Applications of the Topological Derivative Method

Description

The book presents new results and applications of the topological derivative method in control theory, topology optimization and inverse problems. It also introduces the theory in singularly perturbed geometrical domains using selected examples. Recognized as a robust numerical technique in engineering applications, such as topology optimization, inverse problems, imaging processing, multi-scale material design and mechanical modeling including damage and fracture evolution phenomena, the topological derivative method is based on the asymptotic approximations of solutions to elliptic boundary value problems combined with mathematical programming tools. The book presents the first order topology design algorithm and its applications in topology optimization, and introduces the second order Newton-type reconstruction algorithm based on higher order topological derivatives for solving inverse reconstruction problems. It is intended for researchers and students in applied mathematics and computational mechanics interested in the mathematical aspects of the topological derivative method as well as its applications in computational mechanics. Front Matter ....Pages i-xiv Introduction (Antonio André Novotny, Jan Sokołowski, Antoni Żochowski)....Pages 1-12 Theory in Singularly Perturbed Geometrical Domains (Antonio André Novotny, Jan Sokołowski, Antoni Żochowski)....Pages 13-39 Steklov–Poincaré Operator for Helmholtz Equation (Antonio André Novotny, Jan Sokołowski, Antoni Żochowski)....Pages 41-50 Topological Derivatives for Optimal Control Problems (Antonio André Novotny, Jan Sokołowski, Antoni Żochowski)....Pages 51-60 Optimality Conditions with Topological Derivatives (Antonio André Novotny, Jan Sokołowski, Antoni Żochowski)....Pages 61-84 A Gradient-Type Method and Applications (Antonio André Novotny, Jan Sokołowski, Antoni Żochowski)....Pages 85-107 Synthesis of Compliant Thermomechanical Actuators (Antonio André Novotny, Jan Sokołowski, Antoni Żochowski)....Pages 109-128 Synthesis of Compliant Piezomechanical Actuators (Antonio André Novotny, Jan Sokołowski, Antoni Żochowski)....Pages 129-148 Asymptotic Analysis of Variational Inequalities (Antonio André Novotny, Jan Sokołowski, Antoni Żochowski)....Pages 149-164 A Newton-Type Method and Applications (Antonio André Novotny, Jan Sokołowski, Antoni Żochowski)....Pages 165-181 The Electrical Impedance Tomography Problem (Antonio André Novotny, Jan Sokołowski, Antoni Żochowski)....Pages 183-200 Back Matter ....Pages 201-212