nach oben

Structural and Multidisciplinary Optimization

Erschienen in:

26.06.2020 | Research Paper

A most probable point method for probability distribution construction

verfasst von: Yongyong Xiang, Baisong Pan, Luping Luo

Erschienen in: Structural and Multidisciplinary Optimization | Ausgabe 5/2020

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

With nonlinearity and uncertainty existing in engineering problems, it is important to predict the probability distribution of a response of an engineering model. The probability distribution is often constructed without sufficient accuracy due to a high computational cost. In this paper, a most probable point (MPP) method for the probability distribution construction is proposed. First, predictive models of the MPP components are established based on the Gaussian mixture distribution (GMD) and the inverse first-order reliability method. A mixture of first- and second-order reliability methods is then used to calculate discrete points of the cumulative distribution function (CDF). Finally, the CDF of the response is constructed by the GMD. A mathematical example and three engineering examples are used to verify the effectiveness of the proposed method.

Vorheriger Artikel Efficient adaptive Kriging-based reliability analysis combining new learning function and error-based stopping criterion

Nächster Artikel Wheel shape optimization approaches to reduce railway rolling noise

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Andrews LC (1992) Special functions of mathematics for engineers (p 407). McGraw-Hill, New York

Arlinghaus S (1994) Practical handbook of curve fitting. CRC press

Au SK, Beck JL (2001) Estimation of small failure probabilities in high dimensions by subset simulation. Probabilist Eng Mech 16(4):263–277

Basudhar A, Missoum S (2008) Adaptive explicit decision functions for probabilistic design and optimization using support vector machines. Comput Struct 86(19–20):1904–1917

Björck Å (1996) Numerical methods for least squares problems. Society for Industrial and Applied Mathematics, PhiladelphiaMATH

Blatman G, Sudret B (2010) An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis. Probabilist Eng Mech 25(2):183–197

Breitung K (1984) Asymptotic approximations for multinormal integrals. J Eng Mech 110(3):357–366MATH

Bucher CG (1988) Adaptive sampling-an iterative fast Monte Carlo procedure. Struct Saf 5(2):119–126

Clarke SM, Griebsch JH, Simpson TW (2004) Analysis of support vector regression for approximation of complex engineering analyses. J Mech Des 127(6):1077–1087

Dai H, Zhang B, Wang W (2015) A multiwavelet support vector regression method for efficient reliability assessment. Reliab Eng Syst Saf 136:132–139

Dang C, Xu J (2019) Novel algorithm for reconstruction of a distribution by fitting its first-four statistical moments. Appl Math Model 71:505–524MathSciNetMATH

Der Kiureghian A, Zhang Y, Li CC (1994) Inverse reliability problem. J Eng Mech 120(5):1154–1159

Dowding KJ, Pilch M, Hills RG (2008) Formulation of the thermal problem. Comput Methods Appl Mech Eng 197(29–32):2385–2389MATH

Du X (2008) Saddlepoint approximation for sequential optimization and reliability analysis. J Mech Des 130(1):011011

Du X, Chen W (2001) A most probable point-based method for efficient uncertainty analysis. J Des Manuf Autom 4(1):47–66

Du X, Sudjianto A, Huang B (2005) Reliability-based design with the mixture of random and interval variables. J Mech Des 127(6):1068–1076

Ferson S, Oberkampf WL, Ginzburg L (2008) Model validation and predictive capability for the thermal challenge problem. Comput Methods Appl Mech Eng 197(29–32):2408–2430MATH

Ghasemi P, Aslani M, Rollins DK, Williams RC (2019) Principal component analysis-based predictive modeling and optimization of permanent deformation in asphalt pavement: elimination of correlated inputs and extrapolation in modeling. Struct Multidiscip Optim 59(4):1335–1353

Glynn PW, Iglehart DL (1989) Importance sampling for stochastic simulations. Manag Sci 35(11):1367–1392MathSciNetMATH

Gorjian N, Ma L, Mittinty M, Yarlagadda P, Sun Y (2010) A review on degradation models in reliability analysis. In: Engineering Asset Lifecycle Management. Springer, London, pp 369–384

Guest PG, Guest PG (2012) Numerical methods of curve fitting. Cambridge University Press, CambridgeMATH

Guo J, Zhao J, Zeng S (2018) Structural reliability analysis based on analytical maximum entropy method using polynomial chaos expansion. Struct Multidiscip Optim 58(3):1187–1203MathSciNet

Hasofer AM, Lind NC (1974) Exact and invariant second-moment code format. J Eng Mech Div 100(1):111–121

