A stabilized $P_1$ immersed finite element method for the interface elasticity problems

TL;DR

This paper introduces a stabilized immersed finite element method for planar elasticity problems with heterogeneous materials, allowing for non-aligned meshes and ensuring optimal error estimates and locking-free performance.

Contribution

The method innovatively combines stabilized nonconforming finite elements with interface-independent meshes and modified basis functions satisfying interface conditions.

Findings

Achieves optimal $H^1$ and divergence norm error estimates.

Demonstrates robustness across various Lamé parameters.

Proves locking-free behavior as $ extlambda o fty$.

Abstract

We develop a new finite element method for solving planar elasticity problems involving of heterogeneous materials with a mesh not necessarily aligning with the interface of the materials. This method is based on the `broken' Crouzeix-Raviart $P_1$-nonconforming finite element method for elliptic interface problems \cite{Kwak-We-Ch}. To ensure the coercivity of the bilinear form arising from using the nonconforming finite elements, we add stabilizing terms as in the discontinuous Galerkin (DG) method \cite{Arnold-IP},\cite{Ar-B-Co-Ma},\cite{Wheeler}. The novelty of our method is that we use meshes independent of the interface, so that the interface may cut through the elements. Instead, we modify the basis functions so that they satisfy the Laplace-Young condition along the interface of each element. We prove optimal $H^1$ and divergence norm error estimates. Numerical experiments are carried out to demonstrate that the our method is optimal for various Lam\`e parameters $\mu$ and $\lambda$ and locking free as $\lambda\to\infty$.