RKDG method solution for hyperbolic hyperelastic model

Abstract

The natural for solid dynamics applications mathematical model has a form of the first-order hyperbolic system of conservation laws in the Eulerian reference frame. This model is proposed by Godunov and Romenski [1] and has a number of characteristic features from more traditional second-order formulations employed in elasticity. The most important is the presence of a complex structure of contact, shock and rarefaction waves and their interactions. Numerical solution of such problems is a difficult task due to a complex wave structure of the solution. Recently a number of efficient approaches were presented to handle a problem. Most of them are based on linearized HLL type solvers and high order WENO reconstruction techniques to obtain accurate fronts resolution in order to resolve complex solution structure. They also use Eulerian schemes based on solving Riemann problems for solid materials. In this paper we apply high order RKDG method as an alternative to WENO schemes to solve the problem. Up to 5-th order reconstruction is considered with 3-rd order timesteppnig scheme. The Riemann-free-solver approach is used with simple flux reconstruction. As a trial problem one dimensional hyperelastic formulation is presented similar to [1]. The developed computer code utilizes automatic differentiation (AD) approach which allows for an easy switch of underlying EOS and strain measure. The presented results cover discussion of the problem statement, corresponding RKDG approximations and simulation results that demonstrate possibilities of RKDG approach applied for the hyperbolic hyperelastic formulations in the Eulerian reference frame.