Abstract
This paper presents a novel stochastic collocation method based on the equivalent weak form of multivariate function integral to quantify and manage uncertainties in complex mechanical systems. The proposed method, which combines the advantages of the response surface method and the traditional stochastic collocation method, only sets integral points at the guide lines of the response surface. The statistics, in an engineering problem with many uncertain parameters, are then transformed into a linear combination of simple functions’ statistics. Furthermore, the issue of determining a simple method to solve the weight-factor sets is discussed in detail. The weight-factor sets of two commonly used probabilistic distribution types are given in table form. Studies on the computational accuracy and efforts show that a good balance in computer capacity is achieved at present. It should be noted that it’s a non-gradient and non-intrusive algorithm with strong portability. For the sake of validating the procedure, three numerical examples concerning a mathematical function with analytical expression, structural design of a straight wing, and flutter analysis of a composite wing are used to show the effectiveness of the guided stochastic collocation method.
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References
Jonas C, Sven OH, Fred N. Generalizing the safety factor approach. Reliab Eng Syst Safety, 2006, 91(8): 964–973
Gerhart IS, Adriano C, Sebastiaan H J A F, et al. Uncertainty analysis of a large-scale satellite finite element model. J Spacecraft Rockets, 2009, 46(1): 191–202
Stéphane T, Zheng W X, Marion G, et al. Unifying some higher-order statistic-based methods for errors-in-variables model identification. Automatica, 2009, 45(8): 1937–1942
M’hamed B, Driss A. Higher-order statistics based blind estimation of non-Gaussian bidimensional moving average models. Signal Processing, 2006, 86(10): 3031–3042
Yu S N, Chen Y H. Noise-tolerant electrocardiogram beat classification based on higher order statistics of subband components. Artif Intell Med, 2009, 46(2): 165–178
Cho H K, Rhee J. Vibration in a satellite structure with a laminate composite hybrid sandwich panel. Composite Struct, 2011, 93(10): 2566–2574
Hong S K, Bogdan I E, Matthew P C, et al. Parametric reduced-order models for predicting the vibration response of complex structures with component damage and uncertainties. J Sound Vib, 2011, 330(6): 1091–1110
Qiu Z Q, Xia Y Y, Yang J L. The static displacement and the stress analysis of structures with bounded uncertainties using the vertex solution theorem. Comput Methods Appl Mech Eng, 2007, 196(49–52): 4965–4984
Knio O M, Maitre O P L. Uncertainty propagation in CFD using polynomial chaos decomposition. Fluid Dyn Res, 2006, 38(9): 616–640
Loeven G J A, Witteveen J A S, Bijl H. Efficient uncertainty quantification in computational fluid-structure interactions. AIAA Paper, 2006, AIAA-2006-1634
Wang X J, Qiu Z P. Nonprobabilistic interval reliability analysis of wing flutter. AIAA J, 2009, 47(3): 743–748
Rippel M, Choi S K. Alternatives to Taylor series approximation for the variance estimation in robust design. AIAA Paper, 2010, AIAA-2010-9083
Qi W C, Qiu Z P. A collocation interval analysis method for interval structural parameters and stochastic excitation. Sci China-Phys Mech Astron, 2012, 55(1): 66–77
Christos E P, Hoi Y. Uncertainty estimation and Monte Carlo simulation method. Flow Meas Instrum, 2001, 12(4): 291–298
Herrador M A, Gonzalez A G. Evaluation of measurement uncertainty in analytical assays by means of Monte-Carlo simulation. Talanta, 2004, 64(2): 415–422
Kamran S, Pooneh N. A new optimized uncertainty evaluation applied to the Monte-Carlo simulation in platinum resistance thermometer calibration. Measurement, 2010, 43(7): 901–911
Kaveh M, Jay R L. A Monte-Carlo game theoretic approach for multi-criteria decision making under uncertainty. Adv Water Resour, 2011, 34(5): 607–616
Abhishek H, Raktim B. Beyond Monte Carlo: A computational framework for uncertainty propagation in planetary entry, descent and landing. AIAA Paper, 2010, AIAA-2010-8029
Lu Z M, Zhang D X. On importance sampling Monte Carlo approach to uncertainty analysis for flow and transport in porous media. Adv Water Resour, 2003, 26(11): 1177–1188
Marco B. Adaptive importance sampling for simulating copula-based distributions. Insurance Math Econom, 2011, 48(2): 237–245
Hans Janssen. Monte-Carlo based uncertainty analysis: Sampling efficiency and sampling convergence. Reliab Eng Syst Safety, 2013, 109: 123–132
lmpollonia N, Sofi A. A response surface approach for the static analysis of stochastic structures with geometrical nonlinearities. Comput Methods Appl Mech Eng, 2003, 192(37–38): 4109–4129
Li D Q, Chen Y F, Lu W B, et al. Stochastic response surface method for reliability analysis of rock slopes involving correlated non-normal variables. Comput Geotech, 2011, 38(1): 58–68
Liu Y. Application of stochastic response surface method in the structural reliability. Procedia Eng, 2012, 28(2012): 661–664
Xiu D B, Karniadakis G. The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J Sci Comput, 2002, 24(2): 619–644
Knio O M, Maitre O P. Uncertainty propagation in CFD using polynominal chaos decomposition. Fluid Dyn Res, 2006, 38(3): 616–640
Field RV, Grigoriu M. On the accuracy of the polynomial chaos approximation. Probab Eng Mech, 2004, 19(1–2): 65–80
Marc G, Jan-Bart S, Peter V, et al. Time-dependent generalized polynomial chaos. J Comput Phys, 2010, 229(22): 8333–8363
Eduardo H S, Floriane A C, Michel B. Sensitivity study of dynamic systems using polynomial chaos. Reliab Eng Syst Safety, 2012, 104(2012): 15–26
Xiu D B, Hesthaven J S. High order collocation methods for the differential equations with random inputs. SIAM J Sci Comput, 2005, 27(3): 1118–1139
Lionel M, Hussaini M Y. A Stochastic Collocation Algorithm for Uncertainty Analysis. NASA/CR-2003-212153, 2003
Tang T, Zhou T. Convergence analysis for stochastic collocation methods to scalar hyperbolic equations with a random wave speed. Commun Comput Phys, 2010, 8: 226–248
Motamed M, Nobile F, Tempone R. A stochastic collocation method for the second order wave equation with a discontinuous random speed. Numer Math, 2013, 123(3): 493–536
Marzouk Y, Xiu D. A stochastic collocation approach to bayesian inference in inverse problems. Commun Comput Phys, 2009, 6(4): 826–847
Aravind A, Aluru N R. Uncertainty quantification of MEMS using a data-dependent adaptive stochastic collocation method. Comput Methods Appl Mech Eng, 2011, 200(45–46): 3169–3182
Rahman S, Xu H. A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics. Probab Eng Mech, 2004, 19(4): 393–408
Wei D, Rahman S. A multi-point univariate decomposition method for structural reliability analysis. Int J Pressure Vessels Piping, 2010, 87(5): 220–229
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Qi, W., Tian, S. & Qiu, Z. A novel stochastic collocation method for uncertainty propagation in complex mechanical systems. Sci. China Phys. Mech. Astron. 58, 1–8 (2015). https://doi.org/10.1007/s11433-014-5525-y
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DOI: https://doi.org/10.1007/s11433-014-5525-y