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The paper presents a new variant of the H-Φ field formulation for solving 3-D magnetostatic and frequency domain eddy current problems. The suggested formulation uses the vector and scalar tetrahedral elements within conducting and non-conducting domains, respectively. The presented numerical method is capable of solving multiply connected regions and eliminates the need for computing the source current density and the source magnetic field before the actual magnetostatic and eddy current simulations. The obtained magnetostatic results are verified by comparison against the corresponding results of the standard stationary current distribution analysis combined with the Biot-Savart integration. The accuracy of the eddy current results is demonstrated by comparison against the classical A-A-f approach in frequency domain.

The theory and implementation of the new H-Φ magnetostatic and eddy current solver is presented in detail. The method delivers reliable results without the need to compute the source current density and source magnetic field before the actual simulation.

The proposed H-Φ produce radically smaller and considerably better conditioned equation systems than the alternative A-A approach, which usually requires the unphysical regularization in terms of a low electric conductivity value within the nonconductive domain.

The presented numerical method is capable of solving multiply connected regions and eliminates the need for computing the source current density and the source magnetic field before the actual magnetostatic and eddy current simulations.

The purpose of this paper is to improve the multistep algorithm using evolutionary algorithm (EA) for the topology optimization of magnetostatic shielding, and the paper reveals the effectiveness of methodology by comparison with conventional optimization method. Furthermore, the design target is to obtain the novel shape of magnetostatic shielding.

The EAs based on random search allow engineers to define general-purpose objects with various constraint conditions; however, many iterations are required in the FEA for the evaluation of the objective function, and it is difficult to realize a practical solution without island and void distribution. Then, the authors proposed the multistep algorithm with design space restriction, and improved the multistep algorithm in order to get better solution than the previous one.

The variant model of optimized topology derived from improved multistep algorithm is defined to clarify the effectiveness of the optimized topology. The upper curvature of the inner shielding contributed to the reduction of magnetic flux density in the target domain.

Because the converged topology has many pixel element unevenness, the special smoother to remove the unevenness will play an important role for the realization of practical magnetostatic shielding.

The optimized topology will give us useful detailed structure of magnetostatic shielding.

First, while the conventional algorithm could not find the reasonable shape, the improved multistep optimization can capture the reasonable shape. Second, An additional search is attached to the multistep optimization procedure. It is shown that the performance of improved multistep algorithm is better than that of conventional algorithm.

This 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.

A 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.

The 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.

The 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.

To reduce the computational scale for quasi-magnetostatic problems, model order reduction is a good option. Reduced-order modelling techniques based on proper orthogonal decomposition (POD) and centroidal Voronoi tessellation (CVT) have been used to solve many engineering problems. The purpose of this paper is to investigate the computational principle, accuracy and efficiency of the POD-based and the CVT-based reduced-order method when dealing with quasi-magnetostatic problems.

The paper investigates computational features of the reduced-order method based on POD and CVT methods for quasi-magnetostatic problems. Firstly the construction method for the POD and the CVT reduced-order basis is introduced. Then, a reduced model is constructed using high-fidelity finite element solutions and a Galerkin projection. Finally, the transient quasi-magnetostatic problem of the TEAM 21^a model is studied with the proposed reduced-order method.

For the TEAM 21^a model, the numerical results show that both POD-based and CVT-based reduced-order approaches can greatly reduce the computational time compared with the full-order finite element method. And the results obtained from both reduced-order models are in good agreement with the results obtained from the full-order model, while the computational accuracy of the POD-based reduced-order model is a little higher than the CVT-based reduced-order model.

The CVT method is introduced to construct the reduced-order model for a quasi-magnetostatic problem. The computational accuracy and efficiency of the presented approaches are compared.

This paper aims to understand the magnetohydrodynamics (MHD) mechanism in the molten pool under different modes of magnetic field. The comparison focuses on the Lorenz force excitation and its effect on the melt flow and solidification parameters, intending to obtain practical references for the design of magnetic field-assisted laser directed energy deposition (L-DED) equipment.

A three-dimensional transient multi-physical model, coupled with MHD and thermodynamic, was established. The dimension and microstructure of the molten pool under a 0T magnetic field was used as a benchmark for accuracy verification. The interaction between the melt flow and the Lorenz force is compared under a static magnetic field in the X-, Y- and Z-directions, and also an oscillating and alternating magnetic field.

