Calculation of equipotentials and flux lines in axially symmetrical permanent magnet assemblies by computer

Two equivalent theoretical models of permanent magnets are used to develop algorithms for numerically computing the magnetic scalar potential and the magnetic vector potential in the vicinity of an axially symmetric array of pole pieces and permanent magnets. A computer program based on these algorithms calculates equipotential surfaces and flux lines in and around the magnets and pole pieces. In deriving the algorithm for numerically calculating the vector potential a relationship between the magnetic scalar potential and the vector potential was found which enables the program to calculate the vector potential from the scalar potential distribution and thus generate equipotentials and flux lines with only one iterative calculation. An algorithm which calculates the scalar potential of a "floating" pole piece, that is, one on which the scalar potential has not been specified, is developed. The vector potential around the pole piece is determined from the scalar potential calculation, and this information is used to calculate the vector potential and the flux lines within the pole piece. The computer program calculates the coordinates of all points at which the equipotential lines and flux lines cross the Liebmann net. This information is fed to a cathode ray tube plotter which generates a field plot. To deal with systems in which macroscopic currents are present as well as permanent magnets, the iterative Liebmann net calculation of the vector potential is developed, and a method of applying Neumann boundary conditions to the vector potential at high-permeability surfaces is described.     

Finite element scheme for transient 3D eddy currents

A transient 3-D finite-element model is presented. The method is based on the solution of the magnetic scalar potential in nonconducting regions and the magnetic vector potential and an electric scalar potential in eddy-current regions. Multiply connected regions of magnetic scalar can be avoided by extending the region modeled by the magnetic vector potential to fill any holes in the conducting regions. The model was used to simulate the FELIX brick experiment     

An integral equation method for the solution of 3-dimensional,non-linear,magnetostatic problems

A finite element method for computing the resultant magnetic field arising from a given source field in the presence of a magnetic material of variable permeability is described; in this method finite element approximations to the scalar potential of the resulting field and the magnetic susceptibility, in the region occupied by the magnetic material, are determined from the non-linear integral equation for the scalar potential and the constitutive susceptibility relation, using a collocation scheme. The method is used to compute the shielding effect of a thin rectangular plate of variable permeability on a given source field. The plate is subdivided uniformly into brick elements; the resulting translational invariance of the integrals required in the calculations is exploited to achieve major computational savings. A consequence of the thinness of the plate is that the calculation of the requisite integrals by analytic methods leads to considerable loss of accuracy by differencing; this difficulty is overcome by using a scheme which combines both analytic and quadrature techniques. The resulting system of non-linear algebraic equations is solved by Powell's hybrid method; an efficient scheme for calculating an initial approximation to the Jacobian, which utilizes the structure of the equations, is presented. The results of the calculations are discussed.     

An accurate scalar potential finite element method for linear,two-dimensional magnetostatics problems

We have assessed the accuracy of a commercially available computer software package for finite element method calculations of magnetostatic fields. The computer program, MSC/NASTRAN,

1 Available from the MacNeal-Schwendler Corporation, Los Angeles, CA 90041, U.S.A.

is well known for its wide applicability in structural analysis and heat transfer problems. We exploit the fact that the differential equations of magnetostatics are identical to those for heat transfer if the magnetic field problem is formulated with the reduced scalar potential.¹ Consequently, the powerful, optimized numerical routines of NASTRAN can immediately be applied to two- and three-dimensional linear magneto-statics problems. Application of the NASTRAN reduced scalar potential approach to a ‘worst case’ two-dimensional problem for which an analytic solution is available has yielded much better accuracy than was recently reported² for a reduced scalar potential calculation using a different finite element program. Furthermore, our method exhibits completely satisfactory performance with regard to computational expense and accuracy for a linear electromagnet with an air gap. Our analysis opens the way for large three-dimensional magnetostatics calculations at far greater economy than is possible with the more commonly used vector potential and boundary integral methods.     

The H-gradient method for magnetostatic field computations

A new method for approximating magnetostatic field problems is given in this paper. The new method approximates the scalar potential for the magnetic intensity and is based on a volume integral formulation. The corresponding algorithm is similar to that obtained from coupled differential and boundary integral approaches. Convergence results in computations are compared with results for the usual volume integral method used in GFUN3D.     

A finite element method to compute three-dimensional magnetic field distribution in transformer cores

A numerical method is proposed to compute three-dimensional magnetic field distributions in nonlinear nonhomogeneous media, neglecting hysteresis and eddy currents. The magnetic field is derived from a scalar potential satisfying a nonlinear elliptic equation, which is solved by a convergent iterative method. A finite element program has been developed to compute the magnetic field distribution in transformer cores. Some numerical results for a butt and lap corner configuration are discussed.     

