SCIENCE

Solution of resistive meshes by deterministic andMonte Carlo transmission-line modelling

P.B. Johns, Ph.D., M.Sc, B.Sc. (Eng.), C.Eng., M.I.E.E., andT.R. Rowbotham, Ph.D., M.Sc, B.Sc. (Eng.), C.Eng., M.I.E.E.

Indexing terms: Modelling, Simulation, Transmission-line theory

Abstract: The conventional random-walk and conventional adjoint random-walk processes for the solutionof general resistive networks are summarised. Particular attention is paid to resistive meshes which modelpotential and diffusion fields. The paper proceeds to introduce a new single-step simultaneous-displacementsmethod for general resistive networks through the concept of transmission-line modelling (TLM). A demon-stration of the method is given for a potential problem. The Monte Carlo method implied by deterministicTLM is then defined for general resistive networks. This new statistical process is evaluated and discussed inthe context of potential and diffusion fields.

List of Symbols

<D = vector of nodal potentials in resistive mesh£<D = vector of nodal potentials at iteration or time step k<t> = vector of statistical estimates of <DT = iteration matrixn — number of nodes in networkPrs = probability of stepping from node r to node s in a

conventional random walkvrs = arbitrary constant associated with Prs

Qrs = probability of stepping from node r to node s in anadjoint random walk

wrs = arbitrary constant associated with Qrs

fixed voltage boundary on meshnumber of walks in random-walk processscattering matrix at nodes in a transmission-line net-workvector of incident pulses on scattering zonesvector of reflected pulses from scattering zonesvector of sources in transmission-line networkprobability of stepping to node t having stepped fromnode s to node r in Monte Carlo TLMarbitrary constant associated withPrfif

probability of stepping to node t having stepped fromnode s to node r in adjoint Monte Carlo TLMarbitrary constant associated with Qrst

ratio of nodal resistance to characteristic resistance ina homogeneous transmission-line mesh

= particles with positive sign contributing to 0r

= particles with negative sign contributing to <pr

= equivalent number of steps per walk

Introduction

ws

V1

VVs

VrstQrst

v —

<frSe

1

Transmission-line modelling (TLM) of physical processes hasled to a deterministic particle-jumping method which has beenuseful in the solution of electromagnetic wave propagation[1], diffusion [2j and lumped-network [3] problems. Themethod can be considered as a systematic way of takingexactly all possible walks in a random-walk procedure, andBevensee [4] refers to a restricted version of this method as anumber diffusion process.

Paper 1456A, first received 17th December 1980 and in revised form5th May 1981Dr. Johns is with the Department of Electrical & Electronic Engineer-ing, University of Nottingham, Nottingham NG7 2RD, England. Dr.Rowbotham is with the British Telecom Research Centre, MartleshamHeath, Ipswich IPS 7RE, England

TLM can be applied over a wide range of problems, also itoften lends itself to a physical interpretation [5]. It is there-fore natural to ask whether the physical processes modelledby transmission lines, and solved in deterministic form, canalso be solved in a Monte Carlo form.

This question was asked when TLM was first developed forsolving the loss-free wave equation. Monte Carlo results wereeasily obtained for one-dimensional analysis [6] but it wasfound, subsequently, that although the deterministic processworks well in two and three dimensions, the Monte Carloprocess did not.

It is well known, however, that Monte Carlo methods areuseful in the solution of linear simultaneous equations [7, 8] ,particularly those associated with potential fields [4]. Inorder to establish common ground for conventional MonteCarlo methods and Monte Carlo TLM methods it was necess-ary to develop the deterministic TLM method for the solutionof simultaneous equations.

This paper presents this development in general, and con-centrates on the equations associated with resistive meshesmodelling diffusion and potential problems in particular.

This paper also shows that the Jacobi method for thesolution of simultaneous equations is a particular case of themore general deterministic TLM process. A demonstration ofthe rapid convergence properties of deterministic TLM forpotential problems is given.

This general deterministic TLM procedure then indicatesa general Monte Carlo process of which the conventionalMonte Carlo process is a particular case. The properties of thisnew statistical method, termed directed random walks, areexplored in the solution of resistive meshes associated withdiffusion and potential problems. Finally, the physical mean-ing and limitations of conventional random walks and directedrandom walks are discussed.

2 Conventional Monte Carlo methods

Consider a general linear resistive network with sources b andpotentials to datum <D, described by the set of algebraicequations

A*= b (1)

These equations may be solved by the stationary linear iter-ation process [7]

fe + 1 O = Tkd>+(I-T)A-lb (2)

IEEPROC, Vol. 128, Pt. A, No. 6, SEPTEMBER 1981 0143-702X/81I060453 + 10 $01.50/0 453

For the Jacobi routine

T = (I-D^A)

and for the Gauss-Seidel routine

T = I-(D + LTXA

(3)

(4)

where / is the unit matrix and where A = D + L + U. D is thediagonal matrix composed of the diagonal elements of ,4,Z, isthe lower triangular portion of ,4 and Uis the upper triangularportion.

