stringent weight requirements, it is also important to provide guidelines for controlling
the magnitudes of uncertainties for the purpose of reducing structural weight.
Deterministic structural optimization is computationally expensive due to the
need to perform multiple structural analyses. However, reliability-based optimization
adds an order of magnitude to the computational expense, because a single reliability
analysis requires many structural analyses. Commonly used reliability analysis methods
are based on either simulation techniques such as Monte Carlo simulation, or moment-
based methods such as the first-order-reliability-method (e.g., Melchers, 1999). 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 low
failure probability. Monte Carlo simulation also produces a noisy estimate of probability
and hence is difficult to use with gradient-based optimization. Moment-based methods do
not have these problems, but they are not well suited for problems with many competing
critical failure modes.
Response surface approximations solve the two problems of Monte Carlo
simulation, namely simulation cost and noise from random sampling. Response surface
approximations (Khuri and Cornell 1996) typically fit low order polynomials to a number
of response simulations to approximate response. Response surface approximations
usually fit the structural response such as stresses in terms of random variables for
reliability analyses. The probability of failure can then be calculated inexpensively by
Monte Carlo simulation using the fitted response surfaces. Response surface
approximations can also be fitted to probability of failure in terms of design variables,
which replace the reliability constraints in RBDO to filter out numerical noise in the