Inverse Problems and Optimal Design in Electricity and Magnetism P. Neittaanmäki Department of Mathematics, University of Jyväskylä M. Rudnicki Institute of Electrical Engineering, Warsaw and A. Savini Department of Electrical Engineering, University ofpavia CLARENDON PRESS OXFORD
Contents I Mathematical methodology 1. Mathematical preliminaries 3 1.1. Basic notation 3 1.2. Vectors, matrices, norms 4 1.2.1. Matrices 4 1.2.2. Independence, orthogonality, subspaces 6 1.2.3. Special matrices 7 1.2.4. Block matrices 8 1.2.5. Vector norms 9 1.2.6. Matrix norms 11 1.3. Functions 12 1.4. Domains 13 1.5. Function spaces 14 1.6. Classical integral formulae 21 1.7. References 23 2. Boundary-value problems 24 2.1. Variational methods for elliptic problems 24 2.1.1. Classical formulation 24 2.1.2. Variational formulation 25 2.2. Integral-equation formulation for partial differential 33 equations 2.3. References 36 3. Numerical methods for boundary-value problems 38 3.1. Introduction 38 3.2. Finite-element method for elliptic problems 42 3.2.1. Triangulation of the domain 42 3.2.2. Finite-element equations 45 3.3. Boundary-element method 51 3.4. References 54 4. Regularization 56 4.1. Ill-posed problems 56 4.2. Regularization schemes 57 4.3. Discrepancy principle 59 4.4. Linear integral equations of the first kind 61 4.5. Tikhonov regularization 62 4.6. Truncated singular-value decomposition 66 4.7. Regularization parameter 67
x Contents 4.8. Solving Fredholm integral equations of the first 69 kind using regularization 4.9. References 72 Numerical methods for systems of equations 76 5.1. Solving linear systems 76 5.1.1. Elementary iterative methods 77 5.1.2. The conjugate gradient method 78 5.1.3. The preconditioned biconjugate gradient method 79 5.2. Solving nonlinear systems 80 5.2.1. Slope methods 80 5.2.2. Newton-Raphson method 82 5.2.3. Powell's hybrid method 83 5.2.4. Brown-Brent methods 84 5.3. References 87 Unconstrained optimization 89 6.1. Optimality conditions 94 6.2. Search methods 100 6.3. Steepest-descent method 102 6.4. Conjugate gradient method 103 6.5. Newton method with a linear search 105 6.6. Quasi-Newton or variable-metric method 107 6.7. Polytope method 111 6.8. Stochastic optimization 115 6.8.1. Random-search methods 115 6.8.2. Genetic algorithms 116 6.8.3. Simulated annealing 116 6.8.4. Simulated annealing optimization code 119 6.9. Neural computing 123 6.9.1. Introduction 123 6.9.2. Preliminaries 123 6.9.3. Multilayer feedforward networks 124 6.9.4. Optimization by neural networks 128 6.10. References 131 Constrained optimization 137 7.1. Linear programming 137 7.2. Optimality (Karush-Kuhn-Tucker) conditions 139 7.2.1. Smooth case 139 7.2.2. Nonsmooth case 141 7.3. Sequential linear programming 142 7.4. Sequential quadratic programming 147 7.5. Nonsmooth and multicriteria optimization 151 7.5.1. Smooth multicriteria optimization 152 7.5.2. Unconstrained convex optimization 152 7.5.3. Unconstrained nonconvex optimization 153 7.5.4. Constrained optimization 154
Contents xi 7.5.5. Multicriteria optimization 154 7.5.6. Proximal bundle nonsmooth optimization code 155 7.6. Concluding remarks 158 7.7. References 159 Linear least-squares 161 8.1. Overdetermined systems of equations 161 8.2. Normal equations and ill-conditioning 163 8.3. Singular-value decomposition 164 8.4. Constrained linear least-squares 167 8.5. References 170 Nonlinear least-squares 172 9.1. Gauss-Newton method 173 9.2. Levenberg-Marquardt method 175 9.3. Powell's hybrid method 178 9.4. Derivative-free methods 179 9.5. Large-residual problems 180 9.6. Trust-region method 181 9.7. Constrained nonlinear least-squares 183 9.8. References 185 II Fundamentals of electromagnetism 10. Introduction 191 11. Maxwell's equations 192 11.1. Basic equations 192 11.2. Interface conditions 195 11.3. References 197 12. Potential equations in electricity and magnetism 198 