High-Order Spectral Stochastic Finite Element Analysis of Stochastic Elliptical Partial Differential Equations
Abstract: This study presents an experiment of improving the performance of spectral stochastic finite element method using high-order elements. This experiment is implemented through a two-dimensional spectral stochastic finite element formulation of an elliptic partial differential equation having stochastic coefficients. Deriving this spectral stochastic finite element formulation couples a two-dimensional deterministic finite element formulation of an elliptic partial differential equation with generalized polynomial chaos expansions of stochastic coefficients. Further inspection of the performance of resulting spectral stochastic finite element formulation with adopting linear and quadratic (9-node or 8-node) quadrilateral elements finds that more accurate standard deviations of unknowns are surprisingly predicted using quadratic quadrilateral elements, especially under high autocorrelation function values of stochastic coefficients. In addition, creating spectral stochastic finite element results using quadratic quadrilateral elements is not unacceptably time-consuming. Therefore, this study concludes that adopting high-order elements can be a lower-cost method to improve the performance of spectral stochastic finite element method.
Cite this paper: G. Sheu, "High-Order Spectral Stochastic Finite Element Analysis of Stochastic Elliptical Partial Differential Equations," Applied Mathematics, Vol. 4 No. 5, 2013, pp. 18-28. doi: 10.4236/am.2013.45A003.
References

[1]   R. G. Ghanem and P. D. Spanos, “Stochastic Finite Ele ments: A Spectral Approach,” Springer-Verlag, New York, 2012.

[2]   D. Xiu and G. E. Karniadakis, “Modeling Uncertainty in Flow Simulations via Generalized Polynomial Chaos,” Journal of Computational Physics, Vol. 187, No. 1, 2003, pp. 137-167. doi:10.1016/S0021-9991(03)00092-5

[3]   M. Loève, “Probability Theory II (Graduate Text in Mathematics),” Springer-Verlag, Berlin, 1978.

[4]   S. Q. Wu and S. S. Law, “Evaluating the Response Statistics of an Uncertain Bridge-vehicle System,” Mechanical Systems and Signal Processing, Vol. 27, 2012, pp. 576-589. doi:10.1016/j.ymssp.2011.07.019

[5]   V. Papadopoulos and O. Kokkinos, “Variability Response Functions for Stochastic Systems,” Probabilistic Engineering Mechanics, Vol. 28, 2012, pp. 176-184. doi:10.1016/j.probengmech.2011.08.002

[6]   G. Y. Sheu, “Prediction of Probabilistic Settlements via Spectral Stochastic Meshless Local Petrov-Galerkin Method,” Computers and Geotechnics, Vol. 38, No. 4, 2011, pp. 407-415. doi:10.1016/j.compgeo.2011.02.001

[7]   M. F. Ngah and A. Young “Application of the Spectral Stochastic Finite Element Method for Performance Pre diction of Composite Structures,” Composite Structures, Vol. 78, No. 3, 2007, pp. 447-456. doi:10.1016/j.compstruct.2005.11.009

[8]   J. N. Reddy, “An Introduction to the Finite Element Method,” 2nd Edition, McGraw-Hill Company, New York, 1993.

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