Research Papers: Design Automation

High-Dimensional Reliability-Based Design Optimization Involving Highly Nonlinear Constraints and Computationally Expensive Simulations

[+] Author and Article Information
Meng Li

Department of Mechanical Engineering,
Iowa State University,
Ames, IA 50011
e-mail: meng@iastate.edu

Mohammadkazem Sadoughi

Department of Mechanical Engineering,
Iowa State University,
Ames, IA 50011
e-mail: sadoughi@iastate.edu

Chao Hu

Department of Mechanical Engineering and
Department of Electrical and
Computer Engineering,
Iowa State University,
Ames, IA 50011
e-mails: chaohu@iastate.edu;

Zhen Hu

Department of Industrial and
Manufacturing Systems Engineering,
University of Michigan-Dearborn,
Dearborn, MI 48128
e-mail: zhennhu@unich.edu

Amin Toghi Eshghi

Department of Mechanical Engineering,
University of Maryland,
Baltimore County,
Baltimore, MD 21250
e-mail: amint1@umbc.edu

Soobum Lee

Department of Mechanical Engineering,
University of Maryland,
Baltimore County,
Baltimore, MD 21250
e-mail: sblee@umbc.edu

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received June 16, 2018; final manuscript received October 23, 2018; published online January 11, 2019. Assoc. Editor: Ping Zhu.

J. Mech. Des 141(5), 051402 (Jan 11, 2019) (14 pages) Paper No: MD-18-1456; doi: 10.1115/1.4041917 History: Received June 16, 2018; Revised October 23, 2018

Reliability-based design optimization (RBDO) aims at optimizing the design of an engineered system to minimize the design cost while satisfying reliability requirements. However, it is challenging to perform RBDO under high-dimensional uncertainty due to the often prohibitive computational burden. In this paper, we address this challenge by leveraging a recently developed method for reliability analysis under high-dimensional uncertainty. The method is termed high-dimensional reliability analysis (HDRA). The HDRA method optimally combines the strengths of univariate dimension reduction (UDR) and kriging-based reliability analysis to achieve satisfactory accuracy with an affordable computational cost for HDRA problems. In this paper, we improve the computational efficiency of high-dimensional RBDO by pursuing two new strategies: (i) a two-stage surrogate modeling strategy is adopted to first locate a highly probable region of the optimum design and then locally refine the accuracy of the surrogates in this region; and (ii) newly selected samples are updated for all the constraints during the sequential sampling process in HDRA. The results of two mathematical examples and one real-world engineering example suggest that the proposed HDRA-based RBDO (RBDO-HDRA) method is capable of solving high-dimensional RBDO problems with higher accuracy and comparable efficiency than the UDR-based RBDO (RBDO-UDR) and ordinary kriging-based RBDO (RBDO-kriging) methods.