A conventional Monte Carlo method for solving theseequations may be expressed in terms of the iteration matrixT [7, 8] in the following way. Start from any node / in anetwork of n nodes, and step in a random manner from nodeto node such that the probability of stepping from node r tonode s isP r s , where

TP — in

rs ~ vPrs >0, I Prs <

(5)

8 = 1

Trs is the element in row r and column s of T and vrs is anarbitrary amplitude. There is a probability Pr that the walkterminates at node r given by

Pr = 1 - (6)8 = 1

Define a score g for the random walk beginning at node i0 andterminating after N steps on node iN as

g = (7)

It is assumed that the initial potentials 0Oare zero. An esti-mate 0,- for the potential 0,- after W walks is given by

>..gw

W(8)

There is an adjoint conventional Monte Carlo method [8]which may also be expressed in terms of the iteration matrixT as follows. Choose a starting node / with probability Qio=igiven by

n - l^i

(9)Qio=i=

where at are a set of constants chosen to meet the criteria setin eqn. 9.

Starting with the chosen node /, step in a random mannerfrom node to node such that the probability of stepping fromnode r to node s is Qrs, where,

Qrs =

Qrs>0 I Qrs <(10)

s = l

Note that the subscripts of T are reversed in the adjointprocess. Again, there is a probability Qr that the walk termin-ates at node r given by

Qr = 1 - 1 Qrs=l

(11)

Define a score h for the random walk beginning at node i0 andpassing through node iM after M steps as

h =1 . W lM-

Q(12)

lN

where the walk terminates after N steps with N>M.Consider the general resistive mesh of n nodes shown in

Fig. 1. There are no controlled generators, and node r isconnected to node s through resistors Rr8 and R8r whereR

rs =R8r- There is a resistor 2Rrr to datum and a currentgenerator Ir to datum at node r.

The equation associated with the rth row of A is,

2RK + h_Z^L+^

rl 2Rr2 2R,/ r = 0 (13)

Fig. 1 General resistive network

i.e.

n 1

«^ r r_IlL

s=l

where

Yr 2R,(14)

£x 2R,05)

The Jacobi routine associated with the rth row of eqn. 2expresses 0r at the (k + l)th interation as

i k<t>s

Yr 2Rrs Yr

and an element in T for the Jacobi routine is

T = (s =£ r)rs Yr2Rrs

K }

(16)

(17)

The Gauss-Seidel routine associated with the rth row ofeqn. 2 is

and the elements in T are obtained by inversion of (£> + L).Thus the random-walk procedure for the resistive network

is defined, and we note that the quantity vrsPrs or wr8Qrsdepends on the node on which the particle is standing and onthe next node to which it will step. Thus, for a general net-work of n nodes, a storage of n2 is needed to remember eachTrs. The main advantage of random walks in a general net-work, therefore, is the possibility in reduction of computingtime, and this depends on the properties of A [7].

454 IEEPROC, Vol. 128, Pt. A, No. 6, SEPTEMBER 1981

3 Application of conventional random walks to potentialfield problems

Consider a Cartesian mesh of resistors of equal value 2R asshown in Fig. 2. The mesh is a discrete model of the continu-ous equation

- / = oV0

modelled through the branch equations, and

W = 0

modelled through KirchhofPs current law.

(18)

(19)

<

>

|

>

2R

*V

AAW

AAvv

AA

<

>

>

' Vv

>

<

<

<

A |

>•

»•

AA 4

/^~xC

AA

AAvv i

. AA i

>•

>•>•

> •

Fig. 2 Cartesian mesh of resistors of equal value with boundary nodesheld at constant potential

J is the continuous current density which is modelled bythe currents / in the branches. In two dimensions, for a meshlength A/ and material thickness L, the components of/ are

JXAIL = / ,

JyAlL = /y

The continuous conductivity a is modelled by R according to

1o = 2RL

Thus the solution of the resistive network provides an approxi-mate solution to the equation

V(aV0) = 0

For a source-free homogeneous region, eqns. 15 and 17 showthat

r« = ~ — (21)

which is independent of R. Thus one possibility is to setvrs equal to unity, thus keeping the amplitude of the particlein the walk constant. The probability of stepping in each ofthe four co-ordinate directions is 25% and we have the wellknown random-walk procedure for potential problems.Implementation of the Gauss-Seidel routine is not attractivebecause the value of Tr8 depends on the values of r and s.

At a boundary between one medium and another, the value

of the resistors change, and Trs for the Jacobi routine changesaccording to eqn. 17. Thus, at the boundaries, the probabilitiesof stepping in the four different directions are modified. Forexample, at the Dirichlet boundary condition shown by thevoltage generators in Fig. 2, Yr = °° and Trs = 0, and the walkterminates. From eqns. 7 and 14, for v = 1,

The score is $>, again in accordance with the well knownprocedure for random walks in potential problems.

