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We investigate a complex system involving multiple shapes to be optimized ina domain, taking into account geometric constraints on the shapes anduncertainty appearing in the physics. We connect the differential geometry ofproduct shape manifolds with multi-shape calculus, which provides a novelframework for the handling of piecewise smooth shapes. This multi-shapecalculus is applied to a shape optimization problem where shapes serve asobstacles in a system governed by steady state incompressible Navier-Stokesflow. Numerical experiments use our recently developed stochastic augmentedLagrangian method and we investigate the choice of algorithmic parameters usingthe example of this application.

Optimization of piecewise smooth shapes under uncertainty using the example of Navier-Stokes flow