Forward and inverse structural uncertainty propagations under stochastic variables with arbitrary probability distributions. (English) Zbl 1440.74475
Summary: In this study, a general frame of the forward and inverse structural uncertainty propagations (UPs) based on the dimension reduction (DR) method and the derivative lambda probability density function \(( \lambda \)-PDF) is proposed to handle arbitrary probability distribution. For the forward UP, a DR method is applied to decompose a multivariable system into multiple univariate subsystems and a derivative \(\lambda \)-PDF is adopted to transform the arbitrary probability distribution of each subsystem. Then, the statistical moments and a fitting region are mathematically derived to analyze the fitting ability of the derivative \(\lambda \)-PDF. According to whether the kurtosis-skewness point lies in or out the fitting region, two different strategies combining the Gauss-Gegenbauer quadrature are proposed to implement the forward propagation. Compared with the conventional methods, the proposed method has advantages in higher accuracy, stability and efficiency. For the inverse propagation, because the unknown variable may be arbitrary distribution, the general frame based on the derivative \(\lambda \)-PDF and the Gauss-Gegenbauer quadrature are utilized to convert the uncertainty propagation into multiple deterministic calculations. Based on this, optimization method is adopted to accurately obtain the statistical moments and PDFs of the unknown stochastic variables. Five examples are provided to verify the accuracy and efficiency of the proposed general frame for the forward and inverse UPs.
MSC:
| 74S99 | Numerical and other methods in solid mechanics |
| 60-08 | Computational methods for problems pertaining to probability theory |
| 62G05 | Nonparametric estimation |
| 74S60 | Stochastic and other probabilistic methods applied to problems in solid mechanics |
Keywords:
uncertainty propagation; dimension reduction method; derivative \(\lambda \)-PDF; inverse problem; fitting region; arbitrary probability distributionReferences:
| [1] | Meng, Z.; Li, G.; Wang, B. P.; Hao, P., A hybrid chaos control approach of the performance measure functions for reliability-based design optimization, Comput. Struct., 146, 32-43 (2015) |
| [2] | Wang, L.; Cai, Y.; Liu, D., Multiscale reliability-based topology optimization methodology for truss-like microstructures with unknown-but-bounded uncertainties, Comput. Methods Appl. Mech. Engrg., 339, 358-388 (2018) · Zbl 1440.74314 |
| [3] | Jiang, C.; Zhang, Q. F.; Han, X.; Liu, J.; Hu, D. A., Multidimensional parallelepiped model—a new type of non-probabilistic convex model for structural uncertainty analysis, Internat. J. Numer. Methods Engrg., 103, 31-59 (2015) · Zbl 1352.74036 |
| [4] | Liu, H.; Jiang, C.; Jia, X.; Long, X.; Zhang, Z.; Guan, F., A new uncertainty propagation method for problems with parameterized probability-boxes, Reliab. Eng. Syst. Saf., 172, 64-73 (2018) |
| [5] | Wang, L.; Xiong, C.; Yang, Y., A novel methodology of reliability-based multidisciplinary design optimization under hybrid interval and fuzzy uncertainties, Comput. Methods Appl. Mech. Engrg., 337, 439-457 (2018) · Zbl 1440.74272 |
| [6] | Fonseca, J. R.; Friswell, M. I.; Mottershead, J. E.; Lees, A. W., Uncertainty identification by the maximum likelihood method, J. Sound Vib., 288, 587-599 (2005) |
| [7] | Jiang, C.; Ni, B. Y.; Han, X.; Tao, Y. R., Non-probabilistic convex model process: A new method of time-variant uncertainty analysis and its application to structural dynamic reliability problems, Comput. Meth. Appl. Mec. Eng., 268, 656-676 (2014) · Zbl 1295.90005 |
| [8] | Lee, S. H.; Chen, W., A comparative study of uncertainty propagation methods for black-box-type problems, Struct. Multi. Optim., 37, 239-253 (2009) |
