3D finite element solution of complex magnetostatic probleminvolving axial and radial excitation-using a single scalar potential

The fictitious magnetic monopole model (FMMM) of motors having a complex magnetostatic field excited by both axial (permanent) and radial current is established. A single scalar potential is used to derive a precise solution of the 3-D anisotropic nonlinear rotational field in such a motor. The 3-D FEM (finite-element method) solution is used to modify the conventional `2-D field-circuit' calculation model     

Determination of the current-equivalent parameters for thefictitious magnetic monopole model of magnetostatic problems

Some principles for determining the current-equivalent parameters in the fictitious magnetic monopole model (FMMM) of magnetostatic problems are presented that can be used to overcome the difficulty related to applying the FMMM in problems with complex-shaped current-carrying conductors. DC coils with different shapes have been analyzed, and the results have been used directly in the calculation of practical examples. Calculated and experimental results show the validity of these principles     

A new method for the solution of three dimensional magnetostatic fields, using a scalar potential

A new method "Equivalent Magnetized Region Solution" based on scalar potential for solving 3D magnetostatic fields is presented. The current distribution is transformed into a region of magnetic dipoles and only one scalar potential is used to calculate the field. A program "CMF3D" has been developed by finite element method, equipped with program "MESH" for subdividing field region into elements. Three examples by the solution are presented and compared with results from analytical method or experiment.     

An ancillary boundary integral equation for magnetostatic analysis

The boundary element method (BEM) has been established as an effective means for magnetostatic analysis. Direct BEM formulations for the magnetic vector potential have been developed over the past 20 years. There is a less well-known direct boundary integral equation (BIE) for the magnetic flux density which can be derived by taking the curl of the BIE for the magnetic vector potential and applying properties of the scalar triple product. On first inspection, the ancillary boundary integral equation for the magnetic flux density appears to be homogeneous, but it can be shown that the equation is well-posed and non-homogeneous using appropriate boundary conditions. In the current research, the use of the ancillary boundary integral equation for the magnetic flux density is investigated as a stand-alone equation and in tandem with the direct formulation for the magnetic vector potential.     

Nonlinear 3D magnetostatic field calculation by the integralequation method with surface and volume magnetic charges

The authors present a mathematical model for the 3-D nonlinear magnetostatic field based on integral equations with fictitious surface and volume magnetic charges. The solution is performed by the extended boundary element method including surface elements and volume elements. Examples of calculation for both linear and nonlinear magnetic systems are presented. The method has been shown to be accurate and efficient     

A new formulation of the magnetic vector potential method for three dimensional magnetostatic field problems

A novel formulation of the magnetic vector potential method for three dimensional magnetostatic field calculations is derived. Rigorously defining the interface and boundary conditions of the gauge of the vector potential, the new method gives a unique solution to the problem. The new field equation does not contain the gauge condition against the usual formulations[1], [2], [3], and takes the form of the diffusion equation. Computed results are favorably compared with the analytic solution of a test problem. This formulation is directly applicable to three dimensional eddy current problems.     

Computation of 3-D magnetostatic fields using a reduced scalarpotential

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     

Computational methods for solving static field and eddy current problems via Fredholm integral equations

Two-dimensional static field problems can be solved by a method based on Fredholm integral equations (equations of the second kind). This has numerical advantages over the mote commonly used integral equation of the first kind. The method is applicable to both magnetostatic and electrostatic problems formulated in terms of either vector or scalar potentials. It has been extended to the solution of eddy current problems with sinusoidal driving functions. The application of the classical Fredholm equation has been extended to problems containing boundary conditions: 1) potential value, 2) normal derivative value, and 3) an interface condition, all in the same problem. The solutions to the Fredholm equations are single or double (dipole) layers of sources on the problem boundaries and interfaces. This method has been developed into computer codes which use piecewise quadratic approximations to the solutions to the integral equations. Exact integrations are used to replace the integral equations by a matrix equation. The solution to this matrix equation can then be used to directly calculate the field anywhere.     

Three-dimensional magnetostatic field calculation using boundaryelement method

The boundary-element calculation of three-dimensional magnetostatic field problems using the reduced and total magnetic scalar potential formulation is described. The method is based on a boundary integral equation that can be derived from Green's theorem. Two regions, a current-free iron region and an air region including the source domains, are considered. The material properties of the iron are assumed to be linear and either isotropic or anisotropic (orthotropic). Two examples are investigated: a C-shaped magnet and an iron cylinder of finite length immersed in the magnetic field of a cylinder coil     

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.     

