Subdomain Method Research Articles

Efficient RCS Computation Over a Broad Frequency Band Using Subdomain MoM and Chebyshev Approximation Technique

The analysis of the target’s broadband electromagnetic scattering characteristics plays an important role in radar stealth and recognition areas. The traditional method of moments (MoM) needs to calculate the currents at each frequency point when analyzing the target. With the increase of the target electrical size and complexity, the radar cross section (RCS) of the target changes drastically with frequency, it is necessary to calculate the accurate frequency response with a small frequency interval, which is very time-consuming in calculation and not feasible. This paper proposes to combine the subdomain method of moments (SMoM) with the Chebyshev approximation technique (CAT) to efficiently analyze the electromagnetic scattering of arbitrary shaped objects over a broad frequency band. In the SMoM technique, the whole object can be divided into several regions, each of which can be independently solved by the method of moments (MoM). The CAT technique is employed to expand the unknown current coefficients into a Chebyshev series for achieving fast frequency sweeping. Instead of direct point-by-point simulations, it is only necessary to calculate the currents by SMoM at several Chebyshev-Gauss frequency sample points. Furthermore, the Chebyshev series are matched via the Maehly approximation to a rational function to improve the accuracy. Numerical examples show that the hybrid technique can greatly reduce the computation time without loss of accuracy.

2-D Analytical Model for Predicting Magnetic Flux Distribution in Slotless Single-sided Axial Flux Permanent-Magnet Synchronous Machines

This paper estimates the magnetic flux density components in the slotless single-sided axial flux permanent-magnet synchronous machines (SAFPMSMs). For this purpose, a 2-D analytical model based on the sub-domain method is utilized in which the cross-section of the presented machine is divided into the seven sub-regions such as stator side exterior, stator, winding, air-gap, permanent-magnets (PMs), mover and mover side exterior. Based on the Maxwell equations, the related partial differential equations (PDEs) of magnetic flux density components are formed in each sub-region which are identified as the essential step for obtaining the machines quantities. According to the superposition theorem, two separate steps are implemented for calculating the magnetic flux density components. In the first step, open circuit analysis includes various type of magnetization patterns, i.e. parallel, ideal Halbach, 2-segment Halbach and bar magnet in shifting direction is investigated and armature currents are zero and in the second step PMs are inactive and the magnetic flux density components are originated due to only armature reaction. Eventually, 2-D finite element method (FEM) is determined to confirm the accuracy of the presented analytical approach and an acceptable agreement between the analytical and FEM models can be observed.

Exact Analytical Method for Active Magnetic Bearings With Rotor Eccentricity

This article presents an exact analytical method for active magnetic bearings to obtain the magnetic field distribution with rotor eccentricity. Based on the subdomain method and the perturbation method, the active magnetic bearing is divided into two regions: the air gap and stator slots. Meanwhile, the zeroth-order and first-order equations in the polar coordinate are obtained and solved by the method of separation of variables, respectively. The magnetic field distribution is obtained by superimposing the first-order solution into the zeroth-order solution according to the perturbation method. The proposed analytical method has provided an effective way to study the magnetic bearing and other electromagnetic actuators. By using the Maxwell stress tensor theory, the unbalanced magnetic force is calculated to obtain the displacement stiffness and current stiffness. Finally, the consequences of magnetic field distribution and the unbalanced magnetic force are compared with those by the finite-element method. The result validates the correctness of the analytical method.

Depth-integrated wave–current models. Part 1. Two-dimensional formulation and applications

Depth-integrated mathematical models for simulating waves and currents from deep to shallow water are presented. These models are derived from Euler’s equations in the -coordinate system, mapping the total water depth in Cartesian coordinates onto a specified range of -coordinates. The horizontal velocity is approximated as a truncated infinite series of products of prescribed shape functions of and unknown functions of horizontal coordinates and time. Adopting the method of weighted residuals, the new models are obtained by minimizing the residuals of the horizontal momentum equations with either the Galerkin method or the subdomain method. These models’ linear and nonlinear water wave properties are investigated. The new models are implemented numerically. A hierarchy of numerical models with different degree of polynomial approximation is developed and checked against several benchmarked experiments and a new set of experiments of self-focusing wave groups. For both the Galerkin and subdomain models, excellent agreements are observed for both the free surface elevations and the velocity profiles. The new models are superior to the existing Boussinesq-type models for their applicability to a wide range of physical scenarios, including the interactions between a wave package of multiple frequency components and a linearly sheared current. The new Galerkin models have similar characteristics and accuracy as the Green–Naghdi models, but the new models are more efficient computationally. Finally, for the same degree of polynomial approximation the subdomain models perform better than the Galerkin models and require less computational time.

