Subset Simulation

Au, S. K. and Beck, J. L. Estimation of small failure probabilities in high dimensions by subset simulation. Probabilistic Engineering Mechanics 16(4) (2001), 263—277. 

Katafygiotis, L. S., Cheung, S. H. and Yuen, K.-V. Spherical subset simulation (S ) for solving nonlinear dynamical reliabOity problems. International Journal of Reliability and Safety 4(0) (2010), in press. 

Subset simulation and line sampling for advanced Monte Carlo reliability analysis... 

ABSTRACT In this paper, the recently developed Subset Simulation (SS) and Line Sampling (LS) techniques are considered for improving the efficiency of Monte Carlo Simulation (MCS) in the estimation of system failure probability. The SS method is founded on the idea that a small failure probability can be expressed as a product of larger conditional probabilities of some intermediate events with a proper choice of the intermediate events, the conditional probabilities can be made sufficiently large to allow accurate estimation with a small number of samples. The LS method employs lines instead of random points in order to probe the failure domain of interest. An important direction is determined, which points towards the failure domain of interest the high-dimensional reliability problem is then reduced to a number of conditional one-dimensional problems which are solved along the important direction . 

In this respect, effective approaches are offered by Subset Simulation (SS) (Au and Beck 2001,... 

In this paper, the Subset Simulation (SS) and Line Sampling (LS) methods have been considered for improving the efficiency of Monte Carlo Simulation (MCS) in the estimation of system failure probability. A structural reliability model of hterature, i.e. the cracked plate model, has been taken as benchmark to test the two methods. 

Au, S. K. Beck, J. L. 2003. Subset Simulation and its application to seismic risk based on dynamic analysis. J. Eng. Mech.-ASCE 129(7) 1-17. 

Katafygiotis, L. 6c Cheung, S. 2007. Application of spherical subset simulation method and auxiliary domain method on a benchmark reliability study. Structural Safety 29(3), 194—207. 

The assessment of P(F) by Eq. (13) is inefficient in case of low probability events. Therefore, Subset Simulation (Au Beck 2001, Au Beck 2003, Ching et al. 2005a, Ching et al. 2005b) has been applied to assess P(P) by Equation 13. 

The basic idea of Subset Simulation is to express the failure probability as the product of the conditional probabilities of some intermediate failure events P,. Each event must have a larger probability of occurrence P(Pi) than P(P), so that it can be efficiently evaluated. Au Beck (2001) have proposed to define a decreasing sequence of failure events Pi D P2 15 D Pm = P, such that ... 

The effects of damage on the structural reliability are examined by using Subset Simulation. In order to solve the reliability integral, a time history of the ground acceleration is generated for each sample 0 for a given pair of values of M and R. 

Three simulation levels are implemented for Subset Simulation, in addition to the first level, using Monte Carlo simulation. For each level, 500 samples of the uncertain parameters 0 are considered. The value of the probability po for each intermediate failure domain is taken equal to 0.1 (Au Beck 2003). This means that the threshold value of the maximum inter-storey drift di i=l, m — 1) defining the intermediate failure domain F, is the 450-th value of the ranked values of the drift corresponding to each of the 500 samples. There are 50 samples corresponding to the next failure level so only 450 additional samples are obtained for that level. The total number of samples required for the simulation is Nj = 500-1-450-t-450- -450 = 1850. With these choices, it is possible to assess a probability of failure greater than or equal to 10 ". ... 

The probability P(b > bc be) is directly estimated by Subset Simulation the samples bei are obtained by sampling from the distribution p(bc). 

In this example, three limit states have been considered LSI, LS2 and LS3, corresponding to three different values of the capacity bi =0.3%, b = 0.7% and b e = 1.3%. The uncertainty in the limit state has been taken into account by introducing three distributions p b[ ), p(b ) and p bf ). Three Lognormal distributions have been taken, with mean value and standard deviation of the logarithm equal, respectively, to —1.20 and 0.1 for LSI, —0.36 and 0.05 for LS2, and 0.26 and 0.03 for LS3. For any group of uncertain parameters, the acceptance rate in the Subset Simulation algorithm has been taken in the range 20% to 50%. 

Ching, J., Au, S-K. Beck, J.L. 2005a. Reliability estimation for dynamical systems subject to stochastic excitation using Subset Simulation with splitting. Computational Methods in Applied Mechanics and Engineering 194 1557-1579. 

Sibilio, E. Ciampoli, M. 2006. Valutazione dell affidabilita strutturale attraverso tecniche di simulazione Monte Carlo Subset Simulation e Bayesian Updating . Froc. CRASC 06, Messina, CDRom. 

Sibilio, E., Ciampoli, M. Beck, J.L. 2006. Seismicreliability assessment of structures via Subset Simulation and Bayesian updating. 3 International Conference on Advances in Mechanical Engineering and Mechanics, Hammamet, Tunisia, CDRom. 

Reliability analysis for Runway Overrun using subset simulation... 

ABSTRACT Runway overrun is one of the main accident types in airline operations. Nevertheless, due to the high safety levels in the aviation industry, the probability of a runway overrun is small. This motivates the use of structural reliability concepts to estimate this probability. We apply the physically-based model for the landing process of Drees and Holzapfel (2012) in combination with a probabilistic model of the input parameters. Subset simulation is used to estimate the probability of runway overrun for different runway conditions. We also carry out a sensitivity analysis to estimate the influence of each input random variable on the probability of an overrun. Importance measures and parameter sensitivities are estimated based on the samples from subset simulation and concepts of the First-Order Reliability Method (FORM). 

In the next section a short outline to structural reliability in general and subset simulation in particular is given. This is followed by a summary of the RWO model and the related probabilistic model. Thereafter, the results of the reliability analysis are presented for different runway conditions. The influence of the individual input random variables on the probability of a RWO is investigated through different importance and sensitivity measures. In this context, FORM is shortly explained, as some of its theory is applied. The paper concludes with a discussion of the results. 

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