Magnetostatic Problems Research Articles

Quasi 3D modelling and simulation of axial flux machines

Purpose– The purpose of this paper is to investigate the theoretical foundation of the so-called quasi 3D modelling method of axial flux machines, and the means for the simulation of the resulting models.Design/methodology/approach– Starting from the first principles, a 3D magnetostatic problem is geometrically decomposed into a coupled system of 2D problems. Genuine 2D problems are derived by decoupling the system. The construction of the 2D simulation models is discussed, and their applicability is evaluated by comparing a finite element implementation to an existing industry-used model.Findings– The quasi 3D method relies on the assumption of vanishing radial magnetic flux. The validity of this assumption is reflected in a residual gained from the 3D coupled system. Moreover, under a modification of the metric of the 2D models, an axial flux machine can be presented as a family of radial flux machines.Research limitations/implications– The evaluation and interpretation of the residual has not been carried out. Furthermore, the inclusion of eddy currents has not been detailed in the present study.Originality/value– A summary of existing modelling and simulation methods of axial flux machines is provided. As a novel result, proper mathematical context for the quasi 3D method is given and the underlying assumptions are laid out. The implementation of the 2D models is approached from a general angle, strengthening the foundation for future research.

Non-uniform FFT for the finite element computation of the micromagnetic scalar potential

We present a quasi-linearly scaling, first order polynomial finite element method for the solution of the magnetostatic open boundary problem by splitting the magnetic scalar potential. The potential is determined by solving a Dirichlet problem and evaluation of the single layer potential by a fast approximation technique based on Fourier approximation of the kernel function. The latter approximation leads to a generalization of the well-known convolution theorem used in finite difference methods. We address it by a non-uniform FFT approach. Overall, our method scales O(M+N+NlogN) for N nodes and M surface triangles. We confirm our approach by several numerical tests.

Model Order Reduction of Non-Linear Magnetostatic Problems Based on POD and DEI Methods

In the domain of numerical computation, model order reduction approaches are more and more frequently applied in mechanics and have shown their efficiency in terms of reduction of computation time and memory storage requirements. One of these approaches, the proper orthogonal decomposition (POD), can be very efficient in solving linear problems but encounters limitations in the non-linear (NL) case. In this paper, the discrete empirical interpolation method coupled with the POD method is presented. This is an interesting alternative to reduce large-scale systems deriving from the discretization of NL magnetostatic problems coupled with an external electrical circuit.

Computed Basis Functions and the Nonconforming Voxel Finite-Element Method

The nonconforming voxel finite-element method (NVFEM) is an alternative to conventional mesh generation that uses purely rectangular elements. Its main drawback is the need to approximate boundaries that would otherwise pass through the interior of the elements. Using computed basis functions, boundaries inside elements are permitted. Results for magnetostatic problems demonstrate the improved accuracy of the method, compared with conventional NVFEM.

The Adaptive Cross Approximation Technique for a Volume Integral Equation Method Applied to Nonlinear Magnetostatic Problems

Volume integral equation methods are particularly well suited to solve electromagnetic problems, where the air domain is predominant. However, their use leads to the heavy resolution of a dense matrix system. The Adaptive Cross Approximation (ACA) combined with hierarchical matrices (H-matrices) decomposition is an algebraic method allowing the compression of fully populated matrices. This paper presents the ACA technique applied to a volume integral equation to solve nonlinear magnetostatic problems.

静磁場問題の領域分割解析におけるBDD前処理の適用

An iterative domain decomposition method is proposed for numerical analysis of 3-Dimensional (3D) linear magnetostatic problems taking the magnetic vector potential as an unknown function. The iterative domain decomposition method is combined with the Preconditioned Conjugate Gradient (PCG) procedure and the Hierarchical Domain Decomposition Method (HDDM) which is adopted in parallel computing. Our previously employed preconditioner was the Neumann-Neumann preconditioner. Numerical results showed that the method was only effective for smaller problems. In this paper, we consider its improvement with the Balancing Domain Decomposition (BDD) preconditioner.

