Research Papers

Reliability-Based Multidisciplinary Design Optimization Using Probabilistic Gradient-Based Transformation Method

[+] Author and Article Information
Po Ting Lin

Assistant Professor
Department of Mechanical Engineering,
Research and Development Center
for Microsystem Reliability,
Chung Yuan Christian University,
Chungli City,
Taoyuan County, Taiwan 32023
e-mail: potinglin@cycu.edu.tw

Hae Chang Gea

Department of Mechanical and
Aerospace Engineering,
The State University of New Jersey,
Piscataway, NJ 08854
e-mail: gea@rci.rutgers.edu

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the Journal of Mechanical Design. Manuscript received January 16, 2012; final manuscript received October 19, 2012; published online December 10, 2012. Assoc. Editor: Wei Chen.

J. Mech. Des 135(2), 021001 (Dec 10, 2012) (12 pages) Paper No: MD-12-1052; doi: 10.1115/1.4023025 History: Received January 16, 2012; Revised October 19, 2012

Recently, solving the complex design optimization problems with design uncertainties has become an important but very challenging task in the communities of reliability-based design optimization (RBDO) and multidisciplinary design optimization (MDO). The MDO algorithms decompose the complex design problem into the hierarchical or nonhierarchical optimization structure and distribute the workloads to each discipline (or subproblem) in the decomposed structure. The coordination of the local responses is crucial for the success of finding the optimal design point. The problem complexity increases dramatically when the existence of the design uncertainties is not negligible. The RBDO algorithms perform the reliability analyses to evaluate the probabilities that the random variables violate the constraints. However, the required reliability analyses build up the degree of complexity. In this paper, the gradient-based transformation method (GTM) is utilized to reduce the complexity of the MDO problems by transforming the design space to multiple single-variate monotonic coordinates along the directions of the constraint gradients. The subsystem responses are found using the monotonicity principles (MP) and then coordinated for the new design points based on two general principles. To consider the design uncertainties, the probabilistic gradient-based transformation method (PGTM) is proposed to adapt the first-order probabilistic constraints from three different RBDO algorithms, including the chance constrained programming (CCP), reliability index approach (RIA), and performance measure approach (PMA), to the framework of the GTM. PGTM is efficient because only the sensitivity analyses and the reliability analyses require function evaluations (FE). The optimization processes of monotonicity analyses and the coordination procedures are free of function evaluations. Several mathematical and engineering examples show the PGTM is capable of finding the optimal solutions with desirable reliability levels.

