Nonlinear Magnetostatic Problems Research Articles

Domain Decomposition for 3-D Nonlinear Magnetostatic Problems: Newton–Krylov–Schur Versus Schur–Newton–Krylov Methods

Domain Decomposition for 3-D Nonlinear Magnetostatic Problems: Newton–Krylov–Schur Versus Schur–Newton–Krylov Methods

Separated representation of the finite element solution of nonlinear magnetostatic problem based on non-intrusive Proper Generalized Decomposition

The Proper Generalized Decomposition has shown its efficiency to solve parameterized problems in nonlinear system events when it is combined with the Discrete Empirical Interpolation Method. Nevertheless, the solution of finite element model with the Proper Generalized Decomposition framework requires to have access to matrices and vectors of the discretized problem, which makes the method highly intrusive. In this context, based on a set of finite element solutions for a set of input parameters, a surrogate model can be developed applying a non-intrusive Proper Generalized Decomposition approach. The proposed non-intrusive approach is based on a canonical decomposition of the finite element solutions combined with an interpolation method. We then obtain a surrogate model approximating the finite element solutions for a wide range of parameters. The surrogate model, given its evaluation speed, can be used for real-time applications. In this paper, the proposed non-intrusive Proper Generalized Decomposition approach is employed to approximate a nonlinear magnetostatic problem and is applied to a single phase standard transformer and to a three-phase inductance.

Vector Nodal Meshless Method for 3-D Applications

Traditional meshless methods use scalar-based functions and therefore, present difficulties in approximating vector fields. The Vector Nodal Meshless Method (VNMM) constructs its approximations using shape functions based on the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H(curl)</i> spaces and Nédélec’s first type elements polynomial space. In this sense, a set of nodes is distributed in the domain and on its boundary, and each node has an associated unit vector. The VNMM has been applied to solve two-dimensional electromagnetic problems. This paper presents the construction of the VNMM shape functions for use in three-dimensional vectorial problems. The interpolation capacity of shape function and the method convergence rates are shown. Then, the VNMM is applied to an eigenvalue problem and compared to the traditional Edge Finite Element Method (EFEM). The numerical solution for eigenvalues is not corrupted by spurious modes. Finally, a nonlinear magnetostatic problem is solved. A good performance of the method is observed.

MATHEMATICAL MODELLING OF THE MAGNETIC SYSTEM BY A.N. TIKHONOV REGULARIZATION METHOD

A nonlinear magnetostatic inverse problem is investigated for a case when it is needed to create a required magnetic field using conductors which coordinates vary on condition that current value is equal in all the conductors. It is known that the same problems fall under the category of noncorrect problems. Mathematical statement of this type of nonlinear magnetostatic inverse problems is given. The proposed numerical algorithm using the regularization method by A.N.Tikhonov permits to overcome comparatively easily the difficulties connected with badly conditioned equation systems to which usually the magnetostatic inverse problems reduce. This algorithm allows one to calculate the existing winding geometry of superconducting dipole ironless megnet which ensures a magnetic field homogeneity up to 10-5 within a rectangular aperture.

Hierarchical Multilevel Surrogate Model Based on POD Combined With RBF Interpolation of Nonlinear Magnetostatic FE Model

From solutions of finite element (FE) simulations depending on parameters, a surrogate model of the FE solution can be built based on the proper orthogonal decomposition (POD) combined with the radial basis functions (RBFs) interpolation. Then, a fast approximation of an FE solution can be computed for any parameter set. In order to optimize the number of FE solutions required for the construction of the surrogate model and to control the error between the FE and surrogate models, a hierarchical multilevel approach based on an iterative algorithm is investigated. In this article, a hierarchical multilevel (H-POD-RBF) approach is developed for a nonlinear magnetostatic problem. In terms of application, a 3-D three-phase inductance is considered. The results obtained from the surrogate model are compared with those given from the FE model for a current or voltage supply.