| [9] | Ni, B. Y.; Jiang, C.; Han, X., An improved multidimensional parallelepiped non-probabilistic model for structural uncertainty analysis, Appl. Math. Model., 40, 4727-4745 (2016) · Zbl 1459.74008 |
| [10] | Wang, L.; Liu, D.; Yang, Y.; Wang, X.; Qiu, Z., A novel method of non-probabilistic reliability-based topology optimization corresponding to continuum structures with unknown but bounded uncertainties, Comput. Methods Appl. Mech. Engrg., 326, 573-595 (2017) · Zbl 1439.74299 |
| [11] | Choi, K. K.; Lee, I.; Gorsich, D., Dimension reduction method for reliability-based robust design optimization, Comput. Struct., 86, 1550-1562 (2008) |
| [12] | Lecieux, Y.; Schoefs, F.; Bonnet, S.; Lecieux, T.; Lopes, S. P., Quantification and uncertainty analysis of a structural monitoring device: detection of chloride in concrete using DC electrical resistivity measurement, Nondestruct. Test. Eval., 30, 216-232 (2015) |
| [13] | Lim, J.; Lee, B.; Lee, I., Second-order reliability method-based inverse reliability analysis using Hessian update for accurate and efficient reliability-based design optimization, Internat. J. Numer. Methods Engrg., 100, 773-792 (2015) · Zbl 1352.74229 |
| [14] | Khader, G. H.M. H.; Zhuang, X.; Alajlan, N.; Rabczuk, T., Sensitivity and uncertainty analysis for flexoelectric nanostructures, Comput. Meth. Appl. Mech. Eng., 337, 95-109 (2018) · Zbl 1440.74133 |
| [15] | Truong, V. H.; Liu, J.; Meng, X.; Jiang, C.; Nguyen, T. T., Uncertainty analysis on vehicle-bridge system with correlativeinterval variables based on multidimensional parallelepiped model, Int. J. Comput. Math., 1850030 (2017) |
| [16] | Meng, Z.; Zhou, H., New target performance approach for a super parametric convex model of non-probabilistic reliability-based design optimization, Comput. Methods Appl. Mech. Engrg., 339, 644-662 (2018) · Zbl 1440.74270 |
| [17] | Du, X.; Chen, W., A most probable point-based method for efficient uncertainty analysis, J. Design Manuf. Autom., 1, 47-65 (2001) |
| [18] | Chen, J. B.; Li, J., Strategy for selecting representative points via tangent spheres in the probability density evolution method, Internat. J. Numer. Methods Engrg., 74, 1988-2014 (2008) · Zbl 1195.74307 |
| [19] | Hasofer, A. M., Exact and invariant second-moment code format, J. Eng. Mech. Div., 100, 111-121 (1974) |
| [20] | Breitung, .; Wilhelm, K., Asymptotic Approximations for Probability Integrals (1994), Springer-Verlag · Zbl 0814.60020 |
| [21] | Rajabi, M. M.; Ataie-Ashtiani, B.; Janssen, H., Efficiency enhancement of optimized Latin hypercube sampling strategies: Application to Monte Carlo uncertainty analysis and meta-modeling, Adv. Water Resour., 76, 127-139 (2015) |
| [22] | Au, S. K.; Cao, Z. J.; Wang, Y., Implementing advanced Monte Carlo simulation under spreadsheet environment, Struct. Saf., 32, 281-292 (2010) |
| [23] | Lu, Z.; Song, S.; Yue, Z.; Wang, J., Reliability sensitivity method by line sampling, Struct. Saf., 30, 517-532 (2008) |
| [24] | Boer, P. T.D.; Nicola, V. F.; Rubinstein, R. Y., Adaptive importance sampling simulation of queueing networks, (Simulation Conference, 2000. Proceedings, Vol. 641. (2017), Winter), 646-655 |
| [25] | Esposito, R.; Mensitieri, G.; De, N. S., Improved maximum entropy method for the analysis of fluorescence spectroscopy data: evaluating zero-time shift and assessing its effect on the determination of fluorescence lifetimes, Analyst, 140, 8138-8147 (2015) |
| [26] | Slifker, J.; Shapiro, S., The Johnson system: selection and parameter estimation, Technometrics, 22, 239-246 (1980) · Zbl 0447.62020 |
| [27] | Yin, S.; Yu, D.; Yin, H.; Xia, B., A new evidence-theory-based method for response analysis of acoustic system with epistemic uncertainty by using Jacobi expansion, Comput. Meth. Appl. Mech. Eng., 322 (2017) · Zbl 1439.76148 |
| [28] | Zhang, Zhiping Xudong, ZhipingXudong## Zhang Static response analysis of structures with interval parameters using the second-order Taylor series expansion and the DCA for QB, Acta Mech. Sinica, 31, 845-854 (2015) · Zbl 1342.65145 |
