Topological derivatives in shape optimization
type: Item
instance of: scholarly article
title: Topological derivatives in shape optimization
zbMATH Open document ID: 1276.35002
DOI: 10.1007/978-3-642-35245-4
author: A. A. Novotny
author: Jan Sokołowski
published in: Interaction of Mechanics and Mathematics
publication date: 2013-01-17T00:00:00Z
review text: In the very important book, the authors recall the classical approach to shape optimization problems and extend the analysis to some singular perturbations of the reference domains for elliptic boundary value problems. Here, the topological derivative can be considered as the singular limit of the classical shape derivative (this means that the topological derivatives are the main subject of the authors' studies). It is obtained by the asymptotic analysis of classical solutions to the asymptotic analysis of the shape functionals, all together with respect to the small parameter. The closed form of the topological derivatives for shape and topology optimization problems governed by the elliptic boundary value problems is proposed. The authors consider the following aspects of the theory of topological derivatives: Descriptions of mathematical theory of shape sensitivity analysis and of domain variations technique in continuous mechanics. Asymptotic analysis in singularly perturbed domains by the compound asymptotic expansions. Topological derivatives of shape functionals for linear and nonlinear elliptic boundary value problems. Mathematical properties of topological derivatives for nonsmooth variational problems. Topological derivatives with respect to singular domain perturbations close to the boundary. Relationship between shape gradients and topological derivatives. Closed formulae for the topological derivatives of the energy -- based shape functionals. Examples and applications to mathematical physics and structural mechanics. The authors consider the problem of how to derive topological asymptotic expansions of shape functionals associated with elliptic boundary value problems. The calculation of the topological derivative for two-dimensional elliptic problems including scalar (Laplace) and vectorial (Navier) second-order partial differential equations and scalar fourth-order partial differential equations (Kirchhoff). The topological derivative associated wit…
Mathematics Subject Classification ID: 35-02
Mathematics Subject Classification ID: 65-02
Mathematics Subject Classification ID: 35Jxx
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35Q74
zbMATH DE Number: 6126548
zbMATH Keywords: shape transformation
zbMATH Keywords: topological perturbations
zbMATH Keywords: unperturbed boundary value problem
zbMATH Keywords: perturbed boundary value problems
zbMATH Keywords: material derivatives
zbMATH Keywords: elliptic regularity
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: shape functional
zbMATH Keywords: spectral problem in elasticity
zbMATH Keywords: singular domain perturbations
zbMATH Keywords: linear elasticity contact problem
reviewed by: Jan Lovíšek
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-642-35245-4
OpenAlex ID: W2478917285
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description: scientific article; zbMATH DE number 6126548