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Sequential approximate reliability-based design optimization for structures with multimodal random variables

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For practical engineering design problems, random variables tend to follow multimodal probability distributions when working at multiple operating conditions. For example, the structural fatigue stress of a steel bridge carrying both highway and railway traffic obeys the bimodal distribution. The existing popular reliability-based design optimization (RBDO) methods are mainly used to treat random variables with only unimodal distributions, which, therefore, tends to result in relatively large computational errors when multimodal random variables are involved. In this paper, a sequential approximate RBDO method is firstly proposed for engineering design involving multimodal random variables. The probability density function (PDF) of the response function is firstly calculated to assess the reliability of each probabilistic constraint, due to the existence of multimodal random variables. Then, a deterministic optimization problem is established to calculate candidate design points, based on the approximation that the response PDF demonstrates only transitional deformation in the optimization process. Three numerical examples and one engineering application of a thermal laptop design are presented to demonstrate the effectiveness of the proposed method.

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  1. Cheng G, Xu L, Jiang L (2006) A sequential approximate programming strategy for reliability-based structural optimization. Comput Struct 84:1353–1367

  2. Cho TM, Lee BC (2011) Reliability-based design optimization using convex linearization and sequential optimization and reliability assessment method. Struct Saf 33:42–50

  3. Du X, Chen W (2004) Sequential optimization and reliability assessment method for efficient probabilistic design. J Mech Des 126:225–233

  4. Du X, Guo J, Beeram H (2008) Sequential optimization and reliability assessment method for multidisciplinary systems design. Struct Multidiscip Optim 35(2):117–130

  5. Dubourg V, Sudret B, Bourinet J (2011) Reliability-based design optimization using kriging surrogates and subset simulation. Struct Multidiscip Optim 44(5):673–690

  6. Enevoldsen I, Sørensen JD (1994) Reliability-based optimization in structural engineering. Struct Saf 15(3):169–196

  7. Figueiredo MAT, Jian AK (2002) Unsupervised learning of finite mixture models. IEEE Trans Pattern Anal Mach Intell 24(3):381–396

  8. Goswami S, Chakraborty S, Chowdhury R, Rabczuk T (2019) Threshold shift method for reliability-based design optimization. Struct Multidiscip Optim 60:2053–2072

  9. Haider SW, Harichandran RS, Dwaikat MB (2009) Closed-form solutions for bimodal axle load spectra and relative pavement damage estimation. J Transp Eng 135(12):974–983

  10. He J, Guan X, Jha R (2016) Improve the accuracy of asymptotic approximation in reliability problems involving multimodal distributions. IEEE Trans Reliab 65(4):1724–1736

  11. Hu Z, Du X (2017) A mean value reliability method for bimodal distributions, in: Proceedings of the ASME 2017 International Design Engineering Technical Conference & Computers and information in engineering conference, Paper DETC 2017–67279

  12. Huang ZL, Jiang C, Zhou YS, Luo Z, Zhang Z (2016) An incremental shifting vector approach for reliability-based design optimization. Struct Multidiscip Optim 53(3):523–543

  13. Huang ZL, Jiang C, Zhang Z, Zhang W, Yang TG (2019) Evidence-theory-based reliability design optimization with parametric correlations. Struct Multidiscip Optim 60:565–580

  14. Keshtegar B, Hao P (2018) Enhanced single-loop method for efficient reliability-based design optimization with complex constraints. Struct Multidiscip Optim 57:1731–1747

  15. Keshtegar B, Lee I (2016) Relaxed performance measure approach for reliability-based design optimization. Struct Multidiscip Optim 54(6):1439–1454

  16. Liang J, Mourelatos ZP, Tu J (2004) A single-loop method for reliability-based design optimization, in: Proceedings of the ASME Design Engineering Technical Conferences and Computers and Information in Engineering Conference, pp DETC 2004–57255

  17. Lima RS, Kucuk A, Berndt CC (2002) Bimodal distribution of mechanical properties on plasma sprayed nanostructured partially stabilized zirconia. Mater Sci Eng A 327(2):224–232

  18. Liu W, Belytschko T, Mani A (1986) Random field finite elements. Int J Numer Methods Eng 23(10):1831–1845

  19. Moens E, Araújo NA, Vicsek T, Herrmann HJ (2014) Shock waves on complex networks. Sci Rep 4:4949

  20. Ni YQ, Ye XW, Ko JM (2010) Monitoring-based fatigue reliability assessment of steel bridges: analytical model and application. J Struct Eng 136(12):1563–1573