Hastie T, Tibshirani R (1996) Discriminant analysis by Gaussian mixtures. J R Stat Soc Ser B Methodol 58(1):155–176MathSciNetMATH

Helton JC, Davis FJ (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, Oberkampf WL (2004) An exploration of alternative approaches to the representation of uncertainty in model predictions. Reliab Eng Syst Saf 85(1–3):39–71

Hengl T, Heuvelink GB, Rossiter DG (2007) About regression-kriging: from equations to case studies. Comput Geosci 33(10):1301–1315

Hohenbichler M, Rackwitz R (1982) First-order concepts in system reliability. Struct Saf 1(3):177–188

Hu Z, Du X (2015) A random field approach to reliability analysis with random and interval variables. ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part B. Mech Eng 1(4):041005

Hu Z, Du X (2018) Saddlepoint approximation reliability method for quadratic functions in normal variables. Struct Saf 71:24–32

Huang B, Du X (2006) Uncertainty analysis by dimension reduction integration and saddlepoint approximations. J Mech Des 128(1):26–33

Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions. Wiley, New YorkMATH

Jung BC, Park J, Oh H, Kim J, Youn BD (2015) A framework of model validation and virtual product qualification with limited experimental data based on statistical inference. Struct Multidiscip Optim 51(3):573–583

Karian ZA, Dudewicz EJ, Mcdonald P (1996) The extended generalized lambda distribution system for fitting distributions to data: history, completion of theory, tables, applications, the ‘final word’ on moment fits. Commun Stati-Simul C 25(3):611–642MathSciNetMATH

Kaymaz I (2005) Application of kriging method to structural reliability problems. Struct Saf 27(2):133–151

Lebrun R, Dutfoy A (2009) A generalization of the Nataf transformation to distributions with elliptical copula. Probabilist Eng Mech 24(2):172–178

Lee SH, Chen W (2009) A comparative study of uncertainty propagation methods for black-box-type problems. Struct Multidiscip Optim 37(3):239

Li W, Chen W, Jiang Z, Lu Z, Liu Y (2014) New validation metrics for models with multiple correlated responses. Reliab Eng Syst Saf 127:1–11

Liu PL, Der Kiureghian A (1991) Optimization algorithms for structural reliability. Struct Saf 9(3):161–177

Liu J, Meng X, Xu C, Zhang D, Jiang C (2018) Forward and inverse structural uncertainty propagations under stochastic variables with arbitrary probability distributions. Comput Methods Appl Mech Eng 342:287–320MathSciNetMATH

Low YM (2013) A new distribution for fitting four moments and its applications to reliability analysis. Struct Saf 42:12–25

Melchers RE (1989) Importance sampling in structural systems. Struct Saf 6(1):3–10

Metropolis N, Ulam S (1949) The Monte Carlo method. J Am Stat Assoc 44(247):335–341MATH

Moon MY, Choi KK, Lamb D (2019) Target output distribution and distribution of bias for statistical model validation given a limited number of test data. Struct Multidiscip Optim 60:1327–1353

Osuna E, Freund R, Girosit F (1997) Training support vector machines: an application to face detection. In Proceedings of IEEE computer society conference on computer vision and pattern recognition (pp.130-136)

Pearson K (1916) Mathematical contributions to the theory of evolution, XIX. Second supplement to a memoir on skew variation. Philos Trans R Soc Lond A Contain Pap Math Phys Character 216(538–548):429–457MATH

Rackwitz R, Flessler B (1974) Structural reliability under combined random load sequences. Comput Struct 9(5):489–494MATH

Rahman S, Xu H (2004) A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics. Probabilist Eng Mech 19(4):393–408

Rasmussen CE (2000) The infinite Gaussian mixture model. Advances in neural information processing systems pp 554–560

Reynolds D (2015) Gaussian mixture models. Encyclopedia of biometrics pp:827–832

Rosenblatt M (1952) Remarks on a multivariate transformation. Ann Math Stat 23(3):470–472MathSciNetMATH

Rosenblueth E (1975) Point estimates for probability moments. Proc Natl Acad Sci 72(10):3812–3814MathSciNetMATH

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):2131–2144MathSciNetMATH

Rubinstein RY, Kroese DP (2016) Simulation and the Monte Carlo method (Vol.10). Wiley

Sacks J, Welch WJ, Mitchell TJ, Wynn HP (1989) Design and analysis of computer experiments. Stat Sci:409–423