For random-walk procedures to be useful, it is necessary forthe user to be able to estimate the statistical error for a givennumber of walks W. This is achieved by calculating the confi-dence limits for the computed potential. Following Bevensee[4] and Royer [9], the binomial distribution describing thenumber of times a particle hits a certain point is approximatedby the Gaussian distribution. Thus if 0r is the statistical esti-mate of the potential at node r after W walks for a point inthe resistive mesh whose deterministic potential is 0r, then thedistribution of 0r about 0r is given, approximately, by

/(0r) =1

exp<t>r~<Pr 2a?

where

(22)

(23)

and 4> is the potential over one boundary surface within theregion; the other boundaries are at zero potential.

Thus, to be 95% certain (for example) that 0r lies within acertain range, the range is given by

- 1 . 9 6 a ! < ( 0 r - 0 r ) / $ < 1.96(7! (24)

4 Conventional random walks and time-domain diffusion

Suppose that each node in the general network of Fig. 1 has,in addition to the current generator Ir, a capacitor Cr toground. The capacitor may be regarded as adding a current ir

to Ir, where

h ~ ~ ^ dt

Eqn. 14 becomes

Yr 2Rrs r Yr Yr dt

(25)

(20) Integrate this equation by Euler

Yr2Rrs k<Pr Yr YrAt

For the special case of

(26)

YrAt(27)

1 k<Ps , A-Y"7R~ Y

6=1

which is the Jacobi method of eqn. 16.

IEEPROC, Vol. 128, Pt. A, No. 6, SEPTEMBER 1981 455

The introduction of capacitors C into the Cartesian mesh ofFig. 2 modifies the continuous equations modelled as follows:

dt(28)

i.e.

V(aV0) = C ^

which is the diffusion equation.Thus the conventional random-walk procedure not only

solves a potential problem but the transient involved in takingthe random walk models the diffusion equation provided thefollowing condition is met:

YrAt _ 2ArCr ~ RC

= 1 (29)

The process is most easily understood in the adjoint form. Ineqn. 12, the number of steps M associated with each score h isrecorded for each node. M is regarded as a time scale and aplot of scores against time for each node provides an estimateof the diffusion transient. The physical significance of theprocess is easily seen. Suppose that the diffusion processconsists of a crystal lattice of pure material with diffusion ofimpurities taking place on the lattice. The random walkcorresponds exactly to a model where the sources of impurityemit atoms which jump from location to location with anequal probability of jumping in any of the co-ordinate direc-tions.

5 Solution of resistive networks by TLM

Bevensee [4] points out that probabilistic potential theory canbe operated in two ways. The first is the statistical MonteCarlo or random-walk process described in the precedingSections and the second is, termed by Bevensee, a number-diffusion process. In the Monte Carlo method a particle isincident on a node or scattering zone and is reflected into oneof a choice of exit ports with a certain probability Pr8. If vrs isunity, the particle maintains its amplitude throughout thewalk. In the number diffusion method a particle is incident ona scattering zone and is scattered into all possible exit ports.The fraction scattered into any port is the value Trs for thatnode and exit port.

The number-diffusion process can be simulated exactly byan electrical network where nodes are interconnected bytransmission lines as shown in Fig. 3. Particles or pulsestravel down the transmission lines and are incident on the

Fig. 3 General resistive network interconnected by transmission lines

456

nodes where they are scattered by the resistors and the junc-tions. Reflected pulses appear at all the ports and traveltowards neighbouring nodes, where they again become inci-dent pulses, and so the process continues. The electricalprocess is time discrete because the pulses remain at constantamplitude as they travel down the ideal transmission lines. Theprocess of rendering a physical problem time discrete by theintroduction of commensurate transmission lines is termed byus a transmission-line modelling (TLM) process.

Electrically, the only difference between the generalresistive network of Fig. 1 and the general transmission-linenetwork of Fig. 3 is that in Fig. 3 the resistors are interconnec-ted by transmission lines. The final nodal voltages of thetransmission line network, after all the transients have diedaway, is the same as for the resistive network without trans-mission lines. The only proof of this statement offered at thisstage is the common-sense physical fact that the lengths ofwire interconnecting the resistors do not affect the DC sol-ution.

The scattering matrix Sr at node r relates the reflectedpulses to the incident pulses on each of the transmission linesconnecting node r to the other nodes. In a general network ofn nodes, it is assumed that node r is connected to all othernodes, and to itself through the transmission-line stub on therth terminal. Thus

Vr — S! V* + VsVr — arvr T vr

where (30)

Y r Vv r\ Kr2 • • • y rni

are the pulses reflected from node r,

y r L K r l y r2 • • • v rni

are the pulses incident on node r and

vs — \vs. vsn vs ^T

r r Yv rl v r2 • • • r mlare the pulses due to sources, and where the superscript Tdenotes the transpose of the vector.

The node potential 0r may be expressed in terms of theincident pulses

t"#r (31)

where

Qr = [QriQr2-"Qrn]

and

Qrs = (Rrs + Zrs)Yr'

where

y; == 1 Rr< +

(32)s = l

and Zrs is the characteristic resistance of the transmissionline connecting node r to node s.

