\_{\mathbf{F}}\eta^{\mathbf{h}} \nabla N^{d} \, \mathrm{d}V \\ &+ H\_{ab} \int\_{\mathcal{B}} \nabla \left(\frac{N^{b}}{\Pi\_{h}(\partial\_{\tau}u^{\mathbf{h}})}\right) \cdot (\Theta^{\mathbf{h}})^{2} \mathbf{K}^{\mathbf{h}} \nabla \left(\frac{N^{c}}{\Pi\_{h}(\partial\_{\tau}u^{\mathbf{h}})}\right) \, \mathrm{d}V H\_{cd} \partial\_{\tau\_{d}} \mathcal{S} \, . \end{split} \tag{5.48}$$

To summarize, the semi-discrete evolution equations resulting from the discretization in space of the underlying GENERIC-based weak form can be written as

$$\begin{aligned} \dot{\mathbf{q}}\_a &= \partial\_{\mathbf{p}^a} \mathcal{E} \ , \\ \dot{\mathbf{p}}^a &= -\partial\_{\mathbf{q}\_a} \mathcal{E} + \mathbf{l}\_{\cdot b}^a \partial\_{\tau\_b} \mathcal{E} \ , \\ \dot{\tau}\_a &= -\left(\mathbf{l}\_{\cdot a}^b\right)^T \partial\_{\mathbf{p}^b} \mathcal{E} + m\_{ab} \partial\_{\tau\_b} \mathcal{S} \ , \end{aligned} \tag{5.49}$$

where

$$\begin{split} \mathbf{1}\_{\cdot b}^{a} &= \int\_{\mathcal{B}} \frac{N^{d}}{\Pi\_{h}(\partial\_{\tau} \eta^{\mathbf{h}})} \partial\_{\mathbf{F}} \eta^{\mathbf{h}} \nabla N^{a} \, \mathrm{d}V H\_{db} \,, \\ m\_{ab} &= H\_{ac} \int\_{\mathcal{B}} \nabla \left( \frac{N^{c}}{\Pi\_{h}(\partial\_{\tau} u^{\mathbf{h}})} \right) \cdot (\Theta^{\mathbf{h}})^{2} \mathbf{K}^{\mathbf{h}} \nabla \left( \frac{N^{d}}{\Pi\_{h}(\partial\_{\tau} u^{\mathbf{h}})} \right) \, \mathrm{d}V H\_{db} \,, \end{split} \tag{5.50}$$

The evolution equations (5.49) t into the GENERIC framework for nite-dimensional systems provided by (2.1). In particular, (5.49) can be written in the form

$$
\dot{\mathbf{z}} = \mathbf{L}\partial\_\mathbf{z}\mathcal{E} + \mathbf{M}\partial\_\mathbf{z}\mathcal{S} \ ,
$$

where the state vector is given by (5.37). Moreover, the Poisson matrix reads

$$\mathbf{L} = \begin{bmatrix} \mathbf{I} \\ \hline \\ \hline \\ \hline \\ 0 & \mathbf{I} \\ \hline \\ 0 & \begin{bmatrix} -\mathbf{I}\_{\cdot a} \end{bmatrix}^{\mathrm{T}} & 0 \\ \hline \\ \hline \\ 0 & \begin{bmatrix} -\mathbf{I}\_{\cdot a}^{b} \end{bmatrix}^{\mathrm{T}} & 0 \end{bmatrix},\tag{5.51}$$

where I is the identity matrix (with appropriate dimension corresponding to the partitioning of the state vector (5.37)), and matrix [l a· · b ] consists of vectors l a· · <sup>b</sup> dened in (5.50)1. Specically, we have

$$\mathbf{I}\_{\cdot,b}^{a,\cdot} = \begin{bmatrix} \mathbf{1}\_{\cdot,1}^{1} & \cdots & \mathbf{1}\_{\cdot,N}^{1} \\ \vdots & \ddots & \vdots \\ \mathbf{1}\_{\cdot,1}^{N,\cdot} & \cdots & \mathbf{1}\_{\cdot,N}^{N,\cdot} \end{bmatrix} \tag{5.52}$$

The friction matrix is given by

$$\mathbf{M} = \begin{bmatrix} 0 & & & & & & & & & \\ & 0 & & & & & & & & \\ \hline & & 0 & & & & & & & \\ \hline & & & & & & & & & \\ \hline & & & & & & & & & \\ \hline & & & & & & & & & \\ \hline & & & & & & & & & \\ \hline \end{bmatrix}, \tag{5.53}$$

where the coecients mab have been introduced in (5.50)2. It can be easily veried by a straightforward calculation that the two degeneration conditions (2.2) and (2.3) are satised. Thus the validity of (2.4) and (2.5) ensures thermodynamic consistency of the semi-discrete formulation.

Remark 6. Instead of using the interpolation of the velocity eld (cf. (5.23)), one could as well interpolate the linear momentum density eld. This would lead, for example, to minor changes in the partial derivative (5.39)<sup>2</sup> and the Poisson matrix (5.51). However, it would not aect the thermodynamic consistency of the resulting semi-discrete formulation. These observations are valid for constant mass density ρ. We prefer the interpolation of the velocity eld, because (i) it does not put any restrictions on the mass density and, (ii) it leads to nodal velocities as degrees of freedom on which initial and boundary conditions for the velocity eld can be directly imposed.

## 5.2.2. Balance laws

We next give a short summary of the balance laws pertaining to the space-discrete formulation described above. We restate the semi-discrete evolution equations derived in Section 5.2.1 in the form

$$\begin{split} \dot{\mathbf{q}}\_{a} &= \mathbf{v}\_{a} \ , \\ M^{ab}\dot{\mathbf{v}}\_{b} &= \int\_{\mathcal{B}} N^{a}\mathbf{b} \, \mathrm{d}V - \int\_{\mathcal{B}} \partial\_{\mathbf{F}} u^{\mathrm{h}} \nabla N^{a} \, \mathrm{d}V + \int\_{\mathcal{B}} \Theta^{\mathrm{h}} \partial\_{\mathbf{F}} \eta^{\mathrm{h}} \nabla N^{a} \, \mathrm{d}V \ , \\ H^{ab}\dot{\boldsymbol{\tau}}\_{b} &= -\mathbf{v}\_{d} \ , \int\_{\mathcal{B}} \frac{N^{a}}{\Pi\_{h}(\partial\_{\tau}\eta^{\mathrm{h}})} \partial\_{\mathbf{F}} \eta^{\mathrm{h}} \nabla N^{d} \, \mathrm{d}V \ - \int\_{\mathcal{B}} \nabla \left( \frac{N^{a}}{\Pi\_{h}(\partial\_{\tau}u^{\mathrm{h}})} \right) \cdot \mathbf{K}^{\mathrm{h}} \nabla \Theta^{\mathrm{h}} \, \mathrm{d}V \\ &- \int\_{\partial\_{\boldsymbol{\theta}}\mathcal{B}} \frac{N^{a}}{\Pi\_{h}(\partial\_{\tau}u^{\mathrm{h}})} \overline{\eta} \, \mathrm{d}A \ . \end{split} \tag{5.54}$$

For simplicity of exposition we have neglected in (5.54)<sup>2</sup> the standard contribution of the external boundary tractions. The total linear momentum of the discrete system at hand is given by

$$\begin{split} \mathbf{L}^{\mathbf{h}} &= \int\_{\mathcal{B}} \rho \mathbf{v}^{\mathbf{h}} \, \mathrm{d}V \\ &= \int\_{\mathcal{B}} \rho N^{b} \, \mathrm{d}V \mathbf{v}\_{b} \\ &= \sum\_{a=1}^{\mathrm{N}} \int\_{\mathcal{B}} \rho N^{a} N^{b} \, \mathrm{d}V \mathbf{v}\_{b} \\ &= \sum\_{a=1}^{\mathrm{N}} M^{ab} \mathbf{v}\_{b} \, . \end{split} \tag{5.55}$$

Note that use has been made of the completeness condition P<sup>N</sup> <sup>a</sup>=1 N<sup>a</sup> = 1 for the nodal shape functions. Now (5.54)<sup>2</sup> directly leads to the balance law for linear momentum

$$\frac{\mathrm{d}}{\mathrm{d}t} \mathbf{L}^{\mathrm{h}} = \int\_{\mathcal{B}} \mathbf{b} \, \mathrm{d}V \,\,,\tag{5.56}$$

which can be regarded as the semi-discrete counterpart of (5.18). The total angular momentum relative to the origin of the inertial frame is given by

$$\begin{split} \mathbf{J^{h}} &= \int\_{\mathcal{B}} \rho \varphi^{\mathbf{h}} \times \mathbf{v^{h}} \, dV \\ &= M^{ab} \mathbf{q}\_{a} \times \mathbf{v}\_{b} \, . \end{split} \tag{5.57}$$

A straightforward calculation based on (5.54)1,<sup>2</sup> yields

$$\boldsymbol{\xi} \cdot \frac{\mathrm{d}}{\mathrm{d}t} \mathbf{J}^{\mathrm{h}} = \boldsymbol{\xi} \cdot \int\_{\mathcal{B}} \boldsymbol{\varphi}^{\mathrm{h}} \times \mathbf{b} \, \mathrm{d}V \,, \tag{5.58}$$

for arbitrary but constant <sup>ξ</sup> <sup>∈</sup> <sup>R</sup> 3 . In the last equation the following relationship has been taken into account:

$$\begin{split} \boldsymbol{\xi} \cdot \left( \mathbf{q}\_a \times (\partial\_{\mathbf{F}} \boldsymbol{u}^{\mathrm{h}} - \Theta^{\mathrm{h}} \partial\_{\mathbf{F}} \boldsymbol{\eta}^{\mathrm{h}}) \nabla N^a \right) &= \boldsymbol{\hat{\xi}} : \left( (\partial\_{\mathbf{F}} \boldsymbol{u}^{\mathrm{h}} - \Theta^{\mathrm{h}} \partial\_{\mathbf{F}} \boldsymbol{\eta}^{\mathrm{h}}) \nabla N^a \otimes \mathbf{q}\_a \right) \\ &= \boldsymbol{\hat{\xi}} : \left( (\partial\_{\mathbf{F}} \boldsymbol{u}^{\mathrm{h}} - \Theta^{\mathrm{h}} \partial\_{\mathbf{F}} \boldsymbol{\eta}^{\mathrm{h}}) \mathbf{F}^{\mathrm{h} \mathrm{T}} \right) \\ &= \boldsymbol{\hat{\xi}} : \left( \mathbf{F}^{\mathrm{h}} \mathbf{S}^{\mathrm{h}} \mathbf{F}^{\mathrm{h} \mathrm{T}} \right) \\ &= 0 \end{split} \tag{5.59}$$

Here, use has been made of (5.43) and (5.26)1. Moreover, <sup>b</sup><sup>ξ</sup> is skew-symmetric such that <sup>b</sup>ξ<sup>a</sup> <sup>=</sup> <sup>ξ</sup>×<sup>a</sup> for any <sup>a</sup> <sup>∈</sup> <sup>R</sup> 3 . Relationship (5.58) can be viewed as semi-discrete counterpart of the balance law for angular momentum (5.19).

For completeness of exposition we eventually verify the balance laws for energy and entropy, although the GENERIC-consistent space discretization at hand guarantees thermodynamic consistency as has already been shown in Section 5.2.1. Multiplying (5.54)<sup>3</sup> by (∂τu)a, and taking into account (5.28) together with (5.26)<sup>3</sup> yields

$$(\partial\_\tau u)\_a H^{ab} \dot{\tau}\_b = -\mathbf{v}\_d \cdot \int\_{\mathcal{B}} \Theta^\mathbf{h} \partial\_\mathbf{F} \eta^\mathbf{h} \nabla N^d \, \mathrm{d}V - \int\_{\partial\_\mathbf{q} \mathcal{B}} \overline{q} \, \mathrm{d}A \dots$$

With regard to (5.30), the last equation can be rewritten as

$$\int\_{\mathcal{B}} \partial\_{\tau} u^{\mathbf{h}} \dot{\tau}^{\mathbf{h}} \, \mathrm{d}V + \int\_{\mathcal{B}} \Theta^{\mathbf{h}} \partial\_{\mathbf{F}} \eta^{\mathbf{h}} : \dot{\mathbf{F}}^{\mathbf{h}} \, \mathrm{d}V = - \int\_{\partial\_{\mathbf{q}} \mathcal{B}} \overline{q} \, \mathrm{d}A \, . \tag{5.60}$$

Scalar multiplication of (5.54)<sup>2</sup> by v<sup>a</sup> yields

$$\mathbf{v}\_a \cdot M^{ab} \dot{\mathbf{v}}\_b = \mathbf{v}\_a \cdot \left( \int\_{\mathcal{B}} N^a \mathbf{b} \, \mathrm{d}V - \int\_{\mathcal{B}} \partial\_{\mathbf{F}} u^{\mathrm{h}} \nabla N^a \, \mathrm{d}V + \int\_{\mathcal{B}} \Theta^{\mathrm{h}} \partial\_{\mathbf{F}} \eta^{\mathrm{h}} \nabla N^a \, \mathrm{d}V \right) \,,$$

or

$$\frac{1}{2}\frac{d}{dt}\left(\mathbf{v}\_a \cdot M^{ab}\mathbf{v}\_b\right) = \int\_{\mathcal{B}} \mathbf{v}^{\mathbf{h}} \cdot \mathbf{b} \, \mathrm{d}V - \int\_{\mathcal{B}} \partial\_{\mathbf{F}} u^{\mathbf{h}} : \dot{\mathbf{F}}^{\mathbf{h}} \, \mathrm{d}V + \int\_{\mathcal{B}} \Theta^{\mathbf{h}} \partial\_{\mathbf{F}} \eta^{\mathbf{h}} : \dot{\mathbf{F}}^{\mathbf{h}} \, \mathrm{d}V \, \dots$$

Substituting from the last equation into (5.60) leads to

$$\int\_{\mathcal{B}} \left( \partial\_{\tau} u^{\mathbf{h}} \dot{\tau}^{\mathbf{h}} + \partial\_{\mathbf{F}} u^{\mathbf{h}} : \dot{\mathbf{F}}^{\mathbf{h}} \right) \mathrm{d}V + \frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}t} \left( M^{ab} \mathbf{v}\_a \cdot \mathbf{v}\_b \right) - \int\_{\mathcal{B}} \mathbf{v}^{\mathbf{h}} \cdot \mathbf{b} \, \mathrm{d}V = - \int\_{\partial\_{\mathbf{q}} \mathcal{B}} \overline{q} \, \mathrm{d}A \dots$$

The last equation can be rewritten as

$$\frac{d}{dt}\mathcal{E}^{\mathbf{h}} = -\int\_{\partial\_q \mathcal{B}} \overline{q} \, \mathrm{d}A \,, \tag{5.61}$$

113

where the total energy of the discrete system assumes the form

$$\mathcal{E}^{\mathbf{h}} = \int\_{\mathcal{B}} u^{\mathbf{h}} \, \mathrm{d}V + \frac{1}{2} M^{ab} \mathbf{v}\_a \cdot \mathbf{v}\_b - \int\_{\mathcal{B}} \varphi^{\mathbf{h}} \cdot \mathbf{b} \, \mathrm{d}V \, . \tag{5.62}$$

Note that (5.61) can be regarded as semi-discrete version of the balance law for energy (5.21).

To recover the balance of entropy, multiply (5.54)<sup>3</sup> by (∂<sup>τ</sup> η)a, and take into account (5.26)3, to obtain

$$(\partial\_\tau \eta)\_a H^{ab} \dot{\tau}\_b = -\mathbf{v}\_d \cdot \int\_{\mathcal{B}} \partial\_\mathbf{F} \eta^\mathbf{h} \nabla N^d \, \mathrm{d}V - \int\_{\mathcal{B}} \nabla \left(\frac{1}{\Theta^\mathrm{h}}\right) \cdot \mathbf{K}^\mathrm{h} \nabla \Theta^\mathrm{h} \, \mathrm{d}V - \int\_{\partial\_\mathbf{q} \mathcal{B}} \frac{1}{\Theta^\mathrm{h}} \overline{\eta} \, \mathrm{d}A \,, \mathbf{A}$$

The last equation can be rewritten as

$$\underbrace{\int\_{\mathcal{B}} \left(\partial\_{\tau} \boldsymbol{\eta}^{\mathbf{h}} \dot{\boldsymbol{\tau}}^{\mathbf{h}} + \partial \mathbf{r} \boldsymbol{\eta}^{\mathbf{h}} : \dot{\mathbf{F}}^{\mathbf{h}}\right) \mathrm{d}V}\_{=\frac{4}{\mathsf{d}t} \mathcal{S}^{\mathbf{h}}} = \underbrace{\int\_{\mathcal{B}} \nabla \left(\frac{1}{\Theta^{\mathbf{h}}}\right) \cdot (\Theta^{\mathbf{h}})^{2} \mathbf{K}^{\mathbf{h}} \nabla \left(\frac{1}{\Theta^{\mathbf{h}}}\right) \mathrm{d}V}\_{\geq 0} - \int\_{\partial\_{\boldsymbol{q}} \mathcal{B}} \frac{1}{\Theta^{\mathbf{h}}} \overline{q} \, \mathrm{d}A} \, \mathcal{A} \, \mathcal{B} \, \tag{5.63}$$

where the total entropy of the discrete system is dened by

$$\mathcal{S}^{\mathbf{h}} = \int\_{\mathcal{B}} \eta^{\mathbf{h}} \, \mathrm{d}V = \int\_{\mathcal{B}} \eta'(\mathbf{C}^{\mathbf{h}}, \tau^{\mathbf{h}}) \, \mathrm{d}V \, . \tag{5.64}$$

Note that (5.63) can be viewed as semi-discrete version of the balance of entropy (5.22).

## 5.3. Discretization in time

We aim at a second-order accurate, implicit time-stepping scheme which is capable of correctly reproducing the main balance laws outlined above. Due to its structure-preserving properties this type of integrator is called energy-momentum-entropy scheme. To devise such a scheme, we essentially apply the mid-point rule in which the derivatives of the internal energy density and the entropy density, respectively, are replaced by appropriate discrete derivatives.

We focus on a representative time interval [tn, tn+1] with corresponding time-step size ∆t = tn+1 − tn. The discrete approximations at times t<sup>n</sup> and tn+1 of a function f(t) will be denoted by f<sup>n</sup> and fn+1, respectively. Assume that the nodal state variables at time tn, q<sup>a</sup><sup>n</sup> , v<sup>a</sup><sup>n</sup> , and τ<sup>a</sup><sup>n</sup> are given. The associated elds result from the nodal interpolation formulas (5.23) and are denoted by ϕ<sup>h</sup> n , v h n : B 7→ <sup>R</sup> 3 and τ h n : B 7→ <sup>R</sup>, <sup>τ</sup> ∈ {θ, η, u}. We aim at the determination of the corresponding quantities at time tn+1 through the mid-point type discretization of the semi-discrete formulation (5.25) given by

0 = <sup>Z</sup> B w<sup>h</sup> ϕ · ϕ<sup>h</sup> n+1 <sup>−</sup> <sup>ϕ</sup><sup>h</sup> n ∆t − v h n+ <sup>1</sup> 2 dV , 0 = <sup>Z</sup> B w<sup>h</sup> p · ρ v h n+1 − v h n ∆t − b <sup>+</sup> <sup>∇</sup>w<sup>h</sup> <sup>p</sup> : F h n+ <sup>1</sup> 2 S h algo dV − Z ∂<sup>σ</sup> B w<sup>h</sup> <sup>p</sup> · tn+ <sup>1</sup> 2 dA , 0 = <sup>Z</sup> B w h τ τ h n+1 − τ h n ∆t + ∇v h n+ <sup>1</sup> 2 : 2 Πh(D<sup>τ</sup> η h) F h n+ <sup>1</sup> 2 DCη h <sup>d</sup><sup>V</sup> − Z B ∇ w h τ Πh(Dτu h) · Q h algo dV + Z ∂qB w h τ Πh(Dτu h) qn+ <sup>1</sup> 2 dA , (5.65)

where

$$\begin{split} \mathbf{S}\_{\text{algo}}^{\text{h}} &= 2 \left( \mathbf{D}\_{\mathbf{C}} \boldsymbol{u}^{\text{h}} - \boldsymbol{\Theta}\_{\text{algo}}^{\text{h}} \mathbf{D}\_{\mathbf{C}} \boldsymbol{\eta}^{\text{h}} \right), \\ \mathbf{Q}\_{\text{algo}}^{\text{h}} &= (\boldsymbol{\Theta}\_{\text{algo}}^{\text{h}})^{2} \mathbf{K}\_{\text{algo}}^{\text{h}} \nabla \left( \frac{1}{\boldsymbol{\Theta}\_{\text{algo}}^{\text{h}}} \right), \\ \boldsymbol{\Theta}\_{\text{algo}}^{\text{h}} &= \frac{\Pi\_{\text{h}} (\mathbf{D}\_{\tau} \boldsymbol{u}^{\text{h}})}{\Pi\_{\text{h}} (\mathbf{D}\_{\tau} \boldsymbol{\eta}^{\text{h}})}, \end{split} \tag{5.66}$$

and K<sup>h</sup> algo = K<sup>0</sup> (C h n+ <sup>1</sup> 2 , τ <sup>h</sup> n+ <sup>1</sup> 2 ).

The above discretization in time relies on the use of discrete derivatives in the sense of Gonzalez [46]. In particular, the discrete derivatives are applied to the internal energy density and the entropy density, respectively. For example, in the case of the internal energy density, the discrete derivatives are denoted by Dτu h and DCu h , respectively. In particular, Dτu h is dened by

$$\mathbf{D}\_{\tau}u^{\mathbf{h}} = \frac{1}{2} \left( \mathrm{d}u\_{\mathbf{C}\_{\mathrm{n}}^{\mathrm{h}}}(\tau\_{\mathrm{n}}^{\mathrm{h}}, \tau\_{\mathrm{n}+1}^{\mathrm{h}}) + \mathrm{d}u\_{\mathbf{C}\_{\mathrm{n}+1}^{\mathrm{h}}}(\tau\_{\mathrm{n}}^{\mathrm{h}}, \tau\_{\mathrm{n}+1}^{\mathrm{h}}) \right) \,, \tag{5.67}$$

where

$$\mathrm{d}u\_{\mathbf{C}}(\tau\_{\mathrm{n}},\tau\_{\mathrm{n}+1}) = \partial\_{\tau}u'(\mathbf{C},\tau\_{\mathrm{n}+\frac{1}{2}}) + \frac{u'(\mathbf{C},\tau\_{\mathrm{n}+1}) - u'(\mathbf{C},\tau\_{\mathrm{n}}) - \partial\_{\tau}u'(\mathbf{C},\tau\_{\mathrm{n}+\frac{1}{2}})\Delta\tau}{(\Delta\tau)^{2}}\Delta\tau,$$

and ∆τ = τn+1 − τn. Furthermore, DCu h is dened by

$$\mathbf{D}\_{\mathbf{C}} u^{\mathbf{h}} = \frac{1}{2} \left( \mathrm{d}u\_{\tau\_{\mathbf{n}}^{\mathbf{h}}} (\mathbf{C}\_{\mathbf{n}}^{\mathbf{h}}, \mathbf{C}\_{\mathbf{n}+1}^{\mathbf{h}}) + \mathrm{d}u\_{\tau\_{\mathbf{n}+1}^{\mathbf{h}}} (\mathbf{C}\_{\mathbf{n}}^{\mathbf{h}}, \mathbf{C}\_{\mathbf{n}+1}^{\mathbf{h}}) \right) \,, \tag{5.68}$$

where

$$\begin{split} \mathrm{d}u\_{\tau}(\mathbf{C}\_{\mathrm{n}}, \mathbf{C}\_{\mathrm{n}+1}) &= \partial\_{\mathbf{C}}u'(\mathbf{C}\_{\mathrm{n}+\frac{1}{2}}, \tau) \\ &+ \frac{u'(\mathbf{C}\_{\mathrm{n}+1}, \tau) - u'(\mathbf{C}\_{\mathrm{n}}, \tau) - \partial\_{\mathbf{C}}u'(\mathbf{C}\_{\mathrm{n}+\frac{1}{2}}, \tau) : \Delta \mathbf{C}}{\Delta \mathbf{C} : \Delta \mathbf{C}} \Delta \mathbf{C} \,, \end{split}$$

and ∆C = Cn+1−Cn. It can be veried by a straightforward calculation that the discrete derivatives (5.67) and (5.68) satisfy the directionality condition

$$\mathbf{D}\_{\mathbf{C}} u^{\mathbf{h}} : \left( \mathbf{C}\_{\mathbf{n}+1}^{\mathbf{h}} - \mathbf{C}\_{\mathbf{n}}^{\mathbf{h}} \right) + \mathbf{D}\_{\tau} u^{\mathbf{h}} \left( \tau\_{\mathbf{n}+1}^{\mathbf{h}} - \tau\_{\mathbf{n}}^{\mathbf{h}} \right) = u'(\mathbf{C}\_{\mathbf{n}+1}^{\mathbf{h}}, \tau\_{\mathbf{n}+1}^{\mathbf{h}}) - u'(\mathbf{C}\_{\mathbf{n}}^{\mathbf{h}}, \tau\_{\mathbf{n}}^{\mathbf{h}}) \,. \tag{5.69}$$

Analogous considerations apply to the discrete derivatives of the internal entropy density, D<sup>τ</sup> η h and DCη h , respectively. Moreover, the time-average of any quantity (•) is given by 1 2 (•)<sup>n</sup> + (•)n+1 . In particular, this implies

$$\mathbf{C}\_{\mathbf{n}+\frac{1}{2}} = \frac{1}{2} \left( \mathbf{C}\_{\mathbf{n}} + \mathbf{C}\_{\mathbf{n}+1} \right) \,. \tag{5.70}$$

Note that in general Cn+ <sup>1</sup> 2 6= F T n+ <sup>1</sup> 2 Fn+ <sup>1</sup> 2 . In (5.65) and (5.66), Πh(Dτu h ) denotes the L<sup>2</sup> projection of Dτu h into the nite dimensional space spanned by the shape functions N<sup>a</sup> , a = 1, . . . , N. In analogy to (5.30),

$$H^{ab}(\mathbb{D}\_\tau u)\_b = \int\_{\mathcal{B}} N^a \mathbb{D}\_\tau u^\mathbf{h} \, \mathrm{d}V \, \,, \tag{5.71}$$

constitutes a linear system of algebraic equations for the determination of the nodal values (Dτu)b, leading to

$$
\Pi\_h(\mathsf{D}\_\tau u^\mathbf{h}) = N^a (\mathsf{D}\_\tau u)\_a \, . \tag{5.72}
$$

Analogous relations apply for Πh(D<sup>τ</sup> η h ).

