  This brief note aims to introduce the recent paradigm of distributional
robustness in the field of shape and topology optimization. Acknowledging that
the probability law of uncertain physical data is rarely known beyond a rough
approximation constructed from observed samples, we optimize the worst-case
value of the expected cost of a design when the probability law of the
uncertainty is ``close'' to the estimated one up to a prescribed threshold. The
``proximity'' between probability laws is quantified by the Wasserstein
distance, a notion pertaining to optimal transport theory. The combination of
the classical entropic regularization technique in this field with recent
results from convex duality theory allows to reformulate the distributionally
robust optimization problem in a way which is tractable for computations. Two
numerical examples are presented, in the different settings of density-based
topology optimization and geometric shape optimization. They exemplify the
relevance and applicability of the proposed formulation regardless of the
selected optimal design framework.
