Statistical tolerance analysis and synthesis in assemblies subject to loading are of significant importance to optimized manufacturing. Modeling the effects of loads on mechanical assemblies in tolerance analysis typically requires the use of numerical CAE simulations. The associated uncertainty quantification (UQ) methods used for estimating yield in tolerance analysis must subsequently accommodate implicit response functions, and techniques such as Monte Carlo (MC) sampling are typically applied due to their robustness; however, these methods are computationally expensive. A variety of UQ methods have been proposed with potentially higher efficiency than MC methods. These offer the potential to make tolerance analysis and synthesis of assemblies under loading practically feasible. This work reports on the practical application of polynomial chaos expansion (PCE) for UQ in tolerance analysis. A process integration and design optimization (PIDO) tool based, computer aided tolerancing (CAT) platform is developed for multi-objective, tolerance synthesis in assemblies subject to loading. The process integration, design of experiments (DOE), and statistical data analysis capabilities of PIDO tools are combined with highly efficient UQ methods for optimization of tolerances to maximize assembly yield while minimizing cost. A practical case study is presented which demonstrates that the application of PCE based UQ to tolerance analysis can significantly reduce computation time while accurately estimating yield of compliant assemblies subject to loading.
References
1.
Hong
, Y. S.
, and Chang
, T. C.
, 2002
, “A Comprehensive Review of Tolerancing Research
,” Int. J. Prod. Res.
, 40
(11
), pp. 2425
–2459
.10.1080/002075402101282422.
Chase
, K. W.
, and Greenwood
, W. H.
, 1988
, “Design Issues in Mechanical Tolerance Analysis
,” ASME Manuf. Rev.
, 1
(1
), pp. 50
–59
.3.
Michael
, W.
, and Siddall
, J. N.
, 1981
, “The Optimization Problem With Optimal Tolerance Assignment
,” ASME J. Mech. Des.
, 103
(4
), pp. 842
–848
.10.1115/1.32549964.
Shah
, J. J.
, Ameta
, G.
, Shen
, Z.
, and Davidson
, J.
, 2007
, “Navigating the Tolerance Analysis Maze
,” Comput.- Aided Des. Appl.
, 4
(5
), pp. 705
–718
.10.1080/16864360.2007.1073855.
Skowronski
, V. J.
, and Turner
, J. U.
, 1997
, “Using Monte-Carlo Variance Reduction in Statistical Tolerance Synthesis
,” CAD Comput. Aided Des.
, 29
(1
), pp. 63
–69
.10.1016/S0010-4485(96)00050-46.
Spotts
, M. F.
, 1973
, “Allocation of Tolerances to Minimize Cost of Assembly
,” ASME J. Eng. Ind.
, 95
(3
), pp. 762
–764
.10.1115/1.34382227.
Ye
, B.
, and Salustri
, F. A.
, 2003
, “Simultaneous Tolerance Synthesis for Manufacturing and Quality
,” Res. Eng. Des.
, 14
(2
), pp. 98
–106
.8.
Edel
, D. H.
, and Auer
, T. B.
, 1964
, “Determine the Least Cost Combination for Tolerance Accumulations in a Drive Shaft Seal Assembly
,” Gen. Mot. Eng. J.
, 4
, pp. 37
–38
.9.
Wu
, Z.
, El Maraghy
, W. H.
, and El Maraghy
, H. A.
, 1988
, “Evaluation of Cost-Tolerance Algorithms for Design Tolerance Analysis and Synthesis
,” ASME Manuf. Rev.
, 1
(3
), pp. 168
–179
.10.
Cho
, B. R.
, Kim
, Y. J.
, Kimbler
, D. L.
, and Phillips
, M. D.
, 2000
, “An Integrated Joint Optimization Procedure for Robust and Tolerance Design
,” Int. J. Prod. Res.
, 38
(10
), pp. 2309
–2325
.10.1080/0020754005002811511.
Zhang
, C.
, and Wang
, H. P.
, 1993
, “Integrated Tolerance Optimisation With Simulated Annealing
,” Int. J. Adv. Manuf. Technol.
, 8
(3
), pp. 167
–174
.10.1007/BF0174990712.
Chase
, K. W.
, Greenwood
, W. H.
, Loosli
, B. G.
, and Hauglund
, L. F.
, 1990
, “Least Cost Tolerance Allocation for Mechanical Assemblies With Automated Process Selection
,” Manuf. Rev.
, 3
(1
), pp. 49
–59
.13.
Choi
, H.-G. R.
