Magnetostatic Field Calculation Research Articles

Improvement of CNN-Based Anisotropic Magnetostatic Field Computation via Adaptive Data Subset Selection

A numerical issue arises when we extend the convolutional neural network (CNN) U-net to the anisotropic magnetostatic field computation. The output magnetic field has a significant gradient with respect to the input geometry parameter, which introduces inevitable errors in the training process to degrade the performance of deep learning (DL). To address this issue, the subset selection approach is utilized to divide the whole database into several subsets, where the samples are assigned according to the gradient between the input and output. Then these subsets with different sample densities are combined into a global one. Taking the uniform dataset with the same sample size as a comparison, the influence of subset selection on DL is investigated by comparing the performance of CNN on different datasets. Numerical experiments illustrate that the adaptive subset selection can be employed to improve the accuracy and efficiency of the CNN network.

Three-dimensional data-driven magnetostatic field computation using real-world measurement data

PurposeThe purpose of this paper is to present the applicability of data-driven solvers to computationally demanding three-dimensional problems and their practical usability when using real-world measurement data.Design/methodology/approachInstead of using a hard-coded phenomenological material model within the solver, the data-driven computing approach reformulates the boundary value problem such that the field solution is directly computed on raw measurement data. The data-driven formulation results in a double minimization problem based on Lagrange multipliers, where the sought solution must conform to Maxwell’s equations while at the same time being as close as possible to the available measurement data. The data-driven solver is applied to a three-dimensional model of a direct current electromagnet.FindingsNumerical results for data sets of increasing cardinality verify that the data-driven solver recovers the conventional solution. Additionally, the practical usability of the solver is shown by using real-world measurement data. This work concludes that the data-driven magnetostatic finite element solver is applicable to computationally demanding three-dimensional problems, as well as in cases where a prescribed material model is not available.Originality/valueAlthough the mathematical derivation of the data-driven problem is well presented in the referenced papers, the application to computationally demanding real-world problems, including real measurement data and its rigorous discussion, is missing. The presented work closes this gap and shows the applicability of data-driven solvers to challenging, real-world test cases.

Magnetostatic Field Computation in Thin Films Based on k-Space Fast Convolution With Truncated Green’s Function

Magnetostatic Field Computation in Thin Films Based on <i>k</i>-Space Fast Convolution With Truncated Green’s Function

Quantitative assessment of finite-element models for magnetostatic field calculations

We present quantitative means for assessing the numerical accuracy of static magnetic field calculations in finite-element models. Our calculations use the three-dimensional Opera simulation software suite of Dassault Systèmes. Our need to assess the effects of fringe fields requires such a 3D algorithm. While we do discuss and compare our approach to a method of accuracy estimation used in the Opera post-processor, our methods are generally applicable to any model using relaxation techniques in finite-element systems. For purposes of illustration, we present modeling and analysis of two types of quadrupole electromagnets presently in operation in the south arc of the Cornell Electron Storage Ring (CESR). Calculations of field multipole expansion coefficients and numerical deviations from Maxwell’s equations in source-free regions are discussed, with emphasis on the dependence of their accuracy on changes to the finite-element model. Successive refinement steps in the finite-element model for the non-extraction type of CESR south arc quadrupole achieve a reduction in the RMS value of the longitudinal component of the curl vector on the magnet axis by a factor of nearly 70 from 4.38×10−2 T/m to 6.36×10−4 T/m, which is 0.0021% of the field gradient. An accuracy of 2.9% is achieved for a dodecapole coefficient of 2.4×10−4 of that of the quadrupole in the longitudinal integral of the vertical field gradient at a radius of 1 cm.

An Efficient and Robust Mixed-Gauge Scheme Based on Current-Splitting for Nonlinear Magnetostatic Field Computation

In the edge element discretization for magnetostatic problems, a gauge scheme, such as tree gauge, Lagrangian multiplier (LM) gauge, and auto gauge, is usually adopted to handle the singular curl-curl equation in terms of the magnetic vector potential. However, tree-gauge and LM-gauge schemes are not very efficient as a nonlinear problem with many increments has to be resolved. In order to overcome those difficulties, a mixed-gauge scheme based on current-splitting is proposed in this article. First, we make an improvement for LM-gauged formulation and adapt it to calculate the initial increment with a specially designed CG solver; then split the discrete source current density into a compatible part and incompatible part based on discrete Helmholtz decomposition; auto gauge cooperates with the compatible part to calculate the following increments. The mixed-gauge scheme combines the advantages of LM-gauge and auto-gauge schemes, which avoids the step to determine the source term, and at the same time, has high computational efficiency. The mixed-gauge scheme is tested in some examples against some other gauge schemes and found to be efficient and robust, where others possibly converge slowly or fail to produce a reliable solution.

