Skip to main content
SpringerLink
Log in

Reliability-based structural optimization with probability and convex set hybrid models

  • Research Paper
  • Published:
Structural and Multidisciplinary Optimization Aims and scope Submit manuscript

Abstract

For structural systems exhibiting both probabilistic and bounded uncertainties, it may be suitable to describe these uncertainties with probability and convex set models respectively in the design optimization problem. Based on the probabilistic and multi-ellipsoid convex set hybrid model, this paper presents a mathematical definition of reliability index for measuring the safety of structures in presence of parameter or load uncertainties. The optimization problem incorporating such reliability constraints is then mathematically formulated. By using the performance measure approach, the optimization problem is reformulated into a more tractable one. Moreover, the nested double-loop optimization problem is transformed into an approximate single-loop minimization problem by considering the optimality conditions and linearization of the limit-state function, which further facilitates efficient solution of the design problem. Numerical examples demonstrate the validity of the proposed formulation as well as the efficiency of the presented numerical techniques.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Adduri PR, Penmetsa RC (2007) Bounds on structural system reliability in the presence of interval variables. Comput Struct 85:320–329

    Article  Google Scholar 

  • Au FTK, Cheng YS, Tham LG, Zeng GW (2003) Robust design of structures using convex models. Comput Struct 81:2611–2619

    Article  Google Scholar 

  • Ben-Haim Y, Elishakoff I (1990) Convex models of uncertainty in applied mechanics. Elsevier Science, Amsterdam

    MATH  Google Scholar 

  • Berleant DJ, Ferson S, Kreinovich V, Lodwich WA (2005) Combining interval and probabilistic uncertainty: foundations, algorithms, challenges—an overview. In: Proceedings of the 4th international symposium on imprecise probabilities and their applications, Pittsburgh, Pennsylvania

  • Beyer HG, Sendhoff B (2007) Robust optimization—a comprehensive survey. Comput Methods Appl Mech Eng 196:3190–3218

    Article  MATH  MathSciNet  Google Scholar 

  • Buckley JJ (2005) Fuzzy probabilities: new approach and applications. Springer, Berlin

    MATH  Google Scholar 

  • Chen X, Hasselman TK, Neill DJ (1997) Reliability based structural design optimization for practical applications. In: Proceedings of the 38th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference and exhibit, Kissimmee, FL, 7–10 April

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

    Article  Google Scholar 

  • Doltsinis I, Kang Z (2004) Robust design of structures using optimization methods. Comput Methods Appl Mech Eng 193:2221–2237

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  • Du L, Choi KK (2008) An inverse analysis method for design optimization with both statistical and fuzzy uncertainties. Struct Multidisc Optim 37:107–119

    Article  Google Scholar 

  • Du L, Choi KK, Youn BD (2006a) Inverse possibility analysis method for possibility-based design optimization. AIAA J 44(11):2682–2690

    Article  Google Scholar 

  • Du L, Choi KK, Youn BD, Gorsich D (2006b) Possibility-based design optimization method for design problems with both statistical and fuzzy input data. J Mech Des 128:928–935

    Article  Google Scholar 

  • Du X, Sudjianto A, Huang B (2005) Reliability-based design with the mixture of random and interval variables. J Mech Des 127:1068–1076

    Article  Google Scholar 

  • Elishakoff I (1995) Essay on uncertainties in elastic and viscoelastic structures: from A. M. Freudenthal’s criticisms to modern convex modeling. Comput Struct 56:871–895

    Article  MATH  Google Scholar 

  • Elishakoff I (1999) Are probabilistic and anti-optimization approaches compatible? In: Elishakoff I (ed) Whys and hows in uncertainty modelling: probability, fuzziness and anti-optimization. Springer, New York, pp 263–341

    Google Scholar 

  • Elishakoff I, Colombi P (1993) Combination of probabilistic and convex models of uncertainty when scarce knowledge is present on acoustic excitation parameters. Comput Methods Appl Mech Eng 104(2):187–209

    Article  MATH  MathSciNet  Google Scholar 

  • Elseifi MA, Gürdal Z, Nikolaidis E (1999) Convex/probabilistic models of uncertainties in geometric imperfections of stiffened composite panels. AIAA J 37:468–474

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Frangopol DM, Corotis RB (1996) Reliability-based structural system optimization: state-of-the-art versus state-of-the-practice. In: Cheng FY (ed) Proceedings of the 12th conference on analysis and computation, Chicago. Analysis and computation. ASCE, New York, pp 67–76

    Google Scholar 

  • Hall JW, Lawry J (2004) Generation, combination and extension of random set approximations to coherent lower and upper probabilities. Reliab Eng Syst Saf 85:89–101

    Article  Google Scholar 

  • Hasofer AM, Lind NC (1974) Exact and invariant second-moment code format. ASCE J Eng Mech Div 100(1):111–121

    Google Scholar 

  • Jaynes ET (1957) Information theory and statistical mechanics. Phys Rev 108:171–197

