Design optimization problems under random uncertainties are commonly formulated with constraints in probabilistic forms. This formulation, also referred to as reliability-based design optimization (RBDO), has gained extensive attention in recent years. Most researchers assume that reliability levels are given based on past experiences or other design considerations without exploring the constrained space. Therefore, inappropriate target reliability levels might be assigned, which either result in a null probabilistic feasible space or performance underestimations. In this research, we investigate the maximal reliability within a probabilistic constrained space using modified efficient global optimization (EGO) algorithm. By constructing and improving Kriging models iteratively, EGO can obtain a global optimum of a possibly disconnected feasible space at high reliability levels. An infill sampling criterion (ISC) is proposed to enforce added samples on the constraint boundaries to improve the accuracy of probabilistic constraint evaluations via Monte Carlo simulations. This limit state ISC is combined with the existing ISC to form a heuristic approach that efficiently improves the Kriging models. For optimization problems with expensive functions and disconnected feasible space, such as the maximal reliability problems in RBDO, the efficiency of the proposed approach in finding the optimum is higher than those of existing gradient-based and direct search methods. Several examples are used to demonstrate the proposed methodology.
Issue Section:
Research Papers
1.
Allen
, M.
, and Maute
, K.
, 2005, “Reliability-Based Shape Optimization of Structures Undergoing Fluid-Structure Interaction Phenomena
,” Comput. Methods Appl. Mech. Eng.
0045-7825, 194
, pp. 3472
–3495
.2.
Huyse
, L.
, Padula
, S.
, Lewis
, R.
, and Li
, W.
, 2002, “Probabilistic Approach to Free-Form Airfoil Shape Optimization Under Uncertainty
,” AIAA J.
0001-1452, 40
(9
), pp. 1764
–1772
.3.
Gu
, L.
, Yang
, R. -J.
, Tho
, C.
, Makowski
, M.
, Faruque
, O.
, and Li
, Y.
, 2001, “Optimization and Robustness for Crashworthiness of Side Impact
,” Int. J. Veh. Des.
0143-3369, 26
(4
), pp. 348
–360
.4.
Chan
, K. -Y.
, Papalambros
, P.
, and Skerlos
, S.
, 2010, “A Method for Reliability-Based Optimization With Multiple Non-Normal Stochastic Parameters: A Simplified Airshed Management Study
,” Stochastic Environ. Res. Risk Assess.
, 21
(1
), pp. 101
–116
.5.
Du
, X.
, and Chen
, W.
, 2004, “Sequential Optimization and Reliability Assessment Method for Efficient Probabilistic Design
,” ASME J. Mech. Des.
0161-8458, 126
(2
), pp. 225
–233
.6.
Kuo
, W.
, and Wan
, R.
, 2007, “Recent Advances in Optimal Reliability Allocation
,” IEEE Trans. Syst. Man Cybern., Part A. Syst. Humans
1083-4427, 37
(2
), pp. 143
–156
.7.
Taflanidis
, A.
, and Beck
, J.
, 2008, “Stochastic Subset Optimization for Optimal Reliability Problems
,” Probab. Eng. Mech.
0266-8920, 23
, pp. 324
–338
.8.
Royset
, J.
, Kiureghian
, A.
, and Polak
, E.
, 2001, “Reliability-Based Optimal Design of Series Structural Systems
,” J. Eng. Mech.
0733-9399, 127
(6
), pp. 607
–614
.9.
Royset
, J.
, Kiureghian
, A.
, and Polak
, E.
, 2006, “Optimal Design With Probabilistic Objective and Constraints
,” J. Eng. Mech.
0733-9399, 132
(1
), pp. 107
–118
.10.
Bichon
, B.
, Eldred
, M.
, Swiler
, L.
, Mahadevan
, S.
, and McFarland
, J.
, 2008, “Efficient Global Relaiblity Analysis for Nonlinear Implicit Performance Functions
,” AIAA J.
0001-1452, 46
(10
), pp. 2459
–2468
.11.
Hasofer
, A.
, and Lind
, N.
