Project SENSUS

Shape optimization problems, as a subset of structural optimization, are particularly challenging due to the infinite-dimensional, nonlinear structure of shape spaces. One option to overcome this drawback is to choose finite dimensional shape spaces, i.e., to discretize first. Nevertheless, this limits the amount of available tools coming from function spaces, resp. the solution space itself. Therefore, shape optimization problems can be solved using one of two different approaches: discretize-then-optimize or optimize-then-discretize. In general, it is a challenging task to decide whether to use a discretize-then-optimize or optimize-then-discretize approach to solve constrained optimization problems.

In this project, we investigate the benefits and drawbacks of both approaches with respect to accuracy of results and computational performance based on special shape optimization problems. In particular, we consider shape optimization problems constrained by partial differential equations (PDE) as well as variational inequalities (VI). Since classical constraint qualifications for deriving Lagrange multipliers generically fail, shape optimization problem constraints in the form of VIs are highly challenging and, thus, require new, sophisticated algorithms. Here, the choice of adjoints to compute shape sensitivities within the field of structural optimization is a topic of current research and part of this joint research project. A special focus is put on determining and updating the areas in which the inequalities are active, the so-called active sets. Moreover, uncertainties in the shape optimization problem play an important role in order to deal with realistic models. In particular, we aim at creating algorithms that are also robust to data from experimental measurements.

Contact: Tim Suchan