On a Classical Spectral Optimization Problem in Linear Elasticity

Annali di Matematica Pura ed Applicata, 2007

The best Sobloev trace constant is given by the first eigenvalue of a Steklov-like problem. We deal with minimizers of the Rayleigh quotient u 2 H 1 (Ω) / u 2 L 2 (∂Ω) for functions that vanish in a subset A ⊂ Ω, which we call the hole. We look for holes that minimize the best Sobolev trace constant among subsets of Ω with prescribed volume. First, we find a formula for the first variation of the first eigenvalue with respect to the hole. As a consequence of this formula, we prove that when Ω is a ball the symmetric hole (a centered ball) is critical when we consider deformations that preserves volume but is not optimal. Finally, we prove that by the Finite Element Method we can approximate the optimal configuration and, by means of the shape derivative, we design and algorithm to compute the discrete optimal holes.

2012

We consider the problem of finding the optimal constant for the embedding of the space

It is known that the torsional rigidity for a punctured ball, with the puncture having the shape of a ball, is minimum when the balls are concentric and the first eigenvalue for the Dirichlet Laplacian for such domains is also a maximum in this case. These results have been obtained by Ashbaugh and Chatelain (private communication), Harrell et. al., by Kesavan and, by Ramm and Shivakumar. In this paper we extend these results to the case of $p$-Laplacian for $1 < p < \infty$. For proving these results, we follow the same line of ideas as in the aforementioned articles, namely, study the sign of the shape derivative using the moving plane method and comparison principles. In the process, we obtain some interesting new side results such as the Hadamard perturbation formula for the torsional rigidity functional for the Dirichlet $p$-Laplacian, the existence and uniqueness result for a nonlinear pde and some extensions of known comparison results for nonlinear pdes.

SIAM Journal on Control and Optimization, 2021

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.

Georgian Mathematical Journal, 2017

In this paper, we consider an eigenvalue problem for the biharmonic operator that describes the transverse vibrations of the plate. Under the imposed boundary conditions, the eigenvalues of this operator are indeed eigenfrequencies of the clamped plate. The domain of the plate is taken variable and the domain functional, involving an eigenfrequency, is studied. A new formula for an eigenfrequency is proved, the first variation of the functional with respect to the domain is calculated, and the necessary condition for an optimal shape is derived. New explicit formulas are obtained for the eigenfrequency in the optimal domain in some particular cases.

We investigate numerically a 1956 conjecture of Payne, Polya, and Weinberger. The conjecture asserts that the ratio of the first two eigenvalues of the Laplacian on a bounded domainOmega of the plane with Dirichlet boundary conditions reaches its minimum value precisely whenOmega is a disk. A crucial feature of this problem is the loss of smoothness of the objective function at the solution. The following results form the core of our numerical treatment. First, we construct finite dimensional families of deformations of a disk equipped with a uniform triangulation. This permits the formulation of a discrete model of the problem via finite element techniques. Second, we build on the work of M. Overton to derive optimality conditions in terms of Clarke's generalized gradients for nonsmooth functions. These ideas are then combined into an algorithm and implemented in Fortran. Contents 1 Introduction 2 2 Preliminaries 4 3 Max characterization of P k i=1 i for symmetric matrices 8 4 ...

Applied Mathematical Modelling, 2017

Biharmonic eigenvalue problems arise in the study of the mechanical vibration of plates. In this paper, we study the minimization of the first eigenvalue of a simplified model with clamped boundary conditions and Navier boundary conditions with respect to the coefficient functions which are of bang-bang type (the coefficient functions take only two different constant values). A rearrangement algorithm is proposed to find the optimal coefficient function based on the variational formula of the first eigenvalue. On various domains, such as square, circular and annular domains, the region where the optimal coefficient function takes the larger value may have different topologies. An asymptotic analysis is provided when two different constant values are close to each other. In addition, a symmetry breaking behavior is also observed numerically on annular domains.

Mathematical Results in Quantum Mechanics, 1999

We consider Laplace operators and Schrödinger operators with potentials containing curvature on certain regions of nontrivial topology, especially closed curves, annular domains, and shells. Dirichlet boundary conditions are imposed on any boundaries. Under suitable assumptions we prove that the fundamental eigenvalue is maximized when the geometry is round.

Journal of Optimization Theory and Applications, 1991

In this paper, we consider problems of eigenvalue optimization for elliptic boundary-value problems. The coefficients of the higher derivatives are determined by the internal characteristics of the medium and play the role of control. The necessary conditions of the first and second order for problems of the first eigenvalue maximization are presented. In the case where the maximum is reached on a simple eigenvalue, the second-order condition is formulated as completeness condition for a system of functions in Banach space. If the maximum is reached on a double eigenvalue, the necessary condition is presented in the form of linear dependence for a system of functions. In both cases, the system is comprised of the eigenfunctions of the initialboundary value problem. As an example, we consider the problem of maximization of the first eigenvalue of a buckling column that lies on an elastic foundation.

The three-dimensional spectral elasticity problem is studied in an anisotropic and inhomogeneous solid with small defects, i.e., inclusions, voids, and microcracks. Asymptotics of eigenfrequencies and the corresponding elastic eigenmodes are constructed and justified. New technicalities of the asymptotic analysis are related to variable coefficients of differential operators, vectorial setting of the problem, and usage of intrinsic integral characteristics of defects. The asymptotic formulae are developed in a form convenient for application in shape optimization and inverse problems.

2013

In this paper, we analyze an optimization problem for the first (nonlinear) Steklov eigenvalue plus a boundary potential with respect to the potential function which is assumed to be uniformly bounded and with fixed L^1-norm.

2009

In this article we deal with the problem of distributing two conducting materials in a given domain, with their proportions being fixed, so as to minimize the first eigenvalue of a Dirichlet operator. When the design region is a ball, it is known that there is an optimal distribution of materials which does not involve the mixing of the materials. However, the optimal configuration even in this simple case is not known. As a step in the resolution of this problem, in this paper, we develop the shape derivative analysis for this two-phase eigenvalue problem in a general domain. We also obtain a formula for the shape derivative in the form of a boundary integral and obtain a simple expression for it in the case of a ball. We then present some numerical calculations to support our conjecture that the optimal distribution in a ball should consist in putting the material with higher conductivity in a concentric ball at the centre.

ESAIM: Control, Optimisation and Calculus of Variations, 2015

In this paper we study the asymptotic behavior of some optimal design problems related to nonlinear Steklov eigenvalues, under irregular (but diffeomorphic) perturbations of the domain.

2013

Partial Differential Equations and Boundary Value Problems, 1998

This paper describes some properties and applications of Steklov eigenproblems for prototypical second-order elliptic operators on bounded regions in R n. Results are described for Schroedinger and weighted harmonic equations. A variational description of the least eigenvalue leads to optimal L 2-trace inequalities. It is shown that the eigenfunctions provide complete orthonormal bases of certain closed subspaces of H 1 ðOÞ and also of L 2 ð@O; dsÞ. This allows the description, and representation, of solution operators for homogeneous elliptic equations subject to inhomogeneous Dirichlet, Neumann or Robin boundary data. They are also used to describe Robin to Dirichlet and Neumann to Dirichlet operators for these equations, and to describe the spectrum of these operators. The allowable regions are quite general; in particular classes of bounded regions with a finite number of disjoint Lipschitz components for the boundary are allowed.