Engineering Optimization: Theory and Practice, Fourth Edition

14.4 Derivatives of Static Displacements and Stresses 745

By premultiplying Eq. (14.27) by [Y]Twe obtain

[K ̃]

r×r

c

r× 1

= P ̃

r× 1 (14.28)

where

[K ̃]=[Y]T[KN][Y] (14.29)

P ̃=[Y]TP (14.30)

It can be seen that an approximate displacement vectorYNcan be obtained by solv-

ing a smaller (r) system of equations, Eq. (14.28), instead of computing the exact

solutionYNby solving a larger (n)system of equations, Eq. (14.25). The foregoing

method is equivalent to applying the Ritz–Galerkin principle in the subspace spanned

by the set of vectorsY 1 ,Y 2 ,... ,Yr. The assumed modesYi, i = 1 , 2 ,... , r, can be

considered to be good basis vectors since they are the solutions of similar sets of

equations.

Fox and Miura 14.3 applied this method for the analysis of a 124-member,

96-degree-of-freedom space truss (shown in Fig. 14.3). By using a 5-degree-of-freedom

approximation, they observed that the solution of Eq. (14.28) required 0.653 s while

the solution of Eq. (14.25) required 5.454 s without exceeding 1% error in the

maximum displacement components of the structure.

14.4 Derivatives of Static Displacements and Stresses

The gradient-based optimization methods require the gradients of the objective and

constraint functions. Thus the partial derivatives of the response quantities with respect

to the design variables are required. Many practical applications require a finite-element

13

12

6

5

2

3

6

(^119)
8
16
14
22
2028
26
34
35
(^1715)
(^2321)
(^2927)
10
18
24
30
36
33
32
7
1
4
19
25
31
40 in.
40 in.
40 in.
30 in.
x
z
y
Figure 14.3 Space truss [13.3].