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     

A finite boundary method for fluid flow field computations

A powerful new finite boundary concept of seeking field solution, at few selected regions in the solution domain, is introduced. Also a highly economical finite boundary method (FBM) which would greatly reduce the size of the coefficient matrix of the resulting system of simultaneous algebraic equations, requiring lesser computer memory and lesser computing time, is developed. Fluid flow fields governed by the basic elliptic partial differential equations—the Laplace's and the Poisson's equations—in two independent variables are mainly considered. The computational merits of the FBM are shown by solving, as an example, a simple representative flow problem, and the relevant computational finite boundary formulae are given in tabular form. The formulae are numerically derived based on a generalized method presented here. The added feature of the FBM is that it proves to be equally economical even when the solution is sought in the entire flow domain. The problem of steady-state viscous flows governed by the Navier-Stokes equations, the system of two simultaneous partial differential equations—the Poisson's equation and the vorticity transport equation—makes the FBM doubly economical. The possibility of developing an efficient hybrid computational algorithm, for curved problem boundaries, in conjunction with the finite element method, is discussed. The extension of FBM to transient, non-elliptic problems and to three-dimensional problem fields is also indicated. The FBM has been discussed in a more detailed manner so as to clearly bring out the advantages of the new finite boundary concept.     

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.     

Simple boundary element method for three-dimensional magnetostatic problems

For numerical solution of three-dimensional magnetostatic problems the applications of finite-difference (FDM) and finite-element (FEM) methods require long calculating time and large storage capacity. In some cases these requirements can be reduced by the use of boundary element methods (BEM). A simple boundary element solution, and its application to the calculation of magnetization and stray fields of magnetic bodies, is described. The results are compared with experimental stray-field measurements performed using a vibrating pick-up loop magnetometer with high geometrical resolution [14].     

Integral solution of nonlinear magnetostatic field problems

This paper presents a procedure for using integrals to solve nonlinear magnetostatic field problems. An integral equation on magnetized volume, expressed in terms of magnetic field H or magnetization M, is discretized by means of edge elements on a tetrahedral mesh and solved numerically by means of a collocation method. The procedure approaches nonlinear by means of the fixed-point iterative technique. It sets up two different iterative schemes with complementary features. This paper gives details about implementation and presents and discusses results on some test cases     

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.<>     

A review of magnetostatic moment method

This paper proposes a review of the magnetostatic moments method (MoM) applied to model electromagnetic devices. This method is now well-known for its "light weight" and its simplicity of implementation. Its main advantages are the nonrequirement of an air region mesh and a coarse mesh of the ferromagnetic material. It leads to very fast resolution and very accurate field, force, and moment computations. The paper proposes a state of the art of this approach and shows some efficient realizations.     

Further discussion on magnetic vector potential finite-elementformulation for three-dimensional magnetostatic field analysis

A linear soft-iron and current model called the IEE Japan model using a novel vector potential finite-element formulation is examined. Calculated and measured results are in close agreement. For comparison, the same model was calculated by the conventional variational formulation. The divergence of magnetic vector potential equals zero at the boundary of different materials and the values themselves are small enough at the Gaussian quadratural points, which means that uniqueness of the solution is guaranteed. The gauge condition is determined by the formulation, not by the boundary conditions. The new formulation requires less computing time and memory than the conventional variational formulation     

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 combined wavelet-FE method for transient electromagnetic-field computations

A combined wavelet-finite-element (wavelet-FE) method is proposed for the computation of transient electromagnetic (EM) fields. In order to retain the essential mathematical properties, such as consistency and linear independence of the shape functions for the proposed method, bridge scales are used to modify the wavelets. To consider the influence of the external voltage supplies, the proposed wavelet-FE method is coupled with a state-space model to develop an integrated simulation approach. Computer simulation results are reported to demonstrate the feasibility of the proposed method for computations of both steady-state and transient EM fields.     

A magnetostatic 2 1/2- dimensional field calculation program for turbogenerators

A finite difference program is described which calculates the magnetic field strength and flux density in a turbogenerator. The mathematical model incorporates solid or laminated iron components with non-linear permeabilities. The fields are evaluated in the r-z plane and vary periodically in the circumferential direction. The program is therefore well suited to three-dimensional end region problems. Use of the program shows how the leakage fields at the core back of a generator vary when radial cooling ducts are introduced. The importance of this for different duct configurations and generator operating voltage is evaluated.     