The scattering given by eqn. 30 can be expressed as

V; =Pr<pr + rrVri (33)

where

Pr = [Pr\Pr2--.Prn\T

IEEPROC, Vol. 128, Pt. A, No. 6, SEPTEMBER 1981

and

Prs =

and

r r l l

where

• l-OD

D 7

Thus the scattering matrix may be written as

(34)

The scattering matrix Sr relates the reflected pulses to theincident pulses at the rth node. For a network of n nodes,equations like eqn. 34 may be combined to give the scatteringequation for the entire network. Thus, at the fcth iteration,

kVr =

where

Vr =

(35)

k? =

Vi

S is a block diagonal matrix with Si, S2,..., Sn on the diag-onal.

The iteration process is due to reflected pulses from nodesbecoming incident pulses on interconnected nodes one time-step later. This may be expressed as

fe+i V* = ChVr (36)

C is a connection matrix which contains a sparse filling of T sand it describes which terminal of a node is connected towhich terminal of another node. The complete process may beexpressed as

k+1

or

Vl =

V = SChVr+ V

(37)

(38)

In order to compare the TLM algorithm with the Jacobialgorithm it is necessary to write the process out in detail.From eqn. 31

fe0r = f (39)

Here the numbering scheme used is that the 5th terminal onthe rth node connects the rth node to the sth node.

Eqns. 39—41 provide a general algorithm in terms of inci-dent and reflected pulses. Alternatively, the algorithm may beexpressed, like eqn. 37, in terms of the incident pulses andsources only. Thus, substituting for p, q and r, also

fe+l<Pr -

fe + i

ft +1 Vrs " v-' (42)

(43)

Eqns. 42 and 43 provide an explicit single-step simultaneous-displacement linear iteration process in which the choice ofthe Zrs (which determines the rate of convergence) is decidedby the user. The larger the value of Zr8, the faster the processconverges, since losses due to the resistors in the transmissionline network are relatively smaller. If, however, the Zrs aremade too large, the network becomes relatively loss free andthere will be overshoot followed by damped oscillation aboutthe final steady-state solution. Extensive tests on the use ofthis method for general resistive networks have not been made,but a few tests show that the convergence of this simultaneous-displacement method can be as fast as the successive-over-relaxation method. The case where Zr8=Rrs is of specialinterest, since, from eqns. 32 and 15,

Y' = Y1 r J r

(44)

Also, since the stub transmission line Zrr is now matched toits terminating resistor Rrr, then Fr'r = 0.

Eqns. 42 and 43 now become

fe + 1 K ~ 2- ' —s *r Yr 2Rr8 Yrs=l

(45)

Eqn. 45 is the same as eqn. 16, and so, in the particular caseof Zrs = Rrs) the TLM method routine is exactly the same asthe Jacobi routine. For this particular case the TLM routinecan be expressed solely in terms of the node voltages and theroutine is a simple finite difference procedure. This propertyof the TLM method results from the matrix r in eqn. 33 beingzero. If r is not zero the TLM routine cannot be expressed as asingle-step routine involving only the node voltages; it mustuse incident and reflected pulses.

6 Directed random walks for general resistive networks

The deterministic TLM process described in the precedingSection is a number-diffusion process for which there areequivalent Monte Carlo processes. For the general network ofFig. 3 with Rrs = Rsr we define the directed random walk inthe following way. Start from any node i in a network of nnodes and step in a random manner from node to node suchthat having stepped from node s to node r the probability ofstepping to node t is Prst, where

and from eqn. 33

(40)

Since reflected pulses from one node at time k become inci-dent pulses on adjacent nodes at time k + 1 (eqn. 36), then

k+iVf. = kv;r (41)

Prst ~Vret

Prst > 0 2Pnt< iforaUst=i

(46)

Srst is the element in row s and column t of the scatteringmatrix associated with node r and vr8t is an arbitary ampli-

IEEPROC, Vol. 128, Pt. A, No. 6, SEPTEMBER 1981 457

tude. There is a probability that the walk terminates at node rgiven by

r = 1 " I Prstt=l

(47)

Define a score g for the random walk beginning at node /0 andterminating after N steps on node iN as

g =, i0 i2

vi2 J, »3 * W'JV-

It is assumed that the initial potentials and the initial incidentpulses 0V

l are zero.An estimate 0,- for the potential 0,- after W walks is

g l + • -SWW

(49)

The relationship between the Monte Carlo TLM process andthe adjoint Monte Carlo process is similar to the relationshipbetween the conventional processes. Thus, in adjoint MonteCarlo TLM, the walk is from source to solution point, and

cQrst =

In order to illustrate Monte Carlo TLM and adjoint MonteCarlo TLM, suppose that in either case there is a single sourcewhich is situated at node 1, and that we are interested in thevalues of the voltage at node 4 (04). Let us examine the walkwhich steps on nodes 1, 2, 3 and 4 by looking first at thedeterministic TLM process. Since the initial potentials arezero, then the V[ in eqn. 33 are zero and

VI = = P l ^ 7 = V{ (50)

These reflected pulses become incident pulses on neighbouringnodes, one of which is node 2. The pulse incident on node 2scatters, and one of the reflected pulses becomes incident onnode 3, and so on. The pulse incident on node 4 from node 3(K43) is given from eqns. 31 and 33 as

The potential 04 is given by eqn. 31

04 = 443 ^43

Thus

Eqn. 51 may be written as

(51)

(52)

This is the deterministic process. It can now be seen that thedirected random walk using Monte Carlo TLM in adjoint ornonadjoint form, as defined in this Section, provides a statisti-cal estimate for 04 . Note that

• -*Va

(Rr8 + Zr8) Yr Rr8+ Zrs Rrs + Zr

(Rr8 + Zrs)Yr RrtZrt

(53)

(54)

For the particular case of R{j = Z{j, for all / and /, prs = 1/2and rr88 = 0, and so

rt8 2RrsYr= T (55)

Using eqns. 50 and 32, for Ry = Zu, equations 52 becomes

04 = T43T32T2i(l)i

which describes the conventional random walk and the con-ventional adjoint random walk.

