A structure theorem for shape functions defined on submanifolds

TL;DR

This paper presents a new structure theorem for shape functions on submanifolds, generalizing classical results and applying to shape derivatives in differential geometry and fracture mechanics.

Contribution

It introduces a comprehensive structure theorem for shape derivatives on submanifolds, unifying classical and specialized cases within a single framework.

Findings

General structure theorem for shape derivatives on submanifolds

Reformulation of Hadamard-Zolésio theorem within the new framework

Application to shape functions in differential geometry and fracture mechanics

Abstract

In this paper, we study shape functions depending on closed submanifolds. We prove a new structure theorem that establishes the general structure of the shape derivative for this type of shape function. As a special case we obtain the classical Hadamard-Zol\'esio structure theorem, but also the structure theorem for cracked sets can be recast into our framework. As an application we investigate several unconstrained shape functions arising from differential geometry and fracture mechanics.