Article
Full entry |
PDF
(0.8 MB)
Feedback
PDF
(0.8 MB)
Feedback
Keywords:
uncertainty quantification; reliability analysis; probability of failure; safety margin; polynomial chaos expansion; regression method; stochastic collocation method; stochastic Galerkin method; Monte Carlo method
uncertainty quantification; reliability analysis; probability of failure; safety margin; polynomial chaos expansion; regression method; stochastic collocation method; stochastic Galerkin method; Monte Carlo method
Summary:
Recent developments in the field of stochastic mechanics and particularly regarding the stochastic finite element method allow to model uncertain behaviours for more complex engineering structures. In reliability analysis, polynomial chaos expansion is a useful tool because it helps to avoid thousands of time-consuming finite element model simulations for structures with uncertain parameters. The aim of this paper is to review and compare available techniques for both the construction of polynomial chaos and its use in computing failure probability. In particular, we compare results for the stochastic Galerkin method, stochastic collocation, and the regression method based on Latin hypercube sampling with predictions obtained by crude Monte Carlo sampling. As an illustrative engineering example, we consider a simple frame structure with uncertain parameters in loading and geometry with prescribed distributions defined by realistic histograms.
References:
[1] Augustin, F., Gilg, A., Paffrath, M., Rentrop, P., Villegas, M., Wever, U.: An accuracy comparison of polynomial chaos type methods for the propagation of uncertainties. J. Math. Ind. 3 (2013), 24 pages. DOI 10.1186/2190-5983-3-2 | MR 3049138 | Zbl 1275.65004
[2] Babuška, I., Nobile, F., Tempone, R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 45 (2007), 1005-1034. DOI 10.1137/050645142 | MR 2318799 | Zbl 1151.65008
[3] Babuška, I., Tempone, R., Zouraris, G. E.: Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42 (2004), 800-825. DOI 10.1137/S0036142902418680 | MR 2084236 | Zbl 1080.65003
[4] Blatman, G., Sudret, B.: An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis. Probabilistic Engineering Mechanics 25 (2010), 183-197. DOI 10.1016/j.probengmech.2009.10.003
[5] Blatman, G., Sudret, B.: Adaptive sparse polynomial chaos expansion based on least angle regression. J. Comput. Phys. 230 (2011), 2345-2367. DOI 10.1016/j.jcp.2010.12.021 | MR 2764550 | Zbl 1210.65019
[6] Cheng, H., Sandu, A.: Efficient uncertainty quantification with the polynomial chaos method for stiff systems. Math. Comput. Simul. 79 (2009), 3278-3295. DOI 10.1016/j.matcom.2009.05.002 | MR 2549773 | Zbl 1169.65005
[7] Choi, S.-K., Grandhi, R. V., Canfield, R. A., Pettit, C. L.: Polynomial chaos expansion with latin hypercube sampling for estimating response variability. AIAA J. 42 (2004), 1191-1198. DOI 10.2514/1.2220
[8] Ditlevsen, O., Madsen, H. O.: Structural Reliability Methods. John Wiley & Sons, Chichester (1996).
[9] Eigel, M., Gittelson, C. J., Schwab, C., Zander, E.: Adaptive stochastic Galerkin FEM. Comput. Methods Appl. Mech. Eng. 270 (2014), 247-269. DOI 10.1016/j.cma.2013.11.015 | MR 3154028 | Zbl 1296.65157
[10] Eldred, M. S., Burkardt, J.: Comparison of non-intrusive polynomial chaos and stochastic collocation methods for uncertainty quantification. The 47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition, Orlando AIAA 2009-976 (2009), 20. DOI 10.2514/6.2009-976
[11] Elman, H. C., Miller, C. W., Phipps, E. T., Tuminaro, R. S.: Assessment of collocation and Galerkin approaches to linear diffusion equations with random data. Int. J. Uncertain. Quantif. 1 (2011), 19-33. DOI 10.1615/int.j.uncertaintyquantification.v1.i1.20 | MR 2823001 | Zbl 1229.65026
[12] Fülöp, A., Iványi, M.: Safety of a column in a frame. Probabilistic Assessment of Structures Using Monte Carlo Simulation: Background, Exercises and Software P. Marek et al. Institute of Theoretical and Applied Mechanics, Academy of Sciences of the Czech Republic, Praha, CD, Chapt. 8.10 (2003).
