Nonlinear Eddy Current Problems Research Articles

MSFEM With MOR and DEIM to Solve Nonlinear Eddy Current Problems in Laminated Iron Cores

MSFEM With MOR and DEIM to Solve Nonlinear Eddy Current Problems in Laminated Iron Cores

Second-Order Approximation of Nonlinear Eddy-Current Problems by a Cauer Ladder Network

Second-Order Approximation of Nonlinear Eddy-Current Problems by a Cauer Ladder Network

Numerical analysis of a FEM based on a time-primitive of the electric field for solving a nonlinear transient eddy current problem

The aim of this paper is to analyze from a mathematical and a numerical point of view a formulation of a transient eddy current problem in terms of a time-primitive of the electric field in a bounded domain with ferromagnetic conductors. To this aim, we introduce a Lagrange multiplier to impose the free-divergence condition in the isolated domain. Thus, we obtain a nonlinear degenerate parabolic problem in mixed form and prove its well-posedness. Then, we propose a fully-discrete scheme by using an implicit Euler scheme for time-discretization and a finite element method based on edge and nodal elements for the spatial discretization. We prove quasi-optimal error estimates for the approximation and include some numerical examples to confirm the obtained theoretical results.

Semiexplicit time integration of a reduced magnetic vector potential magneto‐quasistatic field formulation

AbstractIn this article, a reduced vector potential formulation for transient magneto‐quasistatic field problems is presented. The finite element method for spatial discretization yields a system of differential algebraic equations (DAE). Applying a generalized Schur complement the DAE system decomposes into a system of ordinary differential equations (ODE) and an algebraic equation. In order to avoid computationally expensive linearization schemes to solve nonlinear eddy current problems the ODE system is integrated in time using a semiexplicit Euler scheme. The vector potential is split into a source vector potential and a reduced vector potential which describes the eddy current and magnetization effects. The reduced formulation is compared with the underlying nonreduced formulation regarding the overall performance. In addition, the proper orthogonal decomposition method is applied to the iterative preconditioned conjugate gradient method to minimize the number of iterations per time step.

MSFEM and MOR to Minimize the Computational Costs of Nonlinear Eddy-Current Problems in Laminated Iron Cores

MSFEM and MOR to Minimize the Computational Costs of Nonlinear Eddy-Current Problems in Laminated Iron Cores

A boundary and finite element coupling for a magnetically nonlinear eddy current problem

The aim of this paper is to provide a mathematical and numerical analysis for a FEM-BEM coupling approximation of a magnetically nonlinear eddy current formulation by using FEM only on the conducting domain, and by imposing the integral conditions on its boundary. The nonlinear relationship between flux density and the magnetic field intensity is given by a physical parameter called magnetic reluctivity, which is assumed to depend on the Euclidean norm of the magnetic induction in the conducting domain. We use the nonlinear monotone operator theory for parabolic equations to show that the continuous formulation obtained for the coupling is a well-posed problem. Furthermore, we use Nédélec edge elements, standard nodal finite elements, and a backward-Euler time scheme, to obtain a fully discrete formulation and to prove quasi-optimal error estimates.

A Posteriori Error Estimation for the p-Curl Problem

We derive a posteriori error estimates for a semidiscrete finite element approximation of a nonlinear eddy current problem arising from applied superconductivity, known as the p-curl problem. In particular, we show the reliability for nonconforming Nédélec elements based on a residual-type argument and a Helmholtz--Weyl decomposition of W^p_0(curl;Omega). As a consequence, we are also able to derive an a posteriori error estimate for a quantity of interest called the AC loss. The nonlinearity for this form of Maxwell's equation is an analogue of the one found in the p-Laplacian. It is handled without linearizing around the approximate solution. The nonconformity is dealt with by adapting error decomposition techniques of Carstensen, Hu, and Orlando. Geometric nonconformities also appear because the continuous problem is defined over a bounded C^1,1 domain, while the discrete problem is formulated over a weaker polyhedral domain. The semidiscrete formulation studied in this paper is often encountered in commercial codes and is shown to be well-posed. The paper concludes with numerical results confirming the reliability of the a posteriori error estimate.