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Cornell, C. A., 1969, “A Probability-Based Structural Code,” J. Am. Concr. Inst., 66(12), pp. 974–985.
Hasofer, A. M., and Lind, N. C., 1974, “Exact and Invariant Second-Moment Code Format,” J. Engrg. Mech. Div., 100(EM1), pp. 111–121.
Rackwitz, R., and Fiessler, B., 1978, “Structural Reliability Under Combined Random Load Sequences,” Comput. Struct., 9, pp. 489–494. [CrossRef]
Kuschel, N., and Rackwitz, R., 1997, “Two Basic Problems in Reliability-Based Structural Optimization,” Math. Methods Oper. Res., 46, pp. 309–333. [CrossRef]
Tu, J., Choi, K. K., and Park, Y. H., 1999, “A New Study on Reliability Based Design Optimization,” J. Mech. Des., 121, pp. 557–564. [CrossRef]
Rackwitz, R., 2001, “Reliability Analysis—A Review and Some Perspectives,” Struct. Saf., 23(4), pp. 365–395. [CrossRef]
Du, X., and Chen, W., 2004, “Sequential Optimization and Reliability Assessment Method for Efficient Probabilistic Design,” J. Mech. Des., 126, pp. 225–233. [CrossRef]
Qu, X., and Haftka, R. T., 2004, “Reliability-Based Design Optimization Using Probabilistic Sufficiency Factor,” Struct. Multidiscip. Optim., 27, pp. 314–325. [CrossRef]
Lin, P. T., Gea, H. C., and Jaluria, Y., 2011, “A Modified Reliability Index Approach for Reliability-Based Design Optimization,” J. Mech. Des., 133(4), 044501. [CrossRef]
Sobieszczanski-Sobieski, J., 1982, “A Linear Decomposition Method for Large Optimization Problems—Blueprint for Development,” Technical Report No. NASA-TM-83248.
Braun, R. D., and Kroo, I. M., 1995, “Development and Application of the Collaborative Optimization Architecture in a Multidisciplinary Design Environment,” ICASE/NASA Langley Workshop on Multidisciplinary Design Optimization, Hampton, Virginia.
Michelena, N., Kim, H. M., and Papalambros, P., 1999, “A System Partitioning and Optimization Approach to Target Cascading,” International Conference on Engineering Design, ICED, Munich, Germany.
Sobieszczanski-Sobieski, J., 1999, “Multidisciplinary Design Optimisation (MDO) Methods: Their Synergy With Computer Technology in the Design Process,” Aeronaut. J., 103(1026), pp. 373–382.
Kim, H. M., Michelena, N., Papalambros, P., and Jiang, T., 2003, “Target Cascading in Optimal System Design,” ASME J. Mech. Des., 125, pp. 474–480. [CrossRef]
Zhang, X.-L., Lin, P. T., Gea, H. C., and Huang, H.-Z., 2011, “Bounded Target Cascading in Hierarchical Design Optimization,” ASME 2011 International Deisgn Engineering Technical Conferences and Computers and Information in Engineering Conference, Washington, DC, USA, Paper No. 48614.
Lin, P. T., and Gea, H. C., 2011, “A Gradient-Based Transformation Method in Multidisciplinary Design Optimization,” 9th World Congress on Structural and Multidisciplinary Optimization, WCSMO9, Shizuoka, Japan.
Michelena, N. F., and Papalambros, P. Y., 1995, “Optimal Model-Based Decomposition of Powertrain System Design,” ASME J. Mech. Des., 117(4), pp. 499–505. [CrossRef]
Braun, R. D., Moore, A. A., and Kroo, I. M., 1996, “Use of the Collaborative Optimization Architecture for Launch Vehicle Design,” Technical Report No. NASA-AIAA-96-4018.
Budianto, I. A., and Olds, J. R., 2000, “A Collaborative Optimization Approach to Design and Deployment of a Space Based Infrared System Constellation,” 2000 IEEE Aerospace Conference, Big Sky, Montana.
Kroo, I. M., 2004, “Distributed Multidisciplinary Design and Collaborative Optimization,” VKI Lecture Series on Optimization Methods and Tools for Multicriteria/Multidisciplinary Design.
Kim, H. M., Kokkolaras, M., Louca, L. S., Delagrammatikas, G. J., Michelena, N. F., Filipi, Z. S., Papalambros, P. Y., Stein, J. L., and Assanis, D. N., 2002, “Target Cascading in Vehicle Redesign: A Class VI Truck Study,” Int. J. Veh. Des., 29(3), pp. 199–225. [CrossRef]
