Abstract
This paper gives an overview of recent advances in the field of non-probabilistic uncertainty quantification. Both techniques for the forward propagation and inverse quantification of interval and fuzzy uncertainty are discussed. Also the modeling of spatial uncertainty in an interval and fuzzy context is discussed. An in depth discussion of a recently introduced method for the inverse quantification of spatial interval uncertainty is provided and its performance is illustrated using a case studies taken from literature. It is shown that the method enables an accurate quantification of spatial uncertainty under very low data availability and with a very limited amount of assumptions on the underlying uncertainty. Finally, also a conceptual comparison with the class of Bayesian methods for uncertainty quantification is provided.
Similar content being viewed by others
References
Abdel-Tawab K, Noor AK (1999) Uncertainty analysis of welding residual stress fields. Comput Methods Appl Mech Eng 179(3–4):327–344. https://doi.org/10.1016/S0045-7825(99)00045-6
Agarwal H, Renaud JE, Preston EL, Padmanabhan D (2004) Uncertainty quantification using evidence theory in multidisciplinary design optimization. Reliab Eng Syst Saf 85(1):281–294
Ahmadian H, Mottershead J, Friswell M (1998) Regularisation methods for finite element model updating. Mech Syst Signal Process 12(1):47–64
Balu A, Rao B (2012) High dimensional model representation based formulations for fuzzy finite element analysis of structures. Finite Elem Anal Des 50:217–230
Barber CB, Dobkin DP, Huhdanpaa H (1996) The quickhull algorithm for convex hulls. ACM Trans Math Softw 22(4):469–483. https://doi.org/10.1145/235815.235821
Beck JL, Au SK (2002) Bayesian updating of structural models and reliability using markov chain monte carlo simulation. J Eng Mech 128(4):380–391
Beck JL, Katafygiotis LS (1998) Updating models and their uncertainties. J Eng Mech 124(4):455–461
Beck JL, Yuen KV (2004) Model selection using response measurements: Bayesian probabilistic approach. J Eng Mech 130(2):192–203
Beer M, Ferson S, Kreinovich V (2013) Imprecise probabilities in engineering analyses. Mech Syst Signal Process 37(1):4–29
Beer M, Ferson S, Kreinovich V (2013) Imprecise probabilities in engineering analyses. Mech Syst Signal Process 37(1):4–29. https://doi.org/10.1016/j.ymssp.2013.01.024
Beer M, Kreinovich V (2013) Interval or moments: Which carry more information? Soft Comput 17(8):1319–1327. https://doi.org/10.1007/s00500-013-1002-1
Beer M, Liebscher M, Möller B (2004) Structural design under fuzzy randomness. In: Proceedings of the NSF workshop on reliable engineering computing, pp 215–234
de Berg M, Cheong O, van Kreveld M, Overmars M (2008) Computational geometry. Springer, Berlin. https://doi.org/10.1007/978-3-540-77974-2
Betz W, Papaioannou I, Straub D (2014) Numerical methods for the discretization of random fields by means of the Karhunen–Loeve expansion. Comput Methods Appl Mech Eng 271:109–129. https://doi.org/10.1016/j.cma.2013.12.010
Biondini F, Bontempi F, Malerba PG (2004) Fuzzy reliability analysis of concrete structures. Comput Struct 82(13–14):1033–1052. https://doi.org/10.1016/j.compstruc.2004.03.011
de Boor C, Ron A (1990) On multivariate polynomial interpolation. Constr Approx 6(3):287–302. https://doi.org/10.1007/BF01890412
Boulkaibet I, Marwala T, Friswell M, Khodaparast HH, Adhikari S (2017) Fuzzy finite element model updating using metaheuristic optimization algorithms. arXiv preprint arXiv:1701.00833
Bulgakov BV (1940) Fehleranhäufung bei kreiselapparaten. Ingenieur-Archiv 11(6):461–469. https://doi.org/10.1007/BF02088988
Bulgakov BV (1946) On the accumulation of disturbances in linear os- cillatory systems with constant coefficients (in russian). Proc USSR Acad Sci 51:343–345
Butlin T (2013) Anti-optimisation for modelling the vibration of locally nonlinear structures: an exploratory study. J Sound Vib 332(26):7099–7122. https://doi.org/10.1016/j.jsv.2013.06.028
Campi MC, Calafiore G, Garatti S (2009) Interval predictor models: identification and reliability. Automatica 45(2):382–392
Catallo L (2004) Genetic anti-optimization for reliability structural assessment of precast concrete structures. Comput Struct 82(13–14):1053–1065. https://doi.org/10.1016/j.compstruc.2004.03.018
Chen L, Rao S (1997) Fuzzy finite-element approach for the vibration analysis of imprecisely-defined systems. Finite Elem Anal Des 27(1):69–83. https://doi.org/10.1016/S0168-874X(97)00005-X
Chen S, Lian H, Yang X (2002) Interval static displacement analysis for structures with interval parameters. Int J Numer Methods Eng 53(2):393–407. https://doi.org/10.1002/nme.281
Cheung SH, Beck JL (2009) Bayesian model updating using hybrid monte carlo simulation with application to structural dynamic models with many uncertain parameters. J Eng Mech 135(4):243–255
Ching J, Chen YC (2007) Transitional markov chain monte carlo method for bayesian model updating, model class selection, and model averaging. J Eng Mech 133(7):816–832
Choi CK, Yoo HH (2017) Stochastic modeling and vibration analysis of rotating beams considering geometric random fields. J Sound Vib 388:105–122. https://doi.org/10.1016/j.jsv.2016.10.030
