Energy-Entropy-Momentum integration of discrete thermo-visco-elastic dynamics

Abstract

A novel time integration scheme is presented for the numerical solution of the dynamics of discrete systems consisting of point masses and thermovisco-elastic springs. Even considering fully coupled constitutive laws for the elements, the obtained solutions strictly preserve the two laws of thermodynamics and the symmetries of the continuum evolution equations. Moreover, the unconditional control over the energy and the entropy growth have the effect of stabilizing the numerical solution, allowing the use of larger time steps than those suitable for comparable implicit algorithms. Proofs for these claims are provided in the article as well as numerical examples that illustrate the performance of the method.

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

“…Eqs. (6)- (8) can be rewritten as a first order partial differential equations system by introducing the momenta p a 2 R d . Moreover, the temperature form of the system evolution equations is achieved by using (4) along with the chain rule and some algebra to give…”

“…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".…”

“…Particularly, discrete thermo-visco-elastodynamics provides a meaningful framework to develop EEM methods for smooth dissipative problems with internal variables which, in addition, could straightforwardly be extended to continuous finite strain deformations, see [14]. In this sense, the formulation in temperature variables becomes central to accommodate Dirichlet boundary conditions and, therefore, special attention will be paid to that formulation in comparison with the entropy approach [8].…”

“…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.…”