Rapid Evaluation of Macromodel Response with the FDTD Method Dmitry A. Gorodetsky and Philip A. Wilsey Department of ECECS, University of Cincinnati, P.O. Box 210030, Cincinnati, OH 45221-0030 USA Abstract-Full wave electromagnetic simulation requires numerically expensive methods such as FDTD. The computation time depends superlinearly on the number of unknowns in the simulation region. In some situations, especially when the results are not needed at every point of the grid, simulation time can be reduced. This reduction can be accomplished by partitioning the grid into macromodels and determining the macromodel impulse response. Then the impulse response can be decomposed into its eigenmodes, some of which can be eliminated because they are non-essential. In this paper we extend our earlier work and describe several strategies with which macromodels can be interconnected, that result in further savings of computation time. Keywords: FDTD methods, Design automation, Reduced order systems, Macromodeling. I. INTRODUCTION One of the most efficient and easy to program methods to solve Maxwell’s equations in the time domain is FDTD [1,10]. FDTD works by recursively solving the Maxwell’s equations in the entire domain during every iteration. The recursive procedure continues until some pre-determined condition is achieved. Schemes of this type are often used when the analytical solution to an electromagnetic problem is prohibitive. Problems to be solved with FDTD are abundant in simulations of aircraft radar cross section at high frequency, microwave ICs, optical pulse propagation, antennas, bioelectromagnetic systems, bodies of revolution, etc. [1]. Real-life problems often require grids with very large numbers of points, due to fine features of the simulated objects and short excitation wavelengths. The simulation time is exacerbated by modeling complex and composite materials, as well as increasing the order of the simulation in space and time to reduce error [2,3]. In some cases the simulation time can become unreasonable which has lead researchers to look for alternatives to conventional approaches. One way with which it may be possible to reduce this simulation time to a more acceptable level is presented in this paper. The generation of macromodels for simulations has attracted some attention recently [4,5,6]. A macromodel (subcell model) is basically an encapsulation of some simulation region. A macromodel establishes an interface between itself and the surrounding region. The authors have described how to create the macromodel impulse response matrix from the state transition matrix that is easily obtained by inspection of the FDTD method [8]. The advantage of such macromodels is that the impulse response history is computed only once and permanently saved as part of the model. In the future, this history can be reused as the surrounding region changes. If the solution within the macromodel is not important, then further savings can be made by computing the solution at a limited number of interface points. Conversion of macromodels to equivalent electrical circuits to model the entire system with circuit simulators when EM simulators are not suitable is straightforward [5]. Conversion to system level is also possible in order to allow for behavioral modeling [7]. The use of a similar method to construct primitive boundary conditions has also been studied [14]. The structure of the paper is as follows. The theoretical derivations of the eigenmodal method are discussed in Section II. In [9] we discussed the operation of a single module. Here we remove this assumption and we extend our earlier work to a system that consists of multiple modules. In Section III we discuss how the proposed system can handle various types of excitation scenarios such as steady state and transient excitations, as well as initial conditions. We consider several alternative methods to evaluate the response of a cascade of multiple modules in Section IV. Our experimental results from waveguide simulation are given and discussed in Section V. Finally section VI concludes the paper with a summary and suggestions for future research. II. THEORY The results of our earlier work describe the advantages and disadvantages of computing the impulse response of macromodels [8]. We have overcome some of the disadvantages with the introduction of the eigenmodal decomposition approach [9]. This involves expressing the inputs to a macromodel (module) as a superposition of the eigenvalues of the state transition matrix. The