Chowdhary K, Najm HN (2016) Bayesian estimation of karhunen-loève expansions; a random subspace approach. J Comput Phys 319:280–293. https://doi.org/10.1016/j.jcp.2016.02.056
Civanlar MR, Trussell HJ (1986) Constructing membership functions using statistical data. Fuzzy Sets Syst 18(1):1–13
Comba JLD, Stolfi J (1993) Affine arithmetic and its applications to computer graphics. In: Anais do VII SIBGRAPI, pp 9–18
Crespo LG, Kenny SP, Giesy DP (2016) Interval predictor models with a linear parameter dependency. J Verif Valid Uncertain Quantif 1(2):021007
Crombecq K, Couckuyt I, Gorissen D, Dhaene T (2009) Space-filling sequential design strategies for adaptive surrogate modelling. In: The first international conference on soft computing technology in civil, structural and environmental engineering
De Munck M, Moens D, Desmet W, Vandepitte D (2009) An efficient response surface based optimisation method for non-deterministic harmonic and transient dynamic analysis. CMES Comput Model Eng Sci 47(2):119–166. https://doi.org/10.3970/cmes.2009.047.119
De Oliveira V, Kedem B, Short DA (1997) Bayesian prediction of transformed gaussian random fields. J Am Stat Assoc 92(440):1422–1433
Degrauwe D, Lombaert G, Roeck GD (2010) Improving interval analysis in finite element calculations by means of affine arithmetic. Comput Struct 88(3–4):247–254. https://doi.org/10.1016/j.compstruc.2009.11.003
Dempster AP (1967) Upper and lower probabilities induced by a multivalued mapping. Ann Math Stat 38: 325–339
Deng Z, Guo Z, Zhang X (2016) Non-probabilistic set-theoretic models for transient heat conduction of thermal protection systems with uncertain parameters. Appl Therm Eng 95:10–17
Deng Z, Guo Z, Zhang X (2017) Interval model updating using perturbation method and radial basis function neural networks. Mech Syst Signal Process 84:699–716
Di W, Wei G (2016) Uncertain static plane stress analysis with interval fields. Int J Numer Methods Eng 110(13):1272–1300. https://doi.org/10.1002/nme.5457
Do DM, Gao W, Song C (2016) Stochastic finite element analysis of structures in the presence of multiple imprecise random field parameters. Comput Methods Appl Mech Eng 300:657–688. https://doi.org/10.1016/j.cma.2015.11.032
Dogan M, Van Dam RL, Liu G, Meerschaert MM, Butler JJ, Bohling GC, Benson DA, Hyndman DW (2014) Predicting flow and transport in highly heterogeneous alluvial aquifers. Geophys Res Lett 41(21):7560–7565
Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46(1):131–150
Donders S, Vandepitte D, Van de Peer J, Desmet W (2004) The short transformation method to predict the FRF of dynamic structures subject to uncertainty. In: Proceedings of ISMA, pp 3043–3054
Donders S, Vandepitte D, de Peer JV, Desmet W (2005) Assessment of uncertainty on structural dynamic responses with the short transformation method. J Sound Vib 288(3):523–549. https://doi.org/10.1016/j.jsv.2005.07.003 Uncertainty in structural dynamicsUncertainty in structural dynamics
Dong W, Shah HC (1987) Vertex method for computing of fuzzy variables. Fuzzy Sets Syst 24:65–78. https://doi.org/10.1016/0165-0114(87)90114-X
Dorigo M, Birattari M, Stutzle T (2006) Ant colony optimization. IEEE Comput Intell Mag 1(4):28–39
Dubois D, Prade H (2012) Possibility theory: an approach to computerized processing of uncertainty. Springer Science & Business Media, New York
Elishakoff I (1990) An idea of uncertainty triangle. Shock Vib Digest 22(10):1
Elishakoff I (2000) Possible limitations of probabilistic methods in engineering. Appl Mech Rev 53(2):19. https://doi.org/10.1115/1.3097337
Elishakoff I (2000) Possible limitations of probabilistic methods in engineering. Appl Mech Rev 53(2):19–36
Elishakoff I, Miglis Y (2012) Novel parameterized intervals may lead to sharp bounds. Mech Res Commun 44:1–8. https://doi.org/10.1016/j.mechrescom.2012.04.004
Elishakoff I, Sarlin N (2016) Uncertainty quantification based on pillars of experiment, theory, and computation. Part I: data analysis. Mech Syst Signal Process 74:54–72. https://doi.org/10.1016/j.ymssp.2015.04.035
Elishakoff I, Sarlin N (2016) Uncertainty quantification based on pillars of experiment, theory, and computation. Part II: theory and computation. Mech Syst Signal Process 2:74
Engl HW, Hanke M, Neubauer A (1996) Regularization of inverse problems, vol 375. Springer Science & Business Media, New York
Faes M (2017) Interval methods for the identification and quantification of inhomogeneous uncertainty in finite element models. Ph.D. thesis, KU Leuven, Department of Mechanical Engineering
Faes M, Broggi M, Beer M, Moens D (2018) Failure probability under uncertain surrogate model predictions. In: Proceedings of the joint ICVRAM ISUMA UNCERTAINTIES conference
Faes M, Broggi M, Patelli E, Govers Y, Mottershead J, Beer M, Moens D (2019) A multivariate interval approach for inverse uncertainty quantification with limited experimental data. Mech Syst Signal Process 118:534–548. https://doi.org/10.1016/j.ymssp.2018.08.050. https://www.sciencedirect.com/science/article/pii/S0888327018305946?via%3Dihub
