INTERNATIONAL JOURNAL OF NUMERICAL MODELLING : ELECTRONIC NETWORKS, DEVICES AND FIELDS, Vol. 6, 161-164 (1993)

SHORT NOTE

A COMPARATIVE STUDY OF TWO TLM NETWORKS FOR THE MODELLING OF DIFFUSION PROCESSES

XIANG GUI* AND PAUL w . WEBB

School of Electronic & Electrical Engineering, University of Birmingham, Birmingham BI.5 2TT, U. K .

SUMMARY A comparative study of two alternative networks for TLM modelling of diffusion processes has been undertaken. Some of their inherent advantages and disadvantages are analysed according to the ratio between the impedance of the lossless transmission lines and the lumped resistance. Their relative accuracy and utility in controlling the timestep automatically are examined.

The transmission-line matrix (TLM) method has been established as an unconditionally stable explicit routine for modelling and numerically solving diffusion problems. A one-dimensional TLM node for this purpose is shown in Figure 1. More realistic multidimensional nodes based on such a standard network topology have been widely and have proved to be extremely useful and reliable. Figure 2 shows an alternative approach obtained by interchanging the nodes and cell boundaries. Both approaches give essentially the same solutions of the telegrapher’s equation to which the diffusion equation can be approximated. The modified TLM network with the node at the mid-points of the transmission lines was proposed as a more appropriate configur- ation in some recent s t ~ d i e s . ~ , ~ In this note, we report a comparative study of these two TLM variants and demonstrate some of their inherent advantages and disadvantages.

It has been reported6 that the standard TLM algorithm may introduce an unphysical artefact of nodal potentials which oscillate between expected values and zero under certain types of situation. Consider an infinite homogeneous bar with an instantaneous Dirac b-function source of quantity M at time t = 0 from unit source area. The analytical one-dimensional solution of the diffusion equation in space and time is given by

1 Figure 1 . A standard one-dimensional TLM node. R is the lumped resistor, Z is the impedance of lossless transmission lines, @ ( n ) is the potential of node n , Vi(V;? is the incident voltage pulse from the left (right), V; ( V a is the scattered

voltage pulse to the left (right), and I is the value of the connected current generator

* Currently with the Department of Electrical Engineering, University of Alberta, Edmonton, Canada; on leave from the Reliability Physics Laboratory, Department of Electronic Engineering, Beijing Polytechnic University, Beijing 100022, China.

0894-33701 931020 161 -04$07.00 @ 1993 by John Wiley & Sons, Ltd.

Received 30 January 1992 Revked 24 July 1992

162 XIANG GUI AND PAUL W. WEBB

-i n

c 1 D

Figure 2. A modified one-dimensional TLM network with the node at the centre of the transmission lines

where D is the diffusion coefficient. The problem was modelled by the TLM method using an elemental length Ax = 0.05 and a

mesh with a large number of nodes. For convenience, D was assumed to be unity as all material parameters may be appropriately normalized. The initial condition in the numerical solution may be satisfied by launching two equal incident voltage pulses VL and VR onto the centre node at the first timestep.

When this was tested using the standard TLM, the solution was found to oscillate between approximately twice the expected values and zero. In order to avoid the occurrence of unpopulated nodes which should be valued, the initial input can be shared with a half value for the first two timestep iterations. This method yields identical nodal potentials within two successive timesteps no matter how long the timestep is, and more accurate results will be generated using a shorter timestep and a smaller spatial resolution.

The same problem was also solved using the modified TLM node. Owing to the rearrangement of the network, the scattering process takes place at the half value of the time step (AtI2) so that reasonable values of nodal potential are obtained at the first and subsequent iterations. For this case, the timestep was chosen in three ways according to its relationship with the spatial resolution, or the relationship between Z and R (ZIR = (2DAt)/Ax2 for one dimension). If Z = R , then the results are exactly the same as those of the standard TLM (except for the first iteration, of course). This can be proved mathematically in a closed algebraic form.' If a smaller timestep is chosen ( Z < R ) , then the modified TLM curve is smoother and looks more elegant but the accuracy seems to be slightly worse than that of the standard TLM. If the timestep is made relatively large such that Z > R then the oscillations of the modified TLM become much more severe compared with those of the standard one. Figure 3 illustrates a comparison between the standard and modified TLM results with the same At of 0.0125 ( Z = 10R). This result is significant as the non-oscillatory nature of consecutive potentials claimed for this modified node cannot be guaranteed where the timestep is to be changed during the course of a transient simulation. If a suitable timestep is employed, the results of the two TLM variants compare very well and are in very good agreement with the analytical expression.

