Engineering Optimization: Theory and Practice, Fourth Edition

760 Practical Aspects of Optimization

Findywhich minimizes

f= 76 , 500 z∗ 1

√

y^2 + 63 + 76 , 500 z∗ 2

√

y^2 + 1

= 109. 2857

y^2 + 63

y

+ 655. 7143

y^2 + 1

y

(E 10 )

subjectto

1 ≤y≤6 andf must be defined

The graph off, given by Eq. (E 10 ), is shown in Fig. 14.7 over the range 1≤y≤ 6

from which the solution can be determined asf∗= 7473 .7 N, y∗=h∗= 2. 4 5 m, z∗ 1 =

A

∗

1 =^3.^7790 ×^10

− (^3) m (^2) , andz∗
2 =A
∗
2 =^9.^2579 ×^10
− (^3) m (^2).

14.9 Parallel Processing

Large-scale optimization problems can be solved efficiently using parallel computers.

Parallel computers are simply multiple processing units combined in an organized

fashion such that multiple independent computations for the same problem could be

performed simultaneously or concurrently, thereby increasing the overall computational

speed. Optimization problems involving extensive analysis, such as a finite-element

analysis, can be solved on parallel computers using the following schemes:

1.A multilevel (decomposition) approach with the subproblems solved in parallel

2.A substructures approach with substructure analyses performed in parallel

3.By implementing the optimization computations in parallel

0

3200

3600

4000

4400

4800

5200

5600

6000

(^123456) y(m)
f(N)
y = 2.45 m
f = 3747.7 N
Figure 14.7 Graphical solution of the second-level problem.