7 Abstract-- A new plane wave time domain (PWTD) scheme is presented that retains the desirable computational complexity of classical PWTD schemes when applied to densely clustered source distributions, i.e., source distributions in which source separations are much smaller than the light time step. The construction of this scheme opens the door to the implementation of fast marching on in time (MOT) based time domain integral equation (TDIE) solvers capable of analysing low frequency electromagnetic phenomena of interest to the electromagnetic compatibility and interference (EMC/EMI) communities. I. INTRODUCTION T his paper describes a new PWTD scheme that is useful in the construction of fast TDIE solvers applicable to the analysis of EMC/EMI problems. Specifically, the proposed adaptive low frequency PWTD kernel permits the efficient and fast analysis of electromagnetic interactions with structures containing fine geometric features. With the advancements in their stability characteristics [1], the main stumbling block in using MOT-based TDIE solvers has been their high computational complexity, which scales as 2 ( ) T S ONN for a problem with T N temporal and S N spatial unknowns. To overcome the problems associated with this high complexity, the multilevel PWTD (MLPWTD) algorithm was introduced in [2]. Since, it has been successfully applied to the analysis of large-scale electromagnetic phenomena [3]. Here, the multilevel PWTD algorithm is extended by means of an adaptive low frequency kernel to facilitate the fast transient analysis of problems involving complex geometries with small and varying geometric scales. The proposed adaptive low frequency PWTD kernel has many applications. We believe, however, that it is especially useful in the analysis of realistic EMI/EMC problems. Such analysis presents significant challenges to all existing state-of-the-art time domain CEM tools. Indeed, varying geometric scales require separate low and high density discretization regions, which limits the applicability of finite difference time domain solvers because of the time step limitations imposed by the Courant-Friedrichs-Lewy condition. The proposed adaptive low frequency PWTD kernel circumvents this problem. II. THE PWTD SCHEME The chief deterrents to using a classical MOT scheme for EMC analysis are its high computational complexity and memory requirements that scale as 2 ( ) T S ONN and 2 ( ) S ON , respectively. To overcome this drawback, we recently introduced the MLPWTD algorithm to complement a classical MOT scheme, and demonstrated that the computational complexity of an MLPWTD enhanced MOT scales as 2 ( log ) T S S ONN N [3]. This reduction in complexity is a consequence of representing the fields in regions outside the source domain as a superposition of plane waves. A succinct outline of the MLPWTD algorithm follows. Further details can be found in [2]. The presentation below assumes that the geometry under consideration is uniformly meshed, with mesh sizes on the order of the light time step. To construct an efficient divide and conquer scheme that reduces the computational complexity of the MOT scheme, consider a cubical box that encloses the structure being analyzed. To implement an 1 l N + level algorithm, this box is recursively subdivided l N times, until the smallest box so obtained contains (1) O sources. At any level, each and every parent box contains 8 child boxes. Next, non-empty boxes at all levels are identified. Finally, the far-field pairs at all levels are tabulated using the following argument: Two boxes ( , ) l l α α ′ are said to be in each others far-field if the distance between their centers is larger than a Low Frequency Plane Wave Time Domain Kernels Kemal Aygün (1) , Balasubramaniam Shanker (2) , and Eric Michielssen (1) (1) Center for Computational Electromagnetics, Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, 1406 West Green Street, Urbana, IL 61801 (2) Department of Electrical and Computer Engineering, 372 Durham Center, Iowa State University, Ames, IA 50011