# Topology optimization with worst-case handling of material uncertainties

## TL;DR

This paper develops a topology optimization method that accounts for material uncertainties in a worst-case scenario, using a minimax formulation with convex relaxation and regularization schemes, demonstrated on additive manufacturing and degradation cases.

## Contribution

It introduces a novel worst-case approach for topology optimization under material uncertainties, with a convex relaxation and regularization schemes for efficient solution.

## Key findings

- The method effectively handles material uncertainties in practical examples.
- Barrier regularization improves the solution process compared to Tikhonov regularization.
- A continuation scheme mitigates over-conservativeness for large uncertainties.

## Abstract

In this article a topology optimization method is developed, which is aware of material uncertainties. The uncertainties are handled in a worst-case sense, i.e. the worst possible material distribution over a given uncertainty set is taken into account for each topology. The worst-case approach leads to a minimax problem, which is analyzed throughout the paper. A conservative convex relaxation for the inner maximization problem is suggested, which allows to treat the minimax problem by minimization of an optimal value function. A Tikhonov type and a barrier regularization scheme are developed, which render the resulting minimization problem continuously differentiable. The barrier regularization scheme turns out to be more suitable for the practical solution of the problem, as it can be closely linked to a highly efficient interior point approach used for the evaluation of the optimal value function and its gradient. Based on this, the outer minimization problem can be approached by a gradient based optimization solver like the method of moving asymptotes. Examples from additive manufacturing as well as material degradation are examined, demonstrating the practical applicability as well as the efficiency of the suggested method. Finally, the impact of the convex relaxation of the inner problem is investigated. It is observed that for large material deviations, the robust solution may become too conservative. As a remedy, a RAMP-type continuation scheme for the inner problem is suggested and numerically tested.