We propose a new second-order accurate lattice Boltzmann formulation for linear elastodynamics that is stable for arbitrary combinations of material parameters under a CFL-like condition. The construction of the numerical scheme uses an equivalent first-order hyperbolic system of equations as an intermediate step, for which a vectorial lattice Boltzmann formulation is introduced. The only difference to conventional lattice Boltzmann formulations is the usage of vector-valued populations, so that all computational benefits of the algorithm are preserved. Using the asymptotic expansion technique and the notion of pre-stability structures we further establish second-order consistency as well as analytical stability estimates. Lastly, we introduce a second-order consistent initialization of the populations as well as a boundary formulation for Dirichlet boundary conditions on 2D rectangular domains. All theoretical derivations are numerically verified by convergence studies using manufactured solutions and long-term stability tests.

中文翻译：

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用于线性弹性动力学的格子玻尔兹曼：周期性问题和狄利克雷边界条件


我们提出了一种新的线性弹性动力学二阶精确晶格玻尔兹曼公式，该公式在类似 CFL 的条件下对材料参数的任意组合是稳定的。数值方案的构造使用等效的一阶双曲方程组作为中间步骤，为此引入了矢量晶格玻尔兹曼公式。与传统的格子玻尔兹曼公式的唯一区别是使用向量值总体，因此保留了算法的所有计算优势。使用渐近展开技术和预稳定结构的概念，我们进一步建立了二阶一致性以及分析稳定性估计。最后，我们引入了种群的二阶一致初始化以及 2D 矩形域上狄利克雷边界条件的边界公式。所有理论推导均通过使用制造解决方案和长期稳定性测试的收敛研究进行数值验证。