A product shape manifold approach for optimizing piecewise-smooth shapes

Abstract

Spaces where each element describes a shape, so-called shape spaces, are of particular interest in shape optimization and its applications. Theory and algorithms in shape optimization are often based on techniques from differential geometry. Challenges arise when an application demands a non-smooth shape, which is commonly-encountered as an optimal shape for fluid-mechanical problems. In order to avoid the restriction to infinitely-smooth shapes of a commonly-used shape space, we construct a space containing shapes in $\mathbb{R}^2$ that can be identified with a Riemannian product manifold but at the same time admits piecewise-smooth curves as elements. We combine the new product manifold with an approach for optimizing multiple non-intersecting shapes. For the newly-defined shapes, adjustments are made in the known shape optimization definitions and algorithms to ensure their usability in applications. Numerical results regarding a fluid-mechanical problem constrained by the Navier-Stokes equations, where the viscous energy dissipation is minimized, show its applicability.