finite element mesh partitioning using neural networks 1996 advances in engineering software

8/19/2019 Finite Element Mesh Partitioning Using Neural Networks 1996 Advances in Engineering Software

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ELSEVIER

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

SO965-9978(96)00011-7

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 meshes s 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

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 inter-

connected 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 net-

works 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

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.

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 determi-

nistic 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


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

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,

S=(q,sz,...,

sN), N is the number of processing units,

and tij represents he 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:

N

Si = sgn

c 1

tijSj - Zi

‘=I

(2)

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.

Buffer

Fig. 2. A general architecture of Hopfield network.

-

0

-

0

0

0

0

0

0

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 he 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 because he 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 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


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Mesh partitioning using neural networks

105

Fig. 4. The MFT representationof the averageof all spins

shown n Fig. 3.

solutions with lower energy. The method carries out

neighbourhood searches n order to find new configura-

tions 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 e-E(S)‘T

(S)

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

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 2ri giving:

Z = C fi lrn dvj Frn &je-E'(v~u~T)

j=l --OO

-i'x

(5)

where C is a constant and n J J refers to multiple

integrals. E’ may be given in the form of:

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

(6)

i=l

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 saddle-

point 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

thus combining eqns (l), (9) and (lo), and assuming

Zi = 0, the MF equation is given by:

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 s 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 anneal-

ing 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

-4.0 -3.0

-2.0

-I .o 0.0 I .o 2.0

3.0 4.0 5.0

Fig. 5. The gain function for different temperatures.


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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, hus storing a complete mesh

requires significant 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 back-

propagation-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 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;

l employ a GA optimization-based procedure using

an objective function z = JGI [([CL1 - lGRl)l - C,

where G is the total number o f elements n 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 meshes n this paper is based on

the recursive bisection carried out in Refs 4 & 7, which

recursively b isects 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 s 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:

E(S) = - i tij, si, sj + F

2

(12)

2=I ]=I

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 n 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’3t was decided,

in this research, o 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 changes n temperature and

the corresponding spins (vi) values are recorded;


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107

Fig. 6. The backgroundmeshbefore he 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 averages s the temperature s

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 e lements

which indicates that the bisection is maximally dis-

ordered. 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

ensures a 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


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value of cx s 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 back-

propagation (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.7V16nput-output

training data were created and applied in the training

procedure which consisted of:

l

the data regarding the geometry of the individual

elements

l the nodal mesh parameters

l 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 o f 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

A

Fig. 9. A squaredomain with in-plane oad.

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

Fig. 10. The initial meshwith 46 elements.


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Table 1 . Comparison of the actual mmher o f generatedelements

per subdomain and the ideal number of elementsper subdomain

usingtbeSGM

Subdomain

Generated

Required Diff. %age diff.

no.

elements (actual) elements

1 99 103 -4 -3.88

2

108 103

5 4.85

3

97

103

-6 -5825

4 108 103

5 4.85

Table 2 . Comparison of the actual number of generatedelements

per sobdomain and the ideal number of elementsper subdomain

using the MFA method

Subdomain

Generated Required D iff. %age diff.

no. elements (actual) elements

1

99 103

-4 -3.88

2

108 103

5 4.85

3 97 103 -6 -5.825

4

108 103

5 4.85

Table 3. Comparison between be SGM and tbe MFA method n

terms of rontimes and tbe total number of interfacing nodesafter

tbe partitioning of example 1

Partitioning method Interfacing nodes

Time (min)

SGM

62

1.067

MFA 62 0.05

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

Fig. 13. The initial mesh with 46 elements divided into four

elements.

partitions using the MFA method.

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 of

eqn (15).

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


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partitioned by the MFA method.

.

