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The purpose of this paper is to propose a two-dimensional (2-D) hybrid analytical model (HAM) in polar coordinates, combining a 2-D exact subdomain (SD) technique and magnetic equivalent circuit (MEC), for the magnetic field calculation in electrical machines at no-load and on-load conditions.

In this paper, the proposed technique is applied to dual-rotor permanent magnet (PM) synchronous machines. The magnetic field is computed by coupling an exact analytical model (AM), based on the formal resolution of Maxwell’s equations applied in subdomains, in regions at unitary relative permeability with a MEC, using a nodal-mesh formulation (i.e. Kirchhoff's current law), in ferromagnetic regions. The AM and MEC are connected in both directions (i.e. r- and theta-edges) of the (non-)periodicity direction (i.e. in the interface between teeth regions and all its adjacent regions as slots and/or air-gap). To provide accurate solutions, the current density distribution in slot regions is modeled by using Maxwell’s equations instead to MEC and characterized by an equivalent magnetomotive force (MMF) located in the slots, teeth and yoke.

It is found that whatever the iron core relative permeability, the developed HAM gives accurate results for both no-load and on-load conditions. Finite element analysis demonstrates the excellent results of the developed technique.

The main objective of this paper is to achieve a direct coupling between the AM and MEC in both directions (i.e. r- and theta-edges). The current density distribution is modeled by using Maxwell’s equations instead to MEC and characterized by an MMF.

This paper aims to propose an improved two-dimensional hybrid analytical method (HAM) in Cartesian coordinates, based on the exact subdomain technique and the magnetic equivalent circuit (MEC).

The magnetic field solution is obtained by coupling an exact analytical model (AM), calculated in all regions having relative permeability equal to unity, with a MEC, using a nodal-mesh formulation (i.e. Kirchhoff’s current law) in ferromagnetic regions. The AM and MEC are connected in both axes (x, y) of the (non-)periodicity direction (i.e. in the interface between the tooth regions and all its adjacent regions as slots and/or air-gap). To provide accuracy solutions, the current density distribution in slot regions is modeled by using Maxwell’s equations instead of the MEC characterized by an equivalent magnetomotive force (MMF) located in slots, teeth and yokes.

It is found that whatever the iron core relative permeability, the developed HAM gives accurate results for no- and on-load conditions. The finite-element analysis demonstrates excellent results of the developed technique.

The main objective of this paper is to make a direct coupling between the AM and MEC in both directions (i.e. x- and y-edges). The current density distribution is modeled by using Maxwell’s equations instead of the MEC and characterized by an MMF.

In semi-analytical modeling of spoke-type permanent-magnet (PM) machines (STPMM), the saturation effect is usually neglected (i.e. iron parts are considered to be infinitely permeable) and the PM magnetization is assumed tangential (i.e. magnetization pattern is considered to be tangential-parallel). This paper aims to present an improved two-dimensional (2D) subdomain technique for STPMM with the PM magnetization orientation in quasi-Cartesian coordinates by using hyperbolic functions considering non-homogeneous Neumann boundary conditions (BCs) in non-periodic regions and by applying the interfaces conditions (ICs) in both directions (i.e. t- and θ edges ICs).

The polar coordinate system is transformed into a quasi-Cartesian coordinate system. The rotor and stator regions are divided into primary subdomains, and a partial differential equation (PDE) is assigned to each subdomain. In the PM region, the magnetization orientation is considered in the equations. By applying BCs, the general solution of the equations is determined, and by applying the ICs, the corresponding coefficients are determined.

Using the proposed coordinate system, the general solution of PDEs and their coefficients can mathematically be simplified. The magnetic field and non-intrinsic unbalanced magnetic forces (UMF) calculations have been performed for three different values of iron core relative permeability (200, 800 and ∞), as well as different magnetization orientation values (135 and 80 degrees). The semi-analytical model based on the subdomain technique is compared with those obtained by the 2D finite-element analysis (FEA). Results disclose that the PM magnetization angle can affect directly the performance characteristics of the STPMM.