Coupling Nonlinear Magneto‐Thermal Problems in Frequency Domain with the Mechanical Field

AbstractThis paper investigates a computationally efficient strategy for simulating induction heating processes, using a multiharmonic ansatz and harmonic balancing for the solution of the nonlinear magnetic eddy current problem in frequency domain. In order to increase the flexibility of the meshing procedure and decrease the number of degrees of freedom, a methodology for nonconforming interfaces and Nédéléc edge elements is presented. The verification of the proposed method is carried out by comparison to the conform (conforming mesh) solution. Furthermore, the results from the eddy current problem are used to simulate the temperature distrubution in a thin steel sheet and compute the resulting stresses.

Stabilized Reduced-Order Model of a Non-Linear Eddy Current Problem by a Gappy-POD Approach

In this paper, a stabilized reduced-order model based on the gappy proper orthogonal decomposition (POD) technique is applied to an eddy-current problem with the non-linear material property. A classical magnetodynamic finite element formulation based on the magnetic vector potential is used as reference and starting point to build up the reduced models. The computational complexity of the non-linear problem might be reduced by the discrete empirical interpolation method-POD technique, but the eddy current effects may make this method unstable. In this paper, we propose an approach which is similar to the gappy method to successfully overcome this instability problem, while keeping the computational cost (time and memory) low.

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.

A Hybrid Parallel Method for 3-D Nonlinear Periodic Eddy Current Problems With Motions

This paper describes a highly robust and efficient hybrid parallel computing method for periodic eddy current problems considering motions. This hybrid parallel approach is composed of two levels of parallelization, with distributed memory parallel based on message passing interface at the higher level and shared memory parallel based on multi-threading at the lower level. Application examples will be presented to demonstrate the effectiveness of this method.

A nonlinear magnetic circuit model for periodic eddy current problems using T,ϕ-ϕ formulation

PurposeA transformer model is used as a benchmark for testing various methods to solve 3D nonlinear periodic eddy current problems. This paper aims to set up a nonlinear magnetic circuit problem to assess the solving procedure of the nonlinear equation system for determining the influence of various special techniques on the convergence of nonlinear iterations and hence the computational time.Design/methodology/approachUsing the T,ϕ-ϕ formulation and the harmonic balance fixed-point approach, two techniques are investigated: the so-called “separate method” and the “combined method” for solving the equation system. When using the finite element method (FEM), the elapsed time for solving a problem is dominated by the conjugate gradient (CG) iteration process. The motivation for treating the equations of the voltage excitations separately from the rest of the equation system is to achieve a better-conditioned matrix system to determine the field quantities and hence a faster convergence of the CG process.FindingsIn fact, both methods are suitable for nonlinear computation, and for comparing the final results, the methods are equally good. Applying the combined method, the number of iterations to be executed to achieve a meaningful result is considerably less than using the separated method.Originality/valueTo facilitate a quick analysis, a simplified magnetic circuit model of the 3D problem was generated to assess how the different ways of solutions will affect the full 3D solving process. This investigation of a simple magnetic circuit problem to evaluate the benefits of computational methods provides the basis for considering this formulation in a 3D-FEM code for further investigation.

Numerical Impact of Using Different $E$ –$J$ Relationships for 3-D Simulations of AC Losses in MgB2Superconducting Wires

AC losses in superconductors are generated every time a time-varying current/field is present. Engineers must be able to predict these losses as accurately as possible during the design phase of power applications. The electrodynamics of superconductors can be formulated as a nonlinear eddy current problem in which the resistivity of the superconducting region is a highly nonlinear function of the current density. In 3-D finite-element simulations, it leads to time-consuming simulations and convergence issues. In this paper, we compare two different $E$ – $J$ constitutive equations, namely: 1) power law model and 2) the percolation model (PM), programmed within both the $H$ – $\phi $ and $T$ – $\phi $ formulations. Based on the 3-D case of a three-filament twisted superconducting wire, the numerical performance of all these formulations/material models is compared in terms of accuracy, computation times, number of time steps, and number of Newton iterations for different relaxation methods. It is shown that the combination of the $T$ – $\phi $ formulation and the $E$ – $J$ PM works fine and should be further developed, as it seems to constitute the best modeling option from both a numerical and physical point of view.