, Park
, M.-H.
, and Salisbury
, E.
, 2000
, “Optimal Tolerance Allocation With Loss Functions
,” ASME J. Manuf. Sci. Eng.
, 122
(3
), pp. 529
–535
.10.1115/1.128591814.
Jeang
, A.
, 1999
, “Optimal Tolerance Design by Response Surface Methodology
,” Int. J. Prod. Res.
, 37
(14
), pp. 3275
–3288
.10.1080/00207549919028415.
Wenzhen
, H.
, Phoomboplab
, T.
, and Ceglarek
, D.
, 2009
, “Process Capability Surrogate Model-Based Tolerance Synthesis for Multi-Station Manufacturing Systems
,” IIE Trans.
, 41
(4
), pp. 309
–322
.10.1080/0740817080251040816.
Rao
, Y.
, Rao
, C. S. P.
, Ranga Janardhana
, G.
, and Vundavilli
, P. R.
, 2011
, “Simultaneous Tolerance Synthesis for Manufacturing and Quality Using Evolutionary Algorithms
,” Int. J. Appl. Evol. Comput. (IJAEC)
, 2
(2
), pp. 1
–20
.10.4018/jaec.201104010117.
Wu
, F.
, Dantan
, J.-Y.
, Etienne
, A.
, Siadat
, A.
, and Martin
, P.
, 2009
, “Improved Algorithm for Tolerance Allocation Based on Monte Carlo Simulation and Discrete Optimization
,” Comput. Ind. Eng.
, 56
(4
), pp. 1402
–1413
.10.1016/j.cie.2008.09.00518.
Franciosa
, P.
, Gerbino
, S.
, and Patalano
, S.
, 2011
, “Simulation of Variational Compliant Assemblies With Shape Errors Based on Morphing Mesh Approach
,” Int. J. Adv. Manuf. Technol.
, 53
(1
), pp. 47
–61
.10.1007/s00170-010-2839-419.
Yu
, K.
, Jin
, S.
, Lai
, X.
, and Xing
, Y.
, 2008
, “Modeling and Analysis of Compliant Sheet Metal Assembly Variation
,” Assem. Autom.
, 28
(3
), pp. 225
–234
.10.1108/0144515081088948420.
Nigam
, S. D.
, and Turner
, J. U.
, 1995
, “Review of Statistical Approaches to Tolerance Analysis
,” CAD Comput. Aided Des.
, 27
(1
), pp. 6
–15
.10.1016/0010-4485(95)90748-521.
Frances
, Y. K.
, and Ian
, H. S.
, 2005
, Lifting the Curse of Dimensionality
.22.
Camelio
, J. A.
, Hu
, S. J.
, and Marin
, S. P.
, 2004
, “Compliant Assembly Variation Analysis Using Component Geometric Covariance
,” ASME J. Manuf. Sci. Eng.
, 126
(2
), p. 355
.10.1115/1.164455323.
Merkley
, K. G.
, 1998
, Tolerance Analysis of Compliant Assemblies
, Brigham Young University
, Provo, UT
.24.
Xie
, K.
, Wells
, L.
, Camelio
, J. A.
, and Youn
, B. D.
, 2007
, “Variation Propagation Analysis on Compliant Assemblies Considering Contact Interaction
,” ASME J. Manuf. Sci. Eng.
, 129
(5
), p. 934
.10.1115/1.275282925.
Singh
, P. K.
, Jain
, P. K.
, and Jain
, S. C.
, 2009
, “Important Issues in Tolerance Design of Mechanical Assemblies. Part 2: Tolerance Synthesis
,” Proc. Inst. Mech. Eng., Part B
, 223
(10
), pp. 1249
–1287
.10.1243/09544054JEM1304B26.
Xiu
, D.
, and Karniadakis
, G. E.
, 2003
, “The Wiener–Askey Polynomial Chaos for Stochastic Differential Equations
,” SIAM J. Sci. Comput.
, 24
(2
), pp. 619
–644
.10.1137/S106482750138782627.
Eldred
, M. S.
, Webster
, C. G.
, and Constantine
, P. G.
, 2008
, “Evaluation of Non-Intrusive Approaches for Wiener–Askey Generalized Polynomial Chaos
,” AIAA
Paper No. 2008-1892.10.2514/6.2008-189228.
Lee
, S. H.
, and Chen
, W.
, 2009
, “A Comparative Study of Uncertainty Propagation Methods
,” Struct. Multidiscip. Optim.
