Finite element mesh partitioning using neural networks

ELSEVIER

SO965-9978(96)00011-7

Advmces in Engineering Software 27 (1996) 103- 115 Copyright 0 1996 Civil-Comp Limited and Elsetier science Liited Printed in Great Britain. All rights reserved 0965~9978/96/$15.00

Finite element mesh partitioning using neural networks A. Bahreininejad, B. H. V. Topping & A. I. Khan Department of Mechanical and Chemical Engineering, Heriot- Watt University, Riccarton, Edinburgh EH14 4AS, UK

This paper examines the application of neural networks to the partitioning of unstructured adaptive meshes for parallel explicit time-stepping finite element analysis. The use of the mean field annealing (MFA) technique,which is basedon

the mean field theory (MFT), for finding approximate solutions to the partitioning of the finite element meshesis investigated.The partitioning is based on the recursive bisection approach. The method of mapping the mesh bisection problem onto the neural network, the solution quality and the convergence times are presented. All computational studies were carried out using a single T800 transputer. Copyright 0 1996 Civil-Comp Limited and Elsevier Science Limited

technique provides a method for finding good solutions to most combinatorial optimization problems, however the algorithm takes a long time to converge. To overcome this problem, MFA was proposed as an approximation to simulated annealing. Quality of solution was traded against the reduced computational time. Several methods4-7 have been developed in order to find good solutions to partitioning graph or mesh problems. These techniques either produce very good solutions in a long time or alternatively produce poor solutions in a short time. The HFA method attempts to find solutions with reasonable accuracy in a short period of time. Thus, MFA strikes a balance between computational time and quality of the resulting solution.

1 INTRODUCTION Combinatorial optimization problems arise in many areas of science and engineering. Unfortunately, due to the NP (non-polynomial) nature of these problems, the computations increase with the size of the problem. Most computational methods that have so far been developed which generally yield good solutions to these problems rely on some form of heuristic. Artificial neural networks (ANNs) make use of highly interconnected networks of simple neurons or processing units which may be programmed to find approximate solutions to these problems. They are also highly parallel systems and have significant potential for parallel hardware implementation. The origin of the optimization neural networks goes back to the work by Hopfield & Tank’ which was a formulation of the travelling salesman problem (TSP). The Hopfield network is a feedback-type of neural network where the output(s) from a processing unit is fed back as the input(s) of other units through their interconnections. This type of network structure is a nonlinear, continuous dynamic system. Figure 1 illustrates a typical feedback neural network. Following the poor performance of Hopfield networks in determining valid solutions to the TSP problem, there followed considerable research effort to improve the performance of this type of network and to find ways of applying it to other optimization problems. At about the same time of the emergency of Hopfield networks, a new optimization method called simulated annealingz,3 was researched and developed. This

2 MEAN FIELD ANNEALING The roots of MFA are in the domain of statistical physics which combines the features of Hopfield neural network and simulated annealing. MFA is a deterministic approach which essentially replaces the discrete degrees of freedom in a simulated annealing problem with their average values as computed by the mean field approximation. 2.1 Hopfield network NP problems may be mapped onto the Hopfield discrete state neural network using the energy function or Liapunov function (in statistical physics) or the 103

A. Bahreininejad, B. H. V. Topping, A. I. Khan

104

0 -

- 0

0-

0

0 0 0

Fig. 1. A simple representation of a feedback network. Hamiltonian (in statistical mechanics) which is defined as:

(1) where S represents the state of the network, Z is the bias, sN), N is the number of processing units, S=(q,sz,..., and tij represents the strength of the synaptic connection between the units. It is assumed that the tij matrix is symmetric and has no self-interaction (i.e. tji = 0). In order to move E(S) downwards on the energy landscape, the network state is modified asynchronously from an initial state by updating each processing unit according to the updating rule:

c 1 N

Si

=

sgn

C

tijSj

-

(2)

Zi

‘=I

where output of ith unit is fed to the input of the jth unit by the connection tijs The symmetry of the matrix tij with zero diagonal elements enables E(S) to decrease monotonically with the updating rule.