| [29] | Huang, B.; Du, X., Uncertainty analysis by dimension reduction integration and saddlepoint approximations, J. Mech. Des., 128, 1143-1152 (2006) |
| [30] | Huang, X.; Zhang, Y., Reliability-sensitivity analysis using dimension reduction methods and saddlepoint approximations, Internat. J. Numer. Methods Engrg., 93, 857-886 (2013) · Zbl 1352.74091 |
| [31] | Pedeli, X.; Davison, A. C.; Fokianos, K., Likelihood estimation for the INAR(p) model by saddlepoint approximation, J. Amer. Statist. Assoc., 110, 1229-1238 (2015) · Zbl 1373.62453 |
| [32] | Acar, E.; Raisrohani, M.; Eamon, C., Reliability estimation using dimension reduction and extended generalized Lambda distribution, Int. J. Reliab. Saf., 4, 166-187 (2010) |
| [33] | Khodaparast, H. H.; Mottershead, J. E.; Friswell, M. I., Perturbation methods for the estimation of parameter variability in stochastic model updating, Mech. Syst. Signal Process., 22, 1751-1773 (2008) |
| [34] | Liu, J.; Hu, Y.; Xu, C.; Jiang, C.; Han, X., Probability assessments of identified parameters for stochastic structures using point estimation method, Reliab. Eng. Syst. Saf., 156, 51-58 (2016) |
| [35] | Blatman, G.; Sudret, B., An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis, Probab. Eng. Mech., 25, 183-197 (2010) |
| [36] | Li, G.; Zhang, K., A Combined Reliability Analysis Approach with Dimension Reduction Method and Maximum Entropy Method (2010), Springer-Verlag New York, Inc. |
| [37] | Xu, H.; Rahman, S., A generalized dimension-reduction method for multidimensional integration in stochastic mechanics, Probab. Eng. Mech., 61, 393-408 (2004) |
| [38] | Ju, B. H.; Lee, B. C., Improved moment-based quadrature rule and its application to reliability-based design optimization, J. Mech. Sci. Technol., 21, 1162-1171 (2007) |
| [39] | Rahman, S., Decomposition methods for structural reliability analysis revisited, Probab. Eng. Mech., 26, 357-363 (2011) |
| [40] | Wu, C. L.; Ma, X. P.; Fang, T., A complementary note on Gegenbauer polynomial approximation for random response problem of stochastic structure, Probab. Eng. Mech., 21, 410-419 (2006) |
| [41] | Pandey, M. D., A numerical method for structural uncertainty response computation, Sci. China Technol. Sci., 54, 3347-3357 (2011) · Zbl 1239.74095 |
| [42] | Tarantola, A., Inverse Problem Theory and Methods for Model Parameter Estimation, xii,342 (2005), Society for Industrial & Applied Mathematics Philadelphia Pa · Zbl 1074.65013 |
| [43] | Moore, E. Z.; Murphy, K. D.; Nichols, J. M., Crack identification in a freely vibrating plate using Bayesian parameter estimation, Mech. Syst. Signal Process., 25, 2125-2134 (2011) |
| [44] | Lam, H. F.; Yang, J. H.; Au, S. K., Bayesian model updating of a coupled-slab system using field test data utilizing an enhanced Markov chain Monte Carlo simulation algorithm, Eng. Struct., 102, 144-155 (2015) |
| [45] | Green, P. L., Bayesian system identification of a nonlinear dynamical system using a novel variant of Simulated Annealing, Mech. Syst. Signal Process., 52-53, 133-146 (2015) |
| [46] | Zhang W, H. X.; Liu, J., A combined sensitive matrix method and statistical approach for engineering inverse problems with insufficient and imprecise information, Comput. Mater. Continua, 26, 201-225 (2011) |
| [47] | Dobrić, J.; Schmid, F., A goodness of fit test for copulas based on Rosenblatt’s transformation, Comput. Statist. Data Anal., 51, 4633-4642 (2007) · Zbl 1162.62343 |
| [48] | Radoslav, L., Sequential Quadratic Programming (2006), Springer US |
| [49] | L. Davis, Handbook of genetic algorithms (1991).; L. Davis, Handbook of genetic algorithms (1991). |
| [50] | Dowding, K. J.; Pilch, M.; Hills, R. G., Formulation of the thermal problem, Comput Meth. Appl. Mech. Eng., 197, 2385-2389 (2011) · Zbl 1138.74020 |
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