Remark 7. The standard mid-point rule can be recovered from (5.65) by simply replacing the discrete derivatives with the mid-point evaluation of the corresponding standard derivatives. That is, in the mid-point rule, instead of (5.67) and (5.68) one has to choose

$$\mathsf{D}\_{\tau}u^{\mathrm{h}} = \partial\_{\tau}u^{\prime}(\mathbf{F}\_{\mathrm{n}+\frac{1}{2}}^{\mathrm{h},\mathrm{T}}\mathbf{F}\_{\mathrm{n}+\frac{1}{2}}^{\mathrm{h}},\tau\_{\mathrm{n}+\frac{1}{2}}^{\mathrm{h}}) \quad \text{and} \quad \mathsf{D}\_{\mathsf{C}}u^{\mathrm{h}} = \partial\_{\mathsf{C}}u^{\prime}(\mathbf{F}\_{\mathrm{n}+\frac{1}{2}}^{\mathrm{h},\mathrm{T}}\mathbf{F}\_{\mathrm{n}+\frac{1}{2}}^{\mathrm{h}},\tau\_{\mathrm{n}+\frac{1}{2}}^{\mathrm{h}}) \dots$$

## 5.3.1. Discrete balance laws

We next show that the present discretization in space and time does indeed yield an EME scheme. In particular, the discrete fulllment of the main balance laws can be shown along the lines of the semi-discrete formulation dealt with in Section 5.2.2. To this end we recast the scheme (5.65) in the form

$$\begin{split} \frac{\mathbf{q}\_{\boldsymbol{a}\_{n+1}} - \mathbf{q}\_{\boldsymbol{a}\_{n}}}{\Delta t} &= \mathbf{v}\_{\boldsymbol{a}\_{n+\frac{1}{2}}},\\ M^{ab} \frac{\mathbf{v}\_{\boldsymbol{b}\_{n+1}} - \mathbf{v}\_{\boldsymbol{b}\_{n}}}{\Delta t} &= \int\_{\mathcal{B}} N^{a} \mathbf{b} \, \mathrm{d}V - \int\_{\mathcal{B}} \mathsf{D} \mathbf{r} \, \mathrm{d}^{\mathbf{b}} \nabla N^{a} \, \mathrm{d}V + \int\_{\mathcal{B}} \Theta^{\mathrm{h}}\_{\mathrm{alg}} \mathsf{D}\_{\mathbf{F}} \eta^{\mathrm{h}} \nabla N^{a} \, \mathrm{d}V, \\\ H^{ab} \frac{\tau\_{\boldsymbol{b}\_{n+1}} - \tau\_{\boldsymbol{b}\_{n}}}{\Delta t} &= -\mathbf{v}\_{\boldsymbol{d}\_{n+\frac{1}{2}}} \cdot \int\_{\mathcal{B}} \frac{N^{a}}{\Pi\_{\mathrm{h}}(\mathsf{D}\_{\mathbf{T}} \eta^{\mathrm{h}})} \mathrm{D} \mathbf{r} \, \eta^{\mathrm{h}} \nabla N^{\mathrm{d}} \, \mathrm{d}V \\\ &\quad - \int\_{\mathcal{B}} \nabla \left( \frac{N^{a}}{\Pi\_{\mathrm{h}}(\mathsf{D}\_{\mathbf{T}} \eta^{\mathrm{h}})} \right) \cdot \mathbf{K}^{\mathrm{h}}\_{\mathrm{alg}} \, \nabla \Theta^{\mathrm{h}}\_{\mathrm{alg}} \, \mathrm{d}V - \int\_{\partial\_{\mathcal{B}} \mathsf{B}} \frac{N^{a}}{\Pi\_{\mathrm{h}}(\mathsf{D}\_{\mathbf{T}} \eta^{\mathrm{h}})} \overline{\boldsymbol{q}}\_{\mathrm{n} + \frac{1}{2}} \mathrm{d} \, \mathrm{d} \,$$

where Θ<sup>h</sup> algo has been introduced in (5.66)3. Moreover, similar to (5.43),

$$\begin{aligned} \mathbf{D}\_{\mathbf{F}} u^{\mathbf{h}} &= 2 \mathbf{F}\_{\mathbf{n} + \frac{1}{2}}^{\mathbf{h}} \mathbf{D}\_{\mathbf{C}} u^{\mathbf{h}} \,, \\ \mathbf{D}\_{\mathbf{F}} \boldsymbol{\eta}^{\mathbf{h}} &= 2 \mathbf{F}\_{\mathbf{n} + \frac{1}{2}}^{\mathbf{h}} \mathbf{D}\_{\mathbf{C}} \boldsymbol{\eta}^{\mathbf{h}} \, . \end{aligned} \tag{5.74}$$

Note that again the standard contribution of external tractions has been neglected in (5.73) for simplicity. Since the total linear momentum is given by L <sup>h</sup> = P<sup>N</sup> <sup>a</sup>=1 Mabvb, it immediately follows from (5.73)2, that

$$\mathbf{L}\_{\mathbf{n}+1}^{\mathbf{h}} - \mathbf{L}\_{\mathbf{n}}^{\mathbf{h}} = \Delta t \int\_{\mathcal{B}} \mathbf{b} \, \mathrm{d}V \, . \tag{5.75}$$

The last equation reects the consistent approximation of the balance law for linear momentum. With regard to (5.57), the total angular momentum relative to the origin of the inertial frame assumes the form J <sup>h</sup> <sup>=</sup> <sup>M</sup>abq<sup>a</sup> <sup>×</sup> <sup>v</sup>b. Then we have

$$\mathbf{J}\_{\mathbf{n}+1}^{\mathbf{h}} - \mathbf{J}\_{\mathbf{n}}^{\mathbf{h}} = M^{ab} \left( \mathbf{q}\_{a\_{\mathbf{n}+\frac{1}{2}}} \times (\mathbf{v}\_{b\_{\mathbf{n}+1}} - \mathbf{v}\_{b\_{\mathbf{n}}}) + (\mathbf{q}\_{a\_{\mathbf{n}+1}} - \mathbf{q}\_{a\_{\mathbf{n}}}) \times \mathbf{q}\_{b\_{\mathbf{n}+\frac{1}{2}}} \right) \,. \tag{5.76}$$

Substituting from (5.73)1,<sup>2</sup> into the last equation leads to

$$\begin{split} \boldsymbol{\xi} \cdot (\mathbf{J}\_{\mathrm{n}+1}^{\mathrm{h}} - \mathbf{J}\_{\mathrm{n}}^{\mathrm{h}}) &= \boldsymbol{\xi} \cdot \left( \mathbf{q}\_{a\_{\mathrm{n}+\frac{1}{2}}} \times M^{ab} (\mathbf{v}\_{b\_{\mathrm{n}+1}} - \mathbf{v}\_{b\_{\mathrm{n}}}) \right) \\ &= \boldsymbol{\xi} \cdot \Delta t \int\_{\mathcal{B}} \varphi\_{\mathrm{n}+\frac{1}{2}}^{\mathrm{h}} \times \mathbf{b} \, dV \,, \end{split} \tag{5.77}$$

for arbitrary but constant <sup>ξ</sup> <sup>∈</sup> <sup>R</sup> 3 . In the last equation it has been taken into account that

$$\begin{split} \boldsymbol{\xi} \cdot \left( \mathbf{q}\_{a\_{n+\frac{1}{2}}} \times (\mathbf{D}\_{\mathbf{F}} u^{\mathbf{h}} - \boldsymbol{\Theta}\_{\mathrm{algo}}^{\mathbf{h}} \mathbf{D}\_{\mathbf{F}} \boldsymbol{\eta}^{\mathbf{h}}) \nabla N^{a} \right) &= \widehat{\boldsymbol{\xi}} \cdot \left( (\mathbf{D}\_{\mathbf{F}} u^{\mathbf{h}} - \boldsymbol{\Theta}\_{\mathrm{algo}}^{\mathbf{h}} \mathbf{D}\_{\mathbf{F}} \boldsymbol{\eta}^{\mathbf{h}}) \mathbf{F}\_{\mathbf{n} + \frac{1}{2}}^{\mathbf{h}, \mathsf{T}} \right) \\ &= \widehat{\boldsymbol{\xi}} \cdot \left( \mathbf{F}\_{\mathrm{n} + \frac{1}{2}}^{\mathbf{h}} \mathbf{S}\_{\mathrm{algo}}^{\mathbf{h}} \mathbf{F}\_{\mathbf{n} + \frac{1}{2}}^{\mathbf{h}, \mathsf{T}} \right) \\ &= 0 \end{split} \tag{5.78}$$

This result follows from the symmetry of S h algo along with the skew-symmetry of <sup>b</sup>ξ. Note that (5.77) conrms the consistent approximation of the balance law for angular momentum.

To recover the discrete version of the balance of energy, multiply (5.73)<sup>3</sup> by (Dτu)<sup>a</sup> and take into account (5.72) together with (5.66)<sup>3</sup> to obtain

$$(\mathsf{D}\_{\tau}u)\_{a}H^{ab}(\tau\_{b\_{n+1}}-\tau\_{b\_{n}}) = -\Delta t \mathbf{v}\_{d\_{n+\frac{1}{2}}} \cdot \int\_{\mathcal{B}} \Theta\_{\mathrm{alg}}^{\mathrm{h}} \mathsf{D}\_{\mathbf{F}} \eta^{\mathrm{h}} \nabla N^{d} \, \mathrm{d}V - \Delta t \int\_{\partial\_{\mathbf{f}}\mathcal{B}} \overline{q}\_{n+\frac{1}{2}} \, \mathrm{d}A \dots$$

Employing (5.71) and (5.73)1, the last equation can be rewritten in the form

$$\int\_{\mathcal{B}} \mathsf{D}\_{\mathsf{F}} u^{\mathbf{h}} (\tau\_{\mathrm{n}+1}^{\mathbf{h}} - \tau\_{\mathrm{n}}^{\mathbf{h}}) \mathrm{d}V + \int\_{\mathcal{B}} \Theta\_{\mathrm{alg}}^{\mathbf{h}} \mathsf{D}\_{\mathsf{F}} \eta^{\mathbf{h}} : (\mathsf{F}\_{\mathrm{n}+1}^{\mathbf{h}} - \mathsf{F}\_{\mathrm{n}}^{\mathbf{h}}) \mathrm{d}V = -\Delta t \int\_{\partial\_{\mathrm{q}} \mathsf{B}} \overline{q}\_{\mathrm{n}+\frac{1}{2}} \, \mathrm{d}A \, . \tag{5.79}$$

Scalar multiplication of (5.73)<sup>2</sup> by v<sup>a</sup>n+ <sup>1</sup> yields

$$\begin{split} \mathbf{v}\_{a\_{n+\frac{1}{2}}} \cdot M^{ab} \frac{\mathbf{v}\_{b\_{n+1}} - \mathbf{v}\_{b\_{n}}}{\Delta t} &= \mathbf{v}\_{a\_{n+\frac{1}{2}}} \cdot \int\_{\mathcal{B}} N^{a} \mathbf{b} - \mathsf{D}\_{\mathbf{F}} u^{\mathbf{h}} \nabla N^{a} \, \mathrm{d}V \\ &+ \mathbf{v}\_{a\_{n+\frac{1}{2}}} \cdot \int\_{\mathcal{B}} \Theta^{\mathbf{h}}\_{\mathrm{algo}} \mathsf{D}\_{\mathbf{F}} \eta^{\mathbf{h}} \nabla N^{a} \, \mathrm{d}V \,, \end{split}$$

2

or

$$\begin{split} \frac{M^{ab}}{2} \left( \mathbf{v}\_{a\_{\mathrm{n}+1}} \cdot \mathbf{v}\_{b\_{\mathrm{n}+1}} - \mathbf{v}\_{a\_{\mathrm{n}}} \cdot \mathbf{v}\_{b\_{\mathrm{n}}} \right) &= \Delta t \int\_{\mathcal{B}} \mathbf{v}\_{\mathrm{n}+\frac{1}{2}}^{\mathrm{h}} \cdot \mathbf{b} \, \mathrm{d}V - \int\_{\mathcal{B}} \mathsf{D}\_{\mathbf{F}} u^{\mathrm{h}} : \left( \mathbf{F}\_{\mathrm{n}+1}^{\mathrm{h}} - \mathbf{F}\_{\mathrm{n}}^{\mathrm{h}} \right) \mathrm{d}V \\ &+ \int\_{\mathcal{B}} \Theta^{\mathrm{h}} \mathsf{D}\_{\mathbf{F}} \eta^{\mathrm{h}} : \left( \mathbf{F}\_{\mathrm{n}+1}^{\mathrm{h}} - \mathbf{F}\_{\mathrm{n}}^{\mathrm{h}} \right) \mathrm{d}V \, . \end{split}$$

Substituting from the last equation into (5.79) and taking into account the relationship

$$\begin{split} \mathsf{D}\_{\mathbf{F}} u^{\mathbf{h}} : (\mathbf{F}\_{\mathbf{n}+1}^{\mathbf{h}} - \mathbf{F}\_{\mathbf{n}}^{\mathbf{h}}) &= 2 \mathbf{F}\_{\mathbf{n}+\frac{1}{2}}^{\mathbf{h}} \mathsf{D}\_{\mathbf{C}} u^{\mathbf{h}} : (\mathbf{F}\_{\mathbf{n}+1}^{\mathbf{h}} - \mathbf{F}\_{\mathbf{n}}^{\mathbf{h}}), \\ &= \mathsf{D}\_{\mathbf{C}} u^{\mathbf{h}} : (\mathbf{C}\_{\mathbf{n}+1}^{\mathbf{h}} - \mathbf{C}\_{\mathbf{n}}^{\mathbf{h}}) \end{split}$$

yields

$$\begin{split} &\int\_{\mathcal{B}} \left( \mathsf{D}\_{\tau} u^{\mathbf{h}} (\tau\_{\mathbf{n}+1}^{\mathbf{h}} - \tau\_{\mathbf{n}}^{\mathbf{h}}) + \mathsf{D}\_{\mathsf{C}} u^{\mathbf{h}} : (\mathsf{C}\_{\mathsf{n}+1}^{\mathbf{h}} - \mathsf{C}\_{\mathsf{n}}^{\mathbf{h}}) \right) \mathrm{d}V + \frac{M^{ab}}{2} \left( \mathbf{v}\_{a\_{\mathsf{n}+1}} \cdot \mathbf{v}\_{b\_{\mathsf{n}+1}} - \mathbf{v}\_{a\_{\mathsf{n}}} \cdot \mathbf{v}\_{b\_{\mathsf{n}}} \right) \\ &= \int\_{\mathcal{B}} \left( \varphi\_{\mathsf{n}+1}^{\mathbf{h}} - \varphi\_{\mathsf{n}+1}^{\mathbf{h}} \right) \cdot \mathbf{b} \, \mathrm{d}V - \Delta t \int\_{\partial\_{\mathsf{t}} \mathfrak{B}} \overline{\eta}\_{\mathsf{n}+\frac{1}{2}} \, \mathrm{d}A \, . \end{split}$$

With regard to the total energy E<sup>h</sup> introduced in (5.62), the last equation can be recast in the form

$$
\mathcal{E}\_{\mathbf{n}+1}^{\mathbf{h}} - \mathcal{E}\_{\mathbf{n}}^{\mathbf{h}} = -\Delta t \int\_{\partial\_{\mathbf{q}}\mathcal{B}} \overline{q}\_{\mathbf{n}+\frac{1}{2}} \,\mathrm{d}A \,\mathrm{d}.\tag{5.80}
$$

Comparison with (5.61) conrms that the present scheme correctly reproduces the balance of energy. Note that the fulllment of the directionality condition (5.69) is of paramount importance for obtaining an energy consistent scheme.

Eventually, we aim at the discrete balance of entropy. To this end, multiply (5.73)<sup>3</sup> by (D<sup>τ</sup> η)<sup>a</sup> and take into account (5.66)3, to obtain

$$\begin{split} (\mathsf{D}\_{\mathsf{r}}\eta)\_{a}H^{ab}\frac{\tau\_{b\_{n+1}}-\tau\_{b\_{n}}}{\Delta t} &= -\mathsf{v}\_{d\_{n+\frac{1}{2}}} \cdot \int\_{\mathcal{B}} \mathsf{D}\_{\mathsf{F}}\eta^{\mathsf{h}} \nabla N^{d} \, \mathrm{d}V \\ &- \int\_{\mathcal{B}} \nabla \left(\frac{1}{\Theta\_{\mathrm{alg}}^{\mathrm{h}}}\right) \cdot \mathbf{K}\_{\mathrm{alg}}^{\mathrm{h}} \nabla \Theta\_{\mathrm{alg}}^{\mathrm{h}} \, \mathrm{d}V - \int\_{\partial\_{\mathsf{q}}\mathcal{B}} \frac{\overline{q}\_{n+\frac{1}{2}}}{\Theta\_{\mathrm{alg}}^{\mathrm{h}}} \, \mathrm{d}A \, \mathrm{d}V \end{split}$$

The last equation can be rewritten as

$$\begin{split} &\int\_{\mathcal{B}} \left( \mathsf{D}\_{\tau} \eta^{\mathrm{h}} (\tau\_{\mathrm{n}+1}^{\mathrm{h}} - \tau\_{\mathrm{n}}^{\mathrm{h}}) + \mathsf{D}\_{\mathsf{C}} \eta^{\mathrm{h}} : (\mathsf{C}\_{\mathrm{n}+1}^{\mathrm{h}} - \mathsf{C}\_{\mathrm{n}}^{\mathrm{h}}) \right) \mathrm{d}V \\ &= \underbrace{\Delta t \int\_{\mathcal{B}} \nabla \left( \frac{1}{\Theta\_{\mathrm{algo}}^{\mathrm{h}}} \right) \cdot (\Theta\_{\mathrm{algo}}^{\mathrm{h}})^{2} \mathbf{K}\_{\mathrm{algo}}^{\mathrm{h}} \nabla \left( \frac{1}{\Theta\_{\mathrm{algo}}^{\mathrm{h}}} \right) \mathrm{d}V}\_{\geq 0} - \Delta t \int\_{\partial\_{\mathrm{q}} \mathcal{B}} \frac{\overline{q}\_{\mathrm{n}+\frac{1}{2}}}{\Theta\_{\mathrm{algo}}^{\mathrm{h}}} \mathrm{d}A \ldots \right. \end{split}$$

The discrete derivatives of the entropy density, namely D<sup>τ</sup> η h and DCη h , respectively, satisfy an associated directionality condition in analogy to (5.69). Accordingly, the lefthand side of the last equation equals S h n+1 − S<sup>h</sup> n , where S h is the total entropy of the discrete system introduced in (5.64). Comparison with (5.63) conrms that the present scheme correctly reproduces the balance of entropy.

## 5.4. Numerical investigations

As has been shown in the last section the newly developed schemes are thermodynamically consistent and comply with the balance laws for linear and angular momentum, respectively. These structure-preserving properties hold for arbitrarily large time-steps. Depending on the specic choice of the thermodynamic state variable τ ∈ {θ, η, u}, we obtain three alternative EME schemes. The respective scheme is denoted as (EME)<sup>τ</sup> . For example, choosing the temperature as state variable leads to the temperature-based (EME)<sup>θ</sup> scheme.

We apply the newly developed (EME)<sup>τ</sup> schemes to representative numerical examples dealing with nite-strain thermo-elastodynamics. In particular, we treat the three examples which have been previously investigated in Section 4.5. In Chapter 4 the time discretization was conned to the mid-point rule which led to a genuine lack of numerical stability. In particular the lack of stability of the mid-point rule is especially pronounced in the case of large time steps. Therefore, in the numerical examples presented below we focus on large time steps to show that the new (EME)<sup>τ</sup> schemes remain stable.

## 5.4.1. Flying L-shaped block

First we consider the example given in Section 4.5.1 and choose the largest time-step size for all three (EME)<sup>τ</sup> schemes. The total linear momentum of the block is a conserved quantity since in the initial loading phase the external forces are equilibrated. As after the loading phase (t > 5s) no external torque is acting on the block the total angular momentum is a conserved quantity as well. All (EME)<sup>τ</sup> integrators under consideration are capable to conserve both momentum maps (up to numerical round-o), independent of the time step size which can be observed from Fig. 5.4, where representative numerical results of the (EME)<sup>u</sup> integrator are shown.

Figure 5.4.: L-shaped block: Algorithmic conservation of linear momentum (EME)<sup>u</sup> scheme (left), Total discrete angular momentum (EME)<sup>u</sup> scheme (right)

During the loading phase (0 ≤ t ≤ 5) the total energy of the block increases, whereas after the loading phase the total energy has to be a conserved quantity. All (EME)<sup>τ</sup> integrators exactly reproduce this conservation law (up to numerical round-o), see Fig. 5.5. This is in sharp contrast to the mid-point based schemes. In particular, as has been shown in Section 4.5.1, all mid-point based schemes exhibit numerical instability even though the formulation in terms of the internal energy density is capable to conserve the energy.

Figure 5.5.: L-shaped block: Total energy (EME)<sup>τ</sup> schemes (left), Incremental change of total energy (EME)<sup>τ</sup> schemes (right)

Figure 5.6.: L-shaped block: total entropy (EME)<sup>τ</sup> schemes (left), Incremental change of total entropy (EME)<sup>τ</sup> schemes (right)

Since the (EME)<sup>τ</sup> integrators are consistent with the second law of thermodynamics (independent of the time-step size), the total entropy is a non-decreasing function over time, as can be observed from Fig. 5.6. The above investigations indicate that, in contrast to the mid-point based schemes (see Section 4.5.1) the (EME)<sup>τ</sup> integrators are numerically stabile even for large time steps. After the loading phase, the environment of the present

example can be classied as thermally perfect in the sense of [146]. Therefore, L introduced in (4.92) plays the role of a Lyapunov function. All of the (EME)<sup>τ</sup> integrators correctly deliver a decreasing value of L over time (Fig. 5.7) thus conrming numerical stability. This is in sharp contrast to the mid-point based schemes dealt with in Section 4.5.1.

Figure 5.7.: L-shaped block: Lyapunov function computed with the (EME)<sup>τ</sup> schemes

Eventually, the motion of the L-shaped block is illustrated in Fig. 5.8 with snapshots at successive points in time. In addition to that, the distribution of the temperature over the block is shown.

Figure 5.8.: L-shaped block: Snapshots of the motion along with the temperature distribution over the block at t ∈ {0, 32, 64, 96, 128, 160, 192, 224}, obtained with the (EME)<sup>θ</sup> scheme and time step ∆t = 0.4s

## 5.4.2. Rotating disc

Next we consider the example given in Section 4.5.2 and choose the largest time-step size for all three (EME)<sup>τ</sup> schemes. As neither external loads (∂σB = ∅) nor displacement boundary conditions (∂ϕB = ∅) are acting on the disc, the thermomechanical system at hand has translational and rotational symmetry. Therefore the corresponding momentum maps are rst integrals of the motion. All (EME)<sup>τ</sup> integrators at hand are capable to conserve the respective momentum map. Representative numerical results of the (EME)<sup>η</sup> integrator are shown in Fig. 5.9. During initial period t ∈ [0, 4]s the total energy of the system is expected to increase due to the prescribed heat ow into the system. After initial period the system is classied as closed and therefore the total energy of the system should stay constant. All (EME)<sup>τ</sup> integrators are capable to correctly reproduce the rst law of thermodynamics (see Fig. 5.10) in contrast to the mid-point based schemes, where only the formulation in the internal energy density was in accordance with the rst law of thermodynamics, see Section 4.5.2. The total entropy of the system is expected to increase due to the prescribed heat ow into the system during the initial period t ∈ [0, 4]s. As the system is closed after the initial period, the total entropy of the system should be a non-decreasing function, whereby the irreversibility is caused by heat conduction.

Figure 5.9.: Rotating disc: Total linear momentum (left) and total angular momentum (right)

Figure 5.10.: Rotating disc: Total energy (left) and incremental change of total energy (right)

All (EME)<sup>τ</sup> integrators correctly reproduce the second law of thermodynamics as can be observed from Fig. 5.11. Again this is in contrast to the mid-point based schemes investigated in Section 4.5.2, where only the formulation in terms of the entropy density was shown to be consistent with the second law.

