Abstract
The probability density evolution method (PDEM) is a new approach for stochastic dynamics whereby the dynamic response and reliability evaluation of multi-degree-of-freedom nonlinear systems could be carried out. The apparent similarity and subtle distinction between the ordinary cubature and PDEM are explored with the aid of the concept of the rank of an integral. It is demonstrated that the ordinary cubature are rank-1 integrals, whereas an rank-∞ integral is involved in PDEM. This interprets the puzzling phenomenon that some cubature formulae doing well in ordinary high-dimensional integration may fail in PDEM. A criterion that the stability index does not exceed unity is then put forward. This distinguishes the cubature formulae by their applicability to higher-rank integrals and the adaptability to PDEM. Several kinds of cubature formulae are discussed and tested based on the criterion. The analysis is verified by numerical examples, demonstrating that some strategies, e.g. the quasi-symmetric point method, are preferred in different scenarios. Problems to be further studied are pointed out.
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References
Au SK, Beck JL (2001) First excursion probabilities for linear systems by very efficient importance sampling. Probab Eng Mech 16: 193–207
Berezin I, Zhidkov N (1965) Computing Methods. Addison-Wesley, Reading, MA
Chen JB, Ghanem R, Li J (2009) Partition of the probability-assigned space in probability density evolution analysis of nonlinear stochastic structures. Probab Eng Mech 24(1): 27–42
Chen JB, Li J (2011) Stochastic harmonic function and spectral representations. Chinese J Theor Appl Mech 43(3):505–513 (in Chinese)
Cools R (1999) Monomial cubature rules since “Stroud”: a comoilation-part 2. J Comput Appl Math 112(1-2): 21–27
Cools R (2003) An encyclopedia of cubature formulas. J Complex 19: 445–453
Dostupov BG, Pugachev VS (1957) The equation for the integral of a system of ordinary differential equations containing random parameters. Automatika i Telemekhanika 18: 620–630
Engels H (1980) Numerical Quadrature and Cubature. Academic Press, New York
Genz A (1986) Fully symmetric interpolatory rules for multiple integrals. SIAM J Numer Anal 23: 1273–1283
Genz A, Keister B (1996) Fully symmetric interpolatory rules for multiple integrals over infinite regions with Gaussian weight. J Comput Appl Math 71: 299–309
Ghanem R, Spanos PD (1990) Polynomial chaos in stochastic finite elements. J Appl Mech 57: 197–202
Hua LK, Wang Y (1981) Applications of number theory to numerical analysis. Springer, Heidelberg Berlin New York
Isaacson E, Keller H (1966) Analysis of Numerical Methods. John Wiley, New York
Kleiber M, Hien TD (1992) The stochastic finite element method. Chishcester, Wiley, New York
Korn GA, Korn TM (1968) Mathematical Handbook for Scientists and Engineers. McGraw-Hill, Inc., New York
Kozin F (1961) On the probability densities of the output of some random systems. J Appl Mech 28(2): 161–164
Li J (1996) Stochastic Structural Systems: Analysis and Modeling. Science Press, Beijing (in Chinese)
Li J, Chen JB (2004) Probability density evolution method for dynamic response analysis of structures with uncertain parameters. Comput Mech 34: 400–409
Li J, Chen JB (2007) The number theoretical method in response analysis of nonlinear stochastic structures. Comput Mech 39(6): 693–708
Li J, Chen JB (2007) The extreme value distribution and dynamic reliability analysis of nonlinear structures with uncertain parameters. Struc Saf 29: 77–93
Li J, Chen JB (2008) The principle of preservation of probability and the generalized density evolution equation. Struc Saf 30: 65–77
Li J, Chen JB (2009) Stochastic Dynamics of Structures. Science Press, John Wiley & Sons
Li J, Chen J, Sun W, Peng Y (2011) Advances of probability density evolution method for nonlinear stochastic systems. Probabilistic Engineering Mechanics, in press doi:10.1016/j.probengmech.2011.08.019
Li J, Yan Q, Chen JB (2012) Stochastic modeling of engineering dynamic excitations for stochastic dynamics of structures. Probabilistic Engineering Mechanics, 27(1):19–28
Lin JH, Zhang YH (2004) Pseudo-excitation Method in Random Vibration (in Chinese). Science Press, Beijing
Lin YK, Cai GQ (1995) Probabilistic Structural Dynamics: Advanced Theory and Applications. McGraw-Hill, Inc., New York
Loève M. (1977) Probability Theory. Springer-Verlag, Berlin
Lu J, Darmofal D (2004) Higher-dimensional integration with Gaussian weight for applications in probabilistic design. Soc Ind Appl Math 26: 613–624
Lutes LD, Sarkani S (2004) Random Vibrations: Analysis of Structural and Mechanical Systems. Elsevier, Amsterdam
Ma F, Zhang H, Bockstedte A, Foliente GC, Paevere P (2004) Parameter analysis of the differential model of hysteresis. J Appl Mech 71: 342–349
Novak E, Ritter K. (1996) High dimensional integration of smooth functions over cubes. Numer Math 75: 79–97
Novak E, Ritter K (1999) Simple cubature formulas with high polynomial exactness. Constr Approx 15: 499–522
Shinozuka M, Deodatis G (1991) Simulation of stochastic processes by spectral representation. Appl Mech Rev 44(4): 191–203
Shinozuka M, Jan CM (1972) Digital simulation of random process and its applications. J Sound Vib 25(1): 111–128
Smolyak S (1963) Quadrature and interpolation formulas for tensor products of certain classes of functions. Soviet Math Dokl 4: 240–243
Stroud H (1957) Remarks on the disposition of points in numerical integration formula. Math Comput 11: 257–261
Stroud H (1967) Some fifth degree integration formulas for symmetric regions II. Numerische Mathematik 9: 60–468
Victoir V (2004) Asymmetric cubature formulae with few points in high dimension for symmetric measures. SIAM J Numer 42(1): 209–227
Wei DL, Cui ZS, Chen J (2008) Uncertainty quantification using polynomial chaos expansion with points of monomial cubature rules. Comput Struc 86: 2102–2108
Wen YK (1976) Method for random vibration of hysteretic systems. J Eng Mech 102(2): 249–263
Xiu DB, Hesthaven JS (2005) High-order collocation methods for differential equations with random inputs. SIAM J Sci Comput 27(3): 1118–1139
Xiu DB (2007) Efficient collocational approach for parametric uncertainty analysis. Commun Comput Phys 2(2): 293–309
Xiu DB (2009) Review article: fast numerical methods for stochastic computations. Commun Comput Phys 5: 242–272
Zhu WQ (2006) Nonlinear stochastic dynamics and control in Hamiltonian formulation. Appl Mech Rev 59: 230–248
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Xu, J., Chen, J. & Li, J. Probability density evolution analysis of engineering structures via cubature points. Comput Mech 50, 135–156 (2012). https://doi.org/10.1007/s00466-011-0678-2
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DOI: https://doi.org/10.1007/s00466-011-0678-2