0 Fig. 3. A simplified illustration of magnetic material described by an Ising model.

In optimization problems, the concept is to associate the Liapunov function (1) with the problem’s objective function by setting the connection weights and input biases appropriately. Figure 2 shows a general topology of Hopfield network. 2.2 Mean field approximation There is a close similarity between Hopfield networks and some simple models of magnetic materials in statistical physics.8 A magnetic material may consist of a set of atomic magnets arranged in a regular lattice. The spin term represents the state of the atomic magnets which may point in various directions. Spin l/2 is a term used when spins can point in one of only two directions. This is represented in an Ising model using a variable si for which each spin may point towards the value 1 if the spin is arranged upward and -1 when it is pointing downward. Figure 3 shows a simplified version of a magnetic material described using an Ising model. In a problem with a large number of interacting spins the solution to the problem is not usually easy to find. This is becausethe evolution of a spin depends on a local field which involves the fluctuation of the spins themselves and therefore finding the exact solution may not be manageable. However, an approximation known as mean field theory may be carried out which consists of replacing the true fluctuation of spins by their average value, which is illustrated in Fig. 4. 2.3 Simulated annealing Simulated

Buffer

Fig. 2. A general architecture of Hopfield network.

annealing

is a probabilistic

hill-climbing

algorithm which attempts to search for the global minimum of the energy function. It carries out uphill moves in order to increase the probability of producing

105

Mesh partitioning using neural networks

and the multiple integrals may be determined using saddle-point expansion of E’ which involves the mean field approximation that is found in Ref. 10. The saddlepoint positions are given by:

Hence using eqns (6)-(g), at the saddle points, gives: wi - tanhui = 0

(9)

and aE(v) x+Ui=O I

Fig. 4. The MFT representationof the averageof all spins

thus combining eqns (l), (9) and (lo), and assuming Zi = 0, the MF equation is given by:

shownin Fig. 3. solutions with lower energy. The method carries out neighbourhood searchesin order to find new configurations using the Boltzmann distribution: e-wIT h(S) = z (3) where T is the temperature of the system and Z is the partition function of the form:

z=c(S)e-E(S)‘T However simulated annealing involves a stochastic search for generating new configurations. In order to reach good solutions, a large number of configurations may have to be searched, which involves the slow lowering of the temperature and therefore is a very CPU time-consuming process. 2.4 Mean field annealing network

Furthermore, the continuous variables, Vi are used as approximations to the discrete variables at a given temperature (i.e. wi x (sJT), thus the final value of vi approximates whether the value of Si is 1 or - 1. Figure 5 illustrates the relationship between the sigmoid tanh function and the variation of temperature change. Equation (11) is applied asynchronously. This is based on updating the value of only one vi at each time-step t + At. Unlike simulated annealing, which is a stochastic hill-climbing method and requires an annealing schedule, MFA is deterministic and an annealing schedule may not be necessary. 3 MEAN FIELD MESH PARTITIONING CONVENTIONAL METHOD

-

Mesh partitioning or domain decomposition, may be

The purpose of the MFA approach is to approximate the stochastic simulated annealing by the average of the stochastic variables with a set of deterministic equations. Peterson & Anderson’ showed that the discrete sum of the partition function (4) may be replaced by a series of multiple nested integrals over the continuous variables ui and 2rigiving: Z

=

C fi j=l

lrn --OO

dvj

Frn

&je-E'(v~u~T)

-i'x

(5)

where C is a constant and n J J refers to multiple integrals. E’ may be given in the form of: -4.0

E’(V, U, T) = E(V)/T + 2 ui’ui - log(coshuJ i=l

-3.0

-2.0

-I .o

0.0

I .o

2.0

3.0

4.0

5.0

(6) Fig. 5. The gain function for different temperatures.