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Tu, J. , Choi, K. K. , and Park, Y. H. , 1999, “ A New Study on Reliability-Based Design Optimization,” ASME J. Mech. Des., 121(4), pp. 557–564. [CrossRef]
Youn, B. D. , Choi, K. K. , and Park, Y. H. , 2003, “ Hybrid Analysis Method for Reliability-Based Design Optimization,” ASME J. Mech. Des., 125(2), pp. 221–232. [CrossRef]
Choi, K. K. , Youn, B. D. , and Du, L. , 2004, “ Enriched Performance Measure Approach (pma+) and Its Numerical Method for Reliability-Based Design Optimization,” AIAA Paper No. AIAA-2004-4401.
Chiralaksanakul, A. , and Mahadevan, S. , 2005, “ First-Order Approximation Methods in Reliability-Based Design Optimization,” ASME J. Mech. Des., 127(5), pp. 851–857. [CrossRef]
Noh, Y. , Choi, K. K. , and Du, L. , 2009, “ Reliability-Based Design Optimization of Problems With Correlated Input Variables Using a Gaussian Copula,” Struct. Multidiscip. Optim., 38(1), pp. 1–16. [CrossRef]
Du, X. P. , and Chen, W. , 2004, “ Sequential Optimization and Reliability Assessment Method for Efficient Probabilistic Design,” ASME J. Mech. Des., 126(2), pp. 225–233. [CrossRef]
Liang, J. , Mourelatos, Z. P. , and Nikolaidis, E. , 2007, “ A Single-Loop Approach for System Reliability-Based Design Optimization,” ASME J. Mech. Des., 129(12), pp. 1215–1224. [CrossRef]
Nguyen, T. H. , Song, J. , and Paulino, G. H. , 2010, “ Single-Loop System Reliability-Based Design Optimization Using Matrix-Based System Reliability Method: Theory and Applications,” ASME J. Mech. Des., 132(1), p. 011005. [CrossRef]
Hu, Z. , and Mahadevan, S. , 2016, “ A Single-Loop Kriging Surrogate Modeling for Time-Dependent Reliability Analysis,” ASME J. Mech. Des., 138(6), p. 061406. [CrossRef]
Lin, P. T. , and Lin, S. P. , 2016, “ An Effective Approach to Solve Design Optimization Problems With Arbitrarily Distributed Uncertainties in the Original Design Space Using Ensemble of Gaussian Reliability Analyses,” ASME J. Mech. Des., 138(7), p. 071403. [CrossRef]
Sadoughi, M. , Li, M. , Hu, C. , MacKenzie, C. , Lee, S. , and Eshghi, A. T. , 2018, “ A High-Dimensional Reliability Analysis Method for Simulation-Based Design Under Uncertainty,” ASME J. Mech. Des., 140(7), p. 071401. [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.
Rahman, S. , and Wei, D. , 2008, “ Design Sensitivity and Reliability-Based Structural Optimization by Univariate Decomposition,” Struct. Multidiscip. Optim., 35(3), pp. 245–261. [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]
Seong, S. , Hu, C. , and Lee, S. , 2017, “ Design Under Uncertainty for Reliable Power Generation of Piezoelectric Energy Harvester,” J. Intell. Mater. Syst. Struct., 28(17), pp. 2437–2449. [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]
Xu, H. , and Rahman, S. , 2005, “ Decomposition Methods for Structural Reliability Analysis,” Probab. Eng. Mech., 20(3), pp. 239–250. [CrossRef]
Bichon, B. J. , Eldred, M. S. , Swiler, L. P. , Mahadevan, S. , and McFarland, J. M. , 2008, “ Efficient Global Reliability Analysis for Nonlinear Implicit Performance Functions,” AIAA J., 46(10), pp. 2459–2468. [CrossRef]
Echard, B. , Gayton, N. , and Lemaire, M. , 2011, “ AK-MCS: An Active Learning Reliability Method Combining Kriging and Monte Carlo Simulation,” Struct. Saf., 33(2), pp. 145–154. [CrossRef]
Wang, Z. , and Wang, P. , 2014, “ A Maximum Confidence Enhancement Based Sequential Sampling Scheme for Simulation-Based Design,” ASME J. Mech. Des., 136(2), p. 021006. [CrossRef]
Sadoughi, M. K. , Hu, C. , MacKenzie, C. A. , Eshghi, A. T. , and Lee, S. , 2017, “ Sequential Exploration-Exploitation With Dynamic Trade-Off for Efficient Reliability Analysis of Complex Engineered Systems,” Struct. Multidiscip. Optim., 56, pp. 1–16. [CrossRef]