Figure 5.11.: Rotating disc: Total entropy (left) and incremental change of total entropy (right)

To investigate the numerical stability of the (EME)<sup>τ</sup> schemes further, we consider again the Lyapunov function L introduced in (4.92). After the initial period the system is closed and therefore L has to decrease with time.

Figure 5.12.: Rotating disc: Lyapunov function (EME)<sup>τ</sup> schemes

All (EME)<sup>τ</sup> integrators correctly reproduce this behaviour as can be observed from Fig. 5.12. This again conrms numerical stability of the EME schemes at hand and is in contrast to the mid-point based schemes investigated in Section 4.5.2.

Figure 5.13.: Rotating disc: Error in the position (left) and error in the velocity (right)

Figure 5.14.: Rotating disc: Error in the absolute temperature

Further we investigate the order of accuracy of the present structure-preserving methods. Therefore we introduce the relative L<sup>2</sup> error norm in the position, the velocity and the absolute temperature, respectively

$$\mathbf{e}\_{\varphi} = \frac{||\varphi - \varphi\_r||\_{L\_2}}{||\varphi\_r||\_{L\_2}}, \quad \mathbf{e}\_{\mathbf{p}} = \frac{||\mathbf{v} - \mathbf{v}\_r||\_{L\_2}}{||\mathbf{v}\_r||\_{L\_2}}, \quad \mathbf{e}\_{\Theta} = \frac{||\Theta - \Theta\_r||\_{L\_2}}{||\Theta\_r||\_{L\_2}},$$

$$||\bullet||\_{L\_2} = \left[\int\_{\mathcal{B}\_r} \langle \bullet, \bullet \rangle \, \mathrm{d}V\right]^{\frac{1}{2}}.$$

In the above equations •<sup>r</sup> with • ∈ {ϕ, v, Θ} is the reference solution at time t = 4.02s calculated with the smallest time step size. We consider the motion of the rotating disk in the time interval [4, 4.02]s. To this end, the simulation is started at time t= 4s by using the data obtained with the (EME)<sup>θ</sup> scheme and ∆t= 0.04s. As can be observed from Figs. 5.13 and 5.14 all schemes exhibit second order accuracy in ϕ, v and Θ. Eventually, the motion of the disc is illustrated in Fig. 5.15 with snapshots at successive points in time. In addition to that, the distribution of the temperature over the disc is shown.

Figure 5.15.: Rotating disc: Snapshots of the motion at successive points in time t ∈ {0, 4, 8, 12, 16, 18, 24, 28}s, and corresponding temperature distribution, calculated with the (EME)<sup>θ</sup> scheme and ∆t = 0.1s

## 5.4.3. Rotating disc in a thermally perfect environment

Finally we examine the example given in Section 4.5.3 and consider only the largest time-step size. Again we focus on the temperature-based (EME)<sup>θ</sup> integrator, which makes possible to impose temperature Dirichlet boundary conditions in a standard manner. As in the previous example the system at hand has both translational and rotational symmetry. Accordingly, the corresponding momentum maps are exactly conserved by the (EME)<sup>θ</sup> integrator, see Fig. 5.16.

Figure 5.16.: Rotating disc in a thermally perfect environment: Total linear momentum (left), total angular momentum (right)

According to the temperature boundary conditions heat is owing out of the disc. The corresponding time-evolution of the total energy is depicted in Fig. 5.17 (left). In addition to that, the total entropy is as well decreasing over time, see Fig. 5.17 (right).

Figure 5.17.: Rotating disc in a thermally perfect environment: Total energy (EME)<sup>θ</sup> scheme (left), and total entropy (right)

By design of the thermally perfect environment, functional L introduced in (4.92) is a Lyapunov function throughout the whole simulation time. Correspondingly, the (EME)<sup>θ</sup> integrator yields a steadily decreasing value of L even for very large time steps (see Fig. 5.18). In particular, in contrast to the mid-point rule (cf. Section 4.5.3), the present scheme does stay numerically stable.

Figure 5.18.: Rotating disc in a thermally perfect environment: Lyapunov function

Eventually, both the motion and the evolution of the temperature distribution are shown in Fig. 5.19 with snapshots at successive points in time.

Figure 5.19.: Rotating disc in a thermally perfect environment: Snapshots of the motion at t ∈ {0, 10, 20, 30, 40, 50, 60, 70}s and corresponding temperature distribution, calculated with the (EME)<sup>θ</sup> scheme and ∆t = 0.0875s

# 6. GENERIC-based numerical methods for large-strain thermo-viscoelasticity<sup>1</sup>

Large-strain thermo-viscoelasticity is described in the framework of GENERIC. To this end, a new material representation of the inelastic part of the dissipative bracket is proposed. The bracket form of GENERIC generates the governing equations for large-strain thermoviscoelasticity including the nonlinear evolution law for the internal variables associated with inelastic deformations. The GENERIC formalism facilitates the free choice of the thermodynamic variable. In particular, one may choose (i) the internal energy density, (ii) the entropy density, or (iii) the absolute temperature as the thermodynamic state variable. A mixed nite element method is proposed for the discretization in space which preserves the GENERIC form of the resulting semi-discrete evolution equations. The GENERICconsistent space discretization makes the design of structure-preserving time-stepping schemes possible. The mid-point type discretization in time yields three alternative schemes. Depending on the specic choice of the thermodynamic variable, these schemes are shown to be partially structure-preserving. Numerical investigations are presented which conrm the theoretical ndings and shed light on the numerical stability of the newly developed schemes.

## 6.1. GENERIC-based formulation of large strain thermo-viscoelasticity

The present Chapter relies on a material description of large-strain thermo-viscoelasticity within the framework of GENERIC. The developments presented herein extend the work in the previous Chapters 4 and 5 on thermo-elasticity to the realm of inelastic deformations. In the GENERIC framework the time-evolution of any functional A is governed by

$$\frac{\mathrm{d}\mathcal{A}}{\mathrm{d}t} = \left\{ \mathcal{A}, \mathcal{E} \right\} + \left[ \mathcal{A}, \mathcal{S} \right] \,. \tag{6.1}$$

<sup>1</sup> This Chapter is based on [4]

The evolution equation (6.1) is valid for closed systems in which the boundaries are disregarded. We consider a continuum body with material points X = XAe<sup>A</sup> in the reference conguration B ⊂ <sup>R</sup> 3 (Fig. 6.1). The material coordinates X<sup>A</sup> refer to canonical base vectors <sup>e</sup><sup>A</sup> <sup>∈</sup> <sup>R</sup> 3 . Here and in the sequel, the summation convention applies to repeated indices.

Figure 6.1.: Reference conguration B and deformed conguration ϕ(B, t) at time t. For now the focus is on the motion of isolated thermomechanically coupled solids. That is, external tractions acting on the boundary as well heat uxes across the boundary are disregarded.

Within the Lagrangian description of continuum mechanics the deformed conguration of the body at time <sup>t</sup> is characterized by the deformation map <sup>ϕ</sup> : B × I 7→ <sup>R</sup> 3 , where I = [0, T] is the time interval of interest. Accordingly, the placement at time t of the material point X ∈ B is given by x = ϕ(X, t). The corresponding velocity eld <sup>v</sup> : B × I 7→ <sup>R</sup> 3 is dened by v = ϕ˙ , where a superposed dot denotes the material time derivative. The deformation gradient is given by

$$\mathbf{F} = \partial\_{\mathbf{X}} \varphi \; , \tag{6.2}$$

or, in components, (F) i · ·<sup>A</sup> <sup>=</sup> ∂x<sup>i</sup>/∂X<sup>A</sup>. Furthermore, the right Cauchy-Green tensor reads

$$\mathbf{C} = \mathbf{F}^{\mathrm{T}} \mathbf{F} \,. \tag{6.3}$$

In the GENERIC (6.1), we consider functionals of the form

$$\mathcal{A} = \mathcal{A}(\varphi, \mathbf{p}, \tau, \mathbf{C}\_p^{-1}) = \int\_{\mathcal{B}} a(\varphi, \mathbf{F}, \mathbf{p}, \tau, \mathbf{C}\_p^{-1}) \, \mathrm{d}V \,, \tag{6.4}$$

where <sup>p</sup> <sup>=</sup> <sup>ρ</sup><sup>v</sup> is the linear momentum density and <sup>ρ</sup> : B 7→ <sup>R</sup><sup>+</sup> is the mass density in the reference conguration. Moreover, <sup>τ</sup> : B × I 7→ <sup>R</sup> is a generalized thermodynamic eld which may be chosen from among three alternatives, τ ∈ {θ, η, u}. The three alternative elds are the absolute temperature θ, the entropy density η, and the internal energy density u. To describe inelastic deformations, we make use of the internal variable C −1 p : B ×I 7→ <sup>R</sup> 3×3 . Since we focus on isotropic inelastic deformations, C −1 p is assumed to be symmetric, i.e. C −T <sup>p</sup> = C −1 p . Therefore the density function a in (6.4) is subject to the same isotropic restrictions as the Helmholtz free energy function, to be introduced in Section 6.2.

GENERIC (6.1) decouples reversible and irreversible processes. In particular, the Poisson bracket ·, · is responsible for reversible processes, while the dissipative (or friction) bracket - ·, · embodies irreversible processes. Accordingly, these two brackets constitute fundamental entities of the GENERIC framework and shall be specied next. Since inelastic deformations are purely irreversible, the Poisson bracket retains its thermo-elastic form (see eq. (4.34))

$$\{\mathcal{A}, \mathcal{B}\} = \int\_{\mathcal{B}} \left(\delta\_{\varphi} a + \text{Div} \left(\frac{\delta\_{\tau} a}{\partial\_{\tau} \eta} \partial\_{\mathsf{F}} \eta \right) \right) \cdot \delta\_{\mathsf{p}} b - \left(\delta\_{\varphi} b + \text{Div} \left(\frac{\delta\_{\tau} b}{\partial\_{\tau} \eta} \partial\_{\mathsf{F}} \eta \right) \right) \cdot \delta\_{\mathsf{p}} a \,\mathrm{d}V \,, \tag{6.5}$$

where A and B are arbitrary functionals of the form (6.4). Furthermore, the functional derivatives in (6.5) are given by

$$\begin{aligned} \delta\_{\mathbf{p}}a &= \partial\_{\mathbf{p}}a - \text{Div}\partial\_{\mathbf{F}}a \ , \\ \delta\_{\mathbf{p}}a &= \partial\_{\mathbf{p}}a \ , \\ \delta\_{\tau}a &= \partial\_{\tau}a \ , \end{aligned} \tag{6.6}$$
 
$$\begin{aligned} \delta\_{\mathbf{C}\_{p}^{-1}}a &= \partial\_{\mathbf{C}\_{p}^{-1}}a \ . \end{aligned} \tag{6.6}$$

In (6.5), η denotes the entropy density giving rise to the total entropy of the system

$$\mathcal{S}(\varphi,\tau,\mathbf{C}\_p^{-1}) = \int\_{\mathcal{B}} \eta(\mathbf{F},\tau,\mathbf{C}\_p^{-1}) \,\mathrm{d}V\,\,. \tag{6.7}$$

The dissipative bracket for the thermo-viscoelastic problem at hand can be additively decomposed into a part due to heat conduction and a part taking into account inelastic deformations:

$$\left[\mathcal{A},\mathcal{B}\right] = \left[\mathcal{A},\mathcal{B}\right]\_{\text{cond}} + \left[\mathcal{A},\mathcal{B}\right]\_{\text{inel}}.\tag{6.8}$$

In analogy to thermo-elasticity with heat conduction, the dissipative bracket accounting for heat conduction is given by (see eq. (4.35))

$$\left[\mathcal{A}, \mathcal{B}\right]\_{\rm cond} = \int\_{\mathcal{B}} \nabla \left(\frac{\delta\_{\tau} a}{\partial\_{\tau} u}\right) \cdot \Theta^{2} \mathbf{K} \nabla \left(\frac{\delta\_{\tau} b}{\partial\_{\tau} u}\right) \mathrm{d}V \,. \tag{6.9}$$

Here, K = K(C, τ ) is the material conductivity tensor, for which we assume that <sup>K</sup><sup>T</sup> <sup>=</sup> <sup>K</sup> and <sup>a</sup> : <sup>K</sup> : <sup>a</sup> <sup>≥</sup> <sup>0</sup> holds for all <sup>a</sup> <sup>∈</sup> <sup>R</sup> 3 . An important element of the GENERIC framework is that the absolute temperature takes the form

$$\Theta(\mathbf{F}, \tau, \mathbf{C}\_p^{-1}) = \frac{\partial\_{\tau} u(\mathbf{F}, \tau, \mathbf{C}\_p^{-1})}{\partial\_{\tau} \eta(\mathbf{F}, \tau, \mathbf{C}\_p^{-1})} \,. \tag{6.10}$$

In (6.9) and (6.10), u denotes the internal energy density which, together with the kinetic energy and the potential of dead loads, constitutes the total energy of the system

$$\mathcal{E}(\varphi, \mathbf{p}, \tau, \mathbf{C}\_p^{-1}) = \int\_{\mathcal{B}} \left( \frac{1}{2} \rho^{-1} \mathbf{p} \cdot \mathbf{p} + u(\mathbf{F}, \tau, \mathbf{C}\_p^{-1}) - \mathbf{b} \cdot \varphi \right) \, \mathrm{d}V \, . \tag{6.11}$$

Here, <sup>b</sup> : B 7→ <sup>R</sup> 3 represent prescribed body forces which, for simplicity, are assumed to be dead loads.

Concerning the contribution of inelastic deformations, we introduce the following form of the dissipative bracket (see Section 6.2 for further details):

$$\begin{split} & \left[ \mathcal{A}, \mathcal{B} \right]\_{\text{inel}} \\ &= \int\_{\mathcal{B}} 2 \left( \frac{\delta\_{\tau} a}{\partial\_{\tau} u} \partial\_{\mathbf{C}\_{p}^{-1} u} \mathbf{C}\_{p}^{-1} - \delta\_{\mathbf{C}\_{p}^{-1}} a \mathbf{C}\_{p}^{-1} \right) : \boldsymbol{\Theta} \mathcal{N} : 2 \left( \frac{\delta\_{\tau} b}{\partial\_{\tau} u} \partial\_{\mathbf{C}\_{p}^{-1} u} \mathbf{C}\_{p}^{-1} - \delta\_{\mathbf{C}\_{p}^{-1}} b \mathbf{C}\_{p}^{-1} \right) \mathrm{d}V \,. \end{split} \tag{6.12}$$

The fourth-order inelastic material ow tensor N has the properties N<sup>T</sup> = N (major symmetry) and M : N : M ≥ 0 (positive semi-deniteness) for all M given in (6.17).

It can be easily veried that the Poisson bracket (6.5) is skew-symmetric ({A, B} = −{B, A}), the dissipative bracket (6.12) is symmetric ([A, B] = [B, A]) and further satises the non-negativity condition [A, A] ≥ 0. Moreover, the two brackets satisfy the fundamental degeneracy (or non-interaction) conditions {A, S} = 0 and [A, E] = 0. In conjunction with these properties of the two brackets, GENERIC (6.1) ensures for closed systems (i) the conservation of total energy (dE/dt = 0), and (ii) a non-decreasing total entropy (dS/dt ≥ 0). Due to these structural properties, the GENERIC-based formulation oers an ideal starting point for the design of structure-preserving numerical methods.

## 6.1.1. Local evolution equations

We provide a short outline of the local evolution equations emanating from GENERIC (6.1). With regard to (6.1) the total energy E generates the reversible contribution through the Poisson bracket {A, E}. Using expression (6.11) for the total energy along with formulas (6.6) for the variational derivatives, Poisson bracket (6.5) yields

$$\left\{\mathcal{A},\mathcal{E}\right\} = \int\_{\mathcal{B}} \left(\delta\_{\varphi}a + \text{Div}\left(\frac{\delta\_{\tau}a}{\partial\_{\tau}\eta}\partial\_{\mathbf{F}}\eta\right)\right) \cdot \rho^{-1}\mathbf{p} + (\mathbf{b} + \text{Div}\mathbf{P}) \cdot \delta\_{\mathbf{p}}a \,\mathrm{d}V\,,,\tag{6.13}$$

where the rst Piola-Kirchho stress tensor P = P(F, τ, C −1 p ) takes the form

$$\mathbf{P} = \partial\_{\mathbf{F}} u - \Theta \partial\_{\mathbf{F}} \eta \; . \tag{6.14}$$

The irreversible contribution to GENERIC (6.1) is generated by the total entropy S through the dissipative bracket [A, S]. Inserting expression (6.7) for the total entropy into the dissipative bracket (6.8) leads to

$$\left[\mathcal{A}, \mathcal{S}\right] = \int\_{\mathcal{B}} \nabla \left(\frac{\delta\_{\tau} a}{\partial\_{\tau} u}\right) \cdot \mathbf{Q} \, dV + \int\_{\mathcal{B}} 2 \left(\frac{\delta\_{\tau} a}{\partial\_{\tau} u} \partial\_{\mathbf{C}\_{p}^{-1}} u \mathbf{C}\_{p}^{-1} - \delta\_{\mathbf{C}\_{p}^{-1}} a \mathbf{C}\_{p}^{-1}\right) : \mathcal{M} : \mathbf{M} \, dV \,, \tag{6.15}$$

where the material heat ux vector Q = Q(F, τ, C −1 p ) is given by

$$\mathbf{Q} = \Theta^2 \mathbf{K} \nabla \left(\frac{1}{\Theta}\right) = -\mathbf{K} \nabla \Theta \ , \tag{6.16}$$

and M = M(F, τ, C −1 p ) denotes the material representation of the Mandel stress tensor, which takes the form

$$\mathbf{M} = 2\left(\partial\_{\mathbf{C}\_p^{-1}} u - \Theta \partial\_{\mathbf{C}\_p^{-1}} \eta \right) \mathbf{C}\_p^{-1} \,. \tag{6.17}$$

It is important to realize that in (6.14), (6.16) and (6.17), Θ is the temperature eld which has been introduced in (6.10). These relationships are representative for the GENERIC-based formulation.

With regard to the left-hand side of GENERIC (6.1) and expression (6.4) for functional A, the time derivative of A can be written as

$$\frac{d}{dt}\mathcal{A} = \int\_{\mathcal{B}} \left( \delta\_{\varphi} a \cdot \dot{\varphi} + \delta\_{\mathbf{p}} a \cdot \dot{\mathbf{p}} + \delta\_{\tau} a \dot{\tau} + \delta\_{\mathbf{C}\_p^{-1}} a : \overline{\mathbf{C}\_p^{-1}} \right) \, \mathrm{d}V \, . \tag{6.18}$$

Substituting (6.18), (6.13) and (6.15) into GENERIC (6.1), we arrive at

$$\begin{split} 0 &= \int\_{\mathcal{B}} \delta\_{\mathcal{B}} a \cdot \left( \dot{\varphi} - \rho^{-1} \mathbf{p} \right) \, \mathrm{d}V \\ &+ \int\_{\mathcal{B}} \delta\_{\mathbf{p}} a \cdot \left( \dot{\mathbf{p}} - \left( \mathbf{b} + \mathrm{Div} \mathbf{P} \right) \right) \, \mathrm{d}V \\ &+ \int\_{\mathcal{B}} \delta\_{\tau} a \left( \dot{\tau} - \frac{2}{\partial\_{\tau} u} \left( \partial\_{\mathbf{C}\_{p}^{-1}} u \mathbf{C}\_{p}^{-1} \right) : \mathcal{M} : \mathbf{M} \right) - \mathrm{Div} \left( \frac{\delta\_{\tau} a}{\partial\_{\tau} \eta} \partial\_{\mathbf{F}} \eta \right) \cdot \rho^{-1} \mathbf{p} \, \mathrm{d}V \\ &- \int\_{\mathcal{B}} \nabla \left( \frac{\delta\_{\tau} a}{\partial\_{\tau} u} \right) \cdot \mathbf{Q} + \delta\_{\mathbf{C}\_{p}^{-1}} a : \left( \overset{\cdot}{\mathbf{C}\_{p}^{-1}} + 2 \left[ \mathcal{M} : \mathbf{M} \right] \mathbf{C}\_{p}^{-1} \right) \, \mathrm{d}V \, . \end{split} \tag{6.19}$$

This equation has to hold for arbitrary functionals A of the form (6.4). Standard arguments now lead to the local evolution equations emanating from GENERIC (6.1):

$$\begin{split} \dot{\boldsymbol{\varphi}} &= \boldsymbol{\rho}^{-1} \mathbf{p} \,, \\ \dot{\mathbf{p}} &= \mathbf{b} + \text{Div} \mathbf{P} \,, \\ \dot{\boldsymbol{\tau}} &= \frac{2}{\partial\_{\tau} u} \left( \partial\_{\mathbf{C}\_{p}^{-1}} u \mathbf{C}\_{p}^{-1} \right) : \boldsymbol{\mathcal{M}} : \mathbf{M} - \frac{1}{\partial\_{\tau} \eta} \partial\_{\mathbf{F}} \boldsymbol{\eta} : \nabla (\boldsymbol{\rho}^{-1} \mathbf{p}) - \frac{1}{\partial\_{\tau} u} \text{Div} \mathbf{Q} \,, \\ \dot{\mathbf{C}}\_{p}^{-1} &= -2 \left( \boldsymbol{\mathcal{N}} : \mathbf{M} \right) \mathbf{C}\_{p}^{-1} \, . \end{split} \tag{6.20}$$

We stress again that in the above formulas the rst Piola-Kirchho stress tensor P, the Mandel stress tensor M and the heat ux vector Q are given by formulas (6.14), (6.17) and (6.16), respectively.

## 6.1.2. GENERIC-based weak form of the IBVP

Starting from the GENERIC-based local evolution equations (6.20), we deduce the weak form of the initial boundary value problem (IBVP) pertaining to nite-strain thermoviscoelasticity. To this end, we decompose the boundary ∂B of the continuum body into a displacement boundary ∂ϕB, on which ϕ = ϕ, and a traction boundary ∂σB, on which PN = t, where ϕ and t are prescribed functions for t ≥ 0 (Fig. 6.2). Moreover, ∂ϕB ∪ ∂σB = ∂B and ∂ϕB ∩ ∂σB = ∅.

Figure 6.2.: Mechanical part of the IBVP. Note that t = PN denotes prescribed external Piola tractions acting on the current boundary expressed per unit area of the reference boundary ∂σB.

Similarly, for the thermal part we consider the subsets of the boundary ∂τB and ∂qB, with the properties ∂τB ∪ ∂qB = ∂B and ∂τB ∩ ∂qB = ∅ (Fig. 6.3). The thermodynamic state variable is prescribed on ∂τB, i.e. τ = τ , whereas the heat ux is prescribed on ∂qB, i.e. Q · N = q. Both τ and q are assumed to be given for t ≥ 0.

Figure 6.3.: Thermal part of the IBVP. Note that q = Q · N is the prescribed rate of heat transfer across the current boundary expressed per unit area of the reference boundary ∂qB.

We aim at the determination of the motion <sup>ϕ</sup> : B × I 7→ <sup>R</sup> 3 , the linear momentum <sup>p</sup> : B × I 7→ <sup>R</sup> 3 , the thermodynamic state variable <sup>τ</sup> : B × I 7→ <sup>R</sup>, and the inelastic deformation C −1 p : B × I 7→ <sup>R</sup> 3×3 . The unknown elds are subject to initial conditions of the form ϕ(·, 0) = X, p(·, 0) = ρV0, τ (·, 0) = τ ini and C −1 p (·, 0) = I in B. Here, V<sup>0</sup> is a prescribed material velocity eld, and τ ini is a prescribed eld of the thermodynamic state variable τ ∈ {θ, η, u}. The unknown elds are determined by applying a space-time discretization to the weak form of the IBVP at hand.

The weak form of the IBVP can be obtained by scalar multiplying the local evolution equations (6.20) by suitable test functions and subsequently integrating over domain B. The standard procedure yields

$$\begin{split} 0 &= \int\_{\mathcal{B}} \mathbf{w}\_{\boldsymbol{\Phi}} \cdot (\boldsymbol{\dot{\varphi}} - \mathbf{v}) \, \mathrm{d}V \\ &+ \int\_{\mathcal{B}} \left( \mathbf{w}\_{\mathbf{p}} \cdot (\rho \mathbf{\dot{v}} - \mathbf{b}) + \mathbf{P} : \nabla \mathbf{w}\_{\mathbf{p}} \right) \, \mathrm{d}V - \int\_{\partial\_{\mathcal{B}} \mathcal{B}} \mathbf{w}\_{\mathbf{p}} \cdot \overline{\mathbf{f}} \, \mathrm{d}A \\ &+ \int\_{\mathcal{B}} \mathbf{w}\_{\tau} \left( \dot{\tau} - \frac{2}{\partial\_{\tau} u} \left( \partial\_{\mathbf{C}\_{p}^{-1}} u \mathbf{C}\_{p}^{-1} \right) : \mathcal{M} : \mathbf{M} + \frac{1}{\partial\_{\tau} \eta} \partial\_{\mathbf{F}} \eta : \nabla \mathbf{v} \right) \, \mathrm{d}V \\ &- \int\_{\mathcal{B}} \nabla \left( \frac{\mathbf{w}\_{\tau}}{\partial\_{\tau} u} \right) \cdot \mathbf{Q} \, \mathrm{d}V + \int\_{\partial\_{\mathcal{B}} \mathcal{B}} \frac{\mathbf{w}\_{\tau}}{\partial\_{\tau} u} \overline{q} \, \mathrm{d}A \\ &+ \int\_{\mathcal{B}} \mathbf{w}\_{\mathbf{C}\_{p}^{-1}} : \left( \overline{\mathbf{C}\_{p}^{-1}} + 2 \left( \mathcal{M} : \mathbf{M} \right) \mathbf{C}\_{p}^{-1} \right) \, \mathrm{d}V \,, \end{split} \tag{6.21}$$

where the velocity eld v = ρ <sup>−</sup>1p has been introduced. In weak form (6.21), wϕ, w<sup>p</sup> : B 7→ <sup>R</sup> 3 , <sup>w</sup><sup>τ</sup> : B 7→ <sup>R</sup> and <sup>w</sup>C−<sup>1</sup> p : B 7→ <sup>R</sup> 3×3 are test functions that have to satisfy the boundary conditions w<sup>ϕ</sup> = 0 and w<sup>p</sup> = 0 on ∂ϕB, and w<sup>τ</sup> = 0 on ∂τB.