Faes M, Cerneels J, Vandepitte D, Moens D (2016) Identification of Interval Fields for Spatial Uncertainty Representation in Finite Element Models. In: Proceedings of the VII European congress on computational methods in applied sciences and engineering (ECCOMAS congress 2016), vol 3, pp 6091–6098. https://doi.org/10.7712/100016.2243.4995. http://www.eccomasproceedia.org/conferences/eccomas-congresses/eccomas-congress-2016/2243
Faes M, Cerneels J, Vandepitte D, Moens D (2017) Identification and quantification of multivariate interval uncertainty in finite element models. Comput Methods Appl Mech Eng 315:896–920. https://doi.org/10.1016/j.cma.2016.11.023
Faes M, Cerneels J, Vandepitte D, Moens D (2017) Influence of measurement data metrics on the identification of interval fields for the representation of spatial variability in finite element models. PAMM 16(1):27–30. https://doi.org/10.1002/pamm.201610008
Faes M, Moens D (2017) Identification and quantification of spatial interval uncertainty in numerical models. Comput Struct 192:16–33. https://doi.org/10.1016/j.compstruc.2017.07.006
Faes M, Moens D (2019) Multivariate dependent interval finite element analysis via convex hull pair constructions and the extended transformation method. Comput Methods Appl Mech Eng 347:85–102. https://doi.org/10.1016/j.cma.2018.12.021. http://www.sciencedirect.com/science/article/pii/S0045782518306200
Fang SE, Zhang QH, Ren WX (2015) An interval model updating strategy using interval response surface models. Mech Syst Signal Process 60–61:909–927. https://doi.org/10.1016/j.ymssp.2015.01.016
Farkas L, Moens D, Donders S, Vandepitte D (2012) Optimisation study of a vehicle bumper subsystem with fuzzy parameters. Mech Syst Signal Process 32:59–68
Farley B, Clark W (1954) Simulation of self-organizing systems by digital computer. Trans IRE Prof Group Inform Theory 4(4):76–84. https://doi.org/10.1109/TIT.1954.1057468
Fedele F, Muhanna RL, Xiao N, Mullen RL (2014) Interval-based approach for uncertainty propagation in inverse problems. J Eng Mech 4(1):1–7. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000815
Ferson S, Ginzburg LR (1996) Different methods are needed to propagate ignorance and variability. Reliab Eng Syst Saf 54(2):133–144. https://doi.org/10.1016/S0951-8320(96)00071-3
Ferson S, Kreinovich V, Ginzburg L, Myers DS, Sentz K (2003) Constructing probability boxes and dempster-shafer structures. Tech. rep., Technical report, Sandia National Laboratories
Ferson S, Moore DR, Van Den Brink P, Estes T, Gallagher K, Connor R, Verdonck F (2010) Bounding uncertainty analyses. Application of uncertainty analysis to ecological risks of pesticides
Ferson S, Siegrist J (2012) Verified computation with probabilities. In: Uncertainty quantification in scientific computing, pp 95–122. Springer
Fisher R (1938) Presidential address by professor ra fisher, SC. D., FRS. Indian J Stat 4(1):14–17
Fleissner F, Haag T, Hanss M, Eberhard P (2011) Analysis of granular chute flow based on a particle model including uncertainties. Trends in computational contact mechanics pp 121–134
Gabriele S, Brancaleoni F, Spina D (2011) Model updating of pescara benchmark: interval versus traditional method. In: Journal of Physics: conference series, vol 305, p 012083. IOP Publishing
Gabriele S, Valente C (2009) An interval-based technique for fe model updating. Int J Reliab Saf 3(1–3):79–103
Gotz M, Leichsenring F, Graf W, Michael K (2018) Four types of dependencies for fuzzy analysis. In: Proceedings of the 6th European conference on computational mechanics (ECCM). ECCOMAS
Govers Y, Haddad Khodaparast H, Link M, Mottershead JE (2015) A comparison of two stochastic model updating methods using the DLR AIRMOD test structure. Mech Syst Signal Process 52–53(1):105–114
Graf W, Gotz M, Kaliske M (2015) Analysis of dynamical processes under consideration of polymorphic uncertainty. Struct Saf. 52:194–201. https://doi.org/10.1016/j.strusafe.2014.09.003. http://www.sciencedirect.com/science/article/pii/S0167473014000861. Engineering Analyses with Vague and Imprecise Information
Gratiet LL, Marelli S, Sudret B (2016) Metamodel-based sensitivity analysis: polynomial chaos expansions and gaussian processes. Handbook of Uncertainty Quantification, pp 1–37
Gull SF (1988) Bayesian inductive inference and maximum entropy. Springer, Berlin, pp 53–74
Haag T, González SC, Hanss M (2012) Model validation and selection based on inverse fuzzy arithmetic. Mech Syst Signal Process 32:116–134
Haag T, Hanss M (2010) Model assessment using inverse fuzzy arithmetic. Information processing and management of uncertainty in knowledge-based systems. applications, pp 461–470
Haag T, Hanss (sup.) M. Moens (cosup.) D, Gaul (cosup.) L (2012) Forward and inverse fuzzy arithmetic for uncertainty analysis with applications to structural mechanics. PhD thesis, Universität Stuttgart
Haag T, Herrmann J, Hanss M (2010) Identification procedure for epistemic uncertainties using inverse fuzzy arithmetic. Mech Syst Signal Process 24(7):2021–2034
Haddad Khodaparast H, Govers Y, Adhikari S, Link M, Friswell MI, Mottershead JE, Sienz J (2014) Fuzzy model updating and its application to the DLR AIRMOD test structure. In: P. Sas, D. Moens, H. Denayer (eds) Proceedings of the international conference on uncertainty in structural dynamics, USD 2014, pp 4509–4522. KU Leuven, Leuven, Belgium
