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Spaces where each element describes a shape, so-called shape spaces, are ofparticular interest in shape optimization and its applications. Theory andalgorithms in shape optimization are often based on techniques fromdifferential geometry. Challenges arise when an application demands anon-smooth shape, which is commonly-encountered as an optimal shape forfluid-mechanical problems. In order to avoid the restriction toinfinitely-smooth shapes of a commonly-used shape space, we construct a spacecontaining shapes in $\mathbb{R}^2$ that can be identified with a Riemannianproduct manifold but at the same time admits piecewise-smooth curves aselements. We combine the new product manifold with an approach for optimizingmultiple non-intersecting shapes. For the newly-defined shapes, adjustments aremade in the known shape optimization definitions and algorithms to ensure theirusability in applications. Numerical results regarding a fluid-mechanicalproblem constrained by the Navier-Stokes equations, where the viscous energydissipation is minimized, show its applicability.

A product shape manifold approach for optimizing piecewise-smooth shapes