Hopfield Network Based Optimization of

Mechanical Design

Ping Wang Hong-iSchool ofAutomation School ofA

Chongqing University of Univ. ofEPosts & Telecommun. Tec]

Chongqing 400065, China ChengduE-mail: wangping(cqupt.edu.cn E-mail: hzhi

Abstract-Most of recent work related to using Hopfieldnetwork to optimization problems focuses on its capabilities insolving linear programming, combinatorial optimization, andso on. This paper discusses the application of Hopfield networkto optimal design of mechanical systems, exploring Hopfieldnetwork's advantages in constrained nonlinear programminginvolved in mechanical design. An example is given to imustratethe efficiency of Hopfield network in solving problems ofmechanical optimal design.

I. INTRODUCTION

As a fully-connected neural network, Hopfield networkplays an important role in evoking a future expectation andexpanding the research fields in the process of developmentof neural network. This network and its learning algorithmwere developed first by J. J. Hopfield in 1982. It was usedby J. J. Hopfield and D. W. Tank in 1985 to solve theTSP[1]. The significant achievements gained by Hopfield interms of neural network optimization not only start the newphase of the applications of neural network in associativememory and optimization calculation, but also tap thepotentials of neural network, which make researches onneural network enter into a fresh and advanced stage.Presently, neural network, which is applied primarily to thefield of mechanical design, is used widely to solving linearprogramming, combinatorial optimization, quadraticprogramming, and so on[2-6].

II. INTRODUCTION THE BASIC PRINCIPLE OF HOPFIELD NEURALNETWORK

Figure 1 shows the topological structure of Hopfieldnetwork, where Ii represents the input signal of nerve-cell i,and Vi represents its steady-state output. Because eachneuron is connected with others by feedbacks, Hopfieldnetwork is a kind of fully-connected feedback network.Hopfield networks are classified into two types: discretetype and continuous type.

Zhong Huang Xu Zhang4echatronics Engn School ofMechanical Engnlectronic Sci. and Dalian University ofIh.of China Technologyt, 610054, China Dalian 116023, Chinauang(uestc.edu.cn E-mail: zhangxdlut(l63.com

Fig. 1. Topological Structure ofHopfield Network

A. Discrete HopfieldNetworkIn discrete Hopfield network, the state of each neuron

takes discrete values, 0 or 1. Suppose that vi (o) representsthe output of neuron i when t=0, then the output signalIi = Vi (O). The topological structure of i-th neuron is shownin Figure 2, which is a multi-input single-output non-lineardevice. Suppose that Vi (t) is the input ofneuron i at time t,and Vi (t + 1) represents the output at time t+l. Therelationship between input and output takes the form

Vi (t +1) = f(E Wij Vj (t) i )

where

0 u<O

(1)

(2)

where Wy is the weight between neuron i and neuron j,and Hi is the threshold ofneuron i.

0-7803-9422-4/05/$20.00 C2005 IEEE1766

we defimeTi = I/Rij

I/Ri = 1/pi + E(/RJ)j=l,j#ei

VI(t)

VAOt Wjt+1 )

V1(t)

Fig. 2. i-th Neuron

It can be proved that when the network meetssymmetrical condition (Wy = Wj ) and has no self-feedback

(W,,= 0), and the state of network has no asynchronousreplacement, the numeration energy function in thetransformation process of the state of network ismonotonically decreasing, which is described as follows[71E

n N n (3)E=-2Y E WUjV'Vj-EOiVii=2 j=l,j*i i=l

A. Continuous Hopfield Networkl8'Note that you need to use nsubsection. Subsection text

goes here, if applicable.This type of network is continuous at the time axis, so

each neuron works in synchronism. The structure of i-thneuron is shown in Figure 3. Suppose that ui represents theinput voltage of neuron i and Vi is its output voltage, the i-thneuron's relationship between input and output can bedescribed as follows

Cdiuu Z 1= _(Vj-ui)+Ii (4)dt Pi i=l j*i Ri

where Ci is the input capacitance of i-th neuron, pirepresents its output resistance, Ii is offset current and Ryrepresents the connective resistance between neuron i andneuron].

