Abstract
Design optimization of structural and multidisciplinary systems under uncertainty has been an active area of research due to its evident advantages over deterministic design optimization. In deterministic design optimization, the uncertainties of a structural or multidisciplinary system are taken into account by using safety factors specified in the regulations or design codes. This uncertainty treatment is a subjective and indirect way of dealing with uncertainty. On the other hand, design under uncertainty approaches provide an objective and direct way of dealing with uncertainty. This paper provides a review of the uncertainty treatment practices in design optimization of structural and multidisciplinary systems under uncertainties. To this end, the activities in uncertainty modeling are first reviewed, where theories and methods on uncertainty categorization (or classification), uncertainty handling (or management), and uncertainty characterization are discussed. Second, the tools and techniques developed and used for uncertainty modeling and propagation are discussed under the broad two classes of probabilistic and non-probabilistic approaches. Third, various design optimization methods under uncertainty which incorporate all the techniques covered in uncertainty modeling and analysis are reviewed. In addition to these in-depth reviews on uncertainty modeling, uncertainty analysis, and design optimization under uncertainty, some real-life engineering applications and benchmark test examples are provided in this paper so that readers can develop an appreciation on where and how the discussed techniques can be applied and how to compare them. Finally, concluding remarks are provided, and areas for future research are suggested.
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References
Acar E (2016) A reliability index extrapolation method for separable limit states. Struct Multidiscip Optim 53:1099–1111
Acar E, Solanki K (2009) System reliability based vehicle design for crashworthiness and effects of various uncertainty reduction measures. Struct Multidiscip Optim 39(3):311–325
Acar E, Kale A, Haftka R, Stroud W (2006) Structural safety measures for airplanes. J Aircr 43(1):30–38
Acar E, Haftka R, Johnson T (2007) Tradeoff of uncertainty reduction mechanisms for reducing structural weight. J Mech Des 129(3):266–274
Acar E, Haftka R, Kim N (2010) Effects of structural tests on aircraft safety. AIAA J 48(10):2235–2248
Agarwal H, Mozumder C, Renaud J, Watson L (2007) An inverse-measure-based unilevel architecture for reliability-based design optimization. Struct Multidiscip Optim 33(3):217–227
Agarwal P, Nayal H (2015) Possibility theory versus probability theory in fuzzy measure theory. Int J Eng Res Appl 5(5):37–43
Ahmad I (1982) Nonparametric estimation of the location and scale parameters based on density estimation. Ann Inst Stat Math 34(1):39–53
Akaike H (1974) A new look at the statistical model identification. IEEE Trans Autom Control 19(6):716–723
Allaire D, Noel G, Willcox K, Cointin R (2014) Uncertainty quantification of an aviation environmental toolsuite. Reliab Eng Syst Saf 126:14–24
Alleman G (2014) Performance-based project management: increasing the probability of project success. Amacom
Allen JK, Panchal J, Mistree F, Singh AK, Gautham B (2015) Uncertainty management in the integrated realization of materials and components. In: Proceedings of the 3rd World Congress on Integrated Computational Materials Engineering (ICME 2015), Springer, pp 339–346
Allen M, Maute K (2004) Reliability-based design optimization of aeroelastic structures. Struct Multidiscip Optim 27(4):228–242
Alyanak E, Grandhi R, Bae H (2008) Gradient projection for reliability-based design optimization using evidence theory. Eng Optim 40(10):923–935
An D, Choi J (2012) Efficient reliability analysis based on Bayesian framework under input variable and metamodel uncertainties. Struct Multidiscip Optim 46(4):533–547
Anderson T, Darling D (1952) Asymptotic theory of certain goodness of fit criteria based on stochastic processes. Ann Math Stat 23(2):193–212
Annis C (2004) Probabilistic life prediction isn’t as easy as it looks. In: Johnson WS, Hillberry BM (eds) Probabilistic aspects of life prediction. ASTM International, West Conshohocken
António CC, Hoffbauer LN (2009) An approach for reliability-based robust design optimisation of angle-ply composites. Compos Struct 90(1):53–59
Aoues Y, Chateauneuf A (2010) Benchmark study of numerical methods for reliability-based design optimization. Struct Multidiscip Optim 41(2):277–294
Arendt P, Apley D, Chen W (2012) Quantification of model uncertainty: calibration, model discrepancy, and identifiability. J Mech Des 134(10):100908
Arslan A, Kaya M (2001) Determination of fuzzy logic membership functions using genetic algorithms. Fuzzy Sets Syst 118(2):297–306
Au S (2005) Reliability-based design sensitivity by efficient simulation. Comput struct 83(14):1048–1061
Au S, Beck J (2001) Estimation of small failure probabilities in high dimensions by subset simulation. Probab Eng Mech 16(4):263–277
Au S, Papadimitriou C, Beck J (1999) Reliability of uncertain dynamical systems with multiple design points. Struct Saf 21(2):113–133
Ayyub B, McCuen R (2016) Probability, statistics, and reliability for engineers and scientists. CRC Press, Boca Raton
Azarkish H, Rashki M (2019) Reliability and reliability-based sensitivity analysis of shell and tube heat exchangers using Monte Carlo simulation. Appl Therm Eng 159:113842
Azevedo CL, Ciuffo B, Cardoso JL, Ben-Akiva ME (2015) Dealing with uncertainty in detailed calibration of traffic simulation models for safety assessment. Transp Res C 58:395–412
Ba-Abbad M, Nikolaidis E, Kapania R (2006) New approach for system reliability-based design optimization. AIAA J 44(5):1087–1096
Bacarreza O, Aliabadi M, Apicella A (2015) Robust design and optimization of composite stiffened panels in post-buckling.structural and multidisciplinary
Bae H, Alyanak E (2016) Sequential subspace reliability method with univariate revolving integration. AIAA J 54(7):2160–2170
Bashtannyk D, Hyndman R (2001) Bandwidth selection for kernel conditional density estimation. Comput Stat Data Anal 36(3):279–298
Basudhar A, Missoum S (2008) Adaptive explicit decision functions for probabilistic design and optimization using support vector machines. Comput Struct 86(19–20):1904
Basudhar A, Missoum S, Sanchez A (2008) Limit state function identification using support vector machines for discontinuous responses and disjoint failure domains. Probab Eng Mech 23(1):1–1
Baudoui V, Klotz P, Hiriart-Urruty J, Jan S, Morel F (2012) Local uncertainty processing (LOUP) method for multidisciplinary robust design optimization. Struct Multidiscip Optim 46(5):711–726
Bayes T (1991) An essay towards solving a problem in the doctrine of chances. Comput Med Pract 8(3):157
Beck J, Katafygiotis L (1998) Updating models and their uncertainties. I: Bayesian statistical framework. J Eng Mech 124(4):455–461
Ben-Haim Y (1994) A non-probabilistic concept of reliability. Struct Saf 14(4):227–245
Ben-Haim Y (2001) Information-gap decision theory: decisions under severe uncertainty. Academic Press, Cambridge
Ben-Haim Y (2006) Information-gap decision theory: decisions under severe uncertainty, 2nd edn. Academic Press, London
Ben-Haim Y, Elishakoff I (1995) Discussion on: a non-probabilistic concept of reliability. Struct Saf 17(3):195–199
Ben-Haim Y, Elishakoff I (2013) Convex models of uncertainty in applied mechanics. Elsevier, Amsterdam
Benner P, Gugercin S, Willcox K (2015) A survey of projection-based model reduction methods for parametric dynamical systems. SIAM Rev 57(4):483–531
Beyer HG, Sendhoff B (2007) Robust optimization-a comprehensive survey. Comput Methods Appl Mech Eng 196(33–34):3190–3218
Bichon B, Eldred M, Swiler L, Mahadevan S, McFarland J (2008) Efficient global reliability analysis for nonlinear implicit performance functions. AIAA J 46(10):2459–2468
Bichon B, Eldred M, Mahadevan S, McFarland J (2013) Efficient global surrogate modeling for reliability-based design optimization. J Mech Des 135(1):011009
Blatman G (2009) Adaptive sparse polynomial chaos expansions for uncertainty propagation and sensitivity analysis
Blatman G, Sudret B (2010a) An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis. Probab Eng Mech 25(2):183–197
Blatman G, Sudret B (2010b) Efficient computation of global sensitivity indices using sparse polynomial chaos expansions. Reliab Eng Syst Saf 95(11):1216–1229
Booker A, Dennis J, Frank P, Serafini D, Torczon V, Trosset M (1999) A rigorous framework for optimization of expensive functions by surrogates. Struct Optim 17(1):1–13
Bowman A (1984) An alternative method of cross-validation for the smoothing of density estimates. Biometrika 71(2):353–360
Breitung K (1984) Asymptotic approximations for multinormal integrals. J Eng Mech 110(3):357–366
