标题： 直接任意拉格朗日-欧拉 ADER-WENO 有限体积格式用于非线性超弹性 HPR 模型 显示英文标题

标题： On direct Arbitrary-Lagrangian-Eulerian ADER-WENO finite volume schemes for the HPR model of nonlinear hyperelasticity

摘要： 本文关注由Peshkov & Romenski提出的连续介质力学统一一阶双曲公式的数值解，该模型基于Godunov & Romenski的非线性超弹性理论。值得注意的是，控制偏微分方程系统是对称双曲的，并且与热力学第一和第二定律完全一致。HPR模型的控制方程非线性系统规模较大，包含刚性源项以及非保守乘积。在本文中，我们首次在多维空间中使用高阶精度单步ADER-WENO有限体积方案，在单元中心直接任意拉格朗日-欧拉（ALE）算法的框架下对该模型进行求解。数值方法基于移动非结构化网格上的WENO多项式重构算子，一个能够处理刚性源项的全离散单步ADER方案，一个具有松弛的节点求解器来确定网格运动，以及Castro & Parès的路径守恒技术用于处理非保守乘积。我们展示了在刚性松弛极限下使用ADER-WENO-ALE方案求解HPR模型得到的数值结果，表明流体（欧拉或纳维-斯托克斯极限），以及纯弹性或弹塑性固体都可以在非线性超弹性框架下进行模拟，使用相同的控制偏微分方程组。所得结果与文献中提供的精确或数值参考解相比，具有良好的一致性。 显示英文摘要

摘要： This paper is concerned with the numerical solution of the unified first order hyperbolic formulation of continuum mechanics proposed by Peshkov & Romenski (HPR model), which is based on the theory of nonlinear hyperelasticity of Godunov & Romenski . 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 HPR model is 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, a nodal solver with relaxation to determine the mesh motion, and a path-conservative technique of Castro & Par\`es for the treatment of non-conservative products. We present numerical results obtained by solving the HPR 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.