Shape Derivative for Some Eigenvalue Functionals in Elasticity Theory

Abstract

This work is the second part of a previous paper which was devoted to scalar problems. Here we study the shape derivative of eigenvalue problems of elasticity theory for various kinds of boundary conditions, that is Dirichlet, Neumann, Robin, and Wentzell boundary conditions. We also study the case of composite materials, having in mind applications in the sensitivity analysis of mechanical devices manufactured by additive printing.The main idea, which rests on the computation of the derivative of a minimum with respect to a parameter, was successfully applied in the scalar case in the first part of this paper and is here extended to more interesting situations in the vectorial case (linear elasticity), with applications in additive manufacturing. These computations for eigenvalues in the elasticity problem for generalized boundary conditions and for composite elastic structures constitute the main novelty of this paper. The results obtained here also show the efficiency of this method for such calculations whereas the methods used previously even for classical clamped or transmission boundary conditions are more lengthy or, are based on various simplifying assumptions, such as the simplicity of the eigenvalue or the existence of a shape derivative.

Shape optimization of a Dirichlet type energy for semilinear elliptic partial differential equations

“…Such semi-differentials are of particular importance when dealing with multiple eigenvalues. A systematic approach was developed by several authors to handle such situations; while we refer to section 1.3 for precise references, let us point to [13], which is the most related to our work as it deals with semi-differentials for the linear elasticity system. The first-order optimality conditions for (5) read: if Ω * is a smooth optimal set for any smooth vector field Φ, there holds…”

“…The relevant formulae will be recalled in the course of this article. In a recent paper, Caubet, Dambrine & Mahadevan [13] developed this semi-differential approach to accomodate the linear elasticity system.…”

“…To write things down in a precise way, we consider a domain Ω whose boundary is of class C 3 . Although λ 1 (Ω) may not be simple, it has finite multiplicity, say N , and the map F is semi-shape differentiable at Ω [20, Theorem 2.1, Chapter 1] (see also [13,Theorem 2.1 and Remark 2.2] and [15]), in the sense that, for any vector field Φ ∈ X , the limit…”