x and d are the random variable and design variable vectors, respectively. They
recommended Latin Hypercube sampling as the statistical design of experiments. The
number of response surface approximations constructed in optimization process is
reduced substantially by introducing design variables into the response surface
approximation formulation.
The selection of RSA approach depends on the limit state function of the problem
and target probability of failure. The global RSA approach is more efficient than local
RSA, but it is limited to problems with relatively high probability or limit state function
that can be well approximated by regression analysis based on simple basis functions. To
avoid the extrapolation problems, RSA generally needs to be constructed around
important region or MPP to avoid large errors in the results of MCS induced by fitting
errors in RS. Therefore, an iterative RSA is desirable for general reliability analysis
problem.
Design response surface approximations (DRS) are fitted to probability of failure to
filter out noise in MCS and facilitate optimization. Based on past experience, high-order
DRS (such as quintic polynomials) are needed in order to obtain a reasonably accurate
approximation of the probability of failure. Constructing highly accurate DRS is difficult
because the probability of failure changes by several orders of magnitude over small
distance in design space. Fitting to safety index /7=-0 (p), where p is the probability of
failure and & is the cumulative distribution function of normal distribution, improves the
accuracy of the DRS to a limited extent. The probabilistic sufficiency factor can be used
to improve the accuracy of DRS approximation.