Inverse Problems and Optimal Design in Electricity and Magnetism

Transcription

1 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

2 Contents I Mathematical methodology 1. Mathematical preliminaries Basic notation Vectors, matrices, norms Matrices Independence, orthogonality, subspaces Special matrices Block matrices Vector norms Matrix norms Functions Domains Function spaces Classical integral formulae References Boundary-value problems Variational methods for elliptic problems Classical formulation Variational formulation Integral-equation formulation for partial differential 33 equations 2.3. References Numerical methods for boundary-value problems Introduction Finite-element method for elliptic problems Triangulation of the domain Finite-element equations Boundary-element method References Regularization Ill-posed problems Regularization schemes Discrepancy principle Linear integral equations of the first kind Tikhonov regularization Truncated singular-value decomposition Regularization parameter 67

3 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 Solving linear systems Elementary iterative methods The conjugate gradient method The preconditioned biconjugate gradient method Solving nonlinear systems Slope methods Newton-Raphson method Powell's hybrid method Brown-Brent methods References 87 Unconstrained optimization Optimality conditions Search methods Steepest-descent method Conjugate gradient method Newton method with a linear search Quasi-Newton or variable-metric method Polytope method Stochastic optimization Random-search methods Genetic algorithms Simulated annealing Simulated annealing optimization code Neural computing Introduction Preliminaries Multilayer feedforward networks Optimization by neural networks References 131 Constrained optimization Linear programming Optimality (Karush-Kuhn-Tucker) conditions Smooth case Nonsmooth case Sequential linear programming Sequential quadratic programming Nonsmooth and multicriteria optimization Smooth multicriteria optimization Unconstrained convex optimization Unconstrained nonconvex optimization Constrained optimization 154

4 Contents xi Multicriteria optimization Proximal bundle nonsmooth optimization code Concluding remarks References 159 Linear least-squares Overdetermined systems of equations Normal equations and ill-conditioning Singular-value decomposition Constrained linear least-squares References 170 Nonlinear least-squares Gauss-Newton method Levenberg-Marquardt method Powell's hybrid method Derivative-free methods Large-residual problems Trust-region method Constrained nonlinear least-squares References 185 II Fundamentals of electromagnetism 10. Introduction Maxwell's equations Basic equations Interface conditions References Potential equations in electricity and magnetism Laplace's equation Poisson's equation Nonlinear magnetostatic fields in R Quasistatic linear electromagnetic fields Electromagnetic fields in linear isotropic media Further elements of electromagnetic theory Energy, power and forces Coulomb's law and the Biot-Savart law References Numerical methods in electromagnetism Introduction Approximation of the static case Least-squares variational methods in 210 electromagnetism Least-squares finite-element methods 212 for electric fields The nonlinear case 213

5 xii Contents Approximation of quasistatic electric and 216 magnetic fields Approximation of the wave equation for electric 221 scalar potentials Concluding remarks References 224 III Inverse problems and optimal design in electromagnetic applications 14. Inverse problems and optimal design Introduction Inverse electromagnetic problems: methodology Optimal design techniques for solving inverse 233 electromagnetic problems Deterministic optimization methods (DOMs) Stochastic optimization methods (SOMs) References Synthesis of sources Synthesis of the magnetic field in a solenoid Synthesis of the magnetic field on a solenoid axis Synthesis of an electric field due to a point charge Synthesis of an electric field due to a surface charge Parallel-plate capacitors Thin conducting plate Synthesis of a magnetic field in a wire system Synthesis of electromagnets References Synthesis of boundary conditions Synthesis of the electric field in a finite domain Synthesis of an electric field due to a boundary 262 potential Synthesis of the magnetic field due to a boundary 266 current References Synthesis of material properties Synthesis of permittivity References Optimal shape synthesis Optimal shape design of a solenoid Optimal shape design of the shim coil of a solenoid Optimal shape design of an electromagnet Optimal shape design of an air-filled capacitor Optimal shape design of a pole profile in a linear 288 H-shaped magnet

6 Contents xiii Optimal shape design of a pole profile in a nonlinear 291 H-shaped magnet References Remarks on inverse and design problems Survey of solved problems References Artificial neural networks (ANNs) for inverse electromagnetic 309 modelling Remarks on artificial neural networks References 312 IV Implementation of the FEM, design-sensitivity and shape design procedures 21. Introduction Implementation of the finite-element method Linear elliptic boundary-value problems Discretization of the problem Isoparametric elements Data structures General program structure A nonlinear FEM solver using a quasi-newton method References Finite-element design-sensitivity analysis Setting of the optimal shape design problem Design-sensitivity analysis for linear problems Sensitivity for the nonlinear potential equation Implementation of optimal shape design procedures Automatic differentiation of computer programs References Subroutine libraries General-purpose software libraries Partial differential equations and electromagnetic 341 software Software libraries for nonsmooth and multicriterion 348 optimization Artificial intelligence tools and software for optimal 349 shape design References 349 Author index 355 Subject index 359