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Robustness-based design optimization under data uncertainty

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Abstract

This paper proposes formulations and algorithms for design optimization under both aleatory (i.e., natural or physical variability) and epistemic uncertainty (i.e., imprecise probabilistic information), from the perspective of system robustness. The proposed formulations deal with epistemic uncertainty arising from both sparse and interval data without any assumption about the probability distributions of the random variables. A decoupled approach is proposed in this paper to un-nest the robustness-based design from the analysis of non-design epistemic variables to achieve computational efficiency. The proposed methods are illustrated for the upper stage design problem of a two-stage-to-orbit (TSTO) vehicle, where the information on the random design inputs are only available as sparse point data and/or interval data. As collecting more data reduces uncertainty but increases cost, the effect of sample size on the optimality and robustness of the solution is also studied. A method is developed to determine the optimal sample size for sparse point data that leads to the solutions of the design problem that are least sensitive to variations in the input random variables.

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References

  • Agarwal H, Mozumder CK, Renaud JE, Watson LT (2007) An inverse-measure-based unilevel architecture for reliability-based design optimization. Struct Multidisc Optim 33:217–227

    Article  Google Scholar 

  • Bichon BJ, McFarland JM, Mahadevan S (2008) Using Bayesian inference and efficient global reliability analysis to explore distribution uncertainty. In: 49th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference, 7–10 April 2008, Schaumburg, IL

  • Bras BA, Mistree F (1993) Robust design using compromise decision support problems. Eng Optim 21:213–239

    Article  Google Scholar 

  • Bras BA, Mistree F (1995) A compromise decision support problem for robust and axiomatic design. ASME J Mech Des 117(1):10–19

    Article  Google Scholar 

  • Bonett DG (2006) Approximate confidence interval for standard deviation of nonnormal distributions. Comput Stat Data Anal 50:775–782

    Article  MathSciNet  MATH  Google Scholar 

  • Cagan J, Williams BC (1993) First-order necessary conditions for robust optimality. In: ASME advances in design automation. Albuquerque, NM, ASME DE-Vol. 65-1

  • Chen W, Allen JK, Mistree F, Tsui K-L (1996) A procedure for robust design: minimizing variations caused by noise factors and control factors. ASME J Mech Des 118:478–485

    Article  Google Scholar 

  • Chen W, Wiecek MM, Zhang J (1999) Quality utility—a compromise programming approach to robust design. J Mech Des (ASME) 121:179–187

    Article  Google Scholar 

  • Chen W, Sahai A, Messac A, Sundararaj GJ (2000) Exploration of the effectiveness of physical programming in robust design. J Mech Des 122:155–163

    Article  Google Scholar 

  • Cheng H, Sandu A (2009) Efficient uncertainty quantification with the polynomial chaos method for stiff systems. Math Comput Simul 79:3278–3295

    Article  MathSciNet  MATH  Google Scholar 

  • Chiralaksanakul A, Mahadevan S (2005) First-order approximation methods in reliability-based design optimization. J Mech Des 127(5):851–857

    Article  Google Scholar 

  • Cojbasic V, Tomovic A (2007) Nonparametric confidence intervals for population variance of one sample and the difference of variances of two samples. Comput Stat Data Anal 51:5562–5578

    Article  MathSciNet  MATH  Google Scholar 

  • Dai Z, Mourelatos ZP (2003) Incorporating epistemic uncertainty in robust design. In: Proceedings of DETC, 2003 ASME design engineering technical conferences, 2–6 September 2003, Chicago, Illinois, USA

  • 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, Beiqing Huang B (2007) Reliability-based design optimization with equality constraints. Int J Numer Methods Eng 72:1314–1331

    Article  MATH  Google Scholar 

  • Du X, Chen W (2000) Towards a better understanding of modeling feasibility robustness in engineering. ASME J Meach Des 122(4):385–394

    Article  Google Scholar 

  • Du X, Sudjianto A, Chen W (2004) An integrated framework for optimization under uncertainty using inverse reliability strategy. ASME

  • Fletcher R (1987) Practical methods of optimization, 2nd edn. Wiley, New York

    MATH  Google Scholar 

  • Ghanem R, Spanos P (1991) Stochastic finite elements: a spectral approach. Springer, New York

    MATH  Google Scholar 

  • Haldar A, Mahadevan S (2000) Probability, reliability and statistical methods in engineering design. Wiley

  • Hong HP (1998) An efficient point estimate method for probabilistic analysis. Reliab Eng Syst Saf 59:261–267

    Article  Google Scholar 

  • Huang B, Du X (2007) Analytical robustness assessment for robust design. Struct Multidisc Optim 34:123–137

    Article  Google Scholar 

  • Johnson NJ (1978) Modified t tests and confidence intervals for assymmetrical populations. J Am Stat Assoc 73(363):536–544

    Article  MATH  Google Scholar 

  • Lee K-H, Park G-J (2001) Robust optimization considering tolerances of design variables. Comput Struct 79:77–86

    Article  Google Scholar 

  • Lee I, Choi KK, Du L, Gorsich D (2008) Dimension reduction method for reliability-based robust design optimization. Comput Struct 86(13–14):1550–1562

    Article  Google Scholar 

  • Marler RT, Arora JS (2004) Survey of multi-objective optimization methods for engineering. Struct Multidisc Optim 26:369–395

    Article  MathSciNet  Google Scholar 

  • Mavrotas G (2009) Effective implementation of the e-constraint method in multi objective mathematical programming problems. Appl Math Comput 213:455–465