106


carried out in order to split a computationally expensive finite element mesh into smaller subsections for parallel computational analysis.“‘12 Mesh generation becomes computationally expensive as the size of the domain increases, thus parallel finite element mesh generation enables the distribution of the domain for the remeshing procedure. Therefore a domain may be partitioned into equal sizes where each subdomain is then placed on a single processor and parallel remeshing is carried out. There is also the problem of memory capacity of the hardware for carrying out parallel computations. As the size of a domain increases,thus storing a complete mesh requiressignificant memory in the central processor(i.e. the ROOT processor in terms of parallel transputer-based computation), therefore an initial large mesh may have to be divided into several subsectionsor subdomains. Khan & Topping’ introduced a domain decomposition method for partitioning meshesusing a predictive backpropagation-based neural network model and a genetic algorithm (GA) optimization approach. The outline of the method which is called the subdomain generation method (SGM) may be summarized as follows: l

l

use a predictive backpropagation neural network to predict the number of triangular elements that are generated within each element of the coarse mesh after an adaptive unstructured remeshing has been carried out; employ a GA optimization-based procedure using an objective function z = JGI- [([CL1- lGRl)l - C, where G is the total number of elements in the final mesh, GL and GR are the number of elements in each bisection and C represents the interfacing edges.The values of GL and GR are provided by the back-propagation-based trained neural net which gives the predicted number of elements that may be generated within each element of the initial coarse mesh after remeshing is made.

The SGM, compared with other mesh-partitioning approaches,4’5 has been shown to be competitive in terms of parallel finite element computations, however the GA like simulated annealing may take a long time to reach a final solution. The MFA approach attempts to exchange the accuracy with execution speed considering the fact that in most cases the MFA method does produce competitive results. Furthermore, the MFA method is highly parallel and should greatly benefit from parallel hardware implementation. The partitioning of meshesin this paper is based on the recursive bisection carried out in Refs 4 & 7, which recursively bisects a mesh into n2 parts, where n is the number of subsections. 3.1 Mapping the problem onto the neural network The problem of bisecting meshesmay be mapped onto a feedback-type neural network and may be defined by an

objective function in the form of the Hopfield energy function given in eqn (1). This is carried out by assigning a binary unit of si = I or si = -1 to each element of the mesh defining which partition the element is to be assigned. The connectivity of the triangular elements is encoded in the fij matrix in the form of: t,, 1 if a pair of elements are connected by an edge I’ { 0 otherwise With this terminology the minimization of the first term of eqn (1) will maximize the connectivity within a bisection while minimizing the connectivity between the bisections. This has the effect of maximizing the boundary. However, using this term alone as the cost function forces all the elements to move into one bisection, thus a penalty term is applied which measures the violation of the equal sized bisection constraint.” Therefore the neural network Liapunov function for the mesh bisection is in the form of: 2

E(S) = - i $ $ tij, si, sj + F 2=I ]=I

(12)

where a is the imbalance parameter which controls the bisection ratio. The mean field equation for the mesh bisection is given by combining eqns (9), (10) and (12), which gives: ui=tanh($(tij-o):)

(13)

This equation is used to compute a new value of Vi asynchronously. Initial values for the vector V are assigned using small continuous random values. The temperature is lowered using a cooling factor (0.9) after one complete iteration of eqn (13). 3.2 Selection of parameters Phase transitions occur in materials as they cool or heat up and they change from one state to another. Sharp changes in the properties of a substance are often visible as the material changes from one state to another. The transition from liquid to gas, solid to liquid or from paramagnet to ferromagnetic are common examples. There is a critical temperature such as the melting or boiling point where the state of a system suddenly changes from a high to a lower energy state. Although there are some analytical methods for determining the critical temperature,8’9Y’3 it was decided, in this research,to determine an approximate value for the critical temperature by using the following procedure: (1) determine an initial bisection using an initial temperature value (for example T 2 2); (2) during the bisection, the changesin temperature and the corresponding spins (vi) values are recorded;

107


Fig. 6. The backgroundmeshbeforethe bisection.

(3) the iteration continues using a temperature reduction factor of 0.9. The programme is terminated either when the number of iterations exceeds 100 or when the system reaches a saturation state which is defined by the term: $ 5 0.999 (4) once the program has terminated, the temperature where a sudden change to the spins has occurred is identified; (5) the MFA bisection is then repeated by initializing the temperature with a value slightly higher than the temperature identified.

Fig. 7. The meshafter the bisection.

