Engineering Optimization: Theory and Practice, Fourth Edition

730 Modern Methods of Optimization

q 3

r 2 r 3 r 4 q 2 w 2

q 4 w 3 w 4 g h

Figure 13.12 Network used to train relationships for a four-bar mechanism. [12.23], reprinted

with permission of Gordon & Breach Science Publishers.

advantage(η)of the mechanism. The network is trained by inputting several possible

combinations of the values ofr 2 , r 3 , r 4 , θ 2 , andω 2 and supplying the corresponding

values ofθ 3 , θ 4 , ω 3 , ω 4 , γ and, η. The difference between the values predicted by the

network and the actual output is used to adjust the various interconnection weights

such that the mean-squared error at the output nodes is minimized. Once trained, the

network provides a rapid and efficient scheme that maps the input into the desired

output of the four-bar mechanism. It is to be noted that the explicit equations relating

r 2 , r 3 , r 4 , θ 2 , andω 2 and the output quantitiesθ 3 , θ 4 , ω 3 , ω 4 , γ and, ηhave not been

programmed into the network; rather, the network learns these relationships during the

training process by adjusting the weights associated with the various interconnections.

The same approach can be used for other mechanical and structural analyses that might

require a finite-element-based computations.

Numerical Results.The minimization of the structural weight of the three-bar

truss described in Section 7.22.1 (Fig. 7.21) was considered with constraints on

the cross-sectional areas and stresses in the members. Two load conditions were

considered withP= 20 ,000 lb,E= 10 × 106 psi,ρ= 0. 1 lb/in^3 , H= 1 00 in.,σmin=

− 15 ,000 psi,σmax= 02 ,000 psi,Ai(l) = 0. 1 in^2 (i = 1 , 2 ),andAi(u)= 5. 0 in^2 (i= 1 , 2 ).

The solution obtained using neural-network-based optimization is [12.23]:

x 1 ∗ = 0. 7 88 in^2 , x∗ 2 = 0. 4 079 in^2 , and f∗= 62 .3716 lb. This can be compared

with the solution given by nonlinear programming:x∗ 1 = 0. 7 745 in^2 , x 2 ∗= 0. 4 499 in^2 ,

andf∗= 62 .4051 lb.

References and Bibliography

13.1 J. H. Holland,Adaptation in Natural and Artificial Systems, University of Michigan

Press, Ann Arbor, MI, 1975.

13.2 I. Rechenberg,Cybernetic Solution Path of an Experimental Problem, Library Transla-

tion 1122, Royal Aircraft Establishment, Farnborough, Hampshire, UK, 1965.

13.3 D. E. Goldberg,Genetic Algorithms in Search, Optimization, and Machine Linearning,

Addison-Wesley, Reading, MA, 1989.