    Article  MathSciNet  MATH  Google Scholar 

  • Messac A (1996) Physical programming effective optimization for computational design. AIAA J 34(1):149–158

    Article  MATH  Google Scholar 

  • Messac A, Ismail-Yahaya A (2002) Multiobjective robust design using physical programming. Struct Multidisc Optim 23:357–371

    Article  Google Scholar 

  • Messac A, Melachrinoudis E, Sukam CP (2001) Mathematical and pragmatic perspectives of physical programming. AIAA J 39(5):885–893

    Article  Google Scholar 

  • Oberkampf WL, Helton JC, Joslyn CA, Wojtkiewicz SF, Ferson S (2004) Challenge problems: uncertainty in system response given uncertain parameters. Reliab Eng Syst Saf 85:11–19

    Article  Google Scholar 

  • Panier ER, Tits AL (1993) On combining feasibility, descent and superlinear convergence in inequality constrained optimization. Math Program 59:261–276

    Article  MathSciNet  MATH  Google Scholar 

  • Park G-J, Lee T-H, Lee KH, Hwang K-H (2006) Robust design: an overview. AIAA J 44(1):181–191

    Article  Google Scholar 

  • Parkinson A, Sorensen C, Pourhassan N (1993) A general approach for robust optimal design. Trans ASME 115:74–80

    Article  Google Scholar 

  • Rahman S, Xu H (2004) A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics. Probab Eng Mech 19:393–408

    Article  Google Scholar 

  • Ramu P, Qu X, Youn BD, Haftka RT, Choi KK (2006) Inverse reliability measures and reliability-based design optimization. Int J Reliab Saf 1(1/2):187–205

    Article  Google Scholar 

  • Robert CP, Casella G (2004) Monte Carlo statistical methods, 2nd edn. Springer, New York

    MATH  Google Scholar 

  • Rosenblueth E (1975) Point estimates for probability moment. Proc Natl Acad Sci U S A 72(10):3812–3814

    Article  MathSciNet  MATH  Google Scholar 

  • Sim M (2004) Robust optimization. PhD dissertation submitted to the Sloan School of Management, Massachusetts Institute of Technology

  • Stevenson MD, Hartong AR, Zweber JV, Bhungalia AA, Grandhi RV (2002) Collaborative design environment for space launch vehicle design and optimization. In: Paper presented at the RTO AVT symposium on “reduction of military vehicle acquisition time and cost through advanced modeling and virtual simulation”, held in Paris, France

  • Sundaresan S, Ishii K, Houser DR (1995) A robust optimization procedure with variations on design variables and constraints. Eng Optim 24(2):101–117

    Article  Google Scholar 

  • Taguchi G (1993) Taguchi on robust technology development: bringing quality engineering upstream. ASME, New York

    Book  Google Scholar 

  • Wei DL, Cui ZS, Chen J (2009) Robust optimization based on a polynomial expansion of chaos constructed with integration point rules. J Mech Eng Sci 223(5):1263–1282 (Part C)

    Google Scholar 

  • Xu H, Rahman S (2004) A generalized dimension-reduction method for multidimensional integration in stochastic mechanics. Int J Numer Methods Eng 61(12):1992–2019

    Article  MATH  Google Scholar 

  • Youn BD, Choi KK, Du L (2007) Integration of possibility-based optimization and robust design for epistemic uncertainty. ASME J Mech Des 129(8):876–882

    Article  Google Scholar 

  • Yu J-C, Ishii K (1998) Design for robustness based on manufacturing variation patterns. Trans ASME 120:196–202

    Article  Google Scholar 

  • Zaman K, Rangavajhala S, McDonald PM, Mahadevan S (2011) A probabilistic approach for representation of interval uncertainty. Reliab Eng Syst Saf 96(1):117–130. Available online

    Article  Google Scholar 

  • Zeleny M (1973) Compromise programming. In: Cochrane JL, Zeleny M (eds) Multiple criteria decision making. University of South Carolina Press, Columbia, SC, pp 262–301

  • Zhang WH (2003) A compromise programming method using multibounds formulation and dual approach for multicriteria structural optimization. Int J Numer Methods Eng 58:661–678

    Article  MATH  Google Scholar 

  • Zhao Y-G, Ono T (2000) New point estimates for probability moments. J Eng Mech 126(4):433–436

    Article  Google Scholar 

  • Zhao Y-G, Ang AH-S (2003) System reliability assessment by method of Moments. J Struct Eng 129(10):1341–1349

    Article  Google Scholar 

  • Zou T, Mahadevan S (2006) Versatile formulation for multiobjective reliability-based design optimization. J Mech Des 128:1217

    Article  Google Scholar 

Download references

Acknowledgement

This study was supported by funds from NASA Langley Research Center under Cooperative Agreement No. NNX08AF56A1 (Technical Monitor: Mr. Lawrence Green). The support is gratefully acknowledged.

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Authors and Affiliations

  1. Vanderbilt University, Nashville, TN, USA

    Kais Zaman, Mark McDonald & Sankaran Mahadevan

  2. NASA Langley Research Center, Hampton, VA, USA

    Lawrence Green

Authors
  1. Kais Zaman
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  2. Mark McDonald
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  3. Sankaran Mahadevan
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  4. Lawrence Green
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Correspondence to Sankaran Mahadevan.

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Zaman, K., McDonald, M., Mahadevan, S. et al. Robustness-based design optimization under data uncertainty. Struct Multidisc Optim 44, 183–197 (2011). https://doi.org/10.1007/s00158-011-0622-2

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  • DOI: https://doi.org/10.1007/s00158-011-0622-2

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