, 37
(3
), pp. 239
–253
.10.1007/s00158-008-0234-729.
Nigam
, S. D.
, and Turner
, J. U.
, 1995
, “Review of Statistical Approaches to Tolerance Analysis
,” Comput.-Aided Des.
, 27
(1
), pp. 6
–15
.10.1016/0010-4485(95)90748-530.
31.
Keramat
, M.
, and Kielbasa
, R.
, 1997
, “Latin Hypercube Sampling Monte Carlo Estimation of Average Quality Index for Integrated Circuits
,” Analog Integr. Circuits Signal Process.
, 14
(1–2
), pp. 131
–142
.10.1023/A:100820711348032.
McKay
, M. D.
, Beckman
, R. J.
, and Conover
, W. J.
, 2000
, “Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output From a Computer Code
,” Technometrics
, 42
(1
), pp. 55
–61
.10.1080/00401706.2000.1048597933.
Taguchi
, G.
, 1978
, “Performance Analysis Design
,” Int. J. Prod. Res.
, 16
(6
), pp. 512
–530
.10.1080/0020754780893004334.
Fiessler
, B.
, Neumann
, H.-J.
, and Rackwitz
, R.
, 1979
, “Quadratic Limit States in Structural Reliability
,” J. Eng. Mech. Div.
, 105
(4
), pp. 661
–676
.35.
Hohenbichler
, M.
, Gollwitzer
, S.
, Kruse
, W.
, and Rackwitz
, R.
, 1987
, “New Light on First- and Second-Order Reliability Methods
,” Struct. Saf.
, 4
(4
), pp. 267
–284
.10.1016/0167-4730(87)90002-636.
Ghanem
, R. G.
, and Spanos
, P. D.
, 2003
, Stochastic Finite Elements: A Spectral Approach
, Dover Publications
10.1007/978-1-4612-3094-6.37.
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
.10.1016/j.probengmech.2004.04.00338.
Liu
, S. C.
, and Hu
, S. J.
, 1997
, “Variation Simulation for Deformable Sheet Metal Assemblies Using Finite Element Methods
,” ASME J. Manuf. Sci. Eng.
, 119
(3
), p. 368
.10.1115/1.283111539.
Xiu
, D.
, and Karniadakis
, G.
, 2003
, “Modeling Uncertainty in Flow Simulations via Generalized Polynomial Chaos
,” J. Comput. Phys.
, 187
(1
), pp. 137
–167
.10.1016/S0021-9991(03)00092-540.
Schoutens
, W.
, 2000
, Stochastic Processes and Orthogonal Polynomials
, Springer
, New York
.10.1007/978-1-4612-1170-941.
Mazur
, M.
, 2013
, Tolerance Analysis and Synthesis of Assemblies Subject to Loading With Process Integration and Design Optimization Tools, in School of Aerospace, Mechanical, and Manufacturing Engineering
, RMIT University
, Melbourne, Australia
.42.
Gerstner
, T.
, and Griebel
, M.
, 1998
, “Numerical Integration Using Sparse Grids
,” Numer. Algorithms
, 18
(3
), pp. 209
–232
.10.1023/A:101912971764443.
Kuo
, F. Y.
, and Sloan
, I. H.
, 2005
, Lifting the Curse of Dimensionality
, Vol. 52
, AMS
, pp. 1320
–1328
.44.
McKay
, M. D.
, Beckman
, R. J.
, and Conover
, W. J.
, 1979
, “A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output From a Computer Code
,” Technometrics (Am. Stat. Assoc.)
, 2
(21
), pp. 239
–245
.10.2307/126852245.
Huntington
, D. E.
, and Lyrintzis
, C. S.
, 1998
, “Improvements to and Limitations of Latin Hypercube Sampling
,” Probab. Eng. Mech.
, 13
(4
), pp. 245
–253
.10.1016/S0266-8920(97)00013-146.
Lee
, S.
, and Chen
, W.
, 2009
, “A Comparative Study of Uncertainty Propagation Methods for Black-Box-Type Problems
,” Struct. Multidiscip. Optim.
, 37
(3
), pp. 239
–253
.10.1007/s00158-008-0234-747.
Eldred
, M.
, and Burkardt
, J.
, 2009
, Comparison of Non-Intrusive Polynomial Chaos and Stochastic Collocation Methods for Uncertainty Quantification
, American Institute of Aeronautics and Astronautics
, Reston, VA
.48.
Wojtkiewicz
, S.
, Eldred
, M. S.
, and Urbina
, A.