Chen, Z. , Qiu, H. , Gao, L. , Li, X. , and Li, P. , 2014, “ A Local Adaptive Sampling Method for Reliability-Based Design Optimization Using Kriging Model,” Struct. Multidiscip. Optim., 49(3), pp. 401–416. [CrossRef]
Li, X. , Qiu, H. , Chen, Z. , Gao, L. , and Shao, X. , 2016, “ A Local Kriging Approximation Method Using MPP for Reliability-Based Design Optimization,” Comput. Struct., 162, pp. 102–115. [CrossRef]
Wang, Z. , Hutter, F. , Zoghi, M. , Matheson, D. , and de Feitas, N. , 2016, “ Bayesian Optimization in a Billion Dimensions Via Random Embeddings,” J. Artif. Intell. Res., 55, pp. 361–387. [CrossRef]
Li, M. , Sadoughi, M. , Hu, Z. , and Hu, C. , 2018, “ Reliability-Based Design Optimization of High-Dimensional Engineered Systems Involving Computationally Expensive Simulations,” AIAA Paper No. AIAA 2018-2171.
Rasmussen, C. E. , 2004, “ Gaussian Processes in Machine Learning,” Advanced Lectures on Machine Learning, Springer, Berlin, pp. 63–71.
Cho, H. , Choi, K. K. , and Lamb, D. , 2017, “ Sensitivity Developments for RBDO With Dependent Input Variable and Varying Input Standard Deviation,” ASME J. Mech. Des., 139(7), p. 071402. [CrossRef]
Rubinstein, R. Y. , and Shapiro, A. , 1993, Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization by the Score Function Method, Wiley, Hoboken, NJ.
Lee, I. , Choi, K. K. , and Zhao, L. , 2011, “ Sampling-Based RBDO Using the Stochastic Sensitivity Analysis and Dynamic Kriging Method,” Struct. Multidiscip. Optim., 44(3), pp. 299–317. [CrossRef]
Paulson, E. J. , and Starkey, R. P. , 2014, “ Development of a Multistage Reliability-Based Design Optimization Method,” ASME J. Mech. Des., 136(1), p. 011007. [CrossRef]
Nocedal, J. , and Wright, S. J. , 2006, Sequential Quadratic Programming, Springer, New York, pp. 529–562.
Shan, S. , and Wang, G. G. , 2008, “ Reliable Design Space and Complete Single-Loop Reliability-Based Design Optimization,” Reliab. Eng. Syst. Saf., 93(8), pp. 1218–1230. [CrossRef]
Liang, J. , Mourelatos, Z. P. , and Tu, J. , 2008, “ A Single-Loop Method for Reliability-Based Design Optimisation,” Int. J. Prod. Dev., 5(1/2), pp. 76–92. [CrossRef]
Dubourg, V. , Sudret, B. , and Bourinet, J. M. , 2011, “ Reliability-Based Design Optimization Using Kriging Surrogates and Subset Simulation,” Struct. Multidiscip. Optim., 44(5), pp. 673–690. [CrossRef]
Eshghi, A. T. , Lee, S. , Sadoughi, M. K. , Hu, C. , Kim, Y. C. , and Seo, J. H. , 2017, “ Design Optimization Under Uncertainty and Speed Variability for a Piezoelectric Energy Harvester Powering a Tire Pressure Monitoring Sensor,” Smart Mater. Struct., 26(10), p. 105037. [CrossRef]
Landau, D. P. , and Binder, K. , 2014, A Guide to Monte Carlo Simulations in Statistical Physics, Cambridge University Press, Cambridge, UK.


Grahic Jump Location
Fig. 1

Overall flowchart of the HDRA method [11]

Grahic Jump Location
Fig. 2

Effect of adding one sample point at the joint region on the prediction accuracy of two LSFs: (a) true and estimated LSFs without the addition of the point and (b) true and estimated LSFs with the addition of the point

Grahic Jump Location
Fig. 3

Identification of samples in the probable LSF region

Grahic Jump Location
Fig. 4

Overall flowchart of the proposed two-stage surrogate modeling strategy for RBDO

Grahic Jump Location
Fig. 5

(a) Comparison of HDRA surrogate with true function after global stage of sample enrichment and (b) error decay of the global surrogate model

Grahic Jump Location
Fig. 6

(a) Comparison of HDRA surrogate with true function after local stage of sample enrichment and (b) error decay of the local surrogate model

Grahic Jump Location
Fig. 7

Schematics of TPMS-EH in example 3: (a) design variables for TPMS-EH and (b) TPMS-EH encased in the housing

Grahic Jump Location
Fig. 8

Decay of the maximum relative error ε at the first (a) and second (b) stages



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