Dispersion Characteristics of 3D Distributed Node and Condensed Node TLM Mesh with Cells of Arbitrary Aspect Ratio

Qi Zhang and Wolfgang J.R. Hoefer

It is well known that the dispersion error affects the accuracy of the TLM method due to the spatial discretization. In order to evaluate the accuracy of the TLM method and quantify the dispersion error, the dispersion relation of the discrete mesh has to be known. Recently the general three-dimensional distributed node and cuboid condensed node TLM mesh with cells of arbitrary aspect ratio have been proposed. The nodes remove the restriction that the cell shape must be cubic. Both nodes are stub-free. In this project we investigate the dispersion characteristics of both nodes. The dispersion analysis of a TLM mesh is based on the application of the Floquet theorem to an infinite three-dimensional mesh. The application can be described by an eigenmatrix equation, and its characteristic is the dispersion relation.

The cuboid distributed node contains a rectangular shunt node and a series node which are in an interlaced arrangement. The dispersion relation is shown in Fig. 1, where alpha=dx/gz=0.5 and dz/lambda. For a coarse discretization dz/lambda the wavelength in the TLM network can no more be considered large compared with the network parameter dz, and the velocity becomes dispersive and depends on the direction of propagation. The maximum dispersion occurs in the axial directions. For the above example dz is larger than dx, and the propagation vector becomes larger along the longer mesh (z-direction) than along the shorter mesh dimension (x-direction). For comparison, the dispersion characteristic of the equivalent cubic mesh has been plotted in Fig. 1. According to the above discussion, we can choose a wave propagation direction for our validation test and thus minimize the dispersion error in the numerical results. Similar work has been done for the cuboid condensed node. Fig. 2 shows its dispersion relation. Comparing with that of the cubic condensed node, the maximum dispersion error does not occur in diagonal direction, and the minimum error can be found at an angle different from 45deg and depending on the cell aspect ratios. In an arbitrary direction the dispersion surface is plotted for the 3D cuboid distributed node and condensed node in Fig. 3 and 4, respectively.


FIGURE 1

Fig. 1 - The dispersion characteristics for a 3D cuboid distributed node in the x-z plane. dx=0.3, dy=dz=0.6

FIGURE 2

Fig. 2 - The dispersion characteristic for a 3D cuboid condensed node in the x-z plane. dx=dy=0.6, dz=0.3

FIGURE 3

Fig. 3 - Plot of the dispersion surface for the cuboid distributed node (dx=0.3, dy=dz=0.6)

FIGURE 4

Fig. 4 - Plot of the dispersion surface for the cuboid condensed


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