The above observations clearly illustrate the importance of the ZIR ratio and corresponding timestep on the nature of the calculated values of consecutive nodal potentials using the two TLM networks. Provided the timestep is short, corresponding to Z < R , then the modified node will give non-oscillatory nodal potentials. This is due to the fact that most of a pulse will be reflected back within one timestep if the values of resistors are large, while there is a delay of two timesteps in the standard TLM node.5 A potential advantage of the non-oscillatory nature of the modified node was the possibility of improving the methods used to control the timestep, where the calculation of aWat and d 2 @ I a f z are r e q ~ i r e d . ~ ? ~ This will clearly be limited to timesteps where Z <R and is an unrealistic condition if the principal advantage of the TLM method, the unconditionally stable nature of its solution for large timesteps, is to be realized.

It is worth noting that in realistic multidimensional problems the current generators representing, for example, heat input are connected at every timestep and there is considerable spatial and

SHORT NOTE 163

1.2

1 Z 0 + Q i- Z W

$ 0.6 0 0

-

0.8

0.4

0 . 2

Modified TLM

Analytical

0.1 0.2 0 .3 0.4 0.5 0.6 0.7 0.8 0.9 1

TIME

Figure 3. Calculated results under the condition of 2 = 10R for the standard TLM approach with an amendment to the initial condition and the modified TLM approach

temporal smoothing of the potentials. A further significant factor in comparing these approaches is the ease with which boundary conditions are incorporated using the standard node. The conditions of constant voltage, constant current, short circuit, open circuit and matched trans- mission lines are not so convenient in the modified approach.

If the timestep in a diffusion simulation is limited to values corresponding to Z = R then, according to Johns and Rowbotham,lo the TLM routine is compatible with a random walk model. The concept has recently been studied by Enders and de Cogan11*6 who termed it a ‘microscopically consistent TLM’. A microscopically small mesh (Ax and At are of the order of the mean free path length and the relaxation time of the material respectively) is generally unrealistic and a coarsened mesh with 2 = R is thought to introduce only mesh-related errors. Coarsening the mesh using this criterion is consistent with the rule (Ax2/At = constant) recommended by Johns and Butler.12 In a diffusion problem using these limiting conditions the accuracy of the two approaches is comparable and there seems to be no reason to favour either of the two nodes.

The maximum timestep that could be used at the start of a typical transient has been found to be of the same order as that determined by making Z = R. For longer timesteps the solution will begin to show oscillations which are clearly incorrect. The relation Z = R corresponds to timesteps which are identical to the stability limit set by an explicit finite difference approach, which is a special case of the TLM method, and there would seem to be little point in using the TLM approach using either node if the condition that Z = R is imposed. Compared with the finite difference and other conventional numerical techniques, the TLM method is very attractive in that its unconditionally stable nature allows the timestep to be varied while retaining reasonable accuracy, thus considerably reducing the computer run time.*,13 It is important to emphasize that for long timesteps ( Z > R ) the modified node has no advantage when compared with the standard one where timestep control is required.

It may be concluded that the modified TLM node has no practical advantage over the standard one except where the timestep is very short (2 < R). Even in this case the only advantage is to produce a smoother simulation but with no increase in numerical accuracy. For many problems, if such short timesteps are required the finite difference approach could be used where the simpler equations would lead to a reduction in solution time.

REFERENCES

1. P. B. Johns, ‘A simple explicit and unconditionally stable numerical routine for the solution of the diffusion equation’,

2. D. de Cogan and M. Henini, ‘TLM modelling of the thermal behaviour of conducting films on insulating substrates’, Int. J . Num. Meth. Eng., 11, 1307-1328 (1977).

J . Phys., D: Appl. PhyS., 20, 1445-1450 (1987).

164 XIANG GUI AND PAUL W. WEBB

3. S. H. Pulko and C. P. Phizacklea, ‘A thermal model for cyclic glass pressing processes using transmission line

4. X. Gui, P. W. Webb and G. B. Gao, ‘Use of the three-dimensional TLM method in the thermal simulation and

5. A. J. Lowery, ‘A study of the static and multigigabit dynamic effects of gain spectra carrier dependence in semiconductor

6. D. de Cogan and P. Enders, ‘Microscopic effects in TLM heat flow modelling’, IEE Colloquium on ‘Transmission

7. P. Enders, Private communication, December 1991. 8. S. H. Pulko, A. Mallik, R . Allen and P. B. Johns, ‘Automatic time stepping in TLM routines for the modelling of

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13. P. W. Webb and X. Gui, ‘Implementation of timestep changes in transmission-line matrix diffusion modelling’, lnt.

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