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A critical evaluation of asymptotic sampling method for highly safe structures

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Abstract

Asymptotic sampling is an efficient simulation-based technique for estimating small failure probabilities of structures. The concept of asymptotic sampling utilizes the asymptotic behavior of the reliability index with respect to the standard deviations of the random variables. In this method, the standard deviations of the random variables are progressively inflated using a scale parameter to obtain a set of scaled reliability indices. The collection of the standard deviation scale parameters and corresponding scaled reliability indices are called support points. Then, least square regression is performed using these support points to establish a relationship between the scale parameter and scaled reliability indices. Finally, an extrapolation is performed to estimate the actual reliability index. The accuracy and performance of the asymptotic sampling method are affected by various factors including the sampling method used, the values of the scale parameters, the number of support points, and the formulation of extrapolation models. The purpose of this study is to make a critical evaluation of the performance of the asymptotic sampling method for highly safe structures, and to provide some guidelines to improve the performance of asymptotic sampling method. A comprehensive numerical procedure is developed, and structural mechanics example problems with varying number of random variables and probability distribution types are used in assessment of the performance of asymptotic sampling method. It is found that generating the random variables by Sobol sequences and using the 6-model mean extrapolation formulation give slightly more accurate results. Besides, the optimum initial scale parameter is approximately around 0.3 and 0.4, and the optimum number of support points is typically four for all problems. As the reliability level increases, the optimum initial scale parameter value decreases, and the optimum number of support points increases.

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Acknowledgements

This paper is written to acknowledge Prof. Raphael (Rafi) T. Haftka’s novel contributions in the field of structural and multidisciplinary design optimization. Rafi was a pioneer in our optimization community and had contributed significantly to the fields of reliability-based design optimization, surrogate-based optimization, structural and multidisciplinary optimization, sensitivity analysis, optimization of laminated composite materials among others. The corresponding author is a PhD student of Rafi, and he is grateful for the guidance and camaraderie of him.

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Authors and Affiliations

  1. Department of Mechanical Engineering, TOBB University of Economics and Technology, Söğütözü, 06560, Ankara, Turkey

    Gamze Bayrak & Erdem Acar

Authors
  1. Gamze Bayrak
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  2. Erdem Acar
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Correspondence to Erdem Acar.

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The results provided herein are replicable. Interested readers can contact the corresponding author to obtain the MATLAB codes used to generate the results.

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Appendices

Appendix A: Reliability levels of the numerical example problems

For all numerical example problems, five different reliability levels are considered by changing a proper term in the LSF (see Table 13). The reliability index values reported in Table 13 are predicted using crude Monte Carlo simulations with a sample size of 107, 108, 109, 1010, and 1011 for reliability indices of 4, 4.5, 5, 5.5, and 6. Note that the reliability indices of 4, 4.5, 5, 5.5, and 6 correspond to the failure probabilities of 3.17 × 10–5, 3.40 × 10–6, 2.87 × 10–7, 1.90 × 10–8, and 9.87 × 10–10, respectively.

Table 13 The reliability levels considered for the example problems
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Appendix B: NFE versus RMSE plots for all example problems and all reliability levels

The NFE values corresponding to different values of f0 for all reliability levels of the example problems are provided in Figs. 19, 20, 21, 22, 23 and 24. It can be realized that the RMSEnor values are greatly increased when f0 = 0.2 at all reliability levels for all example problems. For this reason, we did not investigate the values of f0 below 0.2.

Fig. 19
figure 19

NFE values corresponding to f0 values for cantilever beam problem

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Fig. 20
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NFE values corresponding to f0 values for central crack problem

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Fig. 21
figure 21

NFE values corresponding to f0 values for connection rod problem

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Fig. 22
figure 22

NFE values corresponding to f0 values for Fortini’s clutch problem

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Fig. 23
figure 23

NFE values corresponding to f0 values for I beam problem

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Fig. 24
figure 24

NFE values corresponding to f0 values for roof truss problem

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Appendix C: The effects of dimensionality and nonlinearity on the optimum initial scale parameter and the number of support points

In this appendix, the effects of dimensionality and nonlinearity on the optimum value of the initial scale parameter f0 and the optimum number of support points Ns for the reliability levels β = 4.5, 5.0, and 5.5 are explored. Note that the effects in the reliability levels β = 4.0 and 6.0 are explored in the main text, in Sects. 5.3 and 5.4.

Figure 25 shows for all reliability levels that the dimensionality does not have an important effect on the optimum value of f0, whereas the nonlinearity has a substantial effect. As the nonlinearity of the limit state function increases, the optimum value of f0 also increases

Fig. 25
figure 25

The effects of dimensionality and nonlinearity on the optimum f0 value for reliability levels β = 4.5, 5.0, and 5.5

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Figure 26 shows for the reliability levels β = 4.5 and β = 5.0 that neither dimensionality nor nonlinearity have an important effect on the optimum number of support points Ns. However, for the reliability level β = 5.5, it is seen that the optimum number of support points increases as the nonlinearity of the LSF increases, even though the dimensionality is still insignificant.

Fig. 26
figure 26

The effects of dimensionality and nonlinearity on the optimum Ns value for reliability levels β = 4.5, 5.0, and 5.5

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Bayrak, G., Acar, E. A critical evaluation of asymptotic sampling method for highly safe structures. Struct Multidisc Optim 64, 3037–3061 (2021). https://doi.org/10.1007/s00158-021-03057-0

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