On the stress and electric field produced by dislocations in anisotropic piezoelectric crystals with special attention to the stress function

The author is concerned with the stress and electric field produced by dislocations in an anisotropic piezoelectric crystal, with a proposal on the way of choosing the best triad as stress functions from among the six components of Beltrami’s stress-function tensor. A pair of constitutive equations is assumed to connect the elastic strain and electric displacement with the stress and electric field. The fundamental equations governing the field of stress functions and electric scalar potential are presented, and solved by application of the method of Fourier transform. The stress and electric field are stated in terms of the dislocation density tensor, and by means of the convolution integrals throughout the region where there exist dislocations. The expressions are converted into those for the fields of an infinitely extended straight dislocation, as well as an elliptic dislocation, by Willis’ method. The choice of three stress functions is made on the way of numerical computations so that the line integrals can be achieved by application of Cauchy’s residue theorem. As example, the field of dislocations in a gallium arsenide is evaluated.     

Finite element solution of 3D magnetostatic field problems involving distributed current using a single scalar potential--The modification and application of the fictitious magnetic monopole model

In this paper the authors present a new model for magnetostatic field problems - the modified fictitious magnetic monopole model, in which a new kind of scalar potential is used which is suitable for the whole region, including the distributed current region. In the FMMM the exciting action of the distributed current density has been replaced by that of a distributed fictitious magnetic monopole density, and the problem of loss of precision (subtraction of two large but similar quantities in the computer) has been solved by putting a magnetic shell into the coil and/or current-carrying conductor loop. According to the new model, the formulation of a magnetostatic problem has almost the same form as that of an electrostatic problem, thus the calculation of magnetostatic problems can be simplified significantly. The new model can also be regarded as a modification for the two-scalar potential model or for the T-Q method in magnetostatic cases. Calculation and test results of some examples of 3D magnetostatic problems are given to verify this new method.     

Field calculation in single- and multilayer coaxial cylindricalshells of infinite length by using a coupled T-Ω andboundary element method

A method is presented for the calculation of the electromagnetic field in systems of single-layer or multilayer coaxial cylindrical shells of infinite length excited by an oscillating current source arbitrarily oriented inside the first shell. The electric vector potential T and the magnetic scalar potential Ω are used for the evaluation of the quantities of the problem. The Helmholtz equations for T and Ω are transformed into integral equations by the use of the Green's function method. Applying the boundary element method, three systems of simultaneous equations have to be solved to give the sought field quantity     

Calculation of multivalued potentials in exterior regions

In magnetostatic and magnetodynamic problems, Ampere's law leads to a multivalued scalar magnetic potential. A method is proposed to calculate this potential through a finite elements program or, in a more efficient way, through a program using a boundary integral method in nonconducting exterior regions. In this case, one has only to define "cutting lines" on the boundary instead of cutting surfaces. Reported here are the results obtained with the three-dimensional eddy-current code Trifou, using the finite element method inside the conductors coupled with an integral method outside, in which the method has been incorporated, for a test model where some global values can be obtained by hand and compared with those obtained by the code. A study of the influence of mesh refinement and of the position of cutting lines is given. Good agreement and numerical stability indicate that the method is operational.     

A legendre polynomial BEM for axisymmetric coil systems including Iron

The scalar potential of any system of axisymmetric conductors can be expressed as a Legendre polynomial expansion, and this provides an efficient and convenient method for computing the field variables. A method is presented for augmenting the expansion coefficients to include the effects of iron of constant permeability on the system. The approach is based on the boundary element method (BEM). As well as giving physical insight into the effects of the iron, the use of the coefficients circumvents the problem of expensive field retrieval. Numerical accuracy is assessed by considering two geometries for which an exact solution is known.     

An evaluation of the computational error near the boundary withmagnetostatic field calculation by BEM

A 3-D magnetostatic field is calculated using the reduced scalar potential method for a two-region model: a current-free iron region with linear and isotropic property, and an air region including the source domain. An unstable computational error near the boundary is investigated from the viewpoints of numerical integration and discretization. Specifically it is shown that near the boundary elements, the calculated results of the magnetic flux density often contain an unstable error. The error is affected by the fineness of the discretization, the point number for Gaussian quadrature, and the distance from the boundary elements. It is found that close integration of the internal field calculation effectively removes the unstable computational error     

