Advances in Mathematical Physics

Volume 2016, Article ID 3292487, 9 pages

http://dx.doi.org/10.1155/2016/3292487

## A Consistent Immersed Finite Element Method for the Interface Elasticity Problems

Korea Advanced Institute of Science and Technology, Daejeon 305-701, Republic of Korea

Received 10 November 2015; Accepted 16 February 2016

Academic Editor: Manuel De León

Copyright © 2016 Sangwon Jin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We propose a new scheme for elasticity problems having discontinuity in the coefficients. In the previous work (Kwak et al., 2014), the authors suggested a method for solving such problems by finite element method using nonfitted grids. The proposed method is based on the -nonconforming finite element methods with stabilizing terms. In this work, we modify the method by adding the consistency terms, so that the estimates of consistency terms are not necessary. We show optimal error estimates in and divergence norms under minimal assumptions. Various numerical experiments also show optimal rates of convergence.

#### 1. Introduction

Linear elasticity equations are important governing equations in continuum mechanics, they describe how solid objects are deformed when external forces are applied on them. In particular, elasticity interface problems are important in various fields such as solid mechanics, material sciences, and biological sciences. There are various numerical methods to solve these problems such as finite difference methods, finite volume methods, and finite element methods; see [1–6] and references therein. But the problems involving composite materials lead to the discontinuity in the coefficients of the governing equations. In this case, there are two types of numerical methods from the point of view of mesh generation. In the first type, people solve the problems by using fitted grid which is created to align with the interface [7, 8]; another type is to use an unfitted (uniform) grid which is generated independently of the interface [9–15].

Fitted grid method is well known and the most widely used accurate method but it has weaknesses for the time dependent problems which include moving interface. In moving interface problems, we need to regenerate the appropriate mesh for the interface which changes according to time. Therefore, it is not adequate to apply to the moving interface problems since this mesh generation has considerable computational costs.

However, using a fixed grid has an obvious advantage that we can use the mesh in the previous time step, in the case when the interface changes over time. Therefore, it is suitable for moving interface problems. There are two classes of methods which belong to this type. One is the extended finite element method (XFEM) and the other is the immersed finite element method (IFEM). In the XFEM, they use enrichment basis functions in addition to the standard finite element basis. Therefore, they have more degrees of freedom than the standard FEM. In IFEM, the degrees of freedom are the same as the standard FEM basis; instead we modify the shape functions so as to satisfy the interface conditions along the interior interface. Recently, Lin et al. [9] have suggested a numerical method for solving elasticity problem with an interface using rotated -nonconforming finite element on uniform grids and Kwak et al. [10] proved the optimal error estimate for -nonconforming finite element on triangular grids under an some extra regularity that the stress component belongs to .

In this paper, we modify the numerical scheme studied in [10] for the elasticity problem with an interface. We add consistency terms in the bilinear form to avoid the consistency term estimate. As a result, the extra regularity assumption which is necessary to estimate the consistency term error in the previous work [10] can be avoided. We prove optimal error estimates by the standard framework of finite element error analysis, which includes proving the coercivity and boundedness of the bilinear form.

The rest of our paper is organized as follows. In Section 2, we introduce a problem with interface along which Laplace-Young condition holds. In Section 3, we review the vector basis functions introduced in [10] based on the nonconforming elements satisfying the interface conditions and discretize the problem using (uniform) triangular grids. In Section 4, we prove the approximation property of our finite element space and the coercivity of our variational form. Next, we prove optimal and divergence norm error estimates. Finally, numerical experiments are presented in Section 5 for various Lamé constants and for shape interfaces.

#### 2. Preliminaries

Let be a connected, convex polygonal domain in which is divided into two subdomains and by interface ; see Figure 1. We assume the subdomains and are occupied by two different elastic materials. For a differentiable function and a tensor , we let Then the displacement of the elastic body under an external force satisfies the Navier-Lamé equation as follows:whereare the stress tensor and the strain tensor, respectively, is outward unit normal vector, is the identity tensor, and is the external force. Here are the Lamé constants, satisfying and , and is Young’s modulus and is the Poisson ratio. When the parameter , this equation describes the behavior of nearly incompressible material. Since the material properties are different in each region, we set Lamé constants for .