The numerical results indicate that the chaotic fluctuation of melt flow trends to stable under the magnetostatic field, while a periodically oscillating melt flow could be obtained by applying a nonstatic magnetic field. The Y and Z directional applied magnetostatic field shows the effective damping effect, while the two nonstatic magnetic fields discussed in this paper have almost the same effect on melt flow. Since the heat transfer inside the molten pool is dominated by convection, the application of a magnetic field has a limited effect on the temperature gradient and solidification rate at the solidification interface due to the convection mode of melt flow is still Marangoni convection.

This work provided a deeper understanding of the interaction mechanism between the magnetic field and melt flow inside the molten pool, and provided practical references for magnetic field-assisted L-DED equipment design.

To compare the numerical solutions in primal and dual meshes of magnetostatic problems solved with the finite integration technique.

The development of the whole set of magnetostatic discrete formulations is proposed. Four formulations are computed: two in terms of fields and two in terms of potentials. Moreover, each computation is carried out on the primal and dual mesh. Two applications are presented and the results are analysed and discussed.

The whole set of magnetostatic formulations gives only two solutions. The solutions do not depend of the formulation, but they depend of the choice of the field discretisation in primal or dual mesh.

The computation is carried out on the dual mesh.

Several second order edge elements have been applied to solving magnetostatic problems. The performances of these elements are compared through an example of magnetic circuit. In order to ensure the compatibility of the system equations and hence the convergence, the current density is represented by the curl of a source field. This avoids an explicit gauge condition which is cumbersome in the case of high order elements.

Confronting the unveiled sophisticated structural and physical characteristics of permanent magnets, notably the samarium–cobalt (Sm-Co) alloy, This work aims to introduce a simulation scheme that can link physics-based micromagnetics on the nanostructures and magnetostatic homogenization on the mesoscale polycrystalline structures.

The simulation scheme is arranged in a multiscale fashion. The magnetization behaviors on the nanostructures examined with various orientations are surrogated as the micromagnetic-informed hysterons. The hysteresis behavior of the mesoscale polycrystalline structures with micromagnetic-informed hysterons is then evaluated by computational magnetostatic homogenization.

The micromagnetic-informed hysterons can emulate the magnetization reversal of the parameterized Sm-Co nanostructures as the local hysteresis behavior on the mesostructures. The simulation results of the mesoscale polycrystal demonstrate that the demagnetization process starts from the grain with the largest orientation angle (a) and then propagates to the surrounding grains.

The presented scheme depicts the demand for integrating data-driven methods, as the parameters of the surrogate hysteron intrinsically depend on the nanostructure and its orientation. Further hysteron parameters that help the surrogate hysteron emulate the micromagnetic-simulated magnetization reversal should be examined.

This work provides a novel multiscale scheme for simulating the polycrystalline permanent magnets’ hysteresis while recapitulating the nanoscale mechanisms, such as the nucleation of domains, and domain wall migration and pinning. This scheme can be further extended to simulate the part-level hysteresis considering the mesoscale features.

The purpose of this paper is to calculate forces created by the magnetic leakage field, which are directly applied to tank walls via magnetic shield.

Electromagnetic and mechanical calculations use 3D finite element technology, both applied to materials having constant orthotropic properties. The magnetic solver uses harmonic excitation; the analysis of mechanical deflection is carried out in static conditions. Two types of forces are considered: magnetostatic surface forces and magnetostriction volumetric ones. In measurements, the laser scanning vibrometer was applied.

Electromagnetic calculations must use an FE mesh much denser than that for typical power loss analysis. The magnetic orthotropy of the shield material does not create any important effects and it may be omitted. Magnetostriction forces are similar in value to magnetostatic ones, but their influence on the shield deformation is negligible.

The results obtained for the analysis of the displacement of elements of the tank wall are exemplary – they show the difference between magnetostatic and magnetostriction excitation only. The analysis of the vibration of the transformer tank must include the presence of the oil inside the tank.

The asymmetrical placement of magnetic shields against the transformer core creates the visible differences in the magnitudes of magnetostatic forces applied to particular shields. Therefore, the design of magnetic shielding should also include the vibrational point of view.

A new finite element‐boundary integral procedure for the solution of the magnetostatic field problem has been introduced previously by the author and J.S. Colonias. The approach there involved a reformulation of an integral equation in the iron domain as a weak formulation combining a differential operator in the iron domain and an integral operator on the iron boundary. The reformulated problem is discretized by finite element techniques in solenoidal piecewise polynomial subspaces. In this paper we modify the above formulation to obtain one with a differential operator inside a sphere containing the iron domain and an integral operator on the boundary of the sphere. We then consider a numerical realization of our method in the two dimensional case. This new formulation considerably simplifies the computation of the discrete operator on the boundary.