Sepahvand K, Marburg S, Hardtke HJ (2010) Uncertainty quantification in stochastic systems using polynomial chaos expansion. Inter J Appl Mech 2(02):305–353

Simpson TW, Poplinski JD, Koch PN, Allen JK (2001) Metamodels for computer-based engineering design: survey and recommendations. Eng Comput 17(2):129–150MATH

Slifker JF, Shapiro SS (1980) The Johnson system: selection and parameter estimation. Technometrics 22(2):239–246MATH

Stein M (1987) Large sample properties of simulations using Latin hypercube sampling. Technometrics 29(2):143–151MathSciNetMATH

Turányi T (1990) Sensitivity analysis of complex kinetic systems. Tools and applications. J Math Chem 5(3):203–248MathSciNet

Tvedt L (1983) Two second-order approximations to the failure probability. Veritas Report RDIV/20-004083

Ugray Z, Lasdon L, Plummer J, Glover F, Kelly J, Martí R (2007) Scatter search and local NLP solvers: a multistart framework for global optimization. INFORMS J Comput 19(3):328–340MathSciNetMATH

Wang Z, Bovik AC (2009) Mean squared error: love it or leave it? A new look at signal fidelity measures. IEEE Signal Proc Mag 26(1):98–117

Winterstein SR, Ude TC, Cornell CA, Bjerager P, Haver S (1993) Environmental parameters for extreme response: inverse FORM with omission factors. Proceedings of the ICOSSAR-93, Innsbruck, Austria, pp.551-557

Wu YT, Millwater HR, Cruse TA (1990) Advanced probabilistic structural analysis method for implicit performance functions. AIAA J 28(9):1663–1669

Xi Z, Hu C, Youn BD (2012) A comparative study of probability estimation methods for reliability analysis. Struct Multidiscip Optim 45(1):33–52MathSciNetMATH

Xiong Y, Chen W, Apley D, Ding X (2007) A non-stationary covariance-based Kriging method for metamodeling in engineering design. Int J Numer Meth Eng 71(6):733–756MATH

Xiu D, Karniadakis GE (2003) Modeling uncertainty in flow simulations via generalized polynomial chaos. J Comput Phys 187(1):137–167MathSciNetMATH

Xu J, Dang C (2019) A new bivariate dimension reduction method for efficient structural reliability analysis. Mech Syst Signal Pr 115:281–300

Xue J, Yang K (1997) Upper and lower bounds of stress-strength interference reliability with random strength-degradation. IEEE T Reliab 46(1):142–145

Youn BD, Choi KK, Park YH (2003) Hybrid analysis method for reliability-based design optimization. J Mech Des 125(2):221–232

Youn BD, Xi Z, Wang P (2008) Eigenvector dimension reduction (EDR) method for sensitivity-free probability analysis. Struct Multidiscip Optim 37(1):13–28MathSciNetMATH

Zhang Y, Der Kiureghian A (1995) Two improved algorithms for reliability analysis. In: Reliability and optimization of structural systems. Springer, Boston, pp 297–304

Zhang J, Ma X, Zhao Y (2017) A stress-strength time-varying correlation interference model for structural reliability analysis using copulas. IEEE T Reliab 66(2):351–365

Zhu SP, Huang HZ, Peng W, Wang HK, Mahadevan S (2016) Probabilistic physics of failure-based framework for fatigue life prediction of aircraft gas turbine discs under uncertainty. Reliab Eng Syst Saf 146:1–12

Titel: A most probable point method for probability distribution construction
verfasst von: Yongyong Xiang
Baisong Pan
Luping Luo
Publikationsdatum: 26.06.2020
Verlag: Springer Berlin Heidelberg
Erschienen in: Structural and Multidisciplinary Optimization / Ausgabe 5/2020
Print ISSN: 1615-147X
Elektronische ISSN: 1615-1488
DOI: https://doi.org/10.1007/s00158-020-02623-2

Premium Partner

Marktübersichten

Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen.

Zur Marktübersicht

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Weitere Artikel der Ausgabe 5/2020

An active learning Kriging-assisted method for reliability-based design optimization under distributional probability-box model

Stress-based shape and topology optimization with cellular level set in B-splines

Time-variant reliability analysis via approximation of the first-crossing PDF

Discrete topology optimization in augmented space: integrated element removal for minimum size and mesh sensitivity control

Topology optimization of structural systems based on a nonlinear beam finite element model

Structural reliability analysis via dimension reduction, adaptive sampling, and Monte Carlo simulation

Premium Partner

Marktübersichten