An adaptive method for calculation of iron losses in switched reluctance motors using a minimum number of magnetostatic finite element simulations

PurposeThis paper aims to present an adaptive method based on finite element analysis to calculate iron losses in switched reluctance motors (SRMs). Calculation of iron losses by analytical formulas has limited accuracy. On the other hand, its estimation in rotating electrical machines through fully dynamic simulations with a fine time-step is time-consuming. However, in the initial design process, a quick and sufficiently accurate method, i.e. a value close to that of iron losses, is always welcome. The method presented in this paper is a semi-analytical approach. The main problem is that iron losses depend on d B/d t. Therefore, the accuracy of the calculation of iron losses depends on the accuracy of the calculation of the first derivative of the flux density waveform. When adopting a magnetostatic model to estimate the iron losses, an important question arises: by how many magnetostatic simulations can the iron losses be estimated within the desired accuracy? In the proposed algorithm, the aim is not to accurately calculate the value of iron losses in SRMs. The objective is to find a numerical error criterion to calculate iron losses in SRMs with a minimum number of magnetostatic simulations.Design/methodology/approachA finite element solver is developed by authors in MATLAB to solve the 2 D nonlinear magnetostatic problem using the Newton–Raphson method. A parametric program is developed to create geometry and mesh. The proposed method is implemented in MATLAB using the developed solver. Counterpart simulations are done in the ANSYS Maxwell software to validate the accuracy of the results generated by the developed solver.FindingsThe performance of the proposed method is studied on a 12/8 (500 W) SRM. Three scenarios are studied. The first one is the calculation of iron losses by uniform refinement, and the second one is by adaptive refinement, and the last one is by adaptive refinement started by particular initial points (switching points). According to the results, the proposed method substantially reduces the number of magnetostatic simulations without sacrificing accuracy.Originality/valueThe main novelty of this paper is introducing an error criterion to find the minimum number of magnetostatic simulations that are needed to calculate iron losses with the desired accuracy.

Parametric Geometric Metamodel of Nonlinear Magnetostatic Problem Based on POD and RBF Approaches

A parametric geometric metamodel is built for a nonlinear magnetostatic problem, using proper orthogonal decomposition (POD) approach combined with radial basis functions (RBFs) interpolation. Furthermore, the geometrical variation of the problem is modeled using an RBF interpolation for smooth mesh deformation. The metamodel is applied for a single-phase EI inductance, and the aim is to create precise flux cartographies based on few solutions of the original finite element (FE) model. The results show that the POD-RBF approach can reduce efficiently the evaluation time of a parametric nonlinear magnetostatic problem.

Loop Formulation in Hybrid Analytical Modeling for Solving 2-D Nonlinear Magnetostatic Problems

This article presents a novel formulation in hybrid analytic modeling (HAM) for solving 2-D nonlinear magnetic problems. This novel formulation, referred to as the loop formulation, solves the magnetic equivalent circuit (MEC) through Kirchhoff's voltage law, as opposed to the traditional node formulation, which uses Kirchhoff's current law. This allows one to solve the system iteratively using the Newton-Raphson method instead of the fixed-point method, providing a convergence rate up to four times faster. The novel HAM formulation also provides an excellent tradeoff between accuracy and computational time. It offers approximately one order of magnitude lower computation time for the same accuracy compared to finite-element analysis (FEA) on a slotless geometry.

A hybrid FPM/BEM scalar potential formulation for field calculation in nonlinear magnetostatic analysis of superconducting accelerator magnets

A new hybrid numerical method for the solution of nonlinear magnetostatic problems in accelerator magnets is proposed. The methodology combines the Fragile Points Method (FPM) and the Boundary Element Method (BEM). The FPM is a Galerkin-type meshless method employing for field approximation simple, local, and discontinuous point – based interpolation functions. Because of the discontinuity of these functions, the FPM can be considered as a meshless discontinuous Galerkin formulation where numerical flux corrections are employed for the treatment of local discontinuity inconsistencies. The FPM possesses the advantages of a mesh-free method, evaluates with high accuracy field gradients, and treats nonlinear magnetostatic problems by utilizing, for the same problem, fewer nodal points than the Finite Element Method (FEM). In the present work, the nonlinear ferromagnetic material of an accelerator magnet is treated by the FPM, while the BEM is employed for the infinitely extended, surrounding air space. The proposed hybrid scheme is based on the scalar potential formulation of Mayergouz et.al (1987), also used in the BEM/BEM and FEM/BEM schemes of Rodopoulos et al. (2019, 2020). The applicability of the method is demonstrated with the solution of representative 2-D magnetostatic problems and the obtained numerical results are compared to those provided by the FEM/BEM scheme of Rodopoulos et al. (2020), as well as by the commercial FEM package ANSYS. Finally, the magnetic field utilized for stable bending of the particles’ trajectory in a 16 Tesla dipole magnet design for the Future Circular Collider (FCC) project of CERN is accurately evaluated with the aid of the proposed here FPM/BEM scheme.

