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The purpose of this paper is to apply rectangular Bézier surface patches directly into the mathematical formula used to solve boundary value problems modeled by Laplace's equation. The mathematical formula, called the parametric integral equation systems (PIES), will be obtained through the analytical modification of the conventional boundary integral equations (BIE), with the boundary mathematically described by rectangular Bézier patches.

The paper presents the methodology of the analytic connection of the rectangular patches with BIE. This methodology is a generalization of the one previously used for 2D problems.

In PIES the paper separates the necessity of performing simultaneous approximation of both boundary shape and the boundary functions, as the boundary geometry has been included in its mathematical formalism. The separation of the boundary geometry from the boundary functions enables to achieve an independent and more effective improvement of the accuracy of both approximations. Boundary functions are approximated by the Chebyshev series, whereas the boundary is approximated by Bézier patches.

The originality of the proposed approach lies in its ability to automatic adapt the PIES formula for modified shape of the boundary modeled by the Bézier patches. This modification does not require any dividing the patch into elements and creates the possibility for effective declaration of boundary geometry in continuous way directly in PIES.

In this paper, the Cauchy-type problem for the Laplace equation was solved in the rectangular domain with the use of the Chebyshev polynomials. The purpose of this paper is to present an optimal choice of the regularization parameter for the inverse problem, which allows determining the stable distribution of temperature on one of the boundaries of the rectangle domain with the required accuracy.

The Cauchy-type problem is ill-posed numerically, therefore, it has been regularized with the use of the modified Tikhonov and Tikhonov–Philips regularization. The influence of the regularization parameter choice on the solution was investigated. To choose the regularization parameter, the Morozov principle, the minimum of energy integral criterion and the L-curve method were applied.

Numerical examples for the function with singularities outside the domain were solved in this paper. The values of results change significantly within the calculation domain. Next, results of the sought temperature distributions, obtained with the use of different methods of choosing the regularization parameter, were compared. Methods of choosing the regularization parameter were evaluated by the norm N_max.

Calculation model described in this paper can be applied to determine temperature distribution on the boundary of the heated wall of, for instance, a boiler or a body of the turbine, that is, everywhere the temperature measurement is impossible to be performed on a part of the boundary.

The paper presents a new method for solving the inverse Cauchy problem with the use of the Chebyshev polynomials. The choice of the regularization parameter was analyzed to obtain a solution with the lowest possible sensitivity to input data disturbances.

This paper seeks to present a wavelet Galerkin boundary element method (WGBEM) for solving 2D Laplace's equation in which the coefficient matrices are exactly computed through fast Fourier transform (FFT).

When Daubechies periodized wavelets are used to approximate smooth functions the wavelet coefficients can be exactly computed through FFT, and numerical evaluation of wavelet integrations is discarded. The same methods for computing the expansion coefficients of 2D function and singular function are derived.

For Daubechies periodized wavelets based WGBEM, two remarkable advantages over the existing WGBEM are: the coefficient matrices can be exactly calculated through FFT; a linear system with potential and flux values at dyadic points as unknowns can be gained. Thus, the mixed boundary condition can be enforced with ease. Sparse matrices with only O(N log N) nonzero entries are obtained by performing the fast wavelet transform (FWT) and the truncation of matrix entries.

A wavelet‐integration‐free method for computing the wavelet coefficients is introduced into the WGBEM. The conventional numerical evaluation of wavelet integrations is avoided. Good performance and accuracy are obtained.

In conventional boundary element formulations, the singularities of the fundamental solution are usually located on the problem boundary. This leads to difficulties in evaluating solution quantities on or near the boundary. A method is presented for locating the singularities on an auxiliary boundary outside the problem domain and having this auxiliary boundary location determined automatically via a Galerkin criterion. This automatic generation of the auxiliary boundary results in a highly accurate, adaptive but non‐linear method. The number of singularities can be significantly reduced compared to conventional boundary element formulations which usually require the same number of singularities as the number of boundary elements used. The method is illustrated with three examples involving Laplace's equation in two dimensions. Excellent numerical results are obtained in all cases using only a few singularities.

The purpose of this paper is to study the saturation and nonlinear performance of magnetic field in the air gap of switched reluctance motor (SRM).

The analytical method of sub-domain combined with the saturation compensation method is used to determine the nonlinear distribution of air gap magnetic field in SRM. Also, the resolutions of the two-dimensional (2D) Laplace’s equation and Poisson’s equation in polar coordinates are used to obtain the simplified expression of magnetic flux density.

For verifying the effectiveness of analytical model, the results are compared with those obtained from the 2D finite element method (FEM). The influence of magnetic saturation is taken into account by associating the sub-domain analysis result with the nonlinear B-H properties of stator and rotor iron. The magnetic flux density in radial and tangential direction considering the saturation effect may be calculated accurately. It can be seen that one can easily determine the linear analytical results accurately, whereas it is difficult to determine the magnetic flux density with saturation influence; especially at some local positions, there is a larger difference between analytical and FE model due to the complex boundary conditions.

This paper presents the development and optimization design of high-performance SRM.

The magnetic saturation may be taken into account for the SRM and analytical models support to simulated system performance.

