Abstract
In this paper we compare an intrusive with an non-intrusive method for computing Polynomial Chaos expansions. The main disadvantage of the nonintrusive method, the high number of function evaluations, is eliminated by a special Adaptive Gauss-Quadrature method. A detailed efficiency and accuracy analysis is performed for the new algorithm. The Polynomial Chaos expansion is applied to a practical problem in the field of stochastic Finite Elements.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
W. Loh: On Latin hypercube sampling, Ann. Stat. 24 2058–2080 (1996).
B. L. Fox: Strategies for Quasi-Monte Carlo, Springer, (1999).
N. N. Madras: Lectures on Monte Carlo Methods, American Mathematical Society, (2002).
M. Rajaschekhar, B. Ellingwood: A new look at the response surface approach for reliability analysis, Struc. Safety 123 205–220 (1993).
R. G. Ghanem, P. D. Spanos: Stochastic Finite Elements. A Spectral Approach., Springer, Heidelberg, (1991).
D. Xiu, G. E. Karniadakis: The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations, SIAM Journal on Scientific Computing 24(2) 619–644 (2003).
G. Karniadakis, C.-H. Su, D. Xiu, D. Lucor, C. Schwab, R. Todor: Generalized Polynomial Chaos Solution for Differential Equations with Random Inputs, ETH Zürich, Seminar für Angewandte Mathematik Research Report No. 01 (2005).
M. Schevenels, G. Lombaert, G. Degrande: Application of the stochastic finite element method for Gaussian and non-Gaussian systems, in: ISMA, Katholieke Universiteit Leuven, Belgium, (2004).
M. Jardak, C.-H. Su, G. E. Karniadakis: Spectral Polynomial Chaos Solutions of the Stochastic Advection Equation, Journal of Scientific Computing 17 319–338 (2002).
O. P. L. Maître, O. M. Knio, H. N. Najm, R. G. Ghanem: A Stochastic Projection Method for Fluid Flow I. Basic Formulation, Journal of Computational Physics 173 481–511 (2001).
D. Lucor, G. E. Karniadakis: Noisy Inflows Cause a Shedding-Mode Switching in Flow Past an Oscillating Cylinder, Physical Review Letters 92 154501 (2004).
N. Wiener: The Homogeneous Chaos, Amer. J. Math. 60 897–936 (1938).
R. Cameron, W. Martin: The orthogonal development of nonlinear functionals in series of Fourier-Hermite functionals, Ann. Math. 48 385 (1947).
M. Herzog: Zur Numerik von stochastischen Störungen in der Finiten Elemente Methode, Master’s thesis, TU München (2005).
D. Xiu, G. E. Karniadakis: Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos, Comput. Methods Appl. Mech. Engrg. 191 4927–4948 (2002).
R. Ghanem, S. Masri, M. Pellissetti, R. Wolfe: Identification and prediction of stochastic dynamical systems in a polynomial chaos basis, Comput. Methods Appl. Mech. Engrg. 194 1641–1654 (2005).
M. Paffrath, U. Wever: The Probabilistic Optimizer RODEO, Internal Report Siemens AG (2005).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Herzog, M., Gilg, A., Paffrath, M., Rentrop, P., Wever, U. (2008). Intrusive versus Non-Intrusive Methods for Stochastic Finite Elements. In: Breitner, M.H., Denk, G., Rentrop, P. (eds) From Nano to Space. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74238-8_13
Download citation
DOI: https://doi.org/10.1007/978-3-540-74238-8_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-74237-1
Online ISBN: 978-3-540-74238-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)