A novel worst case approach for robust optimization of large scale structures

Abstract

In robust optimization, an optimum solution of a system is obtained when some uncertainties exist in the system. The uncertainty can be defined by probabilistic characteristics or deterministic intervals (uncertainty ranges or tolerances) that are the main concern in this study. An insensitive objective function is obtained with regard to the uncertainties or the worst case is considered for the objective function within the intervals in robust optimization. A supreme value within the uncertainty interval is minimized. The worst case approach has been extensively utilized in the linear programming (LP) community. However, the method solved only small scale problems of structural optimization where nonlinear programming (NLP) is employed. In this research, a novel worst case approach is proposed to solve large scale problems of structural optimization. An uncertainty interval is defined by a tolerance range of a design variable or problem parameter. A supreme value is obtained by optimization of the objective function subject to the intervals, and this process yields an inner loop. The supremum is minimized in the outer loop. Linearization of the inner loop is proposed to save the computational time for optimization. This technique can be easily extended for constraints with uncertainty intervals because the worst case of a constraint should be satisfied. The optimum sensitivity is utilized for the sensitivity of a supremum in the outer loop. Three examples including a mathematical example and two structural applications are presented to validate the proposed idea.

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

References

  1. [1]

    J. K. Allen, C. Seepersad, H. Choi and F. Mistree, Robust design for multiscale and multidisciplinary applications, ASME Journal of Mechanical Design, 128 (4) (2006) 832–843.

    Article  Google Scholar 

  2. [2]

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

    Article  Google Scholar 

  3. [3]

    H. G. Beyer and B. Sendhoff, Robust optimization-a comprehensive survey, Computer Methods in Applied Mechanics and Engineering, 196 (33) (2007) 3190–3218.

    MathSciNet  Article  MATH  Google Scholar 

  4. [4]

    I. S. Park and R. V. Grandhi, Quantifying multiple types of uncertainty in physics-based simulation using Bayesian model averaging, AIAA Journal, 49 (5) (2011) 1038–1045.

    Article  Google Scholar 

  5. [5]

    W. Yao, X. Chen, W. Luo, M. Van Tooren and J. Guo, Review of uncertainty-based multidisciplinary design optimization methods for aerospace vehicles, Progress in Aerospace Sciences, 47 (6) (2011) 450–479.

    Article  Google Scholar 

  6. [6]

    B. A. Steinfeldt and R. D. Braun, Using dynamical systems concepts in multidisciplinary design, AIAA Journal (2014).

    Google Scholar 

  7. [7]

    K. H. Lee and G. J. Park, Robust optimization considering tolerances of design variables, Computers & Structures, 79 (1) (2001) 77–86.

    Article  Google Scholar 

  8. [8]

    D. B. Parkinson, The application of a robust design method to tolerancing, Transactions-American Society of Mechanical Engineers Journal of Mechanical Design, 122 (2) (2000) 149–154.

    Google Scholar 

  9. [9]

    Y. Tang, J. Chen and J. Wei, A sequential algorithm for reliability-based robust design optimization under epistemic uncertainty, Journal of Mechanical Design, 134 (1) (2012) 014502.

    Article  Google Scholar 

  10. [10]

    L. Du, K. K. Choi, B. D. Youn and D. Gorsich, Possibilitybased design optimization method for design problems with both statistical and fuzzy input data, ASME Journal of Mechanical Design, 128 (4) (2006) 928–935.

    Article  Google Scholar 

  11. [11]

    M. Lombardi and R. T. Haftka, Anti-optimization technique for structural design under load uncertainties, Computer Methods in Applied Mechanics and Engineering, 157 (1–2) (1998) 19–31.

    Article  MATH  Google Scholar 

  12. [12]

    I. Elishakoff, B. Kriegesmann, R. Rolfes, C. Hühne and A. Kling, Optimization and antioptimization of buckling load for composite cylindrical shells under uncertainties, AIAA Journal, 50 (7) (2012) 1513–1524.

