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

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

“…As for the Poisson operator, the general form of the dissipative operator can be obtained from a 'simple' M 0 , with the appropriate change of variables. We note, however, that the previous reasoning does not prove that every pair of operators M, L must be as in (20) and (24), but rather than these kind of expressions simplify the formulation of the individual blocks in the operators.…”

“…A few remarkable methods have been proposed in the past for single, specific problems (for example, [15][16][17][18]) but cannot be employed beyond the boundaries of the problem they were designed for.The only class of structure preserving methods for general (smooth) dissipative solids that have been proposed so far is, to the authors' knowledge, the energy-entropy-momentum (EEM) integrators initially proposed in [19] for finite dimensional thermomechanical problems and later extended to the infinite dimensional case [20,21]. In addition to thermoelastic problems, the EEM method has been applied to phase field modeling [22], discrete thermo-visco-plasticity [23], and thermo-visco-elasticity [24][25][26].Energy-entropy-momentum methods could be applied to a broad class of thermomechanical systems after realizing that many of the latter can be formulated as metriplectic models [27]. This mathematical and geometrical formalism generalizes the ideas of Hamiltonian problems and is able to encompass several dissipative phenomena of interest in thermomechanics.…”

“…The only class of structure preserving methods for general (smooth) dissipative solids that have been proposed so far is, to the authors' knowledge, the energy-entropy-momentum (EEM) integrators initially proposed in [19] for finite dimensional thermomechanical problems and later extended to the infinite dimensional case [20,21]. In addition to thermoelastic problems, the EEM method has been applied to phase field modeling [22], discrete thermo-visco-plasticity [23], and thermo-visco-elasticity [24][25][26].…”

Structure-Preserving Discretization in the Framework of a Discrete Model Problem for Large-Strain Thermo-Viscoelasticity

“…While GENERIC-based numerical methods for continuum formulations of thermo-mechanically coupled solids have been dealt with in [6][7][8][9][10][11], various related discrete model problems have been the subject of previous work. For example, structure-preserving integrators have been developed in the context of a thermoelastic double pendulum [4,12,13], a thermo-viscoelastic element [14,15], a thermo-viscoelastic double pendulum [8], and a thermo-elasto-plastic element [16].…”

A minimizing-movements approach to GENERIC systems

“…These schemes are often discrete-gradient methods, where gradients of functionals are specifically modified in order to fulfill a discrete chain rule and exactly replicate conservation [16,17,19,30]. In the thermomechanical context, structure-preserving discretizations either in terms of the absolute temperature [9,10,15,38] or of the internal energy or the entropy [5,6] have been obtained. The reader is referred to [28] for an approach to open systems.…”