European Congress on Computational Methods in Applied Sciences and Engineering ECCOMAS 2004 P. Neittaanm¨ aki, T. Rossi, S. Korotov, E. O˜ nate, J. P´eriaux, and D. Kn¨ orzer (eds.) Jyv¨ askyl¨ a, 24–28 July 2004
EXPLORING NEW SUBGRID SCALE STABILIZED METHODS FOR ADVECTION-DIFFUSION-REACTION G. Hauke ? , M.H. Doweidar
†
? Departamento
de Mec´anica de Fluidos Centro Polit´ecnico Superior, C/ Maria de Luna 3, 50018, Zaragoza, Spain e-mail:
[email protected], web page: http://www.cps.unizar.es/ ghauke †
Departamento de Mec´anica de Fluidos Centro Polit´ecnico Superior, C/ Maria de Luna 3, 50018, Zaragoza, Spain
Key words: Advection-diffusion-reaction equation, Stabilized methods, Variational multiscale methods, SGS method, Combining stabilized methods. Abstract. Advection-diffusion-reaction problems are receiving much attention lately. Among finite elements, multiscale/adjoint/unusual stabilized methods are one of the most popular techniques to stabilize problems with strong convection or reaction. However, some local instabilities have been detected depending on the position and nature of the boundary layers. Therefore, several augmented formulations are explored in order to remove the above local instabilities. The approaches include combined stabilized methods and higher order multiscale methods.
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1
INTRODUCTION
Stabilized finite element methods have consolidated a primordial position in finite element analysis of flow problems. However, in the presence of strong advection combined with large source terms, some difficulties remain to be solved yet. For example, in [9] it was shown that depending on the nature of the interior or boundary layer, some local instabilities may persist. Therefore, this work examines several alternatives to improve the performance of existing methods in the regime of strongly reacting flows. Two strategies are investigated. The first one consists of using a two parameter method, easily extended into multidimensions, similar to the combined GLS-GGLS method [4, 8, 15]. The second one inspects increasing the accuracy of the subgrid scale method [11] (or adjoint or unusual stabilized method [6]) including higher order terms. 2
THE ADVECTION-DIFFUSION-REACTION MODEL PROBLEM
Let us consider the open spatial domain Ω = (0, 1). The differential form of the model problem can be stated as follows. Find ϕ(x) : Ω → R such that: u ϕ,x = κ ϕ,xx + s ϕ ϕ(0) = g0 ϕ(1) = g1
in Ω (1)
The coefficients of this equation are u, the velocity field, κ ≥ 0, the diffusion coefficient and s, the source parameter, where s > 0 for production and s < 0 for dissipation or absorption. Of particular interest for this problem are the operators advective-diffusivereactive operator, the adjoint operator and the advective operator, which are, respectively, Lϕ = u ϕ,x − κ ϕ,xx − s ϕ −L∗ ϕ = u ϕ,x + κ ϕ,xx + s ϕ Ladv ϕ = u ϕ,x
(2)
The dimensionless numbers that characterize the solution are Pe = U L/κ Da = sL/U
Peclet Damk¨ohler
(3)
where U and L are the characteristic velocity and length scales of the problem. 3
THE VARIATIONAL MULTISCALE METHOD
The exact variational multiscale (VMS) method [11, 14, 12] is a technique that feeds the scales which cannot be resolved by the finite element functions in the solution.
