Research Papers

A Nested Extreme Response Surface Approach for Time-Dependent Reliability-Based Design Optimization

[+] Author and Article Information
Zequn Wang

e-mail: zxwang5@wichita.edu

Pingfeng Wang

e-mail: pingfeng.wang@wichita.edu
Department of Industrial and Manufacturing Engineering,
Wichita State University, Wichita, KS 67260

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the Journal of Mechanical Design. Manuscript received May 12, 2012; final manuscript received October 10, 2012; published online November 15, 2012. Assoc. Editor: Zissimos P. Mourelatos.

J. Mech. Des 134(12), 121007 (Nov 15, 2012) (14 pages) doi:10.1115/1.4007931 History: Received May 12, 2012; Revised October 10, 2012

A primary concern in practical engineering design is ensuring high system reliability throughout a product's lifecycle, which is subject to time-variant operating conditions and component deteriorations. Thus, the capability of dealing with time-dependent probabilistic constraints in reliability-based design optimization (RBDO) is of vital importance in practical engineering design applications. This paper presents a nested extreme response surface (NERS) approach to efficiently carry out time-dependent reliability analysis and determine the optimal designs. This approach employs the kriging model to build a nested response surface of time corresponding to the extreme value of the limit state function. The efficient global optimization (EGO) technique is integrated with the NERS approach to extract the extreme time responses of the limit state function for any given system design. An adaptive response prediction and model maturation (ARPMM) mechanism is developed based on the mean square error (MSE) to concurrently improve the accuracy and computational efficiency of the proposed approach. With the nested response surface of time, the time-dependent reliability analysis can be converted into the time-independent reliability analysis, and existing advanced reliability analysis and design methods can be used. The NERS approach is compared with existing time-dependent reliability analysis approaches and integrated with RBDO for engineered system design with time-dependent probabilistic constraints. Two case studies are used to demonstrate the efficacy of the proposed NERS approach.

