I will discuss some strategies in shape optimization problems implemented to consider the case where external loads on the structure are subject to uncertainty.
The natural approach is deterministic: it consists in a worst case point of view. Nevertheless, it is often overpessimistic. Therefore statisticals point of view can be used. The first idea is to considering a moment of the initial criterion. The objective function (or constraints) can be written as the expectation of a polynomial function of degree $m$. We provide a deterministic expression for the expectation of the polynomial as a function of the first $m$ moments of the random variables describing the uncertainties, as well as a method for calculating its form derivative. In particular, no further assumptions on the distribution of the random variables are required, and the method does not rely on computationally expensive sampling techniques.
In the last part, I will focus on minimizing the probability of failure in the case of a quadratic criterion. Our approach is based on the fact that the integration domain is an ellipsoid in the high-dimensional parameter space when the shape function of interest is quadratic. We derive the corresponding expressions for the shape function and the associated shape gradient. As an example for numerical implementation, we assume that the random charge is a Gaussian random field. In this context, we derive an efficient algorithm for evaluating the shape gradient of the failure probability.