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

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 Diff. %age diff.

no. elements (actual) elements

1

159 168.75 -9.75

-5.78

2

167 168.75 -1.75

-1.03

3 165 168.75 -3.75 -2.22

4

184 168.75 15.25

9-03

Table 5. Compar ison of the actual number of generatedelements

per suBdomainand the ideal number of elementsper subdomain

using the MFA method

Subdomain

Generated Required Diff. %age diff.

no.

elements (actual) elements

1

152 168.75 - 16.75

-9.92

2

184 168.75 15.25

9.03

3

177 168.75 8.25

4.88

4

167 168.75 -1.75

-1.03

Table 6. Comparison between he SGM and the

MFA

method a

terms of nmtimes and the total number of interfacing nodesafter

the partitioning of example 2


Time (min)

SGM

76

3.2

MFA

84

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-shaped domain shown in Fig. 9

was uniformly meshed and the result was an initial mesh


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Fig. 17. The initial meshwith 126elements ivided into four Fig. 19. The initial meshwith 126elements ivided 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.

the partitioning by the SGM and MFA method,

respectively.

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.

The partitions generated by the methods were identical

however the MFA method was considerably faster.

5.2 Example 2

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 remeshed subdomains after

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.


partitioned by the SGM.

Fig. 20. The remeshed subdomains with 666 e lements

partitioned by the MFA method.


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112


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

Figures 17 and 19 show the partitioning of the

generated within each subdomain and the ideal

initial

mesh by the SGM and the MFA method,

number of elements required per subdomain.

respectively.

Table 6 shows the comparison between the com-

Figures 18 and 20 illustrate the remeshed subdomains

putation time and the total number of interfacing nodes

after the partitioning by the SGM and the MFA

for both methods.

method, respectively.


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113

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 same order of accuracy

Table 7. Comparison of the aedal number of generatedelements

per subdomain and the ided number of elementsper subdomain

usingtheSGM

Subdomain Generated

Required Diff. %agediff.

no.

elements actual) elements

1 128 146.5 -18.5 -12.62

2 150 146.5 3.5 2.39

3 150 146.5 3.5 2.39

4 144 146.5 -2.5 -1.707

5 152 146.5 5.5 3.75

6 149 146.5 2.5 1.707

7 150 146.5 3.5 2.39

8 147 146.5 0.5 0.341

Table 8. Comparison of the actual number of generatedelements

per s&domain and the ided mm&r of elementsper subdomain

USillgtkMFAIldlOd

Subdomain

Generated Required Diff. %agediff.

no. elements actual) elements

1 136 146.5 -10.5 -7.16

2 160 146.5 13.5 9.21

3 151 146.5 4.5 3.07

4 157 146.5 10.5 7.16

5 146 146.5 -0.5 -0.34

6 138 146.5 -8.5 -5.80

7 141 146.5 -5.5 -3.75

8 137 146.5 -9.5 -6.48

Table 9. Comparison between he SGM and the MFA method n

terms of runtimes and the total number of interfacing nodesafter

tbe partitioning of example 3


SGM 179

MFA 188

Time (min)

4.267

0.333

with respect o 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.

5.3 Example 3

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.

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 remeshed ubdomainswith 1172elements artitioned by the SGM.


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114


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

Fig. 26. The remeshed ubdomainswith 1172elements artitioned 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

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

acknowledge the useful discussions with other members

of the Structural Engineering Computational Technol-

ogy 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, Uni-

versity 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.

3.

Topping, B. H. V., Khan, A. I. & de Barros Leite, J. P.Topological design of truss structures using simulated

annealing. n Neural Networks and Combinatorial Optimi-

zation in Civil and Structural Engineering, eds B. H. V.

Topping & A. I. Khan. Civil-Comp Press,Edinburgh,

1993,151-165.


13/13


115

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 NP-

complete 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.

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, (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 Zntroduc-

tion. 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 Processing or Structural Mechanics,

eds B. H. V. Topping & M. Papadrakakis. Civil-Comp

Press, Edinburgh. 1994.

finite element mesh partitioning using neural networks 1996 advances in engineering software

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