Application of the virtual work principle to compute magnetic forces with a volume integral method

SUMMARYMagnetic forces are computed in the scope of magnetostatic problems solved by a volume integral method. This method uses the range interactions between magnetizable elements and is particularly well suited to compute the fields without meshing the air domain. This paper proposes an adaptation of the virtual work principle to the framework of the volume integral method in order to compute the magnetic forces. First, its application for electromagnetic devices in the nonlinear case allows to compute the global magnetic force. Second, its local application in the linear case provides a magnetic force density. 3D numerical examples are given, and the results are compared with the analytical solution and the finite element method. Copyright © 2013 John Wiley & Sons, Ltd.

A hybrid method BEM-NRM for magnetostatics problems

This article proposes a new method to solve non-linear magnetostatic problems applied to the modeling of electromagnetic devices. A reduced scalar potential formulation is presented and solved by a hybrid BEM-NRM. It has been implemented and tested in a software called MaGot. We will show that the computation time of this new method is very low. Thus, it could be easily and efficiently used for the pre-sizing of actuators.

Residual and equilibrated error estimators for magnetostatic problems solved by finite element method

In finite element computations, the choice of the mesh is crucial to obtain accurate solutions. In order to evaluate the quality of the mesh, a posteriori error estimators can be used. In this paper, we develop residual-based error estimators for magnetostatic problems with both classical formulations in term of potentials used, as well as the equilibrated error estimator. We compare their behaviors on some numerical applications, to understand the interest of each of them in the remeshing process.

Moment equations for the mixed formulation of the Hodge Laplacian with stochastic loading term

We study the mixed formulation of the stochastic Hodge-Laplace problem dened on a $n$-dimensional domain $D (n ≥ 1)$, with random forcing term. In particular, we focus on the magnetostatic problem and on the Darcy problem in the three dimensional case. We derive and analyze the moment equations, that is the deterministic equations solved by the $m-th$ moment $(m ≥ 1$) of the unique stochastic solution of the stochastic problem. We find stable tensor product finite element discretizations, both full and sparse, and provide optimal order of convergence estimates. In particular, we prove the inf-sup condition for sparse tensor product finite element spaces.

Improvised Open Boundary Conditions for Magnetic Finite Elements

Although sparse exact open boundary methods exist, the implementation of these methods requires capabilities that are not present in all finite-element solvers. This work describes methods for the implementation of accurate higher order asymptotic boundary conditions for 2D planar, axisymmetric, and 3D magnetostatic problems that can be improvised in virtually any finite-element program.

Accurate magnetostatic simulation of step-lap joints in transformer cores using anisotropic higher order FEM

Purpose – The purpose of this paper is to propose a solution strategy for both accurate and efficient simulation of nonlinear magnetostatic problems in thin structures using higher order finite element methods. Special interest is put in the investigation of the step-lap joints of transformer cores, with a focus on the spatial resolution of the field quantities. Design/methodology/approach – The usage of hierarchical finite elements of higher order makes it possible to adapt the local accuracy in different spatial directions in thin steel sheets. Due to explicit representation of gradients in the basis functions, a simple Schwarz-type block preconditioner with a conjugate gradient solver can efficiently solve the arising algebraic system. By adapting the block size automatically according to the aspect ratio, deterioration of convergence in case of thin elements can be prevented. The resulting Newton scheme is accelerated utilizing the hierarchical splitting in a two-level scheme, where an initial guess i...

On the inverse problem of magnetostatics

This work is devoted to solving the inverse problem of the magnetic method for nonde- structive testing (MMNDT). The purpose of the work, frankly speaking and avoiding complicated concepts and formulas, is to identify research directions in MMNDT that would approach solution of the inverse problem in the field of magnetic defectoscopy to the highest extent.

A Hybrid Boundary Element Method-Reluctance Network Method for Open Boundary 3-D Nonlinear Problems

A scalar potential formulation coupling a hybrid boundary element method and a reluctance network method is presented to solve 3-D nonlinear magnetostatic problems. The computation times and the accuracy of this approach are compared with those of the finite element method in the context of force and torque evaluations. It is shown that this method could efficiently be used for the presizing of actuators. Discussion concerning the advantages of this approach compared to a pure reluctance network method is also presented.

Fast Magnetic Field Modeling for Shielding Systems

In this paper, a new iterative algorithm for the analysis of properties of magnetic shielding systems is proposed. Since such shielding systems for low frequency applications are composed of thin-walled shells with magnetic materials of high permeability, the thin shell approximation is adopted and a numerical integral formulation of the ensuing magnetostatic problem is obtained in the form of a convolution equation. This algebraic system is eventually solved iteratively by exploiting the convolution properties of the discrete Fourier transform. The convergence properties of the procedure are analyzed and tested on two case studies where isotropic and anisotropic magnetic shields are used.