A new model for prediction of electromagnetic performances in the STPMM takes into account magnetization direction, and soft magnetic material relative permeability in a pseudo-Cartesian coordinate system by using subdomain technique is presented.

This paper aims to develop an analytical approach to validate the finite element analysis (FEA) results. FEA itself is a powerful tool to evaluate the performance of electrical machines but takes more time and requires more drive storage. To overcome this issue, subdomain modeling (SDM) is used for the proposed machine.

SDM is developed to validate the electromagnetic performance of a new linear hybrid excited flux switching machine (LHEFSM) with ferrite magnets. In SDM, the problem is divided into different physical regions called subdomains. Maxwell's governing equation is solved analytically for each region, where the magnetic flux density (MFD) is generated. From the generated MFD, x and y components are calculated, which are then used to find the useful force along the x-axis.

FEA validates the developed SDM via JMAG v. 20.1. The results obtained show excellent agreement with an accuracy of 95.13%.

The proposed LHEFSM is developed for long stroke applications like electric trains.

The proposed LHEFSM uses low-cost ferrite magnets with DC excitation, which offers better flux regulation capability with improved electromagnetic performance. Moreover, the developed SDM reduces drive storage and computational time by modeling different parts of the machine.

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 purpose of this study is the introduction of finite permeability of ferromagnetic core in analytical approach for slotted spoke-type permanent magnet machine.

A two-dimensional analytical approach of magnetic field distribution is established for spoke-type permanent magnet machine to calculate the flux density distribution in the middle of airgap. The paper presents an analytical subdomain model accounting for stator slotting effect. The governing equations are obtained from Maxwell’s equations by using vector potential in all regions of the machine, i.e. magnet, airgap, stator slots and rotor/stator yoke. The finite element analysis is used to validate the analytical results.

It is found that the developed subdomain model including finite permeability of ferromagnetic core is accurate and is applicable for spoke-type permanent magnet machine for no-load and on-load condition. The analytical results are in accurate agreement with the numerical simulation.

Some assumptions and conditions are presented to improve and simplify the analytical method for analyzing the global saturation for spoke-type permanent magnet machine.

The paper gives the description of boundary element method (BEM) with subdomains for the solution of convection—diffusion equations with variable coefficients and Burgers’ equations. At first, the whole domain is discretized into K subdomains, in which linearization of equations by representing convective velocity by the sum of constant and variable parts is carried out. Then using fundamental solutions for convection—diffusion linear equations for each subdomain the boundary integral equation (in which the part of the convective term with the constant convective velocity is not included into the pseudo‐body force) is formulated. Only part of the convective term with the variable velocity, which is, as a rule, more than one order less than convective velocity constant part contribution, is left as the pseudo‐source. On the one hand, this does not disturb the numerical BEM—algorithm stability and, on the other hand, this leads to significant improvement in the accuracy of solution. The global matrix, similar to the case of finite element method, has block band structure whereas its width depends only on the numeration order of nodes and subdomains. It is noted, that in comparison with the direct boundary element method the number of global matrix non‐zero elements is not proportional to the square of the number of nodes, but only to the total number of nodal points. This allows us to use the BEM for the solution of problems with very fine space discretization. The proposed BEM with subdomains technique has been used for the numerical solution of one‐dimensional linear steady‐state convective—diffusion problem with variable coefficients and one‐dimensional non‐linear Burgers’ equation for which exact analytical solutions are available. It made it possible to find out the BEM correctness according to both time and space. High precision of the numerical method is noted. The good point of the BEM is the high iteration convergence, which is disturbed neither by high Reynolds numbers nor by the presence of negative velocity zones.

The purpose of this paper is as follows: to significantly reduce the computation time (by a factor of 1,000 and more) compared to known numerical techniques for real-world problems with complex interfaces; and to simplify the solution by using trivial unfitted Cartesian meshes (no need in complicated mesh generators for complex geometry).

