Journal of Theoretical
and Applied Mechanics

57, 1, pp. 37-48, Warsaw 2019
DOI: 10.15632/jtam-pl.57.1.37

Time integration of stochastic generalized equations of motion using SSFEM

Mariusz Poński
The paper develops an integration approach to stochastic nonlinear partial differential equations
(SPDE’s) with parameters to be random fields. The methodology is based upon
assumption that random fields are from a special class of functions, and can be described
as a product of two functions with dependent and independent random variables. Such an
approach allows one to use Karhunen-Lo`eve expansion directly, and the modified stochastic
spectral finite element method (SSFEM). It is assumed that a random field is stationary
and Gaussian while the autocovariance function is known. A numerical example of onedimensional
heat waves analysis is shown.
Keywords: spectral stochastic finite element method, time integration, heat waves


Acharjee S., Zabaras N., 2006, Uncertainty propagation in finite deformations – a spectral

stochastic Lagrangian approach, Computer Methods in Applied Mechanics and Engineering, 195, 19, 2289-2312

Acharjee S., Zabaras N., 2007, A non-intrusive stochastic Galerkin approach for modeling

uncertainty propagation in deformation processes, Computational Stochastic Mechanics, 85, 5, 244-254

Al-Nimr, M., 1997, Heat transfer mechanisms during short-duration laser heating of thin metal

films, International Journal of Thermophysics, 18, 5, 1257-1268

Arregui-Mena J.D., Margetts L., Mummery P.M., 2016, Practical application of the stochastic

finite element method, Archives of Computational Methods in Engineering, 23, 1, 171-190

Babuška I., Nobile F., Tempone R., 2007, A stochastic collocation method for elliptic partial

differential equations with random input data, SIAM Journal on Numerical Analysis, 45, 3, 1005-1034

Bargmann S., Favata A., 2014, Continuum mechanical modeling of laser-pulsed heating in polycrystals:

a multi-physics problem of coupling diffusion, mechanics, and thermal waves, ZAMM –

Journal of Applied Mathematics and Mechanics/Zeitschrift f¨ur Angewandte Mathematik und Mechanik, 94, 6, 487-498

Bathe K.J., 1996, Finite Element Procedures, Prentice Hall, New Jersey

Cattaneo M., 1948, Sulla Conduzione de Calor, Mathematics and Physics Seminar, 3, 3, 83-101

Fourier J., 1822, Theorie Analytique de la Chaleur, Chez Firmin Didot, Paris

Ghanem R.G., Spanos P.D., 2003, Stochastic Finite Elements: A Spectral Approach, Dover

Publications, Mineola, USA

Ghosh D., Avery P., Farhat C., 2008, A method to solve spectral stochastic finite element

problems for large-scale systems, International Journal for Numerical Methods in Engineering, 00, 1-6

Hu J., Jin S., Xiu„ D., 2015, A stochastic Galerkin method for Hamilton-Jacobi equations with

uncertainty, SIAM Journal on Scientific Computing, 37, 5, A2246-A2269

Joseph D.D., Preziosi L., 1989, Heat waves, Reviews of Modern Physics, 61, 1, 41-73

Kamiński M., 2013, The Stochastic Perturbation Method for Computational Mechanics, John Wiley

& Sons, Chichester

Le Maitre O.P., Knio O.M., 2010, Spectral Methods for Uncertainty Quantification: with Applications

to Computational Fluid Dynamics, Springer Science & Business Media, Doredrecht

Matthies H.G., Keese A., 2005, Galerkin methods for linear and nonlinear elliptic stochastic

partial differential equations, Computer Methods in Applied Mechanics and Engineering, 194, 2, 1295-1331

Nouy A., 2008, Generalized spectral decomposition method for solving stochastic finite element

equations: invariant subspace problem and dedicated algorithms, Computer Methods in Applied

Mechanics and Engineering, 197, 51, 4718-4736

Nouy A., Le Maitre O.P., 2009, Generalized spectral decomposition for stochastic nonlinear

problems, Journal of Computational Physics, 228, 1, 202-235

Służalec A., 2003, Thermal waves propagation in porous material undergoing thermal loading,

International Journal of Heat and Mass Transfer, 46, 9, 1607-161

Smolyak S.A., 1963, Quadrature and interpolation formulas for tensor products of certain classes

of functions, Doklady Akademii Nauk SSSR, 4, 240-243

Stefanou G., 2009, The stochastic finite element method: past, present and future, Computer

Methods in Applied Mechanics and Engineering, 198, 9, 1031-1051

Stefanou G., Savvas D., Papadrakakis M., 2017, 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

Straughan B., 2011, Heat Waves, Springer, New York

Subber W., Sarkar A., 2014, A domain decomposition method of stochastic PDEs: An iterative

solution techniques using a two-level scalable preconditioner, Journal of Computational Physics, 257, 298-317

Tamma K.K., Zhou X., 1998, Macroscale and microscale thermal transport and thermo-

-mechanical interactions: some noteworthy perspectives, Journal of Thermal Stresses, 21, 3-4, 405-449

Ván P., Fülöp T., 2012, Universality in heat conduction theory: weakly nonlocal thermodynamics,

Annalen der Physik, 524, 8, 470-478

Vernotte P., 1958, Les paradoxes de la theorie continue de l’equation de la chaleur, Comptes

Rendus, 246, 3154-3155

Xiu D., 2010, Numerical Methods for Stochastic Computations: A Spectral Method Approach, Princeton

University Press, Princeton

Xiu D., Hesthaven J.S., 2005, High-order collocation methods for differential equations with

random inputs, SIAM Journal on Scientific Computing, 27, 3, 1118-1139

Xiu D., Karniadakis G.E., 2003, A new stochastic approach to transient heat conduction modeling

with uncertainty, International Journal of Heat and Mass Transfer, 46, 24, 4681-4693

Zakian P., Khaji N., 2016, A novel stochastic-spectral finite element method for analysis of

elastodynamic problems in the time domain, Meccanica, 51, 4, 893-920