7 TLM in potential and diffusion fields

Consider the resistive mesh of Fig. 4 where nodes are inter-connected by transmission lines. Since the lines can be con-sidered merely as connecting wires at DC, the solution of the

Fig. 4 Cartesian resistive mesh interconnected by transmission lines

TLM routine after transients have died away will be preciselythe same as the mesh with no transmission lines. The transientsintroduced by the transmission lines provide a good model fordiffusion, and proof of convergence and stability are given inReference 2. The continuous equation modelled is [2]

20 =V20 =90

(56)

where Ld and Cd are the distributed inductance and capaci-tance per unit length of the transmission lines, respectively,and, for an internodal length A/, Rd =R/Al. The approxi-mation to the diffusion process arises when the term2LdCdd

2<t)/dt2 is small.The scattering matrix associated with a source-free node,

given by eqns. 53 and 54, is

1

2(1 + .)

(57)

where v = R/Z.If v = 1 we have the scattering matrix associated with the

Jacobi routine and the finite difference diffusion process. Ifv is reduced below 1, the TLM process models a faster dif-fusion process [2], although it can be seen from eqn. 56 thatif v is too low the wave term will predominate. In the limit, ifv is zero then R and Rd are zero and the loss-free waveequation is modelled [1].

458 IEEPROC, Vol. 128, Pt. A, No. 6, SEPTEMBER 1981

Table 1 : Convergence of accelerated TLM

Iteration

12

34b6/8Q91011121314151617181920

v = 0.5

1.3333

1.4815

1.3004

1.3242

1.3390

1.3331

1.3328

1.3335

1.3334

1.3333

0.6

1.2500

1.4258

1.3284

1.3281

1.3342

1.3336

1.3333

0.7

1.1 765

1.3759

1.3379

1.3320

1.3332

1.3334

1.3333

0.75

1.1429

1.3528

1.3380

1.3331

1.3332

1.3333

0.8

1.1111

1.3306

1.3358

1.3336

1.3333

0.9

1.0526

1.2888

1.3265

1.3323

1.3322

1.3333

1.0

1.0000

1.2500

1.3125

1.3281

1.3320

1.3330

1.3333

Size = 3 X 3 , output node (1,2)

Table 2: Convergence of accelerated T L M

T

1520253035404550556065707580859095100

y = 0.2

0.97141.02691.02231.01941.02341.02241.02251.02261.02251.0226

0.3

1.01531.01901.02311.02261.02251.0226

0.4

1.02551.02271.02251.0226

0.5

1.01421.02061.02231.02251.0226

0.6

0.99071.01081.01991.02161.02231.02251.02251.02251.0226

0.7

0.96100.99461.01401.01871.02141.02201.02241.02251.02251.02251.0226

1.00

0.86700.92610.97760.99561.01001.01501.01901.02041.02161.02201.02231.02241.02251.02251.02251.02251.02261.0226

Size = 7 X 7 , output node (2,4)

In the solution of potential problems by an iterationprocess the aim is to reach the final steady-state solution in asfew iterations as possible. It is therefore useful to investigatethe effect of reducing v, and numerical experiments have beenperformed on the square coaxial waveguide problem shown inFig. 5. The results are given in Tables 1—3 for various meshsizes indicated by the dimensions of the internal node points.Examination of these Tables shows that as the value of v isreduced the rate of convergence is increased, initially. How-ever, a point is reached when the calculated value overshootsthe correct value, and a damped oscillation occurs. It is at thispoint, of course, that the effect of the wave term in eqn. 56begins to become obvious. Tables 1—3 also show that, as thesize of the problem increased, the optimum value of vdecreased. Thus the factor shows similar properties to anacceleration factor in the successive-overrelaxation (SOR)method, except that the TLM (as described) is a simultaneous-displacement method.

8 Monte Carlo TLM in potential and diffusion fields

The ability of deterministic TLM to speed up diffusionprocesses has been investigated [2] and is being exploited[10]. This paper shows that a statistical estimate of thedeterministic TLM result can be made using Monte Carlo TLM

in the form of a directed random walk. Thus, in diffusionproblems, the directed random walk requires less steps perwalk to cover a physical time period than does the conven-tional random walk.