Large-scale magnetostatic field calculation in finite element micromagnetics withH2-matrices

Magnetostatic field calculations in micromagnetic simulations can be numerically expensive, particularly in the case of large-scale finite element simulations. The established finite element / boundary element method (FEM/BEM) by Fredkin & Koehler involves a densely populated matrix with unacceptable numerical costs for problems involving a large number of degrees of freedom $N$. By using hierarchical matrices of $\mathcal{H}^2$ type, we show that the memory requirements for the FEM/BEM method can be reduced dramatically, effectively converting the quadratic complexity $\mathcal{O}(N^2)$ of the problem to a linear one $\mathcal{O}(N)$. We obtain matrix size reductions of nearly $99\%$ in test cases with more than $10^6$ degrees of freedom, and we test the computed magnetostatic energy values by means of comparison with analytic values. The efficiency of the $\mathcal{H}^2$-matrix compression opens the way to large-scale magnetostatic field calculations in micromagnetic modeling, all while preserving the accuracy of the established FEM/BEM formalism.

2-D Volume Integral Formulations for Nonlinear Magneto-Static Field Computation for Rotating Machines Pre-Design Considering Periodicities

This paper provides an alternative approach to the finite elements method of solving magneto-static field for an efficient pre-design of electrical rotating machines. First, the 2-D volume integral formulation will be presented. Second, this formulation will be extended to periodic systems. Comparisons to the finite elements, methods will be carried out to conclude on the performances of the new method presented.

ТРЕХМЕРНЫЙ ЭЛЕМЕНТ "ГИПЕР-КУБ" И АЛГОРИТМ ФОРМИРОВАНИЯ КОЭФФИЦИЕНТОВ ГЛОБАЛЬНОЙ СЛАУ МЕТОДА КОНЕЧНЫХ ЭЛЕМЕНТОВ ПРИ РАСЧЕТАХ МАГНИТНЫХ ПОЛЕЙ ЭЛЕКТРОТЕХНИЧЕСКИХ УСТРОЙСТВ

The article is devoted to calculation of three-dimensional magnetostatic fields of electrical engineering devices of fixed or with linear motion trajectory of various functional purposes by the finite element method using a regular finite element network. The aim of the research is to create an algorithm for forming coefficients of global SLAE of the finite element method on the basis of three-dimensional finite element hyper-cube when solving magnetostatic problems for obtaining engineering methods. The study subject is magnetostatic three-dimensional vector models of electromagnetic processes of electrical engineering devices, electrical engineering and electromechanical systems and methods of their calculation. The tasks are: to develop three-dimensional elements such as cube and hyper-cube, to obtain recurrent relations for global system of linear algebraic equations (SLAE) coefficients. Research methods are: numerical calculations based on projection-grid method of the Galerkin in combination with the finite element method. The results are: a three-dimensional element of cube and hyper-cub type for a three-dimensional regular network of the finite element method has been offered, and recurrence relations for global SLAE coefficients are developed when working out engineering methods for calculating magnetostatic fields and their power characteristics in electrical engineering devices with minimal costs of the computer dynamic memory.

ON ELECTROSTATIC ANALOGY OF MAGNETOSTATIC FIELD IN INHOMOGENEOUS MAGNETIZED MEDIUM

Purpose. The application in electrostatic analogy of magnetostatics for inhomogeneous magnetized media of two known magnetization models. Methodology. A comparison of basic equations and formulas of electrostatic and magnetostatic field in immoveable isotropic inhomogeneous polarized medium for dipole model and the magnetization model by molecular currents is made. The value-analogues for dipole model of magnetization are established. Results. We have shown that the using of dipole model of magnetization is correct. There is not even formal analogy with electrostatic field in the case of using the magnetization model by molecular currents. The relation between the magnetizations for various models is obtained. It allows us to justify the using magnetization by molecular currents in electrostatic analogy. Originality. The magnetization for dipole model is introduced and the possibility of using magnetization by molecular currents in the formulas for calculating potential magnetostatic field in magnetized medium is substantiated. Practical value. The results allow to obtain correct formulation and solution of the problem of magnetostatic field calculation in inhomogeneous magnetized medium.