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Let us assume that the spatial domain Ω is divided into nel elements Ωe , with size he . The finite element spaces are defined according to: S h = {ϕh | ϕh ∈ C 0 (Ω), ϕh |Ωe ∈ Pk (Ωe ) , ϕh (0) = g0h , ϕh (1) = g1h } V h = {w h | w h ∈ C 0 (Ω), w h |Ωe ∈ Pk (Ωe ) , w h (0) = 0, w h (1) = 0}
(4)
where Pk (Ωe ) are the k-th-order polynomials in each element Ωe . Then, for the advection-diffusion-reaction equation, the smooth variational multiscale method is defined as follows. Find ϕh ∈ S h such that for all w h ∈ V h : Z Z Z h h h h h h w uϕ,x + w,x κϕ,x − w sϕ dΩ + −L∗ w h (x) g(x, y)Lϕh(y) dΩx dΩy = 0 (5) Ω
Ω
Ω
e = Pnel Ωe . The function g(x, y) is the Green’s function of the problem, solution where Ω e=1 of the problem with homogeneous boundary conditions: Find g(x, y) such that: Lg(x, y) = δy g(x, y) = 0
in Ω on Γ
(6)
The function δy (x) is the Dirac delta function. The Green’s function can be represented by its moments, which up to first order contributions are R R e e g(ξ, η) dξdη (7) g00 = Ω RΩ R dξdη Ωe Ωe R R e e ξ g(ξ, η) dξdη g10 = Ω RΩ R (8) e Ωe dξdη Ω R R e e η g(ξ, η) dξdη g01 = Ω RΩ R (9) e Ωe dξdη Ω R R ξη g(ξ, η) dξdη e Re R g11 = Ω Ω (10) dξdη Ωe Ωe An approximation of g(x, y) can be obtained from the element Green’s function g e (x, y), which is computed from the corresponding homogeneous problem within an element: Find g e (x, y) such that: Lg e (x, y) = δy g e (x, y) = 0
3
in Ωe on Γe
(11)
G. Hauke, M.H. Doweidar
u g00
0.4 0.3 0.2 0.1 0 -10
20
u g10
15 10 -8 -6
15
0 -10
α
10 -8 -6 σ
-2
0 u g01
20
5
-4
σ
0.1 0.05
α
5
-4 -2
0.03 20
-0.05 -0.1
u g11
-10 -6 σ
10 α
-8 -6
5
-4
15
0 -10
10 α
-8
20
0.02 0.01
15
σ
-2
Figure 1: Dimensionless ug00 , ug10 , ug01 , ug11
4
5
-4 -2
G. Hauke, M.H. Doweidar
In this case, the element Green’s function in local coordinates ξ, η is found to be 1 1 e 1 (12) exp( λ2 (ξ + 1)h) − exp(h(λ2 − λ1 )) exp( λ1 (ξ + 1)h) g (ξ, η) = Aκ 2 2 1 1 1 exp( λ1 (η + 1)h) − exp( λ2 (η + 1)h) exp(− u(η + 1)h/κ) if ξ > η 2 2 2 e
g (ξ, η) =
1 Aκ
1 1 exp( λ2 (η + 1)h) − exp(h(λ2 − λ1 )) exp( λ1 (η + 1)h) 2 2 1 1 1 exp( λ1 (ξ + 1)h) − exp( λ2 (ξ + 1)h) exp(− u(η + 1)h/κ) 2 2 2
(13) if ξ < η
with A = (exp((λ2 − λ1 )h) − 1)(λ2 − λ1 ) √ u − u2 − 4κs λ1 = √ 2κ u + u2 − 4κs λ2 = 2κ
(14)
The moments are represented in Fig. 1. 4
BASIC STABILIZED METHODS
The poor stability of the Galerkin method for strongly convected or reacting problems can be enhanced by augmenting the variational formulation, for instance, via stabilized methods. The stabilized methods, such as SUPG [1], GLS [13] and SGS [11] can be seen as zeroth order approximations of the multiscale method. The SGS method was firstly introduced as the complete stabilized method [3] and the unusual stabilized method [6, 7]. Then it was further studied in [2, 10, 9]. In particular, these methods can be formulated as follows. Given the solution finite element space S h and weighting space V h in Ω, find ϕh ∈ S h such that for all w h ∈ V h : Z h w h uϕh,x + w,x κϕh,x − w h sϕh dΩ Ω
+
nel Z X e=1
where
Ωe
Lw h τ uϕh,x − κϕh,xx − sϕh dΩ = 0
Ladv ϕ Lϕ = Lϕ −L∗ ϕ 5
SUPG GLS SGS
(15)
(16)
G. Hauke, M.H. Doweidar
α = 10., σ = −10 1
α = 10., σ = −10 1
exact sgs supg galerkin
0.8 0.6
exact sgs supg galerkin
0.8 0.6 φ
0.4
φ
0.4 0.2
0.2
0
0
-0.2
-0.2
-0.4
-0.4 0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
x
0.6
0.8
x
Figure 2: SUPG and SGS solutions for an outflow and an inflow boundary layer