, 2001
, Uncertainty Quantification in Large Computational Engineering Models
, Vol. 14
, American Institute of Aeronautics and Astronautics
, Reston, VA
.49.
Haldar
, A.
, and Mahadevan
, S.
, 2000
, Probability, Reliability, and Statistical Methods in Engineering Design
, Wiley
, New York.50.
51.
Berveiller
, M.
, Sudret
, B.
, and Lemaire
, M.
, 2006
, “Stochastic Finite Element: A Non Intrusive Approach by Regression
,” Rev. Eur. de Mécanique Numér.
, 15
(1–3
), pp. 81
–92
.52.
Wan
, X.
, and Karniadakis
, G. E.
, 2007
, “Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures
,” SIAM J. Sci. Comput.
, 28
(3
), pp. 901
–928
.10.1137/05062763053.
McRae
, G. J.
, and Tatang
, M. A.
, 1995
, Direct Incorporation of Uncertainty in Chemical and Environmental Engineering Systems
, Massachusetts Institute of Technology
.54.
Armen Der Kiureghian
, M.
, and Liu
, P. L.
, 1986
, “Structural Reliability Under Incomplete Probability Information
,” J. Eng. Mech.
, 112
(1
), p. 85
.10.1061/(ASCE)0733-9399(1986)112:1(85)55.
Box
, G. E. P.
, and Cox
, D. R.
, “An Analysis of Transformations
,” J. R. Stat. Soc. Ser B (Methodological)
, pp. 211
–252
.56.
Choi
, S. K.
, Grandhi
, R. V.
, and Canfield
, R. A.
, 2004
, “Structural Reliability Under Non-Gaussian Stochastic Behavior
,” Comput. Struct.
, 82
(13–14
), pp. 1113
–1121
.10.1016/j.compstruc.2004.03.01557.
Hosder
, S.
, Walters
, R. W.
, and Balch
, M.
, 2007
, “Efficient Sampling for Non-Intrusive Polynomial Chaos Applications With Multiple Uncertain Input Variables
,” AIAA
Paper No. 2007-1939.10.2514/6.2007-193958.
Xiu
, D.
, 2010
, Numerical Methods for Stochastic Computations: A Spectral Method Approach
, Princeton University Press
, Princeton, NJ.59.
Hosder
, S.
, Walters
, R. W.
, and Balch
, M.
, 2007
, “Efficient Sampling for Non-Intrusive Polynomial Chaos Applications With Multiple Uncertain Input Variables
,” AIAA
Paper No. 2007-1939.10.2514/6.2007-193960.
Xiu
, D.
, 2007
, “Efficient Collocational Approach for Parametric Uncertainty Analysis
,” Commun. Comput. Phys.
, 2
(2
), pp. 293
–309
.61.
Kalos
, M. H.
, and Whitlock
, P. A.
, 2009
, Monte Carlo Methods
, Wiley-VCH Verlag GmbH & Co. KGaA
, Weinheim, Germany
.10.1002/978352762621262.
Atkinson
, K. E.
, 2009
, An Introduction to Numerical Analysis
, 2nd ed., Wiley India Pvt. Ltd.
63.
Delves
, L. M.
, and Mohamed
, J. L.
, 1988
, Computational Methods for Integral Equations
, Cambridge University Press
.10.1017/CBO978051156960964.
Kovvali
, N.
, 2011
, Theory and Applications of Gaussian Quadrature Methods
, Morgan and Claypool
.65.
Epperson
, J. F.
, 2007
, An Introduction to Numerical Methods and Analysis
, Wiley-Interscience
, Blacksburg, VA.66.
Bungartz
, H. J.
, and Griebel
, M.
, 2004
, “Sparse Grids
,” Acta Numer.
, 13
(1
), pp. 147
–269
.10.1017/S096249290400018267.
Smolyak
, S. A.
, 1963
, “Quadrature and Interpolation Formulas for Tensor Products of Certain Classes of Functions
,” Dokl. Akad. Nauk SSSR
, 4
, pp. 240
–243
.68.
Gerstner
, T.
, and Griebel
, M.
, 2003
, “Dimension–Adaptive Tensor–Product Quadrature
,” Computing
, 71
(1
), pp. 65
–87
.10.1007/s00607-003-0015-569.
Burkardt
, J.
, 2010
, “1D Quadrature Rules for Sparse Grids
,” Interdisciplinary Center for Applied Mathematics and Information Technology Department, Virginia Tech.70.
Crestaux
, T.
