an:06894251
Zbl 1390.76399
Boscheri, Walter; Dumbser, Michael; Loubère, Raphaël
Cell centered direct arbitrary-Lagrangian-Eulerian ADER-WENO finite volume schemes for nonlinear hyperelasticity
EN
Comput. Fluids 134-135, 111-129 (2016).
0045-7930
2016
j
76M12 65M08 74B20
high order direct arbitrary-Lagrangian-Eulerian finite volume schemes; hyperbolic conservation laws with stiff source terms and non-conservative products; high order ADER-WENO schemes on moving unstructured meshes; unified first order hyperbolic formulation of continuum mechanics; symmetric-hyperbolic Godunov-Peshkov-Romenski model (GPR model) of nonlinear hyperelasticity; viscous heat conducting fluids and nonlinear elasto-plastic solids
Summary: This paper is concerned with the numerical solution of the unified first order hyperbolic formulation of continuum mechanics proposed by \textit{I. Peshkov} and \textit{E. Romenski} [Contin. Mech. Thermodyn. 28, No. 1--2, 85--104 (2016; Zbl 1348.76046)], which is based on the theory of nonlinear hyperelasticity of \textit{S. K. Godunov} and \textit{E. I. Romenskii} [``Nonstationary equations of nonlinear elasticity theory in eulerian coordinates'', J. Appl. Mech. Tech. Phys. 13, No. 6, 868--884 (1972; \url{doi:10.1007/bf01200547}); Elements of continuum mechanics and conservation laws. Translation from the 1998 Russian original. New York, NY: Kluwer Academic/Plenum Publishers (2003; Zbl 1031.74004)], 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 [the second author et al., J. Comput. Phys. 227, No. 8, 3971--4001 (2008; Zbl 1142.65070)], 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 [\textit{C. Parés}, SIAM J. Numer. Anal. 44, No. 1, 300--321 (2006; Zbl 1130.65089); \textit{M. Castro} et al., Math. Comput. 75, No. 255, 1103--1134 (2006; Zbl 1096.65082)]. 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.
1348.76046; 1031.74004; 1142.65070; 1130.65089; 1096.65082