Nonlinear Magnetostatic Problems Research Articles

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.

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.

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.

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...

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.

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 Proper Generalized Decomposition Method Applied to the Magnetic Simulation of a SMC Microstructure

Improvement of the magnetic performances of soft magnetic composites (SMC) materials requires to link the microstructure to the macroscopic magnetic behavior law. This can be achieved with the finite-element method using the geometry reconstruction from images of the microstructure. Nevertheless, it can lead to large computational times. In that context, the proper generalized decomposition (PGD), that is an approximation method originally developed in mechanics, and based on a finite sum of separable functions, can be of interest in our case. In this work, we propose to apply the PGD method to the SMC microstructure magnetic simulation. A nonlinear magnetostatic problem with the scalar potential formulation is then solved.

Some Treatments of Fictitious Volume Charges in Nonlinear Magnetostatic Analysis by BIE

The scalar potential formulation by the boundary integral equation approach is attractive for numerical analysis but has fatal drawbacks due to a multi-valued function in current excitation. We derive an all-purpose boundary integral equation with double layer charges as the state variable and apply it to nonlinear magnetostatic problems by regarding the nonlinear magnetization as fictitious volume charges. We investigate two approaches how to treat the fictitious charges. In discretization by the constant volume element, a surface loop current is introduced for the volume charge. By the linear volume element, the fictitious charges are evaluated on the condition that the divergence of the magnetic flux density is zero. We give a comparative study of these two approaches.

Coupled Three Dimensional Numerical Calculation of Forces and Stresses on the End Windings of Large Turbo Generators via Integral Formulation

A novel numerical approach to calculate the time evolution of the three dimensional distribution of the magnetic field and forces in the end winding regions of large turbine generators is presented. The proposed approach is based on an integral formulation for nonlinear magnetostatic problems. Its main advantage is the reduction of the discretization to only the conductors and magnetic materials. In this paper the solution of a coupled magnetostructural problem consisting in the calculation of the mechanical stresses and deformations caused by the electrodynamic forces is presented. The analysis is based on a time stepping simulation where the currents are derived from the integration of a lumped parameter model.

A Discrete Geometric Approach to Solving 2-D Non-Linear Magnetostatic Problems

The aim of this paper is to introduce a Discrete Geometric Approach to solving 2-D non-linear magnetostatic problems. In particular, an efficient algorithm will be presented to solve magnetostatics in isotropic non-linear media by means of a Newton-Raphson scheme, in which the Jacobian is calculated analytically. Results on a reference configuration (TEAM Workshop Problem 25) are reported and discussed.

Accuracy improvement in nonlinear magnetostatic field computations with integral equation methods and indirect total scalar potential formulations

PurposeThe paper seeks to solve nonlinear magnetostatic field problems with the integral equation method and different indirect formulations.Design/methodology/approachTo avoid large cancellation errors in cases where the demagnetizing field is high a difference field concept is used. This requires the computation of sources of the scalar potential of the excitation field.FindingsA new formulation to compute these sources is presented. The improved computational accuracy is demonstrated with numerical examples.Originality/valueThe paper develops a novel formulation for the computation of sources of scalar excitation potential.

Magnetic field computation with integral equation method and energy-controlled relaxation

The magnetic field integral equation method is applied to linear and nonlinear magnetostatic problems. Two drawbacks of this method, the slow convergence rate and the dense system matrix, are tackled. The convergence behavior is improved by a novel algorithm that determines adaptive relaxation factors at every iteration step by an energy minimum principle. The dense matrix is compressed with the fast multipole method

Fast Multipole Method Applied to Volume Integral Equation Method

Abstract. The Volume Integral Equation Method (VIEM) has been used for the solution of three-dimensional non-linear magnetostatic field problems. The number of unknowns is minimal as only the magnetic material has to be discretized. For accurate solutions of problems where the magnetic field is small compared to the excitation field a difference field formulation has been developed. To reduce computational costs the fast multipole method is applied both on compression of the system matrix and during post processing. The efficiency of the formulation is demonstrated in several examples.

