  We investigate a complex system involving multiple shapes to be optimized in
a domain, taking into account geometric constraints on the shapes and
uncertainty appearing in the physics. We connect the differential geometry of
product shape manifolds with multi-shape calculus, which provides a novel
framework for the handling of piecewise smooth shapes. This multi-shape
calculus is applied to a shape optimization problem where shapes serve as
obstacles in a system governed by steady state incompressible Navier-Stokes
flow. Numerical experiments use our recently developed stochastic augmented
Lagrangian method and we investigate the choice of algorithmic parameters using
the example of this application.
