The shape derivative of the Gauss curvature

TL;DR

This paper derives new formulas for the shape derivatives of geometric quantities like Gauss curvature and introduces a Newton-type method for shape optimization, extending previous results to more general surface energies and higher dimensions.

Contribution

It provides the first systematic derivation of the shape derivative of Gauss curvature and generalizes shape derivative formulas for various geometric invariants in any dimension.

Findings

Derived formulas for shape derivatives of Gauss curvature and other invariants.

Developed a Newton-type numerical scheme for shape optimization.

Computed first and second order derivatives of area and Willmore functional.

Abstract

We introduce new results about the shape derivatives of scalar- and vector-valued functions, extending the results from (Dogan-Nochetto 2012) to more general surface energies. They consider surface energies defined as integrals over surfaces of functions that can depend on the position, the unit normal and the mean curvature of the surface. In this work we present a systematic way to derive formulas for the shape derivative of more general geometric quantities, including the Gauss curvature (a new result not available in the literature) and other geometric invariants (eigenvalues of the second fundamental form). This is done for hyper-surfaces in the Euclidean space of any finite dimension. As an application of the results, with relevance for numerical methods in applied problems, we introduce a new scheme of Newton-type to approximate a minimizer of a shape functional. It is a mathematically sound generalization of the method presented in (Chicco-Ruiz et. al. 2016). We finally find the particular formulas for the first and second order shape derivatives of the area and the Willmore functional, which are necessary for the Newton-type method mentioned above.