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A Meshless Method for Electromagnetic Field Computation Based on the Multiquadric Technique

Page 1. IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 4, APRIL 2007 1281 A Meshless Method for Electromagnetic Field Computation Based on the Multiquadric Technique Frederico G. Guimarães 1 , Rodney R. Saldanha ...
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  IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 4, APRIL 2007 1281 A Meshless Method for Electromagnetic FieldComputation Based on the Multiquadric Technique Frederico G. Guimarães    , Rodney R. Saldanha    , Renato C. Mesquita    , David A. Lowther    , and Jaime A. Ramírez   Department of Electrical Engineering, Federal University of Minas Gerais, Belo Horizonte 31270-010, BrazilDepartment of Electrical and Computer Engineering, McGill University, Montreal, QC, H3A 2T5, Canada A meshless method for electromagnetic field computation is developed based on the multiquadric interpolation technique. A globalapproximation to the solution is built based only on the discretization of the domain in nodes and the differential equations describingthe problem in the domain and its boundary. An attractive characteristic of the multiquadric solution is that it is continuous and it hasinfinitely continuous derivatives. This is particularly important to obtain field quantities in electromagnetic analysis. The method is alsocapable of dealing with physical discontinuities present at the interface between different materials. The formulation is presented in theCartesian and polar coordinates, which can be extended to other systems. We applied the formulation in the analysis of an electrostaticmicromotor and a microstrip. The results demonstrate good agreement with other numerical technique, showing the adequacy of theproposed methodology for electromagnetic analysis.  Index Terms— Collocation methods, interface conditions, mesh-free methods, meshless methods, multiquadrics. I. I NTRODUCTION R ECENTLY, numerous meshless methods have arisen forsolving partial differential equations (PDEs) in many con-texts [1], [2], including electromagnetic field computation; see, for instance, [3]–[6]. The attractive characteristic with meshlesstechniques is that they do not need a mesh to divide the domain;onlynodesareused.Itworkswiththeweakformoftheproblem,in a manner similar to that of the finite element method (FEM).Generally, a local approximation is adopted in the vicinity of each node, however, the integration in some cases is more com-plicated than it is in the FEM case [4].In this paper, we employ a different meshless approach basedon the global approximation of the solution by means of ra-dial basis functions (RBF), specifically the multiquadric (MQ)function. The multiquadric function was initially proposed in[7] for scattered data approximation and later reviewed by itsauthor in [8]. Since then, the MQ interpolation technique hasbeen successfully applied for multidimensional interpolation,including electromagnetic optimization problems [9], [10]. In [11],Kansaillustratedtheideaof usingRBFs forsolvingPDEs,introducing the collocation method. The MQ-based approach isa truly mesh-free method, because it does not work with theweak form of the problem and it does not need any local in-tegration cell as required by typical meshless techniques [2],[12]. Theoretical results on using RBFs for solving PDEs canbe found in [13] and [14]. However, since the method develops a global approximation, the matrix of coefficients is full. Somealternatives to overcome this difficulty may be found in litera-tureasdomaindecomposition,matrixpreconditioners,andtrun-cated MQ functions [15]. Digital Object Identifier 10.1109/TMAG.2007.892396 The collocation method is quite explored in many initial andboundary value problems; see, for instance, [14], [16], and thereferencestherein,butveryfewworksarerelatedtoelectromag-netics. The continuously differentiable characteristic of someRBFs is an attractive feature, since the derivatives can be cap-tured with high accuracy. A particular difficulty in using RBFs,for electromagnetic analysis, is to treat physical discontinuitiesat the