Abstract
Product performance varies with respect to time and space in many engineering applications. This paper discusses how to measure and evaluate the robustness of a product or component when its quality characteristics (QCs) are functions of random variables, random fields, temporal variables, and spatial variables. At first, the existing time-dependent robustness metric is extended to the present time- and space-dependent problem. The robustness metric is derived using the extreme value of the quality characteristics with respect to temporal and spatial variables for the nominal-the-better type quality characteristics. Then, a metamodel-based numerical procedure is developed to evaluate the new robustness metric. The procedure employs a Gaussian Process regression method to estimate the expected quality loss that involves the extreme quality characteristics. The expected quality loss is obtained directly during the regression model building process. Four examples are used to demonstrate the robustness analysis method. The proposed method can be used for robustness analysis during robust design optimization (RDO) under time- and space-dependent uncertainty.
References
1.
Taguchi
, G.
, 1993
, Taguchi on Robust Technology Development: Bringing Quality Engineering Upstream
, ASME Press
, New York
.2.
Phadke
, M. S.
, 1995
, Quality Engineering Using Robust Design
, Prentice Hall PTR
, Englewood Cliffs, NJ
.3.
Du
, X.
, and Chen
, W.
, 2000
, “Towards a Better Understanding of Modeling Feasibility Robustness in Engineering Design
,” ASME J. Mech. Des.
, 122
(4
), pp. 385
–394
. 10.1115/1.12902474.
Huang
, B.
, and Du
, X.
, 2007
, “Analytical Robustness Assessment for Robust Design
,” Struct. Multidiscip. Optim.
, 34
(2
), pp. 123
–137
. 10.1007/s00158-006-0068-05.
Mulvey
, J. M.
, Vanderbei
, R. J.
, and Zenios
, S. A.
, 1995
, “Robust Optimization of Large-Scale Systems
,” Oper. Res.
, 43
(2
), pp. 264
–281
. 10.1287/opre.43.2.2646.
Goh
, J.
, and Sim
, M.
, 2010
, “Distributionally Robust Optimization and its Tractable Approximations
,” Oper. Res.
, 58
(4-part-1
), pp. 902
–917
. 10.1287/opre.1090.07957.
Fabozzi
, F. J.
, Huang
, D.
, and Zhou
, G.
, 2010
, “Robust Portfolios: Contributions From Operations Research and Finance
,” Ann. Oper. Res.
, 176
(1
), pp. 191
–220
. 10.1007/s10479-009-0515-68.
Alexandrov
, N. M.
, and Lewis
, R. M.
, 2002
, “Analytical and Computational Aspects of Collaborative Optimization for Multidisciplinary Design
,” AIAA J.
, 40
(2
), pp. 301
–309
. 10.2514/2.16469.
Sobieszczanski-Sobieski
, J.
, and Haftka
, R. T.
, 1997
, “Multidisciplinary Aerospace Design Optimization: Survey of Recent Developments
,” Struct. Optim.
, 14
(1
), pp. 1
–23
. 10.1007/BF0119755410.
Doltsinis
, I.
, and Kang
, Z.
, 2004
, “Robust Design of Structures Using Optimization Methods
,” Comput. Methods Appl. Mech. Eng.
, 193
(23–26
), pp. 2221
–2237
. 10.1016/j.cma.2003.12.05511.
Lagaros
, N. D.
, Plevris
, V.
, and Papadrakakis
, M.
, 2007
, “Reliability Based Robust Design Optimization of Steel Structures
,” Int. J. Simul. Multidiscip. Des. Optim.
, 1
(1
), pp. 19
–29
. 10.1051/ijsmdo:200700312.
Cheng
, J.
, Liu
, Z.
, Wu
, Z.
, Li
, X.
, and Tan
, J.
, 2015
, “Robust Optimization of Structural Dynamic Characteristics Based on Adaptive Kriging Model and CNSGA
,” Struct. Multidiscip. Optim.
, 51
(2
), pp. 423
–437
. 10.1007/s00158-014-1140-913.
Roy
, B. K.
, and Chakraborty
, S.