The first integral is the Galerkin contribution to the variational formulation, while the second one is the stabilizing term, where τ is the intrinsic time scale parameter. Expressions for the stabilizing parameter can be found in [5, 7, 9], which basically are an approximation to τ = g00 meas(Ωe ) ≈ g00 he (17) The numerical solution is characterized by the dimensionless numbers α = |u|he /2κ σ =
she |u|
element Peclet number element Damk¨ohler number
where he is the element size. Although these methods have good global stability, Fig. 2 shows that depending on the type of boundary layer or interior layer some local instability may persists [9]. Next two ideas are investigated to cure this problem. 5
COMBINING STABILIZED METHODS
Following the ideas of combining stabilized methods, such as the GLS-GGLS method [4, 8, 15], we propose to combine the SGS and the gradient SGS (GSGS) methods to yield the SGS-GSGS method. Since there are two free parameters, nodal exactness can be readily achieved. The integral form would be then: Given the solution finite element space S h and
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G. Hauke, M.H. Doweidar
weighting space V h in Ω, find ϕh ∈ S h such that for all w h ∈ V h : Z h w h uϕh,x + w,x κϕh,x − w h sϕh dΩ Ω
+
nel Z X e=1
+
nel Z X e=1
Ωe
Ωe
Lw h τ00 uϕh,x − κϕh,xx − sϕh dΩ
(18)
∇Lw h τ11 ∇ uϕh,x − κϕh,xx − sϕh dΩ = 0
The operator L preserves the meaning of (16) and τ00 τ11 are the two parameters, defined as τ00 =
he t00 u
(he )3 t11 u These parameters can be defined imposing nodal exactness, −1 σ 2 sinh(α) n t00 = −2σ + − cosh(α) + cosh(γ) + σ sinh(α) 3σ σ(3 σ cosh(γ) + (−3 + σ 2 ) sinh(α) 1 2 n −3 − σ + + t11 = 6 σ3 α −2 cosh(α) + 2 cosh(γ) + σ sinh(α) τ11 =
(19) (20)
(21) (22)
with
γ=
p α(−2σ + α)
(23)
But it also can be defined using the Green’s function moments, tg00 = u g00 tg11 = u g11
(24) (25)
The results of both methods are shown for various combinations of parameters in Figs. 4 and 5, respectively. As can be seen, the nodal exact method is indeed nodal exact. Whereas solutions computed from the Green’s function moments have some problems for negative source terms. Therefore, the missing moments play a fundamental role in the stabilization of source terms. Solutions for the GLS-GGLS method [15] are depicted in Fig. 3. The method works perfectly for negative source terms, the design region. However, it is inaccurate in the propagation region. Remark. The nodal exact parameters lack the standard limits in the advective-diffusive region. Therefore, this methods do not converge to GLS (SUPG) or SGS when s → 0. This fact was already pointed out in [15]. 7
G. Hauke, M.H. Doweidar
5 Exact Solution Approximate Solution
4
Exact Solution Approximate Solution
1 0.8
3 φ
φ
0.6
2
0.4
1
0.2 0
0 0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
x
2 1.5 1
Exact Solution Approximate Solution
-5
0.5 0 -0.5 -1
-10
-1.5 -2
0
φ
φ
5
0.2
1
0.4
0.6
0.8
1
Exact Solution Approximate Solution
0
0.2
0.4
x
0.6
0.8
1
x
Exact Solution Approximate Solution
1
0.8
x
10
0
0.6
Exact Solution Approximate Solution
1
0.6
0.6 φ
0.8
φ
0.8
0.4
0.4
0.2
0.2
0
0 0
0.2
0.4
0.6
0.8
1
0
x
0.2
0.4
0.6 x
Figure 3: GLS-GGLS for (σ, α) = (0.15 ,9) ; (9, 0.15) ; (-10, 20)
8
0.8
1
4 3.5 3 2.5 2 1.5 1 0.5 0
Exact Solution Approximate Solution
Exact Solution Approximate Solution
1 0.8 0.6 φ
φ
G. Hauke, M.H. Doweidar
0.4 0.2 0 0
0.2
0.4
0.6
0.8
1
0
0.2
