A Generalized Local Time-Step Scheme for the FVTD Method for Efficient Simulation of Microwave Antennas Christophe Fumeaux, Dirk Baumann, Pascal Leuchtmann, Rüdiger Vahldieck Swiss Federal Institute of Technology, ETHZ, IFH, Gloriastrasse 35, 8092 Zürich, Switzerland Phone: +41 1 632 0601, Email: fumeaux@ifh.ee.ethz.ch Abstract — This paper introduces a new generalized local time-step scheme to improve the computational efficiency of the Finite-Volume Time-Domain (FVTD) method. The new approach exploits the advantages of the FVTD method to use unstructured meshes (which allow inhomogeneity of cell densities) for large electromagnetic circuits with fine structural details and at the same time avoids the disadvantage of using a single time step determined by the smallest cell dimensions in the entire mesh. To illustrate this new scheme a large double-ridged horn antenna excited by a finely resolved coaxial feed is analyzed demonstrating a significant speed-up of the computation. I. INTRODUCTION The Finite-Volume Time-Domain (FVTD) method has been applied to the numerical solution of Maxwell's equations since the end of the 80's [1]-[2]. Since the technique is naturally suited for unstructured meshes, it constitutes a powerful alternative to the Finite-Difference Time-Domain (FDTD) method for problems where conformal meshing is advantageous. A class of problems that still constitute a challenge for time-domain methods are those in which fine structural circuit details compared to the wavelength (in the order of or even smaller) are in proximity to large dimensions. Modeling such geometries with regular structured 3D meshes results in prohibitive computational costs. Therefore, sub-gridding techniques and sub-cell models have been developed for FDTD, at the cost of increased complexity and threat to stability. In contrast the unstructured meshes used in FVTD permit to model accurately fine structural details, avoiding a 3D explosion of the number of cells by locally adapting the cell size to that of the geometrical features. The unpleasant side effect of an unstructured mesh with strongly inhomogeneous cell sizes is the required short time step determined by the smallest cell of the mesh. 100 / In this paper, we propose a novel generalized scheme for introducing local time steps in the FVTD scheme in an unstructured mesh. The proposed technique considers discrete local time steps with lengths given by power-of- two multiples (2 m-1 , m = 1,2,3,…) of the smallest time step necessary for stability in the entire mesh. It is based on an automatic separation of the computational domain in sub-domains where the various discrete local time steps can be applied without compromising stability. During the meshing process, no special time-step considerations are necessary, since the new generalized scheme adapts the time step automatically during preprocessing. Since the density of cells in an inhomogeneous mesh smoothly changes from one region to the other, the described scheme does not require spatial interpolation of field components and fits very naturally in the FVTD time-marching scheme. As illustration, the new local time-step scheme is applied to the simulation of a double-ridged horn antenna including the coaxial feed modeled in detail. In this example, the maximum relative ratio of local time-step lengths in the different areas of the computational domain reaches 16 (m = 5), and the scheme permits to accelerate the computation by a factor close to 5 without reducing the accuracy of the results. II. THE FVTD METHOD In the FVTD scheme, the Maxwell's curl equations in conservative form [3] are integrated over elementary polyhedral volumes V using the divergence theorem. For the FVTD algorithm used here, the elementary cells are tetrahedrons with a typical side length 10 < / . The discretized equations can be written as = = = × = × 4 1 4 1 1 1 k k k k V k k k V F k B n E t V D n H t V F F F (1) where represents the area of a tetrahedron's face, k F k n its normal unit vector, and denotes a spatial mean value. The coupled equations (1) represent the foundation of the FVTD scheme. The integrands of the discretized surface integrals on the right-hand side (RHS) of (1) are interpreted as "fluxes" t rough the cell faces. Inserting the material equations h = D and E μ = B into (1), localizing mean volume values in the barycenter of the cells, and mean area values in the barycenter of the cell faces permit to write discretized coupled equations for the time-dependent H E and H . In the cell-centered FVTD formulation presented here, the electric and magnetic field components are stored at the same location (cell barycenters). 33rd European Microwave Conference - Munich 2003 467