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An improved reliability analysis approach based on combined FORM and Beta-spherical importance sampling in critical region

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Abstract

The reliability analysis of structural systems is generally difficult when limit state function (LSF) is implicitly defined especially by black-box models in engineering. To balance analysis efficiency and accuracy, an improved approach named BIS-FC is proposed in this paper based on the Beta-spherical importance sampling (BIS) framework with the innovative concept of critical region defined by combination with First Order Reliability Method (FORM). The critical region is defined by the hyper-tangent plane of LSF at the Most Probable Point (MPP) and its parallel hyperplanes according to the convex or concave features of LSF, wherein the samples are of both high occurrence probability and high misjudgment risk due to the linearization assumption of FORM. BIS-FC only conducts LSF analysis for the BIS samples located in the critical region, and the other samples are directly identified as safe or failure according to the linearized hyperplanes. Thus large computational cost can be saved compared to the original BIS, and meanwhile, the analysis accuracy can be greatly enhanced compared to FORM. An iterative process is proposed to properly define the critical region, based on which reliability is analyzed sequentially until the stopping criteria for desired estimation error level and stable convergence are satisfied. The algorithms of BIS-FC for both single and multiple MPP situations are developed and testified with six numerical examples and one satellite structural engineering problem. The results demonstrate the effectiveness of BIS-FC regarding good balance between efficiency and accuracy.

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References

  • Ang GL, Ang AHS, Tang WH (2015) Kernel method in importance sampling density estimation. Structural Safety and Reliability

  • Au SK, Beck JL (1999) A new adaptive importance sampling scheme for reliability calculations. Struct Saf 21:135–158

    Article  Google Scholar 

  • Bourinet JM, Deheeger F, Lemaire M (2011) Assessing small failure probabilities by combined subset simulation and support vector machines. Struct Saf 33:343–353

    Article  Google Scholar 

  • Breitung K (1984) Asymptotic approximations for multinormal integrals. J Eng Mech ASCE 110:357–366

    Article  MATH  Google Scholar 

  • Cai X, Qiu H, Gao L, Shao X (2017) Metamodeling for high dimensional design problems by multi-fidelity simulations. Struct Multidiscip Optim 56:151–166

    Article  MathSciNet  Google Scholar 

  • Chen X, Yao W, Zhao Y, Ouyang Q (2016) An extended probabilistic method for reliability analysis under mixed aleatory and epistemic uncertainties with flexible intervals. Struct Multidiscip Optim 54(6):1–12

    Article  MathSciNet  Google Scholar 

  • Cressie N (1990) The origins of kriging mathematical. Geology 22:239–252

    MathSciNet  MATH  Google Scholar 

  • Crestaux T, MaıTre OL, Martinez JM (2009) Polynomial chaos expansion for sensitivity analysis. Reliab Eng Syst Saf 94:1161–1172

    Article  Google Scholar 

  • Du X, Hu Z (2012) First order reliability method with truncated random variables. J Mech Design 134:255–274

    Google Scholar 

  • Echard B, Gayton N, Lemaire M (2011) AK-MCS: an active learning reliability method combining kriging and Monte Carlo simulation. Struct Saf 33:145–154

    Article  Google Scholar 

  • Engelund S, Rackwitz R (1993) A benchmark study on importance sampling techniques in structural reliability. Struct Saf 12:255–276

    Article  Google Scholar 

  • Evans M, Swartz T (1995) Methods for approximating integrals in statistics with special emphasis on Bayesian integration problems. Struct Saf 10:254–272

    MathSciNet  MATH  Google Scholar 

  • Gayton N, Bourinet JM, Lemaire M (2003) CQ2RS: a new statistical approach to the response surface method for reliability analysis. Struct Saf 25:99–121

    Article  Google Scholar 

  • Grooteman F (2008) Adaptive radial-based importance sampling method for structural reliability. Struct Saf 30:533–542

    Article  Google Scholar 

  • Guan XL, Melchers RE (2001) Effect of response surface parameter variation on structural reliability estimates. Struct Saf 23:429–444

    Article  Google Scholar 

  • Harbitz A (1986) An efficient sampling method for probability of failure calculation. Struct Saf 3:109–115

    Article  Google Scholar 

  • Hasofer AM, Lind NC (1974) An exact and invariant first order reliability format. J Eng Mech ASCE

  • Helton JC, Johnson JD, Sallaberry CJ, Storlie CB (2006) Survey of sampling-based methods for uncertainty and sensitivity analysis. Reliab Eng Syst Saf 91:1175–1209

    Article  Google Scholar 

  • Hinrichs A (2010) Optimal importance sampling for the approximation of integrals. J Complex 26:125–134

    Article  MathSciNet  MATH  Google Scholar 

  • Hohenbichler M, Rackwitz R (1982) First-order concepts in system reliability. Struct Saf 1:177–188

    Article  Google Scholar 

  • Hu C, Youn BD (2011) Adaptive-sparse polynomial Chaos expansion for reliability analysis and Design of Complex Engineering Systems. Struct Multidiscip Optim 43:419–442

    Article  MathSciNet  MATH  Google Scholar 

  • Hu Z, Mahadevan S (2015) Global sensitivity analysis-enhanced surrogate (GSAS) modeling for reliability analysis. Struct Multidiscip Optim 53:501–521