This study extends the recently developed optimal local truncation error method (OLTEM) for the Poisson equation with constant coefficients to a much more general case of discontinuous coefficients that can be applied to domains with different material properties (e.g. different inclusions, multi-material structural components, etc.). This study develops OLTEM using compact 9-point and 25-point stencils that are similar to those for linear and quadratic finite elements. In contrast to finite elements and other known numerical techniques for interface problems with conformed and unfitted meshes, OLTEM with 9-point and 25-point stencils and unfitted Cartesian meshes provides the 3-rd and 11-th order of accuracy for irregular interfaces, respectively; i.e. a huge increase in accuracy by eight orders for the new 'quadratic' elements compared to known techniques at similar computational costs. There are no unknowns on interfaces between different materials; the structure of the global discrete system is the same for homogeneous and heterogeneous materials (the difference in the values of the stencil coefficients). The calculation of the unknown stencil coefficients is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy of OLTEM at a given stencil width. The numerical results with irregular interfaces show that at the same number of degrees of freedom, OLTEM with the 9-points stencils is even more accurate than the 4-th order finite elements; OLTEM with the 25-points stencils is much more accurate than the 7-th order finite elements with much wider stencils and conformed meshes.

The significant increase in accuracy for OLTEM by one order for 'linear' elements and by 8 orders for 'quadratic' elements compared to that for known techniques. This will lead to a huge reduction in the computation time for the problems with complex irregular interfaces. The use of trivial unfitted Cartesian meshes significantly simplifies the solution and reduces the time for the data preparation (no need in complicated mesh generators for complex geometry).

It has been never seen in the literature such a huge increase in accuracy for the proposed technique compared to existing methods. Due to a high accuracy, the proposed technique will allow the direct solution of multiscale problems without the scale separation.

The cell‐based method of domain decomposition was first introduced for complex 3D geometries. To further assess the method, the aim is to carry out flow simulation in rectangular ducts to compare the known analytical solutions.

The method is not based on equal subvolumes but on equal numbers of active cells. The variables of the simulation are stored in ordered 1D arrays to replace the conventional 3D arrays, and the domain decomposition of the complex 3D problems therefore becomes 1D. Finally, the 3D results can be recovered using a coordinate matrix. Through the flow simulation in the rectangular ducts how the algorithm of the domain decompositions works was illustrated clearly, and the numerical solution was compared with the exact solutions.

The cell‐based method can find the subdomain interfaces successfully. The parallelization based on the algorithm does not cause additional errors. The numerical results agree well with the exact solutions. Furthermore, the results of the parallelization show again that domains of 3D geometries can be decomposed automatically without inducing load imbalances.

Although, the approach is illustrated with lattice Boltzmann method, it is also applicable to other numerical methods in fluid dynamics and molecular dynamics.

Unlike the existing methods, the cell‐based method performs the load balance first based on the total number of fluid cells and then decomposes the domain into a number of groups (or subdomains). Thus, the task of the cell‐based method is to recover the interface rather than to balance the load as in the traditional methods. This work has examined the celled‐based method for the flow in rectangular ducts. The benchmark test confirms that the cell‐based domain decomposition is reliable and convenient in comparison with the well‐known exact solutions.

Addresses problems in mechanics and physics involving two or more coupled variables of different nature, or a number of distinct domains which interact. For these kinds of problems, considers numerical solution by the coupling of operators appertaining to the individual participating phenomena, or defined in the domains. Reviews the co‐operation of distinct discretized operators in connection with the integration of temporal evolution processes, and the iterative treatment of stationary equations of state. The specification of subtasks complies with the demand for an independent treatment on different processing units arising in parallel computation. Physical subtasks refer to problems of different field variables interacting on the continuum level; their number is usually small. Fine granularity may be achieved by separating the problem region into subdomains which communicate via the boundaries. In multiphysics simulations operators are preferably combined such that subdomains are processed in parallel on different units, while physical phenomena are processed sequentially in the subdomain.