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.

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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

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Fig. 7

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

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Fig. 10

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

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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)

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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))

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Fig. 5

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

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Fig. 6

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

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Fig. 8

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

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Fig. 9

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

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Fig. 11

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



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