Fig. 8. The variation of spin averagesas the temperatureis

decreased. Figure 7 was the result of the bisection of the mesh shown in Fig. 6, where the initial temperature was chosen as 3. The total number of iterations for the bisection was 25. The effect of the decrease in temperature upon individual spins for this example is represented in Fig. 8. This figure shows that at high temperatures the spin average diverges to near zero for all the elements which indicates that the bisection is maximally disordered. As the temperature is decreased, the system reaches a critical temperature where each element starts to move significantly into one or another of two bisections. At sufficiently low temperatures the spins saturate at or near values of 1 or - 1. Therefore, if a net is initialized with a temperature just above or equal to the critical temperature, the fastest convergence to a good global minimum should occur. Thus a bisection for the mesh shown in Fig. 6 was carried out but this time with an initial temperature 0.8. The total number of iterations for this bisection was 11 which produced the same result shown in Fig. 7. The initial imbalance factor o is usually selectedas 1.0 for a balanced bisection. This value in most cases ensuresa balanced bisection but a minimum cutsize may not be produced. Therefore it was decided to carry out an inverse annealing or, in other words, incrementation of the initial value of (Y.This was carried out by selecting a small initial value for IY (for example, 50.1) and once the neural network optimization has started, CI is increased by a factor (for example, 1.5) after one complete iteration of eqn (13) until it reaches 1.0. The

108


value of cx is set at 1.0 for the rest of the optimization procedure. This dynamic implementation of (Y proved to be highly efficient in creating a good minimum bisection interfaces while producing two well-balanced subdomains. 4 MEAN FIELD MESH PARTITIONING PREDICTIVE METHOD

-

MFA mesh partitioning using a predictive neural network model differs from the conventional method described in Section 3. The aim of the new approach is to partition a coarse mesh on the basis of the predicted number of triangular elements which will be generated within each triangle of the coarse mesh after the remeshing. The predicted number of elements is given by a trained neural network based on the backpropagation (BP) algorithm.7’16 4.1 Back-propagation training of finite element meshes Back-propagation neural nets are generally based on multilayered networks which are used to establish a relationship between a set of input and output values. This relationship is stored in the form of a matrix of connecting weight values. Once a network has been trained, if presented with unfamiliar input, the network considers all the learned input-output patterns and checks the one which is most close to the given new input and generalizes the output. For a more detailed discussion concerning this type of network the reader may refer to Refs 14 & 15. The training of a BP network is such that once trained, it may be used to predict the number of elements that may be generated within an element of the coarse or initial mesh. Training is carried out in the following manner. In order to carry out the training several background coarse meshes were chosen and these meshes were analyzed using different point loads.7V16Input-output training data were created and applied in the training procedure which consisted of: l

l l

A Fig. 9. A squaredomain with in-plane load. data consisted of three side lengths, three internal angles and two scaled mesh parameters of each element. The output data consisted of the number of generated triangles in the refined mesh. A network was trained and once a desired trained network is achieved it may be used to predict the number of elements which may be generated within an element of a coarse mesh. 4.2 The predictive mean field Hamiltonian The original equation of the MFA mesh bisection (i.e. eqn 13) was modified in order to accommodate the predicted number of elements which may be generated within an element of the coarse mesh. This equation was thus modified and the MFA bisection equation for using

the data regarding the geometry of the individual elements the nodal mesh parameters number of elements generated in the refined adaptive mesh.

The geometry of a triangular element was represented by the side lengths and the three internal angles. It was noted that the geometry of a triangle may be defined by the length of its sides, thus the three internal angles of the triangular elements need not have been included in the training datafile.i6 The three nodal mesh parameters actually represent the size of the triangle to be generated and they were scaled down to two. Therefore, the input

Fig. 10. The initial meshwith 46 elements.


109

Table 1. Comparison of the actual mmher of generatedelements per subdomain and the ideal number of elementsper subdomain usingtbeSGM Required Generated Subdomain elements (actual) elements no. 1 2 3 4

99 108 97 108

103 103 103 103

Diff.

%age diff.

-4 5 -6 5

-3.88 4.85 -5825 4.85

Table 2. Comparison of the actual number of generatedelements per sobdomain and the ideal number of elementsper subdomain using the MFA method Generated Subdomain no. elements (actual) 99 108 97 108

1 2 3 4

Required elements

Diff.

%age diff.