Convergence Analysis of the Fixed-Point Method With the Hybrid Analytical Modeling for 2-D Nonlinear Magnetostatic Problems

This article presents the convergence analysis of the fixed-point method (FPM) to model the nonlinear magnetic characteristics of a 2-D magnetostatic problem. In this study, FPM is used as the iterative nonlinear solver of the hybrid analytical modeling (HAM) technique for the accurate computation of the magnetic field distribution. The benchmark consists of a stator with excitation windings, an air gap, and a slotless mover. The relative errors between two successive iterations are calculated using different error estimators: the attraction force on the mover, the Fourier coefficients defined in the air gap, the magnetic flux density, and the magnetic scalar potential distributions. The effect of the number of mesh elements and harmonics on the accuracy and computational cost of the model is investigated for different levels of magnetic saturation. It is observed that the maximum rate of change in the relative difference of attraction force during the iterations is found to be 0.52 under the magnetic saturation. In addition, the absolute error of the attraction force between the developed hybrid model with FPM and the finite element method (FEM) is achieved to be 0.18%, while HAM has approximately three times less number of degrees-of-freedom when compared to FEM.

Nonlinear BEM/FEM scalar potential formulation for magnetostatic analysis in superconducting accelerator magnets

Hybrid schemes that combine the Finite Element Method (FEM) and the Boundary Element Method (BEM) have been extensively used for the solution of nonlinear magnetostatic problems. The majority of those schemes rely either on reduced and total scalar or vector potential formulations. One main issue regarding the scalar potential formulations is the cancelation errors and difficulties on the definition of potential jump condition on the surface of a magnet, while in the case of vector potential formulations, the size of the simulated problem increases drastically especially in three dimensions. An alternative formulation has been proposed by Mayergoyz et al. (1987), which is based on scalar potentials and provides assistance in ameliorating the aforementioned problems. In the present work that formulation is implemented by employing a BEM/FEM scheme, for the solution of nonlinear magnetostatic problems. The proposed BEM/FEM formulation is employed for the solution of representative magnetostatic problems dealing with different types of superconducting accelerator magnets, while the provided accuracy is assessed through comparisons made with corresponding results obtained by a well-known commercial FEM package.

Surrogate Model Based on the POD Combined With the RBF Interpolation of Nonlinear Magnetostatic FE Model

The proper orthogonal decomposition (POD) is an interesting approach to compress into a reduced basis numerous solutions obtained from a parametrized finite-element (FE) model. In order to obtain a fast approximation of an FE solution, the POD can be combined with an interpolation method based on radial basis functions (RBFs) to interpolate the coordinates of the solution into the reduced basis. In this article, this POD-RBF approach is applied to a nonlinear magnetostatic problem and is used with a single-phase transformer and a three-phase inductance.

A Thin-Layer Volume Integral Equation Formulation for Linear and Non-Linear Magnetostatic Problems

A thin-layer approximation of the magnetostatic volume integral equation (VIE) method is presented. This method, referred to as the thin-VIE (TVIE), utilizes a thin approximation of Green’s function. The TVIE approximates both the transverse and the normal magnetization in the material represented by a surface mesh, dramatically improving the accuracy of the method. The TVIE is solved via a locally corrected Nystrom formulation, supporting high-order meshing and solution accuracy. A high-order singularity extraction method is used to compute the hypersingular terms required by near-field local corrections, leading to a very efficient algorithm. The method is applied to linear media, as well as non-linear, anisotropic, and hysteretic medium. The method is also hybridized with the VIE method, allowing for mixed meshes for solving general problems.

Topological derivative for the nonlinear magnetostatic problem

The topological derivative represents the sensitivity of a domain-dependent functional with respect to a local perturbation of the domain and is a valuable tool in topology optimization. Motivated by an application from electrical engineering, we derive the topological derivative for an optimization problem which is constrained by the quasilinear equation of two-dimensional magnetostatics. Here, the main ingredient is to establish a sufficiently fast decay of the variation of the direct state at scale 1 as $|x|\rightarrow \infty$. In order to apply the method in a bi-directional topology optimization algorithm, we derive both the sensitivity for introducing air inside ferromagnetic material and the sensitivity for introducing material inside an air region. We explicitly compute the arising polarization matrices and introduce a way to efficiently evaluate the obtained formulas. Finally, we employ the derived formulas in a level-set based topology optimization algorithm and apply it to the design optimization of an electric motor.