, 2015
, “Robust Optimum Design of Base Isolation System in Seismic Vibration Control of Structures Under Random System Parameters
,” Struct. Saf.
, 55
, pp. 49
–59
. 10.1016/j.strusafe.2015.02.00514.
Fang
, J.
, Gao
, Y.
, Sun
, G.
, and Li
, Q.
, 2013
, “Multiobjective Reliability-Based Optimization for Design of a Vehicle Door
,” Finite Elements Anal. Des.
, 67
, pp. 13
–21
. 10.1016/j.finel.2012.11.00715.
Hwang
, K.
, Lee
, K.
, and Park
, G.
, 2001
, “Robust Optimization of an Automobile Rearview Mirror for Vibration Reduction
,” Struct. Multidiscip. Optim.
, 21
(4
), pp. 300
–308
. 10.1007/s00158010010716.
Sun
, G.
, Li
, G.
, Zhou
, S.
, Li
, H.
, Hou
, S.
, and Li
, Q.
, 2011
, “Crashworthiness Design of Vehicle by Using Multiobjective Robust Optimization
,” Struct. Multidiscip. Optim.
, 44
(1
), pp. 99
–110
. 10.1007/s00158-010-0601-z17.
Lee
, T. H.
, and Jung
, J.
, 2006
, “Metamodel-Based Shape Optimization of Connecting rod Considering Fatigue Life
,” Key Eng. Mater.
, 306–308
, pp. 211
–216
. 10.4028/www.scientific.net/KEM.306-308.21118.
Li
, F.
, Meng
, G.
, Sha
, L.
, and Zhou
, L.
, 2011
, “Robust Optimization Design for Fatigue Life
,” Finite Elements Anal. Des.
, 47
(10
), pp. 1186
–1190
. 10.1016/j.finel.2011.05.00919.
Carpinelli
, G.
, Caramia
, P.
, and Varilone
, P.
, 2015
, “Multi-Linear Monte Carlo Simulation Method for Probabilistic Load Flow of Distribution Systems With Wind and Photovoltaic Generation Systems
,” Renew. Energy
, 76
, pp. 283
–295
. 10.1016/j.renene.2014.11.02820.
Li
, F.
, Luo
, Z.
, Sun
, G.
, and Zhang
, N.
, 2013
, “An Uncertain Multidisciplinary Design Optimization Method Using Interval Convex Models
,” Eng. Optim.
, 45
(6
), pp. 697
–718
. 10.1080/0305215X.2012.69087121.
Du
, X.
, Guo
, J.
, and Beeram
, H.
, 2008
, “Sequential Optimization and Reliability Assessment for Multidisciplinary Systems Design
,” Struct. Multidiscip. Optim.
, 35
(2
), pp. 117
–130
. 10.1007/s00158-007-0121-722.
Kim
, N.-K.
, Kim
, D.-H.
, Kim
, D.-W.
, Kim
, H.-G.
, Lowther
, D. A.
, and Sykulski
, J. K.
, 2010
, “Robust Optimization Utilizing the Second-Order Design Sensitivity Information
,” IEEE Trans. Magn.
, 46
(8
), pp. 3117
–3120
. 10.1109/TMAG.2010.204371923.
Papadimitriou
, D.
, and Giannakoglou
, K.
, 2013
, “Third-Order Sensitivity Analysis for Robust Aerodynamic Design Using Continuous Adjoint
,” Int. J. Numer. Methods Fluids
, 71
(5
), pp. 652
–670
. 10.1002/fld.367724.
Li
, M.
, Hamel
, J.
, and Azarm
, S.
, 2010
, “Optimal Uncertainty Reduction for Multi-Disciplinary Multi-Output Systems Using Sensitivity Analysis
,” Struct. Multidiscip. Optim.
, 40
(1–6
), p. 77
. 10.1007/s00158-009-0372-625.
Siddiqui
, S.
, Azarm
, S.
, and Gabriel
, S.
, 2011
, “A Modified Benders Decomposition Method for Efficient Robust Optimization Under Interval Uncertainty
,” Struct. Multidiscip. Optim.
