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Cell centered direct arbitrary-Lagrangian-Eulerian ADER-WENO finite volume schemes for nonlinear hyperelasticity. (English) Zbl 1390.76399
Summary: This paper is concerned with the numerical solution of the unified first order hyperbolic formulation of continuum mechanics proposed by I. Peshkov and 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 S. K. Godunov and E. I. Romenskii [“Nonstationary equations of nonlinear elasticity theory in eulerian coordinates”, J. Appl. Mech. Tech. Phys. 13, No. 6, 868–884 (1972; 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 [C. Parés, SIAM J. Numer. Anal. 44, No. 1, 300–321 (2006; Zbl 1130.65089); 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.

MSC:
76M12 Finite volume methods applied to problems in fluid mechanics
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
74B20 Nonlinear elasticity
Software:
HE-E1GODF
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