Stochastic Finite Element Analysis using Polynomial Chaos

Open access

Abstract

This paper presents a procedure of conducting Stochastic Finite Element Analysis using Polynomial Chaos. It eliminates the need for a large number of Monte Carlo simulations thus reducing computational time and making stochastic analysis of practical problems feasible. This is achieved by polynomial chaos expansion of the displacement field. An example of a plane-strain strip load on a semi-infinite elastic foundation is presented and results of settlement are compared to those obtained from Random Finite Element Analysis. A close matching of the two is observed.

[1] Davies R., Harte D., Tests for Hurst effect, Biometrika, 1987, 74(4), 95–101.

[2] Dembo A., Mallows C., Shepp L., Embedding nonnegative definite Toeplitz matrices in nonnegative definite circulant matrices, with applications to covariance estimation, IEEE Transactions on Information Theory, 1989, 35, 1206–1212.

[3] Dietrich C., Newsam G., A fast and exact simulation for multidimensional Gaussian stochastic simulations, Water Resources Research, 1993, 29(8), 2861–2869.

[4] Dietrich C., Newsam G., Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix, SIAM Journal on Scientific Computing, 1997, 18(4), 1088–1107.

[5] Gneiting T., Power-law correlations, related models for long-range dependence and their simulation, Journal of Applied Probability, 2000, 37(4), 1104–1109.

[6] Gneiting T., Sevčíková H., Percival D., Schlather M., Jiang Y., Fast and exact simulation of large Gaussian lattice systems in R2: Exploring the limits, Journal of Computational and Graphical Statistics, 2006, 15(3), 483–501.

[7] Stein M., Local stationarity and simulation of self-affine intrinsic random functions, IEEE Transactions on Information Theory, 2001, 47(4), 1385–1390.

[8] Stein M., Fast and exact simulation of fractional Brownian surfaces, Journal of Computational and Graphical Statistics, 2002, 11(3), 587–599.

[9] Stein M., Simulation of Gaussian random fields with one derivative, Journal of Computational and Graphical Statistics, 2012, 21(1), 155–173.

[10] Wood A., Chan G., Simulation of stationary Gaussian processes in [0, 1]d, Journal of Computational and Graphical Statistics, 1994, 3(4), 409–432.

[11] Paice G.M., Griffiths D.V., Fenton G.A., Finite element modeling of settlements on spatially random soil, J. Geotech. Eng., 1996, 122(9), 777–779. Smith, I. M., and Griffiths,

[12] Fenton G.A., Griffiths D.V., Statistics of block conductivity through a simple bounded stochastic medium, Water Resour. Res., 1993, 29(6), 1825–1830.

[13] Fenton G.A., Griffiths D.V., Probabilistic foundation settlement on spatially random soil, J. Geotech. Geoenviron. Eng., 2002, 128(5), 381–390.

[14] Fenton G.A., Griffiths D.V., Three-dimensional probabilistic foundation settlement, J. Geotech. Geoenviron. Eng., 2005, 131(2), 232–239.

[15] Fenton G.A., Griffiths D.V., Risk assessment in geotechnical engineering, Wiley, Hoboken, N.J. 2008.

[16] Fenton G.A., Vanmarcke E.H., Simulation of random fields via local average subdivision, J. Eng. Mech., 1990, 116(8), 1733–1749.

[17] Ghanem R.G., Spanos P.D., Stochastic finite elements: A spectral approach, Springer-Verlag, New York 1991.

[18] Griffiths D.V., Fenton G.A., Seepage beneath water retaining structures founded on spatially random soil, Geotechnique, 1993, 43(4), 577–587.

[19] Xiu D, Karniadakis G.E., Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos, Computer Methods in Applied Mechanics and Engineering, 2003, 191 (43), 4927–4948

[20] Gray R.M., Toeplitz and Circulant Matrices: A review, Department of electrical Engineering Stanford University, 2006.

Studia Geotechnica et Mechanica

The Journal of Wrocław University of Science and Technology and AGH University of Science and Technology

Journal Information

CiteScore 2017: 0.14

SCImago Journal Rank (SJR) 2017: 0.131
Source Normalized Impact per Paper (SNIP) 2017: 0.448

Cited By

Metrics

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 155 155 16
PDF Downloads 47 47 5