, 2009
, “Polynomial Chaos Expansion for Sensitivity Analysis
,” Reliab. Eng. Syst. Saf.
, 94
(7
), pp. 1161
–1172
.10.1016/j.ress.2008.10.00871.
Burkardt
, J.
, 2010
, “The Combining Coefficient for Anisotropic Sparse Grids
,” Interdisciplinary Center for Applied Mathematics & Information Technology Department, Virginia Tech.72.
Jakeman
, J. D.
, Archibald
, R.
, and Xiu
, D.
, 2011
, “Characterization of Discontinuities in High-Dimensional Stochastic Problems on Adaptive Sparse Grids
,” J. Comput. Phys.
, 230
(10
), pp. 3977
–3997
.10.1016/j.jcp.2011.02.02273.
Congedo
, P. M.
, Abgrall
, R.
, and Geraci
, G.
, 2011
, “On the Use of the Sparse Grid Techniques Coupled With Polynomial Chaos.
”74.
Debusschere
, B. J.
, Najm
, H. N.
, Pébay
, P. P.
, Knio
, O. M.
, Ghanem
, R. G.
, and Le Maître
, O. P.
, 2005
, “Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes
,” SIAM J. Sci. Comput.
, 26
(2
), pp. 698
–719
.10.1137/S106482750342774175.
Archibald
, R. K.
, Deiterding
, R.
, and Xiu
, D.
, 2012
, “Approximation and Error Estimation in High Dimen-Sional Space for Stochastic Collocation Methods on Ar-Bitrary Sparse Samples
,” Oak Ridge National Laboratory (ORNL); Center for Computational Sciences.76.
Mazur
, M.
, Leary
, M.
, and Subic
, A.
, 2011
, “Computer Aided Tolerancing (CAT) Platform for the Design of Assemblies Under External and Internal Forces
,” Comput.-Aided Des.
, 43
(6
), pp. 707
–719
.10.1016/j.cad.2011.02.00477.
Deb
, K.
, 2004
, Optimization for Engineering Design: Algorithms and Examples
, Prentice-Hall of India
.78.
Flager
, F.
, Soremekun
, G.
, Welle
, B.
, Haymaker
, J.
, and Bansal
, P., 2009
, “Multidisciplinary Process Integration and Design Optimization of a Classroom Building
,” Electron. J. Inf. Technol. Constr.
, 14
, pp. 595
–612
.79.
Hiriyannaiah
, S.
, and Mocko
, G. M.
, 2008
, “Information Management Capabilities of MDO Frameworks
,” ASME
Paper No. DETC2008-49934.10.1115/DETC2008-4993480.
Sobieszczanski-Sobieski
, J.
, and Haftka
, R. T.
, 1997
, “Multidisciplinary Aerospace Design Optimization: Survey of Recent Developments
,” Struct. Multidiscip. Optimization
, 14
(1
), pp. 1
–23
.10.1007/BF0119755481.
Lovasz
, E. C.
, 2012
, Mechanisms, Transmissions, and Applications
, Springer
London, UK
.10.1007/978-94-007-2727-482.
Shigley
, J. E.
, Mischke
, C. R.
, and Budynas
, R. G.
, 2004
, Mechanical Engineering Design
, McGraw-Hill
.83.
Lee
, D.
, Kwon
, K. E.
, Lee
, J.
, Jee
, H.
, Yim
, H.
, Cho
, S. W.
, Shin
, J.-G.
, and Lee
, G.
, 2009
, “Tolerance Analysis Considering Weld Distortion by Use of Pregenerated Database
,” ASME J. Manuf. Sci. Eng.
, 131
(4
), p. 041012
.10.1115/1.316031984.
Pierre
, L.
, Teissandier
, D.
, and Nadeau
, J. P.
, 2009
, “Integration of Thermomechanical Strains Into Tolerancing Analysis
,” Int. J. Interact. Des. Manuf.
, 3
(4
), pp. 247
–263
.10.1007/s12008-009-0058-885.
Shiu
, B.
, Apley
, D. W.
, Ceglarek
, D.
, and Shi
, J.
, 2003
, “Tolerance Allocation for Compliant Beam Structure Assemblies
,” IIE Trans.
, 35
(4
), pp. 329
–342
.10.1080/0740817030437686.
Makelainen
, E.
, Ramseier
, Y.
, Salmensuu
, S.
, Heilala
, J.
, Voho
, P.
, and Vaatainen
, O.
