Séminaire EDPs2: Walter Boscheri A geometrically and thermodynamically compatible finite volume scheme for continuum mechanics on unstructured polygonal meshes

par Walter Boscheri (LAMA, USMB)

vendredi 9 févr. 2024, 10:00 → 11:00 Europe/Paris

In the first part of this talk we will give an overview of our past and present research activity, highlighting the different fields of applied mathematics that have been considered so far. 

In the second part of the talk, I present a novel Finite Volume (FV) scheme on unstructured polygonal meshes that is provably compliant with the Second Law of Thermodynamics and the Geometric Conservation Law (GCL) at the same time. The governing

equations are provided by a subset of the class of symmetric and hyperbolic thermodynamically compatible (SHTC)
models introduced by Godunov in 1961. Specifically, our numerical method discretizes the equations for the conser-
vation of momentum, total energy, distortion tensor and thermal impulse vector, hence accounting in one single unified
mathematical formalism for a wide range of physical phenomena in continuum mechanics, spanning from ideal and
viscous fluids to hyperelastic solids. By means of two conservative corrections directly embedded in the definition of
the numerical fluxes, the new schemes are proven to satisfy two extra conservation laws, namely an entropy balance
law and a geometric equation that links the distortion tensor to the density evolution. As such, the classical mass
conservation equation can be discarded. Firstly, the GCL is derived at the continuous level, and subsequently it is
satisfied by introducing the new concepts of general potential and generalized Gibbs relation. The new potential is
nothing but the determinant of the distortion tensor, and the associated Gibbs relation is derived by introducing a set
of dual or thermodynamic variables such that the GCL is retrieved by dot multiplying the original system with the
new dual variables. Once compatibility of the GCL is ensured, thermodynamic compatibility is tackled in the same
manner, thus achieving the satisfaction of a local cell entropy inequality. The two corrections are orthogonal, meaning
that they can coexist simultaneously without interfering with each other. The compatibility of the new FV schemes
holds true at the semi-discrete level, and time integration of the governing PDE is carried out relying on Runge-Kutta
schemes. A large suite of test cases demonstrates the structure preserving properties of the schemes at the discrete
level as well.