Space-time adaptive ADER discontinuous Galerkin schemes for nonlinear hyperelasticity with material failure

Abstract

We are concerned with the numerical solution of a unified first order hyperbolic formulation of continuum mechanics that originates from the work of Godunov, Peshkov and Romenski (GPR model) and which is an extension of nonlinear hyperelasticity that is able to describe simultaneously nonlinear elasto-plastic solids at large strain, as well as viscous and ideal fluids. The proposed governing PDE system also contains the effect of heat conduction and can be shown to be symmetric and thermodynamically compatible. In this paper we extend the GPR model to the simulation of nonlinear dynamic rupture processes and material fatigue effects, by adding a new scalar variable to the governing PDE system. This extra parameter describes the material damage and is governed by an advection-reaction equation, where the stiff and highly nonlinear reaction mechanisms depend on the ratio of the local von Mises stress to the yield stress of the material. The stiff reaction mechanisms are integrated in time via an efficient exponential time integrator. Due to the multiple space-time scales, the model is solved on space-time adaptive Cartesian meshes using high order discontinuous Galerkin finite element schemes with a posteriori subcell finite volume limiting. A key feature of our new model is the use of a twofold diffuse interface approach that allows the cracks to form anywhere and at any time, independently of the chosen computational grid, without requiring that the geometry of the rupture fault be known a priori. We furthermore make use of a scalar volume fraction function that indicates whether a given point is inside the solid or outside, allowing the description of solids of arbitrarily complex shape.