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Exact and approximate solutions of Riemann problems in non-linear elasticity

机译：非线性弹性黎曼问题的精确解和近似解

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Eulerian shock-capturing schemes have advantages for modelling problems involving complex non-linear wave structures and large deformations in solid media. Various numerical methods now exist for solving hyperbolic conservation laws that have yet to be applied to non-linear elastic theory. In this paper one such class of solver is examined based upon characteristic tracing in conjunction with high-order monotonicity preserving weighted essentially non-oscillatory (MPWENO) reconstruction. Furthermore, a new iterative method for finding exact solutions of the Riemann problem in non-linear elasticity is presented. Access to exact solutions enables an assessment of the performance of the numerical techniques with focus on the resolution of the seven wave structure. The governing model represents a special case of a more general theory describing additional physics such as material plasticity. The numerical scheme therefore provides a firm basis for extension to simulate more complex physical phenomena. Comparison of exact and numerical solutions of one-dimensional initial values problems involving three-dimensional deformations is presented.
机译：欧拉减震方案对于建模涉及复杂非线性波结构和固体介质中大变形的问题具有优势。现在存在用于求解双曲守恒定律的各种数值方法，这些方法尚未应用于非线性弹性理论。在本文中，基于特征跟踪结合高阶单调性保留加权的基本非振荡（MPWENO）重构，研究了此类求解器。此外，提出了一种新的迭代方法，可以找到非线性弹性中黎曼问题的精确解。使用精确的解决方案可以评估数值技术的性能，重点是七波结构的分辨率。控制模型代表了一个更一般的理论的特殊情况，该理论描述了诸如材料可塑性之类的附加物理学。因此，数值方案为扩展以模拟更复杂的物理现象提供了坚实的基础。提出了涉及三维变形的一维初始值问题的精确解和数值解的比较。

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