Energy-consistent time integration for nonlinear viscoelasticity

Abstract

This work is concerned with the numerical solution of the evolution equations of thermomechanical systems, in such a way that the scheme itself satisfies the laws of thermodynamics. Preserving structure integrators are widely developed for conservative (Hamiltonian) systems, being the most representative method the well-known energy-momentum due to Simó and Tarnow [1]. Recently, after the works of Romero I.[2], these ideas are being used in evolution system with irreversible processes, i.e non-conservative, such as those with internal dissipation mechanisms (viscoelasticity, plasticity, damage). In this context, the structure meant to be preserved is understood in the thermodynamical way, namely, the integrator must intrinsically satisfy the laws of thermodynamics as in conservative problems this one preserved the Hamiltonian.Within this framework, we present a novel integration method for the dynamics of an isothermal viscoelastic continuum body. This method exactly preserves the continuum laws of thermodynamics intrinsically, plus equations symmetries. The resulting solutions are physically accurate since they preserve fundamental physical properties of the model. What is more, they show an excellent performance with respect to the methods robustness and stability. Proof for these claims will be provided in the presentation as well as numerical examples that illustrate the method's performance.The viscoelastic behaviour is modeled according to Holzapfel and Simó [3] (regardless of temperature). Such model is based on large-strain deformations. Our approach is based on a double semidiscretisation, spatial (FE-based) and temporal, being on the latter one we do the main contribution.

Energy–entropy–momentum integration schemes for general discrete non‐smooth dissipative problems in thermomechanics

“…Discrete derivatives have been employed in the formulation of conservative implicit methods for classical and solid mechanics. See, among others, [2,9,16,19,22,[41][42][43][44][45][46][47][48]. Although we will not discuss this aspect in the current article, discrete derivatives are also responsible for the preservation of first integrals of the motion, if present in the equations.…”

An energy-consistent integration scheme for flexible multibody systems with dissipation

“…In this section we outline the main steps for proving that the discrete scheme (19) with 20is energy-consistent for a single viscoelastic body. 17 By using the directionality property of the discrete derivative operator (17), one may write the energy balance in any time interval I n as follows:…”

On Energy–Entropy–Momentum integration methods for discrete thermo-visco-elastodynamics

“…However, few works have hitherto addressed the formulation of EEM time-stepping methods involving internal variables. Some remarkable exceptions are the works [8,5] although they were not rigorously EEM methods since no use of the GENERIC formalism was involved to reveal their thermodynamic structure in order to formulate their structure-preserving discrete counterparts. However, they fully correspond to those that would have been achieved within the unified methodology based on GENERIC.…”

“…However, they fully correspond to those that would have been achieved within the unified methodology based on GENERIC. In particular, [8] demonstrates that the choice of the entropy facilitates the comprehension of the geometric structure of coupled thermomechanic dissipative problems whereas [5] shows that isothermal considerations simplify the formulation of structure-preserving methods, as it closely follows the guidelines provided by the Energy-Momentum method. Apart from these works, Krüger et al [19] also addressed the inclusion of internal variables but working with what the authors called ''the enhanced GENERIC formalism".…”

“…These methods have recently reached a considerable maturity after the works [8,5,18], enabling a wide variety of dissipative problems to be formulated within a unified procedure. As a result of the recently proposed temperature-based formulation [4], whose feasibility was a crucial issue since the beginning of the EEM methods, this procedure has currently adopted two different approaches based on the election of the thermodynamic variable: entropy or temperature.…”