Breitung K (2019) The geometry of limit state function graphs and subset simulation: Counterexamples. Reliab Eng Syst Saf 182:98–106
Broemeling L (2011) An account of early statistical inference in Arab cryptology. Am Stat 65(4):255–257
Burnham K, Anderson D (2004) Multimodel inference: understanding AIC and BIC in model selection. Sociol Methods Res 33(2):261–304
Cadini F, Santos F, Zio E (2014) An improved adaptive Kriging-based importance technique for sampling multiple failure regions of low probability. Reliab Eng Syst Saf 131:109–117
Cadini F, Gioletta A, Zio E (2015) Improved metamodel-based importance sampling for the performance assessment of radioactive waste repositories. Reliab Eng Syst Saf 134:188–197
das Chagas Moura M, Zio E, Lins ID, Droguett E (2011) Failure and reliability prediction by support vector machines regression of time series data. Reliab Eng Syst Saf 96:1527–1534
Chakraborty S, Chatterjee T, Chowdhury R, Adhikari S (2017) A surrogate based multi-fidelity approach for robust design optimization. Appl Math Model 47:726–744
Chan K, Skerlos S, Papalambros P (2007) An adaptive sequential linear programming algorithm for optimal design problems with probabilistic constraints. J Mech Des 129(2):140–149
Chatterjee T, Chakraborty S, Chowdhury R (2019) A critical review of surrogate assisted robust design optimization. Arch Comput Methods Eng 26(1):245–274
Chaudhuri A, Haftka R (2013) Separable Monte Carlo combined with importance sampling for variance reduction. Int J Reliab Saf 7(3):201–215
Chaudhuri A, Kramer B, Willcox K (2020) Information reuse for importance sampling in reliability-based design optimization. Reliab Eng Syst Saf 201:106853
Chen G, Fan J, Xu H, Li B (2020) Calculation of hybrid reliability of turbine disk based on self-evolutionary game model with few shot learning. Struct Multidiscip Optim 2020:1–13
Chen S, Yang X (2000) Interval finite element method for beam structures. Finite Elem Anal Des 34(1):75–88
Chen S, Nikolaidis E, Cudney H, Rosca R, Haftka R (1999) Comparison of probabilistic and fuzzy set methods for designing under uncertainty. In: 40th structures, structural dynamics, and materials conference and exhibit, p 1579
Chen X, Hasselman T, Neill D (1997) Reliability-based structural design optimization for practical applications. In: Proceedings of the 38th AIAA structures, structural dynamics, and materials conference, Florida
Chen Z, Qiu H, Gao L, Su L, Li P (2013) An adaptive decoupling approach for reliability-based design optimization. Comput Struct 117:58–66
Chen Z, Qiu H, Gao L, Li X, Li P (2014) A local adaptive sampling method for reliability-based design optimization using Kriging model. Struct Multidiscip Optim 49(3):401–416
Chen Z, Peng S, Li X, Qiu H, Xiong H, Gao L, Li P (2015) An important boundary sampling method for reliability-based design optimization using Kriging model. Struct Multidiscip Optim 52(1):55–70
Chen Z, Li X, Chen G, Gao L, Qiu H, Wang S (2018) A probabilistic feasible region approach for reliability-based design optimization. Struct Multidiscip Optim 57(1):359–372
Chen Z, Wu Z, Li X, Chen G, Gao L, Gan X, Wang S (2019a) A multiple-design-point approach for reliability-based design optimization. Eng Optim 51(5):875–895
Chen Z, Zhou P, Liu Y (2019b) A novel approach to uncertainty analysis using methods of hybrid dimension reduction and improved maximum entropy. Struct Multidiscip Optim 60:1841–1866
Cheng H, Chen J (1997) Automatically determine the membership function based on the maximum entropy principle. Inf Sci 96(3–4):163–182
Cheng J, Liu Z, Qian Y, Zhou Z, Tan J (2020) Non-probabilistic robust equilibrium optimization of complex uncertain structures. J Mech Des 142(2):021405
Chiralaksanakul A, Mahadevan S (2005) First-order approximation methods in reliability-based design optimization. J Mech Des 127:851
Cho H, Choi K, Gaul N, Lee I, Lamb D, Gorsich D (2016a) Conservative reliability-based design optimization method with insufficient input data. Struct Multidiscip Optim 54(6):1609–1630
Cho H, Choi K, Lee I, Lamb D (2016b) Design sensitivity method for sampling-based RBDO with varying standard deviation. J Mech Des 138(1):011405
Cho H, Choi K, Shin J (2020) Iterative most probable point search method for problems with a mixture of random and interval variables. J Mech Des 142(7):071703
Cho S, Jang J, Kim S, Park S, Lee T, Lee M, Hong S (2016) Nonparametric approach for uncertainty-based multidisciplinary design optimization considering limited data. Struct Multidiscip Optim 54(6):1671–1688
Cho T, Lee B (2011) Reliability-based design optimization using convex linearization and sequential optimization and reliability assessment method. Struct Saf 33(1):42–50
Chutia R (2017) Uncertainty quantification under hybrid structure of probability-fuzzy parameters in Gaussian plume model. Life Cycle Reliab Saf Eng 6(4):277–284
Cicala D, Irias X (2014) Utilizing info-gap decision theory to improve pipeline reliability: a case study. In: Pipelines 2014: from underground to the forefront of innovation and sustainability, pp 1749–1760
Civanlar M, Trussell H (1986) Constructing membership functions using statistical data. Fuzzy Sets Syst 18(1):1–13
Constantine P, Emory M, Larsson J, Iaccarino G (2015) Exploiting active subspaces to quantify uncertainty in the numerical simulation of the HyShot II scramjet. J Comput Phys 302:1–20
Coppitters D, De Paepe W, Contino F (2019) Surrogate-assisted robust design optimization and global sensitivity analysis of a directly coupled photovoltaic-electrolyzer system under techno-economic uncertainty. Appl Energy 248:310–320
Council NR et al (2009) Science and decisions: advancing risk assessment. National Academies Press, Washington DC
Degrauwe D, Lombaert G, De Roeck G (2010) Improving interval analysis in finite element calculations by means of affine arithmetic. Comput Struct 88(3–4):247–254
Dempster A (1967) Upper and lower probabilities induced by a multivalued mapping. Ann Math Stat 38(2):325–339
Der Kiureghian A (1996) Structural reliability methods for seismic safety assessment: a review. Eng Struct 18(6):412–424
Der Kiureghian A, Dakessian T (1998) Multiple design points in first and second-order reliability. Struct Saf 20(1):37–49
Dodson M, Parks G (2015) Robust aerodynamic design optimization using polynomial chaos. J Aircr 46(2):635–646
Doltsinis I, Kang Z (2004) Robust design of structures using optimization methods. Comput Methods Appl Mech Eng 193(23–26):2221–2237
Du L, Choi K, Youn B, Gorsich D (2006) Possibility-based design optimization method for design problems with both statistical and fuzzy input data. J Mech Des 128(4):928
Du X, Chen W (2004) Sequential optimization and reliability assessment method for efficient probabilistic design. J Mech Des 126(2):225
Du X, Hu Z (2012) First order reliability method with truncated random variables. J Mech Des 134(9):091005
Du X, Sudjianto A, Chen W (2004) An integrated framework for optimization under uncertainty using inverse reliability strategy. J Mech Des 126(4):562–570
Du X, Sudjianto A, Huang B (2005) Reliability-based design with the mixture of random and interval variables. J Mech Des 127(6):1068
Duan Z, Jung Y, Yan J, Lee I (2020) Reliability-based multi-scale design optimization of composite frames considering structural compliance and manufacturing constraints. Struct Multidiscip Optim 61(6):2401–2421
Dubois D, Prade H (1988) Possibility theory. Plenum, New York
Dubourg V, Sudret B, Bourinet J (2011) Reliability-based design optimization using Kriging surrogates and subset simulation. Struct Multidiscip Optim 44(5):673–690
Dubourg V, Sudret B, Deheeger F (2013) Metamodel-based importance sampling for structural reliability analysis. Probab Eng Mech 33:47–57
Duong P, Yang Q, Park H, Raghavan N (2019) Reliability analysis and design of a single diode solar cell model using polynomial chaos and active subspace. Microelectron Reliab 100:113477
Duong T, Hazelton M (2003) Plug-in bandwidth matrices for bivariate kernel density estimation. J Nonparametr Stat 15(1):17–30
Duong T, Hazelton M (2005) Cross-validation bandwidth matrices for multivariate kernel density estimation. Scand J Stat 32(3):485–506
Echard B, Gayton N, Lemaire M, Relun N (2013) A combined importance sampling and Kriging reliability method for small failure probabilities with time-demanding numerical models. Reliab Eng Syst Saf 111:232–240
El Moçayd N, Mohamed M, Ouazar D, Seaid M (2020) Stochastic model reduction for polynomial chaos expansion of acoustic waves using proper orthogonal decomposition. Reliab Eng Syst Saf 195:106733
Elishakoff I, Bekel Y (2013) Application of Lame’s super ellipsoids to model initial imperfections. J Appl Mech 80(6): 061006
Elishakoff I, Zingales M (2003) Contrasting probabilistic and anti-optimization approaches in an applied mechanics problem. Int J Solids Struct 40(16):4281–4297