Convergence improvement in two-dimensional finite element nonlinear magnetic problems—a fuzzy logic approach

A fuzzy-logic based factor for improving the convergence in nonlinear magnetostatic problems is proposed. The Newton–Raphson method is modified by a relaxation factor which is obtained by a fuzzy logic system. It is shown that the proposed factor improves convergence in nonlinear operating points where the conventional Newton–Raphson algorithm fails. A comparison with other techniques is made to show that the method requires less computing time and less iterations to meet the convergence criteria. The fuzzy relaxation factor was applied to a nonlinear finite element model of an electric power transformer. The set of finite element equations was obtained by applying a nodal formulation, which is based in Ampere's law, and it solves Maxwell's equations. The resulting set of nonlinear equations were solved in terms of the magnetic vector potential. Two finite element meshes were employed and different excitations were also considered. The obtained results show that fuzzy-logic relaxation factor can be successfully applied in improving the Newton–Raphson method.

An Application of the Inexact Newton Method to Nonlinear Magnetostatics

Inexact Newton solvers offer many attractive features for the solution of nonlinear problems in the field of electromagnetics. A critical point for the optimal setup of the solver is the choice of the best algorithm for the evaluation of the approximate solution of the linear systems at each Newton step and the most effective preconditioning strategy. In this paper, the Newton iterative solver method is proposed for the solution of a nonlinear magnetostatic problem. The problem is discretized by means of a finite-element approach. The generalized minimum residuals (GMRES) method is adopted as linear solver, and three preconditioners are tested. The performance of this procedure is evaluated for different magnetostatic problems and compared to the results obtained by means of a Newton-Raphson method.

Nonlinear magnetostatic calculations based on fast multipole method

A globally convergent iterative technique for the solution of nonlinear magnetostatic problems is described. This technique can be naturally implemented by using the fast multipole method. In this technique, all computations are confined to the region occupied by ferromagnetic objects. For this reason, this technique may emerge as a viable alternative to the finite-element method. Numerical examples that highlight the efficiency of the fast multipole implementation are presented.

Comparison between different approaches for fast and efficient 3-D BEM computations

Fast methods like the fast multipole method or the adaptive cross approximation technique reduce the memory requirements and the computational costs of the boundary-element method (BEM) to approximately O(N). In this paper, both fast methods are applied in combination with BEM-finite-element method coupling to nonlinear magnetostatic problems.

A new BEM technique for nonlinear 2D magnetostatics

The paper presents a new boundary element technique for the solution of nonlinear 2D magnetostatic problems. It is based on a recurrent formulation of Poisson's equation where the change in material property between the domains is modelled as a single layer along the interfaces. Then the nonhomogeneity appears as a source term. The nonlinear regions are decomposed into subdomains where the material is assumed homogeneous. The underlying principle lies on the transmission-line modelling (TLM) method that is commonly used for electric network analysis. A new numerical scheme is derived where the solution is obtained iteratively without any system of algebraic equations for open boundary problems. This is an important point with respect to the classical FEM and BEM treatment that always require the assembly and the solution of a set of simultaneous algebraic equations. Numerical results show significant reduction in memory requirements for our method due to the absence of any matrix storage.

A numerical procedure for 3-D nonlinear magnetostatic problems using the magnetic vector potential

A numerical procedure for 3-D nonlinear magnetostatic problems using the magnetic vector potential

A new solution technique for open nonlinear magnetostatic problems

The paper presents a new FEM-BEM technique for the solution of open nonlinear two-dimensional magnetostatic problems. It is based on a boundary integral formulation that leads to a numerical scheme where the field solution is obtained iteratively without any system of algebraic equations. This is an important point with respect to the treatments that always require the assembly and the solution of a set of simultaneous algebraic equations. Numerical results show significant reduction in memory requirements for our method due to the absence of any matrix storage.