  21. Ni YQ, Ye XW, Ko JM (2011) Modeling of stress spectrum using long-term monitoring data and finite mixture distributions. J Eng Mech 138(2):175–183

  22. Nikou C, Galatsanos NP (2007) A class-adaptive spatially variant mixture model for image segmentation. IEEE Trans Image Process 16(4):1121–1130

  23. Papadrakakis M, Lagaros ND (2002) Reliability-based structural optimization using neural networks and Monte Carlo simulation. Comput Methods Appl Mech Eng 191(32):3491–3507

  24. Papadrakakis M, Stefanou G, Papadopoulos V (2010) Computational methods in stochastic dynamics. Springer, Berlin

  25. Reddy MV, Granhdi RV, Hopkins DA (1994) Reliability based structural optimization: a simplified safety index approach. Comput Struct 53(6):1407–1418

  26. Schuöllera GI, Jensen HA (2008) Computational methods in optimization considering uncertainties-an overview. Comput Methods Appl Mech Eng 198(1):2–13

  27. Shan S, Wang G (2008) Reliable design space and complete single-loop reliability-based design optimization. Reliab Eng Syst Saf 93(8):1218–1230

  28. Sobczyk K, Trcebicki J (1999) Approximate probability distributions for stochastic systems: maximum entropy method. Comput Methods Appl Mech Eng 168(168):91–111

  29. Timm DH, Tisdale SM, Turochy RE (2005) Axle load spectra characterization by mixed distribution modelling. J Transp Eng 131(2):83–88

  30. Tu J, Choi KK, Park YH (1999) A new study on reliability-based design optimization. J Mech Des 121(4):557–564

  31. Yi P, Cheng G, Jiang L (2008) A sequential approximate programming strategy for performance-measure-based probabilistic structural design optimization. Struct Saf 30:91–109

  32. Youn BD, Choi KK (2004) A new response surface methodology for reliability-based design optimization. Comput Struct 82(2–3):241–256

  33. Youn BD, Choi KK, Du L (2005) Enriched performance measure approach for reliability-based design optimization. AIAA J 43(4):874–884

  34. Yu X, Chang KH, Choi KK (1998) Probabilistic structural durability prediction. AIAA J 36(4):628–637

  35. Zhang X, Pandey MD (2013) Structural reliability analysis based on the concepts of entropy, fractional moment and dimensional reduction method. Struct Saf 43:28–40

  36. Zhang X, Pandey MD, Zhang Y (2014) Computationally efficient reliability analysis of mechanisms based on a multiplicative dimensional reduction method. J Mech Des 136(6):061006–061006-11

  37. Zhang Z, Jiang C, Han X, Ruan XX (2019) A high-precision probabilistic uncertainty propagation method for problems involving multimodal distributions. Mech Syst Signal Process 126(1):21–41

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This work is supported by the National Natural Science Foundation of China (Grant No. 51805157, Grant No. 51725502 and Grant No. 51490662) and Hunan Natural Science Foundation (Grant No. 2019JJ40015).

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Correspondence to C. Jiang.

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The authors declare that they have no conflict of interest.

Replication of results

In order to facilitate the replication of results presented in this paper, the MATLAB code files of the steel column design in Sect. 4.1 are provided as the supplementary material, and brief descriptions are given to the function of each file, as shown in Table 14. The results of the other examples can be reproduced conveniently, by modifying the characteristics of the problems such as objective function, probabilistic constraints, distribution type, dimensionality, etc

Thirteen Matlab code files are provided to perform the proposed sequential approximate method and double-loop method effectively. “MainProgram.m” is the main program of the proposed method, which consists of eight subprograms, namely, “GaussQuad.m”, “HermitePoly.m”, “UnivarQuad.m”, “GMFun.m”, “ReliabAsm.m”, “PdfFun.m”, “GNonlinear.m”, “ObjFun.m”. “DoubleLoop.m” is the main program of the double-loop method, which consists of one subprogram, namely “DLNonlinear.m”. “GMM_result.mat” is used in the programs of both methods to define the GMM characteristics and “MaxEnt_Newton” is the tool box to calculate the response PDF. The function of each subprogram is illustrated in Table 14.

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Zhang, Z., Deng, W. & Jiang, C. Sequential approximate reliability-based design optimization for structures with multimodal random variables. Struct Multidisc Optim (2020). https://doi.org/10.1007/s00158-020-02507-5

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  • Reliability-based design optimization
  • Multimodal random variable
  • Sequential approximate optimization