Hansen E, Walster GW (2003) Global optimization using interval analysis: revised and expanded, vol 264. CRC Press, New York
Hansen PC (1998) Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion. SIAM, Philadelphia
Hanss M (2002) The transformation method for the simulation and analysis of systems with uncertain parameters. Fuzzy Sets Syst 130(3):277–289. https://doi.org/10.1016/S0165-0114(02)00045-3
Hanss M (2003) An approach to inverse fuzzy arithmetic. In: 22nd International conference of the north American fuzzy information processing society, NAFIPS 2003, pp 474–479. https://doi.org/10.1109/NAFIPS.2003.1226831
Hanss M (2003) The extended transformation method for the simulation and analysis of fuzzy-parameterized models. Int J Uncertain Fuzziness Knowl Based Syst 11(06):711–727. https://doi.org/10.1142/S0218488503002491
Hanss M (2005) Applied fuzzy arithmetic: an introduction with engineering applications. Springer, Berlin
Hanss M, Gauger U, Turrin S (2006) Fuzzy arithmetical robustness analysis of mechanical structures with uncertainties. In: International conference on computational structures technology, Gran Canaria, Spain
Hanss M, Klimke A (2004) On the reliability of the influence measure in the transformation method of fuzzy arithmetic. Fuzzy Sets Syst 143(3):371–390
Hanss M, Turrin S (2010) A fuzzy-based approach to comprehensive modeling and analysis of systems with epistemic uncertainties. Struct Saf 32(6):433–441
Harvey CR, Zhou G (1990) Bayesian inference in asset pricing tests. J Financ Econ 26(2):221–254
Hoffman FO, Hammonds JS (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. https://doi.org/10.1111/j.1539-6924.1994.tb00281.x
Hristopulos DT (2003) Spartan gibbs random field models for geostatistical applications. SIAM J Sci Comput 24(6):2125–2162
Hsu CW, Lin CJ (2002) A comparison of methods for multiclass support vector machines. IEEE Trans Neural Netw 13(2):415–425
Huelsenbeck JP, Ronquist F, Nielsen R, Bollback JP (2001) Bayesian inference of phylogeny and its impact on evolutionary biology. Science 294(5550):2310–2314
Hughes TJ, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: cad, finite elements, nurbs, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194(39):4135–4195
Imholz M, Vandepitte D, Moens D (2015) Analysis of the effect of uncertain clamping stiffness on the dynamical behaviour of structures using interval field methods. In: Applied mechanics and materials, vol 807, pp 195–204. Trans Tech Publ
Imholz M, Vandepitte D, Moens D (2015) Derivation of an input interval field decomposition based on expert knowledge using locally defined basis functions. In: 1st ECCOMAS Thematic conference on international conference on uncertainty quantification in computational sciences and engineering, pp 1–19
Imholz M, Vandepitte D, Moens D (2018) Application of interval fields to fit experimental data on deepdrawn components. In: Proceedings of the joint ICVRAM ISUMA UNCERTAINTIES conference (2)
Jaynes ET (1957) Information theory and statistical mechanics. Phys Rev 106:620–630. https://doi.org/10.1103/PhysRev.106.620
Jaynes ET (1957) Information theory and statistical mechanics. II. Phys Rev 108:171–190. https://doi.org/10.1103/PhysRev.108.171
Jinglai W, Zhen L, Yunqing Z, Nong Z, Liping C (2013) Interval uncertain method for multibody mechanical systems using chebyshev inclusion functions. Int J Numer Methods Eng 95(7):608–630. https://doi.org/10.1002/nme.4525
Kandel S, McCulloch R, Stambaugh RF (1995) Bayesian inference and portfolio efficiency. Rev Financ Stud 8(1):1–53
Katafygiotis LS, Beck JL (1998) Updating models and their uncertainties. II: Model identifiability. J Eng Mech 124(4):463–467
Kennedy J (2011) Particle swarm optimization. In: Encyclopedia of machine learning, pp 760–766. Springer
Kennedy MC, O’Hagan A (2001) Bayesian calibration of computer models. J R Stat Soc Ser B 63(3):425–464
Khodaparast HH, Mottershead JE, Badcock KJ (2011) Interval model updating with irreducible uncertainty using the Kriging predictor. Mech Syst Signal Process 25(4):1204–1206. https://doi.org/10.1016/j.ymssp.2010.10.009
der Kiureghian A, Ditlevsen O (2009) Aleatory or epistemic? Does it matter? Struct Saf 31(2):105–112. https://doi.org/10.1016/j.strusafe.2008.06.020
Klimke A, Nunes RF, Wohlmuth B (2006) Fuzzy arithmetic based on dimension-adaptive sparse grids: a case study of a large-scale finite element model under uncertain parameters. Int J Uncertian Fuzziness Knowl Based Syst 14(05):561–577. https://doi.org/10.1142/S0218488506004199
Kononenko I (1989) Bayesian neural networks. Biol Cybern 61(5):361–370
Köylüoglu H, Elishakoff I (1998) A comparison of stochastic and interval finite elements applied to shear frames with uncertain stiffness properties. Comput Struct 67(1–3):91–98. https://doi.org/10.1016/S0045-7949(97)00160-0
Kulpa Z, Pownuk A, Skalna I (1998) Analysis of linear mechanical structures with uncertainties by means of interval methods. Comput Assist Mech Eng Sci 5(4):443–477
Lava P, Cooreman S, Coppieters S, De Strycker M, Debruyne D (2009) Assessment of measuring errors in DIC using deformation fields generated by plastic FEA. Opt Lasers Eng 47(7–8):747–753. https://doi.org/10.1016/j.optlaseng.2009.03.007