Copyright © 2012 by ASME
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Youn, B. D., Choi, K. K., and Du, L., 2005, “Enriched Performance Measure Approach for Reliability-Based Design Optimization,” AIAA J., 43(4), pp. 874–884. [CrossRef]
Youn, B., and Wang, P., 2009, “Complementary Interaction Method (CIM) for System Reliability Assessment,” J. Mech. Des., 131(4), p. 041004(15). [CrossRef]
Noh, Y., Choi, K. K., Lee, I., Gorsich, D., and Lamb, D., 2011, “Reliability-Based Design Optimization With Confidence Level for Non-Gaussian Distributions Using Bootstrap Method,” J. Mech. Des., 133(9), p. 091001(12). [CrossRef]
Wang, P., Hu, C., and Youn, B. D., 2011, “A Generalized Complementary Intersection Method for System Reliability Analysis and Design,” J. Mech. Des., 133(7), p. 071003(13). [CrossRef]
Du, X., and Chen, W., 2004, “Sequential Optimization and Reliability Assessment Method for Efficient Probabilistic Design,” J. Mech. Des., 126(2), pp. 225–233. [CrossRef]
Youn, B., Hu, C., and Wang, P., 2011, “Resilience-Driven System Design for Complex Engineered Systems,” J. Mech. Des., 133(10), p. 101011(15). [CrossRef]
Liu, D., and Peng, Y., 2012, “Reliability Analysis by Mean-Value Second-Order Expansion,” J. Mech. Des., 134(6), p. 061005(8). [CrossRef]
Youn, B. D., Choi, K., and Du, L., 2005, “Adaptive Probability Analysis Using an Enhanced Hybrid Mean Value Method,” Struct. Multidiscip. Optim., 29(2), pp. 134–148. [CrossRef]
Du, X., 2012, “First Order Reliability Method With Truncated Random Variables,” J. Mech. Des., 134(9), p. 091005(9). [CrossRef]
Rahman, S., and Xu, H., 2004, “A Univariate Dimension-Reduction Method for Multi-Dimensional Integration in Stochastic Mechanics,” Probab. Eng. Mech., 19(4), pp. 393–408. [CrossRef]
Youn, B. D., Xi, Z., and Wang, P., 2008, “Eigenvector Dimension Reduction (EDR) Method for Sensitivity-Free Probability Analysis,” Struct. Multidiscip. Optim., 37(1), pp. 13–28. [CrossRef]
Hazelrigg, G. A., 1998, “A Framework for Decision-Based Engineering Design,” J. Mech. Des., 120(4), pp. 653–658. [CrossRef]
Zhang, R., and Mahadevan, S., 2000, “Model Uncertainty and Bayesian Updating in Reliability-Based Inspection,” Struct. Saf., 22(2), pp. 145–160. [CrossRef]
Youn, B. D., and Wang, P., 2008, “Bayesian Reliability-Based Design Optimization Using Eigenvector Dimension Reduction (EDR) Method,” Struct. Multidiscip. Optim., 36(2), pp. 107–123. [CrossRef]
Wang, P., Youn, B. D., Xi, Z., and Kloess, A., 2009, “Bayesian Reliability Analysis With Evolving, Insufficient, and Subjective Data Sets,” J. Mech. Des., 131(11), p. 111008(11). [CrossRef]
Currin, C., Mitchell, T., Morris, M., and Ylvisaker, D., 1991, “Bayesian Prediction of Deterministic Functions, With Applications to the Design and Analysis of Computer Experiments,” J. Am. Stat. Assoc., 86(416), pp. 953–963. [CrossRef]
Li, J., Chen, J., and Fan, W., 2007, “The Equivalent Extreme-Value Event and Evaluation of the Structural System Reliability,” Struct. Saf., 29(2), pp. 112–131. [CrossRef]
Chen, J. B., and Li, J., 2007, “The Extreme Value Distribution and Dynamic Reliability Analysis of Nonlinear Structures With Uncertain Parameters,” Struct. Saf., 29(2), pp. 77–93. [CrossRef]
Li, J., and Mourelatos, Z. P., 2009, “Time-Dependent Reliability Estimation for Dynamic Problems Using a Niching Genetic Algorithm,” J. Mech. Des., 131(7), p. 071009. [CrossRef]
Lutes, L. D., and Sarkani, S., 2009, “Reliability Analysis of Systems Subject to First-Passage Failure,” NASA Technical Report No. NASA/CR-2009-215782.
Kuschel, N., and Rackwitz, R., 2000, “Optimal Design Under Time-Variant Reliability Constraints,” Struct. Saf., 22(2), pp. 113–127. [CrossRef]
Li, C., and Der Kiureghian, A., 1995, “Mean Out-Crossing Rate of Nonlinear Response to Stochastic Input,” Proceedings of ICASP-7, Balkema, Rotterdam, pp. 295–302.
Schrupp, K., and Rackwitz, R., 1988, “Out-crossing Rates of Marked Poisson Cluster Processes in Structural Reliability,” Appl. Math. Model., 12(5), pp. 482–490. [CrossRef]
Breitung, K., 1994, “Asymptotic Approximations for the Crossing Rates of Poisson Square Waves,” NIST Special Publication SP, pp. 75–75.
Singh, A., Mourelatos, Z. P., and Li, J., 2010, “Design for Lifecycle Cost Using Time-Dependent Reliability,” J. Mech. Des., 132(9), p. 091008. [CrossRef]
Andrieu-Renaud, C., Sudret, B., and Lemaire, M., 2004, “The PHI2 Method: A Way to Compute Time-Variant Reliability,” Reliab. Eng. Syst. Saf., 84(1), pp. 75–86. [CrossRef]
Rackwitz, R., 1998, “Computational Techniques in Stationary and Non-Stationary Load Combination—A Review and Some Extensions,” J. Struct. Eng., 25(1), pp. 1–20.
Sudret, B., 2008, “Analytical Derivation of the Out-Crossing Rate in Time-Variant Reliability Problems,” Struct. Infrastruct. Eng., 4(5), pp. 353–362. [CrossRef]
Zhang, J., and Du, X., 2011, “Time-Dependent Reliability Analysis for Function Generator Mechanisms,” J. Mech. Des., 133(3), p. 031005(9). [CrossRef]
Du, X., 2012, “Toward Time-Dependent Robustness Metrics,” J. Mech. Des., 134(1), p. 011004. [CrossRef]
Son, Y. K., and Savage, G. J., 2007, “Set Theoretic Formulation of Performance Reliability of Multiple Response Time-Variant Systems Due to Degradations in System Components,” Quality Reliab. Eng. Int., 23(2), pp. 171–188. [CrossRef]
Hagen, O., and Tvedt, L., 1991, “Vector Process Out-Crossing as Parallel System Sensitivity Measure,” J. Eng. Mech., 117(10), pp. 2201–2220. [CrossRef]
