which proper DOE must be employed. The DOE issues are discussed in the last section

of this chapter.

 Design Response Surface (DRS) Approximation

 Direct Monte Carlo simulation introduces noise in computed probability of failure

due to limited samples. The noise can be reduced by using a relatively large number of

samples, which is computationally made possible by using response surface

approximation. The noise can also be filtered out by using another response surface

approximation, the design response surface (DRS). DRS fitted to probability of failure P

as a function of design variables d can be shown as

 P(d) = Z(d)T b (3-5)

The use of DRS also reduces the computational cost of RBDO by approximating the

reliability constraint by close-form function.

 The probability of failure is found to change by several orders of magnitude over

narrow bands in design space, especially when the random variables have small

coefficients of variation (Chapter 5). The steep variation of probability of failure requires

DRS to use high-order polynomials for the approximation, such as quintic polynomials,

increasing the required number of probability calculations (Qu et al. 2000). An additional

problem arises when Monte Carlo simulations (MCS) are used for calculating

probabilities. For a given number of simulations, the accuracy of the probability estimates

deteriorates as the probability of failure decreases.

 The numerical problems associated with steep variation of probability of failure led

to consideration of alternative measures of safety. The most common one is to use the