The propagation of uncertainty through a deterministic model is a key issue when one wants to estimate the failure probability of a system. Several methods exist to describe the probability distribution of an output variable from the probabilistic parameters of the random input variables, but these methods usually require a large number of calls to the deterministic model. This is the case of the famous Monte-Carlo method, which provides an unbiased estimate of the failure probability pf, but requires a number of calls larger than 10/pf to provide a correct estimation. For the orders of magnitude of failure probabilities that are of interest to engineers, such a number of calls of the deterministic model is often out of reach for computationnal cost reasons. Several recent probabilistic methods allow a reduction of this computationnal time.
The Response Surface Methodology (RSM) is based on a quadratic approximation of the limit state surface, and consists in determining by successive iterations the position of the so-called design point, defined as the point of most probable failure. When this point is known, the FORM approximation provides a direct estimate of the failure probability. However, RSM presents a few limitations, not the least being the fact that it is only able to provide a failure probability for a failure criterion defined a priori, but that it does not provide any information on the probability distribution of the output variable of the model.
The Collocation-based Stochastic Response Surface Methodology (CSRSM) was introduced in order to overcome the limitations of RSM. CSRSM is also based on an analytical approximation of the deterministic model, but this approximation is more complex and presents the advantage of being defined on a wide domain in the space of the input random variables (and not only in the neighbourhood of the design point as in RSM). When this analytical approximation is properly fitted from a rather limited number of calls to the deterministic model, it provides a so-called meta-model with a negligible computational cost, which can be subtituted to the original deterministic model in any probabilistic method.
Figure 1. Principle of the reliability-based methods : the ellipses ("probabilistic" part) represent the domains of equal probability of occurence of the random input variables, and the limit state curve ("déterministic" part) represents the limit between the domains of failure and of safety. The design point is the point of most probable failure.
Figure 3. Collocation scheme used in the framework of CSRSM. Instead of performing hundreds of calls as in the Monte-Carlo method, the computations are efficiently spread over the domain of the input random variables. From this it is possible to obtain an analytical approximation of the original model with a very small time cost.
Figure 5. Examples of results obtained by CSRSM, in the framework of the analysis of pressurized tunnel face stability (Mollon et al. 2011) : influence of the dispersion of the friction angle and of the cohesion on the probability distribution of the critical collapse pressure and on the collapse probability.
Figure 2. Response Surface Methodology (RSM) in the case of three random variables. The ellipses become ellipsoids, and the limit state surface is approximated by a quadratic funtion.
Figure 4. Comparison between the original model and the meta-model in the framework of CSRSM, for several orders of the method. At the order 2 the approximation is only accurate in the central zone (area of high probability of occurence), while at the order 5 it is satisfactory in the whole domain.