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

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. >

Finite element analysis of 3-D eddy currents

The authors review formulations of three-dimensional (3-D) eddy current problems in terms of various magnetic and electric potentials. The differential equations and boundary conditions are formulated to include the necessary gauging conditions and thus to ensure the uniqueness of the potentials. Different sets of potentials can be used in distinct subregions, thus facilitating an economic treatment of various types of problems. A novel technique for interfacing conducting regions with an electric vector and a magnetic scalar potential to eddy-current-free regions with a magnetic vector potential is described. Finite-element solutions to several large eddy-current problems are presented. >

On the use of the magnetic vector potential in the nodal and edge finite element analysis of 3D magnetostatic problems

An overview of various finite element techniques based on the magnetic vector potential for the solution of three-dimensional magnetostatic problems is presented. If nodal finite elements are used for the approximation of the vector potential, a lack of gauging results in an ill-conditioned system. The implicit enforcement of the Coulomb gauge dramatically improves the numerical stability, but the normal component of the vector potential must be allowed to be discontinuous on iron/air interfaces. If the vector potential is is interpolated with the aid of edge finite elements and no gauge is enforced, a singular system results. It can be solved efficiently by conjugate gradient methods, provided care is taken to ensure that the current density is divergence free. Finally, if a tree-cotree gauging of the vector potential is introduced, the numerical stability depends on how the tree is selected with no obvious optimal choice available.

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. >

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

Computation of 3-D magnetostatic fields using a reduced scalar potential

Some improvements to the finite element computation of static magnetic fields in three dimensions using a reduced magnetic scalar potential are presented. Methods are described for obtaining an edge element representation of the rotational part of the magnetic field from a given source current distribution. When the current distribution is not known in advance, a boundary value problem is set up in terms of a current vector potential. An edge element representation of the solution can be directly used in the subsequent magnetostatic calculation. The magnetic field in a DC arc furnace is calculated by first determining the current distribution in terms of a current vector potential. A 3-D problem involving a permanent magnet as well as a coil is solved, and the magnetic field in some points is compared with measurement results. >

Performance of different vector potential formulations in solving multiply connected 3-D eddy current problems

The authors describe their numerical experiences in applying FEM (finite-element method) solution techniques to a 3-D (three-dimensional) eddy-current problem with a coil-driven multiply connected conductor, the benchmark problem No.7 of the International TEAM Workshops. Several formulations have been tried using a magnetic vector and electric scalar potential or an electric vector and a magnetic scalar in the conductor and a magnetic vector or scalar potential outside. The problem has been solved at two frequencies. The authors briefly describe the formulations used and compare the performance. Magnetic field and current density plots are also compared. The advantages and disadvantages of the various versions are pointed out. The use of a magnetic scalar potential H rather than a magnetic vector potential A outside the conductor and the hole substantially reduces the number of degrees of freedom and thus the computational effort. The versions using it in the conductor yield relatively ill-conditioned systems. Also, at the higher frequency, the conditioning deteriorates considerably. >