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x
0.6
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1
x
10
1.5 Exact Solution Approximate Solution
Exact Solution Approximate Solution
1
5 0
0
φ
φ
0.5
-0.5 -5 -1 -10
-1.5 0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
x
Exact Solution Approximate Solution
1
0.6
0.8
1
x
Exact Solution Approximate Solution
1
0.6
0.6 φ
0.8
φ
0.8
0.4
0.4
0.2
0.2
0
0 0
0.2
0.4
0.6
0.8
1
0
x
0.2
0.4
0.6
0.8
x
Figure 4: SGS-GSGS using tn00 and tn11 for (σ, α) = (0.15 ,9) ; (9, 0.15) ; (-10, 20)
9
1
G. Hauke, M.H. Doweidar
5 Exact Solution Approximate Solution
4
Exact Solution Approximate Solution
1 0.8
3 φ
φ
0.6
2
0.4
1
0.2 0
0 0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
x
2 1.5 1
Exact Solution Approximate Solution 5
-5
0.5 0 -0.5 -1
-10
-1.5 -2
φ
φ
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
Exact Solution Approximate Solution
3 φ
φ
2 1 0 -1 -2 0.2
0.4
0.6
0.8
1
x
5
0
1
Exact Solution Approximate Solution
x
4
0.8
x
10
0
0.6
0.6
0.8
1
1.2 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6
Exact Solution Approximate Solution
0
x
0.2
0.4
0.6
0.8
x
Figure 5: SGS-GSGS using tg00 and tg11 for (σ, α) = (0.15 ,9) ; (9, 0.15) ; (-10, 20)
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G. Hauke, M.H. Doweidar
6
APPROXIMATE HIGHER ORDER MULTISCALE METHODS
The variational multiscale method (5) can be approximated to include up to the linear terms, which are present in reactive flows. Using Taylor series, the differential operators can be expanded in local coordinates as Lϕh (ξ) = (Lϕh )(0) + (∇ξ Lϕh )(0) ξ L∗ w h (ξ) = (L∗ w h )(0) + (∇ξ L∗ w h )(0) ξ
(26) (27)
Application to the stabilizing term, yields Z Z L∗ w h (x) g e (x, y)Lϕh (y) dΩx dΩy = (28) e e ZΩ Z Ω ∗ h (L w )(0) + (∇ξ L∗ w h )(0)ξ g e (ξ, η) ϕh )(0) + (∇η Lϕh )(0)η dΩx dΩy(29) Ωe Ωe Z Z ∗ h e ≈ (L w )(0) g (ξ, η) dΩx dΩy (Lϕh )(0) (30) Ωe Ωe Z Z + (∇ξ L∗ w h )(0) ξ g e (ξ, η) dΩx dΩy (Lϕh )(0) (31) Ωe Ωe Z Z e ∗ h g (ξ, η) η dΩx dΩy (∇η Lϕh )(0) (32) + (L w )(0) e e Ω Ω Z Z ∗ h e + (∇ξ L w )(0) ξ g (ξ, η) η dΩx dΩy (∇η Lϕh )(0) (33) Ωe
Ωe
Respective solutions are shown in Fig. 6, where, naturally, the method is nodally exact. However, this method is difficult to extend into multi-dimensions. 7
CONCLUSIONS
Two methods have been proposed in order to increase the local stability of subgrid scale stabilized methods in the presence of strong source terms. One of the strategies consists of combining the SGS method with the gradient SGS method. The two free parameters allow nodally exact solutions. However, the so derived parameters do not have the standard asymptotic limits. Therefore, in the absence of source terms, the method does not tend to the SGS (or SUPG) method. Despite of this drawback, which is shared by the GLS-GGLS method, the good trait of the method is that it can be readily extended to multi-dimensions. On the other hand, the combined SGS-GSGS method which uses the moments derived from the Green’s function, has also been tested. It turns out that this method does not perform so well for negative source terms, indicating that the cross moments of the Green’s function are important in order to solve problems with large negative source terms. Finally, a higher order variational multiscale method has been derived which keeps all the first order moments, therefore, being exact for linear residuals. The results indeed 11
4 3.5 3 2.5 2 1.5 1 0.5 0