[13] Ghanem, R. G., Spanos, P. D.: Stochastic Finite Elements: A Spectral Approach, Revised Edition. Dover Civil and Mechanical Engineering, Dover Publications (2012). DOI 10.1007/978-1-4612-3094-6 | MR 1083354 | Zbl 0722.73080
[14] Gutiérrez, M., Krenk, S.: Stochastic finite element methods. Encyclopedia of Computational Mechanics E. Stein et al. John Wiley & Sons, Chichester (2004). DOI 10.1002/0470091355.ecm044 | MR 2288276 | Zbl 1190.76001
[15] Heiss, F., Winschel, V.: Likelihood approximation by numerical integration on sparse grids. J. Econom. 144 (2008), 62-80. DOI 10.1016/j.jeconom.2007.12.004 | MR 2439922 | Zbl 06592098
[16] Hosder, S., Walters, R. W., Balch, M.: Efficient sampling for non-intrusive polynomial chaos applications with multiple uncertain input variables. The 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Honolulu AIAA 2007-1939 (2007), 16. DOI 10.2514/6.2007-1939
[17] Hu, C., Youn, B. D.: Adaptive-sparse polynomial chaos expansion for reliability analysis and design of complex engineering systems. Struct. Multidiscip. Optim. 43 (2011), 419-442. DOI 10.1007/s00158-010-0568-9 | MR 2774576 | Zbl 1274.74271
[18] Janouchová, E., Kučerová, A.: Competitive comparison of optimal designs of experiments for sampling-based sensitivity analysis. Comput. Struct. 124 (2013), 47-60. DOI 10.1016/j.compstruc.2013.04.009
[19] Janouchová, E., Kučerová, A., Sýkora, J.: Polynomial chaos construction for structural reliability analysis. Y. Tsompanakis et al. Proceedings of the Fourth International Conference on Soft Computing Technology in Civil, Structural and Environmental Engineering Civil-Comp Press, Stirlingshire (2015), Paper 9. DOI 10.4203/ccp.109.9
[20] Li, J., Li, J., Xiu, D.: An efficient surrogate-based method for computing rare failure probability. J. Comput. Phys. 230 (2011), 8683-8697. DOI 10.1016/j.jcp.2011.08.008 | MR 2845013 | Zbl 1370.65005
[21] Li, J., Xiu, D.: Evaluation of failure probability via surrogate models. J. Comput. Phys. 229 (2010), 8966-8980. DOI 10.1016/j.jcp.2010.08.022 | MR 2725383 | Zbl 1204.65010
[22] Ma, X., Zabaras, N.: An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations. J. Comput. Phys. 228 (2009), 3084-3113. DOI 10.1016/j.jcp.2009.01.006 | MR 2509309 | Zbl 1161.65006
[23] Matthies, H. G.: Uncertainty quantification with stochastic finite elements. Encyclopedia of Computational Mechanics E. Stein et al. John Wiley & Sons, Chichester (2007). DOI 10.1002/0470091355.ecm071 | MR 2288276 | Zbl 1190.76001
[24] Matthies, H. G., Keese, A.: Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Eng. 194 (2005), 1295-1331. DOI 10.1016/j.cma.2004.05.027 | MR 2121216 | Zbl 1088.65002
[25] Najm, H. N.: Uncertainty quantification and polynomial chaos techniques in computational fluid dynamics. Annual Review of Fluid Mechanics 41 S. H. Davis et al. Annual Reviews, Palo Alto (2009), 35-52. DOI 10.1146/annurev.fluid.010908.165248 | MR 2512381 | Zbl 1168.76041
[26] Nobile, F., Tempone, R., Webster, C. G.: A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal. 46 (2008), 2309-2345. DOI 10.1137/060663660 | MR 2421037 | Zbl 1176.65137
[27] Paffrath, M., Wever, U.: Adapted polynomial chaos expansion for failure detection. J. Comput. Phys. 226 (2007), 263-281. DOI 10.1016/j.jcp.2007.04.011 | MR 2356359 | Zbl 1124.65011
[28] Pettersson, M. P., Iaccarino, G., Nordström, J.: Polynomial chaos methods. Polynomial Chaos Methods for Hyperbolic Partial Differential Equations. Numerical Techniques for Fluid Dynamics Problems in the Presence of Uncertainties Mathematical Engineering, Springer, Cham (2015), 23-29. DOI 10.1007/978-3-319-10714-1_3 | MR 3328389 | Zbl 1325.76004
[29] Pulch, R.: Stochastic collocation and stochastic Galerkin methods for linear differential algebraic equations. J. Comput. Appl. Math. 262 (2014), 281-291. DOI 10.1016/j.cam.2013.10.046 | MR 3162322 | Zbl 1301.65090
[30] Stefanou, G.: The stochastic finite element method: Past, present and future. Comput. Methods Appl. Mech. Eng. 198 (2009), 1031-1051. DOI 10.1016/j.cma.2008.11.007 | Zbl 1229.74140
[31] Wiener, N.: The homogeneous chaos. Am. J. Math. 60 (1938), 897-936. DOI 10.2307/2371268 | MR 1507356 | Zbl 0019.35406
[32] Xiu, D.: Fast numerical methods for stochastic computations: A review. Commun. Comput. Phys. 5 (2009), 242-272. MR 2513686 | Zbl 1364.65019
[33] Xiu, D.: Numerical Methods for Stochastic Computations: A Spectral Method Approach. Princeton University Press, Princeton (2010). DOI 10.2307/j.ctv7h0skv | MR 2723020 | Zbl 1210.65002
[34] Xiu, D., Hesthaven, J. S.: High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27 (2005), 1118-1139. DOI 10.1137/040615201 | MR 2199923 | Zbl 1091.65006
Search
Partner of
