Ebook: Topological Derivative in Shape Optimization : Machine Learning in Social Media

The topological derivative is defined as the first term (correction) of the asymptotic expansion of a given shape functional with respect to a small parameter that measures the size of singular domain perturbations, such as holes, inclusions, defects, source-terms and cracks. Over the last decade, topological asymptotic analysis has become a broad, rich and fascinating research area from both theoretical and numerical standpoints. It has applications in many different fields such as shape and topology optimization, inverse problems, imaging processing and mechanical modeling including synthesi. Read more... Title Page; Preface; Contents; Notation; Introduction; The Topological Derivative Concept; Relationship between Shape and Topological Derivatives; The Topological-Shape Sensitivity Method; An Example of Topological Derivative Evaluation; Monograph Organization; Exercises; Domain Derivation in Continuum Mechanics; Material and Spatial Descriptions; Gradient of Scalar Fields; Gradient of Vector Fields; Spatial Description of Velocity Fields; Material Derivatives of Spatial Fields; Derivative of the Gradient of a Scalar Field; Derivative of the Gradient of a Vector Field. Material Derivatives of Integral ExpressionsDomain Integral; Boundary Integral; Summary of the Derived Formulae; The Eshelby Energy-Momentum Tensor; Exercises; Material and Shape Derivatives for Boundary Value Problems; Preliminaries; Sobolev-Slobodetskii Spaces; Elliptic Regularity; Elliptic Problems in Nonsmooth Domains; Shape Derivatives; Material Derivatives for Second Order Elliptic Equations; Weak Material Derivatives for the Dirichlet Problem; Strong Material Derivatives for the Dirichlet Problem; Material Derivatives for the Neumann Problem. Shape Derivatives for Second Order Elliptic EquationsShape Derivatives for the Dirichlet Problem; Shape Derivatives for the Neumann Problem; Material and Shape Derivatives for Elasticity Problems; Problem Formulation; Material Derivatives for Elasticity; Shape Derivatives for Elasticity; Shape Derivatives for Interfaces; Material and Shape Derivatives for Kirchhoff Plates; Problem Formulation; Material Derivatives for the Kirchhoff Plate; Shape Derivatives for the Kirchhoff Plate; Material and Shape Derivatives in Fluid Mechanics; The Adjugate Matrix Concept. Shape Derivatives for the Stationary, Homogeneous Navier-Stokes ProblemMaterial Derivatives for the Stationary, Homogeneous Navier-Stokes Problem; Exercises; Singular Perturbations of Energy Functionals; Second Order Elliptic Equation: The Poisson Problem; Problem Formulation; Shape Sensitivity Analysis; Asymptotic Analysis of the Solution; Topological Derivative Evaluation; Examples with Explicit Form of Topological Derivatives; Additional Comments and Summary of the Results; Second Order Elliptic System: The Navier Problem; Problem Formulation; Shape Sensitivity Analysis. Asymptotic Analysis of the SolutionTopological Derivative Evaluation; Fourth Order Elliptic Equation: The Kirchhoff Problem; Problem Formulation; Shape Sensitivity Analysis; Asymptotic Analysis of the Solution; Topological Derivative Evaluation; Exercises; Configurational Perturbations of Energy Functionals; Second Order Elliptic Equation: The Laplace Problem; Problem Formulation; Shape Sensitivity Analysis; Asymptotic Analysis of the Solution; Topological Derivative Evaluation; Numerical Example; Second Order Elliptic System: The Navier Problem; Problem Formulation