There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

List of Figures. List of Tables. Preface. Foreword. 1. Basic Concepts. 2. Evolutionary Algorithm MOP Approaches. 3. MOEA Test Suites. 4. MOEA Testing and Analysis. 5. MOEA Theory and Issues. 3. MOEA Theoretical Issues. 6. Applications. 7. MOEA Parallelization. 8. Multi-Criteria Decision Making. 9. Special Topics. 10. Epilog. Appendix A: MOEA Classification and Technique Analysis. Appendix B: MOPs in the Literature. Appendix C: Ptrue & PFtrue for Selected Numeric MOPs. Appendix D: Ptrue & PFtrue for Side-Constrained MOPs. Appendix E: MOEA Software Availability. Appendix F: MOEA-Related Information. Index. References.

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Evolutionary algorithms for solving multi-objective problems

: This research organizes, presents, and analyzes contemporary Multiobjective Evolutionary Algorithm (MOEA) research and associated Multiobjective Optimization Problems (MOPs). Using a consistent MOEA terminology and notation, each cited MOEAs' key factors are presented in tabular form for ease of MOEA identification and selection. A detailed quantitative and qualitative MOEA analysis is presented, providing a basis for conclusions about various MOEA-related issues. The traditional notion of building blocks is extended to the MOP domain in an effort to develop more effective and efficient MOEAs. Additionally, the MOEA community's limited test suites contain various functions whose origins and rationale for use are often unknown. Thus, using general test suite guidelines appropriate MOEA test function suites are substantiated and generated. An experimental methodology incorporating a solution database and appropriate metrics is offered as a proposed evaluation framework allowing absolute comparisons of specific MOEA approaches. Taken together, this document's classifications, analyses, and new innovations present a complete, contemporary view of current MOEA "state of the art" and possible future research. Researchers with basic EA knowledge may also use part of it as a largely self-contained introduction to MOEAs.

Multiobjective evolutionary algorithms: classifications, analyses, and new innovations

Various magnetic vector potential formulations for the eddy-current problem are reviewed. The uniqueness of the vector potential is given special attention. The aim is to develop a numerically stable finite-element scheme that performs well at low and high frequencies, does not require an unduly high number of degrees of freedom, and is capable of treating multiple connected conductors. >

On the use of the magnetic vector potential in the finite-element analysis of three-dimensional eddy currents

Edge element formulations of eddy current problems

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.

Stochastic algorithms in electromagnetic optimization

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.

Multiobjective optimization in magnetostatics: a proposal for benchmark problems

A time-domain method is presented yielding the periodic steady-state solution of nonlinear eddy current problems without having to step through the transient process. A novel technique is introduced which, under periodic conditions, in the linear case, decouples the finite element equation systems written for different time steps. Thereupon, a fixed point method is used to iteratively solve the nonlinear equations. The resulting procedure retains the decoupling property valid in the linear case, therefore, it suffices to step through one period only. The efficiency of the method is illustrated by two two-dimensional examples

An efficient time domain method for nonlinear periodic eddy current problems

An edge finite element formulation of eddy current problems using a reduced magnetic vector potential and, in addition, a current vector potential is presented. Multiply connected conducting regions are taken care of. The method is illustrated by a numerical example.

An edge finite element eddy current formulation using a reduced magnetic and a current vector potential

The authors review the theories that they have used to develop two finite-element software packages. One uses a magnetic vector potential combined with an electric scalar potential in conducting regions and only the magnetic vector potential elsewhere. The second formulation is based on an electric vector potential and a magnetic scalar potential whereby only the latter is used in conducting regions. A way to obtain unique solutions by satisfying the divergence condition of the vector potentials is shown. Calculations for several problems are reported, showing the efficiency of the codes. The pros and cons of both formulation are discussed. >

K. Preis

Papers

Stochastic algorithms in electromagnetic optimization

Multiobjective optimization in magnetostatics: a proposal for benchmark problems

An efficient time domain method for nonlinear periodic eddy current problems

An edge finite element eddy current formulation using a reduced magnetic and a current vector potential

Calculation of 3D eddy current problems by finite element method using either an electric or a magnetic vector potential