The use of graded and multigrid techniques in transmission-line modelling

The transmission-line modelling (TLM) method of numerical electromagnetic analysis with the symmetrical condensed node is well established. However, in many systems there is the need to represent very fine features within large volumes and, in 3-dimensions, it is rarely possible to employ a uniform mesh throughout the entire problem space. The traditional approach to this problem is to use a graded mesh. The stub-loaded node was originally used but more recently a hybrid node has been developed which requires fewer stubs. The disadvantage of using a graded mesh is that fine regions cannot be completely localised and there is an upper limit on the timestep which is determined by the smallest node dimensions. In an alternative approach, termed the multigrid method, the principles of connectivity and synchronism are relaxed so that regions of different spatial and temporal resolution can be joined together to form a complete system. Useful work can be done if each region consists of uniform cubic nodes but, in general, each region can be considered as a separate graded mesh. The transfer of information between different regions cannot be exact but good results can be obtained with significant reductions in both run-time and storage. If it is assumed that a uniform mesh cannot be used then the choice is between using a single graded mesh or a multigrid mesh. If the multigrid approach is adopted then there is a choice of using cubic nodes or graded mesh regions and if cubic nodes are selected it may be possible to use a node without stubs