An optimized magnetostatic field solver on GPU using open computing language

SummaryRecent graphic processing units (GPUs) have remarkable raw computing power, which can be used for very computationally challenging problems. Like in micromagnetic simulations, where the magnetostatic field computation to analyze the magnetic behavior at very small time and space scale demands a huge computation time. This paper presents a multidimensional FFT‐based parallel implementation of a magnetostatic field computation on GPUs. We have developed a specialized 3D FFT library for magnetostatic field calculation on GPUs. This made it possible to fully exploit the symmetries inherent in the field calculation and other optimizations specific to the GPUs architecture. We have compared our results with the widely used CPU‐based parallel OOMMF program and with an equivalent serial implementation on CPU. The results have shown a speedup of up to 95x and 8.7x for single and 66x and 4.6x for double precision floating point accuracy against equivalent serial implementation and OOMMF, respectively.

Questions of the theory and calculation of magnetostatic fields in ferromagnets

Questions of calculating the magnetostatic fields in ferromagnets with consideration for a nonlinear dependence of the induction (magnetization) on the field strength are considered. It is shown that nonlinearities of the magnetic properties of a ferromagnet are significant only at some points that are positioned at the ferromagnet-air medium.

Fuzzy logic control of electrodynamic levitation devices coupled to dynamic finite volume method analysis

Electrodynamic levitation devices, which utilizing eddy currents induced in the levitated item to produce the repulsive force, are being involved in many engineering applications due to its fast response. This kind of repulsion is particularly used in electromagnetic launcher, electromagnetic brake and other applications. To analyze and improve the dynamic behavior and performances of such devices, the conventional way is using the finite element method (FEM), due to its ability of using adaptive mesh to handle complex geometries. Nevertheless, it has a serious limitation in efficiency for large number of variables which is reflected by the high cost in terms of computational properties. During the past few years, the finite volume method FVM formulations have gained attention inside the electromagnetic community. The method has been proved its effectiveness in the solution of different kinds of problems, such as in magnetostatic field computation and eddy current nondestructive testing. The FVM method is particularly attractive thanks to its small required storage memory and reduced CPU time. In this paper an FVM model is developed to analyze the dynamic characteristic of the motion of the electrodynamic levitation device TEAM Workshop Problem 28. The dynamic characteristic of the motion is obtained by solving the electromagnetic equation coupled to the mechanical one. The repulsive force applied to the levitated plate of TEAM Workshop Problem 28, is computed by the interaction between eddy current induced in the plate and the magnetic flux density. A comparison between experimental and numerical results is carried out to show the efficiency of the developed model. What’s more, based on the developed FVM model, a fuzzy logic controller FLC is designed and implemented to control the position of the levitated item.

Implementation of a finite‐difference micromagnetic model on GPU hardware

SUMMARYWe have developed a micromagnetic simulator for graphical processing units (GPU), using the CUDA framework. In this paper, we discuss the optimization of the effective field calculation, both from a mathematical and from a hardware‐specific point of view. By using a finite‐difference discretization scheme, the long‐range magnetostatic field can be calculated using fast Fourier transforms, an approach well suited for the GPU. We show how the implementation can be tuned to the GPU hardware and how the performance can be further increased by dealing with the large number of zeros that typically occurs in the micromagnetic field computation. Additionally, we show how the ferromagnetic exchange interaction can be readily included in the magnetostatic field calculation without any additional computational cost. The resulting high‐performance software can be used to run large‐scale simulations that would have been very time‐consuming on regular CPU hardware. As an example, we present a case study on the de‐pinning of domain walls in racetrack memory devices. Copyright © 2012 John Wiley &amp; Sons, Ltd.