A fully implicit splitting method for accurate tidal computations

Often industrial codes for the numerical integration of the 2D shallow water equations are based on an Alternating Direction Implicit method. However, for large time steps these codes suffer from inaccuracies when dealing with a complex geometry or bathymetry. This reduces the performance considerably. In this paper a new method is presented in which these inaccuracies are absent, even for large time steps. The method is a fully implicit time integration method. In order to obtain linear systems that can be solved efficiently, we introduce a time splitting method. The resulting linear systems are solved iteratively by using the preconditioned Conjugate Gradients Squared method.     

Finite element computation of three-dimensional electrostatic and magnetostatic field problems

Three-dimensional field solutions have been of considerable interest in electric machine field problems for a number of years. However, only with the advent of large scale computers, numerical analysis methods, developments in grid generation and graphics display techniques computation of three-dimensional magnetic fields has become somewhat tractable. In this paper, the progress made by the authors' company in three-dimensional field computation for scalar and vector field problems is presented. Several applications and comparison of solutions with test results, where available, and other methods are included. New computational techniques employing mixed scalar and vector formulations are introduced and comparison of results in one case with those obtained by direct methods is presented.     

Alternative vector potential formulations of 3-D magnetostatic field problems

A unified theory of three-dimensional vector potential formulations of magnetostatic field problems is presented. It is shown that existing formulations are based on one of two equivalent boundary value problems. A new formulation is derived, using the concept of projection between the space of arbitrary vector finite elements and the space of non-divergent vector finite elements. The merits of the different approaches are examined.     

Parallelization of lattice Boltzmann method for incompressible flow computations

Parallel computation of the two and three-dimensional decaying homogeneous isotropic turbulence using the lattice Boltzmann method are presented. BGK type approximation for collision term in 9 velocity square lattice model is used. It is found that the lattice Boltzmann method is able to reproduce the dynamics of decaying turbulence and could be an alternative for solving the Navier-Stokes equations. The lattice Boltzmann method is parallelized by using domain decomposition and implemented on a distributed memory computer, Hitachi SR2201. It is found that vertical decomposition gives the highest speedup. In the case of horizontal decomposition the longer the number of lattice units in horizontal direction of each subdomain, the shorter the CPU time. Extension to three-dimension is carried out using 15 velocity cubic lattice model. Compared with the result of two-dimensional case, a higher speedup is obtained than in the three-dimensional simulation. Further investigation is needed on the accuracy and efficiency of cubic lattice BGK model.     

Efficient integral equation method for the solution of 3-D magnetostatic problems

Nonlinear three-dimensional (3-D) magnetostatic field problems are solved using integral equation methods (IEM). Only the nonlinear material itself has to be discretized. This results in a system of nonlinear equations with a relative small number of unknowns. To keep computational costs low the fully dense system matrix is compressed with the fast multipole method. The accuracy of the applied indirect IEM formulation is improved significantly by the use of a difference field concept and a special treatment of singularities at edges. An improved fixed point solver is used to ensure convergence of the nonlinear problem.     

A new computational approach for the linearized scalar potential formulation of the magnetostatic field problem

We consider the linearized scalar potential formulation of the magnetostatic field problem in this paper. Our approach involves a reformulation of the continuous problem as a parametric boundary problem. By the introduction of a spherical interface and the use of spherical harmonics, the infinite boundary condition can also be satisfied in the parametric framework. The reformulated problem is discretized by finite element techniques and a discrete parametric problem is solved by conjugate gradient iteration. This approach decouples the problem in that only standard Neumann type elliptic finite element systems on separate bounded domains need be solved. The boundary conditions at infinity and the interface conditions are satisfied during the boundary parametric iteration.     

Magnetostatic field computations in terms of two-component vector potentials

In this paper the magnetostatic problem is stated in terms of two-component electric and magnetic vector potentials. An associated numerical method, based on the adoption of edge elements, is proposed. This procedure overcomes the cancellation problems and the complexity of the interface conditions encountered by similar approaches in the presence of magnetic inhomogeneities and discontinuities of currents and magnetic fields.     

Time domain boundary element method for dynamic stress intensity factor computations

This paper presents a procedure for transient dynamic stress intensity factor computations using traction singular quarter-point boundary elements in combination with the direct time domain formulation of the Boundary Element Method. The stress intensity factors are computed directly from the traction nodal values at the crack tip. Several examples of finite cracks in finite domains under mode-I and mixed mode dynamic loading conditions are presented. The computed stress intensity factors are represented versus time and compared with those obtained by other authors using different methods. The agreement is very good. The results are reliable and little mesh dependent. These facts allow for the analysis of dynamic crack problems with simple boundary discretizations. The versatile procedure presented can be easily applied to problems with complex geometry which include one or several cracks.