103 103 103 103

-4 5 -6 5

-3.88 4.85 -5.825 4.85

Table 3. Comparison betweentbe SGM and tbe MFA method in terms of rontimes and tbe total number of interfacing nodesafter tbe partitioning of example 1 Partitioning method

Interfacing nodes 62 62

SGM MFA

Time (min) 1.067 0.05

Fig. 12. The remeshed subdomains with 412 elements partitioned by the SGM.

This equation was applied iteratively using the same procedure as the non-predictive asynchronous method described earlier by initializing Uj with small continuous random values. The temperature was lowered by a cooling factor (0.9) after each complete iteration eqn (15).

of

During this optimization there is a strong competition between the cutsize term based on the connectivity of the initial mesh and the imbalance term for the bisections using the number of predicted elements. This makes the selection of the initial parameters more difficult.

The selection of the initial temperature for the predictive mesh bisection may not be as straightforward

Fig. 11. The initial mesh with 46 elements divided into four partitions using the SGM.

the predicted number of elements is given by: vi = tanh \j=l

where Npred represents the predicted number of elements.

Fig. 13. The initial mesh with 46 elements divided into four partitions using the MFA method.

110


Fig. 14. The remeshed subdomains with 412 elements partitioned by the MFA method.

Fig. 16. The initial mesh with 126 elements. Table 4. Comparison of the actual number of generatedelements per subdomain and the ideal number of elementsper subdomain using the SCM

.

Subdomain Generated Required no. elements (actual) elements

Diff.

%age diff.

1 2 3 4

-9.75 -1.75 -3.75 15.25

-5.78 -1.03 -2.22 9-03

159 167 165 184

168.75 168.75 168.75 168.75

Table 5. Comparison of the actual number of generatedelements per suBdomainand the ideal number of elementsper subdomain using the MFA method Subdomain Generated elements (actual) no.

A

kr

Fig. 15. An L-shaped domain with in-plane load.

as the conventional method (without considering the predictive aspect). The method for choosing the initial temperature, which was described for the conventional bisection method, may be carried out as a benchmark for the predictive method, however a few temperature and Q values may have to be tested on a trial-and-error basis. Experienced users will find it easier to estimate from experience close initial values for these parameters. 5 EXAMPLE STUDIES Three examples have been presented as a test-bed for comparative studies. In these examples the performance of the MFA mesh-partitioning method has been compared with the original SGM. The SGM has been

1 2 3 4

152 184 177 167

Required elements

Diff.

%age diff.

168.75 - 16.75 -9.92 168.75 15.25 9.03 168.75 8.25 4.88 168.75 -1.75 -1.03

Table 6. Comparison betweenthe SGM and the MFA method ia terms of nmtimes and the total number of interfacing nodesafter the partitioning of example 2 Partitioning method SGM MFA

Interfacing nodes

Time (min)

76 84

3.2 0.233

compared with two other domain decomposition methods for which the reader may refer to Refs 7 & 17. 5.1 Example 1 In this example a square-shapeddomain shown in Fig. 9 was uniformly meshed and the result was an initial mesh


Fig. 17. The initial meshwith 126elementsdivided into four

Fig. 19. The initial meshwith 126elementsdivided into four

partitions using the SGM.

partitions using the MFA method.

with 46 elements, which is shown in Fig. 10. This initial mesh was then decomposed using the SGM and the MFA method. Tables 1 and 2 show the elements generated within each subdomain and the corresponding ideal number of elements required per subdomain. Table 3 gives a comparison between the computation time and the total number of interfacing nodes for each method. Figures 11 and 13 show the partitioning of the initial mesh by the SGM and the MFA method, respectively. Figure 12 and 14 show the remeshedsubdomains after

the partitioning by the SGM and MFA method, respectively. The partitions generated by the methods were identical however the MFA method was considerably faster.

Fig. 18. The remeshed subdomains with 666 elements partitioned by the SGM.

5.2 Example 2 This example is an L-shaped domain shown in Fig. 15, which is uniformly meshed, and the result is an initial coarse mesh with 126 elements, which is shown in Fig. 16. This initial mesh was then decomposed using the SGM and the MFA method.

Fig. 20. The remeshed subdomains with 666 elements

partitioned by the MFA method.

112

A. Bahreininejad, B. H. V. Topping, A. I. Khan A

Fig. 21. A domain with cut-out and chamfer.