## 6.1.3. Balance laws

We verify the pertinent balance laws of the IBVP at hand. For that purpose it suces to consider the pure Neumann problem (i.e. ∂σB = ∂qB = ∂B). We start with the total linear momentum of the continuum body dened by L = R B p dV . Choosing (wϕ, wp, w<sup>τ</sup> , wC−<sup>1</sup> p ) = (0, <sup>ξ</sup>, <sup>0</sup>, <sup>0</sup>), where <sup>ξ</sup> <sup>∈</sup> <sup>R</sup> 3 is arbitrary but constant, weak form (6.21) leads to

$$\boldsymbol{\xi} \cdot \frac{\mathrm{d}\mathbf{L}}{\mathrm{d}t} = \boldsymbol{\xi} \cdot \left( \int\_{\mathcal{B}} \mathbf{b} \, \mathrm{d}V + \int\_{\partial \mathcal{B}} \overline{\mathbf{f}} \, \mathrm{d}A \right) \, . \tag{6.22}$$

Due to the arbitrariness of <sup>ξ</sup> <sup>∈</sup> <sup>R</sup> 3 , (6.22) coincides with the balance law for linear momentum. Note that the parentheses on the right-hand side of (6.22) contain the resultant external force exerted on the continuum body (see also Fig. 6.2).

The total angular momentum relative to the origin of the inertial frame is dened by J = R <sup>B</sup> <sup>ϕ</sup> <sup>×</sup> <sup>p</sup> <sup>d</sup><sup>V</sup> . Choosing (wϕ, <sup>w</sup>p, <sup>w</sup><sup>τ</sup> , <sup>w</sup>C−<sup>1</sup> p ) = (p × ξ, ξ × ϕ, 0, 0), weak form (6.21) yields

$$\boldsymbol{\xi} \cdot \frac{\mathrm{d} \mathbf{J}}{\mathrm{d}t} = \boldsymbol{\xi} \cdot \left( \int\_{\mathcal{B}} \boldsymbol{\varphi} \times \mathbf{b} \, \mathrm{d}V + \int\_{\partial \mathcal{B}} \boldsymbol{\varphi} \times \overline{\mathbf{t}} \, \mathrm{d}A \right) \, . \tag{6.23}$$

Note that the symmetry condition FP<sup>T</sup> = PF<sup>T</sup> has been employed. Since <sup>ξ</sup> <sup>∈</sup> <sup>R</sup> 3 is arbitrary, (6.23) corresponds to the balance of angular momentum. The parentheses on the right-hand side of (6.23) contain the resultant external torque about the origin (see also Fig. 6.2).

Next, we substitute (wϕ, wp, w<sup>τ</sup> , wC−<sup>1</sup> p ) = (−b, ρ−<sup>1</sup>p, ∂τu, ∂C−<sup>1</sup> p u) into weak form (6.21). A straightforward calculation leads to the balance law for total energy

$$\frac{d}{dt}\int\_{\mathcal{B}} \left(\frac{1}{2}\rho^{-1}\mathbf{p}\cdot\mathbf{p} + u\right) \mathrm{d}V = \int\_{\mathcal{B}} \mathbf{b}\cdot\dot{\varphi}\,\mathrm{d}V + \int\_{\partial\mathcal{B}} \left(\rho^{-1}\mathbf{p}\cdot\overline{\mathbf{t}} - \overline{q}\right) \mathrm{d}A\,\,.\tag{6.24}$$

For the balance of entropy we choose (wϕ, wp, w<sup>τ</sup> , wC−<sup>1</sup> p ) = (0, ρ−<sup>1</sup>p, ∂<sup>τ</sup> η, ∂C−<sup>1</sup> p η) in the weak form (6.21) to obtain

$$\begin{split} 0 &= \int\_{\mathcal{B}} \left( \partial\_{\mathbf{F}} \boldsymbol{\eta} : \dot{\mathbf{F}} + \partial\_{\tau} \boldsymbol{\eta} \dot{\tau} + 2 \partial\_{\mathbf{C}\_{p}^{-1} \boldsymbol{\eta}} : \left( \left( \boldsymbol{\mathcal{M}} : \mathbf{M} \right) \mathbf{C}\_{p}^{-1} \right) - 2 \left( \partial\_{\mathbf{C}\_{p}^{-1} \boldsymbol{u}} \mathbf{C}\_{p}^{-1} \right) : \boldsymbol{\mathcal{M}} : \mathbf{M} \right) \mathrm{d}V \\ &+ \int\_{\mathcal{B}} \partial\_{\mathbf{C}\_{p}^{-1} \boldsymbol{\eta}} : \overset{\cdot}{\mathbf{C}\_{p}^{-1}} - \nabla \left( \frac{\partial\_{\tau} \eta}{\partial\_{\tau} u} \right) \cdot \mathbf{Q} \, \mathrm{d}V + \int\_{\partial \mathcal{B}} \frac{\partial\_{\tau} \eta}{\partial\_{\tau} u} \overline{q} \, \mathrm{d}A \\ &= \int\_{\mathcal{B}} \left( \frac{\mathrm{d} \eta}{\mathrm{d}t} - \frac{1}{\Theta} \mathbf{M} : \mathbf{M} : \mathbf{M} - \nabla \left( \frac{1}{\Theta} \right) \cdot \Theta^{2} \mathbf{K} \nabla \left( \frac{1}{\Theta} \right) \right) \, \mathrm{d}V + \int\_{\partial \mathcal{B}} \frac{1}{\Theta} \overline{q} \, \mathrm{d}A \, . \end{split}$$

Here, use has been made of formula (6.10) for the temperature along with expressions (6.16) for the material heat ux vector and (6.17) for the Mandel stress tensor, respectively. The above equation can be rewritten as

dS dt + Z ∂B 1 Θ q dA = Z B ∇ 1 Θ · Θ <sup>2</sup>K<sup>∇</sup> 1 Θ | {z } =Dcond≥0 + 1 Θ M : N : M | {z } =Dinel≥0 dV ≥ 0 , (6.25)

where Dinel is the local production of entropy due to inelastic deformations and Dcond is the local production of entropy due to heat conduction. The last equation corresponds to the Clausius-Duhem form of the second law of thermodynamics (see, for example, [136, Sec. 5]).

## 6.2. Inelastic part of the dissipative bracket

The present model for large-strain thermo-viscoelasticity relies on the introduction of the internal variable C −1 p : B × I 7→ <sup>R</sup> 3×3 , whose time-evolution accounts for local inelastic deformations. Following [148], C −1 p can be associated with the multiplicative decomposition of the deformation gradient [149]

$$\mathbf{F} = \mathbf{F}\_e \mathbf{F}\_p \,, \tag{6.26}$$

into an elastic part F<sup>e</sup> and an inelastic (or viscous) part Fp. Decomposition (6.26) gives rise to the relationship

$$\mathbf{C}\_p = \mathbf{F}\_p^\mathrm{T} \mathbf{F}\_p \,. \tag{6.27}$$

The restriction to the isotropic case implies that the free energy takes the separable form (see, for example, [150])

$$\Psi = \Psi(\mathbf{F}, \tau, \mathbf{C}\_p^{-1}) = \Psi^{\text{eq}}(\mathbf{F}, \tau) + \Psi^{\text{ne}}(\mathbf{F}, \tau, \mathbf{C}\_p^{-1})\,,\tag{6.28}$$

where Ψeq is the equilibrium part and Ψneq is the non-equilibrium part.

Since inelastic deformations are purely irreversible in nature, they lead to a contribution to the dissipative bracket in GENERIC (6.1), cf. (6.8). To derive the inelastic dissipative bracket (6.12), we resort to the dissipative bracket derived in [99] within a temperature-based framework for GENERIC. In particular, [99] consider functionals of the form

$$\overline{\mathbf{A}}(\varphi, \mathbf{p}, \theta, \mathbf{F}\_p) = \int\_{\mathcal{B}} \overline{\mathbf{a}}(\varphi, \mathbf{F}, \mathbf{p}, \theta, \mathbf{F}\_p) \, \mathrm{d}V \,\, \mathrm{},\tag{6.29}$$

with associated density function a(ϕ, F, p, θ, Fp). The inelastic dissipative bracket from [99] is given by

$$\left[\overline{\mathbf{A}}, \overline{\mathbf{B}}\right]\_{\text{inel}} = \int\_{\mathcal{B}} \left(\frac{\delta\_{\theta}\overline{\mathbf{a}}}{\partial\_{\theta}\overline{\mathbf{u}}} \partial\_{\mathbf{F}\_{p}}\overline{\mathbf{u}} - \delta\_{\mathbf{F}\_{p}}\overline{\mathbf{a}}\right) : \boldsymbol{\Theta}\mathbf{N} : \left(\frac{\delta\_{\theta}\overline{\mathbf{b}}}{\partial\_{\theta}\overline{\mathbf{u}}} \partial\_{\mathbf{F}\_{p}}\overline{\mathbf{u}} - \delta\_{\mathbf{F}\_{p}}\overline{\mathbf{b}}\right) \,\mathrm{d}V\,,\tag{6.30}$$

where u(F, θ, Fp) is the internal energy density and N is a fourth-order inelastic ow tensor which has the properties <sup>N</sup><sup>T</sup> <sup>=</sup> <sup>N</sup> (major symmetry) and <sup>A</sup> : <sup>N</sup> : <sup>A</sup> <sup>≥</sup> <sup>0</sup> for all <sup>A</sup> <sup>∈</sup> <sup>R</sup> 3×3 . In components, these properties read

$$(\mathbf{N})^{\alpha \cdot \beta \cdot}\_{\cdot I \cdot J} = (\mathbf{N})^{\beta \cdot \alpha \cdot}\_{\cdot J \cdot I} \qquad \text{and} \qquad (\mathbf{A})^{\cdot I}\_{\alpha \cdot} (\mathbf{N})^{\alpha \cdot \beta \cdot}\_{\cdot I \cdot J} (\mathbf{A})^{\cdot J}\_{\beta \cdot} \geq 0 \,. \tag{6.31}$$

As has been shown in [99], inelastic dissipative bracket (6.30) comes along with the ow rule

$$
\dot{\mathbf{F}}\_p = -\mathbf{N} : \boldsymbol{\Sigma},
$$

where

$$
\Sigma = \partial\_{\mathbf{F}\_p} \Pi - \Theta \partial\_{\mathbf{F}\_p} \overline{\eta} \ .
$$

In the last equation, η(F, θ, Fp) is the entropy density of the temperature-based formulation.

## 6.2.1. Change of variables

We perform a change of variables to transform the inelastic bracket (6.30) to the present setting which is based on functionals of the form (6.4). In particular, to link the current density functions a(ϕ, F, p, τ, C −1 p ) to those in (6.29), we express the generalized thermodynamic state variable τ ∈ {θ, η, u} in terms of the temperature by inverting relation (6.10) to get

$$
\tau = \overline{\tau}(\mathbf{F}, \theta, \mathbf{C}\_p^{-1}) \,. \tag{6.32}
$$

Moreover, relation (6.27) implies

$$\mathbf{C}\_p^{-1} = \mathbf{F}\_p^{-1} \mathbf{F}\_p^{-\Upsilon} \,. \tag{6.33}$$

Now, the two density functions under consideration can be connected through

$$\overline{\mathbf{a}}(\varphi, \mathbf{F}, \mathbf{p}, \theta, \mathbf{F}\_p) = a(\varphi, \mathbf{F}, \mathbf{p}, \overline{\tau}(\mathbf{F}, \theta, \mathbf{C}\_p^{-1}), \mathbf{C}\_p^{-1}),$$

where relationships (6.32) and (6.33) are employed on the right-hand side of the last equation. A straightforward application of the chain rule to the last equation yields

$$\begin{aligned} \partial\_{\theta} \overline{\mathbf{a}} &= \partial\_{\tau} a \partial\_{\theta} \overline{\tau} \ , \\ \partial\_{\mathbf{F}\_{p}} \overline{\mathbf{a}} &= \partial\_{\mathbf{F}\_{p}} a + \partial\_{\tau} a \partial\_{\mathbf{F}\_{p}} \overline{\tau} \ . \end{aligned} \tag{6.34}$$

Note that δθa = ∂θa, and δ<sup>F</sup><sup>p</sup> a = ∂<sup>F</sup><sup>p</sup> a. Furthermore, with regard to (6.10) and (6.32) we have θ = Θ(F, τ (F, θ, C −1 p ), C −1 p ), from which follows that

$$\begin{aligned} 1 &= \partial\_{\theta}\Theta = \partial\_{\tau}\Theta\partial\_{\theta}\overline{\tau} \ , \\ \mathbf{0} &= \partial\_{\mathbf{F}\_{p}}\Theta = \partial\_{\tau}\Theta\partial\_{\mathbf{F}\_{p}}\overline{\tau} + \partial\_{\mathbf{F}\_{p}}\Theta \ . \end{aligned}$$

We thus obtain

$$\begin{aligned} \partial\_{\theta}\overline{\tau} &= \frac{1}{\partial\_{\tau}\Theta} \,, \\ \partial\_{\mathbf{F}\_{p}}\overline{\tau} &= -\frac{\partial\_{\mathbf{F}\_{p}}\Theta}{\partial\_{\tau}\Theta} \,, \end{aligned} \tag{6.35}$$

Substituting from (6.35) into (6.34) yields

$$\begin{split} \partial\_{\theta} \overline{\mathbf{a}} &= \frac{\partial\_{\tau} a}{\partial\_{\tau} \Theta} \,, \\ \partial\_{\mathbf{F}\_{p}} \overline{\mathbf{a}} &= \partial\_{\mathbf{F}\_{p}} a - \frac{\partial\_{\tau} a}{\partial\_{\tau} \Theta} \partial\_{\mathbf{F}\_{p}} \Theta \ . \end{split} \tag{6.36}$$

Note that δτa = ∂τa and δ<sup>F</sup><sup>p</sup> a = ∂<sup>F</sup><sup>p</sup> a. Making use of (6.36), the terms in the parenthesis of (6.30) can be rewritten as

$$\frac{\delta\_{\theta}\overline{\mathbf{a}}}{\partial\_{\theta}\overline{\mathbf{u}}}\partial\_{\mathbf{F}\_{p}}\overline{\mathbf{u}} - \delta\_{\mathbf{F}\_{p}}\overline{\mathbf{a}} = \frac{\delta\_{\tau}a}{\partial\_{\tau}u}\partial\_{\mathbf{F}\_{p}}u - \delta\_{\mathbf{F}\_{p}}a \,. \tag{6.37}$$

Taking into account the relationship

$$
\partial\_{\mathbf{F}\_p} a = -2\mathbf{F}\_p^{-\sf T} \partial\_{\mathbf{C}\_p^{-1}} a \mathbf{C}\_p^{-1} \,, \tag{6.38}
$$

the inelastic dissipative bracket (6.30) can be recast in the form

$$\begin{split} & \left[\mathcal{A}, \mathcal{B}\right]\_{\text{incl}} \\ &= \int\_{\mathcal{B}} 2 \left( \frac{\delta\_{\tau} a}{\partial\_{\tau} u} \partial\_{\mathbf{C}\_{p}^{-1} u} \mathbf{C}\_{p}^{-1} - \delta\_{\mathbf{C}\_{p}^{-1}} a \mathbf{C}\_{p}^{-1} \right) : \boldsymbol{\Theta} \mathcal{N} : 2 \left( \frac{\delta\_{\tau} b}{\partial\_{\tau} u} \partial\_{\mathbf{C}\_{p}^{-1} u} \mathbf{C}\_{p}^{-1} - \delta\_{\mathbf{C}\_{p}^{-1}} b \mathbf{C}\_{p}^{-1} \right) \mathrm{d}V \, . \end{split}$$

This bracket coincides with the one introduced in (6.12). Note that in the above formula, δC−<sup>1</sup> p a = ∂C−<sup>1</sup> p a. Moreover, N is the material form of the fourth-order ow tensor N in (6.30). In components,

$$(\mathsf{A}\mathsf{V})^{I}\_{\cdot,J}{}^{K}\_{\cdot,L} = (\mathsf{F}\_{p}^{-1})^{I}\_{\cdot,\alpha} (\mathsf{F}\_{p}^{-1})^{K}\_{\cdot,\gamma} (\mathsf{N})^{\alpha}\_{\cdot,J}{}^{\gamma}\_{\cdot,L} \,. \tag{6.39}$$

Accordingly, the material ow tensor inherits symmetry and positive semi-deniteness. That is, in components,

$$(\mathbf{M})\_{\cdot,J}^{I\cdot K\cdot} \overset{\cdot}{L} = (\mathbf{M})\_{\cdot\cdot L}^{K\cdot I\cdot} \qquad \text{and} \qquad (\mathbf{M})\_{I\cdot\cdot}^{\cdot J} (\mathbf{M})\_{\cdot\cdot J\cdot L}^{I\cdot K\cdot} (\mathbf{M})\_{K\cdot\cdot}^{\cdot L} \ge 0 \tag{6.40}$$

for all M introduced in eq. (6.17). As has been shown in Section 6.1.1, the above inelastic dissipative bracket gives rise to ow rules of the form (cf. (6.20)4)

$$\stackrel{\cdot}{\mathbf{C}\_p^{-1}} = -2 \left( \mathcal{N} : \mathbf{M} \right) \mathbf{C}\_p^{-1} \,. \tag{6.41}$$

It is worth noting that this ow rule can be viewed as material version of the viscoelastic evolution equation

$$\mathcal{L}\_v \mathbf{b}\_e = -2 \left( \mathcal{V}^{-1} : \boldsymbol{\tau}^{\rm neq} \right) \mathbf{b}\_e \,, \tag{6.42}$$

derived in [148, 150] and being applied in, e.g., [151]. In the last equation, b<sup>e</sup> = FeF T <sup>e</sup> = FC−<sup>1</sup> <sup>p</sup> F T , Lvb<sup>e</sup> = F ˙ C −1 <sup>p</sup> F T , τ neq = 2F∂CΨneqF T , and V −1 is an isotropic, positive denite fourth-order tensor. Using these relationships, ow rule (6.42) can be recast in the form

$$\stackrel{\cdot}{\mathbf{C}\_p^{-1}} = -2\mathbf{F}^{-1}\left(\mathcal{V}^{-1} : \left(\mathbf{F}^{-\mathbf{T}}\mathbf{M}\mathbf{F}^{\mathrm{T}}\right)\right)\mathbf{F}\mathbf{C}\_p^{-1},\tag{6.43}$$

.

where the relation

$$2\mathbf{C}\partial\_{\mathbf{C}}\Psi^{\text{neq}} = 2\partial\_{\mathbf{C}\_p^{-1}}\Psi^{\text{neq}}\mathbf{C}\_p^{-1} = \mathbf{M} \,, $$

has been employed. Note that in the last equation, denition (6.17) of the Mandel stress tensor has been taken into account. Comparing (6.43) with present ow rule (6.41), leads to the conclusion that the respective fourth-order ow tensors are related by

$$(\mathcal{N})^{I \cdot K}\_{\cdot J \cdot L} = (\mathbf{F}^{-1})^{I \cdot}\_{\cdot a} (\mathbf{F})^{b \cdot}\_{\cdot J} (\mathbf{F}^{-1})^{K \cdot}\_{\cdot c} (\mathbf{F})^{d \cdot}\_{\cdot L} (\mathcal{V}^{-1})^{a \cdot c \cdot}\_{\cdot b \cdot d}$$

We further remark that the present viscoelastic evolution equation (6.41) can also be brought into the form

$$
\dot{\mathbf{C}}\_p = 2 \mathbf{C}\_p \mathcal{N} : \mathbf{M} \dots
$$

This version of the viscoelastic evolution equation has been used in [106], see also [76]. Thus, we conclude that the newly proposed inelastic dissipative bracket (6.12) gives rise to evolution equations for the internal variables that have been previously developed in the context of nite deformation thermo-viscoelasticity.

## 6.3. Discretization in space

Concerning the space discretization of the present GENERIC-based formulation, we essentially apply an isoparametric nite element approach (see, for example, [145]). Our main goal is to achieve a GENERIC-consistent space discretization in the sense of Section 2.1.2. In particular, a GENERIC-consistent space discretization ensures that the discrete formulation inherits the fundamental balance laws for both the energy and the entropy from the underlying continuous formulation (cf. Section 6.1.3).

We rst restate the governing equations of the IBVP to be discretized in space and time. With regard to the GENERIC-based weak form (6.21), we consider the following set of equations:

0 = ϕ˙ − v , 0 = ˙ C −1 <sup>p</sup> + 2 (N : M) C −1 p , 0 =<sup>Z</sup> B w<sup>u</sup><sup>τ</sup> (∂τu − u<sup>τ</sup> ) dV , 0 =<sup>Z</sup> B w<sup>η</sup><sup>τ</sup> (∂<sup>τ</sup> η − η<sup>τ</sup> ) dV , 0 =<sup>Z</sup> B (w<sup>p</sup> · (ρv˙ − b) + P : ∇wp) dV − Z ∂<sup>σ</sup> B w<sup>p</sup> · t dA , 0 =<sup>Z</sup> B w<sup>τ</sup> τ˙ − 2 uτ ∂C−<sup>1</sup> p uC −1 p : N : M + 1 ητ ∂Fη : ∇v dV − Z B ∇ w<sup>τ</sup> uτ · Q dV + Z ∂qB w<sup>τ</sup> uτ q dA dV . (6.44)

Here, the rst equation represents the local form of the kinematic relationship ϕ˙ = v, while the second equation is the viscoelastic evolution equation. The third and fourth equation serve the purpose to introduce the new elds u<sup>τ</sup> and η<sup>τ</sup> . This procedure facilitates a mixed nite element approach which turns out to be crucial for a GENERIC-consistent discrete formulation. Due to the arbitrariness of the test functions w<sup>u</sup><sup>τ</sup> and w<sup>η</sup><sup>τ</sup> , (6.44)<sup>3</sup> and (6.44)<sup>4</sup> impose the conditions u<sup>τ</sup> = ∂τu and η<sup>τ</sup> = ∂<sup>τ</sup> η, respectively. Moreover,

$$\begin{aligned} \mathbf{P} &= \partial\_{\mathbf{F}} u - \Theta \partial\_{\mathbf{F}} \eta \; \; \; \mathbf{\tilde{}} \\ \mathbf{M} &= 2 \left( \partial\_{\mathbf{C}\_p^{-1}} u - \Theta \partial\_{\mathbf{C}\_p^{-1}} \eta \right) \mathbf{C}\_p^{-1} \end{aligned} \tag{6.45}$$

are the GENERIC-specic representations of the rst Piola-Kirchho stress tensor and the material Mandel stress tensor previously introduced in (6.14) and (6.17), respectively. Similarly, the material heat ux vector Q has been introduced in (6.16). We emphasize again that the GENERIC-based formulation is based on expression (6.10) for the temperature eld. In the present mixed formulation this implies

$$
\Theta = \frac{u\_\tau}{\eta\_\tau} \,. \tag{6.46}
$$

The nite element method is based on nite dimensional approximations of the following quantities

$$\begin{aligned} \boldsymbol{\varphi}^{\text{h}} &= \boldsymbol{N}^{a} \mathbf{q}\_{a} \ , \\ \mathbf{v}^{\text{h}} &= \boldsymbol{N}^{a} \mathbf{v}\_{a} \ , \\ \boldsymbol{\tau}^{\text{h}} &= \boldsymbol{N}^{a} \boldsymbol{\tau}\_{a} \ , \end{aligned} \tag{6.47}$$

and

$$\begin{aligned} u\_{\tau}^{\mathbf{h}} &= N^{a} (u\_{\tau})\_{a} \; , \\ \eta\_{\tau}^{\mathbf{h}} &= N^{a} (\eta\_{\tau})\_{a} \; . \end{aligned} \tag{6.48}$$

As before, the summation convention applies, where a = 1, . . . , N, and N denotes the total number of nodes in the nite element mesh. Moreover, N<sup>a</sup> : B → <sup>R</sup> are the nodal shape functions with associated nodal values q<sup>a</sup> , <sup>v</sup><sup>a</sup> <sup>∈</sup> <sup>R</sup> 3 and <sup>τ</sup>a,(u<sup>τ</sup> )a,(η<sup>τ</sup> )<sup>a</sup> <sup>∈</sup> <sup>R</sup>. Analogous approximations are used for the test functions w<sup>u</sup><sup>τ</sup> , w<sup>η</sup><sup>τ</sup> , wp, and w<sup>τ</sup> , denoted by w h u<sup>τ</sup> , w h η<sup>τ</sup> , w<sup>h</sup> p , and w h τ .

