Numerical simulation of random fields using correlated random vector and the Karhunen-Loève expansion

  • Mariusz Poński Czestochowa University of Technology
  • Iwona Pokorska Czestochowa University of Technology

Abstract

This paper presents an approach of one- and two-dimensional random field simulation methods using a correlated random vector and the Karhunen–Loève expansion. Comparison of the authors’ analytical solution of the Fredholm integral equation of the second kind with the numerical solution using the finite element method and the inverse vector iteration technique is presented. Numerical approach and sample realizations of one- and two-dimensional random fields are presented using described techniques as well as generated probability distribution functions for chosen point of the analysed domain.

Keywords

eigenproblem, random field, Lalescu-Picard equation, Karhunen–Loève expansion,

References

[1] K.J. Bathe. Finite Element Procedures. Prentice Hall, New Jersey, 1996.
[2] H. Cho, D. Venturi, G.E. Karniadakis. Karhunen–Loève expansion for multi-correlated stochastic processes. Probabilistic Engineering Mechanics, 34: 157–167, 2013.
[3] R.V. Field. Stochastic Models: Theory and Simulation. Sandia National Laboratories, Raport SAND2008-1365, 2008.
[4] R.G. Ghanem, P.D. Spanos. Stochastic Finite Elements: A Spectral Approach. Dover Publications, Mineola, USA, 2003.
[5] O.P. Le Maˆıtre, O.M. Knio. Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics. Springer Science & Business Media, Dordrecht, 2010.
[6] H.G. Matthies, A. Keese. Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. Computer Methods in Applied Mechanics and Engineering, 194: 1295–1331, 2005.
[7] J. Przewłócki. Problems of Stochastic Soil Mechanics. Reliability Analysis [in Polish: Problemy Stochastycznej Mechaniki Gruntów. Ocena Niezawodnosci]. Dolnoslaskie Wydawnictwo Edukacyjne, Wrocław, Poland, 2006.
[8] G. Stefanou, D. Savvas, M. Papadrakakis. Stochastic finite element analysis of composite structures based on mesoscale random fields of material properties. Computer Methods in Applied Mechanics and Engineering, 326: 319–337, 2017.
[9] E. Vanmarcke. Random Fields: Analysis and Synthesis. World Scientific, 2010.
[10] D. Xiu, J.S. Hesthaven. High-order collocation methods for differential equations with random inputs. SIAM Journal on Scientific Computing, 27: 1118–1139, 2005.
[11] D. Xiu, G.E. Karniadakis. A new stochastic approach to transient heat conduction modeling with uncertainty. International Journal of Heat and Mass Transfer, 46: 4681–4693, 2003.
[12] D. Xiu. Numerical Methods for Stochastic Computations: A Spectral Method Approach. Princeton University Press, Princeton, 2010.
[13] P. Zakian, N. Khaji. A novel stochastic-spectral finite element method for analysis of elastodynamic problems in the time domain. Meccanica, 51: 893–920, 2016.
Published
Feb 17, 2019
How to Cite
POŃSKI, Mariusz; POKORSKA, Iwona. Numerical simulation of random fields using correlated random vector and the Karhunen-Loève expansion. Computer Assisted Methods in Engineering and Science, [S.l.], v. 25, n. 1, p. 47-58, feb. 2019. ISSN 2299-3649. Available at: <https://cames.ippt.pan.pl/index.php/cames/article/view/237>. Date accessed: 26 jan. 2022. doi: http://dx.doi.org/10.24423/cames.237.