P,' fx (x)dx: (6-13)
 g(x} 0

where fx(x) is the joint probability distribution function (JPDF). This integral is hard to

evaluate, because the integration domain defined by g(x) 0 is usually unknown, and

integration in high dimension is difficult. Commonly used probabilistic analysis methods

are either moment-based methods such as the first-order-reliability method (FORM) and

the second-order-reliability method (SORM), or simulation techniques such as Monte

Carlo simulation (MCS) (e.g., Melchers 1999). Monte Carlo simulation is a good method

to use for system reliability analysis with multiple failure modes. The present chapter

focuses on the use of MCS with response surface approximation in RBDO.

 Monte Carlo simulation utilizes randomly generated samples according to the

statistical distribution of the random variables, and the probability of failure is obtained

by calculating the statistics of the sample simulation. Fig. 3 illustrated the Monte Carlo

simulation of a problem with two random variables. The probability of failure of the

problem is calculated as the ratio of the number of samples in the unsafe region over the

total number of samples.

 A small probability requires a large number of samples for MCS to achieve low

relative error. Therefore, for fixed number of simulations, the accuracy of MCS

deteriorates with the decrease of probability of failure. For example, with 106

simulations, a probability estimate of 10 has a relative error of a few percent, while a

probability estimate of 10 has a relative error of the order of 100 percent. In RBDO, the

required probability of failure is often very low, thus the probability (or safety index)

calculated by MCS is inaccurate near the optimum. Furthermore, the probabilities of

failure in some design regions may be so low that they are calculated as zero by MCS.