, 44
(2
), pp. 259
–275
. 10.1007/s00158-011-0631-126.
Siddiqui
, S.
, Gabriel
, S. A.
, and Azarm
, S.
, 2015
, “Solving Mixed-Integer Robust Optimization Problems With Interval Uncertainty Using Benders Decomposition
,” J. Oper. Res. Soc.
, 66
(4
), pp. 664
–673
. 10.1057/jors.2014.4127.
Chatterjee
, T.
, Chowdhury
, R.
, and Ramu
, P.
, 2019
, “Decoupling Uncertainty Quantification From Robust Design Optimization
,” Struct. Multidiscip. Optim.
, 59
(6
), pp. 1969
–1990
. 10.1007/s00158-018-2167-028.
Chatterjee
, T.
, Chakraborty
, S.
, and Chowdhury
, R.
, 2019
, “A Critical Review of Surrogate Assisted Robust Design Optimization
,” Arch. Comput. Methods Eng.
, 26
(1
), pp. 245
–274
. 10.1007/s11831-017-9240-529.
Steuben
, J.
, and Turner
, C. J.
, “Robust Optimization Exploration Using NURBs-Based Metamodeling Techniques
,” Proceedings of the ASME 2010 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
, American Society of Mechanical Engineers Digital Collection
, Montreal, Quebec, Canada
, Aug. 15–18
, pp. 239
–250
.30.
Steuben
, J. C.
, and Turner
, C.
, 2012
, “Robust Optimization of Mixed-Integer Problems Using NURBs-Based Metamodels
,” J. Comput. Inf. Sci. Eng.
, 12
(4
), p. 041010
. 10.1115/1.400798831.
Steuben
, J. C.
, Turner
, C. J.
, and Crawford
, R. H.
, 2013
, “Robust Engineering Design Optimization With Non-Uniform Rational B-Splines-Based Metamodels
,” Eng. Optim.
, 45
(7
), pp. 767
–786
. 10.1080/0305215X.2012.70950932.
Zhou
, H.
, Zhou
, Q.
, Liu
, C.
, and Zhou
, T.
, 2018
, “A Kriging Metamodel-Assisted Robust Optimization Method Based on a Reverse Model
,” Eng. Optim.
, 50
(2
), pp. 253
–272
. 10.1080/0305215X.2017.130735533.
Du
, X.
, 2012
, “Toward Time-Dependent Robustness Metrics
,” ASME J. Mech. Des.
, 134
(1
), p. 011004
. 10.1115/1.400544534.
Du
, X.
, Sudjianto
, A.
, and Chen
, W.
, 2004
, “An Integrated Framework for Optimization Under Uncertainty Using Inverse Reliability Strategy
,” ASME J. Mech. Des.
, 126
(4
), pp. 562
–570
. 10.1115/1.175935835.
Gu
, X.
, Renaud
, J. E.
, Batill
, S. M.
, Brach
, R. M.
, and Budhiraja
, A. S.
, 2000
, “Worst Case Propagated Uncertainty of Multidisciplinary Systems in Robust Design Optimization
,” Struct. Multidiscip. Optim.
, 20
(3
), pp. 190
–213
. 10.1007/s00158005014836.
Tsui
, K.-L.
, 1999
, “Modeling and Analysis of Dynamic Robust Design Experiments
,” IIE Trans.
, 31
(12
), pp. 1113
–1122
. 10.1080/0740817990896991237.
Wu
, F.-C.
, 2009
, “Robust Design of Nonlinear Multiple Dynamic Quality Characteristics
,” Comput. Ind. Eng.
, 56
(4
), pp. 1328
–1332
. 10.1016/j.cie.2008.08.00138.
Goethals
, P. L.
, and Cho
, B. R.
, 2011
, “The Development of a Robust Design Methodology for Time-Oriented Dynamic Quality Characteristics With a Target Profile
,” Qual. Reliab. Eng. Int.
, 27
(4
), pp. 403
–414
. 10.1002/qre.112239.
Wei
, X.
, and Du
, X.
, 2019
, “Uncertainty Analysis for Time-and Space-Dependent Responses With Random Variables
,” ASME J. Mech. Des.
