Nonlinear Magnetostatics Research Articles

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.

An Improved Newton Method Based on Choosing Initial Guess Applied to Scalar Formulation in Nonlinear Magnetostatics

An improved starting point Newton method applied to 3-D scalar formulation in magnetostatics is proposed in this paper. Compared with the classical Newton method, the inexact-Newton and quasi-Newton methods are reported by testing on a benchmark problem as well as an industrial example. Remarkable convergence acceleration using the proposed strategy is observed, and thus, it significantly reduces the computational time.

Comparison of DEIM and BPIM to Speed Up a POD-Based Nonlinear Magnetostatic Model

The Proper Orthogonal Decomposition (POD) combined with the (Discrete) Empirical Interpolation Method (DEIM) has been successfully used to reduce the computation time of nonlinear Finite Element (FE) problems. The DEIM enables to alleviate the complexity cost of the nonlinear terms. In this communication, the Best Points Interpolation Method (BPIM) is compared with the DEIM in the case of nonlinear magnetostatics.

Design Sensitivity Analysis for Shape Optimization of Nonlinear Magnetostatic Systems

This paper discusses the sensitivity analysis for the shape optimization of a nonlinear magnetostatic system, evaluated both by direct and adjoint approaches. The calculations rely on the Lie derivative concept of differential geometry, where the flow is the velocity field associated with the modification of a geometrical parameter in the model. The resulting sensitivity formulas can be expressed naturally in a finite-element setting through a volume integral in the layer of elements connected to the surface undergoing shape modification. The accuracy of the methodology is analyzed on a 2-D model of an interior permanent magnet motor and on a 3-D model of a permanent magnet system.

Topology Optimization of Electric Motor Using Topological Derivative for Nonlinear Magnetostatics

We aim at finding an optimal design for an interior permanent magnet electric motor by means of a sensitivity-based topology optimization method. The gradient-based ON/OFF method has been successfully applied to optimization problems of this form. We show that this method can be improved by considering the mathematical concept of topological derivatives (TDs). TDs for optimization problems constrained by linear partial differential equations (PDEs) are well understood, whereas little is known about TDs in combination with nonlinear PDE constraints. We derive the TD for an optimization problem constrained by the equation of nonlinear 2-D magnetostatics, illustrate its advantages over the sensitivities used in the ON/OFF method, and show numerical results for the optimization of an interior permanent magnet electric motor obtained by a level-set algorithm, which is based on the TD.

Shape Optimization of an Electric Motor Subject to Nonlinear Magnetostatics

The goal of this paper is to improve the performance of an electric motor by modifying the geometry of a specific part of the iron core of its rotor. To be more precise, the objective is to smooth the rotation pattern of the rotor. A shape optimization problem is formulated by introducing a tracking-type cost functional to match a desired rotation pattern. The magnetic field generated by permanent magnets is modeled by a nonlinear partial differential equation of magnetostatics. The shape sensitivity analysis is rigorously performed for the nonlinear problem by means of a new shape-Lagrangian formulation adapted to nonlinear problems.

Complementarity bilateral bounds on forces in magnet systems

PurposeIt is well known that complementarity can provide bilateral bounds on energy, in numerical approximations of nonlinear magnetostatics. Questions whether like force is, up to sign, the derivative of energy, can such bounds also apply to forces, torques, and other concepts?Design/methodology/approachThis is a discussion paper exploring the issues conceptually.FindingsOn the face of it, no, since inequalities are not preserved by differentiation. Yet, useful bounds can be given in some cases, as shown in the discussion, following an idea by Matagne.Originality/valueThe paper aids the understanding of nonlinear magnetostatic systems.

A sequential coupling of optimal topology and multilevel shape design applied to two-dimensional nonlinear magnetostatics

In this paper, a sequential coupling of two-dimensional (2D) optimal topology and shape design is proposed so that a coarsely discretized and optimized topology is the initial guess for the following shape optimization. In between, we approximate the optimized topology by piecewise Bezier shapes via least square fitting. For the topology optimization, we use the steepest descent method. The state problem is a nonlinear Poisson equation discretized by the finite element method and eliminated within Newton iterations, while the particular linear systems are solved using a multigrid preconditioned conjugate gradients method. The shape optimization is also solved in a multilevel fashion, where at each level the sequential quadratic programming is employed. We further propose an adjoint sensitivity analysis method for the nested nonlinear state system. At the end, the machinery is applied to optimal design of a direct electric current electromagnet. The results correspond to physical experiments.

Finite formulation of nonlinear magnetostatics with Integral boundary conditions

This paper describes two hybrid methods coupling finite formulation of electromagnetic fields (FFEF) in a bounded domain to integral boundary conditions taking into account far field conditions. The two hybrid techniques use different boundary conditions: the first formulation is based on Green's function applied to magnetization source inside bounded domain while the other one is based on a boundary-element method on its external surface. Details about the coupling terms are given and handling of different magnetization sources is described, including the fictitious magnetization sources coming from nonlinear solutions. The proposed methods are validated versus different benchmark cases. Comparisons between the two techniques have been performed using different criteria (accuracy and convergence, memory requirements, etc.).