In what follows, we summarize the space-discrete version of (6.44). Nodal collocation of kinematic equation (6.44)<sup>1</sup> yields

$$
\dot{\mathbf{q}}\_a = \mathbf{v}\_a \; , \tag{6.49}
$$

for a = 1, . . . , N. Viscoelastic evolution equation (6.44)<sup>2</sup> is collocated at the integration points X<sup>g</sup> ∈ B used for the numerical evaluation of the volume integrals in (6.44). Accordingly,

$$(\dot{\mathbf{C}}\_p^{-1})\_g = -2 \left( \mathcal{N}\_g : \mathbf{M}\_g \right) (\mathbf{C}\_p^{-1})\_g \,, \tag{6.50}$$

for g = 1, . . . , G, where G denotes the total number of integration points. Here and in the sequel, index g indicates evaluation at the integration point X<sup>g</sup> ∈ B. For example,

$$\begin{aligned} (\mathbf{C}\_p^{-1})\_g &= \mathbf{C}\_p^{-1}(\mathbf{X}\_g, t) \; , \\ \mathbf{M}\_g &= 2 \left( \partial\_{\mathbf{C}\_p^{-1}} u\_g - \Theta\_g \partial\_{\mathbf{C}\_p^{-1}} \eta\_g \right) (\mathbf{C}\_p^{-1})\_g \; , \end{aligned} \tag{6.51}$$

where

$$\begin{aligned} u\_g &= u \left( \mathbf{F}\_g^{\mathbf{h}}, \tau\_g^{\mathbf{h}}, (\mathbf{C}\_p^{-1})\_g \right), \\ \eta\_g &= \eta \left( \mathbf{F}\_g^{\mathbf{h}}, \tau\_g^{\mathbf{h}}, (\mathbf{C}\_p^{-1})\_g \right), \end{aligned} \tag{6.52}$$

are the internal energy and entropy densities at point Xg. Furthermore, the discrete deformation gradient at Xg, F h g , and the discrete generalized thermal variable τ h g , follow from (6.47)1,<sup>3</sup> and thus take the form

$$\begin{aligned} \mathbf{F}\_g^{\text{h}} &= \mathbf{q}\_a(t) \otimes \nabla N^a(\mathbf{X}\_g) \; , \\ \tau\_g^{\text{h}} &= N^a(\mathbf{X}\_g) \tau\_a(t) \; . \end{aligned} \tag{6.53}$$

Similarly, the discrete temperature at X<sup>g</sup> follows from (6.46) and is given by

$$
\Theta\_g = \frac{(u\_\tau^\mathbf{h})\_g}{(\eta\_\tau^\mathbf{h})\_g},
\tag{6.54}
$$

where, in view of interpolations (6.48), (u h τ )<sup>g</sup> = u h τ (Xg, t) and (η h τ )<sup>g</sup> = η h τ (Xg, t). In the discrete setting, the elds u h τ and η h τ are determined through (6.44)3,4. In particular, inserting the approximations (6.48) along with the corresponding formulas for w h u<sup>τ</sup> , w h η<sup>τ</sup> into (6.44)3,4, we obtain

$$\begin{aligned} 0 &= \sum\_{g=1}^{G} N^a(\mathbf{X}\_g) \left( \partial\_\tau u\_g - N^b(\mathbf{X}\_g)(u\_\tau)\_b \right) w\_g \ , \\ 0 &= \sum\_{g=1}^{G} N^a(\mathbf{X}\_g) \left( \partial\_\tau \eta\_g - N^b(\mathbf{X}\_g)(u\_\tau)\_b \right) w\_g \ , \end{aligned} \tag{6.55}$$

for a = 1, . . . , N. To calculate the spatial integrals, appropriate numerical quadrature formulas of the form

$$\int\_{\mathcal{B}} f(\mathbf{X}) \, \mathrm{d}V \approx \sum\_{g=1}^{G} f(\mathbf{X}\_g) w\_g \,, \tag{6.56}$$

have been applied to obtain (6.55) 2 . Here, w<sup>g</sup> play the role of generalized weighting coecients resulting from the specic quadrature rule along with the isoparametric description of reference domain B. Now, (6.55)<sup>1</sup> can be solved for the nodal quantities (u<sup>τ</sup> )a, a = 1, . . . , N. To this end, we introduce the components Hab of the positive denite Gram matrix [Hab],

$$H^{ab} = \sum\_{g=1}^{G} N^a(\mathbf{X}\_g) N^b(\mathbf{X}\_g) w\_g \,, \tag{6.57}$$

so that (6.55)<sup>1</sup> can be rewritten as

$$H^{ab}(u\_\tau)\_b = \sum\_{g=1}^G N^a(\mathbf{X}\_g) \partial\_\tau u\_g w\_g \,. \tag{6.58}$$

The components Hab of the inverse Gram matrix, [Hab] = [Hab] −1 , satisfy the relationship

$$H\_{ab}H^{bc} = \delta\_{a\cdot}^{\cdot c},\tag{6.59}$$

where δ · c <sup>a</sup>· denotes the Kronecker delta. Now, interpolation formula (6.48)<sup>1</sup> along with (6.58) lead to the result

$$u\_{\tau}^{\mathbf{h}} = \left. N^{a} H\_{ab} \sum\_{g=1}^{G} N^{b}(\mathbf{X}\_{g}) \partial\_{\tau} u\_{g} w\_{g} \right. \tag{6.60}$$

<sup>2</sup> The summation over g will always be stated explicitly, so that the summation convention does not apply to index g.

which will be utilized below. Similarly, interpolation formula (6.48)<sup>2</sup> in conjunction with (6.55)<sup>2</sup> yield the result

$$\eta\_{\tau}^{\mathbf{h}} = \left. N^{a} H\_{ab} \sum\_{g=1}^{G} N^{b}(\mathbf{X}\_{g}) \partial\_{\tau} \eta\_{g} w\_{g} \right. \tag{6.61}$$

Next, we turn to the discretization of (6.44)5, which can be done in a straightforward way to obtain

$$M^{ab}\dot{\mathbf{v}}\_b - \sum\_{g=1}^G N^a(\mathbf{X}\_g)\mathbf{b}\_g w\_g + \sum\_{g=1}^G \mathbf{P}\_g \nabla N^a(\mathbf{X}\_g) w\_g = 0 \,. \tag{6.62}$$

For simplicity, we have neglected the contribution of the external tractions which could be easily added to the above equation. In the above equation, rst Piola-Kirchho stress tensor P<sup>g</sup> at point X<sup>g</sup> ∈ B is given by

$$\mathbf{P}\_g = \partial\_{\mathbf{F}} u\_g - \Theta\_g \partial\_{\mathbf{F}} \eta\_g \, . \tag{6.63}$$

Moreover, in (6.62), the components Mab of the mass matrix are given by

$$M^{ab} = \sum\_{g=1}^{G} \rho(\mathbf{X}\_g) N^a(\mathbf{X}\_g) N^b(\mathbf{X}\_g) w\_g \,. \tag{6.64}$$

We further introduce nodal momentum vectors p a conjugate to nodal position vectors q<sup>a</sup> through the standard relation

$$\mathbf{p}^a = M^{ab} \mathbf{v}\_b \,. \tag{6.65}$$

Eventually, we consider the space-discrete version of (6.44)6. Straight-forward application of our approach yields

$$\begin{split} 0 &= H^{ab}\dot{\boldsymbol{\tau}}\_{b} - \sum\_{g=1}^{G} \frac{2N^{a}(\mathbf{X}\_{g})}{(u\_{\tau}^{\mathbf{h}})\_{g}} \left(\partial\_{\mathbf{C}\_{p}^{-1}} u\_{g}(\mathbf{C}\_{p}^{-1})\_{g}\right) : \mathcal{M}\_{g} : \mathbf{M}\_{g} w\_{g} \\ &+ \mathbf{v}\_{c} \cdot \sum\_{g=1}^{G} \frac{N^{a}(\mathbf{X}\_{g})}{(\eta\_{\tau}^{\mathbf{h}})\_{g}} \partial\_{\mathbf{F}} \eta\_{g} \nabla N^{c}(\mathbf{X}\_{g}) w\_{g} - \sum\_{g=1}^{G} \nabla \left(\frac{N^{a}(\mathbf{X}\_{g})}{(u\_{\tau}^{\mathbf{h}})\_{g}}\right) \cdot \mathbf{Q}\_{g} w\_{g} \,, \end{split} \tag{6.66}$$

where the material heat ux vector Q<sup>g</sup> at point X<sup>g</sup> ∈ B is given by

$$\mathbf{Q}\_g = \Theta\_g^2 \mathbf{K}\_g \nabla \left( \frac{(\eta\_\tau^\mathbf{h})\_g}{(u\_\tau^\mathbf{h})\_g} \right) \,\,\,\mathcal{J}$$

For simplicity, in the space-discrete evolution equation (6.66) for the generalized nodal thermal variable τb, the term accounting for heat transfer across the boundary has been neglected. To summarize, the resulting evolution equations for the space-discrete system at hand can be written in the form

$$\begin{split} \dot{\mathbf{q}}\_{a} &= M\_{ab}\mathbf{p}^{b}, \\ \dot{\mathbf{p}}^{a} &= \sum\_{g=1}^{G} N^{a}(\mathbf{X}\_{g}) \mathbf{b}\_{g} w\_{g} - \sum\_{g=1}^{G} \mathbf{P}\_{g} \nabla N^{a}(\mathbf{X}\_{g}) w\_{g} \,, \\ \dot{\tau}\_{a} &= H\_{ab} \sum\_{g=1}^{G} \frac{2N^{b}(\mathbf{X}\_{g})}{(u\_{\tau}^{\mathrm{h}})\_{g}} \left( \partial\_{\mathbf{C}\_{p}^{-1}} u\_{g} (\mathbf{C}\_{p}^{-1})\_{g} \right) : \mathcal{N}\_{g} : \mathbf{M}\_{g} w\_{g} \\ &- \mathbf{v}\_{c} \cdot H\_{ab} \sum\_{g=1}^{G} \frac{N^{b}(\mathbf{X}\_{g})}{(\eta\_{\tau}^{\mathrm{h}})\_{g}} \partial\_{\mathbf{F}} \eta\_{g} \nabla N^{c}(\mathbf{X}\_{g}) w\_{g} + H\_{ab} \sum\_{g=1}^{G} \nabla \left( \frac{N^{b}(\mathbf{X}\_{g})}{(u\_{\tau}^{\mathrm{h}})\_{g}} \right) \mathbf{Q}\_{g} w\_{g}, \\ \dot{\mathbf{C}}\_{p}^{-1})\_{g} &= -2 \left( \mathcal{N}\_{g} : \mathbf{M}\_{g} \right) (\mathbf{C}\_{p}^{-1})\_{g} \,, \tag{6.67} \end{split}$$

for 1, . . . , N and 1, . . . , G. In (6.67)1, Mab stands for the components of the inverse mass matrix satisfying MabMbc = δ · c a· . The set of equations (6.67) constitute nonlinear rstorder ordinary dierential equations for the determination of the unknowns which can be collected in the state vector

$$\mathbf{z} = \left( \mathbf{q}\_1, \dots, \mathbf{q}\_N, \mathbf{p}^1, \dots, \mathbf{p}^N, \tau\_1, \dots \tau\_N, \left( \mathbf{C}\_p^{-1} \right)\_1^{AB}, \dots, \left( \mathbf{C}\_p^{-1} \right)\_G^{AB} \right) \,. \tag{6.68}$$

In the sequel, state vector (6.68) will be viewed as column vector. In particular, this implies that the six independent components (C −1 p ) AB g , g = 1, . . . , G, of the internal variable (C −1 p )<sup>g</sup> (at quadrature point Xg) are arranged in a column vector.

The set of evolution equations (6.67) ts into the GENERIC framework for discrete systems, as shown next.

## 6.3.1. GENERIC-consistent space discretization

(

Our discretization approach presented above is GENERIC-consistent in the sense of [2] and thus can be framed in the context of GENERIC for discrete systems. To see this, the set of evolution equations (6.67) needs to be put into the form

$$\dot{\mathbf{z}} = \mathbf{L}\nabla\mathcal{E}(\mathbf{z}) + \mathbf{M}\nabla\mathcal{S}(\mathbf{z})\,. \tag{6.69}$$

Here, the focus is again on closed systems in which the boundary contributions are disregarded. In analogy to (6.1), the time-evolution of the state vector is decomposed additively into a reversible part generated by the total energy E and an irreversible part generated by the total entropy S. Poisson matrix L needs be skew-symmetric, while friction matrix M needs be symmetric positive semi-denite.

In the above equation, the total energy of the discrete system under consideration is given by

$$\mathcal{E}(\mathbf{z}) = \frac{1}{2} M\_{ab} \mathbf{p}^a \cdot \mathbf{p}^b + \sum\_{g=1}^{G} \left[ u \left( \mathbf{F}\_g^{\mathrm{h}}, \tau\_g^{\mathrm{h}}, (\mathbf{C}\_p^{-1})\_g \right) - \mathbf{q}\_a N^a(\mathbf{X}\_g) \mathbf{b}(\mathbf{X}\_g) \right] w\_g \,\,. \tag{6.70}$$

This is the space-discrete version of total energy (6.11). Similarly, the space-discrete version of total entropy (6.7) takes the form

$$\mathcal{S}(\mathbf{z}) = \sum\_{g=1}^{G} \eta \left( \mathbf{F}\_g^{\mathrm{h}}, \tau\_g^{\mathrm{h}}, (\mathbf{C}\_p^{-1})\_g \right) w\_g \,. \tag{6.71}$$

To get the gradient of the total energy, ∇E(z), we consider the derivative of (6.70) with respect to time. Accordingly, the left-hand side of (6.70) yields

$$\frac{d}{dt}\mathcal{E}(\mathbf{z}) = \partial\_{\mathbf{q}\_a} \mathcal{E} \cdot \dot{\mathbf{q}}\_a + \partial\_{\mathbf{p}^a} \mathcal{E} \cdot \dot{\mathbf{p}}^a + \partial\_{\tau\_a} \mathcal{E} \cdot \dot{\tau}\_a + \sum\_{g=1}^{G} \partial\_{\left(\mathbf{C}\_p^{-1}\right)\_g^{AB}} \mathcal{E} \left(\overline{\mathbf{C}\_p^{-1}}\right)\_g^{AB},\tag{6.72}$$

while the right-hand side of (6.70) gives

$$\begin{split} \frac{\mathbf{d}}{dt} \mathcal{E}(\mathbf{z}) &= \dot{\mathbf{p}}^a \cdot \mathbf{p}^b M\_{ab} + \dot{\mathbf{q}}\_a \cdot \sum\_{g=1}^G \left[ \partial\_{\mathbf{F}} u\_g \nabla N^a(\mathbf{X}\_g) - N^a(\mathbf{X}\_g) \mathbf{b}(\mathbf{X}\_g) \right] w\_g \\ &+ \dot{\tau}\_a \sum\_{g=1}^G N^a(\mathbf{X}\_g) \partial\_{\tau} u\_g w\_g + \sum\_{g=1}^G w\_g \partial\_{\mathbf{C}\_p^{-1}} u\_g : (\overline{\mathbf{C}\_p^{-1}})\_g \ . \end{split} \tag{6.73}$$

Comparing (6.72) with (6.73) yields the following expressions for the respective derivatives of the total energy

$$\begin{split} \partial\_{\mathbf{q}\_a} \mathcal{E} &= \sum\_{g=1}^{G} \left[ \partial\_{\mathbf{F}} u\_g \nabla N^a(\mathbf{X}\_g) - N^a(\mathbf{X}\_g) \mathbf{b}(\mathbf{X}\_g) \right] w\_g \,, \\ \partial\_{\mathbf{p}^a} \mathcal{E} &= M\_{ab} \mathbf{p}^b \,, \\ \partial\_{\tau\_a} \mathcal{E} &= \sum\_{g=1}^{G} N^a(\mathbf{X}\_g) \partial\_{\tau} u\_g w\_g \,, \\ \frac{\partial \mathcal{E}}{\partial(\mathbf{C}\_p^{-1})\_g^{AB}} &= w\_g \frac{\partial u\_g}{\partial(\mathbf{C}\_p^{-1})^{AB}} \, . \end{split} \tag{6.74}$$

In particular, inserting from (6.74)<sup>3</sup> into (6.60), we obtain the important relationship

$$u^{\mathbf{h}}\_{\tau} = \ N^{a} H\_{ab} \partial\_{\tau\_{b}} \mathcal{E} \ . \tag{6.75}$$

Similarly, the derivatives of total entropy (6.71) take the form

$$\begin{split} \partial\_{\mathbf{q}\_{a}}\mathcal{S} &= \sum\_{g=1}^{G} \partial\_{\mathbf{F}} \eta\_{g} \nabla N^{a}(\mathbf{X}\_{g}) w\_{g} \,, \\ \partial\_{\mathbf{p}^{a}}\mathcal{S} &= \mathbf{0} \,, \\ \partial\_{\tau\_{a}}\mathcal{S} &= \sum\_{g=1}^{G} N^{a}(\mathbf{X}\_{g}) \partial\_{\tau} \eta\_{g} w\_{g} \,, \\ \frac{\partial \mathcal{S}}{\partial(\mathbf{C}\_{p}^{-1})\_{g}^{AB}} &= w\_{g} \frac{\partial \eta\_{g}}{\partial(\mathbf{C}\_{p}^{-1})^{AB}} \,, \end{split} \tag{6.76}$$

Inserting from (6.76)<sup>3</sup> into (6.61), we obtain

$$
\eta^{\text{h}}\_{\tau} = \left. N^{a} H\_{ab} \partial\_{\tau\_{b}} \mathcal{S} \right. \tag{6.77}
$$

Now, guided by discrete GENERIC (6.69), the evolution equations in (6.67) can be recast. In particular, taking into account (6.74)2, kinematic relation (6.67)<sup>1</sup> can be rewritten as

$$
\dot{\mathbf{q}}\_a = \partial\_{\mathbf{p}^a} \mathcal{E} \,. \tag{6.78}
$$

Next, evolution equation (6.67)<sup>2</sup> for the nodal momentum vectors together with (6.63) and (6.54) yields

$$\dot{\mathbf{p}}^a = -\sum\_{g=1}^G \left[ \partial\_\mathbf{F} u\_g \nabla N^a(\mathbf{X}\_g) - N^a(\mathbf{X}\_g) \mathbf{b}(\mathbf{X}\_g) \right] w\_g + \sum\_{g=1}^G \frac{(u\_\tau^\mathbf{h})\_g}{(\eta\_\tau^\mathbf{h})\_g} \partial\_\mathbf{F} \eta\_g \nabla N^a(\mathbf{X}\_g) w\_g \dots$$

Taking into account (6.74)<sup>1</sup> and (6.75), we obtain

$$\dot{\mathbf{p}}^a = -\partial\_{\mathbf{q}\_a} \mathcal{E} + \partial\_{\tau\_b} \mathcal{E} \ H\_{bc} \sum\_{g=1}^G \frac{N^c(\mathbf{X}\_g)}{(\eta\_\tau^\mathrm{h})\_g} \partial\_\mathbf{F} \eta\_g \nabla N^a(\mathbf{X}\_g) w\_g \,. \tag{6.79}$$

Concerning the evolution of the nodal thermal variable τa, (6.67)<sup>3</sup> along with (6.51)2, (6.54) and (6.77) lead to

τ˙<sup>a</sup> =HabX G g=1 4N<sup>b</sup> (Xg)N<sup>c</sup> (Xg) (u h τ )g(η h τ )g ∂C−<sup>1</sup> p ug(C −1 p )g : N<sup>g</sup> : ∂C−<sup>1</sup> p ug(C −1 p )g w<sup>g</sup> Hcd∂<sup>τ</sup><sup>d</sup> S <sup>−</sup> <sup>H</sup>ab<sup>X</sup> G g=1 4N<sup>b</sup> (Xg) (η h τ )g ∂C−<sup>1</sup> p ηg(C −1 p )g : N<sup>g</sup> : ∂C−<sup>1</sup> p ug(C −1 p )g w<sup>g</sup> <sup>−</sup> <sup>∂</sup>p<sup>c</sup> E · <sup>H</sup>ab<sup>X</sup> G g=1 N<sup>b</sup> (Xg) (η h τ )g ∂Fηg∇N c (Xg)w<sup>g</sup> + HabX G g=1 Θ 2 <sup>g</sup>∇ N<sup>b</sup> (Xg) (u h τ )g · Kg∇ N<sup>c</sup> (Xg) (u h τ )g w<sup>g</sup> Hcd∂<sup>τ</sup><sup>d</sup> S . (6.80)

149

Eventually, evolution equation (6.67)<sup>4</sup> for the internal variable, together with (6.51)2, (6.54) and (6.77) result in

$$\begin{split} \dot{\mathbf{(C\_{p}^{-1}})}\_{g} &= -\frac{4N^{a}(\mathbf{X}\_{g})}{(\eta\_{\tau}^{\mathrm{h}})\_{g}} \left( \mathcal{N}\_{g} : \left( \partial\_{\mathbf{C\_{p}^{-1}}} u\_{g} (\mathbf{C\_{p}^{-1}})\_{g} \right) \right) (\mathbf{C\_{p}^{-1}})\_{g} \ H\_{ab} \partial\_{\tau\_{b}} \mathcal{S} \\ &+ 4\Theta\_{g} \left( \mathcal{N}\_{g} : \left( \partial\_{\mathbf{C\_{p}^{-1}}} \eta\_{g} (\mathbf{C\_{p}^{-1}})\_{g} \right) \right) (\mathbf{C\_{p}^{-1}})\_{g} \ . \end{split} \tag{6.81}$$

Next, we aim at introducing relation (6.76)<sup>4</sup> into the last term of (6.81). To this end, we introduce the fourth-order tensor M with components

$$(\mathcal{M})^{ABIJ} = (\mathbf{C}\_p^{-1})^{CB} (\mathcal{N})^{A}\_{\cdot \cdot} \stackrel{I}{C}\_K (\mathbf{C}\_p^{-1})^{JK} \,. \tag{6.82}$$

Note that M enjoys major symmetry, (M) ABIJ = (M) IJAB, due to the major symmetry of the material ow tensor N and the symmetry of C −1 p . Now, (6.81) can be recast in index form

$$\mathbf{(\overline{C\_{p}^{-1}})}\_{g}^{AB} = -4(\mathbf{M})\_{g}^{ABIJ} \frac{\partial u\_{g}}{\partial (\mathbf{C\_{p}^{-1}})^{IJ}} \frac{N^{a}(\mathbf{X\_{g}})}{\left(\eta\_{\tau}^{\mathbf{h}}\right)\_{g}} H\_{ab} \,\partial\_{\tau\_{b}} \mathcal{S} + 4 \frac{\Theta\_{g}}{w\_{g}} (\mathbf{M})\_{g}^{ABIJ} \quad \frac{\partial \mathcal{S}}{\partial (\mathbf{C\_{p}^{-1}})\_{g}^{IJ}}.\tag{6.83}$$

Altogether, evolution equations (6.67) pertaining to the state variables of the discrete system at hand can be recast in the form

$$\begin{split} \dot{\mathbf{q}}\_{a} &= \partial\_{\mathbf{p}^{a}} \mathcal{E} \ , \\ \dot{\mathbf{p}}^{a} &= -\partial\_{\mathbf{q}\_{a}} \mathcal{E} + \mathbf{1}\_{b}^{a} \cdot \partial\_{\tau\_{b}} \mathcal{E} \ , \\ \dot{\tau}\_{a} &= -\left(\mathbf{1}\_{a}^{b}\right)^{T} \partial\_{\mathbf{p}^{b}} \mathcal{E} + m\_{ab} \,\partial\_{\tau\_{b}} \mathcal{S} + \sum\_{g=1}^{G} m\_{g,a}^{IJ} \,\frac{\partial \mathcal{S}}{\partial(\mathbf{C}\_{p}^{-1})\_{g}^{IJ}} \ , \\ \dot{\mathbf{C}}\_{p}^{-1}\big)\_{g}^{AB} &= m\_{g,b}^{AB} \,\partial\_{\tau\_{b}} \mathcal{S} + m\_{g}^{ABIJ} \,\frac{\partial \mathcal{S}}{\partial(\mathbf{C}\_{p}^{-1})\_{g}^{IJ}} \ , \end{split} \tag{6.84}$$

where

(

$$\begin{split} \mathbf{I}\_{b}^{a} &= H\_{bc} \sum\_{g=1}^{G} \frac{N\_{g}^{c}}{(\eta\_{\rm r}^{\rm{I}})\_{g}} \partial\_{\mathbf{F}} \eta\_{g} \nabla N\_{g}^{d} w\_{g} \,, \\ m\_{ab} &= H\_{ac} \sum\_{g=1}^{G} \Theta\_{g}^{2} \nabla \left( \frac{N\_{g}^{c}}{(u\_{\rm r}^{\rm{I}})\_{g}} \right) \cdot \mathbf{K}\_{g} \nabla \left( \frac{N\_{g}^{d}}{(u\_{\rm r}^{\rm{I}})\_{g}} \right) w\_{g} \, H\_{db} \\ &+ H\_{ac} \sum\_{g=1}^{G} \frac{4N\_{g}^{c} N\_{g}^{d}}{(u\_{\rm r}^{\rm{I}})\_{g} (\eta\_{\rm r}^{\rm{I}})\_{g}} \frac{\partial u\_{g}}{\partial (\mathbf{C}\_{p}^{-1})^{AB}} (\mathcal{M})\_{g}^{ABIJ} \frac{\partial u\_{g}}{\partial (\mathbf{C}\_{p}^{-1})^{IJ}} w\_{g} \, H\_{ab} \,, \\ m\_{g,b}^{AB} &= -4 (\mathbf{M} \mathbf{M})\_{g}^{ABIJ} \frac{\partial u\_{g}}{\partial (\mathbf{C}\_{p}^{-1})^{IJ}} \frac{N\_{g}^{d}}{(\eta\_{\rm r}^{\rm{I}})\_{g}} H\_{ab} \,, \\ m\_{g}^{ABIJ} &= 4 \frac{\Theta\_{g}}{w\_{g}} (\mathbf{M})\_{g}^{ABIJ} \, . \end{split} \tag{6.85}$$

For simplicity of exposition, in (6.85), N<sup>a</sup> <sup>g</sup> = N<sup>a</sup> (Xg). Note that the properties mab = mba and mABIJ <sup>g</sup> = mIJAB <sup>g</sup> hold.

