Christian Magele

Stochastic algorithms in electromagnetic optimization

This paper gives an overview of some stochastic optimization strategies, namely, evolution strategies, genetic algorithms, and simulated annealing, and how these methods can be applied to problems in electrical engineering. Since these methods usually require a careful tuning of the parameters which control the behavior of the strategies (strategy parameters), significant features of the algorithms implemented by the authors are presented. An analytical comparison among them is performed. Finally, results are discussed on three optimization problems.

Multiobjective optimization in magnetostatics: a proposal for benchmark problems

A proposal for benchmark problems to test electromagnetic optimization methods, relevant to multiobjective optimization of a solenoidal superconducting magnetic energy storage with active and passive shielding is presented. The system has been optimized by means of different optimization procedures based on the global search algorithm, evolution strategies, simulated annealing and the conjugate gradient method, all coupled to integral or finite element codes. A comparison of results is performed and the features of the problem as a test of optimization procedures are discussed.

Numerical analysis of 3D magnetostatic fields

Formulations of three-dimensional magnetostatic fields are reviewed for their finite-element analysis. Partial differential equations and boundary conditions are set up for various kinds of potentials. Besides the method using two scalar potentials, several vector potential formulations are also discussed. Galerkin techniques combined with the finite element method are applied for the numerical solution of the boundary value problems. The effect of gauging the vector potential upon the numerical performance is investigated. Solutions by different formulations to a simple test problem and a benchmark problem involving relatively thin saturated iron plates are presented. The latter is compared to measured results. >

Different finite element formulations of 3D magnetostatic fields

Several finite-element formulations of three-dimensional magnetostatic fields are reviewed. Both nodal and edge elements are considered. The aim is to suggest remedies to some shortcomings of widely used methods. Various formulations are compared based on results for Problem No. 13 of the TEAM Workshops, a nonlinear magnetostatic problem involving thin iron plates. >

Global optimization methods for computational electromagnetics

Both higher-order (pseudo)deterministic and zeroth-order probabilistic optimization methods have been analyzed and tested for solving the global optimization problems arising in computational electromagnetics. Previously recommended, but seemingly independent schemes (evolution strategies, simulated annealing, Monte Carlo iteration) have been unified into a robust general method: the global evolution strategy (GES). Regularization techniques, the stability of solutions, and nonlinear phenomena are shown to be topics closely related to global optimization and inverse problems. The speed of convergence is evaluated for different optimization methods. A real-world application (from nuclear magnetic resonance and magnetic resonance imaging) demonstrates the favorable behavior of GES in the context of the finite element method. >

FEM and evolution strategies in the optimal design of electromagnetic devices

The application of evolution strategies to the optimal design of electromagnetic devices is investigated. The corresponding field analysis is performed by the finite element method. The strategies involve a simplified simulation of biological evolution. They are especially advantageous in the case of strongly nonlinear optimization problems. The three strategies used are the (1+1), the

Managing uncertainties in electromagnetic design problems with robust optimization

Numerical optimization techniques are widely used for the design of electromagnetic devices. In practical implementations of such devices, the problem parameters may be subject to tolerances and uncertainties. Optimal designs should be insensitive to parameter variations. The demand for robustness is often neglected during the optimization process. A formulation of robust nonlinear design problems is proposed in this paper. The influence of uncertainties on the target performance and the feasibility of a solution is assessed and incorporated into the optimization strategy. Methods for the solution of robust problems are introduced. Special emphasis is put on keeping the computational effort as small as possible. The proposed robust optimization method is applied to a standard benchmark optimization problem.

Comparison of different optimization strategies in the design of electromagnetic devices

Both higher-order deterministic optimization methods (steepest-descent, conjugate gradient, quasi-Newton) and zeroth-order stochastic optimization methods (( mu + lambda )-evolution strategy) have been tested for designing electromagnetic devices. The underlying field analyses have involved the finite element method in 2-D and 3-D, both linear and nonlinear. Numerical experiments indicate some preference for the ( mu + lambda )-evolution strategy not only because of its robustness and generality, but also its speed of convergence. >

Higher order evolution strategies for the global optimization of electromagnetic devices

Basic evolution strategies (ESs) utilizing simplified features of biological evolution like mutation and selection are applied to solve problems of parameter identification for the optimal design of electromagnetic devices. Their main advantage lies in their stable convergence behavior and their self-adapting stepwidth sigma . Higher order ( mu / rho , lambda ) evolution strategies can be successfully applied to multimodel problems. To ensure at least a result very near the global optimum, a theory of disaster can be introduced. After a certain number of standard generations, the stepwidth sigma is increased greatly for a specified number of generations. Descendants arrived at during these generations are more likely to escape a local minimum than ordinary descendants. ( mu / rho , lambda ) evolution strategies can be easily implemented on parallel computers, helping to reduce the real-time requirements for one generation. >

Fast calculation of the sensitivity matrix in magnetic induction tomography by tetrahedral edge finite elements and the reciprocity theorem

Magnetic induction tomography of biological tissue is used to reconstruct the changes in the complex conductivity distribution by measuring the perturbation of an alternating primary magnetic field. To facilitate the sensitivity analysis and the solution of the inverse problem a fast calculation of the sensitivity matrix, i.e. the Jacobian matrix, which maps the changes of the conductivity distribution onto the changes of the voltage induced in a receiver coil, is needed. The use of finite differences to determine the entries of the sensitivity matrix does not represent a feasible solution because of the high computational costs of the basic eddy current problem. Therefore, the reciprocity theorem was exploited. The basic eddy current problem was simulated by the finite element method using symmetric tetrahedral edge elements of second order. To test the method various simulations were carried out and discussed.