RESEARCH PAPER
Time integration of stochastic generalized equations of motion using SSFEM
 
 
More details
Hide details
1
Czestochowa University of Technology, Department of Civil Engineering, Częstochowa, Poland
Publish date: 2019-01-20
Submission date: 2018-01-31
Acceptance date: 2018-06-17
 
Journal of Theoretical and Applied Mechanics 2019;57(1):37–48
KEYWORDS
ABSTRACT
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.
 
REFERENCES (30)
1.
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.
 
2.
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.
 
3.
Al-Nimr, M., 1997, Heat transfer mechanisms during short-duration laser heating of thin metal films, International Journal of Thermophysics, 18, 5, 1257-1268.
 
4.
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.
 
5.
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.
 
6.
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.
 
7.
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.
 
8.
Fourier J., 1822, Theorie Analytique de la Chaleur, Chez Firmin Didot, Paris.
 
9.
Ghanem R.G., Spanos P.D., 2003, Stochastic Finite Elements: A Spectral Approach, Dover Publications, Mineola, USA.
 
10.
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.
 
11.
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.
 
12.
Joseph D.D., Preziosi L., 1989, Heat waves, Reviews of Modern Physics, 61, 1, 41-73.
 
13.
Kamiński M., 2013, The Stochastic Perturbation Method for Computational Mechanics, John Wiley & Sons, Chichester.
 
14.
Le Maitre O.P., Knio O.M., 2010, Spectral Methods for Uncertainty Quantification: with Applications to Computational Fluid Dynamics, Springer Science & Business Media, Doredrecht.
 
15.
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.
 
16.
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.
 
17.
Nouy A., Le Maitre O.P., 2009, Generalized spectral decomposition for stochastic nonlinear problems, Journal of Computational Physics, 228, 1, 202-235.
 
18.
Służalec A., 2003, Thermal waves propagation in porous material undergoing thermal loading, International Journal of Heat and Mass Transfer, 46, 9, 1607-161.
 
19.
Smolyak S.A., 1963, Quadrature and interpolation formulas for tensor products of certain classes of functions, Doklady Akademii Nauk SSSR, 4, 240-243.
 
20.
Stefanou G., 2009, The stochastic finite element method: past, present and future, Computer Methods in Applied Mechanics and Engineering, 198, 9, 1031-1051.
 
21.
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.
 
22.
Straughan B., 2011, Heat Waves, Springer, New York.
 
23.
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.
 
24.
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.
 
25.
Ván P., Fülöp T., 2012, Universality in heat conduction theory: weakly nonlocal thermodynamics, Annalen der Physik, 524, 8, 470-478.
 
26.
Vernotte P., 1958, Les paradoxes de la theorie continue de l’equation de la chaleur, Comptes Rendus, 246, 3154-3155.
 
27.
Xiu D., 2010, Numerical Methods for Stochastic Computations: A Spectral Method Approach, Princeton University Press, Princeton.
 
28.
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.
 
29.
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.
 
30.
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.
 
eISSN:2543-6309
ISSN:1429-2955