Structural and Multidisciplinary Optimization

, Volume 57, Issue 4, pp 1533–1551 | Cite as

Structural optimization oriented time-dependent reliability methodology under static and dynamic uncertainties

  • Lei Wang
  • Xiaojun Wang
  • Di Wu
  • Menghui Xu
  • Zhiping Qiu


Uncertainty with characteristics of time-dependency, multi-sources and small-samples extensively exists in the whole process of structural design. Associated with frequent occurrences of material aging, load varying, damage accumulating, traditional reliability-based design optimization (RBDO) approaches by combination of the static assumption and the probability theory will be no longer applicable when dealing with the design problems for lifecycle structural models. In view of this, a new non-probabilistic time-dependent RBDO method under the mixture of time-invariant and time-variant uncertainties is investigated in this paper. Enlightened by the first-passage concept, the hybrid reliability index is firstly defined, and its solution implementation relies on the technologies of regulation and the interval mathematics. In order to guarantee the stability and efficiency of the optimization procedure, the improved ant colony algorithm (ACA) is then introduced. Moreover, by comparisons of the models of the safety factor-based design as well as the instantaneous RBDO design, the physical means of the proposed optimization policy are further discussed. Two numerical examples are eventually presented to demonstrate the validity and reasonability of the developed methodology.


Non-probabilistic time-dependent RBDO method The mixture of time-invariant and time-variant uncertainties The first-passage approach The improved ant colony algorithm (ACA) The safety factor-based design 



The authors would like to thank the National Nature Science Foundation of China (No. 11372025, 11432002, 11602012), the 111 Project (No. B07009), the Defense Industrial Technology Development Program (No. JCKY2016601B001, JCKY2016205C001), and the China Postdoctoral Science Foundation (No. 2016 M591038) for the financial supports. Besides, the authors wish to express their many thanks to the reviewers for their useful and constructive comments.


