Topological derivatives in shape optimization
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 with three-dimensional linear elasticity problems is also considered. The full mathematical justification for the asymptotic expansion associated with the singular domain perturbations is presented for the semilinear elliptic boundary value problems and the topological derivatives in linear elasticity contact problem are provided, too. Main results: Here, the results are presented in their closed form, which are useful for numerical methods in shape optimization. Some exercises are proposed at the end of each chapter. The shape derivatives which describe the dependence of solutions to elliptic boundary value problems with respect to the boundary variations of the geometrical domains are investigated. The variations of boundary value problems are considered in the scale of Sobolev spaces. The material derivatives are obtained for weak and strong solutions of the elliptic boundary value problems. The elliptic regularity of shape and material derivatives which depends on the regularity of the data, including the domain, sources and boundary conditions. The main aim of the presented book is the presentation of topological derivatives for a wide class of elliptic problems with the applications in numerical methods of shape and topology optimization. This volume is primarily addressed to applied mathematicians working in the field of partial differential equations and their applications, especially those concerned with numerical aspects. However, the book will also be useful for applied scientists from engineering and physics.
scientific article; zbMATH DE number 6126548