Legault J, Langley R, Woodhouse J (2012) Physical consequences of a nonparametric uncertainty model in structural dynamics. J Sound Vib 331(25):5469–5487. https://doi.org/10.1016/j.jsv.2012.07.017
Manson G (2005) Calculating frequency response functions for uncertain systems using complex affine analysis. J Sound Vib 288(3):487–521. https://doi.org/10.1016/j.jsv.2005.07.004
de Marsily G, Delay F, Teles V, Schafmeister MT (1998) Some current methods to represent the heterogeneity of natural media in hydrogeology. Hydrogeol J 6(1):115–130. https://doi.org/10.1007/s100400050138
Martin JD, Simpson TW (2005) Use of kriging models to approximate deterministic computer models. AIAA J 43(4):853–863. https://doi.org/10.2514/1.8650
Marwala T, Mdlazi L, Sibisi S (2007) Finite element model updating using bayesian approach. arXiv preprint arXiv:0705.2515
Massa F, Ruffin K, Tison T, Lallemand B (2008) A complete method for efficient fuzzy modal analysis. J Sound Vib 309(1–2):63–85. https://doi.org/10.1016/j.jsv.2007.06.004
Massa F, Tison T, Lallemand B (2006) A fuzzy procedure for the static design of imprecise structures. Comput Methods Appl Mech Eng 195(9–12):925–941. https://doi.org/10.1016/j.cma.2005.02.015
McCulloch WS, Pitts W (1943) A logical calculus of the ideas immanent in nervous activity. Bull Math Biophys 5(4):115–133. https://doi.org/10.1007/BF02478259
McDonald DB, Grantham WJ, Tabor WL (2005) Response surface development for global/local optimization using radial basis functions. In: Aiaa 2000-4776. AIAA, Long Beach, CA, USA
McWilliam S (2001) Anti-optimisation of uncertain structures using interval analysis. Comput Struct 79(4):421–430. https://doi.org/10.1016/S0045-7949(00)00143-7
Mehrez L, Doostan A, Moens D, Vandepitte D (2012) Stochastic identification of composite material properties from limited experimental databases, Part II: Uncertainty modelling. Mech Syst Signal Process 27(1):484–498. https://doi.org/10.1016/j.ymssp.2011.09.001
Missoum S, Lacaze S, Amabili M, Alijani F (2017) Identification of material properties of composite sandwich panels under geometric uncertainty. Compos Struct. https://doi.org/10.1016/j.compstruct.2017.07.020
Modares M, Venkitaraman S (2015) Reliable condition assessment of structures using hybrid structural measurements and structural uncertainty analyses. Struct Saf 52:202–208
Moens D, De Munck M, Desmet W, Vandepitte D (2011) Numerical dynamic analysis of uncertain mechanical structures based on interval fields. In: IUTAM symposium on the vibration analysis of structures with uncertainties, pp 71–83. Springer
Moens D, De Munck M, Vandepitte D (2007) Envelope frequency response function analysis of mechanical structures with uncertain modal damping characteristics. Comput Model Eng Sci 22(2):129–149
Moens D, Hanss M (2011) Non-probabilistic finite element analysis for parametric uncertainty treatment in applied mechanics: recent advances. Finite Elem Anal Des 47(1):4–16. https://doi.org/10.1016/j.finel.2010.07.010
Moens D, Vandepitte D (2004) An interval finite element approach for the calculation of envelope frequency respons functions. Int J Numer Methods Eng 61(14):2480–2507
Moens D, Vandepitte D (2006) Recent advances in non-probabilistic approaches for non-deterministic dynamic finite element analysis. Arch Comput Methods Eng 13(3):389–464. https://doi.org/10.1007/BF02736398
Moens D, Vandepitte D (2007) Interval sensitivity theory and its application to frequency response envelope analysis of uncertain structures. Comput Methods Appl Mech Eng 196(21–24):2486–2496. https://doi.org/10.1016/j.cma.2007.01.006
Möller B (2004) Fuzzy randomness-a contribution to imprecise probability. ZAMM J Appl Math Mech 84(10–11):754–764
Möller B, Beer M (2013) Fuzzy randomness: uncertainty in civil engineering and computational mechanics. Springer Science & Business Media, New York
Möller B, Beer M, Graf W, Sickert JU (2006) Time-dependent reliability of textile-strengthened RC structures under consideration of fuzzy randomness. Comput Struct 84(8):585–603
Möller B, Beer M, Reuter U (2005) Theoretical basics of fuzzy randomness-application to time series with fuzzy data. In: CD Proceedings of 9th international conference on structural safety and reliability ICOSSAR, vol 5
Möller B, Graf W, Beer M (2000) Fuzzy structural analysis using alpha-level optimization. Comput Mech 26(6):547–565. https://doi.org/10.1007/s004660000204
Möller B, Graf W, Beer M (2000) Fuzzy structural analysis using α-level optimization. Comput Mech 26(6):547–565
Möller B, Graf W, Beer M (2003) Safety assessment of structures in view of fuzzy randomness. Comput Struct 81(15):1567–1582
Monelli D, Mai P (2008) Bayesian inference of kinematic earthquake rupture parameters through fitting of strong motion data. Geophys J Int 173(1):220–232
Moore RE (1962) Interval arithmetic and automatic error analysis in digital computing. Ph.D. thesis, Stanford University, Department of Mathematics
Moore RE (1979) Methods and applications of interval analysis. SIAM, Philadelphia
Moore RT (1966) Interval analysis, vol 4. Prentice Hall, Englewood Cliffs. https://doi.org/10.1016/0016-0032(67)90590-X