Fig. 22. The initial mesh with 153 elements.

Fig. 23. The initial mesh with 153 elements divided into eight partitions using the SGM.

Tables 4 and 5 shows the number of elements generated within each subdomain and the ideal number of elements required per subdomain. Table 6 shows the comparison between the computation time and the total number of interfacing nodes for both methods.

Figures 17 and 19 show the partitioning of the mesh by the SGM and the MFA method, respectively. Figures 18 and 20 illustrate the remeshed subdomains after the partitioning by the SGM and the MFA method, respectively. initial

113

Mesh partitioning using neural networks Figures 18 and 20 illustrate the remeshed subdomains after the partitioning by the SGM and the MFA, method respectively. The partitions for this example were not identical for each method but they were of the sameorder of accuracy

with respectto the partition sixes(0.0%) and the number of interface nodes (105%). The MFA method took only 7.3% of the computational time of the SGM.

Table 7. Comparison of the aedal number of generatedelements per subdomain and the ided number of elementsper subdomain usingtheSGM

The domain shown in Fig. 21 was selected for the final example study and is uniformly meshed. This provided an initial mesh with 153 elements and is shown in Fig. 22. This initial mesh was then decomposed using both the SGM and the MFA method. Tables 7 and 8 show the elements generated within each subdomain and the ideal number of elements which is desired per subdomain. Table 9 shows a comparison between the computation time and the total number of interfacing nodes for each method. Figures 23 and 25 show the partitioning of the initial mesh by the SGM and the MFA method, respectively. Figures 24 and 26 show the remeshed subdomains after the partitioning by the SGM and the MFA method, respectively. The partitions generated by the methods were different. The maximum positive imbalance in the mesh partitions was 3.75 and 9.21% for the SGM and the MFA method, respectively. There was a 50% difference in the number of interfacing nodes in favour of the SGM. The MFA method took less than 8% of the computational time of the SGM.

Required Diff. %agediff. Subdomain Generated elements(actual) elements no. 1 128 146.5 -18.5 -12.62 2 150 146.5 3.5 2.39 3 150 146.5 3.5 2.39 4 5 6 7 8

144 152

146.5 146.5

149

146.5

150 147

146.5 146.5

-2.5 5.5 2.5 3.5 0.5

-1.707 3.75 1.707

2.39 0.341

Table 8. Comparison of the actual number of generatedelements per s&domain and the ided mm&r of elementsper subdomain USillgtkMFAIldlOd

Generated Required Diff. %agediff. Subdomain no. elements(actual) elements 1 136 146.5 -10.5 -7.16 2 160 146.5 13.5 9.21 3 4 5 6 7 8

151 157 146 138 141 137

146.5 146.5 146.5 146.5 146.5 146.5

4.5 10.5 -0.5 -8.5 -5.5

-9.5

3.07 7.16 -0.34 -5.80 -3.75 -6.48

Table 9. Comparison betweenthe SGM and the MFA method in terms of runtimes and the total number of interfacing nodesafter tbe partitioning of example 3

Partitioning method SGM MFA

Interfacing nodes 179 188

Time (min) 4.267 0.333

5.3 Example 3

6 CONCLUDING

REMARKS

From the examples presented it is clear that partitioning of the initial coarse mesh using the MFA method may be achieved with much less computational effort. The number of interacting nodes and the number of elements generated per subdomain after the remeshing of each partition produced by the MFA partitioning method are competitive with those produced by the SGM.

Fig. 24. The remeshedsubdomainswith 1172elementspartitioned by the SGM.

114


Fig. 25. The initial meshwith 153elementsdivided into eight partitions using the MFA method.

Fig. 26. The remeshedsubdomainswith 1172elementspartitioned by the MFA method. This paper has demonstrated the efficient use of neural networks in the partitioning of finite element meshes. The method is so efficient it appears apparent that it might be applied directly to the refined meshes without using the predictive model. The partitioning using the predictive neural network model has more complex energy criteria which may consist of many local minima. Perhaps the only drawback of using the predictive MFA partitioning method is the high degree of parameter sensitivity (i.e. temperature and a). In general, good partitions may be generated for both conventional and predictive MFA partitioning with little computational expense. The method also has a high potential for parallel implementation which would increase the performance of the network in terms of convergence.