The ability of TLM to accelerate convergence in potentialproblems has been demonstrated in the preceding Section.Again, this means that the directed random walk for thepotential problem will require fewer steps per walk. It shouldbe noted that in the definition of conventional and directedrandom walks no mention is made of a limit to the number ofsteps per walk. In conventional random walks the particle isquenched at a Dirichlet boundary but, even so, a limit islikely to be placed on the length of a walk according to thedegree of convergence indicated by the deterministic process.In directed random walks there is a nonzero probability thata particle is reflected at a Dirichlet boundary as indicated byeqn. 53. To avoid unnecessary work it is therefore importantto impose such a limit on the number of steps per walk, and,indeed, the only advantage of directed random walks inpotential problems is that this limit can be considerablylower than in conventional random walks.

In the TLM Monte Carlo method, as in the conventionalmethod, there is a choice to be made between the amplitudeof a particle proceeding in a given direction and the prob-ability of it proceeding in that direction. This choice is

IEEPROC, Vol. 128, Pt. A, No. 6, SEPTEMBER 1981 459

Table 3: Convergence of accelerated TLM

Step

1520253035404550556065707580859095

100

y = 0.2

0.95410.88340.88710.88570.88920.88950.88910.8890

i> = 0.3

0.84650.88350.88860.88900.88910.8890

v = 0A

0.75170.83610.86090.87810.88320.88680.88780.88850.88880.88890.8890

i/=0.5

0.67080.77940.81980.85420.86700.87800.88200.88550.88680.88790.88830.88870.88880.88890.8889

f = 0.6

0.60220.72370.77450.82300.84330.86260.87070.87850.88170.88480.88610.88730.88780.88830.88850.88870.88880.8889

v = 0.7

0.54370.67190.72940.78840.81510.84240.85470.86740.87310.87900.88160.88440.88560.88690.88740.88800.88830.8885

Size = 1 1 X 11, output node (3,6)

expressed by eqns. 5 and 46 in the conventional and TLMcases, respectively.

In the conventional case for potential and diffusion prob-lems it is usual to take vrs = 1. However, in the TLM case thechoice is not so obvious because particles are scattered differ-ently in different directions. Nevertheless it is shown inAppendix 12, for homogeneous field problems, that vr8t = 1 isthe choice which minimises statistical noise. Intuitively thisseems to be correct also. Suppose that, for particular values ofr, s and t, Srst is small, then it can be argued that it is wastefulto carry around a small amplitude contribution with the sameprobability as a much more significant contribution given bya larger Sr8t. Again, from a physical point of view, the choiceseems correct because a particle conserves its contribution forVrst= I-

There is, however, a fundamental limitation on MonteCarlo TLM. If vrst > 0 then v cannot be less than 0.5. If i isless than 0.5 then the diagonal in S becomes negative (eqn.57), requiring negative probabilities, and this is physically andmathematically impossible. Thus we have the interestingresult that a directed random walk with a constant-amplitudeparticle cannot be used to solve potential, diffusion or lossy-wave problems with t»<0.5. In particular, the techniquecannot be used for loss-free wave problems where v = 0. Theremainder of this paper is restricted therefore to resistivemeshes modelling homogeneous fields for which y > 0.5 andv=\.

If v =£ 1 in directed random walks then the particle jumps

1AV

| ov

into the direction from which it was incident with a differentprobability to the other directions. This is because r is nonzeroin eqn. 34 for v =£ 1, and it is for this reason that the walksare termed directed random walks. Klahr [11] and Bevensee[4] both refer to the ability to have Pnt*l/4, Bevenseereferring to Klahr. Klahr treats the elliptic equation withprst = 1/4 but both discuss only the case where the walksquench on a boundary, and the adjoint method is not referredto. In TLM there is reflection at a Dirichlet boundary forv < 1 and the physical interpretation of TLM relies heavily onthe adjoint process. Klahr's method also assumes that approxi-mate results are known; such a requirement is not necessary inMonte Carlo TLM. Thus it is evident that the method ofweights referred to by Bevensee is not a Monte Carlo TLMprocess.

In directed random walks the number of steps per walk isreduced, but it is still necessary to determine the number ofwalks required for a given accuracy. This number of walks canmost easily be calculated as a ratio of the number of conven-tional random walks (y= 1) required to achieve the chosenmean-square deviation in eqns. 22 and 23. The overall measureof performance for comparison of different values of meshresistance for a given value of deviation is the number ofsteps per walk multiplied by the number of walks, i.e. the totalnumber of steps. The deviation ox for the v=\ case is givenby eqn. 23. Only approximate results are required and so inthe interest of simplicity it will be assumed that <j>r is smallcompared with 3>. Thus, for W\ walks

* ? = (58)

Fig. 5 Mesh for approximate solution of square coaxial waveguide,size 3x3

The reason that the necessary number of walks will vary as afunction of mesh resistance, despite the fact that the samevalue of 0r results, can be seen by noticing how 0r is obtainedas steps progress in the deterministic TLM process. For v = 1the particles spread out isotropically from each node and areabsorbed at the boundary. For v < 1 more particles will arriveearlier at the node of interest because the diagonal term ineqn. 57 is less. This faster build up of 0r is convergent onto thecorrect <pr owing to negative-amplitude impulses returningfrom the boundaries. That these reflected particles are negativeand have nonzero magnitude is inevitable because the bound-ary resistors are not matched but of lower resistance than thecharacteristic resistance of the transmission-lines. Thus thefinal 0r is the sum of positive and negative contributions,whereas the sum of their magnitudes, which affects a2, isgreater than 0r. The disadvantage is compensated by the fewer

