Multiobjective Shape Design in Electricity and Magnetism

Abstract

Preface. 1 INTRODUCTION. 2 INVERSE PROBLEMS AND ERROR MINIMISATION. 2.1 A Copernican revolution: direct and inverse problems. 2.2 Insidiousness of inverse problems. 2.3 Classification of inverse problems. 2.4 Green formula and Fredholm equation. 2.5 Solving inverse problems by minimising a functional. 2.6 Constrained minimisation. 2.7 Local vs global search. 2.8 Evolutionary computing. 2.9 Solving inverse problems by means of rectangular systems of algebraic equations. 3 A PARETIAN APPROACH TO MOSD THEORY. 3.1 Need of a multiobjective formulation. 3.2 Multiobjective formulation of a design problem. 3.3 Paretian optimality. 4 FIELD MODELS AND SHAPE DESIGN. 4.1 Maxwell equations in differential form. 4.2 Wave, diffusion and steady-state equations in terms of vectors. 4.3 Wave, diffusion and steady-state equations in terms of potentials. 4.4 Boundary and transmission conditions. 4.5 Insidiousness of direct problems. 4.6 Field-based inverse problems. 4.7 More insidious difficulties. 4.8 A unifying view of analysis and synthesis. 5 SOLVING MULTIOBJECTIVE OPTIMISATION PROBLEMS. 5.1 Classical methods of multiobjective optimisation. 5.2 Classical vs Paretian formulation. 5.3 Evolutionary methods of multiobjective optimisation. 5.4 Multi-objective evolution strategy (MOESTRA). 5.5 The gradient-balance method for 2D problems. 6 A FIELD-BASED BENCHMARK. 6.1 A twofold meaning of benchmarking. 6.2 Test problem: shape design of a magnetic pole. 6.3 The test problem simplified. 6.4 Criticism to Pareto optimality in the static case. 7 STATIC MOSD. 7.1 A bibliographic insight. 7.2 FEM-assisted optimal design. 7.3 Test problem: a priori analysis of the objective space. 7.4 Optimisation strategies and results. 7.5 Processing clusters. 7.6 The test problem solved by means of the GB method. 7.7 An industrial case study: permanent-magnet alternator. 8 MOVING ALONG THE PARETO FRONT. 8.1 John optimality. 8.2 Reconsidering the industrial case study. 8.3 Exploring the Pareto front. 8.4 Optimising along the front. 9 SENSITIVITY ANALYSIS AND MOSD. 9.1 Discrete sets and perturbation domains. 9.2 Case study: superconducting magnetic-bearing design. 9.3 Design optimisation of the PM-HTSC interaction. 9.4 An inexpensive evaluation of sensitivity. 9.5 Results. 10 NON-CONFLICTING MULTIPLE OBJECTIVES. 10.1 Case study: a system for magnetic induction tomography. 10.2 Design problem. 10.3 Analysis problem. 10.4 Optimal shape design of the MIT antenna. 11 HIGHER-ORDER DIMENSIONALITY. 11.1 Case study: an electrostatic micromotor. 11.2 Field analysis: doubly-connected domain. 11.3 Field synthesis and rotor shape design. 11.4 Results. 11.5 A criterion for decision making. 12 MULTI-SCALE EVOLUTION STRATEGY. 12.1 Industrial electromagnetic design. 12.2 A multi-scale evolutionary search. 12.3 Permanent-magnet alternator design. 12.4 Results. 13 GAME THEORY AND MOSD. 13.1 From Pareto front to Nash equilibrium. 13.2 Theoretical background. 13.3 Analytical validation. 13.4 Numerical implementation. 13.5 Case study: permanent-magnet motor design. 14 DYNAMIC MOSD. 14.1 From static to dynamic conditions. 14.2 Theoretical background. 14.3 An analytical benchmark. 14.4 Criticism to dynamic Pareto optimality. 14.5 Numerical benchmark. 14.6 Direct problem. 14.7 Design problem. 14.8 Auxiliary inverse problems. 14.9 Main inverse problem: synthesising the device geometry. 14.10 Computational aspects. 14.11 Results I. 14.12 The design problem revisited: recovering steady state from time evolution. 14.13 Results II. 15 AN INTRODUCTION TO BAYESIAN PROBABILITY THEORY. 15.1 Bayesian conception of probability. 15.2 Prior distributions. 15.3 Bayesian inference vs maximum likelihood. 15.4 Bayesian non-parametric problems. 15.5 Model choice. 16 A BAYESIAN APPROACH TO MULTIOBJECTIVE OPTIMISATION. 16.1 Reasons for a new approach. 16.2 Weak regularity. 16.3 Local Bayesian formulation. 16.4 Integral Bayesian formulation. 16.5 Computation of the Bayesian terms. 16.6 Bayesian imaging. 17 BAYESI