A novel way of computing the shape derivative for a class of non-smooth PDEs and its impact on deriving necessary conditions for locally optimal shapes

TL;DR

This paper introduces a new method for computing shape derivatives in non-smooth PDEs, enabling the derivation of necessary conditions for optimal shapes without domain extensions or approximations.

Contribution

It presents a novel sensitivity analysis technique within the functional variational approach for non-smooth PDEs, advancing shape optimization methods.

Findings

New shape derivative computation method for non-smooth PDEs

Ability to handle pointwise and distributed observations in shape optimization

Established necessary conditions for locally optimal shapes in non-smooth settings

Abstract

We derive necessary conditions for locally optimal shapes of a design problem governed by a non-smooth PDE. The main particularity of the state system is the lack of differentiability of the nonlinearity. We work in the framework of the functional variational approach (FVA), which has the capacity to transfer geometric optimization problems into optimal control problems, the set of admissible shapes being parametrized by a large class of continuous mappings. In the FVA setting, we introduce a sensitivity analysis technique that is novel even for smooth PDEs. We emphasize that we do not resort to extensions on the hold-all domain or any kind of approximation of the original PDE. The computation of the directional derivative of the state w.r.t. functional variations results in a new way of computing the shape derivative. The presented approach allows us to handle in the objective pointwise observation and derivatives of the state on an observation set as well as distributed observation terms. In addition, we introduce the concept of locally optimal shapes and we put into evidence its connection to locally minimizers of the corresponding control problem. With directional differentiability results for the control-to-state map at our disposal, we can then state necessary conditions for locally optimal shapes in general non-smooth settings.