Shape optimization of a thermoelastic body under thermal uncertainties

We consider a shape optimization problem in the framework of the thermoelasticity model under uncertainty. The uncertainty is supposed to be located in the Robin boundary condition of the heat equation. The purpose of considering this model is to account for thermal residual stresses or thermal deformations, which may hinder the mechanical properties of the final design in case of a high environmental temperature. In this situation, the presence of uncertainty in the external temperature must be taken into account to ensure the correct manufacturing and performance of the device of interest. The objective functional under consideration is based on volume minimization in the presence of an inequality constraint for a quadratic shape functional. Exemplarily, we consider the $L^2$-norm of the von Mises stress and demonstrate that the robust constraint and its derivative are completely determined by low order moments of the random input, thus computable by means of low-rank approximation. The resulting shape optimization problem is discretized by using the finite element method for the underlying partial differential equations and the level-set method to represent the sought domain. Numerical results for a model case in structural optimization are given.