A Self-Adaptive Model-Order Reduction Algorithm for Nonlinear Eddy-Current Problems Based on Quadratic–Bilinear Modeling

The finite-element time-domain simulation of nonlinear eddy-current problems requires the iterative solution of a large, sparse system of equations at every time-step. Model-order reduction is a powerful tool for reducing the computational effort for this task. In this paper, an adaptive order-reduction methodology with error control is proposed. In contrast to previous approaches, it treats the nonlinearity without simplification, by rewriting the original equations as a quadratic–bilinear system.

Fixed-Point Harmonic-Balanced Method for Nonlinear Eddy Current Problems

A new method to optimally determine the fixed-point reluctivity is presented to ensure the stable and fast convergence of harmonic solutions. Nonlinear system matrix is linearized by using the fixed-point technique, and harmonic solutions can be decoupled by the diagonal reluctivity matrix. The 1-D and 2-D non-linear eddy current problems under DC-biased magnetization are computed by the proposed method. The computational performance of the new algorithm proves the validity and efficiency of the new algorithm. The corresponding decomposed method is proposed to solve the nonlinear differential equation, in which harmonic solutions of magnetic field and exciting current are decoupled in harmonic domain.

Computation and Analysis of DC-Biased Eddy Current Problems by an Efficient Fixed-Point Technique in the Harmonic Domain

A new method to optimally determine the fixed-point reluctivity is presented to ensure the stable and fast convergence of harmonic solutions. The diagonal fixed-point reluctivity matrix decouples the harmonic solutions and significantly reduces the memory requirement in computation. The nonlinear eddy current problem under dc-biased magnetization is computed by the fixed-point harmonic-balanced finite-element method. The computational performance of the new algorithm proves the validity and the efficiency of the proposed method.

Subspace Correction Method for Computing Magnetic Shields in Large Power Transformers

In a large power transformer, magnetic shields of the oil tank are made of grain-oriented silicon steel laminations. The thickness of a lamination is 0.3 mm and the thickness of its coating film is only 4 μm. Accurate solution of non-linear eddy current problem in such a multi-scale structure is very challenging. This paper proposes a subspace correction method (SCM) to solve the nonlinear eddy current problem in the lamination system. The method can compute 3-D eddy currents in the steel laminations and is very efficient for large-scale simulations. Numerical experiments on the TEAM benchmark model P21 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sup> -M1 show good agreements between the calculated and measured data and demonstrate the competitive behavior of the SCM.

Steady-State Algorithms for Nonlinear Time-Periodic Magnetic Diffusion Problems Using Diagonally Implicit Runge–Kutta Methods

A general framework is presented for the formulation of steady-state simulation algorithms for magnetically nonlinear eddy-current problems using implicit Runge-Kutta (RK) methods. A close analogy is drawn between equations discretized using the backward-Euler method and fully implicit RK methods. Detailed formulations of the time-periodic finite-element method (TPFEM) and the shooting-Newton method (SNM) are derived using the popular family of diagonally implicit RK (DIRK) methods. Both algorithms employ the generalized minimum residual method to solve the linear equations arising at each Newton iteration. The benefits of higher-order DIRK methods are demonstrated by simulating a surface mount permanent magnet synchronous machine. The effects of using a solid versus a laminated rotor back iron on the simulation time are examined. Simulation results indicate that the performance of TPFEM and SNM is quite similar and much faster than transient analysis.

Novel Approach to Model Order Reduction for Nonlinear Eddy-Current Problems

A novel model order reduction approach is proposed for nonlinear eddy-current problems in which the $B$ – $H$ curves are approximated by neural networks. Such approach stems from rewriting the eddy-current equations in an equivalent way in which only quadratic nonlinearities occur and allows to directly construct compact models by projection. The numerical results show that, using such compact models, the whole space–time distribution of the electromagnetic field can be accurately approximated at low computational cost.

Validation of Finite Element Solutions of Nonlinear, Periodic Eddy Current Problems

Abstract An industrial application is presented to validate a finite element analysis of 3-dimensional, nonlinear eddy-current problems with periodic excitation. The harmonic- balance method and the fixed-point technique are applied to get the steady state solution using the finite element method. The losses occurring in steel reinforcements underneath a reactor due to induced eddy-currents are computed and compared to measurements.