Research Papers: Combustion and Reactive Flows

Galerkin Solution of Stochastic Reaction-Diffusion Problems

[+] Author and Article Information
C. R. Ávila da Silva, Jr.

Federal Technological University of Paraná,
Department of Mechanical Engineering,
Av. Sete de Setembro 3165,
80230-901, Curitiba, PR, Brazil
e-mail: avila@utfpr.edu.br

André Teófilo Beck

Department of Structural Engineering,
University of São Paulo,
Av. Trabalhador Sancarlense, 400,
13566-590, São Carlos, SP, Brazil
e-mail: atbeck@sc.usp.br

Admilson T. Franco

Federal Technological University of Paraná,
Department of Mechanical Engineering,
Av. Sete de Setembro 3165,
80230-901, Curitiba, PR, Brazil
e-mail: admilson@utfpr.edu.br

Oscar A. de Suarez

Department of Mechanical Engineering,
University of Caxias, do Sul,
95070-560, Caxias do Sul, RS, Brazil
e-mail: oscar.alfredo.garcia@gmail.com

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received April 29, 2012; final manuscript received February 4, 2013; published online June 17, 2013. Assoc. Editor: Darrell W. Pepper.

J. Heat Transfer 135(7), 071201 (Jun 17, 2013) (12 pages) Paper No: HT-12-1197; doi: 10.1115/1.4023938 History: Received April 29, 2012; Revised February 04, 2013

In this paper, the Galerkin method is used to obtain numerical solutions to two-dimensional steady-state reaction-diffusion problems. Uncertainties in reaction and diffusion coefficients are modeled using parameterized stochastic processes. A stochastic version of the Lax–Milgram lemma is used in order to guarantee existence and uniqueness of the theoretical solutions. The space of approximate solutions is constructed by tensor product between finite dimensional deterministic functional spaces and spaces generated by chaos polynomials, derived from the Askey–Wiener scheme. Performance of the developed Galerkin scheme is evaluated by comparing first and second order moments and probability histograms obtained from approximate solutions with the corresponding estimates obtained via Monte Carlo simulation. Results for three example problems show very fast convergence of the approximate Galerkin solutions. Results also show that complete probability densities (histograms) of the responses are correctly approximated by the developed Galerkin basis.

Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.


