Shape optimization for quadratic functionals and states with random right-hand sides

TL;DR

This paper studies shape optimization problems involving quadratic functionals with random inputs, providing explicit formulas for the objective and gradient, and proposing an efficient deterministic algorithm for robust optimization.

Contribution

It introduces a method to explicitly compute the robust objective and gradient using low-order moments, enabling efficient shape optimization under uncertainty.

Findings

Explicit formulas for the robust objective and gradient.

A computationally cheap deterministic optimization algorithm.

Application to structural optimization problems.

Abstract

In this work, we investigate a particular class of shape optimization problems under uncertainties on the input parameters. More precisely, we are interested in the minimization of the expectation of a quadratic objective in a situation where the state function depends linearly on a random input parameter. This framework covers important objectives such as tracking-type functionals for elliptic second order partial differential equations and the compliance in linear elasticity. We show that the robust objective and its gradient are completely and explicitly determined by low-order moments of the random input. We then derive a cheap, deterministic algorithm to minimize this objective and present model cases in structural optimization.