Topology optimization with worst-case handling of material uncertainties

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 concave 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 efficiency of the suggested method. Finally, the impact of the concave relaxation of the inner problem is investigated. In order to test the conservatism of the latter, a RAMP-type continuation scheme providing a lower bound for the inner problem is suggested and numerically tested.