PhD on Topology Optimization of uncertain physical systems

Topology Optimization (TO) is a powerful technology which allows to compute optimized designs of physical systems  in a fully automated manner, without any assumed parameterization of the geometry. From the mathematical standpoint, it consists of solving nonlinear optimization problems constrained by partial differential equations. This technology allows to improve the performance in various physical settings such as structural design, heat exchangers, or nanophotonic crystals.

In many industrial problems, the optimized design need to be robust with respect to various forms of uncertainties coming e.g. from the load conditions or from the manufacturing process. The optimized design needs to perform well in a continuum of (many) possible scenarios. This is especially challenging because solving physical models for all these possible situations (and accounting them in the optimization program) is not computationally tractable for realistic problems.

The goal of this PhD thesis is to develop efficient methods for taking into account arbitrary sources of uncertainties in Topology Optimization problems. A particular focus will be given on the treatment of uncertainties present in the optimization constraints, and on the use of efficient low-rank methods for capturing uncertainty.  The work will involve a good balance between  rigorous mathematical analysis and numerical implementation.