Eq. (4) is transformed into the following formdu. n V (Pi in

Iivi

Pi

Fig. 3. Structure of i-th Neuron

wheref(u) = 1/(1+ e-au) (7)

where a is a constant. Based on the above circuit, theenergy function can be defined as

i n N n n IE=-Z- T,j IVi - VIIi+E : f -1 (xd (8)

i=a j=i l i=f i

Aiming at the ideal amplifier, Eq. (8) is simplified as

i n N n

E= --Z-2IETyiij - ViIiI i=l j=l i=l

(9)

It can be proved by strict mathematical deduction thatthese two types of Hopfield network models are equivalent,and their energy functions are monotonically decreasing[81.With the time passing, the network will converge at a stablebalance point, which is the minimum value of energyfunction.

III. APPLICATION OF HOPFIELD NETWORK TO MECHANICALOPTIMAIL DESIGN

In terms of solving optimization problem by usingHopfield network, if the objective function of solvingproblem can be transformed into the energy fimction of thenetwork, in other words, if the problem's variables matchthe state of network units, the optimization solution can beobtained when the energy function of the network convergesat the minimum value.

1200mm

C 2 A

\1.96x105N

00 ~ 60m

Fig. 4. Four-bar Truss

In practical situations, most of mechanical systems or

components optimization are constrained optimization, thus

1767

(6)

we only study the application of Hopfield network tomechanical constrained optimization in this paper. As shownin Figure 4, the four-bar truss bears the vertical load1.96x105N at point A. We need to determine each bar'scross-sectional area so as to minimize the weight of thefour-bar truss system, which the vertical displacement ofpoint A is 5mm[91. The material densityp =l.OxlO4kg/m3 , and Young's modulusE = 196GN/m2 .Suppose that xi (i= 1, 2, 3, 4) represents i-th bar's

cross-sectional area, this optimization problem can bedescribed as

X= [XI,X2,X3,X4]Tminf(X)=x1 +1.2X2 +X3 +0.6x41.5625 0.6750 1.5625 1.3500s.t. + + + =0.5

x1 X2 X3 X4X1 20,X2 >O,X3 20,x4 .0

Introducing penalty function into the above equation anddefining x. (i= 1, 2, 3, 4), the above constraint

IVioptimization problem can be transformed into anon-constraint one, which takes the form

V =[VV2 V3 IV4]Tmm~..v, rk = +1.2 1 +.

(VI V2 V3 V4 )10(V, +V2+V3+V4)r +(1.5625V,+ (10)0.675V2 +1.5625V3 +1.35V4 -0.5)2/+f[

When minimizing 'D by using the penalty factor, rk, whichdecreases step-by-step, the minimum of the aboveunrestricted optimization problem will converge at thesolution of original problem, X*.

According to the above descriptions, objective functionshould be transferred into energy function of network firstlywhen using neural network to search the solution. Theenergy function of the Hopfield network with 4 neurons isdescribed as

4 4 4

-2 jI= j=l i=4We define

Tl= = - 4.8828/~rkT12 =T23 =-2.1094/~fkT22 = - 0.9112/1FT44 = - 3.645/.~F

To = - 4.8828/grkT14 =T34 =-4.2188/.rkT24 = -1.8225/frk

(1 1)

1 0.0625 1.5625V2 V1rEI1 -+-+ rk--k

(1.2 0.0625 0.675H + +rk

L 1 0.0625 1.5625tV3 V3 k

L0.6 0.0625 1.35

V24 Vf k

Apparently, Eq. (10) is equivalent to Eq. (11). The statefunction ofneuron is as follow

