Direct Generation of Non-Gaussian Weighted Integrals
Publication: Journal of Engineering Mechanics
Volume 126, Issue 1
Abstract
A simple numerical algorithm is proposed for directly generating realizations of weighted integrals (or, similarly, local averages) of non-Gaussian random fields for use in simulation-based stochastic finite-element analyses. The method uses a Gaussian quadrature integration rule to numerically evaluate an individual weighted integral (or local average), thus reducing the computation of the integral to a summation of a small number of properly weighted non-Gaussian random variables. Consequently, the need to generate actual realizations of the non-Gaussian random field is eliminated. The vector of non-Gaussian random variables is obtained from a nonlinear mapping of a vector of properly correlated Gaussian random variables, which in turn is obtained from a vector of uncorrelated Gaussian random variables using modal decomposition. The “proper” correlation structure of the Gaussian random variables is established a priori from the correlation structure of the non-Gaussian random variables, which itself is established a priori from the known or desired correlation structure of the non-Gaussian random field. Numerical results are provided to demonstrate the statistical equivalence of weighted integrals (or local averages) generated using the proposed approach to those computed using conventional numerical integration of actual realizations of the non-Gaussian random field.
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Benaroya, H., and Rehak, M. (1998). “Finite element methods in probabilistic structural analysis: A selective review.” Appl. Mech. Rev., 41(5), 201–213.
2.
Brenner, C., and Schuëller, G. (1998). “Stochastic finite elements—under the perspectives on the assumptions on material properties.” Proc., 12th ASCE Engrg. Mech. Conf., ASCE, Reston, Va.
3.
Bucher, C., and Brenner, C. (1992). “Stochastic response of uncertain systems.” Archive of Appl. Mech., 62, 507–516.
4.
Chang, C., and Yang, H. T. (1991). “Random vibration of flexible, uncertain beam element.”J. Engrg. Mech., ASCE, 117(10), 2329–2350.
5.
Cherng, R.-H., and Wen, Y. (1994). “Reliability of uncertain nonlinear trusses under random excitation. I.”J. Engrg. Mech., ASCE, 120(4), 733–747.
6.
Deodatis, G. (1990a). “Bounds on response variability of stochastic finite element systems.”J. Engrg. Mech., ASCE, 116(3), 565–585.
7.
Deodatis, G. (1990b). “Bounds on response variability of stochastic finite element systems: Effect of statistical dependence.” Probabilistic Engrg. Mech., 5(2), 88–98.
8.
Deodatis, G. (1991). “Weighted integral method. I: Stochastic stiffness matrix.”J. Engrg. Mech., ASCE, 117(8), 1851–1864.
9.
Deodatis, G., and Shinozuka, M. (1989). “Bounds on response variability of stochastic systems.”J. Engrg. Mech., ASCE, 115(11), 2543–2563.
10.
Deodatis, G., and Shinozuka, M. (1991). “Weighted integral method. II: Response variability and reliability.”J. Engrg. Mech., ASCE, 117(8), 1865–1877.
11.
Der Kiureghian, A., and Ke, J.-B. (1988). “The stochastic finite element method in structural reliability.” Probabilistic Engrg. Mech., 3(2), 83–91.
12.
Ghanem, R., and Spanos, P. (1991a). “Spectral stochastic finite-element formulation for reliability analysis.”J. Engrg. Mech., ASCE, 117(10), 2351–2372.
13.
Ghanem, R., and Spanos, P. (1991b). . Stochastic finite elements: A spectral approach. Springer, New York.
14.
Graham, L. ( 1996). “Variability response functions and the weighted integral method in stochastic finite element analysis,” PhD thesis, Princeton University, Princeton, N.J.
15.
Graham, L., and Deodatis, G. (1996a). “Variability response functions for plane elasticity problems with multiple stochastic material/geometric properties.” Probabilistic Mech. & Struct. Reliability: Proc., 7th Spec. Conf., D. Frangopol and M. Grigoriu, eds., ASCE, New York, 174–177.
16.
Graham, L., and Deodatis, G. (1996b). “Analysis of eigenvalue variability for 2d stochastic structural systems using variability response functions.” Probabilistic Mech. & Struct. Reliability: Proc., 7th Spec. Conf., D. Frangopol and M. Grigoriu, eds., ASCE, New York, 600–603.
17.
Graham, L., and Deodatis, G. (1998). “Variability response functions for stochastic plate bending problems.” Struct. Safety, 20, 167–188.
18.
Graham, L., and Deodatis, G. (1999). “Response and eigenvalue analysis of stochastic finite element systems with multiple correlated material and geometric properties.” Probabilistic Engrg. Mech., accepted for publication.
19.