Field strength computation at edges in nonlinear magnetostatics

Magnetostatic problems including iron components can be solved by a nonlinear indirect volume integral equation. Its unknowns are scalar field sources. They are evaluated iteratively. In doing so the integral representation of fields has to be calculated. At edges singularities occur. Following a method to calculate the field strength on charged surfaces a way out is presented.

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.

Topology optimization of nonlinear magnetostatics

The topology optimization of nonlinear magnetostatic systems is studied using the finite-element method. The topology design sensitivity formulation of nonlinear magnetostatics is derived using the adjoint variable method. A computer program is developed using object-orient programming and applied to the topology optimization of a C-core actuator. A numerical study shows the effect of saturation by comparing linear and nonlinear topology optimization.

Error estimation of finite element solution in nonlinear magnetostatic 2D problems

This paper presents an error estimator which enables control of the quality of finite element solutions in nonlinear magnetostatics. This method, based on the works of P. Ladeveze [1983, 1991] in mechanics, consists of constructing complementary admissible field from the one calculated by the finite element method. This method has been transposed in the cases of nonlinear vector and scalar potential 2D formulations. The results given by the proposed estimator are compared to the one based on two finite element solutions on two examples.

Symmetry and TLM method in nonlinear magnetostatics

Exploitation of geometrical symmetry in magnetostatic problems with asymmetry in excitation is generally solved by using a symmetric decomposition technique derived from group theory. However, the nonlinearity makes this method invalid since the material property no longer shares this symmetry. A method that makes the exploitation of symmetry feasible is proposed. It consists of replacing the nonlinear material by a linear and homogeneous one, the nonlinearity is expressed as fictitious sources in an iterative algorithm via the transmission line modeling (TLM) method. The symmetric decomposition of the problem may then be performed at each step of the process. Large computational cost savings are obtained as demonstrated by numerical results.

A new vector element in the volume integral equation method for nonlinear magnetostatics

To calculate nonlinear magnetostatics a new vector element in the volume integral equation has been developed. The magnetization vector in an element is expressed by using co- and contra-variant vectors corresponding to the each surface of the hexahedron element. To accelerate the conversion process, the iteration procedure has been divided into initialization and convergence parts. The method has been applied to TEAM workshop problem 13 of IEEE. Its calculated results agree with the measured data within 2%, indicating the method provides good accuracy. >

Some improvements in nonlinear 3D magnetostatics

It is pointed out that nonlinear magnetostatic field problems must be solved by an iterative approach, either by direct iteration or by the Newton-Raphson method or by a combination of both. Some improvements are described here to decrease the number of iterations that are necessary to arrive at reliable results. CPU-time requirements with and without these improvements are compared. The improved software package is applied to an inverse problem, in which the form of the pole of a C-shaped nonlinear magnet is to be optimized. >

Adaptively meshed boundary integral equation method for nonlinear magnetostatics

A general approach to the treatment of nonlinear problems in magnetostatics using boundary integral equation methods is discussed. Adaptive meshing is used to obtain a more optimal distribution of boundary elements using the dependent variable as the criterion. Permanent magnets are shown to be accurately modeled using this method. Example solutions are presented for a ring magnet in the vicinity of a ferromagnetic sphere. Forces are easily calculated using Maxwell stress and are seen to compare very well against measurement. >

Recent advances in MAGNUS computational technology for three-dimensional nonlinear magnetostatics

Presents recent advances of the computer program MAGNUS. MAGNUS can solve numerically any general problem of nonlinear magnetostatics in three dimensions. The problem is formulated in a domain with boundary conditions of the following types: Dirichlet, Neumann (field confinement), or periodicity. The domain can contain conductors of any shape in space, nonlinear magnetic materials with magnetic properties specified by magnetization tables, and nonlinear permanent magnets with any given demagnetization curve. MAGNUS uses the two-scalar-potentials formulation of magnetostatics and the finite-element method, has an automatic 3D mesh generator, and advanced post-processing features that include graphics on a variety of supported devices, tabulation, and calculation of design quantities required in magnetic engineering. Because of its generality, MAGNUS has found applications in the design of various vacuum electronic devices that include accelerator magnets and spectrometers, steering magnets, wigglers for free-electron lasers, light sources for lithography and microtrons as well as magnets for NMR and medical applications and recording heads. This paper deals with the latest extensions of MAGNUS.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The iterative boundary element method for nonlinear electromagnetic field calculations

The iterative boundary-element method for the calculation of nonlinear magnetostatic and eddy-current problems in 2-D is treated. Boundary-integral-equation formulations in terms of the vector potential and its normal scheme that can assure the solution convergence is described. Two examples are given to show the efficiency and accuracy of the present method. One consists of modeling the nonlinear magnetostatics in terms of the magnetic field vector potential; the other is the problem of computing the nonlinear T-E-mode eddy current. >

Implementation of MB algorithm for nonlinear magnetostatics

The iteration strategy in the MB formulation for nonlinear magnetostatic problems is discussed. The difference between MH and MB iteration is examined, and two methods for implementating the constitutive M(B) relation are presented.