    Article  MathSciNet  Google Scholar 

  • Jiang C, Qiu H, Gao L et al (2017) An adaptive hybrid single-loop method for reliability-based design optimization using iterative control strategy. Struct Multidiscip Optim 56:1271–1286

    Article  MathSciNet  Google Scholar 

  • Kahn H, Marshall AW (1953) Methods of reducing sample size in Monte Carlo computations. J Oper Res Soc Am 1:263–278

    MATH  Google Scholar 

  • Keshtegar B, Chakraborty S (2018) A hybrid self-adaptive conjugate first order reliability method for robust structural reliability analysis. Appl Math Model 53:319–332

    Article  MathSciNet  Google Scholar 

  • Keshtegar B, Hao P (2018) A hybrid descent mean value for accurate and efficient performance measure approach of reliability-based design optimization. Comput Methods Appl Mech Eng 336:237–259

    Article  MathSciNet  Google Scholar 

  • Kiureghian AD, Dakessian T (1998) Multiple design points in first and second-order reliability. Struct Saf 20:37–49

    Article  Google Scholar 

  • Lee I, Choi KK, Gorsich D (2010) System reliability-based design optimization using MPP-based dimension reduction method. Struct Multidiscip Optim 41:823–839

    Article  Google Scholar 

  • Lee I, Shin J, Choi KK (2013) Equivalent target probability of failure to convert high-reliability model to low-reliability model for efficiency of sampling-based RBDO. Struct Multidiscip Optim 48:235–248

    Article  MathSciNet  Google Scholar 

  • Li S, Chen L, Chen X, Zhao Y, Bai Y (2017) Long-range AIS message analysis based on the TianTuo-3 micro satellite. Acta Astronaut 136:159–165

    Article  Google Scholar 

  • Lv Z, Lu Z, Wang P (2015) A new learning function for kriging and its applications to solve reliability problems in engineering. Comput Math Appl 70:1182–1197

    Article  MathSciNet  Google Scholar 

  • Melchers RE (1990) Search-based importance sampling. Struct Saf 9:117–128

    Article  Google Scholar 

  • Noh, Y., Choi, K. K., & Lee, I. (2008). MPP-based dimension reduction method for RBDO problems with correlated input variables. AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference

  • Papaioannou I, Papadimitriou C, Straub D (2016) Sequential importance sampling for structural reliability analysis. Struct Saf 62:66–75

    Article  Google Scholar 

  • Rosenblatt M (1952) Remarks on a multivariate transformation. Ann Math Stat 23:470–472

    Article  MathSciNet  MATH  Google Scholar 

  • Song BF (1990) A numerical integration method for computing structural system reliability. Comput Struct 36:65–70

    Article  MATH  Google Scholar 

  • Sun Z, Wang J, Li R, Tong C (2017) LIF: a new kriging based learning function and its application to structural reliability analysis. Reliab Eng Syst Saf 157:152–165

    Article  Google Scholar 

  • Wang Z, Wang P (2016) Accelerated failure identification sampling for probability analysis of rare events. Struct Multidiscip Optim 54:1–13

    Article  MathSciNet  Google Scholar 

  • Yao W, Chen X, Huang Y, Van TM (2013a) An enhanced unified uncertainty analysis approach based on first order reliability method with single-level optimization. Reliab Eng Syst Saf 116:28–37

    Article  Google Scholar 

  • Yao W, Chen X, Luo W, Tooren MV, Guo J (2011) Review of uncertainty-based multidisciplinary design optimization methods for aerospace vehicles. Prog Aerosp Sci 47:450–479

    Article  Google Scholar 

  • Yao W, Chen X, Ouyang Q, Tooren MV (2013b) A reliability-based multidisciplinary design optimization procedure based on combined probability and evidence theory. Struct Multidiscip Optim 48:339–354

    Article  MathSciNet  Google Scholar 

  • Yun W, Lu Z, Jiang X (2018) A modified importance sampling method for structural reliability and its global reliability sensitivity analysis. Struct Multidiscip Optim 57:1625–1641

    Article  MathSciNet  Google Scholar 

  • Zou T, Mahadevan S, Mourelatos Z, Meernik P (2002) Reliability analysis of automotive body-door subsystem. Reliab Eng Syst Saf 78:315–324

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported in part by National Natural Science Foundation of China under Grant No.51675525 and 11725211.

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Authors and Affiliations

  1. Chinese Academy of Military Science, National Innovation Institute of Defense Technology, Beijing, 100071, China

    Wen Yao & Xiaoqian Chen

  2. College of Aerospace Science and Engineering, National University of Defense Technology, Changsha, 410073, China

    Guijian Tang & Ning Wang

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  1. Wen Yao
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  2. Guijian Tang
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  3. Ning Wang
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  4. Xiaoqian Chen
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Correspondence to Wen Yao.

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Responsible Editor is KK Choi

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Yao, W., Tang, G., Wang, N. et al. An improved reliability analysis approach based on combined FORM and Beta-spherical importance sampling in critical region. Struct Multidisc Optim 60, 35–58 (2019). https://doi.org/10.1007/s00158-019-02193-y

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  • DOI: https://doi.org/10.1007/s00158-019-02193-y

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