Shape Optimization under Uncertainty from a Stochastic Programming Point of View
We consider an elastic body subjected to internal and external forces which are uncertain. Simply averaging the possible loadings will result in a structure that might not be robust for the individual loadings at all. Instead, we apply techniques from level set based shape optimization and two-stage stochastic programming: In the first stage, the non-anticipative decision on the shape has to be taken. Afterwards, the realizations of the random forces are observed, and the variational formulation of the elasticity system takes the role of the second-stage problem. Taking advantage of the PDE's linearity, we are able to compute solutions for an arbitrary number of scenarios without increasing the computational effort significantly. The deformations are described by PDEs that are solved efficiently by Composite Finite Elements. The objective is, e.g., to minimize the compliance. A gradient method using the shape derivative is used to solve the problem. Results for 2D instances are shown. The obtained solutions strongly depend on the initial guess, in particular its topology. To overcome this issue, we included the topological derivative into our algorithm as well. The stochastic programming perspective also allows us to incorporate risk measures into our model which might be a more appropriate objective in many practical applications.