Adjoint based methods to compute higher order topological derivatives with an application to elasticity

TL;DR

This paper reviews methods for computing higher order topological derivatives for PDE-constrained shape functionals, comparing three adjoint-based approaches and applying them to elasticity models in 2D and 3D.

Contribution

It provides a comprehensive comparison of three adjoint-based methods for higher order topological derivatives and demonstrates their application to elasticity problems.

Findings

Compared three methods: Amstutz, averaged adjoint, Delfour.

Computed first and second order derivatives for elasticity models.

Analyzed applicability and efficiency of each method.

Abstract

The goal of this paper is to give a comprehensive and short review on how to compute the first and second order topological derivative and potentially higher order topological derivatives for PDE constrained shape functionals. We employ the adjoint and averaged adjoint variable within the Lagrangian framework and compare three different adjoint based methods to compute higher order topological derivatives. To illustrate the methodology proposed in this paper, we then apply the methods to a linear elasticity model. We compute the first and second order topological derivative of the linear elasticity model for various shape functionals in dimension $2$ and $3$ using Amstutz' method, the averaged adjoint method and Delfour's method. In contrast to other contributions regarding this subject, we not only compute the first and second order topological derivative, but additionally give some insight on various methods and compared their applicability and efficiency with respect to the underlying problem formulation.