Elishakoff I, Elisseeff P, Glegg S (1994a) Nonprobabilistic, convex-theoretic modeling of scatter in material properties. AIAA J 32(4):843–849
Elishakoff I, Haftka R, Fang J (1994b) Structural design under bounded uncertainty-optimization with anti-optimization. Comput Struct 53(6):1401–1405
Ellingwood B (1980) Development of a probability based load criterion for American National Standard A58: Building code requirements for minimum design loads in buildings and other structures, vol 13. National Bureau of Standards, US Department of Commerce
Engelund S, Rackwitz R (1993) A benchmark study on importance sampling techniques in structural reliability. Struct Saf 12:255–276
Fan X, Wang P, Hao F (2019) Reliability-based design optimization of crane bridges using Kriging-based surrogate models. Struct Multidiscip Optim 59(3):993–1005
Fang J, Gao Y, Sun G, Xu C, Li Q (2015) Multiobjective robust design optimization of fatigue life for a truck cab. Reliab Eng Syst Saf 135:1–8
Ferson S, Ginzburg L (1996) Different methods are needed to propagate ignorance and variability. Reliab Eng Syst Saf 54(2–3):133–144
Ferson S, Joslyn C, Helton J, Oberkampf W, Sentz K (2004) Summary from the epistemic uncertainty workshop: consensus amid diversity. Reliab Eng Syst Saf 85(1–3):355–369
Freund R (2003) Model reduction methods based on Krylov subspaces. Acta Numer 12:267–319
Gao W, Wu D, Song C, Tin-Loi F, Li X (2011) Hybrid probabilistic interval analysis of bar structures with uncertainty using a mixed perturbation Monte-Carlo method. Finite Elem Anal Des 47(7):643–652
Gersem D, Hilde DM, Desmet W, Vandepitte D (2006) Non-probabilistic uncertainty assessment in finite element models with superelements. In: 47th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference 14th AIAA/ASME/AHS adaptive structures conference 7th, p 2072
Ghanem R, Spanos P (1991) Stochastic finite element method: response statistics. Stochastic finite elements: a spectral approach. Springer, New York, pp 101–119
Ghanem R, Higdon D, Owhadi H (2017) Handbook of uncertainty quantification. Springer, New York
Ghisu T, Parks GT, Jarrett JP, Clarkson PJ (2011) Robust design optimization of gas turbine compression systems. J Propul Power 27(2):282–295
Giles M (2008) Multilevel Monte Carlo path simulation. Oper Res 56(3):607–617
Goel T, Haftka R, Shyy W, Queipo N (2007) Ensemble of surrogates. Struct Multidiscip Optim 33(3):199–216
Gomes HM, Awruch AM, Lopes PAM (2011) Reliability based optimization of laminated composite structures using genetic algorithms and artificial neural networks. Struct Saf 33(3):186–195
Grujicic M, Arakere G, Bell W, Marvi H, Yalavarthy H, Pandurangan B, Haque I, Fadel G (2010) Reliability-based design optimization for durability of ground vehicle suspension system components. J Mater Eng Perform 19(3):301–313
Guo S, Lu Z (2015) A non-probabilistic robust reliability method for analysis and design optimization of structures with uncertain-but-bounded parameters. Appl Math Model 39(7):1985–2002
Guyonnet D, Bourgine B, Dubois D, Fargier H, Co me B, Chilès JP, (2003) Hybrid approach for addressing uncertainty in risk assessments. J Environ Eng 129(1):68–78
Hájek A (2019) Interpretations of probability, the stanford encyclopedia of philosophy
Håkansson A (2019) Estimating convective heat transfer coefficients and uncertainty thereof using the general uncertainty management (GUM) framework. J Food Eng 263:53–62
Hao P, Wang Y, Liu C, Wang B, Wu H (2017) A novel non-probabilistic reliability-based design optimization algorithm using enhanced chaos control method. Comput Methods Appl Mech Eng 318:572–593
Hasofer A, Lind N (1974) Exact and invariant second-moment code format. J Eng Mech Div 100(1):111–121
Hassan R, Crossley W (2008) Spacecraft reliability-based design optimization under uncertainty including discrete variables. J Spacecr Rocket 45(2):394–405
Hasuike T, Katagiri H (2016) Construction of an appropriate membership function based on size of fuzzy set and mathematical programming. In: Proceedings of the international multiconference of engineers and computer scientists, vol 2
Hawchar L, El Soueidy CP, Schoefs F (2018) Global Kriging surrogate modeling for general time-variant reliability-based design optimization problems. Struct Multidiscip Optim 58:955–968
He W, Zeng Y, Li G (2020) An adaptive polynomial chaos expansion for high-dimensional reliability analysis. Struct Multidiscip Optim 62:2051–2067
Helton J, Davis F (2003) Latin hypercube sampling and the propagation of uncertainty in analyses of complex systems. Reliab Eng Syst Saf 81(1):23–69
Helton JC, Johnson JD, Sallaberry CJ, Storlie CB (2006) Survey of sampling-based methods for uncertainty and sensitivity analysis. Reliab Eng Syst Saf 91(10–11):1175–1209
Hess P, Bruchman D, Assakkaf I, Ayyub B (2002) Uncertainties in material and geometric strength and load variables. Nav Eng J 114(2):139–166
Hoffman F, Hammonds J (1994) Propagation of uncertainty in risk assessments: the need to distinguish between uncertainty due to lack of knowledge and uncertainty due to variability. Risk Anal 14(5):707–712
Hong T, Lee C (1996) Induction of fuzzy rules and membership functions from training examples. Fuzzy Sets Syst 84(1):33–47
Hora S (1996) Aleatory and epistemic uncertainty in probability elicitation with an example from hazardous waste management. Reliab Eng Syst Saf 54(2–3):217–223
Hosder S, Watson L, Grossman B, Mason W, Kim H, Haftka R, Cox S (2001) Polynomial response surface approximations for the multidisciplinary design optimization of a high speed civil transport. Optim Eng 2(4):431–452
Hoseyni S, Pourgol-Mohammad M, Tehranifard A, Yousefpour F (2014) A systematic framework for effective uncertainty assessment of severe accident calculations; hybrid qualitative and quantitative methodology. Reliab Eng Syst Saf 125:22–35
Hosseinzadeh Y, Taghizadieh N, Jalili S (2018) A new structural reanalysis approach based on the polynomial-type extrapolation methods. Struct Multidiscip Optim 58(3):1033–1049
Hu C, Youn BD (2011) Adaptive-sparse polynomial chaos expansion for reliability analysis and design of complex engineering systems. Struct Multidiscip Optim 43(3):419–442
Hu W, Choi K, Cho H (2016) Reliability-based design optimization of wind turbine blades for fatigue life under dynamic wind load uncertainty. Struct Multidiscip Optim 54(4):953–970
Hu X, Parks G, Chen X, Seshadri P (2015) Discovering a one-dimensional active subspace to quantify multidisciplinary uncertainty in satellite system design. Adv Space Res 57:1268
Hu X, Chen X, Zhao Y, Tuo Z, Yao W (2017) Active subspace approach to reliability and safety assessments of small satellite separation. Acta Astronaut 131:159–165
Hu Z, Du X (2013a) A sampling approach to extreme value distribution for time-dependent reliability analysis. J Mech Des 135:071003
Hu Z, Du X (2013b) Time-dependent reliability analysis with joint up-crossing rates. Struct Multidiscip Optim 48:893–907
Hu Z, Du X (2015) First order reliability method for time-variant problems using series expansions. Struct Multidiscip Optim 51:1–21
Huang B, Du X (2006) Uncertainty analysis by dimension reduction integration and saddlepoint approximations
Huang X, Li Y, Zhang Y, Zhang X (2018) A new direct second-order reliability analysis method. Appl Math Model 55:68–80
Huang Z, Jiang C, Zhou Y, Luo Z, Zhang Z (2016) An incremental shifting vector approach for reliability-based design optimization. Struct Multidiscip Optim 53(3):523–543
Iooss B, Le Gratiet L (2019) Uncertainty and sensitivity analysis of functional risk curves based on Gaussian processes. Reliab Eng Syst Saf 187:58–66
Isight (2021) Simulia execution engine—-dassault systèmes\(\textregistered \). https://www.3ds.com/products-services/simulia/products/isight-simulia-execution-engine/
Ito M, Kim N, Kogiso N (2018) Conservative reliability index for epistemic uncertainty in reliability-based design optimization. Struct Multidiscip Optim 57(5):1919–1935
Jalota H, Thakur M, Mittal G (2017) A credibilistic decision support system for portfolio optimization. Appl Soft Comput 59:512–528
Jang J (1993) Anfis: adaptive-network-based fuzzy inference system. IEEE Trans Syst Man Cybern 23(3):665–685
Jensen H, Valdebenito M, Schuëller G, Kusanovic D (2009) Reliability-based optimization of stochastic systems using line search. Comput Methods Appl Mech Eng 198(49–52):3915–3924
Jeong S, Park G (2017) Single loop single vector approach using the conjugate gradient in reliability based design optimization. Struct Multidiscip Optim 55(4):1329–1344
Ji W, Ren Z, Marzouk Y, Law C (2019) Quantifying kinetic uncertainty in turbulent combustion simulations using active subspaces. Proc Combust Inst 37(2):2175–2182
Jiang C, Han X, Li W, Liu J, Zhang Z (2012a) A hybrid reliability approach based on probability and interval for uncertain structures. J Mech Des 134(3):031001