interface of different media. The discontinuity of the so-lution derivatives is not easily handled with RBF meshless. Infact, it is not possible to capture discontinuities using a continu-ously differentiable global solution. In this paper, we propose aspecific methodology to deal with physical discontinuities withthe collocation MQ method, making it suitable in the contextof electromagnetic field computation problems. Two numericalexamples demonstrate the usefulness and potential of the tech-nique to electromagnetic analysis.II. M ULTIQUADRIC -B ASED  M ESHLESS  T ECHNIQUE Let us assume the following boundary problem:(1)(2)where and represent linear differential operators, andare known functions, is the position vector, and and are,respectively, the domain and its boundary. The closed domainis given by . More specifically, the boundary can bedefined as , in which is the boundary associatedwithDirichletconditionsand istheboundaryassociatedwithCauchy conditions. Thusin (3)in (4) 0018-9464/$25.00 © 2007 IEEE  GUIMARÃES  et al. : MESHLESS METHOD FOR ELECTROMAGNETIC FIELD COMPUTATION 1283 Fig. 1. Solution        of the 1-D example.Fig. 2. The mse as a function of     and for          , and      . The solution of this problem is(15)and it is illustrated in Fig. 1. This function is continuous overthe domain, but its derivative is discontinuous at .Let represent thenumber ofpoints uniformlyspacedin theregions and . The distance between twoconsecutive nodes is then equal to , where . Theshape parameter is set as(16)where is a multiplying factor of the mean distance of nodes.The mean squared error (mse) between the MQ solution andthe analytical solution is shown in Fig. 2 as a function of andhence the shape parameter.ThecurvesinFig.2showthatthereisanoptimalvaluefortheshape parameter for a given number of nodes. If the parameteris too small, the error is low, but often unacceptable. An optimalvalue may be found close to the limit value of . If it istoo big (greater than , for the present case), the errorgreatly increases due to the ill-conditioning of the matrix.This behavior is well known for the MQ method in the contextof function interpolation and PDE approximation problems [8],[11].Fig.3showstheconvergenceofthemsewithrespecttothenumber of nodes . By increasing , it is possible to decreasethe mse convergence curve as shown in Fig. 3, until the limitvalue makes the system ill-conditioned. Fig. 3. Convergence of the mse as a function of     and for      and      . IV. R ESULTS  A. Analysis of an Electrostatic Micromotor  First, we illustrate the MQ method in the analysis of an elec-trostatic micromotor with 12 poles at the stator and eight polesat the rotor; see Fig. 4. The problem is solved in polar coordi-nates and one MQ solution is used for the entire domain. Thetooth width is 20 , the stator radius is 100 m, the rotor radiusis 50 m, the slot radius is 30 m, and the gap is of 2 m. Weemploy a polynomial basis of degree , hence .Thus, the following set of equations is defined:(17)(18)(19)(20)giving the system of equations as in (7). Fig. 4 also shows the regions , and . The equipotential lines of the MQsolution are depicted in Fig. 4, which also presents a detailedperspective of the region of interest.  B. Microstrip Finally, we analyze the microstrip in Fig. 5 in Cartesian coor-dinates and with two different media, to illustrate our approachto deal with interface conditions. We have 0 V at and 10 Vat . A reference solution is generated with the FEM using avery fine mesh with 5910 nodes and first-order elements. Fig. 6shows the solutions across the line crossing the two dielectricsat mm, obtained by the MQ method with three dif-ferent nodal distributions. The solution with 1527 nodes is al-most undistinguishable from the reference solution.  