Monte Carlo Simulation Using Response Surface Approximation

 Monte Carlo simulation is easy to implement, robust, and accurate with sufficiently

large samples, but it requires a large number of analyses to obtain a good estimate of

small failure probabilities. Monte Carlo simulation also produces a noisy response and

hence is difficult to use in optimization. Response surface approximations solve the two

problems, namely simulation cost and noise from random sampling.

 Response surface approximations fit a closed-form approximation to the limit state

function to facilitate reliability analysis. Therefore, response surface approximation is

particularly attractive for computationally expensive problems such as those requiring

complex Einite element analyses. Response surface approximations usually fit low-order

polynomials to the structural response in terms of random variables

 g(x) = Z(x)T b (6-17)

where g(x) denotes the approximation to the limit state function g(x), Z(x) is the basis

function vector that usually consists of monomials, and b is the coefficient vector

estimated by least square regression. The probability of failure can then be calculated

inexpensively by Monte Carlo simulation or moment-based methods using the fitted

polynomials.

 Response surface approximations (RSA) can be used in different ways. One

approach is to construct local RSA around the Most Probable Point (MPP) that

contributes most to the probability of failure of the structure. The statistical design of

experiment (DOE) of this approach is iteratively performed to approach the MPP on the

failure boundary. For example, Bucher and Bourgund (1990), and Sues (1996, 2000)

constructed progressively refined local RSA around the MPP by an iterative method. This