A Topology-Preserving Level Set Method for Shape Optimization

TL;DR

The paper addresses enforcing topology and geometric constraints in shape optimization using level-set methods. It introduces a logarithmic barrier penalty $H(\phi)$ built from boundary offsets $I_d$ and $E_l$ that weakly enforces a fixed number of components, minimum component size, and distance constraints, leading to the penalized objective $F_\varepsilon(\phi)=F(\phi)+\varepsilon H(\phi)$. An approximate descent direction $u$ is derived from the derivative of $F_\varepsilon$ and extended off the boundary, with step-size safeguards ensuring topology preservation; the method is implemented in 2-D and remains grid-size independent for fixed $d,l$. Numerical examples demonstrate topology preservation while minimizing perimeter or a quadratic area-weighted objective under a fixed area constraint; results show components do not merge or vanish when constraints are active. The approach offers a practical, grid-resilient framework for constrained shape optimization that complements existing level-set techniques and can extend to higher dimensions.

Abstract

The classical level set method, which represents the boundary of the unknown geometry as the zero-level set of a function, has been shown to be very effective in solving shape optimization problems. The present work addresses the issue of using a level set representation when there are simple geometrical and topological constraints. We propose a logarithmic barrier penalty which acts to enforce the constraints, leading to an approximate solution to shape design problems.

Figures (4)

• Figure 1: The set $I_d$ (dashed curves) and $E_l$ (dotted curves).
• Figure 2: A blow-up of Fig. \ref{['fig1']}.
• Figure 3: The initial and optimized shape for example 1.
• Figure 4: The initial and optimized shape for example 2.