General treatment of tlm node with embedded structures

This contribution is an extension of the authors’ work in modelling multi-scale systems which is of great importance in many applications such as Electromagnetic Compatibility (EMC). By its very nature EMC analysis often requires consideration of systems characterized by many different physical scales, such as when thin wires, gaps and slots are present in large structures. In the Transmission Line Modelling (TLM) method of analysis, nodes that enable fine structures to be embedded within them have been demonstrated for special cases where the fields local to the object may be expanded in terms of analytically known local solutions [1-6]. The approach presented here differs in that it is based on purely numerical calculations of these local expansion and it therefore allows arbitrary structures for which analytical representations of local fields are unavailable, to be incorporated into TLM nodes. This enables the presence of general sub cell objects to be correctly and efficiently modelled in coarse grids. Results are presented for a canonical scattering problem in order to demonstrate the proposed procedure. Introduction Accurate and efficient modelling of electromagnetic fields becomes very challenging with increasing complexity and volume of the simulated systems. Particularly difficult to deal with are structures containing diverse physical geometries in which small structures have a significant effect on the overall response of the system. Such a multi-scale configuration is typical for Electromagnetic Compatibility (EMC) applications when fine features e.g. thin wires, slots and gaps are embedded within physically large bodies. In order to reliably characterise such a system these features must be included in the simulation models. The classical approach using dense meshing is often inefficient, if not beyond the available computational resources and therefore over the years substantial effort has been made to develop more suitable techniques. One approach to describing fine features that has already been given much attention is to use a special node modelling the effect of sub cell structure into a coarse mesh model [1-6]. A variety of examples for EMC predictions have already been shown using such techniques. A node with a straight metal wire placed in the centre has been demonstrated for the 2D [1,2] and 3D TLM methods [3]. Further extension in the 2D case has been demonstarted, mapping a node containing an arbitrarily positioned thin metal or dielectric post within a single cell into a TLM network [4]. Recently a 2D node that embeds an arbitrary number of wires of various dimensions and characteristics coupled by their near fields within a single cell has 0-7803-9544-1/05/$20.00 ©2005 IEEE 52 been described [5]. The important case of short lengths of thin wires which are obliquely oriented to the cell faces has also been illustrated [6] making it possible to simulate piecewise linear models of arbitrarily routed curved wires in 3D TLM. The embedding process demonstrated in the above examples is based on a suitable set of local solutions to Maxwell’s equations in the vicinity of the enclosed objects. By appropriately sampling the fields at the boundary of the cell, these solutions have then been interfaced with the conventional numerical TLM algorithm. It should be noted that in this previous work the local fields have been obtained analytically. However, in many practical situations suitable analytical solutions may be impossible to identify in which case numerical methods must be employed. This contribution explores the development of sub cell models of arbitrarily shaped objects using numerical algorithms, i.e. the local fields in close proximity to the objects are calculated numerically using a fine mesh at the pre-processing stage and then mapped into a coarse grid. This concept must yield an overall algorithm that is computationally stable and, as it will be shown below, this requires that the local field solutions must be sampled in a physically consistent manner. Although the examples presented in this paper are based upon the TLM method as a well-established time domain technique for electromagnetic modelling, it should be recognised that there is no fundamental restriction of the general technique to use with TLM. Theoretical Formulations The basis of this approach is to identify a suitable set of local frequency domain solutions to Maxwell’s equations within the vicinity of an object. By appropriate sampling of these fields on the boundary of a cell, a scattering formulation is derived which will readily interface to the overarching numerical algorithm. Consider a small object bounded by a cubic surface, as shown in Figure 1a. For the TLM method it is required to map the tangential electric and magnetic fields on the faces of the cube to port voltages and currents in an equivalent electrical network. Enforcing voltage and current continuity between adjacent cells in the grid enables the node to be interfaced to the rest of the solution. The fields inside the cube are represented as a superposition of local field solutions, en and hn weighted by expansion coefficients Xn. Each solution satisfies the boundary conditions on the enclosed object. ∑ = = n T n n X e X e E and ∑ = = n T n n X h X H H (1) As presented in [6] the port voltage vector is defined by: ∫∫ ⋅ = p S p EdS f V , where fp is a set of orthonormal vector basis functions at port “p” positioned on surface Sp. Substituting equation (1) leads to the definition of a general voltage vector as: X u X dS e f V