Now, evolution equations (6.84) give rise to specic forms of Poisson matrix L and friction matrix M in GENERIC (6.69). In particular, the Poisson matrix takes the form

$$\mathbf{L} = \begin{bmatrix} \mathbf{I} & \mathbf{0} & \mathbf{I} & \mathbf{0} & \mathbf{0} \\ \hline \\ \hline \\ 0 & \begin{bmatrix} \mathbf{I}^{\mathbf{a}\_{\cdot}^{\cdot}} & \mathbf{0} & \begin{bmatrix} \mathbf{I}^{\mathbf{a}\_{\cdot}^{\cdot}} \end{bmatrix} & \mathbf{0} \\ \hline \\ \hline \\ 0 & \begin{bmatrix} \mathbf{I}^{\mathbf{b}\_{\cdot}^{\cdot}} \end{bmatrix} & \mathbf{0} & \mathbf{0} \\ \hline \\ \hline \\ 0 & 0 & \mathbf{0} & \mathbf{0} \\ \end{bmatrix}, \end{bmatrix}, \qquad \text{(6.86)}$$

where I is the identity matrix (with appropriate dimension corresponding to the partitioning of the state vector (6.68)), and matrix [l a· · b ] consists of vectors l a· · <sup>b</sup> dened in (6.85)1. Specically, we have

$$\begin{bmatrix} \mathbf{I}\_{:,b}^{a} \end{bmatrix} = \begin{bmatrix} \mathbf{1}\_{:,1}^{1} & \cdots & \mathbf{1}\_{:,N}^{1} \\ \vdots & \ddots & \vdots \\ \mathbf{1}\_{:,1}^{N\_{\cdot,1}} & \cdots & \mathbf{1}\_{:,N\_{\cdot,N}}^{N\_{\cdot,\cdot}} \end{bmatrix} \tag{6.87}$$

It can be observed that Poisson matrix (6.86) is skew-symmetric. Furthermore, the friction matrix is given by

$$\mathbf{M} = \begin{bmatrix} 0 & \begin{array}{c|c|c|c} 0 & & & 0 & & 0 \\ \hline \\ 0 & 0 & & 0 & & 0 \\ \hline \\ 0 & 0 & & \begin{bmatrix} m\_{ab} \end{bmatrix} & \begin{bmatrix} \mathbf{m}\_{g,a}^{IJ} \end{bmatrix} \\\\ \hline \\ 0 & 0 & \begin{bmatrix} \mathbf{m}\_{g,b}^{AB} \end{bmatrix}^{\mathrm{T}} & \begin{bmatrix} \mathbf{m}\_{g}^{ABIJ} \end{bmatrix} \end{bmatrix},\tag{6.88}$$

Here, the block matrices [mab] <sup>∈</sup> <sup>R</sup> <sup>N</sup>×<sup>N</sup> , [mIJ g,b] <sup>∈</sup> <sup>R</sup> <sup>N</sup>×6<sup>G</sup>, and [mABIJ g ] contain the components dened in (6.85)<sup>2</sup>−<sup>4</sup> and take the form

$$\begin{aligned} \begin{bmatrix} m\_{ab} \end{bmatrix} = \begin{bmatrix} m\_{11} & \cdots & m\_{1N} \\ \vdots & \ddots & \vdots \\ m\_{N1} & \cdots & m\_{NN} \end{bmatrix}, \qquad \begin{bmatrix} \mathbf{m}\_{g,a}^{IJ} \end{bmatrix} = \begin{bmatrix} [m\_{1,1}^{IJ}] & \cdots & [m\_{G,1}^{IJ}] \\ \vdots & \ddots & \vdots \\ [m\_{1,N}^{IJ}] & \cdots & [m\_{G,N}^{IJ}] \end{bmatrix} \end{aligned}$$

and

$$\begin{bmatrix} \mathbf{m}\_g^{ABIJ} \end{bmatrix} = \begin{bmatrix} [m\_1^{ABIJ}] & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & [m\_G^{ABIJ}] \end{bmatrix} \cdot$$

Note that the last matrix is block-diagonal and symmetric. It can be easily observed that friction matrix (6.88) is symmetric and positive-semidenite.

## 6.3.2. Conservation properties

We next verify the conservation properties of the semi-discrete formulation of the closed system dealt with in Section 6.3.1. Since the evolution equations pertaining to the semidiscrete formulation can be brought into GENERIC form (6.69), (i) conservation of total energy, and (ii) non-decreasing total entropy are automatically satised. To see this, we rst consider the time-derivative of the total energy,

$$\begin{split} \frac{d}{dt}\mathcal{E}(\mathbf{z}) &= \nabla \mathcal{E}(\mathbf{z})\dot{\mathbf{z}} \\ &= \nabla \mathcal{E}(\mathbf{z})\mathbf{L}\nabla \mathcal{E}(\mathbf{z}) + \nabla \mathcal{E}(\mathbf{z})\mathbf{M}\nabla \mathcal{S}(\mathbf{z}) \end{split} \tag{6.89}$$

where (6.69) has been used. The rst term on the right-hand side of the last equation vanishes due to the skew-symmetry of Poisson matrix (6.86). The second term vanishes too, since the non-interaction condition

$$\mathbf{M}\nabla\mathcal{E}(\mathbf{z}) = \mathbf{0}\,. \tag{6.90}$$

holds. The last equation can be easily veried by a straight-forward calculation. Accordingly, (6.89) yields the conservation law dE/dt = 0. Similarly, the time-derivative of the total entropy yields

$$\begin{split} \frac{d}{dt} \mathcal{S}(\mathbf{z}) &= \nabla \mathcal{S}(\mathbf{z}) \dot{\mathbf{z}} \\ &= \nabla \mathcal{S}(\mathbf{z}) \mathbf{L} \nabla \mathcal{E}(\mathbf{z}) + \nabla \mathcal{S}(\mathbf{z}) \mathbf{M} \nabla \mathcal{S}(\mathbf{z}) \,. \end{split} \tag{6.91}$$

It can be veried by a straight-forward calculation that the second non-interaction condition

$$\mathbf{L}\nabla \mathcal{S}(\mathbf{z}) = \mathbf{0} \,. \tag{6.92}$$

is satised. In addition to that, the positive semi-deniteness of friction matrix (6.88) implies the result dS/dt ≥ 0. In particular, this result represents the semi-discrete version of (6.25) (apart from the boundary term in (6.25) originating from heat ux across the boundary). Thus, the total entropy of the closed system at hand is non-decreasing, due to the irreversible nature of heat conduction and visco-elastic deformations.

Since the material response is assumed to be frame-indierent (or objective), specic symmetry properties are inherent to the discrete system under consideration. In particular, invariance under rigid rotations implies satisfaction of the following conditions (see Appendix C.1 for further details):

$$\begin{aligned} \mathbf{0} &= \mathbf{q}\_a \times \partial\_{\mathbf{q}\_a} \mathcal{E} \ , \\ \mathbf{0} &= \mathbf{q}\_a \times \mathbf{l}\_{.b}^{a.} \ , \end{aligned} \tag{6.93}$$

for all b = 1, . . . , N. We tacitly assume that no resultant external torque is acting on the system (i.e. the right-hand side of (6.23) is assumed to vanish). The semi-discrete version of the angular momentum relative to the origin is given by

$$\mathbf{J}^{\mathbf{h}} = \int\_{\mathcal{B}} \rho \boldsymbol{\varphi}^{\mathbf{h}} \times \mathbf{v}^{\mathbf{h}} \,\mathrm{d}V = \mathbf{q}\_a \times \mathbf{p}^a \,, \tag{6.94}$$

where formulas (6.47)1,<sup>2</sup> along with denition (6.65) of the nodal momentum vectors p a have been used. The time-derivative of (6.94) reads

$$\begin{split} \frac{d}{dt} \mathbf{J}^{\mathbf{h}} &= \dot{\mathbf{q}}\_a \times \mathbf{p}^a + \mathbf{q}\_a \times \dot{\mathbf{p}}^a \\ &= M\_{ab} \mathbf{p}^b \times \mathbf{p}^a + \mathbf{q}\_a \times \left( -\partial\_{\mathbf{q}\_a} \mathcal{E} + \mathbf{l}\_{\cdot b}^{a \cdot} \partial\_{\tau b} \mathcal{E} \right) \end{split} \tag{6.95}$$

where (6.67)<sup>1</sup> and (6.84)<sup>2</sup> have been used. Now the right-hand side of the last equation vanishes due to (i) the symmetry of Mab together with the skew-symmetry of the vector cross product, and (ii) symmetry conditions(6.93). Result dJ h /dt = 0 implies conservation of total angular momentum.

## 6.3.3. Choice of the thermodynamic state variable

We shortly outline the impact of the specic choice of the thermodynamic state variable, τ ∈ {θ, η, u}, on the structure of GENERIC (6.69). For simplicity, in this section we neglect body forces, that is, b = 0 and still focus on closed systems.

Choosing the internal energy density as thermodynamic state variable, i.e. τ = u, the total energy (6.70) takes a particularly simple form given by

$$\hat{\mathcal{E}}(\mathbf{z}) = \frac{1}{2} M\_{ab} \mathbf{p}^a \cdot \mathbf{p}^b + \sum\_{g=1}^G u\_g^{\text{h}} w\_g \, . \tag{6.96}$$

As before, u h g stands for u h (Xg, t). In particular, interpolation formula (6.53)<sup>2</sup> gives rise to u h <sup>g</sup> = N<sup>a</sup> (Xg)ua(t). The derivatives of the total energy in (6.74) simplify to

$$\begin{aligned} \partial\_{\mathbf{q}\_a} \hat{\mathcal{E}} &= \mathbf{0} \,, \\ \partial\_{\mathbf{p}^a} \hat{\mathcal{E}} &= M\_{ab} \mathbf{p}^b \,, \\ \partial\_{u\_a} \hat{\mathcal{E}} &= \sum\_{g=1}^G N^a(\mathbf{X}\_g) w\_g \,, \\ \frac{\partial \hat{\mathcal{E}}}{\partial(\mathbf{C}\_p^{-1})\_g^{AB}} &= 0 \, . \end{aligned} \tag{6.97}$$

Consequently,

$$\begin{split} \frac{d}{dt}\mathring{\mathcal{E}}(\mathbf{z}) &= \nabla \mathring{\mathcal{E}}(\mathbf{z}) \dot{\mathbf{z}} \\ &= M\_{ab} \mathbf{p}^a \cdot \dot{\mathbf{p}}^b + \sum\_{g=1}^G N^a(\mathbf{X}\_g) \dot{u}\_a w\_g \,. \end{split} \tag{6.98}$$

Moreover, for the choice τ = u friction matrix (6.88) attains a particularly simple blockdiagonal form, since mAB g,b = 0, and coecients mab only contain contributions due to heat conduction (cf. (6.85)).

Choosing the internal entropy density as thermodynamic state variable, i.e. τ = η, yields a particularly simple form of the total entropy (6.71) given by

$$\bar{\mathcal{S}}(\mathbf{z}) = \sum\_{g=1}^{G} \eta\_g^{\mathcal{h}} w\_g \,, \tag{6.99}$$

where interpolation formula (6.53)<sup>2</sup> gives rise to η h <sup>g</sup> = N<sup>a</sup> (Xg)ηa(t). Consequently, the derivatives of the total entropy in (6.76) simplify to

$$\begin{aligned} \partial\_{\mathbf{q}\_a} \tilde{\mathcal{S}} &= \mathbf{0} \,, \\ \partial\_{\mathbf{p}^a} \tilde{\mathcal{S}} &= \mathbf{0} \,, \\ \partial\_{\eta\_a} \tilde{\mathcal{S}} &= \sum\_{g=1}^G N^a(\mathbf{X}\_g) w\_g \,, \\ \frac{\partial \tilde{\mathcal{S}}}{\partial(\mathbf{C}\_p^{-1})\_g^{AB}} &= 0 \,. \end{aligned} \tag{6.100}$$

Thus

$$\begin{split} \frac{d}{dt}\tilde{\mathcal{S}}(\mathbf{z}) &= \nabla \tilde{\mathcal{S}}(\mathbf{z}) \dot{\mathbf{z}} \\ &= \sum\_{g=1}^{G} N^{a}(\mathbf{X}\_{g}) \dot{\eta}\_{a} w\_{g} \, . \end{split} \tag{6.101}$$

Moreover, the choice τ = η leads to l a· · <sup>b</sup> <sup>=</sup> <sup>0</sup> (see (6.85)), so that Poisson matrix (6.86) yields a particularly simple form.

In contrast to the above considerations, selecting the total temperature as thermodynamic state variable, i.e. τ = θ, essentially does not lead to any simplications. We eventually remark that these conclusions also aect the discretization in time, which will be treated next.

## 6.4. Discretization in time

We now turn to the discretization in time of the semi-discrete GENERIC-consistent evolution equations derived in Section 6.3.1. To this end, we focus on a representative time interval [tn, tn+1] with corresponding time-step size ∆t = tn+1 − tn. We aim at secondorder accurate, implicit time-stepping schemes based on the mid-point rule. Application of the mid-point rule to (6.69) yields

$$\mathbf{z}\_{\mathrm{n}+1} - \mathbf{z}\_{\mathrm{n}} = \Delta t \mathbf{L}(\mathbf{z}\_{\mathrm{n}+\frac{1}{2}}) \nabla \mathcal{E}(\mathbf{z}\_{\mathrm{n}+\frac{1}{2}}) + \Delta t \mathbf{M}(\mathbf{z}\_{\mathrm{n}+\frac{1}{2}}) \nabla \mathcal{S}(\mathbf{z}\_{\mathrm{n}+\frac{1}{2}}) \,. \tag{6.102}$$

Here, z<sup>n</sup> stands for the discrete vector of state variables at time tn, and

$$\mathbf{z}\_{\mathbf{n}+\frac{1}{2}} = \frac{1}{2} \left( \mathbf{z}\_{\mathbf{n}} + \mathbf{z}\_{\mathbf{n}+1} \right) \dots$$

Provided that the state variables z<sup>n</sup> are given, the state variables zn+1 can be determined by solving (6.102).

## 6.4.1. Partially structure-preserving schemes

Next, we check whether, or under what conditions, structure-preserving properties hold in the discrete setting. In this connection, we shall see that the specic choice of the thermodynamic state variable τ ∈ {θ, η, u} plays a crucial role. It can be easily veried that non-interaction conditions (6.90) and (6.92) are still satised in the sense that

$$\begin{aligned} \mathbf{M}(\mathbf{z}\_{\mathbf{n}+\frac{1}{2}}) \nabla \mathcal{E}(\mathbf{z}\_{\mathbf{n}+\frac{1}{2}}) &= \mathbf{0} \\ \mathbf{L}(\mathbf{z}\_{\mathbf{n}+\frac{1}{2}}) \nabla \mathcal{S}(\mathbf{z}\_{\mathbf{n}+\frac{1}{2}}) &= \mathbf{0} \end{aligned} \tag{6.103}$$

To see whether the fundamental balance laws are correctly reproduced in the discrete formulation, we proceed along the lines of the time-continuous formulation in Section 6.3.2. In particular, concerning the balance of energy, similar to (6.89), we consider

$$\begin{split} \nabla \mathcal{E}(\mathbf{z}\_{\mathbf{n}+\frac{1}{2}}) \cdot (\mathbf{z}\_{\mathbf{n}+1} - \mathbf{z}\_{\mathbf{n}}) &= \Delta t \nabla \mathcal{E}(\mathbf{z}\_{\mathbf{n}+\frac{1}{2}}) \cdot \mathbf{L}(\mathbf{z}\_{\mathbf{n}+\frac{1}{2}}) \nabla \mathcal{E}(\mathbf{z}\_{\mathbf{n}+\frac{1}{2}}) \\ &+ \Delta t \nabla \mathcal{E}(\mathbf{z}\_{\mathbf{n}+\frac{1}{2}}) \cdot \mathbf{M}(\mathbf{z}\_{\mathbf{n}+\frac{1}{2}}) \nabla \mathcal{S}(\mathbf{z}\_{\mathbf{n}+\frac{1}{2}}) \,, \end{split}$$

where (6.102) has been used. In analogy to the time-continuous case the right-hand side of the above equation vanishes due to the skew-symmetry of L(zn+ <sup>1</sup> 2 ) and non-interaction condition (6.103)1. Thus

$$\nabla \mathcal{E}(\mathbf{z}\_{\mathbf{n}+\frac{1}{2}}) \cdot (\mathbf{z}\_{\mathbf{n}+1} - \mathbf{z}\_{\mathbf{n}}) = 0 \ .$$

On the other side,

$$\nabla \mathcal{E}(\mathbf{z}\_{\mathbf{n}+\frac{1}{2}}) \cdot (\mathbf{z}\_{\mathbf{n}+1} - \mathbf{z}\_{\mathbf{n}}) \neq \mathcal{E}(\mathbf{z}\_{\mathbf{n}+1}) - \mathcal{E}(\mathbf{z}\_{\mathbf{n}}) \tag{6.104}$$

in general. This inequality complies with the well-known fact that the mid-point rule is not capable to conserve nonlinear rst integrals in general. However, there exists the exceptional case related to the choice τ = u, for which (6.104) turns into an equality. This is due to the fact that for <sup>τ</sup> <sup>=</sup> <sup>u</sup> the total energy takes the form (6.96) and thus <sup>E</sup>ˆ(z) is merely quadratic. Since the mid-point rule preserves quadratic rst integrals (see [6, Sec. 4.4.2]), the choice τ = u yields a structure-preserving scheme which is capable to conserve total energy.

Concerning the evolution of total entropy, guided by (6.91), we consider

$$\begin{split} \nabla \mathcal{S}(\mathbf{z}\_{\mathbf{n}+\frac{1}{2}}) \cdot (\mathbf{z}\_{\mathbf{n}+1} - \mathbf{z}\_{\mathbf{n}}) &= \Delta t \nabla \mathcal{S}(\mathbf{z}\_{\mathbf{n}+\frac{1}{2}}) \cdot \mathbf{L}(\mathbf{z}\_{\mathbf{n}+\frac{1}{2}}) \nabla \mathcal{E}(\mathbf{z}\_{\mathbf{n}+\frac{1}{2}}) \\ &+ \Delta t \nabla \mathcal{S}(\mathbf{z}\_{\mathbf{n}+\frac{1}{2}}) \cdot \mathbf{M}(\mathbf{z}\_{\mathbf{n}+\frac{1}{2}}) \nabla \mathcal{S}(\mathbf{z}\_{\mathbf{n}+\frac{1}{2}}) \,, \end{split}$$

where again (6.102) has been used. Employing non-interaction condition (6.103)2, we obtain

$$
\nabla \mathcal{S}(\mathbf{z}\_{\mathbf{n}+\frac{1}{2}}) \cdot (\mathbf{z}\_{\mathbf{n}+1} - \mathbf{z}\_{\mathbf{n}}) = \Delta t \nabla \mathcal{S}(\mathbf{z}\_{\mathbf{n}+\frac{1}{2}}) \cdot \mathbf{M}(\mathbf{z}\_{\mathbf{n}+\frac{1}{2}}) \nabla \mathcal{S}(\mathbf{z}\_{\mathbf{n}+\frac{1}{2}}) \ge 0,\tag{6.105}
$$

where the positive semi-deniteness of friction matrix M(zn+ <sup>1</sup> 2 ) has been taken as a basis. On the other hand, in analogy to (6.104), we have

$$
\nabla \mathcal{S}(\mathbf{z}\_{\mathbf{n}+\frac{1}{2}}) \cdot (\mathbf{z}\_{\mathbf{n}+1} - \mathbf{z}\_{\mathbf{n}}) \neq \mathcal{S}(\mathbf{z}\_{\mathbf{n}+1}) - \mathcal{S}(\mathbf{z}\_{\mathbf{n}}) \,. \tag{6.106}
$$

This implies that, despite the encouraging result (6.105), the mid-point scheme in general does not guarantee a non-decreasing entropy. However, there again is an exception related to the choice τ = η. For this particular case, the total entropy takes the form (6.99) and thus <sup>S</sup>˜(z) is merely linear. Accordingly, the choice <sup>τ</sup> <sup>=</sup> <sup>η</sup> turns inequality (6.106) into an equality and the entropy-based mid-point scheme is therefore capable to correctly reproduce the second law of thermodynamics in the discrete setting.

We eventually verify that all mid-point-based schemes under consideration are capable to conserve angular momentum. The incremental change of angular momentum (6.94) can be written in the form

$$\mathbf{J}\_{\mathbf{n}+1}^{\mathbf{h}} - \mathbf{J}\_{\mathbf{n}}^{\mathbf{h}} = \mathbf{q}\_{a\_{\mathbf{n}+\frac{1}{2}}} \times \left(\mathbf{p}\_{\mathbf{n}+1}^{a} - \mathbf{p}\_{\mathbf{n}+1}^{a}\right) - \mathbf{p}\_{\mathbf{n}+\frac{1}{2}}^{a} \times \left(\mathbf{q}\_{a\_{\mathbf{n}+1}} - \mathbf{q}\_{a\_{\mathbf{n}}}\right) \,. \tag{6.107}$$

Mid-point scheme (6.102) gives rise to

$$\begin{split} \mathbf{q}\_{a\_{n+1}} - \mathbf{q}\_{a\_{n}} &= \Delta t M\_{ab} \mathbf{p}\_{n+\frac{1}{2}}^{b}, \\ \mathbf{p}\_{n+1}^{a} - \mathbf{p}\_{n+1}^{a} &= \Delta t \left( -\partial\_{\mathbf{q}\_{a}} \mathcal{E}(\mathbf{z}\_{n+\frac{1}{2}}) + \mathbf{l}\_{-b}^{a}(\mathbf{z}\_{n+\frac{1}{2}}) \, \partial\_{\tau\_{b}} \mathcal{E}(\mathbf{z}\_{n+\frac{1}{2}}) \right) . \end{split} \tag{6.108}$$

Inserting from (6.108) into (6.107) yields

$$\mathbf{J}\_{\mathbf{n}+1}^{\mathbf{h}} - \mathbf{J}\_{\mathbf{n}}^{\mathbf{h}} = \Delta t \mathbf{q}\_{a\_{\mathbf{n}+\frac{1}{2}}} \times \left( -\partial\_{\mathbf{q}\_{\mathbf{a}}} \mathcal{E}(\mathbf{z}\_{\mathbf{n}+\frac{1}{2}}) + \mathbf{I}\_{-b}^{a}(\mathbf{z}\_{\mathbf{n}+\frac{1}{2}}) \, \partial\_{\tau\_{b}} \mathcal{E}(\mathbf{z}\_{\mathbf{n}+\frac{1}{2}}) \right) - \Delta t M\_{ab} \mathbf{p}\_{\mathbf{n}+\frac{1}{2}}^{a} \times \mathbf{p}\_{\mathbf{n}+\frac{1}{2}}^{b} \, . \tag{6.109}$$

Symmetry conditions (6.93) imply

$$\begin{aligned} \mathbf{0} &= \mathbf{q}\_{a\_{\mathrm{n}+\frac{1}{2}}} \times \partial\_{\mathbf{q}\_{a}} \mathcal{E}(\mathbf{z}\_{\mathrm{n}+\frac{1}{2}}) \,, \\ \mathbf{0} &= \mathbf{q}\_{a\_{\mathrm{n}+\frac{1}{2}}} \times \mathbf{l}^{a}\_{\cdot,b}(\mathbf{z}\_{\mathrm{n}+\frac{1}{2}}) \,. \end{aligned} \tag{6.110}$$

Inserting from (6.110) into (6.109) and taking into account the symmetry of Mab together with the skew-symmetry of the vector product leads to the result J h n+1 = J h n .

To summarize, depending on the choice for the thermodynamic state variable we get three alternative mid-point schemes which are partially structure-preserving. Correspondingly, the resulting schemes are denoted by (EM)<sup>u</sup> (energy-momentum scheme related to τ = u), (ME)<sup>η</sup> (momentum-entropy scheme associated with τ = η), and M<sup>θ</sup> (momentum scheme related to τ = θ).

## 6.5. Numerical investigations

In this section, the alternative mid-point type schemes newly developed in the present Chapter are applied to representative numerical examples dealing with nite-strain thermoviscoelastodynamics. Depending on the specic choice for the thermodynamic state variable τ ∈ {θ, η, u}, the following methods are applied:


Following the previous Chapters we shall focus on momentum maps associated with symmetries of the mechanical system at hand, and the balance laws associated with the two fundamental laws of thermodynamics. In this connection we also consider the functional

$$
\mathcal{L} = \mathcal{E} - \theta\_0 \mathcal{S} \,. \tag{6.111}
$$

According to [146], for certain types of environments, L is a natural Lyapunov function and thus qualies as estimate for the numerical stability of the schemes under consideration.

In each time step, the schemes emanating from (6.102) generate a system of nonlinear algebraic equations for the determination of the state variables zn+1. To this end, we apply a Multilevel-Newton algorithm, see [152] and references therein for more details.

