Stochastic Dominance Constraints in Elastic Shape Optimization

TL;DR

This paper introduces a novel stochastic shape optimization framework for elastic materials that incorporates dominance constraints to manage risk, demonstrated through numerical experiments with variable loads.

Contribution

It extends stochastic dominance concepts to shape optimization, enabling risk-averse design via constraints rather than objective modification.

Findings

Effective handling of high variability in loads.

Demonstrated feasibility of stochastic dominance constraints.

Potential for improved risk management in shape optimization.

Abstract

This paper deals with shape optimization for elastic materials under stochastic loads. It transfers the paradigm of stochastic dominance, which allows for flexible risk aversion via comparison with benchmark random variables, from finite-dimensional stochastic programming to shape optimization. Rather than handling risk aversion in the objective, this enables risk aversion by including dominance constraints that single out subsets of nonanticipative shapes which compare favorably to a chosen stochastic benchmark. This new class of stochastic shape optimization problems arises by optimizing over such feasible sets. The analytical description is built on risk-averse cost measures. The underlying cost functional is of compliance type plus a perimeter term, in the implementation shapes are represented by a phase field which permits an easy estimate of a regularized perimeter. The analytical description and the numerical implementation of dominance constraints are built on risk-averse measures for the cost functional. A suitable numerical discretization is obtained using finite elements both for the displacement and the phase field function. Different numerical experiments demonstrate the potential of the proposed stochastic shape optimization model and in particular the impact of high variability of forces or probabilities in the different realizations.