Arkadi Berezovski : Thermodynamic interpretation of finite volume algorithms
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Europe/Budapest
Description
Numerical methods are problem-dependent. For quasi-static problems in
solid mechanics the best choice is the finite element method, while in
hydrodynamics finite differences or finite volumes are preferable.
Division of a body into a finite number of computational cells
suggests the description of all fields inside the cells as well as the
interaction between neighboring cells. It is desired that the
corresponding description should be thermodynamically consistent. If a
cell is considered as a thermodynamic system, its state should be
clearly defined. In the local equilibrium approximation, all the
notions of classical thermodynamics are valid for quantities averaged
over the cell. Averaging of wanted fields inside the cell leads to
discontinuity in their values at the boundaries between cells. The
difference between exact and approximate values of the fields is
represented by excess quantities. The values of excess quantities at
cell boundaries can be related to fluxes that describe the interaction
between neighboring cells. This provides the representation of
numerical schemes in terms of averaged and excess quantities.
Classical numerical schemes are normally thermodynamically consistent
at smooth solutions. This consistency fails, however, at moving
discontinuities (like cracks and martensitic phase-transition fronts)
because of the absence of appropriate jump conditions at the moving
discontinuities. The formulation of the jump conditions in terms of
excess quantities provides the thermodynamic consistency at moving
discontinuities in numerical simulations.