  1. Aoues Y, Chateauneuf A (2010) Benchmark study of numerical methods for reliability-based design optimization. Struct Multidiscip Optim 41:277–294MathSciNetCrossRefzbMATHGoogle Scholar
  2. Babykina G, Brînzei N, Aubry JF (2016) Modeling and simulation of a controlled steam generator in the context of dynamic reliability using a stochastic hybrid automaton. Reliab Eng Syst Saf 152:115–136CrossRefGoogle Scholar
  3. Ben-Haim Y (1994) Convex models of uncertainty: Applications and implications. Erkenntnis 41:139–156CrossRefGoogle Scholar
  4. Chun JH, Song JH, Paulino GH (2015) Structural topology optimization under constraints on instantaneous failure probability. Struct Multidiscip Optim 53:773–799MathSciNetCrossRefGoogle Scholar
  5. Ditlevsen OD, Madsen HO (1996) Structural reliability methods. John Wiley & Sons, ChichesterGoogle Scholar
  6. Du XP, Sudjianto A, Huang BQ (2005) Reliability-based design with the mixture of random and interval variables. J Mech Des 127:1068–1076CrossRefGoogle Scholar
  7. Elishakoff I, Haftka RT, Fang J (1994) Structural design under bounded uncertainty – Optimization with anti-optimization. Comput Struct 53:1401–1405CrossRefzbMATHGoogle Scholar
  8. Frangopol DM, Corotis RB, Rackwitz R (1997) Reliability and optimization of structural systems. Pergamon, New YorkGoogle Scholar
  9. Ge R, Chen JQ, Wei JH (2008) Reliability-based design of composites under the mixed uncertainties and the optimization algorithm. Acta Mech Solida Sin 21:19–27CrossRefGoogle Scholar
  10. Hu Z (2014) Probabilistic engineering analysis and design under time-dependent uncertainty. In: Mechanical and Aerospace Engineering, Missouri University of Science and TechnologyGoogle Scholar
  11. Hu Z, Du XP (2013) Time-dependent reliability analysis with joint upcrossing rates. Struct Multidiscip Optim 48:893–907MathSciNetCrossRefGoogle Scholar
  12. Hu Z, Du XP (2014) Lifetime cost optimization with time-dependent reliability. Eng Optim 46:1389–1410MathSciNetCrossRefGoogle Scholar
  13. Hu Z, Du XP (2015) Reliability-based design optimization under stationary stochastic process loads. Eng Optim:1–17Google Scholar
  14. Hu Z, Li HF, Du XP, Chandrashekhara K (2013) Simulation-based time-dependent reliability analysis for composite hydrokinetic turbine blades. Struct Multidiscip Optim 47:765–781CrossRefGoogle Scholar
  15. Jiang C, Bai YC, Han X, Ning HM (2010) An efficient reliability-based optimization method for uncertain structures based on non-probability interval model. Comput Mater Continua 18:21–42Google Scholar
  16. Jiang C, Zhang Q, Han X, Li D, Liu J (2011) An interval optimization method considering the dependence between uncertain parameters. Comput Model Eng Sci 74:65–82Google Scholar
  17. Jiang C, Ni BY, Han X, Tao YR (2014) Non-probabilistic convex model process: A new method of time-variant uncertainty analysis and its application to structural dynamic reliability problems. Comput Methods Appl Mech Eng 268:656–676MathSciNetCrossRefzbMATHGoogle Scholar
  18. Kang Z, Luo YJ (2009) Non-probabilistic reliability-based topology optimization of geometrically nonlinear structures using convex models. Comput Methods Appl Mech Eng 198:3228–3238MathSciNetCrossRefzbMATHGoogle Scholar
  19. Kang Z, Luo YJ, Li A (2011) On non-probabilistic reliability-based design optimization of structures with uncertain-but-bounded parameters. Struct Saf 33:196–205CrossRefGoogle Scholar
  20. Kayedpour F, Amiri M, Rafizadeh M, Nia AS (2016) Multi-objective redundancy allocation problem for a system with repairable components considering instantaneous availability and strategy selection. Reliab Eng Syst Saf 160:11–20CrossRefGoogle Scholar
  21. Kharmanda G, Olhoff N, Mohamed A, Lemaire M (2004) Reliability-based topology optimization. Struct Multidiscip Optim 26:295–307CrossRefGoogle Scholar
  22. Kuschel N (2000) Time-variant reliability-based structural optimization using sorm. Optimization 47:349–368MathSciNetCrossRefzbMATHGoogle Scholar
  23. Kuschel N, Rackwitz R (2000) Optimal design under time-variant reliability constraints. Struct Saf 22:113–127CrossRefzbMATHGoogle Scholar
  24. Li XK, Qiu HB, Chen ZZ, Gao L, Shao XY (2016) A local Kriging approximation method using MPP for reliability-based design optimization. Comput Struct 162:102–115CrossRefGoogle Scholar
  25. Liu X, Zhang ZY, Yin LR (2017) A multi-objective optimization method for uncertain structures based on nonlinear interval number programming method. Mech Based Des Struct Mach 45:25–42CrossRefGoogle Scholar
  26. Luo YJ, Li A, Kang Z (2011) Reliability-based design optimization of adhesive bonded steel – concrete composite beams with probabilistic and non-probabilistic uncertainties. Eng Struct 33:2110–2119CrossRefGoogle Scholar
  27. Madsen PH, Krenk S (1984) An integral equation method for the first-passage problem in random vibration. J Appl Mech 51:674–679MathSciNetCrossRefzbMATHGoogle Scholar
