Cell centered direct Arbitrary-Lagrangian-Eulerian ADER-WENO finite volume schemes for nonlinear hyperelasticity

This paper is concerned with the numerical solution of the unified first order hyperbolic formulation of continuum mechanics proposed by Peshkov & Romenski [Peshkov I, Romenski E. A hyperbolic model for viscous Newtonian flows. Continuum Mechanics and Thermodynamics 2016;28:85-104.], which is based on the theory of nonlinear hyperelasticity of Godunov & Romenski [67][Godunov S, Romenski E. Nonstationary equations of the nonlinear theory of elasticity in Euler coordinates. Journal of Applied Mechanics and Technical Physics 1972;13:868-885.][Godunov S., Romenski E., Elements of Continuum Mechanics and Conservation Laws. Kluwer Academic/ Plenum Publishers; 2003.], further denoted by GPR model. Notably, the governing PDE system is symmetric hyperbolic and fully consistent with the first and the second principle of thermodynamics. The nonlinear system of governing equations of the GPR model is overdetermined, large and includes stiff source terms as well as non-conservative products. In this paper we solve this model for the first time on moving unstructured meshes in multiple space dimensions by employing high order accurate one-step ADER-WENO finite volume schemes in the context of cell-centered direct Arbitrary-Lagrangian-Eulerian (ALE) algorithms. The numerical method is based on a WENO polynomial reconstruction operator on moving unstructured meshes, a fully-discrete one-step ADER scheme that is able to deal with stiff sources [Dumbser M., Enaux C., Toro E., Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. Journal of Computational Physics 2008a;227:3971-4001.], a nodal solver with relaxation to determine the mesh motion, and a path-conservative technique of Castro & Parés for the treatment of non-conservative products [Parés C., Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM Journal on Numerical Analysis. 2006;44:300-321.][Castro M, Gallardo J, Parés C. High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. applications to shallow-water systems. Mathematics of Computation 2006;75:1103-1134.]. We present numerical results obtained by solving the GPR model with ADER-WENO-ALE schemes in the stiff relaxation limit, showing that fluids (Euler or Navier-Stokes limit), as well as purely elastic or elasto-plastic solids can be simulated in the framework of nonlinear hyperelasticity with the same system of governing PDE. The obtained results are in good agreement when compared to exact or numerical reference solutions available in the literature.