A product shape manifold approach for optimizing piecewise-smooth shapes

TL;DR

The paper addresses optimizing piecewise-smooth shapes within PDE-constrained shape optimization by building a shape space with a Riemannian product-manifold structure that can represent glued-together, kink-containing shapes. It defines a gradient-based optimization framework on the novel space $M_s(U^N)$ using a multi-pushforward to obtain a multi-shape gradient and updates via a retraction-based gradient descent, tailored for piecewise-smooth shapes. The approach is demonstrated on a steady Navier-Stokes problem where the objective $j(u)=\int_D \frac{\mu}{2} \nabla v : \nabla v \, dx$ is minimized under PDE and geometric constraints, achieving substantial objective reduction and convergence of the geometry within tight feasibility. The work broadens shape-optimization applicability to non-smooth, multi-shape configurations in fluid dynamics and related PDE applications, with future directions including analysis of kink overlaps and convergence guarantees.

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.

Figures (3)

• Figure 1: Sketch of two shapes $u_1$, $u_2$ surrounded by a domain $D_{\bm{u}}\subset \mathbb{R}^2$.
• Figure 2: Optimization results: objective functional (left) and $H^1$-norm of the mesh deformation (right).
• Figure 3: Fluid velocity magnitude at different stages of the optimization. Figure \ref{['fig:NumericResults_velocity_iterEnd']} has an increased objective functional value in comparison to figure \ref{['fig:NumericResults_velocity_iter2000']} and \ref{['fig:NumericResults_velocity_iter15000']}, however it fulfills the geometrical constraints while the others do not yet.

Theorems & Definitions (3)

• Definition 2.1
• Definition 2.2
• Definition 2.3