Mikhailov, M. D., and Ozisik, M. N., 1984, Unified Analysis and Solutions of Heat and Mass Diffusion, John Wiley & Sons, New York.
Bejan, A., 1995, Convection Heat Transfer, 2nd ed., Wiley InterScience, New York.
Matthies, H. G., Brenner, C. E., Bucher, C. G., and SoaresC. G., 1997, “Uncertainties in Probabilistic Numerical Analysis of Structures and Solids—Stochastic Finite Elements,” Struct. Safety, 19, pp. 283–336. [CrossRef]
Xiu, D., and KarniadakisG. E., 2003, “A New Stochastic Approach to Transient Heat Conduction Modeling With Uncertainty,” Int. J. Heat Mass Tran., 46, pp. 4681–4693. [CrossRef]
Emery, A. F., 2004, “Solving Stochastic Heat Transfer Problems,” Eng. Anal. Bound. Elem., 28, pp. 279–291. [CrossRef]
Spanos, P. D., and GhanemR., 1989, “Stochastic Finite Element Expansion for Random Media,” J. Eng. Mech.125, pp. 26–40.
Georgiadis, J. G., 1991, “On the Approximate Solution of Non-Deterministic Heat and Mass Transport Problems,” Int. J. Heat Mass Tran., 34, pp. 2097–2105. [CrossRef]
Madera, A. G., 1994, “Simulation of Stochastic Heat Conduction Processes,” Int. J. Heat Mass Tran., 37, pp. 2571–2577. [CrossRef]
Hien, T. D., and Kleiber, M., 1997, “Stochastic Finite Element Modeling in Linear Transient Heat Transfer,” Comput. Method. Appl. M., 144, pp. 111–124. [CrossRef]
Kaminski, M., and Hien, T. D., 1999, “Stochastic Finite Element Modeling of Transient Heat Transfer in Layered Composites,” Int. Commun. Heat Mass Tran., 26, pp. 801–810. [CrossRef]
Deb, M. K., Babuška, I., and Oden, J. T., 2008, “Solution of Stochastic Partial Differential Equations Using Galerkin Finite Element Techniques,” Comput. Method. Appl. M., 190, pp. 6359–6372. [CrossRef]
Jin, B., and ZouJ., 2008, “Inversion of Robin Coefficient by a Spectral Stochastic Finite Element Approach,” J. Comput. Phys., 227, pp. 3282–3306. [CrossRef]
Jardak, M., Su, C. H., and KarniadakisG. E., 2002, “Spectral Polynomial Chaos Solutions of the Stochastic Advection Equation,” SIAM J. Sci. Comput., 17, pp. 319–338. [CrossRef]
Xiu, D., and Shen, J., 2007, “An Efficient Spectral Method for Acoustic Scattering From Rough Surfaces,” Commun. Comput. Phys., 1, pp. 54–72.
Grzywinski, M., and Sluzalec, A., 2000, “Stochastic Convective Heat Transfer Equations in Finite Differences Method,” Int. J. Heat Mass Tran., 43, pp. 4003–4008. [CrossRef]
Engblom, S., Ferm, L., Hellander, A., and Lötstedt, P., 2009, “Simulation of Stochastic Reaction-Diffusion Processes on Unstructured Meshes,” SIAM J. Sci. Comput., 31, pp. 1774–1797. [CrossRef]
Phongthanapanich, S., and Dechaumphai, P., 2009 “Combined Finite Volume Element Method for Singularly Perturbed Reaction–Diffusion Problems,” Appl. Math. Comput., 209, pp. 177–185. [CrossRef]
Babuška, I., Tempone, R., and Zouraris, G. E., 2005, “Solving Elliptic Boundary Value Problems With Uncertain Coefficients by the Finite Element Method: The Stochastic Formulation,” Comput. Method. Appl. M., 194, pp. 1251–1294. [CrossRef]
Grigoriu, M., 1995, Applied Non-Gaussian Processes: Examples, Theory, Simulation, Linear Random Vibration, and Matlab Solutions, Prentice-Hall, Englewood Cliffs, NJ.
Wiener, N., 1938, “The Homogeneous Chaos,” Am. J. Math., 60, pp. 897–936. [CrossRef]
Askey, R., and Wilson, J., 1985, Some Basic Hypergeometric Polynomials That Generalize Jacobi Polynomials, Vol. 319, Mem. Amer. Math., Providence, RI.
Ogura, H., 1972, “Orthogonal Functionals of the Poisson Process,” IEEE Trans. Inform. Theory, 18, pp. 473–481. [CrossRef]
Cameron, R. H., and Martin, W. T., 1947, “The Orthogonal Development of Nonlinear Functionals in Series of Fourier–Hermite Functionals,” Ann. Math., 48, pp. 385–392. [CrossRef]
Janson, S., 1997, Gaussian Hilbert Spaces, Cambridge University Press, Cambridge, UK.
Frauenfelder, P., Schwab, C., and Todor, R. A., 2005, “Finite Elements for Elliptic Problems With Stochastic Coefficients,” Comput. Method. Appl. M., 194, pp. 205–228. [CrossRef]
Matthies, H. G., and Keese, A., 2005, “Galerkin Methods for Linear and Nonlinear Elliptic Stochastic Partial Differential Equations,” Comput. Method. Appl. M., 194, pp. 1295–1331. [CrossRef]
Brenner, S. C., and Scott, L. R., 1994, The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York.
Olsson, A. M., and Sandberg, G. E., 2002, “Latin Hypercube Sampling for Stochastic Finite Element Analysis,” J. Eng. Mech., 128, pp. 121–125. [CrossRef]
Silva, C. R. A., Jr., 2004, “Application of the Galerkin Method to Stochastic Bending of Kirchhoff Plates,” Doctoral thesis, Department of Mechanical Engineering, Federal University of Santa Catarina, Florianópolis, SC, Brazil (in Portuguese).


Grahic Jump Location
Fig. 1

Sparseness of the stiffness matrix of example 1: (a) for m = 4, n = 5, p = 1; (b) for m = 4, n = 140, p = 3

Grahic Jump Location
Fig. 2

Realizations of the response field for example 1

Grahic Jump Location
Fig. 3

(a) Convergence in expected value for u((Lx/2),(Ly/2),ω); (b) convergence in variance for u((Lx/2),(Ly/2),ω)

Grahic Jump Location
Fig. 4

(a) Expected value of the response; (b) covariance function of the response process evaluated at (x,y*,t,y*)=(x,(Ly/2),t,(Ly/2))

Grahic Jump Location
Fig. 5

(a) Expected value of the random response; (b) relative error in expected value

Grahic Jump Location
Fig. 6

(a) Covariance function h(x)=Covu(x,(Ly/2),x,(Ly/2)); (b) relative error in covariance

Grahic Jump Location
Fig. 7

(a) Histograms and (b) cumulative probability distribution of random variable u(x*,y*,ω)

Grahic Jump Location
Fig. 8

10,000 realizations of the random response process for example 2

Grahic Jump Location
Fig. 9

(a) Expected value of the random response; (b) relative error in expected value

Grahic Jump Location
Fig. 10

(a) Covariance function h=h(x); (b) relative error in covariance approximation

Grahic Jump Location
Fig. 11

(a) Histograms and (b) cumulative probability distribution of random variable u(x*,y*,ω)




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In