The Lattice Boltzmann Method (LBM), e.g. in [ 1] and [2 ], can be interpreted as an alternative method for the numerical solution of partial differential equations. Consequently, although the LBM is usually applied to solve fluid flows, the above interpretation of the LBM as a general numerical tool, allows the LBM to be extended to solid mechanics as well. In this spirit, the LBM has been studied in recent years. First publications [3], [4] presented an LBM scheme for the numerical solution of the dynamic behavior of a linear elastic solid under simplified deformation assumptions. For so-called anti-plane shear deformation, the only non-zero displacement component is governed by a two-dimensional wave equation. In this work, an existing LBM for the two-dimensional wave equation is extended to more general plane strain problems. The proposed algorithm reduces the plane strain problem to the solution of two separate wave equations for the volume dilatation and the non-zero component of the rotation vector, respectively. A particular focus is on the implementation of types of boundary conditions that are commonly encountered in engineering practice for solids: Dirichlet and Neumann boundary conditions. Last, several numerical experiments are conducted that highlight the performance of the new LBM in comparison to the Finite Element Method.
翻译:Lattice Boltzmann方法(LBM),例如[1]和[2]),可以解释为部分差异方程数字解决方案的一种替代方法,因此,尽管LBM通常用于解决液体流,但LBM通常用于解决液体流,上述LBM作为一般数字工具的解释使LBM也能够扩大到固体机械学。本着这一精神,近年来对LBM进行了研究。第一本出版物[3],[4]提出了在简化变形假设下线状弹性固态动态行为的数字解决方案。对于所谓的反板切变形,唯一的非零异位化部分由二维波方程式调节。在这项工作中,二维波方方方程式的现有LBMM(LBM)模型扩展至更为一般的平面压力问题。拟议的算法将飞机压力问题降低到两个不同的波方程式的解决方案。在简化变形假设假设下,特别侧重于执行在工程实践中常见的边界条件类型:DirichIBLFIM(IB)和NUI(FIB)模式(LIL)进行的若干次测试。