Eddy currents are investigated in a ferromagnetic bar exposed to a transverse magnetic field. Such an open boundary field problem is analysed applying the Galerkin finite element method coupled with a separation of variables. A steady state is considered, introducing time periodic conditions. The resultant system of nonlinear equations is solved using an iterative procedure based on Brouwer's fixed‐point theorem referred to the nonlinear material reluctivity. Numerical results are presented for a massive conductor made of cast steel and cast iron. The eddy‐current distribution and characteristics of power losses are illustrated in a graphic form.

The purpose of this study is to propose an analytical model with consideration of the permeability of soft-magnetic materials, which can predict the magnetic field distribution more accurately and facilitate the initial design and parameter optimization of the machine.

This paper proposes an analytical model of stator yokeless radial flux dual rotor permanent magnet synchronous machine (SYRFDR-PMSM) with the consideration of magnetic saturation of soft-magnetic material. The analytical model of SYRFDR-PMSM is divided into seven regions along the radial direction according to the different excitation source and magnetic medium, and the iron permeability in each region is considered based on the Maxwell–Fourier method and Cauchy’s product theorem. The magnetic vector potential of each region is obtained by the Laplace’s or Poisson’s equation, and the magnetic field solution is determined using the boundary conditions of adjacent regions.

The inner and outer air-gap flux density, flux linkage, output torque, etc., of SYRFDR-PMSM are predicted by analytical model, resulting in good agreement with that of finite element model. Additionally, the SYRFDR-PMSM prototype is manufactured and the correctness of analytical model is further verified by experiments on no-load back electromotive force and current–torque curve. Reasonable design of the slot opening width and pole arc coefficient can improve the average output torque and reduce output torque ripple.

The analytical model proposed in this paper assumes that the permeability of soft-magnetic material is a fixed value. However, the actual iron’s permeability varies nonlinearly; thus, the prediction results of the analytical model will have some deviations from the actual machine.

The main contribution of this paper is to propose an accurate magnetic field analytical model of SYRFDR-PMSM. It takes into account the permeability of soft-magnetic material and slot opening, which can quickly and accurately predict the electromagnetic performance of SYRFDR-PMSM. It can provide assistance for the initial design and optimization of SYRFDR-PMSM.

This study aims to accurately calculate the magnetic field distribution, which is a prerequisite for pre-design and optimization of electromagnetic performance. Accurate calculation of magnetic field distribution is a prerequisite for pre-design and optimization.

This paper proposes an analytical model of permanent magnet machines with segmented Halbach array (SHA-PMMs) to predict the magnetic field distribution and electromagnetic performance. The field problem is divided into four subdomains, i.e. permanent magnet, air-gap, stator slot and slot opening. The Poisson’s equation or Laplace’s equation of magnetic vector potential for each subdomain is solved. The field’s solution is obtained by applying the boundary conditions. The electromagnetic performances, such as magnetic flux density, unbalanced magnetic force, cogging torque and electromagnetic torque, are analytically predicted. Then, the influence of design parameters on the torque is explored by using the analytical model.

The finite element analysis and prototype experiments verify the analytical model’s accuracy. Adjusting the design parameters, e.g. segments per pole and air-gap length, can effectively increase the electromagnetic torque and simultaneously reduce the torque ripple.

The main contribution of this paper is to develop an accurate magnetic field analytical model of the SHA-PMMs. It can precisely describe complex topology, e.g. arbitrary segmented Halbach array and semi-closed slots, etc., and can quickly predict the magnetic field distribution and electromagnetic performance simultaneously.

The numerical solution of problems involving two‐dimensional flow in an infinite or a semi‐infinite channel is considered. Beyond a certain finite region, where the flow and geometry may be general, a “tail” region is assumed where the flow is potential and the channel is uniform. This situation is typical in many cases of fluid‐structure interaction and flow around obstacles in a channel. The unbounded domain is truncated by means of an artificial boundary B, which separates between the finite computational domain and the “tail.” On B, special boundary conditions are devised. In the finite computational domain, the problem is solved using a finite element scheme. Both non‐local and local artificial boundary conditions are considered on B, and their performance is compared via numerical examples.

Non‐intrusive magnetic measurements in AC machines are possible with small flat coils stuck on the external surface of the housing of a running motor. The aim of the paper consists in determining transmission coefficients able to give a direct relationship between the weak external flux density and the airgap one.

An experimental approach shows that the decoupling principle can be applied. Transmission coefficients are determined separately for the stator yoke, the motor housing and the external air.

For low frequencies and a housing made of steel, eddy current can be neglected. The transmission coefficient depends strongly of the mode (number of poles) of the rotating field. Conversely, for higher harmonic ranks, the additional attenuation caused by eddy currents in the housing does not practically depend on the mode but is strongly dependant on the frequency.

The transmission coefficients are determined considering a 2D electromagnetic model and several simplifying hypothesis. Experiments prove the validity of the proposed approach up to 550 Hz.

Up to now, many fault detection systems are based on the presence of additional harmonics in the external magnetic field spectrum. With the knowledge of simple transmission coefficients, an analysis of the variation of the magnitude of critical spectrum lines is now possible for a more precise fault detection in AC machines.

To the authors' knowledge, the only alternative way for the interpretation of external field measurements consist in using a numerical method with a full model of the machine which takes a lot of computation time. The proposed transmission coefficients provide a faster method valid for most of the interesting spectrum lines.