    Article  Google Scholar 

  13. [13]

    J. S. Arora, Introduction to optimum design, Academic Press (2011).

    Google Scholar 

  14. [14]

    A. Ben-Tal, L. E. Ghaoui and A. Nemirovski, Robust optimization, Princeton University Press (2009).

    Google Scholar 

  15. [15]

    I. Elishakoff and M. Ohsaki, Optimization and antioptimization of structures under uncertainty, Imperial College Press (2010).

    Google Scholar 

  16. [16]

    I. Elishakoff, R. Haftka and J. Fang, Structural design under bounded uncertainty—optimization with antioptimization, Computers & Structures, 53 (6) (1994) 1401–1405.

    Article  MATH  Google Scholar 

  17. [17]

    O. Amir and I. Elishakoff, Intricate interrelation between robustness and probability in the context of structural optimization, ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part B: Mechanical Engineering, 1 (3) (2015) 031003.

    Article  Google Scholar 

  18. [18]

    G. J. Park, Analytic methods for design practice, Springer London (2007).

    Google Scholar 

  19. [19]

    J. Han and B. Kwak, Robust optimization using a gradient index: MEMS applications, Structural and Multidisciplinary Optimization, 27 (6) (2004) 469–478.

    Article  Google Scholar 

  20. [20]

    M. Li, S. Azarm and A. Boyars, A new deterministic approach using sensitivity region measures for multiobjective robust and feasibility robust design optimization, ASME Journal of Mechanical Design, 128 (4) (2006) 874–883.

    Article  Google Scholar 

  21. [21]

    J. F. Barthelemy and J. Sobieszczanski Sobieski, Extrapolation of optimum design based on sensitivity derivatives (1983).

    Google Scholar 

  22. [22]

    G. N. Vanderplaats, Numerical optimization techniques for engineering design, Vanderplaats Research & Development, Incorporated (2005).

    Google Scholar 

  23. [23]

    G. N. Vanderplaats and N. Yoshida, Efficient calculation of optimum design sensitivity, AIAA Journal, 23 (11) (1985) 1798–1803.

    MathSciNet  Article  MATH  Google Scholar 

  24. [24]

    S. K. Choi, R. Grandhi and R. A. Canfield, Reliabilitybased structural design, Springer London (2006).

    Google Scholar 

  25. [25]

    A. Belegundu and S. Zhang, Robustness of design through minimum sensitivity, ASME Journal of Mechanical Design, 114 (6) (1992) 213–217.

    Article  Google Scholar 

  26. [26]

    B. Ramakrishnan and S. S. Rao, A general loss function based optimization procedure for robust design, Engineering Optimization, 25 (4) (1996) 255–276.

    Article  Google Scholar 

  27. [27]

    S. Sundaresan, K. Ishii and D. R. Houser, A robust optimization procedure with variations on design variables and constraints, Engineering Optimization+ A35, 24 (2) (1995) 101–117.

    Article  Google Scholar 

  28. [28]

    J. E. Renaud, Automatic differentiation in robust optimization, AIAA Journal, 35 (6) (1997) 1072–1079.

    Article  MATH  Google Scholar 

  29. [29]

    W. Chen, M. M. Wiecek and J. Zhang, Quality utility—a compromise programming approach to robust design, ASME Journal of Mechanical Design, 121 (2) (1999) 179–187.

    Article  Google Scholar 

  30. [30]

    I. Doltsinis and Z. Kang, Robust design of structures using optimization methods, Computer Methods in Applied Mechanics and Engineering, 193 (23) (2004) 2221–2237.

    Article  MATH  Google Scholar 

  31. [31]

    Z. P. Mourelatos and J. Liang, A methodology for tradingoff performance and robustness under uncertainty, ASME Journal of Mechanical Design, 128 (4) (2006) 856–863.

    Article  Google Scholar 

  32. [32]

    A. Parkinson, C. Sorensen and N. Pourhassan, A general approach for robust optimal design, ASME Journal of Mechanical Design, 115 (1) (1993) 74–80.

    Article  Google Scholar 

  33. [33]

    K. H. Lee and G. J. Park, A global robust optimization using Kriging based approximation model, JSME International Journal Series C Mechanical Systems, Machine Elements and Manufacturing, 49 (3) (2006) 779–788.