On three-dimensional boundary element methods for magnetostatics invector variables

The boundary-element solution of three-dimensional magnetostatic fields is dealt with. The formulations are based on the magnetic vector potential and the magnetic flux density. The proper boundary-conditions of the problem are discussed, and vector boundary integral equations are presented. An isoparametric boundary element method is used for the solution. Numerical examples are given for both of the formulations     

A new boundary meshfree method with distributed sources

A new boundary meshfree method, to be called the boundary distributed source (BDS) method, is presented in this paper that is truly meshfree and easy to implement. The method is based on the same concept in the well-known method of fundamental solutions (MFS). However, in the BDS method the source points and collocation points coincide and both are placed on the boundary of the problem domain directly, unlike the traditional MFS that requires a fictitious boundary for placing the source points. To remove the singularities of the fundamental solutions, the concentrated point sources can be replaced by distributed sources over areas (for 2D problems) or volumes (for 3D problems) covering the source points. For Dirichlet boundary conditions, all the coefficients (either diagonal or off-diagonal) in the systems of equations can be determined analytically, leading to very simple implementation for this method. Methods to determine the diagonal coefficients for Neumann boundary conditions are discussed. Examples for 2D potential problems are presented to demonstrate the feasibility and accuracy of this new meshfree boundary-node method.     

A 3-D Magnetic Charge Finite-Element Model of an Electrodynamic Wheel

When a magnetic rotor is both rotated and translationally moved above a conductive, nonmagnetic, guideway eddy currents are induced that can simultaneously create lift, thrust, and lateral forces. In order to model these forces, a 3D finite-element model with a magnetic charge boundary has been created. The modeling of the rotational motion of magnets by using a fictitious complex magnetic charge boundary enables fast and accurate steady-state techniques to be used. The conductive regions have been modeled using the magnetic vector potential and nonconducting with the magnetic scalar potential. The steady-state model has been validated by comparing it with a Magsoft Flux 3D transient model (without translational velocity) and with experimental results. The 3D model is also compared with a previously presented 2D steady-state complex current sheet model.     

Synthesis of magnetic fields

The paper deals with some problems of magnetic fields synthesis, depending on determination of the current density distribution, which generates the required magnetic field in the investigated region. Such problems can be reduced to the linear, or nonlinear Fredholm integral equations of the first kind, or to the set of these equations. Fredholm integral equation of the first kind belongs to the class of the ill-posed problems, and for its solving the method of regularisation has been used. In the paper there are given some useful results of synthesis of magnetic fields in few practical configurations.     

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     

Numerical analysis for eddy current characteristics of magnetically saturated ferromagnetic tubes

Magnetic saturation is applied to ferromagnetic tubes inspected by the encircling or inner coil because suppressing magnetic noise is important for the eddy current testing technique. Eddy current signal characteristics in magnetically saturated tubes are different from those in nonmagnetic tubes. In ferromagnetic tubes, defect signal phase angle is not useful for estimating defect depth because it does not depend on the defect depth. In this paper, numerical eddy current analysis has been done in order to explain the relationship between the defect depth and the phase angle in magnetically saturated tubes. This analysis is performed as follows: 1) The magnetic permeability near the defect is calculated as the non-linear magnetostatic problem. 2) The eddy current distribution is calculated as the linear magnetodynamic problem using the incremental permeability value calculated in step 1. The numerical analysis results reveal that the permeability around the defect remains inhomogeneous and it causes the unique eddy current characteristics. Based on these calculated results, a quantitative evaluation method of determining defect depth is proposed. After determining the defect shape by using signal characteristics obtained from the strongly magnetizing state, the defect depth can be estimated by using the signal amplitude.     

基于附加源波叠加法的声辐射计算研究

针对单极子波叠加法在特征波数处声场解的非唯一性问题,采用一种通过添加附加源克服解非唯一性的方法-附加源波叠加法,即在单极子波叠加法的基础上添加一定数量附加源从而获得声场全波数域内的唯一解。本文给出了具有解析解的脉动球源、振荡球源及无解析解的立方箱体结构三个数值算例。计算结果表明：对于脉动球源,添加一个附加源就可较好解决声场解的非唯一性问题;对于振荡球源,增加附加源个数可解决声场解的非唯一性问题,但会降低声场解的精度,但通过增加单极子源个数可以很好提高计算精度;该方法计算效率略低于复数矢径波叠加法,但较三极子波叠加法效率更高;对于立方箱体结构,确定了最佳的附加源个数,保证了声场解的唯一性。     

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.