Mottershead J, Friswell M (1993) Model updating in structural dynamics: a survey. J Sound Vib 167(2):347–375
Muhanna RL, Mullen RL (2001) Uncertanty in mechanics problems- interval based approach. J Eng Mech 127(6):557–566
Muhanna RL, Mullen RL, Zhang H (2005) Penalty-based solution for the interval finite-element methods. J Eng Mech 131:1102–1112. https://doi.org/10.1061/(ASCE)0733-9399(2005)131:10(1102)
Mullen RLMHZRL (2007) Combined axial and bending stiffness in interval finite-element methods. J Struct Eng 133(12):1700–1709. https://doi.org/10.1061/(ASCE)0733-9445(2007)133:12(1700)
Muscolino G, Santoro R, Sofi A (2014) Explicit sensitivities of the response of discretized structures under stationary random processes. Probab Eng Mech 35:82–95
Muscolino G, Santoro R, Sofi A (2015) Explicit reliability sensitivities of linear structures with interval uncertainties under stationary stochastic excitation. Struct Saf 52:219–232
Muscolino G, Sofi A (2012) Stochastic analysis of structures with uncertain-but-bounded parameters via improved interval analysis. Probab Eng Mech 28:152–163. https://doi.org/10.1016/j.probengmech.2011.08.011
Muscolino G, Sofi A (2013) Bounds for the stationary stochastic response of truss structures with uncertain-but-bounded parameters. Mech Syst Signal Process 37(1):163–181
Nicolaï BM, Egea JA, Scheerlinck N, Banga JR, Datta AK (2011) Fuzzy finite element analysis of heat conduction problems with uncertain parameters. J Food Eng 103(1):38–46. https://doi.org/10.1016/j.jfoodeng.2010.09.017
Nocedal J, Wright SJ (1999) Numerical optimization. Springer, New York. https://doi.org/10.1007/b98874
Oberkampf W, DeLand S, Rutherford B, Diegert K, Alvin K (1999) A new methodology for the estimation of total uncertainty in computational simulation. In: A new methodology for the estimation of total uncertainty in computational simulation, pp 3061–3083
Oberkampf WL, Trucano TG, Hirsch C (2004) Verification, validation, and predictive capability in computational engineering and physics. Appl Mech Rev 57(5):345–384
Otto K, Antonsson E (1995) Imprecision in engineering design. ASME J Mech Des 117:25–32
Patelli E, Govers Y, Broggi M, Gomes HM, Link M, Mottershead JE (2017) Sensitivity or bayesian model updating: a comparison of techniques using the dlr airmod test data. Arch Appl Mech pp 1–21. https://doi.org/10.1007/s00419-017-1233-1
Pota M, Esposito M, Pietro GD (2013) Transforming probability distributions into membership functions of fuzzy classes: a hypothesis test approach. Fuzzy Sets Syst 233:52–73. https://doi.org/10.1016/j.fss.2013.03.013
Qiu Z, Elishakoff I (1998) Antioptimization of structures with large uncertain-but-non- random parameters via interval analysis. Comput Methods Appl Mech Eng 7825(96):361–372
Qiu Z, Wang X (2009) Vertex solution theorem for the upper and lower bounds on the dynamic response of structures with uncertain-but-bounded parameters. Acta Mech Sin 25(3):367–379. https://doi.org/10.1007/s10409-008-0223-5
Rao MR, Mullen RL, Muhanna RL (2011) A new interval finite element formulation with the same accuracy in primary and derived variables. Int J Reliab Saf 5(3/4):336. https://doi.org/10.1504/IJRS.2011.041184
Rao S, Berke L (1997) Analysis of uncertain structural systems using interval analysis. AIAA J 35(4):727–735. https://doi.org/10.2514/3.13572
Rao SS, Chen LI (1998) Numerical solution of fuzzy linear equations in engineering analysis. Int J Numer Methods Eng 846(March 1997):829–846
Rao SS, Sawyer JP (1995) Fuzzy finite element approach for the analysis of imprecisely defined systems. AIAA J 33(12):2364–2370
Rossi PE, Allenby GM (2003) Bayesian statistics and marketing. Mark Sci 22(3):304–328
Rupert C, Miller C (2007) An analysis of polynomial chaos approximations for modeling single-fluid-phase flow in porous medium systems. J Comput Phys 226(2):2175–2205. https://doi.org/10.1016/j.jcp.2007.07.001
Sadeghi J, De Angelis M, Patelli E (2018) Frequentist history matching with interval predictor models. Appl Math Model 61:29–48
Scionti A, Lardeur P (2006) Experimental and numerical study of the intra/inter variability of an acoustic windscreen. In: Proceedings of the international conference on noise and vibration engineering ISMA 2006, pp 1999–2003. Leuven, Belgium
Sentz K, Ferson S (2002) Combination of evidence in Dempster–Shafer theory, vol 4015. Citeseer, New York
Serhat Erdogan Y, Gundes Bakir P (2013) Inverse propagation of uncertainties in finite element model updating through use of fuzzy arithmetic. Eng Appl Artif Intell 26(1):357–367. https://doi.org/10.1016/j.engappai.2012.10.003
Shafer G et al (1976) A mathematical theory of evidence, vol 1. Princeton University Press, Princeton
Sim J, Qiu Z, Wang X (2007) Modal analysis of structures with uncertain-but-bounded parameters via interval analysis. J Sound Vib 303:29–45. https://doi.org/10.1016/j.jsv.2006.11.038
Simoen E, Roeck GD, Lombaert G (2015) Dealing with uncertainty in model updating for damage assessment: a review. Mech Syst Signal Process 56–57:123–149. https://doi.org/10.1016/j.ymssp.2014.11.001
Singh P, Deschrijver D, Dhaene T (2013) A balanced sequential design strategy for global surrogate modeling. In: Simulation conference (WSC), 2013 Winter, pp 2172–2179. IEEE