Jiang C, Lu G, Han X, Liu L (2012b) A new reliability analysis method for uncertain structures with random and interval variables. Int J Mech Mater Des 8(2):012–9184
Jiang C, Bi R, Lu G, Han X (2013a) Structural reliability analysis using non-probabilistic convex model. Comput Methods Appl Mech Eng 254:83–98
Jiang C, Zhang Q, Han X, Liu J, Hu D (2015) Multidimensional parallelepiped model-a new type of non-probabilistic convex model for structural uncertainty analysis. Int J Numer Methods Eng 103(1):31–59
Jiang C, Qiu H, Gao L, Cai X, Li P (2017) An adaptive hybrid single-loop method for reliability-based design optimization using iterative control strategy. Struct Multidiscip Optim 56(6):1271–1286
Jiang C, Hu Z, Liu Y, Mourelatos ZP, Gorsich D, Jayakumar P (2020) A sequential calibration and validation framework for model uncertainty quantification and reduction. Comput Methods Appl Mech Eng 368:113172
Jiang H, Deng H, He Y (2008) Determination of fuzzy logic membership functions using extended ant colony optimization algorithm. In: 2008 Fifth international conference on fuzzy systems and knowledge discovery, IEEE, vol 1, pp 581–585
Jiang Z, Li J (2017) High dimensional structural reliability with dimension reduction. Struct Saf 69:35–46
Jiang Z, Chen W, Fu Y, Yang R (2013b) Reliability-based design optimization with model bias and data uncertainty. SAE Int J Mater Manuf 6(3):502–516
Jiao G, Moan T (1990) Methods of reliability model updating through additional events. Struct Saf 9(2):139–153
Jo H, Lee K, Lee M, Jung Y, Lee I (2021) Optimization-based model calibration of marginal and joint output distributions utilizing analytical gradients. Struct Multidiscip Optim 63:1–16
Ju B, Lee B (2008) Reliability-based design optimization using a moment method and a Kriging metamodel. Eng Optim 40(5):421–438
Jung Y, Cho H, Lee I (2019a) MPP-based approximated DRM (ADRM) using simplified bivariate approximation with linear regression. Struct Multidiscip Optim 59(5):1761–1773
Jung Y, Cho H, Lee I (2019b) Reliability measure approach for confidence-based design optimization under insufficient input data. Struct Multidiscip Optim 60(5):1967–1982
Jung Y, Cho H, Duan Z, Lee I (2020a) Determination of sample size for input variables in RBDO through bi-objective confidence-based design optimization under input model uncertainty. Struct Multidiscip Optim 61(1):253–266
Jung Y, Cho H, Lee I (2020b) Intelligent initial point selection for MPP search in reliability-based design optimization. Struct Multidiscip Optim 62:1–12
Jung Y, Kang K, Cho H, Lee I (2021) Confidence-based design optimization for a more conservative optimum under surrogate model uncertainty caused by gaussian process. J Mech Des 143(9):091701
Kale A, Haftka R (2008) Tradeoff of weight and inspection cost in reliability-based structural optimization. J Aircr 45(1):77–85
Kang HY, Kwak BM (2009) Application of maximum entropy principle for reliability-based design optimization. Struct Multidiscip Optim 38(4):331–346
Kang K, Qin C, Lee B, Lee I (2019) Modified screening-based Kriging method with cross validation and application to engineering design. Appl Math Model 70:626–642
Kang S, Park J, Lee I (2017a) Accuracy improvement of the most probable point-based dimension reduction method using the Hessian matrix. Int J Numer Methods Eng 111(3):203–217
Kang Y, Hong J, Lim O, Noh Y (2017b) Reliability analysis using parametric and nonparametric input modeling methods. J Comput Struct Eng Inst Korea 30(1):87–94
Kang Y, Noh Y, Lim O (2018) Kernel density estimation with bounded data. Struct Multidiscip Optim 57(1):95–113
Kang Y, Noh Y, Lim O (2019) Integrated statistical modeling method: part I-statistical simulations for symmetric distributions. Struct Multidiscip Optim 60(5):1719–1740
Kang Z, Bai S (2013) On robust design optimization of truss structures with bounded uncertainties. Struct Multidiscip Optim 47(5):699–714
Kang Z, Luo Y, Li A (2011) On non-probabilistic reliability-based design optimization of structures with uncertain-but-bounded parameters. Struct Saf 33(3):196–205
Kanno Y (2019) A data-driven approach to non-parametric reliability-based design optimization of structures with uncertain load. Struct Multidiscip Optim 60(1):83–97
Kanno Y, Takewaki I (2006) Robustness analysis of trusses with separable load and structural uncertainties. Int J Solids Struct 43(9):2646–2669
Kaufman J, Prager M (1990) Marine structural steel toughness data bank. In: National materials property data network, Columbus OH, abridged edn
Kaymaz I, McMahon C (2005) A response surface method based on weighted regression for structural reliability analysis. Probab Eng Mech 20:11–17
Keane AJ, Voutchkov II (2020) Robust design optimization using surrogate models. J Comput Des Eng 7(1):44–55
Kennedy MC, O’Hagan A (2001) Bayesian calibration of computer models. J R Stat Soc 63(3):425–464
Keshtegar B, Hao P (2017) A hybrid self-adjusted mean value method for reliability-based design optimization using sufficient descent condition. Appl Math Model 41:257–270
Kim N, Wang H, Queipo N (2006) Efficient shape optimization under uncertainty using polynomial chaos expansions and local sensitivities. AIAA J 44(5):1112–1116
Kim T, Lee G, Youn B (2019) Uncertainty characterization under measurement errors using maximum likelihood estimation: cantilever beam end-to-end UQ test problem. Struct Multidiscip Optim 59(2):323–333
Knight FH (1921) Risk, uncertainty and profit, vol 31. Houghton Mifflin, Boston
Kolmogoroff A (1941) Confidence limits for an unknown distribution function. Ann Math Stat 12(4):461–463
Kolmogorov A (1933) Sulla determinazione empirica di une legge di distribuzione. Giornale dell’Istituto Italiano degli Attuari 4:83–91
Konečná K, Horová I (2019) Maximum likelihood method for bandwidth selection in kernel conditional density estimate. Comput Stat 34(4):1871–1887
Kumar R, Ali S, Jeyaraman S, Gupta S (2020) Uncertainty quantification of bladed disc systems using data driven stochastic reduced order models. Int J Mech Sci 190:106011
Kumar S, Pippy R, Acar E, Kim N, Haftka R (2009) Approximate probabilistic optimization using exact-capacity-approximate-response-distribution (ECARD). Struct Multidiscip Optim 38:613–626
Laplace P (1812) Analytical theory of probability. Courier, Paris
Lee D, Kim N, Kim H (2016) Validation and updating in a large automotive vibro-acoustic model using a P-box in the frequency domain. Springer-Verlag, New York
Lee G, Kim W, Oh H, Youn B, Kim N (2019a) Review of statistical model calibration and validation-from the perspective of uncertainty structures. Struct Multidiscip Optim 60(4):1619–1644
Lee I, Choi K, Du L, Gorsich D (2008a) Dimension reduction method for reliability-based robust design optimization. Comput Struct 86(13–14):1550–1562
Lee I, Choi K, Du L, Gorsich D (2008b) 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):14–27
Lee I, Choi K, Gorsich D (2010) System reliability-based design optimization using the MPP-based dimension reduction method. Struct Multidiscip Optim 41(6):823–839
Lee I, Choi K, Noh Y, Zhao L, Gorsich D (2011) Sampling-based stochastic sensitivity analysis using score functions for RBDO problems with correlated random variables. J Mech Des. https://doi.org/10.1115/DETC2010-28591
Lee I, Noh Y, Yoo D (2012) A novel second-order reliability method (SORM) using noncentral or generalized chi-squared distributions. J Mech Des 134(10):100912
Lee I, Choi K, Noh Y, Lamb D (2013) Comparison study between probabilistic and possibilistic methods for problems under a lack of correlated input statistical information. Struct Multidiscip Optim 47(2):175–189
Lee J, Kwak B (1995) Reliability-based structural optimal design using the Neumann expansion technique. Comput Struct 55(2):287–296
Lee KH, Park GJ (2001) Robust optimization considering tolerances of design variables. Comput Struct 79(1):77–86
Lee S, Chen W (2009) A comparative study of uncertainty propagation methods for black-box-type problems. Struct Multidiscip Optim 37(3):239
Lee T, Jung J (2008) A sampling technique enhancing accuracy and efficiency of metamodel-based RBDO: constraint boundary sampling. Comput Struct 86(13–14):1463–1476
Lee U, Kang N, Lee I (2019) Selection of optimal target reliability in RBDO through reliability-based design for market systems (RBDMS) and application to electric vehicle design. Struct Multidiscip Optim 60(3):949–963
Lee U, Kang N, Lee I (2020a) Shared autonomous electric vehicle design and operations under uncertainties: a reliability-based design optimization approach. Struct Multidiscip Optim 61(4):1529–1545
Lee U, Park S, Lee I (2020b) Robust design optimization (rdo) of thermoelectric generator system using non-dominated sorting genetic algorithm II (nsga-II). Energy 196:117090
Li G, Zhang K (2011) A combined reliability analysis approach with dimension reduction method and maximum entropy method. Struct Multidiscip Optim 43:121–134