Analytical Model of Permanent Magnet Linear Synchronous Machines Considering End Effect and Slotting Effect

An effective analytical model of permanent magnet linear synchronous machine (PMLSM) is proposed in this paper, which considers the end effect by adding virtual slots and the slotting effect by adopting relative air-gap permeance. To employ this model, the open-circuit air-gap flux density of a slotless PMLSM with virtual slots is firstly predicted by Subdomain (SD) method, and then the complex relative air-gap permeance is introduced based on conformal mapping. By multiplying these two quantities, the open-circuit air-gap flux density of actual slotted PMLSM can be obtained. The phase flux linkage, back-EMF and detent force can be predicted accordingly. Moreover, the detent force is more easily depressed by optimizing the end teeth and slots via the proposed method. The predicted results are validated by 2-D nonlinear finite-element (FE) and prototype measurements.

Parameters evaluation of a permanent magnet synchronous generator with modular stator

This paper presents the evaluation of the parameters of a surface-mounted permanent magnet synchronous generator designed for wind generation in urban areas, and manufactured in a ring shaped around the turbine with modular stator. In the project, the analytical subdomain method and finite element analysis provides the inductances, both in two dimensions, and compared with experimental results in order to quantify the end winding leakage inductance. The analytic and numeric results indicate a reduction of inductances with increases of slots per winding phase. The experimental measurement allows identify the end winding leakage inductance that, due to the unconventional dimension proportions, is approximately 30% of total.

Subdomain Method for Layout Optimization of Piles in a Composite Pile Foundation

Subdomain Method for Layout Optimization of Piles in a Composite Pile Foundation

Characteristic Analysis of the Peak Braking Force and the Critical Speed of Eddy Current Braking in a High-Speed Maglev

In the eddy current braking system of high-speed maglev, the peak braking force and the critical speed are key factors determining the performance of eddy current braking force. In this paper, the analytical formula of eddy current braking force is derived by a subdomain method considering the skin effect of the induction plate, and, subsequently, the characteristics of peak braking force and critical speed are analyzed. The analytical model is set up in a 2D Cartesian coordinate system. The Poisson equations in each subdomain are listed by treating the vector magnetic potential as a variable. By combining the boundary conditions between two adjacent subdomains, the expressions of eddy current density and magnetic density in the induction plate are obtained. Then, the analytical formula of the eddy current braking force is obtained by the Ampere force formula. The results of finite-element analysis confirm the validity of the analytical calculation. The methods of improving the performance of eddy current braking force under high speed are proposed by parametric analysis of peak braking force and critical speed, which provides guidance for the design of the eddy current braking system in high-speed maglev.

An Analytical Study on Iron Pole Shape Optimization in High-Speed Interior Permanent Magnet Machines

Interior permanent magnet (IPM) motors are of extensive interest in high-speed applications, where high efficiency, high power density and robustness for these motors are a challenge. However, the noise and vibration caused due to cogging torque, electromagnetic torque ripple and reluctance torque ripple can greatly affect the machine performance. The main contribution of this paper is to develop a hyperbolic form analytical expression for optimal rotor iron pole curve in IPM motors to mitigate pulsating torque components. The analytical method is based on the resolution of Laplace equation in air gap sub-domain in polar coordinate by using the sub-domain method. The proposed method is applied to the performance computation of a prototype IPM machine, i.e., a three-phase 18S-8P motor. The analytical results are validated through FEA method and experimental tests.

