%0 Journal Article
%T A Numerical Method for Solving 3D Elasticity Equations with Sharp-Edged Interfaces
%A Liqun Wang
%A Songming Hou
%A Liwei Shi
%J International Journal of Partial Differential Equations
%D 2013
%R 10.1155/2013/476873
%X Interface problems occur frequently when two or more materials meet. Solving elasticity equations with sharp-edged interfaces in three dimensions is a very complicated and challenging problem for most existing methods. There are several difficulties: the coupled elliptic system, the matrix coefficients, the sharp-edged interface, and three dimensions. An accurate and efficient method is desired. In this paper, an efficient nontraditional finite element method with nonbody-fitting grids is proposed to solve elasticity equations with sharp-edged interfaces in three dimensions. The main idea is to choose the test function basis to be the standard finite element basis independent of the interface and to choose the solution basis to be piecewise linear satisfying the jump conditions across the interface. The resulting linear system of equations is shown to be positive definite under certain assumptions. Numerical experiments show that this method is second order accurate in the norm for piecewise smooth solutions. More than 1.5th order accuracy is observed for solution with singularity (second derivative blows up). 1. Introduction The macroscopic mechanical response of most elastic materials, such as wood, metal, rubber, rock, and bone, can be described by the elasticity theory. However, the numerical solution for an elasticity problem with interface is highly nontrivial, because the coefficients in the partial differential equation for elasticity system are discontinuous, and the solution to the PDE system also needs to satisfy the jump conditions across each material interface. Examples of elasticity interface problem include the minimum-compliance-design problems [1], the microstructural evolution [2], and the atomic interactions [3]. Designing an accurate and efficient method for these problems is a difficult job, especially when the interface is not smooth. In this paper, we consider a three-dimensional elasticity problem with sharp-edged interfaces and matrix coefficients. The model of interest is as follows. Consider an open bounded domain . Let be an interface of codimension , which divides into disjoint open subdomains, and , hence . Assume that the boundary and the boundary of each subdomain are Lipschitz continuous as submanifolds. Since are Lipschitz continuous, so is . A unit normal vector of can be defined as a.e. on ; see Section 1.5 in [4]. We seek solutions of the variable coefficient elliptic equation away from the interface given by in which denotes the spatial variables and is the gradient operator. The coefficient is assumed to be a
%U http://www.hindawi.com/journals/ijpde/2013/476873/