, 2001
, “Assembly Process Level Tolerance Analysis for Electromechanical Products
,” Proceedings of the IEEE International Symposium on Assembly and Task Planning (ISATP2001
), Fukuoka, May 28–29, pp. 405
–410
.10.1109/ISATP.2001.92906887.
Taguchi
, G.
, 1989
, Introduction to Quality Engineering
, Asian Productivity Organization, Unipub
, New York
.88.
Phadke
, M. S.
, 1989
, Quality Engineering Using Robust Design
, Prentice Hall
.89.
Feigenbaum
, A. V.
, 2012
, Total Quality Control, 4th ed.: Achieving Productivity, Market Penetration, and Advantage in the Global Economy
, McGraw-Hill
.90.
Dong
, Z.
, Hu
, W.
, and Xue
, D.
, 1994
, “New Production Cost-Tolerance Models for Tolerance Synthesis
,” ASME J. Eng. Ind.
, 116
(2
), pp. 199
–205
.10.1115/1.290193191.
Dong
, Z.
, and Soom
, A.
, 1990
, “Automatic Optimal Tolerance Design for Related Dimension Chains
,” Manuf. Rev.
, 3
, pp. 262
–271
.92.
Pearn
, W. L.
, and Kotz
, S.
, 2006
, Encyclopedia and Handbook of Process Capability Indices: A Comprehensive Exposition of Quality Control Measures (Series on Quality, Reliability and Engineering Statistics)
, World Scientific Publishing
, Hackensack, NJ
.93.
Feng
, C. X.
, and Kusiak
, A.
, 1997
, “Robust Tolerance Design With the Integer Programming Approach
,” ASME. J. Manuf. Sci. Eng.
, 119
(4A
), pp. 603
–610
.10.1115/1.283119394.
Nocedal
, J.
, and Wright
, S. J.
, 1999
, Numerical Optimization
, Springer Verlag
, New York
.10.1007/b9887495.
Voß
, S.
, 2001
, Meta-Heuristics: The State of the Art, in Local Search for Planning and Scheduling
, Springer
, pp. 1
–23
.96.
Zitzler
, E.
, and Thiele
, L.
, 1999
, “Multiobjective Evolutionary Algorithms: A Comparative Case Study and the Strength Pareto Approach
,” IEEE Evol. Comput., Trans.
, 3
(4
), pp. 257
–271
.10.1109/4235.79796997.
Bäck
, T.
, 1996
, Evolutionary Algorithms in Theory and Practice: Evolution Strategies, Evolutionary Programming, Genetic Algorithms
, Oxford University Press
.98.
Fonseca
, C. M.
, and Fleming
, P. J.
, 1995
, “An Overview of Evolutionary Algorithms in Multiobjective Optimization
,” Evol. Comput.
, 3
(1
), pp. 1
–16
.10.1162/evco.1995.3.1.199.
Forouraghi
, B.
, 2002
, “Worst-Case Tolerance Design and Quality Assurance Via Genetic Algorithms
,” J. Optim. Theory Appl.
, 113
(2
), pp. 251
–268
.10.1023/A:1014826824323100.
Kumar
, M. S.
, and Kannan
, S.
, 2007
, “Optimum Manufacturing Tolerance to Selective Assembly Technique for Different Assembly Specifications by Using Genetic Algorithm
,” Int. J. Adv. Manuf. Technol.
, 32
(5
), pp. 591
–598
.10.1007/s00170-005-0337-x101.
Leary
, M.
, Mazur
, M.
, Mild
, T.
, and Subic
, A.
, 2011
, Benchmarking and Optimisation of Automotive Seat Structures
.102.
Williams
, J. A.
, 1994
, Engineering Tribology
, Cambridge University Press
.103.
Adams
, B. M.
, Bohnhoff
, W. J.
, Dalbey
, K. R.
, Eddy
, J. P.
, Eldred
, M. S.
, Gay
, D. M.
, Haskell
, K.
, Hough
, P. D.
, and Swiler
, L. P.
, 2011
, “DAKOTA, A Multilevel Parallel Object-Oriented Framework for Design Optimization, Parameter Estimation, Uncertainty Quantification, and Sensitivity Analysis
,” Sandia Technical Report No. SAND2010-2183.104.
Deb
, K.
, Pratap
, A.
, Agarwal
, S.
, and Meyarivan
, T.
, 2002
, “A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II
,” IEEE Trans. Evol. Comput.
, 6
(2
), pp. 182
–197
.10.1109/4235.996017Copyright © 2015 by ASME
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