Multipole expansion technique for the magnetostatic field computation in patterned magnetic films

This paper deals with the calculation of the magnetostatic field in magnetic antidot arrays with variable hole size. Boundary effects are taken into account by considering films with finite size, without introducing infinite periodicity assumption. To speed-up the computation, a macroscale grid with resolution equal to the spatial period is overlapped to the microscopic one, approximating the long-range contributions by a multipole expansion technique. Accuracy and efficiency are evaluated by varying the extension of the near-interaction zone and the expansion order.

Speedup of FEM Micromagnetic Simulations With Graphical Processing Units

We have adapted our finite element micromagnetic simulation software to the massively parallel architecture of graphical processing units (GPUs) with double-precision floating point accuracy. Using the example of Standard Problem #4 with different numbers of discretization points, we demonstrate the high speed performance of a single GPU compared with an OpenMP-parallelized version of the code using eight CPUs. The adaption of both the magnetostatic field calculation and the time integration of the Landau-Lifshitz-Gilbert equation routines can lead to a speedup factor of up to four. The gain in computation performance of the GPU code increases with increasing number of discretization nodes. The computation time required for high-resolution micromagnetic simulations of the magnetization dynamics in large magnetic samples can thus be reduced effectively by employing GPUs.

On the accuracy of FFT based magnetostatic field evaluation schemes in micromagnetic hysteresis modeling

Micromagnetic hysteresis models for large, bulk like samples are useful for the identification of relations between microscopic material properties and macroscopic magnetic behavior. To bridge the gap between the nanometer space scale of the micromagnetic theory and the large sample dimensions, time and memory efficient numerical schemes are needed. In micromagnetic computations, fast Fourier transforms (FFTs) have been widely adopted to speed up magnetostatic field computations. In this paper, two FFT schemes are compared. The first scheme evaluates the magnetostatic field directly starting from the magnetization and has a large accuracy, while in the second scheme the magnetostatic field is derived from the scalar magnetic potential resulting in a reduced accuracy but also in a CPU time reduction for a magnetostatic field evaluation to 65% and a reduction of memory requirements to 55%. The influence of the low accuracy evaluations on the simulated macroscopic hysteresis behavior is studied. Therefore, comparison is made with the influence of thermal effects in hysteresis simulations. It is found that the resulting changes in macroscopic hysteresis behavior are of the same order of magnitude as the ones obtained when thermal fluctuations are taken into account in the high accuracy computations.

Calculation of Magnetostatic Fields in Nanostructured Ellipsoidal Cells of Exchange-Coupled Trilayers

Calculation of Magnetostatic Fields in Nanostructured Ellipsoidal Cells of Exchange-Coupled Trilayers

Analytical and numerical calculation of magnetic field distribution in the slotted air-gap of tangential surface permanent-magnet motors

This paper deals with the analytical and numerical analysis of the flux density distribution in the slotted air gap of permanent magnet motors with surface mounted tangentially magnetized permanent magnets. Two methods for magnetostatic field calculations are developed. The first one is an analytical method in which the effect of stator slots is taken into account by modulating the magnetic field distribution by the complex relative air gap permeance. The second one is a numerical method using 2-D finite element analysis with consideration of Dirichlet and anti-periodicity (periodicity) boundary conditions and Lagrange Multipliers for simulation of movement. The results obtained by the analytical method are compared to the results of finite-element analysis.

Influence of Space Harmonics on Starting Performance of a Single-Phase Line Start Permanent-Magnet Motor

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> In this paper, we investigate the effects of space harmonics of single-phase line start permanent-magnet (LSPM) motors on starting performance. We construct an equivalent electric circuit considering the harmonics for the analysis, in which the magnetostatic field computation is coupled with finite-element modeling. We analyze the starting characteristics by circuit simulation and show that the results compare well with those of time-step finite-element analysis and of experiments. </para>

Fast computation of magnetostatic fields by nonuniform fast Fourier transforms

The bottleneck of micromagnetic simulations is the computation of the long-ranged magnetostatic fields. This can be tackled on regular N-node grids with fast Fourier transforms in time N log N, whereas the geometrically more versatile finite element methods (FEMs) are bounded to N4/3 in the best case. We report the implementation of a nonuniform fast Fourier transform algorithm, which brings a N log N convergence to FEM, with no loss of accuracy in the results.