A Linear View on Shape Optimization

TL;DR

This paper investigates the linear structure of shape deformations to reformulate shape optimization as an optimal control problem, introducing a novel linear second shape derivative and related Newton-type algorithms.

Contribution

It proposes a new linear perspective on shape optimization, including a linear second shape derivative and algorithms based on this framework.

Findings

Introduces a linear second shape derivative.

Develops shape Newton-type algorithms.

Highlights numerical challenges of the linear approach.

Abstract

Shapes do not define a linear space. This paper explores the linear structure of deformations as a representation of shapes. This transforms shape optimization to a variant of optimal control. The numerical challenges of this point of view are highlighted and a novel linear version of the second shape derivative is proposed leading to particular algorithms of shape Newton type.