, 1974, “Exact and Invariant Second-Moment Code Format
,” J. Engrg. Mech. Div.
0044-7951, 100
, (EM1), pp. 111
–121
.12.
Hohenbichler
, M.
, and Rackwitz
, R.
, 1983, “First-Order Concepts in System Reliability
,” Struct. Safety
0167-4730, 1
(3
), pp. 177
–188
.13.
Breitung
, K.
, 1984, “Asymptotic Approximations for Multinormal Integrals
,” J. Eng. Mech.
0733-9399, 110
(3
), pp. 357
–366
.14.
Fiessler
, B.
, Neumann
, H. -J.
, and Rackwitz
, R.
, 1979, “Quadratic Limit States in Structural Reliability
,” J. Engrg. Mech. Div.
0044-7951, 105
(4
), pp. 661
–676
.15.
Mitteau
, J. -C.
, 1996, “Error Estimates for FORM and SORM Computations of Failure Probability
,” Proceedings of the Seventh Specialty Conference on Probabilistic Mechanics and Structural Reliability
, Aug. 7–9, ASCE, pp. 562
–565
.16.
Halton
, J.
, 1970, “Retrospective and Prospective Survey of the Monte Carlo Method
,” SIAM Rev.
0036-1445, 12
(1
), pp. 1
–63
.17.
Rief
, H.
, Gelbard
, E.
, Schaefer
, R.
, and Smith
, K.
, 1986, “Review of Monte Carlo Techniques for Analyzing Reactor Perturbations
,” Nucl. Sci. Eng.
0029-5639, 92
(2
), pp. 289
–297
.18.
Turner
, J.
, Wright
, H.
, and Hamm
, R.
, 1985, “A Monte Carlo Primer for Health Physicists
,” Health Phys.
0017-9078, 48
(6
), pp. 717
–733
.19.
Melchers
, R.
, 2000, Structural Reliability Analysis and Prediction
, Wiley
, New York
.20.
Fishman
, G.
, 1996, Monte Carlo: Concepts, Algorithms, and Applications
, Springer
, New York
.21.
Hofer
, E.
, 1999, “Sensitivity Analysis in the Context of Uncertainty Analysis for Computationally Intensive Models
,” Comput. Phys. Commun.
0010-4655, 117
(1–2
), pp. 21
–34
.22.
Jones
, D.
, Schonlau
, M.
, and Welch
, W.
, 1998, “Efficient Global Optimization of Expensive Black-Box Functions
,” J. Global Optim.
0925-5001, 13
(4
), pp. 455
–492
.23.
Schonlau
, M.
, Welch
, W.
, and Jones
, D.
, 1998, “Global Versus Local Search in Constrained Optimization of Computer Models
,” New Developments and Applications in Experimental Design
, N.
Flournoy
, W.
Rosenberger
, and W.
Wong
, eds., Institute of Mathematical Statistics
, Beachwood, OH
.24.
Schonlau
, M.
, 1997, “Computer Experiments and Global Optimization
,” Ph.D. thesis, University of Waterloo, Waterloo, ON, Canada.25.
Sasena
, M.
, 2002, “Flexibility and Efficiency Enhancements for Constrained Global Design Optimization With Kriging Approximations
,” Ph.D. thesis, University of Michigan, Ann Arbor, MI.26.
Jones
, D.
, Perttunen
, C.
, and Stuckman
, B.
, 1993, “Lip-Schitzian Optimization Without the Lipschitz Constant
,” J. Optim. Theory Appl.
0022-3239, 79
(1
), pp. 157
–181
.27.
Matheron
, G.
, 1963, “Principles of Geostatistics
,” Econ. Geol.
0361-0128, 58
, pp. 1246
–1266
.28.
Chen
, W.
, Allen
, J.
, Schrage
, D.
, and Mistree
, F.
, 1997, “Statistical Experimentation Methods for Achieving Affordable Concurrent System Design
,” AIAA J.
0001-1452, 35
(5
), pp. 893
–900
.29.
Montgomery
, D.