Breitung, K., 1988, “Asymptotic Crossing Rates for Stationary Gaussian Vector Processes,” Stochastic Process. Appl., 29(2), pp. 195–207. [CrossRef]
Xu, H., and Rahman, S., 2005, “Decomposition Methods for Structural Reliability Analysis,” Probab. Eng. Mech., 20(3), pp. 239–250. [CrossRef]
Youn, B. D., and Xi, Z., 2009, “Reliability-Based Robust Design Optimization Using the Eigenvector Dimension Reduction (EDR) Method,” Struct. Multidiscip. Optim., 37(5), pp. 475–492. [CrossRef]
Xu, H., and Rahman, S., 2004, “A Generalized Dimension-Reduction Method for Multidimensional Integration in Stochastic Mechanics,” Int. J. Numer. Methods Eng., 61(12), pp. 1992–2019. [CrossRef]
Jones, D. R., Schonlau, M., and Welch, W. J., 1998, “Efficient Global Optimization Of Expensive Black-Box Functions,” J. Global Optim., 13(4), pp. 455–492. [CrossRef]
Schonlau, M., 1997, “Computer Experiments and Global Optimization,” Ph.D. dissertation, University of Waterloo, Waterloo, Ontario, Canada.
Stuckman, B. E., 1988, “A Global Search Method for Optimizing Nonlinear Systems,” IEEE Trans. Syst., Man Cybern., 18(6), pp. 965–977. [CrossRef]
Žilinskas, A., 1992, “A Review of Statistical Models for Global Optimization,” J. Global Optim., 2(2), pp. 145–153. [CrossRef]
Koehler, J., and Owen, A., 1996, “Computer Experiments” Handbook of Statistics, 13: Design and Analysis of Experiments, S.Ghosh and C. R.Rao, eds., pp. 261–308, Elsevier, Amsterdam.
Sacks, J., Welch, W. J., Mitchell, T. J., and Wynn, H. P., 1989, “Design and Analysis of Computer Experiments,” Stat. Sci., 4(4), pp. 409–423. [CrossRef]
Mockus, J., Tiesis, V., and Zilinskas, A., 1978, “The Application of Bayesian Methods for Seeking the Extreme,” Towards Global Optimization, Vol. 2, L. C. W.Dixon and G. P.Szego, eds., pp. 117–129.
Haftka, R. T., and Watsonft, L. T., 1999, “Response Surface Models Combining Linear and Euler Aerodynamics for Supersonic Transport Design,” J. Aircraft, 36(1), pp. 75–86. [CrossRef]
Madsen, J. I., Shyy, W., and Haftka, R. T., 2000, “Response Surface Techniques for Diffuser Shape Optimization,” AIAA J., 38(9), pp. 1512–1518. [CrossRef]
Welch, W. J., Buck, R. J., Sacks, J., Wynn, H. P., Mitchell, T. J., and Morris, M. D., 1992, “Screening, Predicting, and Computer Experiments,” Technometrics, 34(1), pp. 15–25. [CrossRef]
Wang, G. G., Dong, Z., and Aitchison, P., 2001, “Adaptive Response Surface Method—A Global Optimization Scheme for Approximation-Based Design Problems,” Eng. Optim., 33(6), pp. 707–733. [CrossRef]
Keane, A. J., and Nair, P. B., 2005, Computational Approaches for Aerospace Design, John Wiley & Sons, Ltd., West Sussex, p. 582.
Simpson, T. W., Mauery, T. M., Korte, J. J., and Mistree, F., 1998, “Comparison of Response Surface and Kriging Models for Multidisciplinary Design Optimization,” AIAA, Paper No. 98, 4758(7).
Paciorek, C. J., 2003, “Nonstationary Gaussian Processes for Regression and Spatial Modelling,” Ph.D. dissertation, Carnegie Mellon University, Pittsburgh, PA.
Farhang-Mehr, A., and Azarm, S., 2005, “Bayesian Metamodeling of Engineering Design Simulations: A Sequential Approach With Adaptation to Irregularities in the Response Behavior,” Int. J. Numer. Methods Eng., 62(15), pp. 2104–2126. [CrossRef]
Qin, S., and Cui, W., 2003, “Effect of Corrosion Models on the Time-Dependent Reliability of Steel Plated Elements,” Marine Struct., 16(1), pp. 15–34. [CrossRef]
Madsen, H. O., Krenk, S., and Lind, N. C., 2006, Methods of Structural Safety, Dover Publications, New York.
Choi, H. G. R., Park, M. H., and Salisbury, E., 2000, “Optimal Tolerance Allocation With Loss Functions,” J. Manuf. Sci. Eng., 122(3), pp. 529–535. [CrossRef]
Xue, W., and Pyle, R., 2004, “Optimal Design of Roller One Way Clutch for Starter Drives,” SAE Technical Paper No. 2004-01-1151.


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Fig. 1

Time-dependent performance function

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Fig. 2

Global extreme response surface of time

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Fig. 3

Flowchart of NERS approach for reliability analysis

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Fig. 4

Time-dependent limit state of mathematical example

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Fig. 5

(a) First EGO iteration for extreme response identification and (b) ninth EGO iteration for extreme response identification

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Fig. 6

Flowchart of ARPMM mechanism

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Fig. 7

Instantaneous limit states for time interval [0, 5]

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Fig. 8

Initial NTPM for mathematical example

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Fig. 9

(a) Mean square error, e(x) for initial NTPM and (b) mean square error, e(x) for updated NTPM

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Fig. 10

Flowchart of time-dependent RBDO with NERS approach

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Fig. 11

Limit state functions

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Fig. 12

Reliabilities of constraints in iterative RBDO process

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Fig. 13

Schematic of roller clutch

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Fig. 14

(a) Cost function during iterative design process and (b) constraints reliabilities during iterative design process




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