Tosserams, S., Kokkolaras, M., Etman, L. F. P., and Rooda, J. E., 2010, “A Nonhierarchical Formulation of Analytical Target Cascading,” ASME J. Mech. Des., 132(5), p. 051002. [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, pp. 221–232. [CrossRef]
Lin, P. T., and Gea, H. C., 2012, “A Gradient-Based Transformation Method in Multidisciplinary Design Optimization,” Struct. Multidiscip. Optim. (in press). [CrossRef]
Papalambros, P. Y., and Wilde, D. J., 2000, Principles of Optimal Design, Cambridge University Press, New York.
Michelena, N. F., and Papalambros, P. Y., 1997, “A Hypergraph Framework for Optimal Model-Based Decomposition of Design Problems,” Comput. Optim. Appl., 8(2), pp. 173–196. [CrossRef]
Michelena, N., Park, H., and Papalambros, P. Y., 2003, “Convergence Properties of Analytical Target Cascading,” AIAA J., 41(5), pp. 897–905. [CrossRef]
Allison, J., 2004, “Complex System Optimization: A Review of Analytical Target Cascading, Collaborative Optimization, and Other Formulations,” Master's thesis, University of Michigan, Ann Arbor, MI.
Allison, J., Kokkolaras, M., Zawislak, M., and Papalambros, P. Y., 2005, “On the Use of Analytical Target Cascading and Collaborative Optimization for Complex System Design,” 6th World Congress on Structural and Multidisciplinary Optimization, Rio de Janeiro, Brazil.
Michalek, J. J., and Papalambros, P. Y., 2005, “Weights, Norms, and Notation in Analytical Target Cascading,” ASME J. Mech. Des., 127(3), pp. 499–501. [CrossRef]
Michelena, N., Papalambros, P., Park, H. A., and Kulkarni, D., 1999, “Hierarchical Overlapping Coordination for Large-Scale Optimization by Decomposition,” AIAA J., 37(7), pp. 890–896. [CrossRef]
Dill, E. H., 2006, Continuum Mechanics: Elasticity, Plasticity, Viscoelasticity, CRC Press, Boca Raton.
Youn, B. D., and Choi, K. K., 2004, “An Investigation of Nonlinearity of Reliability-Based Design Optimization Approaches,” ASME J. Mech. Des., 126(3), pp. 403–411. [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, pp. 1–16. [CrossRef]
Noh, Y., Choi, K. K., and Lee, I., 2010, “Identification of Marginal and Joint CDFs Using Bayesian Method for RBDO,” Struct. Multidiscip. Optim., 40, pp. 35–51. [CrossRef]
Rubinstein, R. Y., 1981, Simulation and The Monte Carlo Method, Wiley, New York, NY.
Charnes, A., and Cooper, W. W., 1959, “Chance Constrained Programming,” Manage. Sci., 6, pp. 73–79. [CrossRef]
Davidson, J. W., Felton, L. P., and Hart, G. C., 1977, “Optimum Design of Structures with Random Parameters,” Comput. Struct., 7, pp. 481–486. [CrossRef]
Rao, S. S., 1980, “Structural Optimization by Chance Constrained Programming Techniques,” Comput. Struct., 12, pp. 777–781. [CrossRef]
Jozwiak, S. F., 1986, “Minimum Weight Design of Structures With Random Parameters,” Comput. Struct., 23, pp. 481–485. [CrossRef]
Madsen, H. O., Krenk, S., and Lind, N. C., 1986, Methods of Structural Safety, Prentice-Hall, Englewood Cliffs, NJ.
Der Kiureghian, A., 2004, Engineering Design Reliability Handbook, CRC Press, Boca Raton, FL.
Nikolaidis, E., and Burdisso, R., 1988, “Reliability Based Optimization: A Safety Index Approach,” Comput. Struct., 28(6), pp. 781–788. [CrossRef]
Frangopol, D. M., and Corotis, R. B., 1996, Analysis and Computation: Proceedings of the 12th Conference Held in Conjunction With Structures Congress XIV, pp. 67–78.
Wu, Y. T., and Wang, W., 1996, “Probabilistic Mechanics & Structural Reliability,” Proceedings of the 7th Special Conference, pp. 274–277.
Carter, A. D. S., 1997, Mechanical Reliability and Design, Wiley, New York.
Youn, B. D., and Choi, K. K., 2004, “Selecting Probabilistic Approaches for Reliability-Based Design Optimization,” AIAA J., 42(1), pp. 124–131. [CrossRef]
Youn, B. D., and Choi, K. K., 2004, “A New Response Surface Methodology for Reliability-Based Design Optimization,” Comput. Struct., 82, pp. 241–256. [CrossRef]
Youn, B. D., Choi, K. K., Yang, R. J., and Gu, L., 2004, “Reliability-Based Design Optimization for Crashworthiness of Vehicle Side Impact,” Struct. Multidiscip. Optim., 26(3-4), pp. 272–283. [CrossRef]