460 IEEPROC, Vol. 128, Pt. A, No. 6, SEPTEMBER 1981

Table 4: Equivalent steps (Se) in convergence of directed random walks

Step

1520253035404550556065707580859095100

v = 03

1.01421.02061.02231.02251.02261.02261.0226

i

Se <i

19.458 (25.59639.95345.15753.64762.078

; = 0.6b

J.99071.01081.01991.01131.02231.02251.02251.02351.0226

Se

18.33425.68733.58040.97148.48251.775162.41170.00977.0946

y=0.7<*>

0.96100.99461.01401.01871.02141.02201.02241.02251.02251.00251.0226

Se

17.36323.93330.87737.50944.23150.75557.27963.721370.16876.57182.9630

v= 1.00

0.82600.92610.97760.99561.01001.01501.01901.02041.02161.02201.02231.02241.02251.02251.02251.02251.0225

Se

15.0020.0025.0030.0035.0040.0045.0050.0055.0060.0065.0070.0075.0080.0085.0090.0095.00

Size = 7 X 7 , output node (2,4)

number of steps required to obtain 0r for v < 1. Let <t>+r and

<f>r be the particles contributing to 0r for v < 1, i.e.

0r — <fr~ (59)

If Wv walks are made on a mesh with v < 1, and assuming thatthe statistical variations in the positive and negative contri-butions are independant, then the standard deviation av forthe v < 1 mesh is given by

(60)

The ratio of the number walks for the statistical noise to beequal in both cases is, from eqns. 58 and 59,

Wy(61)

This factor shows the ratio of walks that are required if v isless than unity, for the same confidence in a level of accuracy,compared with the walks required for v = 1. Using this factor,another factor Se has been devised and called the equivalentnumber of steps per walk. If Si steps are required for a certainaccuracy in the correct result 0r for v = 1 and Sv steps arerequired for the same accuracy in <pr for v<\ (Su<Si),then S* is defined as

— >JIJ

The ratio Se/Si represents the saving in total number of stepsobtained using a directed random walk compared with aconventional random walk. The equivalent number of stepsper walk Se has been calculated from numerical results usingdeterministic TLM for the square coaxial waveguide problemof Fig. 5 using a 7 x 7 mesh. The results corresponding to thepotential at node (2,4) are shown in Table 4. Here it canbe seen that as v changes from 1.0 to 0.5 the equivalentnumber of steps falls from about 95.0 to 53.6 if convergence

to within 0.0001 in 0r is required. For an error of 0.001 in0r, the results are similar, with equivalent steps reducing fromabout 75 to 45.2 for v = 0.5.

For nodes adjacent to a zero-potential boundary theadvantage obtained is not so significant. Whereas a morecentral node requires considerably less steps as v decreases andthe probability distribution of 0r widens only slightly to offsetthis rapid convergence, the boundary node has almost all ofits convergence advantage eliminated by the increasing devi-ation of the distribution. Table 5 compares the results for Se

for node (2,4) with Se for node (2,1) for 0r converged to0.0001. The trend is clear; the directed random-walk tech-nique is quicker than conventional random walks for thenode in the centre of the field, but only comparable for anouter node. The clue to this phenomenon lies in the compara-tive values for the <p* and <fr. There is a larger differencebetween 0* and <fr for node (2,4) than for node (2,1). Thusthe low potential (0.2707) of node (2,1) is due to the poten-tial difference between two relatively larger numbers each ofwhich makes a relatively large contribution to the statisticalnoise.

A similar set of results was obtained for the square coaxialwaveguide using an 11x11 mesh and the same conclusioncan be made.

The physical interpretation of directed random walks isinteresting. In conventional random walks (y= 1), eqn. 57shows that the probabilities of a particle jumping into the fourco-ordinate directions of a two-dimensional problem areequal. If v > 1, in a directed random walk, then the prob-ability of jumping into a location just vacated is higher thanfor jumping into a new location. This is a common phenom-enon in solid-state physics [12], and the probabilities can bequantified. Numerically speaking, v > 1 is not attractive sincethe associated diffusion process is slower. The case for 0.5 <v < 1 is also physically possible since, within the material, theparticle is persuaded to jump into new locations rather thanjump back into old ones. At a boundary, however, the particlehas to change sign, i.e. start making negative contributions to

Table 5: Equivalent steps Se and contributions <pp and <£y in convergence of directed random walks

V

0.50.60.71.0

Node (2,4)step

35556595

Se

53.64777.094682.963095.00

0+r1.29501.22801.16391.0226

0r

0.27240.20540.14130.00

Nodestep

30455580

(2,1)Se78.20897.5692.75080.00

0.48820.41980.36360.2707

0.21750.14910.09290.00

Size = 7 X 7 , output nodes (2,4) and (2,1)

IEEPROC, Vol. 128, Pt. A, No. 6, SEPTEMBER 1981 461

locations within the material. From a physical point of viewthis is less realistic but, of course, from a numerical point ofview it is attractive. For y < 0 . 5 the particle is not onlyrequired to have the ability to change sign at each location,but it must also grow in its contribution as time elapses. Notonly is this difficult to imagine physically, but also numericalexperiments indicate that enormous numbers of walks must bemade for a realistic estimate. Thus, although deterministicTLM has been used with considerable success for the solutionof loss-free wave problems [13], the Monte Carlo TLM processfor such problems does not have realistic physical meaning,nor can it be used numerically.

