Abstract:When solving 2D linear elastodynamic equations in homogeneous isotropic media, a Helmholtz decomposition of the displacement field decouples the equations into two scalar wave equations that only interact at the boundary. It is then natural to look for numerical schemes that independently solve the scalar equations and couple the solutions at the boundary. The case of rigid boundary condition was treated by Burel [Ph.D. thesis, Université Paris Sud-Paris XI (2014)] and Burel et al. [Numer. Anal. Appl. 5 (2012), pp. 136–143]. However the case of traction free boundary condition was proven by Martinez et al. [J. Sci. Comput. 77 (2018), pp. 1832–1873] to be unstable if a straightforward approach is used. Then an adequate functional framework as well as a time domain mixed formulation to circumvent these issues was presented. In this work we first review the formulation presented by Martinez et al. [J. Sci. Comput. 77 (2018), pp. 1832–1873] and propose a subsequent discretised formulation. We provide the complete stability analysis of the corresponding numerical scheme. Numerical results that illustrate the theory are also shown.

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中文翻译：

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用势能和有限元解带牵引自由边界条件的二维线性各向同性弹性动力学方法的数值分析

摘要：在均质各向同性介质中求解二维线性弹性动力学方程时，位移场的亥姆霍兹分解将方程解耦为两个仅在边界处相互作用的标量波动方程。然后自然会寻找独立求解标量方程并在边界处耦合求解的数值方案。刚性边界条件的情况由Burel [Ph.D. 论文，巴黎巴黎南部巴黎大学（2014）]和Burel等人。[数字。肛门 应用 5（2012），第136–143页]。但是，马丁内兹等人证明了无牵引边界条件的情况。[J. 科学 计算 77（2018），第1832–1873页]如果使用直接方法，则不稳定。然后提出了一个适当的功能框架以及时域混合公式来规避这些问题。在这项工作中，我们首先回顾一下Martinez等人提出的公式。[J. 科学 计算 77（2018），1832-1873页]，并提出了随后的离散化表述。我们提供了相应数值方案的完整稳定性分析。还显示了说明该理论的数值结果。

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