Discussion on: Applications of asymptotic sampling on high dimensional structural dynamic problems: M.T. Sichani, S.R.K. Nielsen and C. Bucher, Structural Safety, 33 (2011) 305–316

Abstract

Article history: Available online 19 October 2013 2013 Elsevier Ltd. All rights reserved. C. Bucher originally proposed the asymptotic sampling (AS) for high dimensional reliability analysis with lower failure probability [1]. Further, M.T. Sichani, S.R.K. Nielsen and C. Bucher developed the AS for the evaluation of high dimensional structural dynamic problems [2]. In their method, the lower failure probability is extrapolated by several high failure probabilities on the asymptotic behavior of the failure probability in n-dimensional independent and identically distributed (i.i.d) Gaussian space with respect to the standard deviation of the variables. The detailed steps of their methods can be summarized as follows: Step 1. Transform the random variables X of the original problem into the i.i.d Gaussian U space, i.e. N, using Rosenblatt transformation or Nataf transformation T: X ? U. Step 2. Choose the parameters fi 2 [0, 1] (i = 1, 2, . . . ,m), and increase the standard deviation of N but keep the mean of N invariant, and then obtain the scaled random variables Nfi . If N NðlN;r2Þ, then Nfi N lN;rN=f 2 i . Step 3. Perform the standard Monte Carlo simulation with Nfi and calculate the scaled failure probability Pf(fi). Step 4. Evaluate the scaled reliability index as b(fi) = U (1 Pf(fi)). Step 5. Obtain all the b(fi) (i = 1, 2, . . . ,m), for different fi to construct a set of (fi, b(fi)) which will be regarded as ‘‘support points’’. Step 6. Estimate coefficients A and B of Eq. (1) using the set of (fi, b(fi)) (i = 1, . . . ,m) by regression analysis. bðfiÞ 1⁄4 Af i þ B fi ð1Þ Step 7. Estimate the un-scaled reliability index b(1) = A + B and the lower failure probability Pf(1) = 1 U(b(1)). As revealed by Steps 1–7, we can find that the main computational cost of the AS lies in the sum of the time for all the b(fi) (i = 1, 2, . . . ,m) by the standard Monte Carlo simulation. Admittedly, the sum of time for b(fi) is generally less than that of the direct evaluation of b(1) and the AS consequentially reduces the computational cost. In addition, this method behaves well when it comes to problems with high nonlinearity and lower failure probability. However, after conducting a detailed investigation into the AS method, we find that there are still three issues necessary to be discussed. The first is that the computational cost in AS may be still prohibitive due to the fact that m independent evaluations of b(fi) are required to obtain the ‘‘support points’’. The second is that sometimes the individual regression analysis using Eq. (1) leads to remarkable discrepancies compared with the accurate result. The third is that the AS is not able to give the confidence interval of the result. In this paper, we discuss these issues for further improving the accuracy and efficiency of the AS method. 1. Discussion to the first issue It is assumed that the performance function is y = G(x). For Pf(fi) (i = 1, 2, . . . ,m), we first construct the same importance sampling probability density function (IS PDF) f (x), then Pf(fi) (i = 1, 2, . . . ,m) can be estimated by the same importance sampling (IS) method through only one set of input–output samples (xk,G(xk)) (k = 1, . . . ,N), where N is the number of samples. Pf 1⁄4 Z F f ðxÞdx 1⁄4 Z IFðxÞ f ðxÞ f ðxÞ f ðxÞdx 1⁄4 E IFðxÞxðxÞ 1⁄2 ð2Þ