## 6.5.1. Material model

In order to particularize the Helmholtz free energy density (6.28) used in the numerical examples we start from a temperature-based description. In particular, we consider

$$\begin{aligned} \overline{\boldsymbol{\psi}}(\mathbf{C}, \boldsymbol{\theta}, \mathbf{C}\_p^{-1}) &= \overline{\boldsymbol{\psi}}^{\text{eq}}(\mathbf{C}, \boldsymbol{\theta}) + \overline{\boldsymbol{\psi}}^{\text{neq}}(\mathbf{C}, \boldsymbol{\theta}, \mathbf{C}\_p^{-1}) \ , \\ \overline{\boldsymbol{\psi}}^{\text{eq}}(\mathbf{C}, \boldsymbol{\theta}) &= \psi\_1(\mathbf{C}) + \psi\_2(\boldsymbol{\theta}) - (\boldsymbol{\theta} - \boldsymbol{\theta}\_0)\psi\_3(J) \ , \\ \overline{\boldsymbol{\psi}}^{\text{neq}}(\mathbf{C}, \boldsymbol{\theta}, \mathbf{C}\_p^{-1}) &= \psi\_{1, \text{visc}}(\mathbf{C}, \mathbf{C}\_p^{-1}) \ , \end{aligned} \tag{6.112}$$

where

$$\begin{aligned} \psi\_1(\mathbf{C}) &= \frac{\mu}{2} \left( \mathbf{C} : \mathbf{I} - 3 - 2 \log J - \frac{2}{3} \left( J - 1 \right)^2 \right) + W\_{\text{vol}}(J) \,, \\ \psi\_{1, \text{visc}}(\mathbf{C}, \mathbf{C}\_p^{-1}) &= \frac{\mu\_e}{2} \left( \mathbf{C} : \mathbf{C}\_p^{-1} - 3 - 2 \log J\_e - \frac{2}{3} \left( J\_e - 1 \right)^2 \right) + W\_{\text{vol}, \text{visc}}(J\_e) \,, \\ \psi\_2(\theta) &= c \left( \theta - \theta\_0 - \theta \log(\theta/\theta\_0) \right) \,, \\ \psi\_3(J) &= 3 \beta W\_{\text{vol}}(J) \,, \\ W\_{\text{vol}}(J) &= \frac{\lambda + \frac{2}{3} \mu}{4} \left( (\log J)^2 + (J - 1)^2 \right) \,, \\ W\_{\text{vol,visc}}(J\_e) &= \frac{\lambda\_e + \frac{2}{3} \mu\_e}{4} \left( (\log J\_e)^2 + (J\_e - 1)^2 \right) \, . \end{aligned}$$

Here, J = √ q det C is the determinant of the deformation gradient and J<sup>e</sup> = det (CC<sup>−</sup><sup>1</sup> p ). Further µ, µe, λ and λ<sup>e</sup> are prescribed parameters, c > 0 is the specic heat capacity at constant deformation, β is the coecient of thermal expansion, and θ<sup>0</sup> is the reference temperature. We refer to [1] and the references therein for a detailed investigation of the thermoelastic part of the specic Helmholtz free energy density (6.112).

Further, for the viscoelastic part we refer to [74] and the references therein. For simplicity we assume incompressible material behavior. Quasi incompressible material models for nite-strain thermo-viscoelasticy are considered in, e.g., [142, 153]. It is now a straightforward exercise to calculate further quantities emanating from (6.112), depending on the specic choice for the thermodynamic variable τ ∈ {u, θ, η} (see also [1] for additional details). In particular, the temperature-based formulation yields

$$\begin{aligned} \overline{\eta}(\mathbf{C}, \boldsymbol{\theta}, \mathbf{C}\_p^{-1}) &= c \log(\boldsymbol{\theta}/\theta\_0) + \psi\_3(J) \; , \\ \overline{u}(\mathbf{C}, \boldsymbol{\theta}, \mathbf{C}\_p^{-1}) &= \psi\_1(\mathbf{C}) + \psi\_{1, \text{visc}}(\mathbf{C}, \mathbf{C}\_p^{-1}) + c(\boldsymbol{\theta} - \boldsymbol{\theta}\_0) + \theta\_0 \psi\_3(J) \; . \end{aligned}$$

The formulation based on the entropy density gives

$$\begin{aligned} \widetilde{\boldsymbol{\eta}}(\mathbf{C}, \boldsymbol{\eta}, \mathbf{C}\_p^{-1}) &= \boldsymbol{\eta} \ , \\ \widetilde{\boldsymbol{u}}(\mathbf{C}, \boldsymbol{\eta}, \mathbf{C}\_p^{-1}) &= \psi\_1(\mathbf{C}) + \psi\_{1, \text{visc}}(\mathbf{C}, \mathbf{C}\_p^{-1}) + c \left( \theta\_0 e^{\frac{\boldsymbol{\eta} - \psi\_3(J)}{c}} - \theta\_0 \right) + \theta\_0 \psi\_3(J) \ . \end{aligned}$$

Moreover, the formulation based on the internal energy density leads to

$$\begin{aligned} \widehat{\eta}(\mathbf{C}, u, \mathbf{C}\_p^{-1}) &= c \log \left( 1 + \frac{u - \psi\_1(\mathbf{C}) - \psi\_{1, \text{visc}}(\mathbf{C}, \mathbf{C}\_p^{-1})}{c \theta\_0} - \frac{\psi\_3(J)}{c} \right) + \psi\_3(J) \,, \\\widehat{u}(\mathbf{C}, u, \mathbf{C}\_p^{-1}) &= u \; . \end{aligned}$$

Concerning the constitutive equation for the material heat ux vector, we assume thermally isotropic material, with material conductivity tensor given by

$$\mathbf{K} = kJ\mathbf{C}^{-1}.\tag{6.114}$$

Here, k is a prescribed coecient of thermal conductivity and, as before, J = p det(C). Finally, the constitutive equation for the isotropic fourth-order material inelastic ow tensor is given by (see [69] or [148] and references therein for the spatial representation of the isotropic fourth-order inelastic ow tensor)

$$\mathcal{N} = \frac{1}{2\nu\_D} \left( \left[ \mathbf{C}^{-1} \odot \mathbf{I} \right] \mathbf{C} - \frac{1}{3} \mathbf{I} \otimes \mathbf{I} \right) + \frac{1}{9\nu\_V} \mathbf{I} \otimes \mathbf{I} \,, \tag{6.115}$$

where ν<sup>D</sup> > 0 and ν<sup>V</sup> > 0 represent the deviatoric and volumetric viscosities, respectively, and where (A  B) A · CD · <sup>B</sup> · · = 1 2 - (A) AC · · (B) · D <sup>B</sup> · + (A) AD · · (B) · C B · .

## 6.5.2. Flying L-shaped block

The rst numerical example deals with the L-shaped block depicted in Fig. 6.4.

Figure 6.4.: L-shaped block: Discretised block with initial temperature distribution and mechanical boundary conditions (left), load function over time (right)

The spatial discretization of the block relies on 117 tri-linear nite elements leading to 224 nodes. The initial temperature eld is varying linearly over the height (x<sup>3</sup> direction) of the block. In particular, at x<sup>3</sup> = 0, the initial temperature is prescribed as θa, while at x<sup>3</sup> = h, the temperature is prescribed as θb. The whole block is assumed to be thermally insulated (<sup>q</sup> = 0 on <sup>∂</sup>q<sup>B</sup> <sup>=</sup> <sup>∂</sup>B). Starting at rest, Piola traction vectors ¯t<sup>a</sup> and ¯t<sup>b</sup> are acting on two parts of the boundary surface of the block (Fig. 6.4). The external loads are applied in the form of a hat function over time. In particular, the traction vectors are given by

$$\begin{array}{ll} \mathsf{i}\mathsf{t}\_{a} = -\mathsf{i}\_{b} = f(t) \begin{pmatrix} 256/9 \\ 512/9 \\ 768/9 \end{pmatrix} \text{Pa}, \quad f(t) = \begin{cases} t & \text{for} \quad 0 \le t \le 2\text{s} \\ 4-t & \text{for} \quad 2\text{s} \le t \le 4\text{s} \\ 0 & \text{for} \quad t > 4\text{s} \end{cases}, \end{array} \tag{6.116}$$

Table 6.1 provides a summary of the data used in the simulations. During the loading phase (t ≤ 4s) the time step size for all simulations is ∆t = 0.05, such that all systems start from the same energy-and entropy level directly after vanishing external loads. No Dirichlet boundary conditions are applied. Since in the initial loading phase the external forces are equilibrated, the total linear momentum of the block is a conserved quantity. In addition to that, after the loading phase (t ≥ 4s) no external torque is acting on the block.


Table 6.1.: L-shaped block: Data used in the simulations

Consequently, the total angular momentum is conserved as well. All of the integrators under consideration are capable to exactly conserve both momentum maps (up to numerical round-o), independent of the chosen time step. This can be observed from Fig. 6.5, where representative numerical results of the (EM)<sup>u</sup> integrator are shown. After the loading phase the total energy must be a conserved quantity. As expected, the (EM)<sup>u</sup> scheme does exactly reproduce this conservation law (up to numerical round-o), see Fig. 6.6. In contrary, the schemes (M)<sup>θ</sup> and (ME)<sup>η</sup> are not capable of conserving the total energy for larger time step sizes. Depending on the time step size, both schemes tend to increase the energy which can be observed from Fig. 6.7. Typically, such energy blow-ups eventually lead to a failure of the iterative (Newton-Raphson) solution procedure. In the diagrams the break down of the simulation is indicated by vertical lines.

Figure 6.5.: L-shaped block: Algorithmic conservation of linear momentum (EM)<sup>u</sup> scheme (left), Total discrete angular momentum (EM)<sup>u</sup> scheme (right)

Figure 6.6.: L-shaped block: Total energy (EM)<sup>u</sup> scheme (left), Incremental change of total energy (EM)<sup>u</sup> scheme (right)

Figure 6.7.: L-shaped block: Total energy (M)<sup>θ</sup> scheme (left), Total energy (ME)<sup>η</sup> scheme (right)

Regardless of the capability of the (EM)<sup>u</sup> scheme to conserve the total energy, the simulation still breaks down at about the same point in time as the break down of (M)<sup>θ</sup> and (ME)<sup>η</sup> occurs. The numerical instability of the (EM)<sup>u</sup> is accompanied by a nonphysical decrease of the total entropy as can be observed from Fig. 6.8. In fact, the total entropy ought to be a non-decreasing function of time. In contrast to (EM)<sup>u</sup> and (M)θ, (ME)<sup>η</sup> is capable to correctly adhere to the second law of thermodynamics, independent of the time step (Fig. 6.9).

Figure 6.8.: L-shaped block: Total entropy (EM)<sup>u</sup> scheme (left), Total entropy (M)<sup>θ</sup> scheme (right)

Indeed, in each time step the total entropy does increase, as can be observed from Fig. 6.9.

Figure 6.9.: L-shaped block: total entropy (ME)<sup>η</sup> scheme (left), Incremental change of total entropy (ME)<sup>η</sup> scheme (right)

For sucient small time step sizes, the incremental change of total entropy is governed by

$$\mathcal{S}\_{\mathrm{n}+1}^{\mathrm{h}} - \mathcal{S}\_{\mathrm{n}}^{\mathrm{h}} = \underbrace{\Delta t \int\_{\mathcal{B}} \mathcal{D}\_{\mathrm{cond}\_{\mathrm{n}+\frac{1}{2}}}^{\mathrm{h}} \mathrm{d}V + \Delta t \underbrace{\int\_{\mathcal{B}} \mathcal{D}\_{\mathrm{inel}\_{\mathrm{n}+\frac{1}{2}}}^{\mathrm{h}} \mathrm{d}V}\_{\text{Contrivial due to } \mathcal{D}\_{\mathrm{cond}}} \mathrm{d}V}\_{\text{Contrivial due to } \mathcal{D}\_{\mathrm{ind}}} \mathrm{d}V},\tag{6.117}$$

for all mid-point based schemes. The two contributions to the discrete evolution of the incremental change of total entropy are visualized in Fig. 6.10 and Fig. 6.11.

Figure 6.10.: L-shaped block: Contributions to incremental change of entropy (EM)<sup>u</sup> scheme (left), Contributions to incremental change of entropy (M)<sup>θ</sup> scheme (right)

Figure 6.11.: L-shaped block: Contributions to incremental change of entropy (ME)<sup>η</sup> scheme

After the loading phase, the environment of the present example can be characterized as thermally perfect in the sense of [146]. Thus L dened in (6.111) plays the role of a Lyapunov function that has to decrease with time. However, the partially structurepreserving schemes (EM)u, (ME)<sup>η</sup> and (M)<sup>θ</sup> do not correctly reproduce this behavior, as can be seen from Figs. 6.12 and 6.13. That is, depending on the time step and the duration

of the simulation, all of the schemes inevitably exhibit numerical instabilities characterized by increasing values of L.

Figure 6.12.: L-shaped block: Lyapunov function (EM)<sup>u</sup> scheme (left), Lyapunov function (M)<sup>θ</sup> scheme (right)

Figure 6.13.: L-shaped block: Lyapunov function (ME)<sup>η</sup> scheme

Eventually, the motion of the L-shaped block is illustrated in Fig. 6.14 with snapshots at successive points in time. In addition to that, the distribution of the temperature over the block is shown.

Figure 6.14.: L-shaped block: Snapshots of the motion along with the temperature distribution over the block at t ∈ {0, 40, 80, 120, 160, 200, 240, 280}s, obtained with the (EM)<sup>u</sup> scheme and time step ∆t = 0.05s

## 6.5.3. Rotating disc

The second example deals with a rotating disc subjected to prescribed heat ow over part of the boundary surface (Fig. 6.15). The spatial discretization of the disc is based on 200 tri-linear nite elements leading to a total of 360 nodes.

Figure 6.15.: Rotating disc: Initial conguration and thermal boundary conditions (left), function f(t) for the prescribed heat ow over part of the boundary surface (right)

The initial velocity distribution over the disc results from a prescribed angular velocity <sup>ω</sup><sup>0</sup> <sup>∈</sup> <sup>R</sup> 3 and is given by

$$\mathbf{v}\_0(\mathbf{X}) = \omega\_0 \times \mathbf{X}, \qquad \omega\_0 = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \frac{1}{\mathbf{s}} \ .$$

The initial temperature of the disc is homogeneously distributed and equal to the reference temperature θ0. In an initial period of time, t ∈ [0, 4]s, heat ow is prescribed over one quarter of the lateral boundary surface (Fig. 6.15). In particular, the heat ow into the disc is described by

$$\overline{q} = -\frac{2000W}{\pi \text{m}^2} f(t), \qquad f(t) = \begin{cases} \sin(\frac{\pi}{4s}t) & \text{for} \quad 0 \le t \le 4s \text{ ,} \\ 0 & \text{for} \quad t > 4s \text{ .} \end{cases}$$

A plot of function f(t) can be found in Fig. 6.15. The rest of the boundary surface of the disc is assumed to be thermally insulated (q = 0). Note that the prescribed heat ow vanishes after t = 4s. For t > 4s, the complete boundary surface of the disc is assumed to be thermally insulated (q = 0 on ∂qB = ∂B). Then the environment of the disc is thermally perfect in the sense of [146]. A summary of the data used in the simulations of the rotating disc can be found in Table 6.2. During the loading phase (t ≤ 4s) the time step size for all simulations is ∆t = 0.04s, such that all systems start from the same energy-and entropy level directly after vanishing external loads.


Table 6.2.: Rotating disc: Data used in the simulations

Due to the fact that neither external loads act on the disc, nor displacement boundary conditions are imposed, the mechanical system at hand has translational and rotational symmetry. Consequently, the corresponding momentum maps are rst integrals of the mo-

tion. All integrators under consideration are capable to conserve the respective momentum map. Representative numerical results are shown in Fig. 6.16.

Figure 6.16.: Rotating disc: Algorithmic conservation of linear momentum (EM)<sup>η</sup> scheme (left), Total discrete angular momentum (EM)<sup>η</sup> scheme (right)

Since heat ow into the system is prescribed in the initial time period [0, 4]s, the total energy is expected to increase. For t > 4s, the system is closed and the total energy should stay constant. Again the (EM)<sup>u</sup> scheme is capable to correctly reproduce the rst law (Fig. 6.17).

Figure 6.17.: Rotating disc: Total energy (EM)<sup>u</sup> scheme (left), Incremental change of total energy (EM)<sup>u</sup> scheme (right)

However, despite the algorithmic energy conservation (for t > 4s), the (EM)<sup>u</sup> scheme is not devoid of numerical instabilities, depending on the time step. The corresponding point in time of the break down of the simulation is indicated with a vertical line in the diagrams. At about the same points in time, (ME)<sup>η</sup> and (M)<sup>θ</sup> break down as well (Fig. 6.18). For these schemes the break down is accompanied by a sudden increase of the total energy leading to the divergence of the Newton-Raphson iterations. For the considered duration of the simulation (t ∈ [0, 30]s), a time step of ∆t = 0.04s is small enough to retain numerical stability of the three partially structure-preserving schemes at hand.

Figure 6.18.: Rotating disc: Total energy (M)<sup>θ</sup> scheme (left), Total energy (ME)<sup>η</sup> scheme (right)

Due to the prescribed heat ow into the disc, the total entropy of the disc is expected to increase in the initial time period [0, 4]s. For t > 4s, the system is closed and the total entropy should be a non-decreasing function of time. The (EM)<sup>u</sup> scheme does not correctly reproduce the second law as can be seen from Fig. 6.19. Accordingly, the divergence of the iterative solution procedure is accompanied by a nonphysical decrease of the total entropy. The (M)<sup>θ</sup> closely adheres to the second law as can be observed from Fig. 6.19.

Figure 6.19.: Rotating disc: Total entropy (EM)<sup>u</sup> scheme (left), Total entropy (M)<sup>θ</sup> scheme (right)

As expected, the (ME)<sup>η</sup> scheme is capable to exactly satisfy the second law, independent of the time step. This can be observed from Fig. 6.20. In particular, Fig. 6.20 (right) conrms that the change per time step of the total entropy is always positive.

Figure 6.20.: Rotating disc: total entropy (ME)<sup>η</sup> scheme (left), Incremental change of total entropy (ME)<sup>η</sup> scheme (right)

In addition, the two contributions to the discrete evolution of the incremental change of total entropy are visualized in Fig. 6.21 and Fig. 6.22 for the mid-point based schemes. Most of the contribution is due to conduction of heat, only about 5.9% is due to inelastic deformations in the given example.

Figure 6.21.: Rotating disc: Contributions to incremental change of entropy (EM)<sup>u</sup> scheme (left), Contributions to incremental change of entropy (M)<sup>θ</sup> scheme (right)

Figure 6.22.: Rotating disc: Contributions to incremental change of entropy (ME)<sup>η</sup> scheme

To shed further light on the numerical stability of the present schemes, we consider the Lyapunov function L dened in (6.111). For t > 4s the system is closed and the function L should decrease with time. As expected, the partially structure-preserving schemes (EM)u, (ME)<sup>η</sup> and (M)<sup>θ</sup> can not guarantee to correctly reproduce this behavior, depending on the size of the time step and the duration of the simulation (Figs. 6.23 and 6.24). In particular, it can be seen that the smallest time step, ∆t = 0.04s, yields a stable numerical

simulation, at least in the considered time interval [0, 30]s. However, for larger time steps ∆t = 0.08s and ∆t = 0.1s, the numerical instability of each scheme becomes visible through the increase of L.

Figure 6.23.: Rotating disc: Lyapunov function (EM)<sup>u</sup> scheme (left), Lyapunov function (M)<sup>θ</sup> scheme

Figure 6.24.: Rotating disc: Lyapunov function (ME)<sup>η</sup> scheme

Eventually, the motion of the disc is illustrated in Fig. 6.25 with snapshots at successive points in time. In addition to that, the distribution of the temperature over the disc is shown.

Figure 6.25.: Rotating disc: Snapshots of the motion along with the temperature distribution over the block at t ∈ {0, 4, 8, 12, 16, 20, 24, 28}s, obtained with the (ME)<sup>η</sup> scheme and time step ∆t = 0.04s

# 7. Summary and outlook

## 7.1. Summary<sup>1</sup>

This thesis has addressed the formulation, analysis, implementation, and verication of second-order accurate structure-preserving schemes by utilizing the GENERIC structure.

First, we have analyzed in Chapter 3 energy and momentum conserving schemes in the context of molecular dynamics. Conserving schemes have been employed for more than four decades for the accurate and robust approximation of Hamiltonian problems in mechanics. However, in the eld of molecular dynamics, where time integrators are routinely employed and are key to their usefulness, the use of conserving schemes has barely been explored. These second-order implicit methods are an interesting alternative to other commonly used integrators, and we have proven that, in addition to exhibiting exact energy conservation, they are more robust than the midpoint rule, the canonical second-order implicit method. We focused on the design of energy-momentum (EM) schemes for molecular dynamics in the view of three issues that are characteristic and unique to these problems: (i) the numerical treatment of dynamics in periodic domains, (ii) the discretization of three-body potentials, and (iii) the study of interatomic functional potentials. None of these three topics had been previously studied in the context of conserving schemes, to the authors' knowledge. However, the three of them are key for their implementation and clearly have an impact on their performance. Some of the most practical results of this Chapter are new expressions for EM approximations in fairly general problems in molecular dynamics. These approximations account for the three key issues mentioned before and can be shown to preserve linear momentum and energy exactly, while exhibiting a remarkable robustness.

In addition, in Chapter 4, we developed a GENERIC-based variational formulation of the IBVP for large-strain thermo-elastodynamics. The weak form of the problem directly emanates from the GENERIC evolution equation for open systems. This formulation makes possible the free choice of the thermodynamic state variable among the three options: (i) the absolute temperature θ, (ii) the entropy density η, and (iii) the internal energy density u. The GENERIC-based weak form is particularly well suited for the design of structure-preserving numerical schemes. This observation already holds true for the discretization in time employing of the standard mid-point rule. Depending on the specic

<sup>1</sup> The summary is based on [1–4]

choice of the thermodynamic state variable, specic balance laws are correctly reproduced in the discrete setting, independent of the size of the time-step. In particular, by using the internal energy density as the thermodynamic state variable, the resulting scheme is capable of consistently reproducing the balance of energy and thus has been termed energy-momentum (or (EM)u) scheme. Similarly, choosing the entropy density as state variable leads to an entropy consistent method, termed momentum-entropy (or (ME)η) scheme. However, none of the three newly developed mid-point-based schemes does preserve all of the key balance laws under discretization. In particular, it was shown that numerical instabilities can occur if the time-step size is too large for a prescribed duration of the simulation. This leads to the conclusion that all major balance laws should be maintained in the discrete setting in order to enhance the numerical stability.

Consequently, novel fully structure-preserving numerical methods for nite-strain thermoelasticity with heat conduction were developed in Chapter 5, starting from the newly developed GENERIC-based weak form. The proposed EME schemes enable the free choice of the thermodynamic state in the same manner. Each choice of the thermodynamic variable τ ∈ {η, θ, u} leads to a corresponding (EME)<sup>τ</sup> scheme. We have shown that two modications of the mid-point-based approach are crucial for obtaining numerically stable EME schemes: (i) the introduction of specic projections in the wake of the space discretization and, (ii) the replacement of the standard derivatives of the internal energy density and the entropy density, respectively, by appropriate discrete derivatives. The in Chapter 2 introduced notion GENERIC-consistent space discretization is crucial with regard to the discretization in space. This notion asserts that a consistent space discretization should t into the GENERIC framework for discrete systems. In particular, a GENERIC consistent space discretization automatically inherits the balance laws for energy and entropy, respectively, from the underlying continuous formulation. We have seen that the discretization in space, relying on standard Lagrangian shape functions, necessitates specic projections to reach a GENERIC-consistent space discretization. Even though the structure-preserving properties of the EME schemes are independent of the specic choice for τ ∈ {η, θ, u}, the absolute temperature can be regarded as the preferred choice for the constitutive description. In addition to that, temperature Dirichlet boundary conditions can be applied in the standard manner by using the temperature-based formulation. It was shown that the EME schemes lead to a signicant improvement in the numerical stability when compared to mid-point type schemes.

Finally, in Chapter 6 the GENERIC-based approach was extended to more involved coupled thermomechanical problems, which also account for viscous (inelastic) deformations. Starting from a GENERIC-based formulation of large-strain thermo-viscoelasticity, we have developed alternative structure-preserving schemes based on the implicit mid-point rule. Depending on the choice of the thermodynamic variable, the plain mid-point rule yields partially structure-preserving schemes just as in the case for thermoelasticity. Also, in analogy to this case as well, these schemes cannot prevent numerical instabilities, as was shown in the numerical examples.

Upon summarizing the ndings of the newly proposed GENERIC-based schemes, the following features are eminent:


## 7.2. Outlook

Based on the work contained within this thesis, some of the following lines of research are currently in progress or seem worthy of investing:


irreversible transformations is not transferred to the mechanical eld.<sup>2</sup> A rst attempt in this direction can be found in [124] for a thermo-elastoplastic spring, but the transcription to innitesimal dimensional systems is, to the author's knowledge, yet to be made.


<sup>2</sup> Solely energy-consistent schemes correctly balance the amount of energy contained within the system but are generally not concerned about the direction of energy transfer and thus do not generally satisfy the second law of thermodyamics (e.g., see results of (EM)<sup>u</sup> methods in Sections 4.5 and 6.5).

