is the vector of random variables and d is the vector of design variables. By including
design variables in the response surface formulation, the efficiency of the RBDO is
improved drastically for certain problems. The ARS is fitted to the response (limit state
function) in terms of both design variables and random variables
G(x) = Z(x, d)T b (3 -4)
The ARS approach combines probabilistic analyses with design optimization. Using the
ARS, the probability of failure at every design point in the design optimization process
can be calculated inexpensively by Monte Carlo simulation based on the fitted
polynomials.
The number of analyses required for ARS depends on the total number of random
variables and design variables. Because the ARS fits an approximation in terms of both
random variables and design variables it requires more analyses than SRS. For our
applications, where the number of random variables is large (around 10) and the number
of design variables is small (around four), this typically results in an ARS that is less than
three times as expensive to construct as an SRS, which is due to the use of Latin
Hypercube sampling that can generate an arbitrary number of design points for RSA
construction (explained in last section of this chapter and demonstrated in chapter 5).
This compares with a large number (order of 10 to 100) of SRS approximation required
in the course of optimization. For a large number of variables (more than 20 to 30), the
construction of ARS is hindered by the curse of dimensionality. SRS might be used to
reduce the dimensionality of the problem. Besides the computational cost issue, the
inclusion of design variables may increase the nonlinearity of the response surface
approximation. It might be necessary to use RSA of order higher than quadratic, for