Recovering elastic inclusions by shape optimization methods with immersed finite elements

Abstract

This article presents a finite element method on a fixed mesh for solving a group of inverse geometric problems for recovering the material interface of a linear elasticity system. A partially penalized immersed finite element method is used to discretize both the elasticity interface problems and the objective shape functionals accurately regardless of the shape and location of the interface. Explicit formulas for both the velocity fields and the shape derivatives of IFE shape functions are derived on a fixed mesh and they are employed in the shape sensitivity framework through the discretized adjoint method for accurately and efficiently computing the gradients of objective shape functions with respect to the parameters of the interface curve. The shape optimization for solving an inverse geometric problem is therefore accurately reduced to a constrained optimization that can be implemented efficiently within the IFE framework together with a standard optimization algorithm. We demonstrate features and advantages of the proposed IFE-based shape optimization method by several typical inverse geometric problems for linear elasticity systems.