12.1. Laplace's equation 198 12.2. Poisson's equation 200 12.3. Nonlinear magnetostatic fields in R 2 201 12.4. Quasistatic linear electromagnetic fields 202 12.5. Electromagnetic fields in linear isotropic media 205 12.6. Further elements of electromagnetic theory 207 12.6.1. Energy, power and forces 207 12.6.2. Coulomb's law and the Biot-Savart law 208 12.7. References 209 13. Numerical methods in electromagnetism 210 13.1. Introduction 210 13.2. Approximation of the static case 210 13.2.1. Least-squares variational methods in 210 electromagnetism 13.2.2. Least-squares finite-element methods 212 for electric fields 13.2.3. The nonlinear case 213
xii Contents 13.3. Approximation of quasistatic electric and 216 magnetic fields 13.4. Approximation of the wave equation for electric 221 scalar potentials 13.5. Concluding remarks 223 13.6. References 224 III Inverse problems and optimal design in electromagnetic applications 14. Inverse problems and optimal design 231 14.1. Introduction 231 14.2. Inverse electromagnetic problems: methodology 232 14.3. Optimal design techniques for solving inverse 233 electromagnetic problems 14.3.1. Deterministic optimization methods (DOMs) 235 14.3.2. Stochastic optimization methods (SOMs) 235 14.4. References 236 15. Synthesis of sources 237 15.1. Synthesis of the magnetic field in a solenoid 237 15.2. Synthesis of the magnetic field on a solenoid axis 239 15.3. Synthesis of an electric field due to a point charge 243 15.4. Synthesis of an electric field due to a surface charge 245 15.4.1. Parallel-plate capacitors 245 15.4.2. Thin conducting plate 247 15.5. Synthesis of a magnetic field in a wire system 249 15.6. Synthesis of electromagnets 256 15.7. References 258 16. Synthesis of boundary conditions 259 16.1. Synthesis of the electric field in a finite domain 259 16.2. Synthesis of an electric field due to a boundary 262 potential 16.3. Synthesis of the magnetic field due to a boundary 266 current 16.4. References 268 17. Synthesis of material properties 269 17.1. Synthesis of permittivity 270 17.2. References 275 18. Optimal shape synthesis 276 18.1. Optimal shape design of a solenoid 276 18.2. Optimal shape design of the shim coil of a solenoid 279 18.3. Optimal shape design of an electromagnet 282 18.4. Optimal shape design of an air-filled capacitor 285 18.5. Optimal shape design of a pole profile in a linear 288 H-shaped magnet
Contents xiii 18.6. Optimal shape design of a pole profile in a nonlinear 291 H-shaped magnet 18.7. References 297 19. Remarks on inverse and design problems 298 19.1. Survey of solved problems 298 19.2. References 304 20. Artificial neural networks (ANNs) for inverse electromagnetic 309 modelling 20.1. Remarks on artificial neural networks 309 20.2. References 312 IV Implementation of the FEM, design-sensitivity and shape design procedures 21. Introduction 315 22. Implementation of the finite-element method 316 22.1. Linear elliptic boundary-value problems 316 22.1.1. Discretization of the problem 316 22.1.2. Isoparametric elements 317 22.1.3. Data structures 320 22.1.4. General program structure 322 22.2. A nonlinear FEM solver using a quasi-newton method 322 22.3. References 324 23. Finite-element design-sensitivity analysis 325 23.1. Setting of the optimal shape design problem 325 23.2. Design-sensitivity analysis for linear problems 327 23.3. Sensitivity for the nonlinear potential equation 330 23.4. Implementation of optimal shape design procedures 333 23.5. Automatic differentiation of computer programs 335 23.6. References 338 24. Subroutine libraries 339 24.1. General-purpose software libraries 339 24.2. Partial differential equations and electromagnetic 341 software 24.3. Software libraries for nonsmooth and multicriterion 348 optimization 24.4. Artificial intelligence tools and software for optimal 349 shape design 24.5. References 349 Author index 355 Subject index 359