A novel subdomain level set method for structural topology optimization and its application in graded cellular structure design

A novel subdomain structural topology optimization method is proposed for the minimum compliance problem based on the level sets with the parameterization of radial basis function (RBF). In this method, the level set function evolves on each subdomain separately and independently according to the requirements of objective functions and additional constraints. This makes the parameterization in the proposed subdomain method much faster and more cost-effective than that in the classical global method, as well as the evolution of the level set function since it can be achieved on each subdomain in parallel. In addition, the microstructures on arbitrary two adjacent subdomains can be connected perfectly, without any mismatch around the interfaces of the microstructures. Several typical examples are conducted to verify the correctness and effectiveness of the developed subdomain method. The effects of some factors on the optimized results are also investigated in detail, such as the RBF types, the connectivity types of microstructures, and the size of subdomain division. Without scale separation assumption, several layered graded cellular structures are successfully designed by employing the proposed method under the condition of corresponding repetition constraints. To improve the computational efficiency, a multi-node extended multiscale finite element method (EMsFEM) is used to solve the structural static equilibrium equation for the three-dimensional layered structure optimization problems. Furthermore, a MATLAB code is also provided in the Appendix for readers to reproduce the results of the two-dimensional problems in this work.

A Subdomain Method for Mapping the Heterogeneous Mechanical Properties of the Human Posterior Sclera.

Although strongly correlated with elevated intraocular pressure, primary open-angle glaucoma (POAG) occurs in normotensive eyes. Mechanical properties of the sclera around the optic nerve head (ONH) may play a role in this disparity. The purpose of this study is to present an automated inverse mechanics based approach to determine the distribution of heterogeneous mechanical properties of the human sclera as derived from its surface deformations arising from pressure inflation experiments. The scleral shell of a 78 year old European Descent male donor eye was utilized to demonstrate the method; the sclera was coated with a speckle pattern on the outer surface and was subjected to inflation pressures of 5, 15, 30, and 45 mmHg. The speckle pattern was imaged at each pressure, and a displacement field was calculated for each pressure step using a previously described sequential digital image correlation (S-DIC) technique. The fiber splay and fiber orientation of the sclera collagen were determined experimentally, and the thickness across the scleral globe was determined using micro CT images. The displacement field from the inflation test was used to calculate the strain and also used as an input for inverse mechanics to determine the heterogeneity of material properties. The scleral geometry was divided into subdomains using the first principal strain. The Holzapfel anisotropic material parameters of matrix and fiber stiffness were estimated within each individual subdomain using an inverse mechanics approach by minimizing the sum of the square of the residuals between the computational and experimental displacement fields. The mean and maximum error in displacement across all subdomains were 8.9 ± 3.0 μm and 13.2 μm, respectively. The full pressure-inflation forward mechanics experiment was done using subdomain-specific mechanical properties on the entire scleral surface. The proposed approach is effective in determining the distribution of heterogeneous mechanical properties of the human sclera in a user-independent manner. Our research group is currently utilizing this approach to better elucidate how scleral stiffness influences those at high risk for POAG.

Asynchronous multi-splitting method for linear and pseudo-linear problems

This paper deals with the solution of fluid-structure interaction problems using an algorithm consisting in the coupling between two solvers, each one related to the fluid and the structure respectively. Due to the difficult decomposition of the domain into sub-domains, we consider for each environment a parallel multi-splitting algorithm which corresponds to an unified presentation of sub-domain methods with or without overlapping. Such method combines several contractant fixed point mappings and, we show that, under appropriate assumptions, each fixed point mapping are contractive in finite dimensional spaces normed by Hilbertian and non-Hilbertian norms. Moreover, we show in a novel way that such study is valid for the parallel synchronous and more generally asynchronous large scale linear systems arising in the solution of fluid-structure interaction problems and can be extended to the case where the displacement of the structure is submitted to some constraints. We perform parallel simulations for a fluid-structure test case on different clusters, considering blocking and non-blocking communications. The performances of the parallel simulations are presented and analyzed.