, 141
(2
), p. 021402
. 10.1115/1.404142940.
Törn
, A.
, and Žilinskas
, A.
, 1989
, Global Optimization
, Springer
, New York
.41.
Strongin
, R. G.
, and Sergeyev
, Y. D.
, 1992
, “Global Multidimensional Optimization on Parallel Computer
,” Parallel Comput.
, 18
(11
), pp. 1259
–1273
. 10.1016/0167-8191(92)90069-J42.
Strongin
, R. G.
, and Sergeyev
, Y. D.
, 2013
, Global Optimization With Non-Convex Constraints: Sequential and Parallel Algorithms
, Springer Science & Business Media
, New York
.43.
Sasena
, M. J.
, Papalambros
, P.
, and Goovaerts
, P.
, 2002
, “Exploration of Metamodeling Sampling Criteria for Constrained Global Optimization
,” Eng. Optim.
, 34
(3
), pp. 263
–278
. 10.1080/0305215021175144.
Williams
, C. K.
, and Rasmussen
, C. E.
, 2006
, Gaussian Processes for Machine Learning
, MIT Press
, Cambridge, MA
.45.
Lophaven
, S. N.
, Nielsen
, H. B.
, and Søndergaard
, J.
, 2002
, DACE: A Matlab Kriging Toolbox
, Citeseer
.46.
Zhu
, Z.
, and Du
, X.
, 2016
, “Reliability Analysis with Monte Carlo Simulation and Dependent Kriging Predictions
,” ASME J. Mech. Des.
, 138
(12
), p. 121403
. 10.1115/1.403421947.
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
. 10.1115/1.403342848.
Sudret
, B.
, and Der Kiureghian
, A.
, 2000
, Stochastic Finite Element Methods and Reliability: A State-of-the-Art Report
, Department of Civil and Environmental Engineering, University of California
, Berkeley, CA
.49.
Gabrel
, V.
, Murat
, C.
, and Thiele
, A.
, 2014
, “Recent Advances in Robust Optimization: An Overview
,” Eur. J. Oper. Res.
, 235
(3
), pp. 471
–483
. 10.1016/j.ejor.2013.09.03650.
Bovier
, A.
, 2005
, Extreme Values of Random Processes
, Lecture Notes Technische Universität Berlin
, Bonn
.51.
Kotz
, S.
, and Nadarajah
, S.
, 2000
, Extreme Value Distributions: Theory and Applications
, World Scientific
, London
.52.
Mooney
, C. Z.
, 1997
, Monte Carlo Simulation
, Sage Publications
, Thousand Oaks, CA
.53.
Zienkiewicz
, O. C.
, Taylor
, R. L.
, Nithiarasu
, P.
, and Zhu
, J.
, 1977
, The Finite Element Method
, McGraw-Hill
, London
.54.
Hu
, Z.
, and Du
, X.
, 2015
, “Mixed Efficient Global Optimization for Time-Dependent Reliability Analysis
,” ASME J. Mech. Des.
, 137
(5
), p. 051401
. 10.1115/1.402952055.
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
. 10.1023/A:100830643114756.
Chen
, W.
, Tsui
, K.-L.
, Allen
, J. K.
, and Mistree
, F.
, 1995
, “Integration of the Response Surface Methodology With the Compromise Decision Support Problem in Developing a General Robust Design Procedure
,” ASME Design Engineering Technical Conference
, Boston, MA
, Sept. 17–21
, pp. 485
–492
.57.
Hosder
, S.
, Walters
, R.
, and Balch
, M.
, 2007
, “Efficient Sampling for Non-Intrusive Polynomial Chaos Applications With Multiple Uncertain Input Variables
,” Proceedings of the 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference
, p. 1939
.58.
Wang
, Z.
, and Wang
, P.
, 2012
, “A Nested Extreme Response Surface Approach for Time-Dependent Reliability-Based Design Optimization
,” ASME J. Mech. Des.
, 134
(12
), p. 121007
. 10.1115/1.4007931Copyright © 2019 by ASME
You do not currently have access to this content.