Sofi A (2015) Structural response variability under spatially dependent uncertainty: stochastic versus interval model. Probab Eng Mech 42:78–86. https://doi.org/10.1016/j.probengmech.2015.09.001
Sofi A, Muscolino G (2015) Static analysis of Euler–Bernoulli beams with interval Young’s modulus. Comput Struct 156:72–82. https://doi.org/10.1016/j.compstruc.2015.04.002
Sofi A, Muscolino G, Elishakoff I (2015) Natural frequencies of structures with interval parameters. J Sound Vib 347:79–95. https://doi.org/10.1016/j.jsv.2015.02.037
Sofi A, Muscolino G, Elishakoff I (2015) Static response bounds of Timoshenko beams with spatially varying interval uncertainties. Acta Mech 226(11):3737–3748. https://doi.org/10.1007/s00707-015-1400-9
Sofi A, Romeo E (2016) A novel interval finite element method based on the improved interval analysis. Comput Methods Appl Mech Eng 311:671–697. https://doi.org/10.1016/j.cma.2016.09.009
Soize C (2008) Construction of probability distributions in high dimension using the maximum entropy principle: applications to stochastic processes, random fields and random matrices. Int J Numer Methods Eng 76:1583–1611. https://doi.org/10.1002/nme
Soize C (2010) Generalized probabilistic approach of uncertainties in computational dynamics using random matrices and polynomial chaos decompositions. Int J Numer Methods Eng 81(8):939–970. https://doi.org/10.1002/nme.2712
Soize C (2011) A computational inverse method for identification of non-gaussian random fields using the bayesian approach in very high dimension. Comput Methods Appl Mech Eng 200(45):3083–3099
Spanos P, Ghanem R (1989) Stochastic finite element expansion for random media. J Eng Mech 115(5):1035–1053
Sraj I, Maître OPL, Knio OM, Hoteit I (2016) Coordinate transformation and polynomial chaos for the bayesian inference of a gaussian process with parametrized prior covariance function. Comput Methods Appl Mech Eng 298:205–228. https://doi.org/10.1016/j.cma.2015.10.002
Stefanou G (2009) The stochastic finite element method: past, present and future. Comput Methods Appl Mech Eng 198(9–12):1031–1051. https://doi.org/10.1016/j.cma.2008.11.007
Stein M, Beer M, Kreinovich V (2013) Bayesian approach for inconsistent information. Inf Sci 245:96–111
Sudret B (2012) Meta-models for structural reliability and uncertainty quantification. arXiv preprint arXiv:1203.2062
Sunaga T (1958) Theory of an interval algebra and its application to numerical analysis. Jpn J Ind Appl Math 26(2):125–143
Taylor BN, Kuyatt CE (1994) Guidelines for evaluating and expressing the uncertainty of NIST measurement results. US Department of Commerce, Technology Administration, National Institute of Standards and Technology Gaithersburg, MD
Teichert WH (1998) Reasons for Uncertainty and their Consequence. In: Proceedings of the 23rd international conference on noise and vibration engineering. ISMA, Leuven, pp 961–966
Thacker BH, Doebling SW, Hemez FM, Anderson MC, Pepin JE, Rodriguez EA (2004) Concepts of model verification and validation. Tech. rep., Los Alamos National Lab., Los Alamos, NM
Titurus B, Friswell M (2008) Regularization in model updating. Int J Numer Methods Eng 75(4):440–478
Tonon F, Bernardini A (1998) A random set approach to the optimization of uncertain structures. Comput Struct 68(6):583–600
Troffaes M, Destercke S (2011) Probability boxes on totally preordered spaces for multivariate modelling. Int J Approx Reason 52(6):767–791. https://doi.org/10.1016/j.ijar.2011.02.001
Turrin S, Hanss M, Selvadurai A (2009) An approach to uncertainty analysis of rockfall simulation. CMES Comput Model Eng Sci 52(3):237–258
Van Der Herten J, Deschrijver D, Dhaene T (2014) Fuzzy local linear approximation-based sequential design. In: IEEE Symposium on computational intelligence for engineering solutions (CIES), 2014, pp 17–21
Vandepitte D, Moens D (2011) Quantification of uncertain and variable model parameters in non-deterministic analysis. In: IUTAM symposium on the vibration analysis of structures with uncertainties, vol 27, pp 15–28. Saint Petersburg. https://doi.org/10.1007/978-94-007-0289-9
Vanmarcke EH, Grigoriu M (1983) Stochastic finite element analysis of simple beams. J Eng Mech 109(5):1203–1214. https://doi.org/10.1061/(ASCE)0733-9399(1983)109:5(1203)
Verhaeghe W, Desmet W, Vandepitte D, Joris I, Seuntjens P, Moens D (2011) Application of interval fields for uncertainty modeling in a geohydrological case. In: ECCOMAS thematic conference: COMPDYN 2011: 3rd International conference on computational methods in structural dynamics and earthquake engineering: an IACM special interest conference, programme. Corfu, Greece
Verhaeghe W, Desmet W, Vandepitte D, Joris I, Seuntjens P, Moens D (2013) Application of interval fields for uncertainty modeling in a geohydrological case. In: Computational methods in stochastic dynamics, pp 131–147. Springer
Verhaeghe W, Desmet W, Vandepitte D, Moens D (2011) Uncertainty assessment in random field representations: an interval approach. In: Annual conference of the north American fuzzy information processing society–NAFIPS, pp 1–6. El Paso, TX. https://doi.org/10.1109/NAFIPS.2011.5752048
Versteeg HK, Malalasekera W (2007) An introduction to computational fluid dynamics: the finite volume method. Pearson Education