Li H, Cao Z (2016) Matlab codes of subset simulation for reliability analysis and structural optimization. Struct Multidiscip Optim 54(2):391–410
Li H, Cho H, Sugiyama H, Choi K, Gaul NJ (2017) Reliability-based design optimization of wind turbine drivetrain with integrated multibody gear dynamics simulation considering wind load uncertainty. Struct Multidiscip Optim 56(1):183–201
Li J, Jiang C, Ni B, Zhan L (2019a) Uncertain vibration analysis based on the conceptions of differential and integral of interval process. Int J Mech Mater Des 16:225
Li L, Wan H, Gao W, Tong F, Li H (2019b) Reliability based multidisciplinary design optimization of cooling turbine blade considering uncertainty data statistics. Struct Multidiscip Optim 59(2):659–673
Li M, Wang Z (2018) Confidence-driven design optimization using Gaussian process metamodeling with insufficient data. J Mech Des 140(12):121405
Li M, Wang Z (2019) Surrogate model uncertainty quantification for reliability-based design optimization. Reliab Eng Syst Saf 192:106432
Li M, Wang Z (2020) Deep learning for high-dimensional reliability analysis. Mech Syst Signal Process 139:106399
Li W, Gao L, Xiao M (2020) Multidisciplinary robust design optimization under parameter and model uncertainties. Eng Optim 52(3):426–445
Li X, Qiu H, Chen Z, Gao L, Shao X (2016) A local Kriging approximation method using MPP for reliability-based design optimization. Comput Struct 162:102–115
Li X, Gong C, Gu L, Jing Z, Fang H, Gao R (2019c) A reliability-based optimization method using sequential surrogate model and Monte Carlo simulation. Struct Multidiscip Optim 59(2):439–460
Li X, Meng Z, Chen G, Yang D (2019d) A hybrid self-adjusted single-loop approach for reliability-based design optimization. Struct Multidiscip Optim 60(5):1867–1885
Li Y, Chen J, Feng L (2012) Dealing with uncertainty: a survey of theories and practices. IEEE Trans Knowl Data Eng 25(11):2463–2482
Liang J, Mourelatos Z, Nikolaidis E (2007) A single-loop approach for system reliability-based design optimization. J Mech Desi 129(12):1215
Liang J, Mourelatos Z, Tu J (2008) A single-loop method for reliability-based design optimisation. Int J Prod Dev 5(1–2):76–92
Lim J, Lee B, Lee I (2014) Second-order reliability method-based inverse reliability analysis using Hessian update for accurate and efficient reliability-based design optimization. Int J Numer Meth Eng 100(10):773–792
Lin P, Gea HC, Jaluria Y (2011) A modified reliability index approach for reliability-based design optimization. J Mech Des 133(4):044501
Lin Q, Xiong F, Wang F, Yang X (2020) A data-driven polynomial chaos method considering correlated random variables. Struct Multidiscip Optim 62(4):2131–2147
Liu H, Jiang C, Jia X, Long X, Zhang Z, Guan F (2018a) A new uncertainty propagation method for problems with parameterized probability-boxes. Reliab Eng Syst Saf 172:64–73
Liu H, Ong Y, Cai J (2018b) A survey of adaptive sampling for global metamodeling in support of simulation-based complex engineering design. Struct Multidiscip Optim 57(1):393–416
Liu H, Jiang C, Liu J (2019) Uncertainty propagation analysis using sparse grid technique and saddlepoint approximation based on parameterized p-box representation. Struct Multidiscip Optim 59:61–74
Liu J, Sun X, Meng X, Li K, Zeng G, Wang X (2016) A novel shape function approach of dynamic load identification for the structures with interval uncertainty. Int J Mech Mater Des 12(3):375–386
Liu P, Der Kiureghian A (1991) Optimization algorithms for structural reliability. Struct Saf 9:161–178
Liu X, Wu Y, Wang B, Ding J, Jie H (2017) An adaptive local range sampling method for reliability-based design optimization using support vector machine and Kriging model. Struct Multidiscip Optim 55(6):2285–2304
Lopez RH, Lemosse D, de Cursi JES, Rojas J, El-Hami A (2011) An approach for the reliability based design optimization of laminated composites. Eng Optim 43(10):1079–1094
Luo Z, Wang X, Shi Q, Liu D (2021) Ubc-constrained non-probabilistic reliability-based optimization of structures with uncertain-but-bounded parameters. Struct Multidiscip Optim 63(1):311–326
Madsen H, Krenk S, Lind N (2006) Methods of structural safety. Courier Corporation
Mahadevan S, Zhang R, Smith N (2001) Bayesian networks for system reliability reassessment. Struct Saf 23(3):231–251
Makhloufi A, Aoues Y, El Hami A (2016) Reliability based design optimization of wire bonding in power microelectronic devices. Microsyst Technol 22(12):2737–2748
Mansour R, Olsson M (2014) A closed-form second-order reliability method using noncentral chi-squared distributions. J Mech Des 136(10):10402
Marelli S, Sudret B (2014) Uqlab: A framework for uncertainty quantification in matlab. The 2nd International conference on vulnerability and risk analysis and management (ICVRAM 2014). University of Liverpool, United Kingdom, pp 2554–2563
Martin N, England J (1981) Mathematical theory of entropy. Addison-Wesley, Reading
McAllister CD, Simpson TW (2003) Multidisciplinary robust design optimization of an internal combustion engine. J Mech Des 125(1):124–130
McDonald M, Mahadevan S (2008) Design optimization with system-level reliability constraints. J Mech Des 130(2):021403
McFarland J, Mahadevan S (2008) Error and variability characterization in structural dynamics modeling. Comput Methods Appl Mech Eng 197(29–32):2621–2631
Melchers R (1989) Importance sampling in structural systems. Struct Saf 6:3–10
Meng D, Li Y, Huang H, Wang Z, Liu Y (2015a) Reliability-based multidisciplinary design optimization using subset simulation analysis and its application in the hydraulic transmission mechanism design. J Mech Des 137(5):051402
Meng Z, Li G, Wang B, Hao P (2015b) A hybrid chaos control approach of the performance measure functions for reliability-based design optimization. Comput Struct 146:32–43
Meng Z, Zhou H, Li G, Yang D (2016) A decoupled approach for non-probabilistic reliability-based design optimization. Comput Struct 175:65–73
Meng Z, Zhang D, Liu Z, Li G (2018) An adaptive directional boundary sampling method for efficient reliability-based design optimization. J Mech Des 140(12):121406
Meng Z, Zhang D, Li G, Yu B (2019) An importance learning method for non-probabilistic reliability analysis and optimization. Struct Multidiscip Optim 59(4):1255–1271
Mischke CR (1987) Prediction of stochastic endurance strength. J Vib Acoust Stress Reliab Des 109(1):113–114
modeFRONTIER (2021) Robust design and reliability—- www.esteco.com. https://www.esteco.com/technology/robust-design-and-reliability/
Moens D, Vandepitte D (2005) A survey of non-probabilistic uncertainty treatment in finite element analysis. Comput Methods Appl Mech Eng 194(12–16):1527–1555
Mohsine A, Kharmanda G, El-Hami A (2006) Improved hybrid method as a robust tool for reliability-based design optimization. Struct Multidiscip Optim 32(3):203–213
Moon M, Choi K, Cho H, Gaul N, Lamb D, Gorsich D (2017) Reliability-based design optimization using confidence-based model validation for insufficient experimental data. J Mech Des 139(3):031404
Moon M, Cho H, Choi K, Gaul N, Lamb D, Gorsich D (2018) Confidence-based reliability assessment considering limited numbers of both input and output test data. Struct Multidiscip Optim 57(5):2027–2043
Moon M, Choi K, Gaul N, Lamb D (2019) Treating epistemic uncertainty using bootstrapping selection of input distribution model for confidence-based reliability assessment. J Mech Des. https://doi.org/10.1115/1.4042149
Moore R (1966) Interval analysis, vol 4. Prentice-Hall, Englewood Cliffs
Moore RE, Kearfott RB, Cloud MJ (2009) Introduction to interval analysis. SIAM
Motta RdS, Afonso SM (2016) An efficient procedure for structural reliability-based robust design optimization. Struct Multidiscip Optim 54(3):511–530
Mourelatos Z, Zhou J (2006) A design optimization method using evidence theory. J Mech Des 128(4):901
Muhanna R, Mullen R, Zhang H (2005) Penalty-based solution for the interval finite-element methods. J Eng Mech 131(10):1102–1111
Mukhopadhyay S, Khodaparast H, Adhikari S (2016) Fuzzy uncertainty propagation in composites using gram-schmidt polynomial chaos expansion. Appl Math Model 40(7–8):4412–4428
Nagel J, Rieckermann J, Sudret B (2020) Principal component analysis and sparse polynomial chaos expansions for global sensitivity analysis and model calibration: application to urban drainage simulation. Reliab Eng Syst Saf 195:106737
Nannapaneni S, Hu Z, Mahadevan S (2016) Uncertainty quantification in reliability estimation with limit state surrogates. Struct Multidiscip Optim 54(6):1509–1526
Nataf A (1962) Determination des distribution don’t les marges sont donnees. Comptes Rendus de l Academie des Sciences 225:42–43
das Neves Carneiro G, António CC, (2019) Reliability-based robust design optimization with the reliability index approach applied to composite laminate structures. Compos Struct 209:844–855
