Comparison between shape optimization and volumic level set approximation for geometrical functionals

TL;DR

This paper compares shape optimization and volumic level set methods for differentiating curvature functionals, demonstrating their equivalence in results through theoretical analysis.

Contribution

It introduces a comparison framework for two differentiation approaches of curvature functionals, showing their equivalence.

Findings

Both approaches yield the same derivative results.

The methods are theoretically equivalent for curvature functionals.

Provides insights into the relationship between shape optimization and level set methods.

Abstract

We propose to differentiate a general curvature functional with two different approaches. In the first one we compute the derivative with the tools of shape optimization and in the second one we compute the derivative of a volumic approximation of the functional with respect to a level set function. We show that the two previous approaches give the same result.