, 2005, Design and Analysis of Experiments
, 6th ed., Wiley
, Hoboken, NJ
.30.
van Dam
, E. R.
, Husslage
, B.
, Hertog
, D. D.
, and Melis-sen
, H.
, 2007, “Maximin Latin Hypercube Designs in Two Dimensions
,” Oper. Res.
0030-364X, 55
(1
), pp. 158
–169
.31.
Sacks
, J.
, Schiller
, S.
, and Welch
, W.
, 1989, “Designs for Computer Experiments
,” Technometrics
0040-1706, 31
(1
), pp. 41
–47
.32.
Chilès
, J. -P.
, and Delfiner
, P.
, 1999, Geostatistics: Modeling Spatial Uncertainty
, Wiley
, New York
.33.
Cressie
, N.
, 1985, “Fitting Variogram Models by Weighted Least Squares
,” Math. Geol.
0882-8121, 17
(5
), pp. 563
–586
.34.
Martin
, J.
, 2009, “Computational Improvements to Estimating Kriging Metamodel Parameters
,” ASME J. Mech. Des.
0161-8458, 131
(8
), p. 084501
.35.
Li
, M.
, Li
, G.
, and Azarm
, S.
, 2008, “A Kriging Meta-Model Assisted Multi-Objective Genetic Algorithm for Design Optimization
,” ASME J. Mech. Des.
0161-8458, 130
(3
), p. 031401
.36.
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.
0162-1459, 86
(416
), pp. 953
–963
.37.
Warnes
, J.
, and Ripley
, B.
, 1987, “Problems With Likelihood Estimation of Covariances Function of Spatial Gaussian Processes
,” Biometrika
0006-3444, 74
(3
), pp. 640
–642
.38.
Davis
, G.
, and Morris
, M.
, 1997, “Six Factors Which Affects the Condition Number of Matrices Associated With Kriging
,” Math. Geol.
0882-8121, 29
(5
), pp. 669
–683
.39.
Cressie
, N.
, 1993, Statistics for Spatial Data
, Wiley
, New York
.40.
Au
, S.
, and Beck
, J.
, 1999, “A New Adaptive Importance Sampling Scheme for Reliability Calculations
,” Struct. Safety
0167-4730, 21
(2
), pp. 135
–158
.41.
Papalambros
, P.
, and Wilde
, D.
, 2000, Principles of Optimal Design
, 2nd ed., Cambridge University Press
, New York
.42.
Dixon
, L.
, and Szegö
, G.
, 1978, Towards Global Optimisation 2
, North-Holland
, Amsterdam
.43.
Gutmann
, H.
, 2001, “A Radial Basis Function Method for Global Optimization
,” J. Global Optim.
0925-5001, 19
, pp. 201
–227
.44.
Goel
, T.
, Haftka
, R.
, Shyy
, W.
, and Queipo
, N.
, 2007, “Ensemble of Surrogates
,” Struct. Multidiscip. Optim.
1615-147X, 33
, pp. 199
–216
.45.
Youn
, B.
, and Choi
, K.
, 2004, “An Investigation of Non-Linearity of Reliability-Based Design Optimization Approaches
,” ASME J. Mech. Des.
0161-8458, 126
(3
), pp. 403
–411
.46.
Chan
, K. -Y.
, Skerlos
, S.
, and Papalambros
, P.
, 2007, “An Adaptive Sequential Linear Programming Algorithm for Optimal Design Problems With Probabilistic Constraints
,” ASME J. Mech. Des.
0161-8458, 129
(2
), pp. 140
–149
.47.
Zou
, T.
, Mourelatos
, Z.
, Mahadevan
, S.
, and Tu
, J.
, 2008, “An Indicator Response Surface Method for Simulation-Based Reliability Analysis
,” ASME J. Mech. Des.
0161-8458, 130
(7
), p. 071401
.48.
Merwade
, V.
, Maldment
, D.
, and Goff
, J.
, 2006, “Anisotropic Considerations While Interpolating River Channel Bathymetry
,” J. Hydrol.
0022-1694, 331
(3–4
), pp. 731
–741
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