Engineering Optimization: Theory and Practice, Fourth Edition

14.3 Fast Reanalysis Techniques 741

Neglecting the term [K]Y 2 , Eq. (14.17) can be used to obtain the second approxi-
mation toY, Y 2 , as

Y 2 = −[K 0 ]−^1 ( [K]Y 1 ) (14.18)

From Eq. (14.16),Ycan be written as

Y=

∑^2

i= 1

Yi (14.19)

This process can be continued andYcan be expressed, in general, as

Y=

∑∞

i= 1

Yi (14.20)

whereYiis found by solving the equations

[K 0 ]Yi= − [K]Yi− 1 (14.21)

Notethat the series given by Eq. (14.20) may not converge if the change in the
design vector,X, is not small. Hence it is important to establish the validity of the
procedure for each problem, by determining the step sizes for which the series will
converge, before using it. The iterative process is usually stopped either by specify-
ing a maximum number of iterations and/or by prescribing a convergence criterion
such as

||Yi||

∣

∣

∣

∣

∣

∣

∣

∣

∑i

j= 1

Yj

∣

∣

∣

∣

∣

∣

∣

∣

≤ε (14.22)

where||Yi|| isthe Euclidean norm of the vectorYiand εis a small number on
the order of 0.01.

Example 14.1 Consider the crane (planar truss) shown in Fig. 14.2. Young’s modulus
of membereis equal toEe= 03 × 106 psi (e= 1 , 2 , 3 , 4 ), and the other data are
shown in Table 14.1. Assuming the base design to beA 1 =A 2 = in. 2 2 andA 3 =
A 4 = in. 1 2 , and perturbations to beA 1 = A 2 = 0. 4 in.^2 and A 3 = A 4 = 0. 2
in.^2 , determine (a) the exact displacements of nodes 3 and 4 at the base design, (b) the
displacements of nodes 3 and 4 at the perturbed design using the exact procedure, and,
(c) the displacements of nodes 3 and 4 at the perturbed design using the approximation
method.

SOLUTION The stiffness matrix of a typical element e is given by

[K(e)]=

AeEe

le











li^2 j lijmij −l^2 ij −lijmij

lijmij m^2 ij −lijmij −m^2 ij

−li^2 j −lijmij l^2 ij lijmij

−lijmij −m^2 ij lijmij m^2 ij











(E 1 )