Grigoriu, M. (1995). Applied non-Gaussian processes: Examples, theory, simulation, linear random vibration, and MATLAB solutions. Prentice-Hall, Englewood Cliffs, N.J.
20.
Grigoriu, M. (1998). “Simulation of stationary non-Gaussian translation processes.”J. Engrg. Mech., ASCE, 124(2), 121–126.
21.
Gurley, K., and Kareem, A. (1998). “Simulation of non-Gaussian processes.” Proc., 3rd Int. Conf. on Computational Stochastic Mech., P. Spanos, ed., Balkema, Rotterdam, The Netherlands, 11–20.
22.
Keeping, E. (1962). Introduction to statistical inference. Van Nostrand Reinhold, New York.
23.
Lawrence, M. (1987). “Basis random variables in finite element analysis.” Int. J. Numer. Methods in Engrg., 24, 1849–1863.
24.
Li, C.-C., and Der Kiureghian, A. (1993). “Optimal discretization of random fields.”J. Engrg. Mech., ASCE, 119(6), 1136–1154.
25.
Liu, W., Mani, A., and Belytschko, T. (1987). “Finite element methods in probabilistic mechanics.” Probabilistic Engrg. Mech., 2(4), 201–213.
26.
Liu, W. K., Belytschko, T., and Mani, A. (1986). “Random field finite elements.” Int. J. Numer. Methods in Engrg., 23, 1831–1845.
27.
Micaletti, R. ( 1999). “Bounds on probabilistic indicators of the response of stochastic systems with large variations of their material properties,” PhD thesis, Princeton University, Princeton, N.J.
28.
Press, W., Vetterling, W., Teukolsky, S., and Flannery, B. (1992). NUMERICAL RECIPES in FORTRAN the art of scientific computing, 2nd Ed., Cambridge University Press, New York.
29.
Shinozuka, M., and Deodatis, G. (1988). “Response variability of stochastic finite element systems.”J. Engrg. Mech., ASCE, 114(3), 499–519.
30.
Spanos, P., and Ghanem, R. (1989). “Stochastic finite element expansion for random media.”J. Engrg. Mech., ASCE, 115(5), 1035–1053.
31.
Takada, T. (1990). “Weighted integral method in stochastic finite element analysis.” Probabilistic Engrg. Mech., 5(3), 143–156.
32.
Vanmarcke, E. (1983). Random fields: Analysis and synthesis. MIT Press, Boston.
33.
Vanmarcke, E., and Grigoriu, M. (1983). “Stochastic finite element analysis of simple beams.”J. Engrg. Mech., ASCE, 109(5), 1203–1214.
34.
Vanmarcke, E., Shinozuka, M., Nakagiri, S., Schuëller, G., and Grigoriu, M. (1986). “Random fields and stochastic finite elements.” Struct. Safety, 3, 143–166.
35.
Wall, F. (1994). “Probabilistic response of stochastic 2d-systems under static loading.” Tech. Rep., Dept. of Civ. Engrg. and Operations Res., Princeton University, Princeton, N.J.
36.
Wall, F., and Deodatis, G. (1994). “Variability response functions of stochastic plane stress/strain problems.”J. Engrg. Mech., ASCE, 120(9), 1963–1982.
37.
Yamazaki, F. (1987). “Simulation of stochastic fields and its application to finite element analysis.” Tech. Rep. ORI 87-04, Ohsaki Research Institute, Inc., Japan.
38.
Yamazaki, F., and Shinozuka, M. (1988). “Digital simulation of non-Gaussian stochastic fields.”J. Engrg. Mech., ASCE, 114(7), 1183–1197.
39.
Yamazaki, F., and Shinozuka, M. (1990). “Simulation of stochastic fields by statistical preconditioning.”J. Engrg. Mech., ASCE, 116(2), 268–287.
40.
Zeldin, B., and Spanos, P. (1998). “On random field discretization in stochastic finite elements.” J. Appl. Mech., 65(2), 320–327.
41.
Zhang, Y., and Der Kiureghian, A. (1997). “Finite element reliability methods for inelastic structures.” Tech. Rep. to National Science Foundation, Dept. of Civ. and Envir. Engrg., University of California, Berkeley, Berkeley, Calif.
42.
Zhang, J, and Ellingwood, B. (1994). “Orthogonal series expansions of random fields in reliability analysis.”J. Engrg. Mech., ASCE, 120(12), 2660–2677.
43.
Zhu, W., Ren, Y., and Wu, W. (1992). “Stochastic fem based on local averages of random vector fields.”J. Engrg. Mech., ASCE, 118(3), 496–511.
Information & Authors
Information
Published In
History
Received: Feb 9, 1999
Published online: Jan 1, 2000
Published in print: Jan 2000
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.