Analytical calculation of magnetic field in surface-mounted permanent-magnet machines with air-gap eccentricity

PurposeThis paper aims to present an analytical method, which combines the complex permeance (CP) and the superposition concept, to predict the air-gap magnetic field distribution in surface-mounted permanent-magnet (SMPM) machines with eccentric air-gap.Design/methodology/approachThe superposition concept is used twice; first, to predict the magnetic field distribution in slot-less machine with eccentric air-gap, the machine is divided into a number of sections. Then, for each section, an equivalent air-gap length is determined, and the magnetic field distribution is predicted as a concentric machine model. The air-gap field in the slot-less machine with eccentricity can be combined from these concentric models. Second, the superposition concept is used to find the CP under eccentricity fault. At this end, the original machine is divided into a number of sections which may be different from the one for slot-less magnetic field prediction, and for each section, the CP is obtained by equivalent air-gap length of that section. Finally, the air-gap magnetic field distribution is predicted by multiplying the slot-less magnetic field distribution and the obtained CP.FindingsThe radial and tangential components of the air-gap magnetic flux density are obtained using the proposed method analytically. The finite element analysis is used to validate the proposed method results, showing good agreements with the analytical results.Originality/valueThis paper addresses the eccentricity fault impact upon the air-gap magnetic field distribution of SMPM machines. This is done by a combined analysis of the complex permeance (CP) method and the superposition concept. This contrasts to previous studies which have instead focused on the subdomain method.

Sloshing of fluid in a baffled rectangular aqueduct considering soil-structure interaction

The dynamic characteristics and responses of a baffled rectangular aqueduct supported on the flexible soil under the horizontal seismic input acceleration are studied. The linear potential theory and subdomain method are adopted in the analysis of fluid motion. An equivalent mechanical model is innovatively established for liquid sloshing. The dynamic responses of the fluid-aqueduct coupling system due to the horizontal excitation is obtained, and an equivalent mechanical model is established on the basis of equivalent principle. The nested lumped parameter model is used to represent the dynamic stiffness function with the frequency dependence. Based on the substructure method, the soil-aqueduct-fluid coupling system is constructed. Dynamic responses of the coupling system are acquired by the Newmark-β method. The numerical results are compared with other results from available literatures. Excellent agreements are observed. Finally, parametric studies are carried out to estimate the effects of baffle parameters and soil property on the natural characteristics and dynamic responses of aqueduct. The results indicate that the shear wave velocity and baffle have significant influence on the sloshing of fluid in a baffled aqueduct.

An equivalent mechanical model for fluid sloshing in a rigid cylindrical tank equipped with a rigid annular baffle

• A mass-spring mechanical model is developed. • The model has clear physical meaning and high accuracy. • The model has excellent assemblability. • The complicated liquid-structure system can be solved with small computational cost. A mass-spring mechanical model for linear sloshing of fluid in a rigid cylindrical tank with a rigid annual baffle is developed. By means of the subdomain method, the complicated fluid domain is divided into several subdomains with pure boundary conditions and interface conditions . Combined with the continuity conditions of velocity and pressure on interfaces, the velocity potential of fluid in each subdomain is analytically solved. The mass-spring model for fluid sloshing is established by generating the same hydrodynamic shear force and overturning moment as those from the exact solutions when the tank is subjected to horizontal excitations. The present model gives all the mechanical parameters including the convective mass-spring oscillators and the impulsive mass as well as their positions. Comparative studies between the present solutions and the available results verify the correctness of the mechanical model. Based on the normalized equivalent masses and spring stiffnesses , the dynamic responses of fluid in tanks are discussed with respect to the fluid height, the baffle position and the inner radius of the baffle, respectively. The present model is especially suitable for dealing with complicated liquid-structure systems. The motivation and novelty of this study are confirmed by an application of the present model to a multi-tank system.

Augmentation of passive magnetic bearing performance by using air or iron intervals

PurposeThis paper aims to suggest the use of air or iron intervals between axially magnetized rings to increase the forces and stiffness of permanent magnet passive magnetic bearings (PMBs). The paper calculates the stiffness of such bearings through an analytical method and optimizes the dimensions of the magnets for achieving maximum stiffness.Design/methodology/approachFor determining the magnetic fields distribution, forces and stiffness of the bearings, a 2D analytical method is used, based on the subdomain method. For the sake of generalization, all of the parameters are normalized and optimized for maximum normalized stiffness per magnet volume ratio.FindingsThe optimum sizes of the magnets as well as the optimum dimensions of the air or iron intervals are calculated in this paper. The optimum sizes of the magnets are around the air gap length and it is very difficult to realize them. Using iron intervals can improve the stiffness to the extremely high values in practical dimensions of the magnets.Originality/valueThis paper presents a novel configuration for improving the performance of PMBs with alternately axially magnetized rings.