1284 IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 4, APRIL 2007 Fig. 4. Geometry of the micromotor, showing the boundary conditions and theequipotentiallinesoftheapproximatesolutionobtainedbytheMQmethod.Theupper right figure zooms the region close to the air gap.Fig. 5. Geometry of the microstrip problem.Fig. 6. Solutions for the microstrip problem. The reference solution is dot-lined. V. F INAL  R EMARKS This paper shows the application of the MQ collocationmethod to electromagnetics in Cartesian and polar coordinates.The MQ meshless method has the great advantage of being atruly meshless method, without the need for complicated inte-gration steps and working directly with the strong form of theproblem. Additionally, the method works well with scatteredpoints and it is very simple to implement. The disadvantagesare the need to tune the shape parameter and the generationof a full matrix (unless MQ with compact support is used).Moreover, we proposed a novel methodology for treatingphysical discontinuities with the MQ method, thus makingit suitable for electromagnetic analysis. We expect that thiswork may contribute to the recent developments in meshlesstechniques.Two problems of small to moderate size have been inves-tigated. In these cases, the method is an efficient and flexibletechnique. However, for large scale problems, the utilization of full matrices is prohibitive, mainly due to storage requirements.Therefore, additional strategies like domain decomposition andadoption of compactly supported RBFs are needed for morecomplex problems.A CKNOWLEDGMENT This work was supported by the National Council of Scien-tific and Technologic Development-CNPq, Brazil, under Grants308260/2003-1 and 141731/2004-4.R EFERENCES[1] J. T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, and P. Krysl,“Meshless methods: An overview and recent development,”  Comput. Methods Appl. Mech. Eng. , vol. 139, no. 1–4, pp. 3–47, 1996.[2] T. P. Fries and H. G. Matthies, “Classification and overview of mesh-free methods,” Brunswick, Germany, Tech. Rep. 2003-3, 2003, revisedin 2004.[3] V. Cingoski, N. Miyamoto, and H. Yamashita, “Element-free galerkinmethod for electromagnetic field computations,”  IEEE Trans. Magn. ,vol. 34, no. 5, pp. 3236–3239, Sep. 1998.[4] Y. Maréchal, “Some meshless methods for electromagnetic field com-putations,”  IEEE Trans. Magn. , vol. 34, no. 5, pp. 3351–3354, Sep.1998.[5] S. A. Viana and R. C. Mesquita, “Moving least square reproducingkernel method for electromagnetic field computation,”  IEEE Trans. Magn. , vol. 35, no. 3, pp. 1372–1375, May 1999.[6] C. Hérault and Y. Maréchal, “Boundary and interface conditions inmeshless methods,”  IEEE Trans. Magn. , vol. 35, no. 3, pp. 1450–1453,May 1999.[7] R.L.Hardy,“Multiquadricequationsoftopographyandotherirregularsurfaces,”  Geophys. Res. , vol. 176, pp. 1905–1915, 1971.[8] ——, “Theory and applications of the multiquadric biharmonicmethod: 20 years of discovery,”  Comput. Math. Appl. , vol. 19, no. 8/9,pp. 163–208, 1990.[9] P. Alotto, A. Caiti, G. Molinari, and M. Repetto, “A multiquadrics-based algorithm for the acceleration of simulated annealing optimiza-tion procedures,”  IEEE Trans. Magn. , vol. 32, no. 3, pp. 1198–1201,May 1996.[10] A. Canova, G. Gruosso, and M. Repetto, “Magnetic design optimiza-tion and objective function approximation,”  IEEE Trans. Magn. , vol.39, no. 5, pp. 2154–2162, Sep. 2003.[11] E. J. Kansa, “Multiquadrics—A scattered data approximation schemewith applications to computational fluid dynamics–Part ii: Solutionsto parabolic, hyperbolic and elliptic partial differential equations,” Comput. Math. Appl. , vol. 19, no. 8/9, pp. 147–161, 1990.[12] G. R. Liu  , Mesh Free Methods . Boca Raton, FL: CRC Press, 2004.[13] C. Franke and R. Schaback, “Solving partial differential equations bycollocation using radial basis functions,”  Appl. Math. Comput. , vol. 93,no. 1, pp. 73–82, 1998.[14] A. H. D. Cheng, M. A. Golberg, E. J. Kansa, and G. Zammito, “Expo-nential convergence and h-c multiquadric collocation method for par-tial differential equations,”  Numer. Meth. Partial Diff. Eq. , vol. 19, no.5, pp. 571–594, Sep. 2003.[15] E. J. Kansa and Y. C. Hon, “Circumventing the ill-conditioningproblem with multiquadric radial basis functions: Applications toelliptic partial differential equations,”  Comput. Math. Appl. , vol. 39,no. 7/8, pp. 123–137, 2000.[16] X.Zhang,K.Z.Song,M.W.Lu,andX.Liu,“Meshlessmethodsbasedoncollocation with radialbasis functions,” Comput. Mech. ,vol. 26, no.4, pp. 333–343, Oct. 2000.Manuscript received April 24, 2006 (e-mail:
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