    Google Scholar 

  34. [34]

    F. A. C. Viana, T. W. Simpson, V. Balabanov and V. Toropov, Metamodeling in multidisciplinary design optimization: How far have we really come?, AIAA Journal, 52 (4) (2014) 670–690.

    Article  Google Scholar 

  35. [35]

    S. Zhang, P. Zhu, W. Chen and P. Arendt, Concurrent treatment of parametric uncertainty and metamodeling uncertainty in robust design, Structural and Multidisciplinary Optimization, 47 (1) (2013) 63–76.

    MathSciNet  Article  MATH  Google Scholar 

  36. [36]

    Y. Zhang and S. Hosder, Robust design optimization under mixed uncertainties with stochastic expansions, Journal of Mechanical Design, 135 (8) (2013) 081005.

    Article  Google Scholar 

  37. [37]

    M. S. Phadke, Quality engineering using robust design, Prentice Hall (1989).

    Google Scholar 

  38. [38]

    X. Du and W. Chen, Efficient uncertainty analysis methods for multidisciplinary robust design, AIAA Journal, 40 (3) (2002) 545–581.

    Article  Google Scholar 

  39. [39]

    B. D. Youn, K. K. Choi, L. Du and D. Gorsich, Integration of possibility-based optimization and robust design for epistemic uncertainty, Journal of Mechanical Design, 129 (8) (2007) 876–882.

    Article  Google Scholar 

  40. [40]

    M. Li and S. Azarm, Multiobjective collaborative robust optimization with interval uncertainty and interdisciplinary uncertainty propagation, ASME Journal of Mechanical Design, 130 (8) (2008) 081402.

    Article  Google Scholar 

  41. [41]

    Korean industrial standards: General tolerances-part 1, KS B 0412.

  42. [42]

    MatWeb, https://doi.org/www.matweb.com.

  43. [43]

    J. S. Arora and C. H. Tseng, User's manual for program IDESIGN version 3.5 for prime and apollo computers, Optimal Design Laboratory, College of Engineering, Univ. (1986).

    Google Scholar 

  44. [44]

    Vanderplaats research and development (VR&D) Inc DOT version 5.7 user's manual, Colorado Springs, CO (2001).

  45. [45]

    S. J. Lee, Robust optimization using supremum of functions for nonlinear programming problems, Ph.D. Thesis, Hanyang University, Seoul, Korea (2014) doi:10.3795/KSME-A.2014.38.5.535 [In Korean].

    Google Scholar 

  46. [46]

    Vanderplaats research and development (VR&D) Inc GENESIS version 12.0 user's manual, Colorado Springs, CO (2010).

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Gyung-Jin Park.

Additional information

Recommended by Associate Editor Gang-Won Jang

Gyung-Jin Park received the B.S. degree from Hanyang University, Korea in 1980, M.S. degree from KAIST, Korea, in 1982, and the Ph.D. from the University of Iowa, USA, in 1986. In 1986–1988, he worked as an Assistant Professor at Purdue University at Indianapolis, USA. His research focuses on Structural Optimization, machine design, design theory and MDO. His work has yielded over 4 books and 360 technical papers. He is currently a Professor in the Department of Mechanical Engineering at Hanyang University, Ansan City, Korea.

Se-Jung Lee received the B.S. degree in mechanical engineering from Soon-ChunHyang University, Korea, in 2003, M.S. degree from Hanyang University, Korea, in 2009, and the Ph.D. from Hanyang University, Korea, in 2014. Her research interests include design methodology and robust design.

Min-Ho Jeong received the B.S. degree in mechanical engineering from Hanyang University, Korea in 2015. He is currently pursuing the Ph.D. degree at Hanyang University. His research interests include design methodology, structural optimization and robust design.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Lee, SJ., Jeong, MH. & Park, GJ. A novel worst case approach for robust optimization of large scale structures. J Mech Sci Technol 32, 4255–4269 (2018). https://doi.org/10.1007/s12206-018-0824-2

Download citation

Keywords

  • Robust optimization
  • Supremum of a function
  • Worst case approach