IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 1, 2002 207 A Dispersion Analysis for the Finite-Element Method in Time Domain With Triangular Edge Elements Agostino Monorchio, Member, IEEE, Enrica Martini, Member, IEEE, Giuliano Manara, Senior Member, IEEE, and Giuseppe Pelosi, Fellow, IEEE Abstract—A numerical dispersion analysis for the finite-element (FE) method in time domain (TD) is presented. The dispersion re- lation is analitically derived by considering a time-harmonic plane wave propagating through an infinite uniform mesh consisting of equilateral triangular elements. The effect of the time step on the numerical dispersion is investigated and it is shown that, if linear tangential-linear normal (LT-LN) edge-basis functions are used, there exists a time-step value that minimizes the deviation of the dispersion relation from the ideal linear case. In particular, the analysis performed shows that this optimum time step holds for any propagation direction of the plane wave within the mesh and, virtu- ally, for any frequency, strongly enhancing numerical accuracy of the FE-TD method. As a working example, we choose to compare the numerically computed TE modes of two-dimensional guiding structures with the corresponding analytical values; to this end, an efficient procedure for the computation of the eigenfrequencies is proposed, allowing us to avoid TD data processing. Index Terms—Discrete time systems, finite-element time-domain (FE-TD) method, numerical dispersion, wave propagation. I. INTRODUCTION I T is well known that an electromagnetic wave propagating through a finite-element (FE) mesh will experience some numerical dispersion, yielding a progressive phase error in the problem solution. Such a phenomenon has been analyzed for the finite-element method (FEM) in frequency domain for both nodal [1], [2] and edge [2]–[4] basis functions; some results rele- vant to the FE solution of first-order Maxwell equations in time domain (TD) have also been presented [5], [6]. However, al- though a TD analysis is required to simulate transient electro- magnetic fields, the numerical dispersion in the TD-FEs has not been thoroughly investigated. In this letter, we analyze the numerical dispersion in two-dimensional (2-D) FE-TD procedures using edge basis functions. Edge elements have revealed particularly suited for solving electromagnetic problems automatically ensuring the continuity of the tangential components of the fields. A comprehensive survey of the properties of edge elements can be found in [7], where curl-conforming edge basis functions of different orders are considered. Manuscript received November 6, 2002; revised December 5, 2002. A. Monorchio and G. Manara are with the Department of Information Engineering, University of Pisa, I-56126 Pisa, Italy (e-mail: a.monorchio@ iet.unipi.it; g.manara@iet.unipi.it). E. Martini is with the Department of Information Engineering, University of Siena, I-53100, Siena, Italy (e-mail: martini@dii.unisi.it). G. Pelosi is with the Department of Electronics and Telecommunications, University of Florence, I-50134 Florence, Italy (pelosi@det.unifi.it). Digital Object Identifier 10.1109/LAWP.2002.807962 The analysis performed in this paper reveals that if linear tangential-linear normal (LT-LN) edge basis functions are used on a uniform hexagonal mesh, there exists a time step value that allows us to increase the convergence rate of the disper- sion relation at any propagation direction of the wave within the mesh and for any frequency, strongly enhancing numerical accuracy of the FE-TD method. This is not the case if constant tangential-linear normal (CT-LN) edge basis functions are used. From a practical point of view, for a given TD formulation, the time step can be used as an additional parameter, with respect to its frequency formulation counterpart, to control the accuracy of the procedure. Numerical results relevant to the analysis of closed guiding structures are presented to verify the theory. II. FE-TD FORMULATION The specific FE-TD formulation adopted relies on the discretization of the second-order vector wave equation with proper boundary conditions. The equation describes a time-harmonic electric field in an isotropic and loss-free region (1) where and are the permeability and the permittivity of the medium, respectively, and is an impressed current density. The region of interest is subdivided into discrete elements, where the field is expanded by using a proper set of vector basis functions and a Galerkin testing procedure is employed. By assuming perfectly electric or magnetic conducting boundaries and by setting , the above equation is finally numerically expressed as T S (2) where denotes the unknown coefficients vector, T ,S , and are the vector basis functions used for expanding the field. As far as the discretization in time is concerned, we resort to the Beta–Newmark formulation [8], for which the above equa- tion can be cast in the following form: M N P (3) where M T S N T S P M (4) 1536-1225/02$17.00 © 2002 IEEE