Ng L, Willcox K (2014) Multifidelity approaches for optimization under uncertainty. Int J Numer Meth Eng 100(10):746–772
Nguyen T, Song J, Paulino G (2010) Single-loop system reliability-based design optimization using matrix-based system reliability method: theory and applications. J Mech Des 132(1):011005
Nikbay M, Kuru M (2013) Reliability based multidisciplinary optimization of aeroelastic systems with structural and aerodynamic uncertainties. J Aircr 50(3):708–715
Nikolaidis E, Chen S, Cudney H, Haftka RT, Rosca R (2004) Comparison of probability and possibility for design against catastrophic failure under uncertainty. J Mech Des 126(3):386–394
Nikolaidis E, Ghiocel D, Singhal S (2004) Engineering design reliability handbook. CRC Press, Boca Raton
Noh Y, Choi K, Du L (2009) Reliability-based design optimization of problems with correlated input variables using a Gaussian copula. Struct Multidiscip Optim 38(1):1–16
Noh Y, Choi K, Lee I (2010) Identification of marginal and joint CDFs using Bayesian method for RBDO. Struct Multidiscip Optim 40(1–6):35
Noh Y, Choi K, Lee I, Gorsich D, Lamb D (2011a) Reliability-based design optimization with confidence level for non-Gaussian distributions using bootstrap method. J Mech Des 133(9):091001
Noh Y, Choi K, Lee I, Gorsich D, Lamb D (2011b) Reliability-based design optimization with confidence level under input model uncertainty due to limited test data. Struct Multidiscip Optim 43(4):443–458
Oberguggenberger M, Fellin W (2008) Reliability bounds through random sets: nonparametric methods and geotechnical applications. Comput Struct 86(10):1093–110
Oberkampf W, DeLand S, Rutherford B, Diegert K, Alvin K (2002) Error and uncertainty in modeling and simulation. Reliab Eng Syst Saf 75(3):333–357
Olivier GDABCMVLA, Shields M (2020) Uqpy: a general purpose python package and development environment for uncertainty quantification. J Comput Sci 47:101204
Omizegba E, Adebayo G (2009) Optimizing fuzzy membership functions using particle swarm algorithm. In: 2009 IEEE international conference on systems. man and cybernetics, IEEE, pp 3866–3870
OmniQuest (2021) Fesoftware. https://omniquest.world/
OptiSLang (2021) Ansys optislang. https://www.ansys.com/en-in/products/platform/ansys-optislang/
Paiva R, Crawford C, Suleman A (2014) Robust and reliability-based design optimization framework for wing design. AIAA J 52(4):711–724
Pan H, Xi Z, Yang R (2016) Model uncertainty approximation using a copula-based approach for reliability based design optimization. Struct Multidiscip Optim 54(6):1543–1556
Papaioannou I, Betz W, Zwirglmaier K, Straub D (2015) MCMC algorithms for subset simulation. Probab Eng Mech 41:89–103
Papaioannou I, Breitung K, Straub D (2018) Reliability sensitivity estimation with sequential importance sampling. Struct Saf 75:24–34
Park J, Lee I (2018) A study on computational efficiency improvement of novel SORM using the convolution integration. J Mech Des 140(2):025401
Park J, Cho H, Lee I (2020) Selective dimension reduction method (DRM) to enhance accuracy and efficiency of most probable point (MPP)-based DRM. Struct Multidiscip Optim 61(3):999–1010
Parsons S, Hunter A (1998) A review of uncertainty handling formalisms. In: Applications of uncertainty formalisms, Springer, pp 8–37
Paté-Cornell M (1996) Uncertainties in risk analysis: six levels of treatment. Reliab Eng Syst Saf 54(2–3):95–111
Paulson J, Buehler E, Mesbah A (2017) Arbitrary polynomial chaos for uncertainty propagation of correlated random variables in dynamic systems. IFAC-PapersOnLine 50(1):3548–3553
Pearl J (2014) Probabilistic reasoning in intelligent systems: networks of plausible inference. Elsevier, Amsterdam
Peherstorfer B, Cui T, Marzouk Y, Willcox K (2016) Multifidelity importance sampling. Comput Methods Appl Mech Eng 300:490–509
Peherstorfer B, Willcox K, Gunzburger M (2018) Survey of multifidelity methods in uncertainty propagation, inference, and optimization. SIAM Rev 60(3):550–591
Periçaro G, Santos S, Ribeiro A, Matioli L (2015) HLRF-BFGS optimization algorithm for structural reliability. Appl Math Model 39(7):2025–2035
Picheny V, Kim N, Haftka R, Queipo N (2008) Conservative predictions using surrogate modeling. In: 49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference. In: 16th AIAA/ASME/AHS adaptive structures conference, 10th aiaa non-deterministic approaches conference, 9th AIAA gossamer spacecraft forum, 4th AIAA multidisciplinary design optimization specialists conference
Platz R, Götz B (2017) Non-probabilistic uncertainty evaluation in the concept phase for airplane landing gear design. Model validation and uncertainty quantification, vol 3. Springer, Cham, pp 161–169
Qiu Z, Wang X (2003) Comparison of dynamic response of structures with uncertain-but-bounded parameters using non-probabilistic interval analysis method and probabilistic approach. Int J Solids Struct 40(20):5423–5439
Qiu Z, Wang X (2005) Parameter perturbation method for dynamic responses of structures with uncertain-but bounded parameters based on interval analysis. Int J Solids Struct 42(18–19):4970
Qiu Z, Yang D, Elishakoff I (2008) Probabilistic interval reliability of structural systems. Int J Solids Struct 45(10):2850–2860
Qu X, Haftka R, Venkataraman S, Johnson T (2003) Deterministic and reliability-based optimization of composite laminates for cryogenic environments. AIAA J 41(10):2029–2036
Rackwitz R (2001) Reliability analysis-a review and some perspectives. Struct Saf 23(4):365–395
Radaideh M, Kozlowski T (2020) Surrogate modeling of advanced computer simulations using deep Gaussian processes. Reliab Eng Syst Saf 195:106731
Rahman S, Wei D (2006) A univariate approximation at most probable point for higher-order reliability analysis. Int J Solids Struct 43(9):2820–2839
Rahman S, Xu H (2004) A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics. Probab Eng Mech 19(4):393–408
Rajabi M (2019) Review and comparison of two meta-model-based uncertainty propagation analysis methods in groundwater applications: polynomial chaos expansion and Gaussian process emulation. Stoch Env Res Risk Assess 33(2):607–631
Rajan A, Luo FJ, Kuang YC, Bai Y, Ooi MPL (2020) Reliability-based design optimisation of structural systems using high-order analytical moments. Struct Saf 86:101970
Ramakrishnan B, Rao S (1996) A general loss function based optimization procedure for robust design. Eng Optim 25(4):255–276
RAMDO (2021) Reliability analysis, and design optimization software—-ramdo. https://www.altair.com/ramdo/
Ramu P, Qu X, Youn B, Haftka R, Choi K (2006) Inverse reliability measures and reliability-based design optimisation. Int J Reliab Saf 1(1–2):187–205
Ranjbar A, Mahjouri N (2019) Multi-objective freshwater management in coastal aquifers under uncertainty in hydraulic parameters. Nat Resour Res 29:1–22
Rao S, Berke L (1997) Analysis of uncertain structural systems using interval analysis. AIAA J 35(4):727–735
Rao SS (1992) Reliability-based design. McGraw-Hill Companies, New York
Romero V, Swiler L, Giunta A (2004) Construction of response surfaces based on progressive-lattice-sampling experimental designs with application to uncertainty propagation. Struct Saf 26(2):201–219
Ronold KO, Larsen GC (2000) Reliability-based design of wind-turbine rotor blades against failure in ultimate loading. Eng Struct 22(6):565–574
Rowe W (1994) Understanding uncertainty. Risk Anal 14(5):743–750
Roy CJ, Oberkampf WL (2011) A comprehensive framework for verification, validation, and uncertainty quantification in scientific computing. Comput Methods Appl Mech Eng 200(25–28):2131–2144
Sandgren E, Cameron TM (2002) Robust design optimization of structures through consideration of variation. Comput Struct 80(20–21):1605–1613
Sankararaman S, Mahadevan S (2011) Likelihood-based representation of epistemic uncertainty due to sparse point data and/or interval data. Reliab Eng Syst Saf 96(7):814–824
Santosh T, Saraf R, Ghosh A, Kushwaha H (2006) Optimum step length selection rule in modified HL-RF method for structural reliability. Int J Press Vessels Pip 83(10):742–748
Schueller G, Pradlwarter H (2007) Benchmark study on reliability estimation in higher dimensions of structural systems—an overview. Struct Saf 29:167–182
Schuëller GI, Jensen HA (2008) Computational methods in optimization considering uncertainties-an overview. Comput Methods Appl Mech Eng 198(1):2–13
Schwarz G (1978) Estimating the dimension of a model. Ann Stat 6(2):461–464
Šehić K, Karamehmedović M (2020) Estimation of failure probabilities via local subset approximations. arxiv:200305994
Shafer G (1976) A mathematical theory of evidence, vol 42. Princeton University Press, Princeton
Shahraki AF, Noorossana R (2014) Reliability-based robust design optimization: a general methodology using genetic algorithm. Comput Ind Eng 74:199–207
Shan S, Wang G (2008) Reliable design space and complete single-loop reliability-based design optimization. Reliab Eng Syst Saf 93(8):1218–1230