ACKNOWLEDGEMENTS

acknowledge the useful discussions with other members of the Structural Engineering Computational Technology Research Group (SECT) in the Department of Mechanical and Chemical Engineering, at Heriot-Watt University. In particular Janos Sziveri, Janet Wilson, Biao Cheng, Joao Leite and Jsrgen Stang. We are also grateful for the contact with Mattias Ohlsson, University of Lund, Sweden; Arun Jagota, State University of New York at Buffalo, NY; and Tal Grossman, Los Alamos National Laboratory, TX. We are grateful for their response over the Internet.

REFERENCES 1. Hopfield, J. J. & Tank, D. W. Neural computation of decisions in optimization problems. Biol. Cybernetics, 1985,52, 141-152. 2. Kirkpatrick, S., Gelatt Jr, C. D. & Vecchi, M. P. Optimization by simulated annealing. Science, 1983, 2204598, 671-680.

The research described in this paper was supported by Marine Technology Directorate Limited (MTD) research grant no. SERC/GR/J33191. The authors wish to thank MTD for their support of this and other research work. The authors would like to

3.

Topping, B. H. V., Khan, A. I. & de Barros Leite, J. P. Topological design of truss structures using simulated annealing.In Neural Networks and Combinatorial Optimization in Civil and Structural Engineering, eds B. H. V. Topping & A. I. Khan. Civil-Comp Press,Edinburgh, 1993,151-165.

Mesh partitioning 4. Simon, H. D. Partitioning of unstructured problems for parallel processing. Comput. Syst. Engng, 1991, 2(2/3), 135-138. 5. Farhat, C. A simple and efficient automatic FEM domain decomposer. Comput. Struct., 1988, 28(5), 579-602. 6. Kemighan, B. W. & Lin, S. An efficient heuristic procedure for partitioning graphs. Bell Syst. Tech. J., 1970,49, 291-307. 7. Khan, A. I. & Topping, B. H. V. Subdomain generation for parallel finite element analysis. Comput. Syst. Engng, 1993, 4(4-6), 473-488. 8. Hertz, J., Krogh, A. & Palmer, R. G. Introduction to the Theory of Neural Computing. Addison-Wesley, Reading, MA, 1991. 9. Peterson, C. & Anderson, J. R. A mean field learning algorithm for neural networks. Complex Syst., 1987, 1, 995-1019. 10. Peterson, C. & Anderson, J. R. Neural networks and NPcomplete optimization problems; a performance study on the graph bisection problem. Complex Syst., 1988, 2(l), 59-89. 11. Khan, AI. & Topping, B. H. V. Parallel adaptive mesh generation. Comput. Syst. Engng, 1991, 2(l), 75-102.

using neural networks

115

12. Topping, B. H. V. & Khan, A. I. Parallel Finite Element Computations. Saxe-Coburg, Edinburgh, 1996. 13. Van den Bout, D. E. & Miller III, T. K. Graph partitioning using annealing neural networks. IEEE Trans. Neural Networks, 1990, l(2), 192-203. 14. Rumelhart, D. E., Hinton, G. E. & Williams, R. J. Learning internal representation by error propagation. In Parallel Distributed Processing:Explorations in the Microstructure of Cognition, Vol. I: Foundations, eds D. E. Rumelhart & J. L. McClelland. MIT Press,Boston, MA, 1986. 15. Beale, R. & Jackson, T. Neural Networks - An Zntroduction. IOP, Bristol, UK, 1990. 16. Khan, A. I., Topping, B. H. V. & Bahreininejad, A. Parallel training of neural networks for finite element mesh generation. In Neural Networks and Combinatorial Optimization in Civil and Structural Engineering, eds B. H. V. Topping & A. I. Khan. Civil-Comp Press, Edinburgh, 1993, pp. 81-94. 17. Topping, B. H. V., Khan, A. I. & Wilson, J. K. Parallel dynamic relaxation and domain decomposition. In Parallel and Vector Processingfor Structural Mechanics, eds B. H. V. Topping & M. Papadrakakis. Civil-Comp Press, Edinburgh. 1994.