    Article  MathSciNet  Google Scholar 

  • Jiang C, Han X, Liu GR (2007) Optimization of structures with uncertain constraints based on convex model and satisfaction degree of interval. Comput Methods Appl Mech Eng 196:4791–4800

    Article  MATH  Google Scholar 

  • Karanki DR, Kushwaha HS, Verma AK, Ajit S (2009) Uncertainty analysis based on probability bounds (p-box) approach in probabilistic safety assessment. Risk Anal 29(5):662–675

    Article  Google Scholar 

  • Kreinovich V, Xiang G, Starks SA et al (2006) Towards combining probabilistic and interval uncertainty in engineering calculations: algorithms for computing statistics under interval uncertainty, and their computational complexity. Reliab Comput 12:1385–3139

    MathSciNet  Google Scholar 

  • Kuschel N, Rackwitz R (2000) A new approach for structural optimization of series systems. In: Melchers RE, Stewart MG (ed) Applications of statistics and probability. Balkema, Rotterdam, pp 987–994

    Google Scholar 

  • Lawrence C, Zhou JL, Tits AL (1997) User’s guide for CFSQP version 2.5. Available at http://www.aemdesign.com

  • Lee JO, Yang S, Ruy WS (2002) A comparative study on reliability-index and target-performance-based probabilistic structural design optimization. Comput Struct 80:257–269

    Article  Google Scholar 

  • Liang J, Mourelatos ZP, Nikolaidis E (2007) A single-loop approach for system reliability-based design optimization. J Mech Des 129:1215–1225

    Article  Google Scholar 

  • Luo Y, Kang Z, Luo Z et al (2009) Continuum topology optimization with non-probabilistic reliability constraints based on multi-ellipsoid convex model. Struct Multidisc Optim 39:297–310

    Article  MathSciNet  Google Scholar 

  • Melchers RE (1999) Structural reliability analysis and prediction, 2nd edn. Wiley, Chichester

    Google Scholar 

  • Minguez R, Castillo E (2009) Reliability-based optimization in engineering using decomposition techniques and FORMS. Struct Saf 31:214–223

    Article  Google Scholar 

  • Moens D, Vandepitte D (2005) A survey of non-probabilistic uncertainty treatment in finite element analysis. Comput Methods Appl Mech Eng 194:1527–1555

    Article  MATH  Google Scholar 

  • Möller B, Beer M (2004) Fuzzy randomness: uncertainty in civil engineering and computational mechanics. Springer, Berlin

    MATH  Google Scholar 

  • Möller B, Beer M (2008) Engineering computation under uncertainty—capabilities of non-traditional models. Comput Struct 86:1024–1041

    Article  Google Scholar 

  • Mourelatos ZP, Zhou J (2005) Reliability estimation and design with insufficient data based on possibility theory. AIAA J 43(8):1696–1705

    Article  Google Scholar 

  • Neumaier A, Pownuk A (2007) Linear systems with large uncertainties, with applications to truss structures. Reliab Comput 13:149–172

    Article  MATH  MathSciNet  Google Scholar 

  • Pantelides CP, Ganzerli S (1998) Design of trusses under uncertain loads using convex models. J Struct Eng 124:318–329

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Penmetsa RC, Grandhi RV (2002) Efficient estimation of structural reliability for problems with uncertain intervals. Comput Struct 80:1103–1112

    Article  Google Scholar 

  • Qiu Z, Elishakoff I (1998) Antioptimization of structures with large uncertain-but-non-random parameters via interval analysis. Comput Methods Appl Mech Eng 152:361–372

    Article  MATH  Google Scholar 

  • Qiu Z, Yang D, Elishakoff I (2008) Probabilistic interval reliability of structural systems. Int J Solids Struct 45:2850–2860

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Royset JO, Der Kiureghian A, Polak E (2001) Reliability-based optimal structural design by the decoupling approach. Reliab Eng Syst Saf 73:213–221

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Soize C (2001) Maximum entropy approach for modeling random uncertainties in transient elastodynamics. J Acoust Soc Am 109:1979–1996

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Utkin LV, Kozine IO (2005) Computing system reliability given interval-value characteristics of the components. Reliab Comput 11:19–34

    Article  MATH  MathSciNet  Google Scholar 

  • Vietor T (1997) Stochastic optimization for mechanical structures. In: Marti K (ed) Special issue on structural reliability and stochastic structural optimization. Math Methods Oper Res 46:377–408

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian, 116024, China

    Zhan Kang

  2. School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’an, 710072, China

    Yangjun Luo

Authors
  1. Zhan Kang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yangjun Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhan Kang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kang, Z., Luo, Y. Reliability-based structural optimization with probability and convex set hybrid models. Struct Multidisc Optim 42, 89–102 (2010). https://doi.org/10.1007/s00158-009-0461-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00158-009-0461-6

Keywords

Search

Navigation