Shi L, Lin S (2016) A new RBDO method using adaptive response surface and first-order score function for crashworthiness design. Reliab Eng Syst Saf 156:125–133
Shi Y, Lu Z (2019) Dynamic reliability analysis model for structure with both random and interval uncertainties. Int J Mech Mater Des 15(3):521–537
Shin J, Lee I (2014) Reliability-based vehicle safety assessment and design optimization of roadway radius and speed limit in windy environments. J Mech Des 136(8):081006
Sim J, Qiu Z, Wang X (2007) Modal analysis of structures with uncertain-but-bounded parameters via interval analysis. J Sound Vib 303(1–2):29–45
Simon C, Bicking F (2017) Hybrid computation of uncertainty in reliability analysis with p-box and evidential networks. Reliab Eng Syst Saf 167:629–638
Simon D (2002) Sum normal optimization of fuzzy membership functions. Int J Uncertain Fuzz Knowl-Based Syst 10(04):363–384
Smarslok B, Haftka R, Carraro L, Ginsbourger D (2010) Improving accuracy of failure probability estimates with separable Monte Carlo. Int J Reliab Saf 4(4):393–414
SmartUQ (2021) Uncertainty propagation—-smartuq. https://www.smartuq.com/software/uncertainty-propagation/
Smirnoff N (1939) Sur les écarts de la courbe de distribution empirique. Matematicheskii Sbornik 48(1):3–26
Sohouli A, Yildiz M, Suleman A (2018) Efficient strategies for reliability-based design optimization of variable stiffness composite structures. Struct Multidiscip Optim 57(2):689–704
Son H, Lee G, Kang K, Kang Y, Youn B, Lee I, Noh Y (2020) Industrial issues and solutions to statistical model improvement: a case study of an automobile steering column. Struct Multidiscip Optim 61(4):1739–1756
Song J, Kang W (2009) System reliability and sensitivity under statistical dependence by matrix-based system reliability method. Struct Saf 31(2):148–156
Soroudi A, Keane A (2015) Risk averse energy hub management considering plug-in electric vehicles using information gap decision theory. Plug in electric vehicles in smart grids. Springer, Singapore, pp 107–127
Soroudi A, Rabiee A, Keane A (2017) Information gap decision theory approach to deal with wind power uncertainty in unit commitment. Electr Power Syst Res 145:137–148
Soundappan P, Nikolaidis E, Haftka R, Grandhi R, Canfield R (2004) Comparison of evidence theory and Bayesian theory for uncertainty modeling. Reliab Eng Syst Saf 85(1–3):295–311
Sun G, Li G, Zhou S, Li H, Hou S, Li Q (2011) Crashworthiness design of vehicle by using multiobjective robust optimization. Struct Multidiscip Optim 44(1):99–110
Sun G, Zhang H, Fang J, Li G, Li Q (2017) Multi-objective and multi-case reliability-based design optimization for tailor rolled blank (TRB) structures. Struct Multidiscip Optim 55(5):1899–1916
Taflanidis A, Beck J (2008) An efficient framework for optimal robust stochastic system design using stochastic simulation. Comput Methods Appl Mech Eng 198(1):88–101
Taflanidis A, Beck J (2008) Stochastic subset optimization for optimal reliability problems. Probab Eng Mech 23(2–3):324–338
Tang Y, Chen J, Wei J (2012) A sequential algorithm for reliability-based robust design optimization under epistemic uncertainty. J Mech Des 134(1):014502
Teckentrup A, Jantsch P, Webster C, Gunzburger M (2015) A multilevel stochastic collocation method for partial differential equations with random input data. SIAM/ASA J Uncertain Quantif 3(1):1046–1074
Thom H (1960) Distributions of extreme winds in the united states. Trans Am Soc Civ Eng 126(2):450–462
Toft HS, Sørensen JD (2011) Reliability-based design of wind turbine blades. Struct Saf 33(6):333–342
Tonon F, Bernardini A, Elishakoff I (2001) Hybrid analysis of uncertainty: probability, fuzziness and anti-optimization. Chaos Solitons Fract 12(8):1403–1414
Tripathy R, Bilionis I (2018) Deep UQ: learning deep neural network surrogate models for high dimensional uncertainty quantification. J Comput Phys 375:565–588
Tripathy R, Bilionis I, Gonzalez M (2016) Gaussian processes with built-in dimensionality reduction: applications to high-dimensional uncertainty propagation. J Comput Phys 321:191–223
Tu J, Choi K, Park Y (1999) A new study on reliability-based design optimization. J Mech Des 121(4):557
Tu J, Choi K, Park Y (2001) Design potential method for robust system parameter design. AIAA J 39(4):667–677
UQWorld (2021) Various uncertainty quantification software tools. https://uqworld.org/t/various-uncertainty-quantification-software-tools/137/
Valdebenito M, Schuëller G (2010) A survey on approaches for reliability-based optimization. Struct Multidiscip Optim 42(5):645–663
Viana F, Haftka R, Steffen V (2009) Multiple surrogates: how cross-validation errors can help us to obtain the best predictor. Struct Multidiscip Optim 39(4):439–457
Viana F, Picheny V, Haftka R (2010) Using cross validation to design conservative surrogates. AIAA J 48(10):2286–2298
Volpi S, Diez M, Gaul N, Song H, Iemma U, Choi K, Stern F (2015) Development and validation of a dynamic metamodel based on stochastic radial basis functions and uncertainty quantification. Struct Multidiscip Optim 51(2):347–368
Wand M, Jones M (1994) Multivariate plug-in bandwidth selection. Comput Stat 9(2):97–116
Wang C, Matthies H (2019) Novel model calibration method via non-probabilistic interval characterization and Bayesian theory. Reliab Eng Syst Saf 183:84–92
Wang C, Duan Q, Tong CH, Di Z, Gong W (2016) A gui platform for uncertainty quantification of complex dynamical models. Environ Modell Softw 76:1–12. https://doi.org/10.1016/j.envsoft.2015.11.004
Wang C, Zhang H, Beer M (2018) Computing tight bounds of structural reliability under imprecise probabilistic information. Comput Struct 208:92–104
Wang F, Xiong F, Chen S, Song J (2019) Multi-fidelity uncertainty propagation using polynomial chaos and Gaussian process modeling. Struct Multidiscip Optim 60(4):1583–1604
Wang L, Beeson D, Wiggs G (2004) Efficient and accurate point estimate method for moments and probability distribution estimation. In: 10th AIAA/ISSMO multidisciplinary analysis and optimization conference, p 4359
Wang L, Wang X, Li Y, Hu J (2019a) A non-probabilistic time-variant reliable control method for structural vibration suppression problems with interval uncertainties. Mech Syst Signal Process 115:301–322
Wang X, Wang Y (2015a) Nonparametric multivariate density estimation using mixtures. Stat Comput 25(2):349–364
Wang X, Wang L, Elishakoff I, Qiu Z (2011) Probability and convexity concepts are not antagonistic. Acta Mech 219(1–2):45–64
Wang Y (2007) On fast computation of the non-parametric maximum likelihood estimate of a mixing distribution. J R Stat Soc Ser B 69(2):185–198
Wang Z, Chen W (2017) Confidence-based adaptive extreme response surface for time-variant reliability analysis under random excitation. Struct Saf 64:76–86
Wang Z, Wang P (2012) A nested extreme response surface approach for time-dependent reliability-based design optimization. J Mech Des 134:121007
Wang Z, Wang P (2014) A maximum confidence enhancement based sequential sampling scheme for simulation-based design. J Mech Des 136(2):021006
Wang Z, Wang P (2015b) An integrated performance measure approach for system reliability analysis. J Mech Des 137(2):021406
Wang Z, Wang Z, Yu S, Zhang K (2019b) Time-dependent mechanism reliability analysis based on envelope function and vine-copula function. Mech Mach Theory 134:667–684
Wang Z, Li H, Chen Z, Li L, Hong H (2020) Sequential optimization and moment-based method for efficient probabilistic design. Struct Multidiscip Optim 62:1–18
Weinmeister J, Xie N, Gao X, Krishna Prasad A, Roy S (2018) Analysis of a polynomial chaos-Kriging metamodel for uncertainty quantification in aerospace applications. In: 2018 AIAA/ASCE/AHS/ASC structures, structural dynamics, and materials conference, p 0911
Wu X, Mui T, Hu G, Meidani H, Kozlowski T (2017) Inverse uncertainty quantification of trace physical model parameters using sparse gird stochastic collocation surrogate model. Nucl Eng Des 319:185–200
Wu X, Kozlowski T, Meidani H (2018) Kriging-based inverse uncertainty quantification of nuclear fuel performance code bison fission gas release model using time series measurement data. Reliab Eng Syst Saf 169:422–436
Wu Y, Y S, Sues R, Cesare M (2001) Safety factor based approach for probability–based design optimization. In: Proceedings of 42nd AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics and materials conference, Seattle, WA
Wunsch D, Hirsch C, Nigro R, Coussement G (2015) Quantification of combined operational and geometrical uncertainties in turbo-machinery design. In: Turbo expo: power for land, sea, and air. American Society of Mechanical Engineers, vol 56659, p V02CT45A018
Xi Z (2019) Model-based reliability analysis with both model uncertainty and parameter uncertainty. J Mech Des 141(5):051404
Xiao M, Zhang J, Gao L (2020) A system active learning Kriging method for system reliability-based design optimization with a multiple response model. Reliab Eng Syst Saf 199:106935