C dt = -(4.8828V, + 2.1094V2 +4.88281V3

+4.2188V47+I,-I 1

d2 = -(2.1094V o+ 0.9112V2+2.1094V3dt

+ 1.8225V4)/#[+ 22

C3 d3 -(4.8828V, +2.1094V2 +4.8828V3dt

+ 4.2188V4 )/F+ I3 -__J?3

C4 d -(4.2188V1+1.8225V2+ 4.2188V3

+ 3.6450V4)/$ +1 U4

where C1 = C2 = C3 = C4 = 100

R-UIVI2X2=u2 J/1.2, R3u3V32, R4 =u4V/0.6.Active function adopts a S type function, and takes the

form

Vi= u)= l+e351 , (i = 1,2,3,4)The initial state of neuron is defined as a random number

which is uniformly distributed in the region -0.5-0. 7.The result of simulation calculation indicates that when rk

becomes small enough, an approximate optimal solution,x of the above problem can be obtained by this neuralnetwork based optimization method, which is shown inTable 1. In order to explain the availability of thisoptimization method, Table 1 also gives the optimalsolutions, YX and XY obtained by penalty functionmethod and dynamic programming method. Comparingwith the solutions obtained by penalty function method, therelative error of this method is only 0.0008.

1768

TABLE I

THE OPTIMAL SOLUTION OF FOUR-BAR TRUSS

i 1 2 3 4 f(X)

X; 10.73 6.76 10.62 12.58 37.01

X; 10.98. 6.46 10.63 12.68 36.97

X; 10.76 6.47 10.73 12.88 36.98

[5] X. G. Shen, T. Q. Zhang, and X. J. Shen, "Application of neuralnetwork to machine element design," Chinese Journal of MechanicalEngineering, vol. 28 pp. 82-85, 1992.

[6] J. G.Lu, J Yu, and H. Wang, "Application of neural network to FEM,"Science in China, vol. 24 pp. 653-658, 1994.

[7] Z. Q. Zhuang, X. W. Wang, and D. S. Wang, Neural Network andNeural Computer. Beijing: Science Press, 1994.

[8] L. C. Jiao, Neural Network System Theory, Xi'an: Xiduan UniversityPress, 1992.

[9] S. S. Rao, Optimization Theory and Application (Second Edition), NewYork: Wiley Eastem Limited, 1984.

IV. CONCLUSIONS

Comparing with traditional optimization method ofmechanical design, the Hopfield network optimizationmethod has the following advantages:

1) Stability: We need to make the objective functions ofoptimization problem match certain energy functions ofHopfield network by using Hopfield network to solveoptimization problem. Due to the transformation of Hopfieldnetwork is the process of which energy function ismonotone decreasing, objective function can converge at theminimum when the network becomes stable status..

2) Availability: Hopfield network can not only processinformation with parallel computation mode, but also can beimplemented by hardware. This method has a potentialavailability to solve large-scale and complex optimizationproblems ofmechanical system.

3) Global Convergence: Generally speaking, only localoptimum of nonconvex programming problems can beobtained by using conventional optimization methods, suchas Powell methods, conjugate gradient method and dynamicprogramming method etc. Combining with simulatedannealing algorithms, neural network can be used to solvingthis kind of problems, and the global optimal solution maybe obtained.

ACKNOWLEDGMENT

This research was partially supported by the NationalExcellent Doctoral Dissertation Special Foundation of Chinaunder the contract number 200232 and the Civic NaturalScience Foundation of Chongqing, China.

REFERENCES

[1] P. Jin, J. B. Han, and Y. D. Tan, Neural Network andNeural Computer.Chengdu: Southwest Jiaotong University Press, 1991.

[2] H. Z. Huang, and W. P. Huang, "Neural network and application tomechanical engineering," Mechanical Science and Technology, vol.4,pp. 97-103, 1995.

[3] H. Z. Huang, and W. P. Huang, "An approach to the weights analysis offeedforward neural network and its application to the analysis ofmechanical structures," Mechanical Science and Technology, vol. 15,pp. 855-858, 1996.

[4] H. Z. Huang, and W. P. Huang, "Decomposition method ofoptimization of mechanical system based on neural network," ChineseJournal ofMechanical Engineering, vol. 33, pp. 31-37, 1997.

1769