Resolution of Nonlinear Magnetostatic Problems With a Volume Integral Method Using the Magnetic Scalar Potential

An integral method using the magnetic scalar potential to solve nonlinear magnetostatic problems is developed. This method uses the range interactions between magnetizable elements and it is particularly well suited to compute field in the air domain which do not need to be meshed. The collocation and Galerkin approaches are presented and compared to solve the nonlinear magnetostatic equation. Both methods need the construction of full interaction matrices which may be computed with analytical formulae. A Newton-Raphson method, in which the interaction matrix must be built at each solver iteration, is used to solve the nonlinear formulation. A modified fixed point scheme, in which the interaction matrix is built only once, is also proposed. 3-D numerical examples are given and results of the different methods are compared.

Nonlinear Magnetostatic Analysis by Unified BIE Utilizing Potential Gap Due to Loop Currents

Since the line loop current is equivalent to the double layer charge, it gives an integral form of scalar potential. The segmental loop current on the interface between magnetic materials produces a potential gap, which works to give a boundary integral equation (BIE). By virtue of another potential gap due to a fictitious circulating current along the contour of cut-surface in the material, the excitation potential becomes single valued and the BIE becomes applicable to generic problems without any restriction. Regarding the nonlinear magnetic material as composed of segmental materials with different values of permeability, we get the same BIE for the nonlinear analysis as for the linear analysis. In order to check the adequacy and effectiveness of the nonlinear BIE, we solve a typical magnetostatic problem and compare the computed results with those by the conventional magnetic moment method.

Finite element approximation of nonlinear transient magnetic problems involving periodic potential drop excitations

This paper deals with the computation of nonlinear 2D transient magnetic fields when the data concerning the electric current sources involve potential drop excitations. In the first part, a mathematical model is stated, which is solved by an implicit time discretization scheme combined with a finite element method for space approximation. The second part focuses on developing a numerical method to compute periodic solutions by determining a suitable initial current which avoids large simulations to reach the steady state. This numerical method leads to solve a nonlinear system of equations which requires to approximate several nonlinear and linear magnetostatic problems. The proposed methods are first validated with an axisymmetric example and sinusoidal source having an analytical solution. Then, we show the saving in computational effort that this methodology offers to approximate practical problems specially with pulse-width modulation (PWM) voltage supply.

Construction of a Finite Element Basis of the First de Rham Cohomology Group and Numerical Solution of 3D Magnetostatic Problems

We devise an efficient algorithm for the finite element approximation of harmonic fields and the numerical solution of three-dimensional magnetostatic problems. In particular, we construct a finite element basis of the first de Rham cohomology group of the computational domain. The proposed method works for general topological configurations and does not need the determination of “cutting” surfaces.

Optimal Control of Quasilinear $\boldsymbol{H}(\mathbf{curl})$-Elliptic Partial Differential Equations in Magnetostatic Field Problems

This paper examines the mathematical and numerical analysis for optimal control problems governed by quasilinear $\boldsymbol{H}(\mathbf{curl})$-elliptic partial differential equations. We consider a mathematical model involving isotropic materials with magnetic permeability depending strongly on the magnetic field. Due to the physical and mathematical nature of the problem, it is necessary to include divergence-free constraints on the state and the control. The divergence-free control constraint is treated as an explicit variational equality constraint, whereas a Lagrange multiplier is included in the state equation to deal with the divergence-free state constraint. We investigate the sensitivity analysis of the control-to-state operator and establish the associated optimality conditions. Here, the key tool for proving the KKT theory is the Helmholtz decomposition. An important consequence of the optimality system is a higher regularity result for the optimal control, which we prove under the assumption of a nonmagnetic control region. The second part of the paper deals with the finite element analysis based on the edge elements of Nédélec's first family for the control and state discretization, and the continuous piecewise linear ansatz for the Lagrange multiplier discretization. The discrete Helmholtz decomposition and the discrete compactness property of the Nédélec edge elements are the main tools for the finite element analysis. Our final results include the convergence and a priori error estimates for the finite element approximation. Numerical results illustrating the theoretical findings are presented.