Meshless Eigenvalue Analysis for Resonant Structures Based on the Radial Point Interpolation Method Thomas Kaufmann #1 , Christophe Fumeaux *2 , Christian Engstr¨ om # , Ruediger Vahldieck # # Laboratory for Electromagnetic Fields and Microwave Electronics (IFH), ETH Zurich, Gloriastrasse 35, CH-8092 Zurich, Switzerland 1 thomas.kaufmann@ifh.ee.ethz.ch ∗ School of Electrical & Electronic Engineering, The University of Adelaide, Adelaide, South Australia 5005, Australia 2 cfumeaux@eleceng.adelaide.edu.au Abstract— Meshless methods are a promising field of numerical methods recently introduced to computational electromagnetics. The potential of conformal and multi-scale modeling and the pos- sibility of dynamic grid refinements are very attractive features that appear more naturally in meshless methods than in classical methods. The Radial Point Interpolation Method (RPIM) uses radial basis functions for the approximation of spatial derivatives. In this publication an eigenvalue solver is introduced for RPIM in electromagnetics. Eigenmodes are calculated on the example of a cylindrical resonant cavity. It is demonstrated that the computed resonance frequencies converge to the analytical values for increasingly fine spatial discretization. The computation of eigenmodes is an important tool to support research on a time- domain implementation of RPIM. It allows a characterization of the method’s accuracy and to investigate stability issues caused by the possible occurrence of non-physical solutions. Index Terms— Meshless Methods, Eigenvalues and eigenfunc- tions, Radial Basis Functions, Radial Point Interpolation Method. I. I NTRODUCTION Meshless Methods for Computational Electromagnetics (CEM) gained attention recently as a new versatile tech- nique for solving the Maxwell equations [1]. The underlying principle avoids an explicit mesh structure for the numerical solution of the differential equations. Instead a set of node locations is selected depending on the physical model at hand. The flexibility in the node distribution allows for conformal and multi-scale modeling. The class of meshless methods is established in other fields of computational physics, e.g. in fluid dynamics or computational mechanics [2]. In CEM in the time domain, recent approaches include Smooth Particle Hydrodynamics for Electromagnetics (SPEM) [3], [4] and the Radial Point Interpolation Time-Domain (RPITD) Method [5], [6]. All these approaches have in common that a node-based interpolation scheme describes the connectivity between the nodes of the computational domain. The method presented in this publication is a domain discretization collocation method with interpolations based on radial basis functions. It has been introduced as the Kansa RBF method [7], or the Radial Point Interpolation Method (RPIM) [2]. The primary objective of the work of the authors focuses on a time-domain implementation of the RPIM method for electromagnetics. As a matter of fact, the advantages over classical methods, such as geometrical versatility and Fig. 1. Support domain of a given node ×, showing the area of influence that only includes nodes within radius ds. The average node distance dc is used for normalization of radial basis functions. the dynamic grid adaptation will be best expressed in time- domain applications to simulate transient and possibly multi- physics effects. In this scope an eigenvalue analysis presents a crucial tool to examine accuracy and investigate the presence of spurious modes which eventually can lead to instabilities in the time iteration. Furthermore, eigenmode solvers are an essential tool to characterize arbitrary shaped ports. The following two sections firstly introduce the interpolation method used in RPIM and secondly their implementation for an eigenvalue problem. Subsequently the approach is analyzed in terms of accuracy and convergence in the example of a two- dimensional cylindrical resonator. II. RADIAL POINT I NTERPOLATION The interpolation scheme used in RPIM is based on local radial basis functions (RBFs). This type of basis functions provides excellent interpolation accuracy and an approxima- tion of the spatial derivations can be obtained relatively simply. Only field values in the vicinity of each node inside a local support domain are considered for interpolation, leading to fast local calculations. Fig. 1 depicts the local support domain with radius d s . The interpolation of the spatial derivatives are used to obtain the curl operators of the Maxwell equations in differential form. The used method is thoroughly described in [2] and there- fore only briefly summarized here. A field component u(x) at position x is interpolated as 〈u(x)〉 = N  n=1 a n r n (x)+ M  m=1 b m p m (x)= r(x) T a + p T (x)b. (1) The radial basis functions r n (x) = exp  −α c |x n − x| 2 d 2 c  (2) 978-1-4244-2802-1/09/$25.00 ©2009 IEEE