Numerical Schemes for a Unified First Order Hyperbolic System of Continuum Mechanics

Abstract

In the first part of this talk we present the unified first order hyperbolic formulation of Newtonian continuum mechanics proposed by Godunov, Peshkov and Romenski (GPR). The governing PDE system can be derived from a variational principle and belongs to the class of symmetric hyperbolic and thermodynamically compatible systems (SHTC), which have been studied for the first time by Godunov in 1961 and later in a series of papers by Godunov & Romenski. An important feature of the model is that the propagation speeds of all physical processes, including dissipative processes, are finite. The GPR model is a geometric approach to continuum mechanics that is able to describe the behavior of nonlinear elasto-plastic solids at large deformations, as well as viscous Newtonian and non-Newtonian fluids within one and the same governing PDE system. This is achieved via appropriate relaxation source terms in the evolution equations for the distortion field and the thermal impulse. It can be shown that the GPR model reduces to the compressible Navier-Stokes equations in the stiff relaxation limit, i.e. when the relaxation times tend to zero. The unified system is also able to describe material failure, such as crack generation and fatigue. In the second part of the talk a family of high order ADER discontinuous Galerkin finite element schemes with a posteriori subcell finite volume limiter is introduced and applied to the GPR model. Computational results for nonlinear elasto-plastic solids with material failure are shown, as well as results in the fluid limit. In the absence of source terms, the homogeneous part of the GPR model is endowed with involutions, namely the distortion field A and the thermal impulse J need to remain curl-free. In the third part of the talk we therefore present a new structure-preserving scheme that is able to preserve the curl-free property of both fields exactly also on the discrete level. This is achieved via the definition of appropriate and compatible discrete gradient and curl operators on a judiciously chosen staggered grid. Furthermore, the pressure terms are discretized implicitly, in order to capture the low Mach number limit of the equations properly, while all other terms are discretized explicitly. In this manner, the resulting pressure system is symmetric and positive definite and can be solved with efficient iterative solvers like the conjugate gradient method. Last but not least, the new staggered semi-implicit scheme is asymptotic-preserving and thus also able to reproduce the stiff relaxation limit of the governing PDE system properly, recovering an appropriate discretization of the compressible Navier-Stokes equations.