# A. Appendix to Chapter 4<sup>1</sup>

## A.1. Linear thermoelasticity

We verify that the Helmholtz free energy density (4.36) yields an expression for the linearized stress tensor which is consistent with the theory of linear thermoelasticity. According to Section 4.2.4, the temperature-based expression for the rst Piola-Kirchho stress tensor reads

$$\overline{\mathbf{P}} = \partial\_{\nabla} \varphi \,\overline{u} - \theta \partial\_{\nabla} \varphi \,\overline{\eta}$$

Taking into account the functions η(∇ϕ, θ), and u(∇ϕ, θ) summarized in Section 4.2.3, the rst Piola-Kirchho stress tensor can be recast in the form

$$\overline{\mathbf{P}}(\mathbf{F}, \theta) = D\psi\_1(\mathbf{F}) - (\theta - \theta\_0)\psi\_3'(J)\text{cof}(\mathbf{F}) \tag{A.1}$$

It can be easily veried that the reference conguration is stress-free in the sense that P(I, θ0) = 0. To perform the linearization of the stress tensor (A.1) about the stress-free reference conguration, consider one-parameter families of the deformation gradient and the temperature of the form

$$\begin{aligned} \mathbf{F}^{\varepsilon} &= \mathbf{I} + \varepsilon \nabla \mathbf{u} \\ \theta^{\varepsilon} &= \theta\_0 + \varepsilon (\theta - \theta\_0) \end{aligned}$$

Here, the displacement eld is denoted by u. Note that P(F 0 , θ<sup>0</sup> ) = 0. The linearized stress tensor can now be calculated via

$$\boldsymbol{\sigma} := \left. \frac{d}{d\varepsilon} \overline{\mathbf{P}} (\mathbf{F}^{\varepsilon}, \theta^{\varepsilon}) \right|\_{\varepsilon=0}$$

Using (A.1) along with the relationships

$$\begin{aligned} \frac{d}{d\varepsilon} \det(\mathbf{F}^{\varepsilon})\bigg|\_{\varepsilon=0} &= \text{tr}(\nabla \mathbf{u})\\ \frac{d}{d\varepsilon} \text{cof}(\mathbf{F}^{\varepsilon})\bigg|\_{\varepsilon=0} &= \text{tr}(\nabla \mathbf{u})\mathbf{I} - \nabla \mathbf{u}^{\mathrm{T}} \end{aligned}$$

a straightforward calculation yields

$$\boldsymbol{\sigma} = \mu \left( \nabla \mathbf{u} + \nabla \mathbf{u}^{\mathrm{T}} \right) + \lambda \text{tr}(\nabla \mathbf{u}) \mathbf{I} - \beta (\theta - \theta\_0)(3\lambda + 2\mu) \mathbf{I}$$

<sup>1</sup> This Appendix is based on [1]

Introducing the innitesimal strain tensor ε = 1 2 (∇u+ ∇u <sup>T</sup>), the linearized stress tensor can be rewritten as

$$\boldsymbol{\sigma} = 2\mu \boldsymbol{\varepsilon} + \lambda \text{tr}(\boldsymbol{\varepsilon}) \mathbf{I} - 3K\beta (\boldsymbol{\theta} - \theta\_0) \mathbf{I} \tag{A.2}$$

Expression (A.2) for the stress tensor coincides with the generalized Duhamel-Neumann form of Hookes's law for isotropic behavior in the framework of linear thermoelasticity (cf. [159, Sec. 6.2]). Accordingly, λ and µ can be identied as the Lamé constants, K = λ+ 2 3 µ is the bulk modulus, and β is the linear coecient of thermal expansion.

## A.2. Lyapunov function

Following [146] we consider the functional

$$
\mathcal{L}' = \mathcal{E}' - \theta\_0 \mathcal{S}'\tag{A.3}
$$

where θ<sup>0</sup> is the reference temperature. Note that the total energy, E 0 , and the total entropy, S 0 , have been dened in (4.39) and (4.41), respectively. The time derivative of (A.3) is given by

$$\frac{\mathrm{d}\mathcal{L}'}{\mathrm{d}t} = \frac{\mathrm{d}\mathcal{E}'}{\mathrm{d}t} - \theta\_0 \frac{\mathrm{d}\mathcal{S}'}{\mathrm{d}t}$$

Taking into account the results of Section 4.3.2, the last equation can be recast in the form

$$\begin{split} \frac{d\mathcal{L}'}{dt} &= -\left( \left[ \mathcal{E}', \mathcal{E}' \right]\_{\text{bound}} + \left[ \mathcal{E}', \mathcal{S}' \right]\_{\text{bound}} \right) - \theta\_0 \left( \left[ \mathcal{S}', \mathcal{S}' \right] - \left[ \mathcal{S}', \mathcal{S}' \right]\_{\text{bound}} \right) \\ &= \int\_{\partial \mathcal{B}} \rho^{-1} \mathbf{p} \cdot \mathbf{P} \mathbf{N} \, \mathrm{d}A - \int\_{\partial \mathcal{B}} \mathbf{N} \cdot \mathbf{Q} \, \mathrm{d}A - \theta\_0 \left( \left[ \mathcal{S}', \mathcal{S}' \right] - \int\_{\partial \mathcal{B}} \frac{1}{\theta} \mathbf{N} \cdot \mathbf{Q} \, \mathrm{d}A \right) \\ &= \mathcal{P}\_{\text{ext}} - \theta\_0 \left( \left[ \mathcal{S}', \mathcal{S}' \right] + \mathcal{G}\_e \right) \end{split}$$

Here, Pext = R ∂B ρ <sup>−</sup><sup>1</sup><sup>p</sup> · <sup>t</sup> <sup>d</sup><sup>A</sup> is the power expended on <sup>B</sup> by external Piola tractions t = PN acting on ∂B. Moreover,

$$\mathcal{G}\_e = \int\_{\partial \mathcal{B}} \frac{\theta - \theta\_0}{\theta\_0 \theta} \mathbf{N} \cdot \mathbf{Q} \,\mathrm{d}A \tag{A.4}$$

is the entropy produced at the interface between the body and the environment. The environment is called thermally perfect in the sense of [146], if G<sup>e</sup> = 0. In the numerical examples we consider the case that G<sup>e</sup> = 0 and Pext = 0, so that

$$\frac{\mathrm{d}\mathcal{L}'}{\mathrm{d}t} = -\theta\_0 \left[ \mathcal{S}', \mathcal{S}' \right] \le 0$$

Then L 0 is non-increasing over time and thus a natural Lyapunov function.

## A.3. Convergence criteria

To terminate the iterative solution procedure we use the norm of the residual vector as convergence criterion for both the (ME)<sup>η</sup> scheme and the (M)<sup>θ</sup> scheme. Since the (EM)<sup>u</sup> scheme is capable to satisfy the balance of energy, we dene an alternative termination criterion based on the discrete balance of energy:

$$|(\mathcal{E}\_{\mathrm{n}+1}^{\mathrm{h}} - \mathcal{E}\_{\mathrm{n}}^{\mathrm{h}}) - \Delta t \left(\mathcal{P}\_{\mathrm{n}+\frac{1}{2}}^{\mathrm{h},\mathrm{ext}} + \mathcal{Q}\_{\mathrm{n}+\frac{1}{2}}^{\mathrm{h},\mathrm{int}}\right)| \leq \varepsilon \tag{A.5}$$

Here, ε is the prescribed tolerance, E h n is the discrete version of the total energy (4.39) at time tn, P h,ext = R ∂B ρ <sup>−</sup><sup>1</sup>p h n+ <sup>1</sup> 2 · t h n+ <sup>1</sup> 2 dA is the power expended on the body by external nodal forces due to external tractions, and <sup>Q</sup>h,int <sup>=</sup> <sup>−</sup> R ∂B q h n+ <sup>1</sup> 2 dA is the heat ux across the boundary.

# B. Appendix to Chapter 5<sup>1</sup>

## B.1. Notes on the implementation

We provide a short outline of the implementation of the present EME schemes. To this end, the residual vector associated with node a can be written in the form

$$\mathbf{R}^{a^{T}} = \left(\mathbf{R}^{a^{T}}\_{\varphi}, \quad \mathbf{R}^{a^{T}}\_{\mathbf{p}}, \quad R^{a}\_{\tau}, \quad R^{a}\_{u}, \quad R^{a}\_{\eta}\right) \tag{\text{B.1}}$$

The nodal contributions to (B.1) emanate from weak form (5.65) by inserting the nite element approximations for the test functions leading to the expressions

R a <sup>ϕ</sup> = Z B N a ϕ<sup>h</sup> n+1 <sup>−</sup> <sup>ϕ</sup><sup>h</sup> n ∆t − v h n+ <sup>1</sup> 2 dV R a <sup>p</sup> = Z B N a ρ v h n+1 − v h n ∆t − b + F h n+ <sup>1</sup> 2 S h algo∇N a dV − Z ∂<sup>σ</sup> B N a t dA R a <sup>τ</sup> = Z B N a τ h n+1 − τ h n ∆t + ∇v h n+ <sup>1</sup> 2 : 2 Πh(D<sup>τ</sup> η h) F h n+ <sup>1</sup> 2 DCη h <sup>d</sup><sup>V</sup> − Z B ∇ N<sup>a</sup> Πh(Dτu h) · Q h algo dV + Z ∂qB N<sup>a</sup> Πh(Dτu h) q dA R a <sup>u</sup> <sup>=</sup>Hab(Dτu)<sup>b</sup> <sup>−</sup> Z B N <sup>a</sup>Dτu <sup>h</sup> dV R a <sup>η</sup> <sup>=</sup>Hab(D<sup>τ</sup> <sup>η</sup>)<sup>b</sup> <sup>−</sup> Z B N <sup>a</sup>D<sup>τ</sup> η <sup>h</sup> dV (B.2)

where, according to (5.66),

$$\begin{split} \mathbf{S}\_{\text{algo}}^{\text{h}} &= 2 \left( \mathbf{D}\_{\mathbf{C}} u^{\text{h}} - \Theta\_{\text{algo}}^{\text{h}} \mathbf{D}\_{\mathbf{C}} \eta^{\text{h}} \right) \\ \mathbf{Q}\_{\text{algo}}^{\text{h}} &= (\Theta\_{\text{algo}}^{\text{h}})^{2} \mathbf{K}\_{\text{algo}}^{\text{h}} \nabla \left( \frac{1}{\Theta\_{\text{algo}}^{\text{h}}} \right) = -\mathbf{K}\_{\text{algo}}^{\text{h}} \nabla \Theta\_{\text{algo}}^{\text{h}} \\ \Theta\_{\text{algo}}^{\text{h}} &= \frac{\Pi\_{h} (\mathbf{D}\_{\tau} u^{\text{h}})}{\Pi\_{h} (\mathbf{D}\_{\tau} \eta^{\text{h}})} = \frac{N^{a} (\mathbf{D}\_{\tau} u)\_{a}}{N^{b} (\mathbf{D}\_{\tau} \eta)\_{b}} \end{split}$$

<sup>1</sup> This Appendix is based on [2]

Note that eventually (B.2) leads to a nonlinear algebraic system of equations, R <sup>a</sup> = 0, a = 1, . . . , N, for the determination of the nodal state variables qan+1 , van+1 , and τan+1 along with the nodal quantities (Dτu)a, (D<sup>τ</sup> η)<sup>a</sup> related to the projections Πh(Dτu h ) and Πh(D<sup>τ</sup> η h ), cf. (5.72). As usual, we apply Newton's method for that purpose. The consistent linearization of R a can be performed in a straightforward way. In this connection, the projection formulas Πh(Dτu h ) and Πh(D<sup>τ</sup> η h ) do not pose any additional diculties. For example, the linearization of the algorithmic temperature assumes the form

$$
\Delta \Theta\_{\text{algo}}^{\text{h}} = \Theta\_{\text{algo}}^{\text{h}} N^a \left( \frac{\Delta (\mathsf{D}\_{\tau} u)\_a}{\Pi\_h (\mathsf{D}\_{\tau} u^{\text{h}})} - \frac{\Delta (\mathsf{D}\_{\tau} \eta)\_a}{\Pi\_h (\mathsf{D}\_{\tau} \eta^{\text{h}})} \right),
$$

Moreover, the material gradient of the algorithmic temperature is given by

$$\nabla \Theta^{\rm h}\_{\rm algo} = \Theta^{\rm h}\_{\rm algo} \nabla N^a \left( \frac{(\mathsf{D}\_{\tau} u)\_a}{\Pi\_h (\mathsf{D}\_{\tau} u^{\rm h})} - \frac{(\mathsf{D}\_{\tau} \eta)\_a}{\Pi\_h (\mathsf{D}\_{\tau} \eta^{\rm h})} \right)$$

leading to the corresponding linearized form

$$\begin{split} \Delta(\nabla\Theta\_{\mathrm{algo}}^{\mathrm{h}}) &= \Theta\_{\mathrm{algo}}^{\mathrm{h}}\nabla N^{a} \left(\delta\_{a}^{b} - \frac{(\mathsf{D}\_{\tau}\eta)\_{a}}{\Pi\_{h}(\mathsf{D}\_{\tau}\eta^{\mathrm{h}})}N^{b}\right) \frac{\Delta(\mathsf{D}\_{\tau}u)\_{b}}{\Pi\_{h}(\mathsf{D}\_{\tau}u^{\mathrm{h}})} \\ &- \Theta\_{\mathrm{algo}}^{\mathrm{h}}\nabla N^{a} \left(\delta\_{a}^{b} - 2\frac{(\mathsf{D}\_{\tau}\eta)\_{a}}{\Pi\_{h}(\mathsf{D}\_{\tau}\eta^{\mathrm{h}})}N^{b} + \frac{(\mathsf{D}\_{\tau}u)\_{a}}{\Pi\_{h}(\mathsf{D}\_{\tau}u^{\mathrm{h}})}N^{b}\right) \frac{\Delta(\mathsf{D}\_{\tau}\eta)\_{b}}{\Pi\_{h}(\mathsf{D}\_{\tau}\eta^{\mathrm{h}})} \end{split}$$

where δ b a is the Kronecker delta. Concerning the termination criterion for the Newton iterations we apply the nonstandard form

$$\left| \mathcal{E}\_{\mathbf{n}+1}^{\mathbf{h}} - \mathcal{E}\_{\mathbf{n}}^{\mathbf{h}} + \Delta t \left( \int\_{\partial\_{\mathbf{q}}\mathcal{B}} \overline{q}\_{\mathbf{n}+\frac{1}{2}} \, \mathrm{d}A - \int\_{\partial\_{\sigma}\mathcal{B}} \mathbf{v}\_{\mathbf{n}+\frac{1}{2}}^{\mathbf{h}} \cdot \overline{\mathbf{t}}\_{\mathbf{n}+\frac{1}{2}} \, \mathrm{d}A \right) \right| \leq \varepsilon \tag{\text{B.3}}$$

which is made possible due to the energy consistency of the present (EME)<sup>τ</sup> schemes. In (B.3), ε denotes the prescribed numerical tolerance for the Newton iterations. The lefthand side of (B.3) can be directly linked to the discrete balance of energy (5.80). Note that here, the standard contribution of the external tractions has been taken into account.

# C. Appendix to Chapter 6<sup>1</sup>

## C.1. Rotational symmetry

Due to invariance under superposed rigid rotations the present problem satises symmetry properties (6.93). To see this, we consider a rotation tensor Q <sup>ε</sup> = exp(εbξ), expressed through the Rodrigues formula [24]. Here, <sup>ε</sup> is a scalar parameter and <sup>b</sup><sup>ξ</sup> is a skewsymmetric tensor with associated vector <sup>ξ</sup> <sup>∈</sup> <sup>R</sup> 3 , such that <sup>b</sup>ξ<sup>a</sup> <sup>=</sup> <sup>ξ</sup> <sup>×</sup> <sup>a</sup> for all <sup>a</sup> <sup>∈</sup> <sup>R</sup> 3 . Now, a superposed rigid motion gives rise to rotated nodal position vectors

$$\mathbf{q}\_a^\varepsilon = \mathbf{Q}^\varepsilon \mathbf{q}\_a \,. \tag{C.1}$$

Note that q 0 <sup>a</sup> = q<sup>a</sup> and <sup>d</sup> dεQ ε <sup>ε</sup>=0 <sup>=</sup> <sup>b</sup>ξ. Nodal pattern (C.1) of the rigidly rotated discrete system gives rise to the corresponding expression for the discrete deformation gradient (cf. (6.53)1)

$$\mathbf{F}^{\mathbf{h},\varepsilon} = \mathbf{q}\_a^{\varepsilon} \otimes \nabla N^a = \mathbf{Q}^{\varepsilon} \mathbf{F}^{\mathbf{h}}.\tag{\text{C.2}}$$

Furthermore, the corresponding right Cauchy-Green deformation tensor follows from (6.3) and is given by

$$\mathbf{C}^{\mathbf{h},\varepsilon} = (\mathbf{F}^{\mathbf{h},\varepsilon^{\mathsf{T}}})\mathbf{F}^{\mathbf{h},\varepsilon} = \mathbf{C}^{\mathbf{h}}\,,\tag{\text{C.3}}$$

where property (Q ε ) <sup>T</sup> = (Q ε ) <sup>−</sup><sup>1</sup> of the rotation tensor has been taken into account. A frame-indierent formulation implies that the internal energy density takes the form u(F, τ, C −1 p ) = u(C, τ, C −1 p ). Thus, taking the derivative of u(F h,ε , τ, C −1 p ) = u(C h , τ, C −1 p ) with respect to parameter ε and subsequently setting ε = 0 yields

$$\begin{split} 0 &= \left. \frac{d}{d\varepsilon} \right|\_{\varepsilon=0} u\left(\mathbf{F}\_g^{\mathbf{h},\varepsilon}, \tau\_g^{\mathbf{h}}, (\mathbf{C}\_p^{-1})\_g \right) \\ &= \partial\_{\mathbf{F}} u\_g : \left. \frac{d}{d\varepsilon} \right|\_{\varepsilon=0} \mathbf{q}\_a^{\varepsilon} \otimes \nabla N\_g^a \\ &= \partial\_{\mathbf{F}} u\_g : \left( \widehat{\xi} \mathbf{q}\_a \otimes \nabla N\_g^a \right) \\ &= \left( \xi \times \mathbf{q}\_a \right) \cdot \partial\_{\mathbf{F}} u\_g \nabla N\_g^a \\ &= \xi \cdot \left( \mathbf{q}\_a \times \partial\_{\mathbf{F}} u\_g \nabla N\_g^a \right) \,, \end{split} \tag{C.4}$$

<sup>1</sup> This Appendix is based on [4]

where <sup>u</sup><sup>g</sup> is given by (6.52)1. Due to the arbitrariness of <sup>ξ</sup> <sup>∈</sup> <sup>R</sup> 3 (and vanishing body forces), the last equation coincides with symmetry condition (6.93)1, that is

$$\mathbf{q}\_a \times \partial\_{\mathbf{q}\_a} \mathcal{E} = \mathbf{0} \ .$$

Note that this condition holds as well for the specic choice τ = u, due to (6.97)1.

Similarly, symmetry condition (6.93)<sup>2</sup> results from the frame-indierence of the entropy density function η(F, τ, C −1 p ) = η(C, τ, C −1 p ), or from (6.100)<sup>1</sup> in the case τ = η.

# Declaration of Authorship

and

**1**

**INTRODUC**

**TION**

shed

thermo-mechanically

light

energy-momentum-entropy

on

coupled

schemes,

problems

of

structure-preserving

the

newly

numerical

developed

methods,

#### Citation:

Schiebl, M. & Betsch, P. Structure-preserving space-time discretization of large-strain thermoviscoelasticity in the framework of GENERIC. International Journal for Numerical Methods in Engineering, 1–41 (2021)

#### Declaration:

Mark Schiebl (MS) worked on the theory together with Peter Betsch (PB). MS was responsible for the numerical implementation. Moreover, MS performed the numerical verication of the newly developed formalism. Furthermore, MS and PB co-worked on the manuscript. Finally, the article was reviewed by all authors (MS, PB).


#### Citation:

Schiebl, M. & Romero, I. Energy-momentum conserving integration schemes for molecular dynamics. Computational Mechanics 67, 915–935 (2021)

#### Declaration:

Mark Schiebl (MS) worked on the theory together with Ignacio Romero (IR). MS was responsible for the numerical implementation. Moreover, MS performed the numerical verication of the newly developed formalism. Furthermore, MS and IR co-worked on the manuscript. Finally, the article was reviewed by all authors (MS, IR).


## Citation:

Betsch, P. & Schiebl, M. GENERIC-based formulation and discretization of initial boundary value problems for nite strain thermoelasticity. Computational Mechanics 65, 503–531 (2020)

#### Declaration:

Mark Schiebl (MS) worked on the theory together with Peter Betsch (PB). MS was responsible for the numerical implementation. Moreover, MS performed the numerical verication of the newly developed formalism. Furthermore, MS and PB co-worked on the manuscript. Finally, the article was reviewed by all authors (MS, PB).

#### Citation:

Betsch, P. & Schiebl, M. Energy-momentum-entropy consistent numerical methods for large-strain thermoelasticity relying on the GENERIC formalism. International Journal for Numerical Methods in Engineering 119, 1216–1244 (2019)

#### Declaration:

Mark Schiebl (MS) worked on the theory together with Peter Betsch (PB). MS was responsible for the numerical implementation. Moreover, MS performed the numerical verication of the newly developed formalism. Furthermore, MS and PB co-worked on the manuscript. Finally, the article was reviewed by all authors (MS, PB).

#### Received: 18 September 2017 Revised: 23 February 2018 Accepted: 4 April 2018 DOI: 10.1002/nme.5816 **RESEARCH ARTICLE An energy momentum consistent integration scheme using apolyconvexity-basedframeworkfornonlinear thermo-elastodynamics** M. Franke A. Janz M. Schiebl P. Betsch Institute of Mechanics, Karlsruhe Institute of Technology, Karlsruhe, Germany **Correspondence** M. Franke, Institute of Mechanics, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany. Email: marlon.franke@kit.edu **Summary** The present contribution provides a new approach to the design of energy momentum consistent integration schemes in the field of nonlinear thermo-elastodynamics. The method is inspired by the structure of polyconvex energy density functions and benefits from a tensor cross product that greatly simplifies the algebra. Furthermore, a temperature-based weak form is used, which facilitates the design of a structure-preserving time-stepping scheme for coupled thermoelastic problems. This approach is motivated by the general equation for nonequilibrium reversible-irreversible coupling (GENERIC) framework for open systems. In contrast to complex projection-based discrete derivatives, a new form of an algorithmic stress formula is proposed. The spatial discretization relies on finite element interpolations for the displacements and the temperature. The superior performance of the proposed formulation is shown within representative quasi-static and fully transient numerical examples. **KEYWORDS** finite element method, nonlinear thermo-elastodynamics, polyconvexity-based framework, structure-preserving discretization, tensor cross product **1 INTRODUCTION** The present contribution aims for the consistent discretization of nonlinear thermo-elastodynamics. The emphasis of this paper is on both theoretical and numerical aspects. In recent decades, thermomechanical constitutive models have been addressed in numerous works (see, eg, the works of Miehe,1 Holzapfel and Simo,2 and Reese and Govindjee3 as well as textbooks by Holzapfel4 and Gonzalez and Stuart5 ). Classically, the hyperelastic Helmholtz free-energy density function depends only on the deformation gradient and the temperature. Furthermore, the weak form is deduced from its strong form. Dependent on the chosen material model, eg, for a Mooney-Rivlin model, the consistent linearization may lead to cumbersome expressions. In contrast to the classic approach, the present work is inspired by the concept of polyconvexity (see, eg, the works of Ball6 and Ciarlet7 ), where the Helmholtz free energy is a convex function of the deformation gradient, its cofactor, and its determinant and is concave with respect to absolute temperature. In addition to the polyconvexity-based framework, the present work relies on the cross product between second-order tensors, as introduced in the work of de Boer8 (see also Appendix B 4.9.3 in another work of de Boer9 ). This tensor cross product has been used in the context of large-strain *Int J Numer Methods Eng*. 2018;**115**:549–577. wileyonlinelibrary.com/journal/nme Copyright © 2018 John Wiley & Sons, Ltd. **549**

#### Citation:

Franke, M., Janz, A., Schiebl, M. & Betsch, P. An energy momentum consistent integration scheme using a polyconvexity-based framework for nonlinear thermoelastodynamics. International Journal for Numerical Methods in Engineering 115, 549–577 (2018)

#### Declaration:

Mark Schiebl (MS) worked on the theory together with Marlon Franke (MF) and Alexander Janz (AJ). MS, MF and AJ were responsible for the numerical implementation. Moreover, MS, MF and AJ performed the numerical verication of the newly developed formalism. Furthermore, MS, MF and AJ co-worked on the manuscript supported by Peter Betsch (PB). Finally, the article was reviewed by all authors (MS, MF, AJ, PB).

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## **Schriftenreihe des Instituts für Mechanik ISSN 2363-4936**

Hrsg. von: Prof. Dr.-Ing. habil. Peter Betsch, Prof. Dr.-Ing. habil. Thomas Seelig


# Band 7

Schriftenreihe des Instituts für Mechanik Karlsruher Institut für Technologie (KIT)

HERAUSGEBER Prof. Dr.-Ing. habil. Peter Betsch Prof. Dr.-Ing. habil. Thomas Seelig

ISSN 2363-4936 ISBN 978-3-7315-1117-5 MARK GEORG SCHIEBL

EME consistent space-time discretization of non-isothermal mechanical systems