Design Response Surfaces
The six quadratic ARS were used to calculate the probabilities of failure with
Monte Carlo simulation. Because the fitting errors in design response surfaces (DRS) are
generally larger than the random errors from finite sampling in probability calculation,
Monte Carlo simulation needs only to be performed until relatively small errors estimated
confidence intervals are achieved. Therefore, a sample size of 1,000,000 was employed.
The design points of DRS combine Face Center Central Composite Design (FCCCD) and
LHS. Table 5-5 compares the three DRS.
Table 5-5. Design response surfaces for probability of failure (Probability calculated by
Monte Carlo simulation with a sample size of 1,000,000)
FCCCD 25 points LHS 252 points L 2 pit
Error Statistics + FCCCD 25 points
quadratic 5th order 5th order
R2adj 0.6855 0.9926 0.9982
RMSE Predictor 0.00053 0.000003 0.000012
Mean of
0.00032 0.000016 0.000044
Response
The accuracy of the quadratic response surface approximation is unacceptable. The
accuracy of fifth order response surface (with 126 unknown coefficient before stepwise
regression) was improved by using a reciprocal transformation on the thickness tl and t2,
since the probability of failure, like most structural response, is inversely correlated with
the stack thickness. We found that LHS might fail to sample points near some corners of
the design space, leading to poor accuracy around these corners. We therefore combined
LHS with FCCCD that includes all the vertices of the design space. The accuracy of DRS
based on LHS combined with FCCCD is slightly worse than that of DRS based on LHS
alone, because the probabilities at the corners of the design space are usually extremely
low or high, presenting a greater fitting difficulty than without FCCCD. But the