Wang Bao, Xia Kelin, Wei Guo-Wei
Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA.
Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA ; Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI 48824, USA ; Center for Mathematical Molecular Biosciences, Michigan State University, East Lansing, MI 48824, USA.
J Comput Phys. 2015 Aug 1;294:405-438. doi: 10.1016/j.jcp.2015.03.053.
Elastic materials are ubiquitous in nature and indispensable components in man-made devices and equipments. When a device or equipment involves composite or multiple elastic materials, elasticity interface problems come into play. The solution of three dimensional (3D) elasticity interface problems is significantly more difficult than that of elliptic counterparts due to the coupled vector components and cross derivatives in the governing elasticity equation. This work introduces the matched interface and boundary (MIB) method for solving 3D elasticity interface problems. The proposed MIB elasticity interface scheme utilizes fictitious values on irregular grid points near the material interface to replace function values in the discretization so that the elasticity equation can be discretized using the standard finite difference schemes as if there were no material interface. The interface jump conditions are rigorously enforced on the intersecting points between the interface and the mesh lines. Such an enforcement determines the fictitious values. A number of new techniques has been developed to construct efficient MIB elasticity interface schemes for dealing with cross derivative in coupled governing equations. The proposed method is extensively validated over both weak and strong discontinuity of the solution, both piecewise constant and position-dependent material parameters, both smooth and nonsmooth interface geometries, and both small and large contrasts in the Poisson's ratio and shear modulus across the interface. Numerical experiments indicate that the present MIB method is of second order convergence in both and error norms for handling arbitrarily complex interfaces, including biomolecular surfaces. To our best knowledge, this is the first elasticity interface method that is able to deliver the second convergence for the molecular surfaces of proteins..
弹性材料在自然界中无处不在，是人造设备和装备中不可或缺的组成部分。当一个设备或装备涉及复合或多种弹性材料时，弹性界面问题就会出现。由于控制弹性方程中的耦合矢量分量和交叉导数，三维（3D）弹性界面问题的求解比椭圆型对应问题要困难得多。这项工作介绍了用于求解三维弹性界面问题的匹配界面和边界（MIB）方法。所提出的MIB弹性界面方案利用材料界面附近不规则网格点上的虚拟值来代替离散化中的函数值，以便可以使用标准有限差分格式对弹性方程进行离散化，就好像没有材料界面一样。在界面与网格线的交点处严格执行界面跳跃条件。这种执行确定了虚拟值。已经开发了许多新技术来构建有效的MIB弹性界面方案，以处理耦合控制方程中的交叉导数。所提出的方法在解的弱和强不连续性、分段常数和位置相关的材料参数、光滑和非光滑的界面几何形状以及界面两侧泊松比和剪切模量的小和大对比度方面都得到了广泛验证。数值实验表明，对于处理任意复杂的界面，包括生物分子表面，目前的MIB方法在 和 误差范数上均具有二阶收敛性。据我们所知，这是第一种能够为蛋白质分子表面提供二阶收敛性的弹性界面方法。