Walker WE, Harremoës P, Rotmans J, van der Sluijs JP, van Asselt MB, Janssen P, Krayer von Krauss MP (2003) Defining uncertainty: a conceptual basis for uncertainty management in model-based decision support. Integr Assess 4(1):5–17
Wang C, Qiu Z, Wang X, Wu D (2014) Interval finite element analysis and reliability-based optimization of coupled structural-acoustic system with uncertain parameters. Finite Elem Anal Des 91:108–114
Wang C, Qiu Z, Wang X, Wu D (2014) Interval finite element analysis and reliability-based optimization of coupled structural-acoustic system with uncertain parameters. Finite Elem Anal Des 91:108–114. https://doi.org/10.1016/j.finel.2014.07.014. http://www.sciencedirect.com/science/article/pii/S0168874X14001553
Wang M, Huang Q (2016) A new hybrid uncertain analysis method for structural-acoustic systems with random and interval parameters. Comput Struct 175:15–28. https://doi.org/10.1016/j.compstruc.2016.07.001. http://www.sciencedirect.com/science/article/pii/S0045794916305466
Warmus M (1956) Calculus of approximations. Bulletin de l’Academie Polonaise de Sciences 4(5):253–257
Wasfy TM, Noor AK (2000) Multibody dynamic simulation of the next generation space telescope using finite elements and fuzzy sets. Comput Methods Appl Mech Eng 190(5–7):803–824. https://doi.org/10.1016/S0045-7825(99)00445-4
Weyl H (1934) Elementare Theorie der konvexen Polyeder. Comentarii Mathematici Helvetici 7(1):290–306
Witteveen JA, Bijl H (2009) Effect of randomness on multi-frequency aeroelastic responses resolved by unsteady adaptive stochastic finite elements. J Comput Phys 228(18):7025–7045. https://doi.org/10.1016/j.jcp.2009.06.013
Wu B, Gao W, Wu D, Song C (2017) Probabilistic interval geometrically nonlinear analysis for structures. Struct Saf 65:100–112. https://doi.org/10.1016/j.strusafe.2017.01.002
Wu D, Gao W (2017) Hybrid uncertain static analysis with random and interval fields. Comput Methods Appl Mech Eng 315:222–246. https://doi.org/10.1016/j.cma.2016.10.047
Xia B, Yu D (2015) Optimization based on reliability and confidence interval design for the structural-acoustic system with interval probabilistic variables. J Sound Vib 336:1–15. https://doi.org/10.1016/j.jsv.2014.10.012
Xiao N, Fedele F, Muhanna R (2014) Interval-based parameter identification for structural static problems. arXiv preprint arXiv:1408.3430
Xiaojun W, Zhiping Q (2008) Interval finite element analysis of wing flutter. Chin J Aeronaut 21(2):134–140
Xu M, Du J, Wang C, Li Y (2017) Hybrid uncertainty propagation in structural-acoustic systems based on the polynomial chaos expansion and dimension-wise analysis. Comput Methods Appl Mech Eng 320:198–217. https://doi.org/10.1016/j.cma.2017.03.026. http://www.sciencedirect.com/science/article/pii/S004578251631461X
Xu M, Qiu Z (2014) A dimension-wise method for the static analysis of structures with interval parameters. Sci China Phys Mech Astron 57(10):1934–1945. https://doi.org/10.1007/s11433-014-5445-x
Yan WJ, Katafygiotis LS (2015) A novel bayesian approach for structural model updating utilizing statistical modal information from multiple setups. Struct Saf 52:260–271. https://doi.org/10.1016/j.strusafe.2014.06.004 Engineering Analyses with Vague and Imprecise Information
Yin H, Yu D, Lü H, Yin S, Xia B (2015) Hybrid finite element/statistical energy method for mid-frequency analysis of structure- acoustic systems with interval parameters. J Sound Vib 353:181–204
Yu M, Bao H, Ye J, Chi Y (2017) The effect of random porosity field on supercritical carbonation of cement-based materials. Construct Build Mater 146:144–155. https://doi.org/10.1016/j.conbuildmat.2017.04.060
Zadeh L (1965) Fuzzy sets. Inf Control 8(3):338–353. https://doi.org/10.1016/S0019-9958(65)90241-X
Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning-III. Inf Sci 9(1):43–80
Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning-III. Inf Sci 8(3):199–249
Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning-III. Inf Sci 8(4):301–357
Zadeh LA et al (1975) Calculus of fuzzy restrictions. University of California, Electronics Research Laboratory
Zhu H, Zhang L, Xiao T, Li X (2017) Generation of multivariate cross-correlated geotechnical random fields. Comput Geotech 86:95–107. https://doi.org/10.1016/j.compgeo.2017.01.006
Zhu Y, Zhang L (2009) Finite element model updating based on least squares support vector machines. Adv Neural Netw 2009:296–303
Zienkiewicz OC, Taylor RL, Taylor RL (1977) The finite element method, vol 3. McGraw-hill, London
Acknowledgements
The authors would like to acknowledge the financial support of the Flemish Research Foundation in the context of the research grant HIDIF (High dimensional interval fields) under grant number G0C2218N, as for the post-doctoral research grant 12P3519N of Matthias Faes.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Rights and permissions
About this article
Cite this article
Faes, M., Moens, D. Recent Trends in the Modeling and Quantification of Non-probabilistic Uncertainty. Arch Computat Methods Eng 27, 633–671 (2020). https://doi.org/10.1007/s11831-019-09327-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11831-019-09327-x