  28. Nikolaidis E, Burdisso R (1988) Reliability based optimization: A safety index approach. Comput Struct 28:781–788CrossRefzbMATHGoogle Scholar
  29. Qiu ZP, Elishakoff I (2001) Anti-optimization technique – A generalization of interval analysis for nonprobabilistic treatment of uncertainty. Chaos, Solitons Fractals 12:1747–1759MathSciNetCrossRefzbMATHGoogle Scholar
  30. Qiu ZP, Wang XJ, Xu MH (2013) Uncertainty-based design optimization technology oriented to engineering structures. Science Press, BeijingGoogle Scholar
  31. Rice SO (1944) Mathematical analysis of random noise. Bell Syst Tech J 23:282–332MathSciNetCrossRefzbMATHGoogle Scholar
  32. Sickert JU, Graf W, Reuter U (2005) Application of fuzzy randomness to time-dependent reliability. Proc ICOSSAR:1709–1716Google Scholar
  33. Singh A, Mourelatos ZP, Li J (2010) Design for lifecycle cost and preventive maintenance using time-dependent reliability. Adv Mater Res 118-120:10–16CrossRefGoogle Scholar
  34. Song J, Kiureghian AD (2006) Joint first-passage probability and reliability of systems under stochastic excitation. J Eng Mech 132:65–77CrossRefGoogle Scholar
  35. Spence SMJ, Gioffrè M (2011) Efficient algorithms for the reliability optimization of tall buildings. J Wind Eng Ind Aerodyn 99:691–699CrossRefGoogle Scholar
  36. Wang ZQ, Wang PF (2012) A nested extreme response surface approach for time-dependent reliability-based design optimization. J Mech Des 134:67–75Google Scholar
  37. Wang BY, Wang XG, Zhu LS, Lu H (2011a) Time-dependent reliability-based robust optimization design of components structure. Adv Mater Res 199-200:456–462CrossRefGoogle Scholar
  38. Wang XJ, Wang L, Elishakoff I, Qiu ZP (2011b) Probability and convexity concepts are not antagonistic. Acta Mech 219:45–64CrossRefzbMATHGoogle Scholar
  39. Wang Y, Zeng SK, Guo JB (2013) Time-dependent reliability-based design Optimization utilizing nonintrusive polynomial chaos. J Appl Math 2013:561–575zbMATHGoogle Scholar
  40. Wang XJ, Wang L, Qiu ZP (2014a) A feasible implementation procedure for interval analysis method from measurement data. Appl Math Model 38:2377–2397MathSciNetCrossRefGoogle Scholar
  41. Wang L, Wang XJ, Xia Y (2014b) Hybrid reliability analysis of structures with multi-source uncertainties. Acta Mech 225:413–430CrossRefzbMATHGoogle Scholar
  42. Wang L, Wang XJ, Chen X, Wang RX (2015) Time-variant reliability model and its measure index of structures based on a non-probabilistic interval process. Acta Mech 226:3221–3241MathSciNetCrossRefGoogle Scholar
  43. Wang L, Wang XJ, Wang RX, Chen X (2016a) Reliability-based design optimization under mixture of random, interval and convex uncertainties. Arch Appl Mech 2016:1–27Google Scholar
  44. Wang L, Wang XJ, Li YL, Lin GP, Qiu ZP (2016b) Structural time-dependent reliability assessment of the vibration active control system with unknown-but-bounded uncertainties. Struct Control Health Monit 24:e1965CrossRefGoogle Scholar
  45. Wang L, Wang XJ, Su H, Lin GP (2016c) Reliability estimation of fatigue crack growth prediction via limited measured data. Int J Mech Sci 121:44–57CrossRefGoogle Scholar
  46. Wei ZP, Li T (2011) Non-probabilistic time-dependent reliability model of a structure based on strength degradation analysis. Mech Sci Technol Aerosp Eng 30:1397–1401Google Scholar
  47. Xu B, Zhao L, Li WY, He JJ, Xie YM (2016) Dynamic response reliability based topological optimization of continuum structures involving multi-phase materials. Compos Struct 149:134–144CrossRefGoogle Scholar
  48. Yang C, Lu ZX (2017) An interval effective independence method for optimal sensor placement based on non-probabilistic approach. Sci China Technol Sci 60:186–198CrossRefGoogle Scholar
  49. Yi XJ, Lai YH, Dong HP, Hou P (2016) A reliability optimization allocation method considering differentiation of functions. Int J Comput Methods 13:1–18MathSciNetCrossRefzbMATHGoogle Scholar
  50. Yoon JT, Youn BD, Wang PF, Hu C, (2013) A time-dependent framework of resilience-driven system design and its application to wind turbine system design. In: World Congress on Structural and Multidisciplinary OptimizationGoogle Scholar
  51. Zhang JF, Wang JG, Du XP (2011) Time-dependent probabilistic synthesis for function generator mechanisms. Mech Mach Theory 46:1236–1250CrossRefzbMATHGoogle Scholar
  52. Zhang DQ, Han X, Jiang C, Liu J, Long XY (2015) The interval PHI2 analysis method for time-dependent reliability. Sci Sin Phys Mech Astron 45:054601CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Lei Wang
    • 1
  • Xiaojun Wang
    • 1
  • Di Wu
    • 2
  • Menghui Xu
    • 3
  • Zhiping Qiu
    • 1
  1. 1.Institute of Solid MechanicsBeihang UniversityBeijingChina
  2. 2.China Academy of Launch Vehicle Technology R&D CenterBeijingChina
  3. 3.Faculty of Mechanical Engineering & MechanicsNingbo UniversityZhejiangChina

Personalised recommendations