Improved Line Sampling Reliability Analysis Method and its Application

For reliability analysis of implicit limit state function, an improved line sampling method is presented on the basis of sample simulation in failure region. In the presented method, Markov Chain is employed to simulate the samples located at failure region, and the important direction of line sampling is obtained from these simulated samples. Simultaneously, the simulated samples can be used as the samples for line sampling to evaluate the failure probability. Since the Markov Chain samples are recycled for both determination of the important direction and calculation of the failure probability, the computational cost of the line sampling is reduced greatly. The practical application in reliability analysis for low cycle fatigue life of an aeronautical engine turbine disc structure under 0-takeoff-0 cycle load shows that the presented method is rational and feasible. Introduction Line sampling evolved from the need to analyze high dimensional reliability problem with small failure probability [1-3]. The efficiency of the line sampling depends on the sampling direction, usually named as the important direction. When the important direction consists with the direction from coordinate origin to maximum likelihood point (MLP) of failure region in the standard normal space, the efficiency of the line sampling reaches optimal state, therefore the direction from coordinate origin to MLP is defined as the optimal important direction. In the worst case that the sampling direction is orthogonal to the optimal one, the efficiency of the line sampling is at least as good as that of direct Monte Carlo simulation. The closer to the optimal one the sampling direction is, the more efficient the line sampling is. The optimal important direction is not easy to be searched as the limit state function is implicit. Ref.[3] presented a finite difference based method to search for the optimal important direction, but the method usually offers an important direction significantly departing from the optimal one due to the problem of the finite difference. In this paper, Markov Chain is employed to simulate the samples distributed in failure region [4], and these samples are used both to search for the optimal important direction accurately and to evaluate the failure probability, which leads to the low computation cost. The following sections demonstrate the basic concept of the presented method and the practical application in reliability analysis for low cycle fatigue life of an aeronautical engine turbine disc structure applied by 0-takeoff-0 cycle load. Line Sampling based on Markov Chain Simulation The key steps of the line sampling concern with the searching of the optimal important direction and the drawing of the samples for failure probability. In the paper, these two key steps are completed by the samples distributed in the failure region, and the samples distributed in the failure region are simulated by Markov Chain simulation. Determination of the Important Direction α . Since correlated non-normal variables can be transformed into independent standard normal variables by applying Rosenblatt’s transformation or Key Engineering Materials Vols. 353-358 (2007) pp 1001-1004 online at http://www.scientific.net © (2007) Trans Tech Publications, Switzerland Online available since 2007/Sep/10 All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of the publisher: Trans Tech Publications Ltd, Switzerland, www.ttp.net. (ID: 130.203.133.34-14/04/08,12:30:14) Nataf’s transformation, we discuss the d -dimensional independent standard normal vector x with joint probability density function (PDF) ( ) f x for the sake of simplification. ( ) g x denotes the limit state function in the standard normal space, and F denotes the failure region, then F { : ( ) 0} g = ≤ x x . In order to determine the optimalα , which consists with the direction from the coordinate origin to the MLP of F in the standard normal space, we simulate the samples distributed in F by Markov Chain simulation. ( | F) q x , the PDF of x distributed in F , can be expressed by Eq.1. F ( | F) ( ) ( ) / f q I f P = x x x . (1) where F ( ) I x is an indicator, F ( ) I x =1 if F ∈ x and F ( ) I x =0 otherwise. f P is the probability of F , i.e. F ... ( ) f P f d = ∫ ∫ x x . Markov Chain simulation can accelerate the efficiency of exploring the failure region[4], therefore, using ( | F) q x as the stationary distribution of Markov Chain, we can obtain the samples distributed in F effectively and efficiently by the following procedure, and we can further determine the optimal α accurately by use of these samples. (1) Select 0 x , an initial state of Markov Chain: 0 x should be in the failure region because the stationary distribution of Markov Chain is ( | F) q x , it can be determined by engineering judgment economically. (2) Determine the j th state j x of Markov Chain: a proposal density function ' 1 ( | ) j f − x x with the symmetry property and centered at the (j-1)th state 1 j− x is selected to generate a candidate state ' x . Compute the ratio ' 1 ( | F) / ( | F) j r q q − = x x . Set j x = ' x with probabilitymin(1, ) r , and set j x = 1 j− x with the remaining probability 1-min(1, ) r . (3) Repeat step (2) until N Markov Chain states ( 1, , ) j j N = L x are generated. Since the stationary distribution of Markov Chain is ( | F) q x , ( 1, , ) j j N = x L obey ( | F) q x approximately. We can utilize the average of the unit vectors / || || j j x x ( 1, , ) j N = L to calculate α , which is shown in Fig.1.