Thermodynamically consistent time‐stepping algorithms for non‐linear thermomechanical systems

Abstract

SUMMARYWe present the basic theory for developing novel monolithic and staggered time-stepping algorithms for general non-linear, coupled, thermomechanical problems. The proposed methods are thermodynamically consistent in the sense that their solutions rigorously comply with the two laws of thermodynamics: for isolated systems they preserve the total energy and the entropy never decreases. Furthermore, if the governing equations of the problem have symmetries, the proposed integrators preserve them too. The formulation of such methods is based on two ideas: expressing the evolution equation in the socalled General Equations for Non-Equilibrium Reversible Irreversible Coupling format and enforcing from their inception certain directionality and degeneracy conditions on the discrete vector fields. The new methods can be considered as an extension of the energy-momentum integration algorithms to coupled thermomechanical problems, to which they reduce in the purely Hamiltonian case. In the article, the new ideas are applied to a simple coupled problem: a double thermoelastic pendulum with symmetry. Numerical simulations verify the qualitative features of the proposed methods and illustrate their excellent numerical stability, which stems precisely from their ability to preserve the structure of the evolution equations they discretize.

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

“…The format of the free energy function is taken from (Romero, 2009) and accounts for nonlinear (logarithmic) deformations, a thermomechanical Gough-Joule type coupling and dilatation in the first, second and third terms respectively. The term c\ is assumed to have a linear dependence with the temperature, given by C\{ff) = c w -Cu{6 -9 r ), with c w ,Cn > 0.…”

“…In particular, several recent works have tried to broaden the ideas of the EnergyMomentum methods to specific dissipative methods such as frictional contact (Ortiz et al, 2000), plasticity (Meng and Laursen, 2002), viscoelasticity (Grofi and Betsch, 2010), etc. To the authors' knowledge, the first attempt to systematically generalize Energy-Momentum to dissipative problems has been presented in (Romero, 2009(Romero, , 2010b. In these works, a general methodology was presented that allows the formulation of time integration methods that strictly preserve the two laws of thermodynamics, as well as the symmetries, for general thermomechanical problems.…”

Computational Phase‐Field Modeling

“…The operator DW is usually referred to as discrete variational derivative. The use of discrete variational derivatives is ubiquitous in the design of exact energy preserving schemes and symplectic methods; see e.g., (Furihata and Matsuo, 2011;Simo et al, 1992;Romero, 2009). …”

Variational Integrators for Thermo‐Viscoelastic Discrete Systems

“…The thermoelastic double pendulum is a minimalistic model, commonly used [5,6] for testing numerical time integrators. For brevity only the enhancements of the previous stage [1] are detailed here.…”