9 Conclusions

The iterative scheme for resistive networks produced byinserting transmission lines is a generalisation of the wellknown Jacobi method. The potential problem used as ademonstration indicates that TLM significantly increases theconvergence rate, while still remaining a single-step simul-taneous-displacements method.

Directed random walks, the Monte Carlo version of TLM,also have an advantage over conventional walks. However, theadvantage observed in the deterministic process is partiallyoffset in the statistical process by an increase in statisticalerror.

Directed random walks are useful for the solution ofpotential problems and of the transients associated withdiffusion and lossy wave problems. The limitations on directedrandom walks mean that they are not realistic for loss-freewave problems. Thus, from a physical point of view, while itis very likely that many diffusion processes are due to themovement of constant-amplitude particles taking random ordirected random walks, it is very unlikely that loss-free wavesare propagated by this mechanism.

10 Acknowledgment

Some of the background of this paper arose from discussionwith A. Wexler while one of the authors (P.B. Johns) was onsabbatical leave at the University of Manitoba, Winnipeg,Canada. This contribution is gratefully acknowledged.

11 References

1 JOHNS, P.B.: 'The solution of inhomogeneous waveguide problemsusing a transmission-line matrix', IEEE Trans., 1974, MTT-22,pp. 209-215

2 JOHNS, P.B.: 'A simple explicit and unconditionally stable numeri-cal routine for the solution of the diffusion equation', Int. J.Numer. Methods Eng., 11, pp. 1307-1328

3 JOHNS, P.B., and O'BRIEN, M.: 'Use of the transmission-linemodelling (TLM) method to solve non-linear lumped networks',Radio & Electron. Eng., 1980, 50, pp. 59-70

4 BEVENSEE, R.M. 'Probabilities potential theory applied to electri-cal engineering problems', Proc. IEEE, 1973, 61, pp. 423-437

5 JOHNS, P.B.: 'The art of modelling', Electron. & Power, 1979,25, pp. 565-569

6 ROWBOTHAM, T.R., and JOHNS, P.B.: 'Waveguide analysis byrandom walks', Electron. Lett., 1972, 8, 10, pp. 251-253 May1972

7 WESTLAKE, J.R.: 'A handbook of numerical matrix inversion andsolution of linear equations' (Wiley, 1968)

8 HAMMERSLEY, J.M., and HANDSCOMB, D.C.: 'Monte Carlomethods' (Methuen Monographs, 1964)

9 ROYER, G.M.: 'A Monte Carlo procedure for potential theoryproblems', IEEE Trans., 1971, MTT-19, pp. 813-818

10 BUTLER, G., and JOHNS, P.B.: 'The solution of moving boundaryheat problems using the TLM method of numerical analysis' inLEWIS, R.W., and MORGAN, K. (Eds.): 'Numerical methods inthermal problems' (Pineridge Press, Swansea, 1979)

11 KLAHR, R.N.: 'A Monte Carlo method for the solution of ellipticpartial differential equations' in RALSTON, A.R., and WILF,H.S. (Eds.): 'Mathematical methods for digital computers' (Wiley,1962)

12 TUCK, B.: 'Introduction to diffusion in semiconductors' (PeterPeregrinus, 1974), Chap. 5

13 AKHTARZAD, S., and JOHNS, P.B.: Three-dimensional trans-mission-line matrix analysis of microstrip resonators', IEEE Trans.,1975, MTT-23, pp. 990-997

12 Appendix

A two-dimensional homogeneous field is modelled by aCartesian mesh of resistors of equal value R connected toeach other by transmission lines of characteristic resistanceRfv as shown in Fig. 4. Consider a node r with a voltage pulseincident on terminal 4, say, and leaving from terminal s. Overa large number of walks W the deviation from the expectedvalue, using a result similar to eqn. 23, is

°s wThere are four exit directions and, provided W is large, thedeviations can be added. Thus, the total deviation a for thenode is given by

2 =Wo2 =8=1

Using eqn.

From eqn.

Sr14 =

and

5"r44 +

Taking

46

s=l

57

5r24 =

3SrU =

j 4 ) 2 ( 1 _ ^ ^

p

SrM

1

(62)

and substituting in eqn. 62

2 _Wo2 =<S'r

1-3P,(63)

rl4

Differentiating eqn. 63 with respect to P r l 4 shows that theminimum deviation is given when

Thus

Vrst =

462 IEEPROC, Vol. 128, Pt. A, No. 6, SEPTEMBER 1981