Использование анизотропной магнитной проницаемости при решении нелинейных задач магнитостатики для конструкций с шихтованной сталью

The solution of nonlinear magnetostatic problems for structures with laminated steel is considered by setting the anisotropic magnetic permeability coefficient. It is shown that the solution with actually specified steel layers converges to the solution of the anisotropic problem with increasing number of layers.

An edge-based smoothed finite element method for nonlinear magnetostatic and eddy current analysis

Abstract In this work, an edge-based smoothed finite element method (ES-FEM) is presented to solve electromagnetic field problems. First, the analysis domain is discretized into a set of tetrahedron cells that can be much easily generated automatically for complicated domains, and edge-based smoothing domains are further constructed associated with edges of tetrahedron elements. The Smoothed Galerkin Weak form is then evaluated based on edge-based smoothing domains. The ES-FEM is implemented here to solve nonlinear magnetostatic and eddy current problems, and to observe its properties using linear tetrahedral elements, in which relatively poor result is carried with standard finite element method (FEM). Numerical examples demonstrate that results obtained by the present ES-FEM are much more accurate than ones by standard FEM, and agree well with the experimental ones. It possesses potentials in the successful applications of electromagnetic problems.

Hybrid Model: Permeance Network and 3-D Finite Element for Modeling Claw-Pole Synchronous Machines

Designing a claw-pole synchronous machine implies solving many 3D nonlinear magnetostatic problems which makes the computation (CPU) time very long. In our model, the mesh is refined to reach the desired level of precision on global quantities such as torque. Since the airgap is very thin (around 0.3 mm for a 100 mm diameter) and a Newton Raphson algorithm requires several iterations to converge, CPU time may be too high. Nowadays, many researches are ongoing to reduce the CPU time, while preserving an acceptable accuracy. One of the most efficient methods is permeance networks but this method is not suitable for complex geometries. Our main contribution is to use a permeance network in the areas where flux lines are easy to guess and to solve a 3D FEM problem in complex geometry areas of the magnetic device: the claw poles and the air gap for example. Moreover, current sources belong to the permeance network model, so that there are no current sources in the 3D FEM problem. Then, a 3D scalar magnetic potential formulation can be used easily. The two classical magnetostatic formulations (magnetic scalar potential formulation (Um – hs) and vector potential formulation (a – j)) are presented in this paper. Then, the hybridization of 3D FEM formulation and the permeance network, is presented. Numerical results are compared with experimental measurements and a good agreement is obtained while reducing the CPU time. Index Terms—Claw-pole synchronous machine, 3D FEM, permeance networks, hybrid models.

A Magnetic Vector Potential Volume Integral Formulation for Nonlinear Magnetostatic Problems

Volume integral method (VIM) has been known as an interesting alternative to the finite-element method for electromagnetic field computation. Since only the active regions have to be meshed, VIM is very efficient for modeling of electromagnetic devices containing a predominated air volume. It is also adapted for multistatic studies with motion or optimization strategies. The aim of this paper is to propose a magnetic vector potential volume integral formulation in order to deal with the 3-D nonlinear magnetostatic problems. The main advantage of this formulation is that the convergence of nonlinear material solution can be easily reached after a few iterations without any relaxation. Moreover, the use of coarse meshes can lead to accurate results. Computed results for the TEAM Workshop problem 13 and for an actuator are reported.

Application of the PGD and DEIM to Solve a 3-D Non-Linear Magnetostatic Problem Coupled With the Circuit Equations

Among the model order reduction techniques, the proper generalized decomposition (PGD) has shown its efficiency to solve static and quasistatic problems in the time domain. However, the introduction of non-linearity due to ferromagnetic materials, for example, has never been addressed. In this paper, the PGD technique combined with the discrete empirical interpolation method (DEIM) is applied to solve a non-linear magnetostatic problem coupled with the circuit equations. To evaluate the reduction technique, the transient state of a three-phase transformer at no load is studied using the full finite-element model and the PGD–DEIM model.

Time-Periodicity Condition of Nonlinear Magnetostatic Problem Coupled With Electric Circuit Imposed by Waveform Relaxation Method

In numerical computation, the finite-element method associated with external electric circuits is often used to evaluate electromagnetic devices with voltage sources. To study the solution of the steady state, the computation time can be prohibitive due to a large transient state compared with the time step used to discretize the time domain. In this paper, a method based on the waveform relaxation method is developed in order to impose the steady state of the solution in the case of a nonlinear magnetostatic problem coupled with electric circuit equations.