Youn, B. D., Choi, K. K., and Du, L., 2005, “Adaptive Probability Analysis Using an Enhanced Hybrid Mean Value Method,” Struct. Multidiscip. Optim., 29(2), pp. 134–148. [CrossRef]
Yang, R. J., and Gu, L., 2004, “Experience With Approximate Reliability-Based Optimization Methods,” Struct. Multidiscip. Optim., 26(2), pp. 152–159. [CrossRef]
Lin, P. T., Gea, H. C., and Jaluria, Y., 2009, “A Modified Reliability Index Approach for Reliability-Based Design Optimization,” 2009 ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (IDETC/CIE), San Diego, CA, Paper No. DETC2009-87804.
Lee, I., Choi, K. K., Du, L., and Gorsich, D., 2008, “Inverse Analysis Method Using Mpp-Based Dimension Reduction for Reliability-Based Design Optimization of Nonlinear and Multi-Dimensional Systems,” Comput. Methods Appl. Mech. Eng., 198(1), pp. 14–27. [CrossRef]
Lin, P. T., 2010, “Parametric Modeling and Optimization of Thermal Systems With Design Uncertainties,” Ph.D. thesis, Rutgers University, New Brunswick, New Jersey.
Lin, P. T., Jaluria, Y., and Gea, H. C., 2009, “Parametric Modeling and Optimization of Chemical Vapor Deposition Process,” J. Manuf. Sci. Eng., 131(1), Paper No. 011011.
Michalek, J. J., and Papalambros, P. Y., 2006, “BB-ATC: Analytical Target Cascading Using Branch and Bound for Mixed-Integer Nonlinear Programming,” Proceedings of ASME 2006 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Philadelphia, Pennsylvania, Paper No. DETC2006/DAC-99040.
Tosserams, S., Etman, L. F. P., Papalambros, P. Y., and Rooda, J. E., 2006, “An Augmented Lagrangian Relaxation for Analytical Target Cascading Using the Alternating Direction Method of Multipliers,” Struct. Multidiscip. Optim., 31(3), pp. 176–189. [CrossRef]
Han, J., and Papalambros, P. Y., 2010, “A Sequential Linear Programming Coordination Algorithm for Analytical Target Cascading,” J. Mech. Des., 132(2), 021003. [CrossRef]
Han, J., and Papalambros, P. Y., 2010, “An SLP Filter Algorithm for Probabilistic Analytical Target Cascading,” Struct. Multidiscip. Optim., 41(6), pp. 935–945. [CrossRef]
Michalek, J. J., and Papalambros, P. Y., 2005, “An Efficient Weighting Update Method to Achieve Acceptable Consistency Deviation in Analytical Target Cascading,” ASME J. Mech. Des., 127(2), pp. 206–214. [CrossRef]


Grahic Jump Location
Fig. 1

Different MDO optimization structures: (a) Hierarchical; (b) nonhierarchical; and (c) GTM [24]

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Fig. 2

Probabilistic design optimization of each subsystem in the hierarchical and nonhierarchical MDO structure

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Fig. 3

The unit constraint gradients and the first-order probabilistic constraints for (a) CCP, (b) PMA, and (c) RIA

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Fig. 4

Flow chart of the PGTM

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Fig. 5

Iteration histories of the design variables in three different PGTM approaches: (a) x1 in PGTM-CCP, (b) x2 in PGTM-CCP, (c) x1 in PGTM-RIA, (d) x2 in PGTM-RIA, (e) x1 in PGTM-PMA, and (f) x2 in PGTM-PMA

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Fig. 6

Iteration processes of the example 2 using (a) PGTM-CCP, (b) PGTM-RIA, and (c) PGTM-PMA

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Fig. 7

Vertical impinging CVD of silicon [55]

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Fig. 8

Performance functions of silicon deposition: (a) MDR, (b) RMS, and (c) PWA [54]

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Fig. 9

Reliability-based optimal experimental parameters for the maximum performance and the sustainable design

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Fig. 10

The structure of the anchor design [29,56-59]



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