A simplified derivation technique of topological derivatives for quasi-linear transmission problems

Abstract

In this paper we perform the rigorous derivation of the topological derivative for optimization problems constrained by a class of quasi-linear elliptic transmission problems. In the case of quasi-linear constraints, techniques using fundamental solutions of the differential operators cannot be applied to show convergence of the variation of the states. Some authors succeeded showing this convergence with the help of technical computations under additional requirements on the problem. Our main objective is to simplify and extend these previous results by using a Lagrangian framework and a projection trick. Besides these generalisations the purpose of this manuscript is to present a systematic derivation approach for topological derivatives.

arXiv comment17 pages, 1 figure

References in corpus (3)

• A Note on Linear Elliptic Systems on ${\mathbb{R}}^d$
• Toplogical derivative for nonlinear magnetostatic problem
• Asymptotic analysis and topological derivative for 3D quasi-linear magnetostatics

Cited by in corpus (4)

• A multi-material topology optimization algorithm based on the topological derivative
• Adjoint based methods to compute higher order topological derivatives with an application to elasticity
• Topological derivative for PDEs on surfaces
• Asymptotic analysis and topological derivative for 3D quasi-linear magnetostatics