K.R. Richter

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.

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. >

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. >

CAD in Electromagnetism

Publisher Summary Computer-aided design (CAD) software packages for the field analysis methods of three-dimensional models are less widespread. This chapter attempts to develop robust and reliable numerical field analysis methods for three-dimensional models. A largely unified field analysis approach is proposed for various types of problems: static fields, eddy current fields, and general electromagnetic fields. The uniformity is attained by using similar potential functions to describe the field in each particular case. This allows for a great degree of generality with respect to material properties, because the continuity of the potentials is sufficient to ensure the satisfaction of the interface conditions on surfaces where the material characteristics change abruptly. The Coulomb gauge is invariably applied to ensure the uniqueness of the vector potentials, which are necessary in three-dimensional analysis. This results in great numerical stability and in the lack of any spurious solutions when the finite element method is employed. The robustness of the methods is shown by some illustrative examples in the chapter. When induced conductive currents are considered with the neglected displacement currents, the equations of eddy current fields are obtained. This case of the quasi-stationary limit is also discussed in the chapter. The chapter also presents the application of a special form of nodal finite elements by means of several examples of computer-aided analysis.

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. >

Computation of 3-D current driven skin effect problems using a current vector potential

A finite element formulation of current-driven eddy current problems in terms of a current vector potential and a magnetic scalar potential is developed. Since the traditional T- Omega method enforces zero net current in conductors, an impressed current vector potential T/sub 0/ is introduced in both conducting and nonconducting regions, describing an arbitrary current distribution with the prescribed net current in each conductor. The function T/sub 0/ is represented by edge elements, while nodal elements are used to approximate the current vector potential and the magnetic scalar potential. The tangential component of T is set to zero on the conductor-nonconductor interfaces. The method is validated by computing the solution to an axisymmetric problem. Problems involving a coil with several turns wound around an iron core are solved. >

Parameters of lossy cavity resonators calculated by the finite element method

The calculation of the resonance frequency and quality factor of closed or aperture coupled cavity resonators with volume and wall losses by an edge finite element method is discussed. An efficient solver is developed to solve the complex nonlinear eigenvalue problem. The effect of the roughness of the walls on the quality factor is taken approximately into account.

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. >

A BEM code for 3-D eddy current calculations

A software package for solving time-harmonic 3-D eddy current problems using the boundary-element method (BEM) is described. The package solves coupled boundary integral equations for the magnetic vector potential and the electric scalar potential within the conductors, and for the magnetic vector potential in non-eddy-current regions. The authors describe the boundary integral formulation and its BEM realization. As an example, the eddy current field of a nonmagnetic conducting cube immersed in a homogeneous magnetic field with harmonic time variation is calculated. Additional numerical results are presented for a multiply connected eddy-current problem (benchmark problem No. 7 defined by the International TEAM workshops). The boundary-element solutions are compared with finite-element results and show excellent agreement. >