# meshless - IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 4,...

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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 I , Rodney R. Saldanha I , Renato C. Mesquita I , David A. Lowther P , and Jaime A. Ramírez I Department of Electrical Engineering, Federal University of Minas Gerais, Belo Horizonte 31270-010, Brazil Department of Electrical and Computer Engineering, McGill University, Montreal, QC, H3A 2T5, Canada A meshless method for electromagnetic Feld computation is developed based on the multiquadric interpolation technique. A global approximation to the solution is built based only on the discretization of the domain in nodes and the differential equations describing the problem in the domain and its boundary. An attractive characteristic of the multiquadric solution is that it is continuous and it has inFnitely continuous derivatives. This is particularly important to obtain Feld quantities in electromagnetic analysis. The method is also capable of dealing with physical discontinuities present at the interface between different materials. The formulation is presented in the Cartesian and polar coordinates, which can be extended to other systems. We applied the formulation in the analysis of an electrostatic micromotor and a microstrip. The results demonstrate good agreement with other numerical technique, showing the adequacy of the proposed methodology for electromagnetic analysis. Index Terms— Collocation methods, interface conditions, mesh-free methods, meshless methods, multiquadrics. I. INTRODUCTION R ECENTLY, numerous meshless methods have arisen for solving partial differential equations (PDEs) in many con- texts [1], [2], including electromagnetic ±eld computation; see, for instance, [3]–[6]. The attractive characteristic with meshless techniques is that they do not need a mesh to divide the domain; only nodes are used. It works with the weak form of the problem, in a manner similar to that of the ±nite 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 based on the global approximation of the solution by means of ra- dial basis functions (RBF), speci±cally the multiquadric (MQ) function. The multiquadric function was initially proposed in [7] for scattered data approximation and later reviewed by its author in [8]. Since then, the MQ interpolation technique has been successfully applied for multidimensional interpolation, including electromagnetic optimization problems [9], [10]. In [11], Kansa illustrated the idea of using RBFs for solving PDEs, introducing the collocation method. The MQ-based approach is a truly mesh-free method, because it does not work with the weak form of the problem and it does not need any local in- tegration cell as required by typical meshless techniques [2],

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## This note was uploaded on 06/08/2011 for the course ELECTRICAL 124 taught by Professor Ghjk during the Spring '11 term at Institute of Technology.

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meshless - IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 4,...

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