Continuum mechanics and thermodynamics in the Hamilton and the Godunov-type formulations

Abstract

SpoilerContinuum mechanics with dislocations, with the Cattaneo type heat conduction, with mass transfer, and with electromagnetic fields is put into the Hamiltonian form and into the form of the Godunov type system of the first order, symmetric hyperbolic partial differential equations (SHTC equations). The compatibility with thermodynamics of the time reversible part of the governing equations is mathematically expressed in the former formulation as degeneracy of the Hamiltonian structure and in the latter formulation as the existence of a companion conservation law. In both formulations the time irreversible part represents gradient dynamics. The Godunov type formulation brings the mathematical rigor (the well-posedness of the Cauchy initial value problem) and the possibility to discretize while keeping the physical content of the governing equations (the Godunov finite volume discretization).

GENERIC guide to the multiscale dynamics and thermodynamics

“…The discretized time evolution equations are again conservation laws but in discrete space elements and in discrete time. The limitation of the Godunov numerical method to local conservation laws is not severe since many time evolution equations (including for example many GENERIC equations arising in the context of complex fluids) can be cast, after appropriately extending them, into the form of the Godunov local conservation laws [66].…”

A new continuum model for general relativistic viscous heat-conducting media

“…those for a, b = 0, while the other 12 are the so-called involution constraints, e.g. see [27,4], that are pure spatial constraints, conserved along the particle trajectories.…”

“…In this Section, we give a variational formulation for the reversible part of the relativistic version of the SHTC heat conduction equations [8,4]. As it was shown in [4], SHTC equations can be viewed as a particular realization of the GENERIC formulation for non-equilibrium thermodynamics. It is therefore not surprising that the same heat conduction equation was also proposed by Öttinger [32,33,34] within the GENERIC approach.…”

“…The left hand-side in (40) represents the reversible part of the time evolution derived in Sec.2.1 and 2.3, while the newly added relaxation source terms on the right hand-side of (40) represent the irreversible part and can be viewed as gradients of the quadratic dissipation potential [4]. The total energy E is left unspecified so far.…”

“…We have presented a general relativistic formulation for viscous/elastoplastic heat-conducting continuous medium. Such a formulation is a generalization of the non-relativistic unified formulation for fluid and solid dynamics advanced recently in [28,16,44] and relies on the theory of Symmetric Hyperbolic Thermodynamically Compatible (SHTC) equations [4,5,7,8].…”