Optimal Pole Number for Magnetic Noise Reduction in Variable-Speed Permanent Magnet Synchronous Machines With Fractional-Slot Concentrated Windings

Increasing the number of pole pairs leads to a lower electromagnetic yoke, and therefore, lower vibration and magnetic noise occur. In this paper, the influence of different numbers of pole pairs on the vibroacoustic design aspects of the machine is studied for the first time using a multislice subdomain method (MS-SDM) while considering the natural frequencies under a variable speed analysis. This paper aims to determine the optimal number of pole pairs for a low-speed, high-torque permanent magnet synchronous generator (PMSG) with double-layer, fractional-slot, nonoverlapping, concentrated windings (FSCWs) for wind turbine applications. First, all possible slots per pole per phase combinations, which offer the use of double-layer FSCW, are studied through a magnetomotive force (MMF) harmonic analysis. Second, the MS-SDM of the PMSG is studied to examine the vibroacoustic performance under a variable speed analysis. Finally, all affected major parameters are compared in order to find the optimal pole number of the PMSG. To verify the MS-SDM-based results, both 3-D finite element analysis and experimental investigations are employed.

Theoretical and numerical analysis of selected hybrid Trefftz method with the use of method of subdomains coupling based on “frame” concept

The aim of this paper is theoretical and numerical analysis of Trefftz method in Galerkin version, which symbol is O-S;T-T.The analysis was conducted on the example of two-dimensional Poisson’s problem. Domain of interest was divided into homogeneous subdomains. In this way hybrid method was obtained, that is a combination of the boundary method with conventional finite element known from the Finite Element Method. The method based on idea of “frames” was used to couple subdomains. Finally, the results of numerical experiments obtained for two boundary value problems were presented. The accuracy of the method was analyzed depending on the number of Trefftz coefficients.

Development and performance of a 1D–2D coupled shallow water model for large river and lake networks

A one-dimensional (1D) – two-dimensional (2D) coupled numerical model for shallow-water flows is developed, inspired by a large river-lake network. The model includes an improved method of coupling subdomains as part of the water stage prediction-correction (WSPC) approach, exploiting the characteristic curves of the shallow-water equations. Compared to other coupling methods, the one presented here results in better computational efficiency, convergence and stability. The coupling method uses parameters with a clear physical meaning, which can be set directly without appealing to empirical values. With the help of the WSPC approach, individual submodels are solved separately with their coupling parameters being consistent with each other at the coupling boundaries. The coupling method proposed herein allows large time-stepping for the coupled model. Practical approaches to achieve large time steps are discussed and analysed by comparing two different solution algorithms for the 2D submodel. The proposed method and model are examined by applying to a real-life case in China, and the results indicate their good validity, practicality and computational superiority.

The generalized finite difference method for in-plane crack problems

This paper documents the first attempt to apply the generalized finite difference method (GFDM), a recently-developed meshless method, for the numerical solution of problems with cracks in general anisotropic materials. To solve the resulting second-order elliptic partial differential equations with mixed boundary conditions, the explicit formulae for the partial derivatives of unknown functions in the equations are derived by using the Taylor series expansions combining with the moving-least squares approximation in this meshless GFD method. To deal with the strong discontinuous crack-faces, some special treatments are applied to modify this GFDM. The node distributions are locally refined in the vicinity of the crack-tips. By dividing the crack domain into two parts, the sub-domain method is also used for comparing with the single-domain method. The direct displacement extrapolation method, the path-independent J-integral and the interaction integral methods are, respectively, used to compute the stress intensity factors and compared. Finally, some classical crack examples are presented to show the effectiveness and accuracy of the proposed meshless method for crack problems.