Stochastic Finite Element and Satellite Structure Design - Finite Element Analysis for Satellite Structures - page 220

analysis and FORM for the reliability analysis with implicit performance functions
does not suffer from the drawbacks of Monte Carlo simulation or response surface
approach. It uses the information about the actual value and the actual gradient of the
performance function at each iteration of the search for the design point and uses an
optimization scheme to converge to the minimum distance point.
In case the performance function Z can be defined as Z ¼ gX 1 ; X 2 ; ... ; X n ð Þ ,

the forward difference approach can be used to compute the derivatives

, as follows:

X 1 ; X 2 ; ... ; X n oZ = oX 1 ; oZ = oX 2 ; ... ; oZ = oX n at the point

1. First Compute Z 0 ¼ gX 1 ; X 2 ; ... ; X n .
2. Change the value of X 1 to X 1 þ DX 1 , where DX 1 is a small number (pertur-
bation value). All other variables stay at the same value. Compute the new
value of Z as Z 1 ¼ gX 1 þ DX 1 ; X 2 ; ... ; X n

and the change in its value

DZ ¼ Z 1 Z 0 .
3. Compute the approximate derivative of Z with respect to X 1 as DZ = DX 1 .
4. Repeat steps 2 and 3 for each variable X 2 to X n . It is common to use pertur-
bation values in proportion size of one-tenth of the standard deviation for each
variable.

The numerical values of the derivatives computed above are valid only at the
mean values of the random variables. During the iterations of the FORM method,
the derivatives need to be recalculated at each iteration.
The steps of SFEM-based reliability analysis are as follows. FORM method
requires the value of the performance function G(Y), and its gradient r G ð Y Þ :

1. Using the parameters of the structure, assemble the global stiffness matrix
K and the global nodal load vector F.
2. Solve for the displacement, U, using the finite element equation,

KU ¼ F ð 5 : 18 Þ

3. Compute the vector of desired response quantities S (e.g., stress) from the
computes displacement using a transformation of the form

S ¼ Q t U þ S 0

ð 5 : 19 Þ

where Q t is a transformation matrix relating U and S, and S 0 is the reference vector
for U = 0.

4. Compute the performance function

g ðÞ¼ gR ðÞ; S ðÞ f g ð 5 : 20 Þ