3D multidomain BEM for solving the Laplace equation

An efficient 3D multidomain boundary element method (BEM) for solving problems governed by the Laplace equation is presented. Integral boundary equations are discretized using mixed boundary elements. The field function is interpolated using a continuous linear function while its derivative in a normal direction is interpolated using a discontinuous constant function over surface boundary elements. Using a multidomain approach, also known as the subdomain technique, sparse system matrices similar to the finite element method (FEM) are obtained. Interface boundary conditions between subdomains leads to an over-determined system matrix, which is solved using a fast iterative linear least square solver. The accuracy and robustness of the developed numerical algorithm is presented on a scalar diffusion problem using simple cube geometry and various types of meshes. Efficiency is demonstrated with potential flow around the complex geometry of a fighter airplane using tetrahedral mesh with over 100,000 subdomains on a personal computer.     

Three-dimensional eddy current analysis by the boundary elementmethod using vector potential

A boundary-element method using a magnetic vector potential for eddy-current analysis is described. For three-dimensional (3-D) problems, the tangential and normal components of the vector potential, tangential components of the magnetic flux density, and an electric scalar potential on conductor surfaces are chosen as unknown variables. When the approximation is introduced so that the conductivity of the conductor is very large in comparison with the conductivity of air, the number of unknowns can be reduced; also, for axisymmetric models the scalar potential can be eliminated from the unknown variables. The formulation of the boundary-element method using the vector potential, and computation results by the proposed method, are presented     

Boundary element analysis of 3-D magnetostatic problems usingscalar potentials

A boundary element formulation for 3-D nonlinear magnetostatic field problems using the total scalar potential and its normal derivative as unknowns is described. The boundary integral equation is derived from a differential equation for the total scalar potential where a nonlinear operator term can be separated from a linear term. The nonlinear term leads to a volume integral which can be treated as a known forcing function within an iterative solution process. An additional forcing term results from the magnetic excitation coil system. It is shown that the line integral of the magnetic source field which can be defined outside of the current-carrying regions as a gradient of a scalar potential acts as an excitation term. The proposed method is applied to a test problem where an iron cube immersed in the magnetic field of a cylindrical coil is investigated. The numerical results for different saturation stages are compared with finite element method (FEM) calculations. The comparison with FEM calculations shows a good agreement only in highly saturated iron parts     

EDDY CURRENT AND TEMPERATURE SIMULATION IN THIN MOVING METAL STRIPS

The paper presents a special finite element formulation for the computer simulation of an inductive heating device for thin moving metal strips. The calculation includes both the electromagnetic field and the temperature distribution resulting from the losses of the induced currents. The electromagnetic field is described by an electric vector potential and a magnetic scalar potential, the induced eddy currents are represented by a special boundary condition for the magnetic scalar potential along the surface of the current-carrying strip. This avoids the necessity to discretize the thin secondary region. The thermal model includes the movement of the strip as well as convection and radiation as its surfaces. The field equations are discretized using the Galerkin variant of the Method of Weighted Residuals. The mutually coupled electromagnetic and temperature fields are then calculated using an iterative, staggered solution scheme. Comparisons between calculated and measured temperature profiles show the validity of the presented approach.     

Vector potential boundary element method for three dimensional magnetostatic

A three-dimensional magnetostatic field computation method is presented. It uses the vector potential formulation, which is valid for any topology in opposition to the scalar formulation needing special cuts that are not always obvious. Attention is given to the calculation of strongly singular integrals using the Cauchy principal value. Two examples show the accuracy of this method.<>     

On the use of the total scalar potential on the numerical solution of fields problems in electromagnetics

The paper summarizes the formulation of a set of computer algorithms for the solution of the three-dimensional non-linear Poisson field problem. Results are presented that were obtained by applying algorithms to the analysis of two-dimensional magnetostatic fields. Scalar and vector potentials were used, and it is shown that the convenient single valued scalar potential associated with the induced sources gives severe accuracy problems in permeable regions. The results become as good as those obtained using vector potential if the scalar potential associated with the total field is used for permeable regions. The combination of two scalar potentials has a significant advantage for three-dimensional problems.     

A few formulation of the magnetic vector potential method in 3-Dmultiply connected regions

Formulations using the continuous and discontinuous magnetic vector potential for 3-D eddy current calculation in both the simply and multiply connected regions are presented. The formulation using the continuous magnetic vector potential does not need cutting for the multiply-connected-region problem, but it needs the electric scalar potential. The formulation using the discontinuous magnetic vector potential does not need the electric scalar potential for the simply-connected-region problem, but it needs cutting for the multiply-connected region problem, so additional computation is required. Thus the former may suite the multiply-connected-region problem, while the latter may suit the simply-connected region problem