Automated computation of topological derivatives with application to nonlinear elasticity and reaction-diffusion problems

TL;DR

This paper introduces a systematic, Lagrangian-based method for computing topological derivatives applicable to various PDE-constrained problems, including nonlinear elasticity and reaction-diffusion, with verified accuracy and novel application to nonlinear elasticity.

Contribution

It provides the first systematic approach for topological derivatives in nonlinear elasticity and extends the method to a broad class of PDE-constrained topology optimization problems.

Findings

Numerical results show good agreement with known formulas.

The method accurately predicts topological derivatives for new problems.

Application to nonlinear elasticity is demonstrated for the first time.

Abstract

While topological derivatives have proven useful in applications of topology optimisation and inverse problems, their mathematically rigorous derivation remains an ongoing research topic, in particular in the context of nonlinear partial differential equation (PDE) constraints. We present a systematic yet formal approach for the computation of topological derivatives of a large class of PDE-constrained topology optimization problems with respect to arbitrary inclusion shapes. Scalar and vector-valued as well as linear and nonlinear elliptic PDE constraints are considered in two and three space dimensions including a nonlinear elasticity model and nonlinear reaction-diffusion problems. The systematic procedure follows a Lagrangian approach for computing topological derivatives. For problems where the exact formula is known, the numerically computed values show good coincidence. Moreover, by inserting the computed values into the topological asymptotic expansion, we verify that the obtained values satisfy the expected behaviour also for other, previously unknown problems, indicating the correctness of the procedure. We present a systematic approach for the computation of topological derivatives that is applicable to a large class of problems. Most notably, our approach covers the topological derivative for a nonlinear elasticity problem, which has not been reported in the literature.