Exact Solution Approximate Solution
Exact Solution Approximate Solution
1 0.8 0.6 φ
φ
G. Hauke, M.H. Doweidar
0.4 0.2 0 0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
x
0.6
0.8
1
x
10
1.5 Exact Solution Approximate Solution
Exact Solution Approximate Solution
1
5 0
0
φ
φ
0.5
-0.5 -5 -1 -10
-1.5 0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
x
Exact Solution Approximate Solution
1
0.6
0.8
1
x
Exact Solution Approximate Solution
1
0.6
0.6 φ
0.8
φ
0.8
0.4
0.4
0.2
0.2
0
0 0
0.2
0.4
0.6
0.8
1
0
x
0.2
0.4
0.6
0.8
1
x
Figure 6: Approximate variational multiscale using tg00 , tg01 , tg10 and tg11 with one quadrature point for (σ, α) = (0.15 ,9) ; (9, 0.15) ; (-10, 20)
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show that the solution is nodally exact for problems with linear shape functions in all the parameters range, but the method is complicated and expensive to use in multidimensional flows.
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REFERENCES [1] A.N. Brooks, T.J.R. Hughes, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible NavierStokes equations, Comput. Meth. Appl. Mech. Engrng., 32, 199–259 (1982). [2] R. Codina, Comparison of some finite element methods for solving the diffusionconvection-reaction equation, Comput. Meth. Appl. Mech. Engrng., 156 185–210 (1998). [3] J. Douglas, J. Wang, An absolutely stabilized finite element method for the Stokes problem, Math. Comp., 52, 495–508 (1989). [4] L.P. Franca, E.G. Dutra do Carmo, The Galerkin/gradient least sqsuares, Comput. Meth. Appl. Mech. Engrng., 74, 41-54 (1989). [5] L.P. Franca, S.L. Frey, T.J.R. Hughes, Stabilized finite element methods: I. Application to the advective-diffusive model, Comput. Meth. Appl. Mech. Engrng., 95, 253-276 (1992). [6] L.P. Franca and C. Farhat, Bubble functions prompt unusual stabilized finite element methods, Comput. Meth. Appl. Mech. Engrng., 123 299–308 (1995). [7] L.P. Franca, F. Valentin, On an improved unusual stabilized finite element method for the advective-reactive-diffusive equation, Comput. Meth. Appl. Mech. Engrng., 190, 1785-1800, (2000). [8] I. Harari, T.J.R. Hughes, Stabilized finite element emthods for steady advectiondiffusion with production, Comput. Meth. Appl. Mech. Engrng., 115, 165-191 (1994). [9] G. Hauke, A simple stabilized method for the advection-diffusion-reaction equation, Comput. Meth. Appl. Mech. Engrng., 191, 2925-2947 (2002). [10] G. Hauke, A. Garcia-Olivares, Variational Subgrid Scale Formulations for the Advection-Diffusion-Reaction Equation, Comput. Meth. Appl. Mech. Engrng., 190, 6847-6865 (2001). [11] T.J.R. Hughes, Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods, Comput. Meth. Appl. Mech. Engrng., 127, 387–401 (1995). [12] T.J.R. Hughes, G.R. Feijoo, L. Mazzei, J.B. Quincy, The variational multiscale method: A paradigm for computational mechanics, Comput. Meth. Appl. Mech. Engrng., 166, 3–24 (1998).
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[13] T.J.R. Hughes, L.P. Franca, G. Hulbert, A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advectivediffusive equations, Comput. Meth. Appl. Mech. Engrng., 73, 173–189 (1989). [14] T.J.R. Hughes, J.R. Stewart, A space-time formulation for multiscale phenomena, Journal of Comput. Appl. Math., 74, 217–229 (1996). [15] F. Valentin, L.P. Franca, Combining stabilized finite element methods, Comput. Appl. Math., 14, 285–300 (1995).
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