Giampaolo Tardioli, Lucia Cascio, Mario Righi, and Wolfgang J.R. Hoefer
Space discrete methods such as Finite Difference Frequency Domain (FDFD), Finite Difference Time Domain (FDTD), Transmission Line Matrix (TLM) and Finite Element Methods (FEM), are currently used for solving a wide variety of fields problems. The computational domain is discretized in a finite number of elementary cells where the electromagnetic field is assumed to have a simple behavior, very often linear. This assumption fails to accurately model sharp features, where highly nonuniform fields are present. This is typically the case of corners and edges, where the electromagnetic fields are singular. We refer to the resulting error as "coarseness error". In time domain methods, a comparison between the dispersion and the coarseness error reveals that the coarseness error is the dominant source of inaccuracies in most of the practical cases and represents the most severe limitation to the maximum admissible cell size. A direct solution to reduce the coarseness error is to use an extremely fine mesh, but this quickly leads to unacceptable memory and time requirements. A better approach is to use a variable or multigrid mesh, so that a higher resolution can be obtained in that region. In this case the resources would be still larger than those of a uniform coarser mesh fixed by the dispersion error only. We present a novel approach to incorporate knowledge of the static field behavior in the vicinity of singularities in a three-dimensional TLM mesh. The procedure is systematic and does not require optimization of the correcting elements. As a result, relatively coarse TLM meshes may be used to obtain highly accurate results, within the dispersion error, across a wide frequency range.
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Further details can be found in: G. Tardioli, L. Cascio, and W. J. R. Hoefer, "Special 3D-TLM Corner Nodes for Singular Field Regions", in 1997 IEEE MTT Symposium Dig., pp. 317-320, Denver, Colorado, June 9-13, 1997.