Xiao Z, Han X, Jiang C (2016) An efficient uncertainty propagation method for parameterized probability boxes. Acta Mech 227:633–649
Xiong Y, Chen W, Tsui K, Apley D (2009) A better understanding of model updating strategies in validating engineering models. Comput Methods Appl Mech Eng 198(15–16):1327–1337
Xu H, Rahman S (2004) A generalized dimension-reduction method for multidimensional integration in stochastic mechanics. Int J Numer Meth Eng 61(12):1992–2019
Xu J, Wang D (2019) Structural reliability analysis based on polynomial chaos, Voronoi cells and dimension reduction technique. Reliab Eng Syst Saf 185:329–340
Yadav OP, Bhamare SS, Rathore A (2010) Reliability-based robust design optimization: a multi-objective framework using hybrid quality loss function. Qual Reliab Eng Int 26(1):27–41
Yang D (2010) Chaos control for numerical instability of first order reliability method. Commun Nonlinear Sci Numer Simul 15(10):3131–3141
Yang M, Zhang D, Han X (2020) New efficient and robust method for structural reliability analysis and its application in reliability-based design optimization. Comput Methods Appl Mech Eng 366:113018
Yang R, Gu L (2004) Experience with approximate reliability-based optimization methods. Struct Multidiscip Optim 26(1–2):152–159
Yang X, Liu Y, Mi C, Wang X (2018) Active learning Kriging model combining with kernel-density-estimation-based importance sampling method for the estimation of low failure probability. J Mech Des 140:051402
Yoo D, Lee I (2014) Sampling-based approach for design optimization in the presence of interval variables. Struct Multidiscip Optim 49(2):253–266
Yoo D, Lee I, Cho H (2014) Probabilistic sensitivity analysis for novel second-order reliability method (SORM) using generalized chi-squared distribution. Struct Multidiscip Optim 50(5):787–797
Youn B, Choi K (2004) An investigation of nonlinearity of reliability based design optimization approaches. J Mech Des 126(3):403–411
Youn B, Wang P (2008) Bayesian reliability-based design optimization using eigenvector dimension reduction (EDR) method. Struct Multidiscip Optim 36(2):107–123
Youn B, Choi K, Park Y (2003) Hybrid analysis method for reliability-based design optimization. J Mech Des 125(2):221
Youn B, Choi K, Yang R, Gu L (2004) Reliability-based design optimization for crashworthiness of vehicle side impact. Struct Multidiscip Optim 26:272–283
Youn B, Choi K, Du L (2005a) Adaptive probability analysis using an enhanced hybrid mean value method. Struct Multidiscip Optim 29(2):134–148
Youn BD, Xi Z (2009) Reliability-based robust design optimization using the eigenvector dimension reduction (edr) method. Struct Multidiscip Optim 37(5):475–492
Youn BD, Choi KK, Yi K (2005b) Performance moment integration (pmi) method for quality assessment in reliability-based robust design optimization. Mech Based Des Struct Mach 33(2):185–213
Youn BD, Choi KK, Du L, Gorsich D (2007) Integration of possibility-based optimization and robust design for epistemic uncertainty
Zadeh L (1965) Fuzzy sets. J Inf Control 8:338–353
Zadeh L (1973) Outline of a new approach to the analysis of complex systems and decision processes. IEEE Trans Syst Man Cybern 1:28–44
Zadeh L (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst 1(1):3–28
Zafar T, Wang Z (2020) Time-dependent reliability prediction using transfer learning. Struct Multidiscip Optim 62:147–158
Zaman K, Mahadevan S (2013) Robustness-based design optimization of multidisciplinary system under epistemic uncertainty. AIAA J 51(5):1021–1031
Zaman K, Mahadevan S (2017) Reliability-based design optimization of multidisciplinary system under aleatory and epistemic uncertainty. Struct Multidiscip Optim 55(2):681–699
Zang C, Friswell M, Mottershead J (2005) A review of robust optimal design and its application in dynamics. Comput Struct 83(4–5):315–326
Zhang D, Han X, Jiang C, Liu J, Li Q (2017) Time-dependent reliability analysis through response surface method. J Mech Des 139:041404
Zhang H, Mullen R, Muhanna R (2010a) Finite element structural analysis using imprecise probabilities based on p-box representation. In: The 4th international workshop on reliable engineering computing. Professional Activities Centre, National University of Singapore
Zhang H, Mullen R, Muhanna R (2010b) Interval Monte Carlo methods for structural reliability. Struct Saf 32(3):183–190
Zhang H, Mullen R, Muhanna R (2011) Structural analysis with probability-boxes. Int J Reliab Saf 6(1–3):110–129
Zhang J (2011) Adaptive normal reference bandwidth based on quantile for kernel density estimation. J Appl Stat 38(12):2869–2880
Zhang J, Du X (2010) A second-order reliability method with first-order efficiency. J Mech Des 132(10):101006
Zhang J, Taflanidis A (2019) Multi-objective optimization for design under uncertainty problems through surrogate modeling in augmented input space. Struct Multidiscip Optim 59(2):351–372
Zhang X, King M, Hyndman R (2006) A Bayesian approach to bandwidth selection for multivariate kernel density estimation. Comput Stat Data Anal 50(11):3009–3031
Zhang X, Wang L, Sørensen J (2020) AKOIS: an adaptive Kriging oriented importance sampling method for structural system reliability analysis. Struct Saf 82:10876
Zhang Z, Wang J, Jiang C, Huang Z (2019) A new uncertainty propagation method considering multimodal probability density functions. Struct Multidiscip Optim 60(5):1983–1999
Zhao L, Choi K, Lee I, Gorsich D (2013) Conservative surrogate model using weighted Kriging variance for sampling-based RBDO. J Mech Des 135(9):091003
Zheng Y, Qiu Z (2018) Non-probabilistic stability reliability analysis of composite laminated panels in supersonic flow with uncertain-but-bounded parameters. In: 2018 AIAA non-deterministic approaches conference, p 0438
Zhou T, Peng Y (2020) Structural reliability analysis via dimension reduction, adaptive sampling, and Monte Carlo simulation. Struct Multidiscip Optim 62(5):2629–2651
Zhou XY, Ruan X, Gosling P (2019a) Robust design optimization of variable angle tow composite plates for maximum buckling load in the presence of uncertainties. Compos Struct 223:110985
Zhou Y, Lu Z (2019) Active polynomial chaos expansion for reliability-based design optimization. AIAA J 57(12):5431–5446
Zhou Y, Lu Z, Cheng K (2019b) Sparse polynomial chaos expansions for global sensitivity analysis with partial least squares and distance correlation. Struct Multidiscip Optim 59(1):229–247
Zhu P, Shi L, Yang R, Lin S (2015) A new sampling-based RBDO method via score function with reweighting scheme and application to vehicle designs. Appl Math Model 39(15):4243–4256
Zhu Z, Du X (2016) Reliability analysis with Monte Carlo simulation and dependent Kriging predictions. J Mech Des 138(12):121403
Zimmermann H (2001) Fuzzy analysis. In: Fuzzy set theory and its applications. Springer, Dordrecht
Zio E, Pedroni N (2013) Literature review of methods for representing uncertainty. FonCSI
Zou T, Mahadevan S (2006) A direct decoupling approach for efficient reliability-based design optimization. Struct Multidiscip Optim 31(3):190
Zougab N, Adjabi S, Kokonendji C (2014) Bayesian estimation of adaptive bandwidth matrices in multivariate kernel density estimation. Comput Stat Data Anal 75:28–38
Zuev K, Beck J, Au S, Katafygiotis L (2012) Bayesian post-processor and other enhancements of subset simulation for estimating failure probabilities in high dimensions. Comput Struct 92:283–296
Acknowledgements
The authors dedicate this paper to Prof. Raphael T. Haftka, who worked extensively on topics related to uncertainties for over 3 decades leading to more than 100 contributions in applications spanning from structural composites to turbomachines and material models. He was a prolific collaborator and worked with numerous colleagues from other universities and countries. In that regard, this paper reflects such an effort with collaborators from three different countries/universities.
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EA—Entire manuscript curation, writing Uncertainty modeling, review & editing entire manuscript, resources; GB—Writing, review & editing Uncertainty modeling; YJ—Writing, review & editing Design optimization under uncertainties; IL—Entire manuscript curation, writing Design optimization under uncertainties, review & editing entire manuscript, resources; PR—Entire manuscript curation, writing Uncertainty analysis, review & editing entire manuscript, resources; SSR—Writing, review & editing Uncertainty analysis.
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Acar, E., Bayrak, G., Jung, Y. et al. Modeling, analysis, and optimization under uncertainties: a review. Struct Multidisc Optim 64, 2909–2945 (2021). https://doi.org/10.1007/s00158-021-03026-7
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DOI: https://doi.org/10.1007/s00158-021-03026-7