An abstract lagrangian framework for computing shape derivatives

TL;DR

This paper develops an abstract Lagrangian framework using the implicit function theorem to efficiently compute shape derivatives in PDE-constrained shape optimization, providing practical formulas and insights into duality.

Contribution

It introduces a novel abstract framework for shape derivatives that unifies and extends existing methods, applicable to various PDE problems.

Findings

Provides practical formulas for shape derivatives and adjoint states.

Demonstrates the framework's application to elliptic and parabolic PDEs.

Shows the duality between material and adjoint derivatives.

Abstract

In this paper we study an abstract framework for computing shape derivatives of functionals subject to PDE constraints. We revisit the Lagrangian approach using the implicit function theorem in an abstract setting tailored for applications to shape optimization. This abstract framework yields practical formulae to compute the derivative of a shape functional, the material derivative of the state, and the adjoint state. Furthermore, it allows to gain insight on the duality between the material derivative of the state and the adjoint state. We show several applications of our main result to the computation of distributed shape derivatives for problems involving linear elliptic, nonlinear elliptic, parabolic PDEs and distributions. We compare our approach with other techniques for computing shape derivatives including the material derivative method and the averaged adjoint method.