MONOLITHIC APPROACH OF STOKES-DARCY COUPLING FOR …bruchon/conferences/Abouorm_ECCOMAS2012.pdf · structures with organic matrix, especially for large pieces in aeronautics. Resin

European Congress on Computational Methodsin Applied Sciences and Engineering (ECCOMAS 2012)

J. Eberhardsteiner et.al. (eds.)Vienna, Austria, September 10-14, 2012

MONOLITHIC APPROACH OF STOKES-DARCY COUPLING FORTHE SIMULATION OF LIQUID INFUSION PROCESS

Lara Abouorm1, Sylvain Drapier1, Julien Bruchon1, Nicolas Moulin1

1 Centre SMS - UMR CNRS 5146Ecole Nationale Superieure de Mines de Saint-Etienne

Saint Etienne, Francee-mail: [email protected], [email protected], [email protected], [email protected]

Keywords: Darcy, Stokes, coupling, stabilization, multiscale method.

Abstract. The aim of this work is to propose a numerical multiphysical model to simulate withthe finite element method, composite manufacturing processes. This model allows to representthe flow of a liquid resin into fibrous preforms undergoing large deformations. The numericalmodel is based on the coupling between the resin flow within a porous medium (Darcy) and apurely domain (Stokes). The weak formulation is obtained by summing up the variational formsof the Stokes and the Darcy equations over the whole domain. In both the purely fluid domainand the porous medium, the Ladyenskaya-Brezzi-Babuska stability condition is not satisfied,we use P1/P1 element stabilized with ASGS (Alegbraic Subgrid Scale) method. The originalityof the model consists in using one single unstructed mesh to represent the Stokes and Darcysubdomains (monolithic approach). A level set context is used to represent the Stokes-Darcyinterface and to capture the moving flow front. The monolithic approach, stabilized with ASGS,can now be used to solve 2D and 3D problems.

Lara Abouorm, Sylvain Drapier, Julien Bruchon, Nicolas Moulin

1 INTRODUCTION

Manufacturing processes by resin infusion are competitive routes to elaborate compositestructures with organic matrix, especially for large pieces in aeronautics. Resin infusion pro-cesses consist in infusing a liquid resin through the thickness of fibrous preforms rather in theirplane. The resin is brought into a flow and infused through the preforms due to the pressureapplied on the top of the mold (Figure 1). The physical and mechanical properties of the finalpart (the final thickness and the final fiber volume fraction) are hardly predictable. To help tothe control of this process, a numerical model is developped. This model is based on the cou-pling between the flow of the resin in a highly permeable medium and in a porous medium (thefibrous preforms). This paper proposes a robust finite element solution for coupling flows inboth purely fluid region, ruled by Stokes equations, and fibrous preforms region governed byDarcy flow. The Stokes-Darcy coupled problem has been studied by many ressearchers in manyfield of engineering. Both a decoupled approach as proposed by [1], and a monolithic approach,as proposed by [3] are investigated in severe regimes. The decoupled strategy consists of usingtwo different element spaces to solve the Stokes and Darcy equations, whereas the unified strat-egy consists in using the same finite element space. In litterature, flows are solved using mixedfinite elements stabilized with hierarchical-based bubble function, i.e P1+/P1 finite element inStokes domain and HVM (Hughes Variational Multiscale) for stabilization in Darcy [2, 3]. Inthis paper we use a robust approach which yields improvements compared to this previous ap-proach, [3]. The robustness of the approach, which is assesed in this paper, is ensured by usingASGS method (Algebraic Subgrid Scale) [4, 5] to stabilize velocity and pressure approximatedby linear and continous elements in Stokes and Darcy domains. Signed functions are used torepresent the Stokes-Darcy interface and to capture the moving flow front. In this paper, wefocus on Stokes-Darcy coupling without taking into consideration either the evolution of thefront of the resin or the deformation of preforms.

Figure 1: Shematic of manufacturing process by resin infusion and 3 zones representation

2 MATHEMATICAL MODEL

In order to model Stokes-Darcy coupling, we consider a bounded domain Ω formed by twonon overlapping subdomains Ωs and Ωd separated by a surface Γ = ∂Ωs ∩ ∂Ωd. Index s isused to denote everything that concerns the purely fluid part (Stokes’ domain) and index d formodelling porous medium (Darcy’s domain). Ωs is the region occupied by the fluid, the motionof which is described by the Stokes equations. The incompressible newtonian fluid flow in Ωd,the porous medium with low permeability, is governed by a Darcy law (Figure 2).

The Stokes equations are then expressed as: find the velocity vs and the pressure ps defined

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on Ωs solution of

− div(2µε(vs)) +∇ps = fs in Ωs

−div vs = hs in Ωs

vs = v1 on Γs,D (1)σ.ns = t on Γs,N

where ns is the unit vector normal to the boundary of Ωs and t is the stress vector to be pre-scribed on Γs,N , vs and ps are the velocity and pressure fields, µ denotes dynamic viscosity, ε isthe Eulerian strain rate tensor, and σs is the Cauchy stress tensor. If the fluid is incompressiblethen hs = 0.

Figure 2: Computational domain.

The Darcy equations are expressed as: find the velocity vd and the pressure pd defined onΩd, solution of

µ

kvd +∇pd = fd in Ωd

−div vd = hd in Ωd (2)vd.nd = 0 on Γd,D

pd = pext on Γd,N

where k is the permeability tensor reduced to a scalar in the isotropic case considered here, pextis a pressure to be prescribed on Γd,N and nd is the outward unit vector normal to the boundaryof Ωd.

In Stokes-Darcy coupled problem, boudary conditions must be considered on the interface Γof normal n = ns. These conditions are:Continuity of normal velocity:

vs.ns + vd.nd = 0 on Γ (3)

Continuity of the fluid normal stress:

n.σs.n = n.σd.n on Γ (4)

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Beaver-Joseph-Saffman condition:

2n.ε(vs).τj =α√k

(vs.τj) (5)

where α is a dimensionless parameter, so-called slip coefficient and τj are the tangentiel vectorson the interface.

3 WEAK FORMULATION

In order to solve the Stokes-Darcy coupled problem by a finite element method, the weak for-mulation has to be established. We present the weak formulation of Stokes and Darcy separatly.The weak formulation of the coupled problem is obtained by summing up the weak formulationof Stokes and Darcy taking into consideration interface conditions described in Section 2. Forthe sake of simplicity, we choose to write the L2 inner product in Ωd,s as <,>.

The spaces of velocity, pressures and test functions are defined by :

L2(Ωi) = q,∫

Ωi

q2dΩ <∞

H1(Ωi)m = q ∈ L2(Ωi)

m,∇q ∈ L2(Ωi)m×m

H1Γi,D

(Ωi)m = q ∈ H1(Ωi)

m | q = 0 sur Γi,Dwith i = s or i = d and m is the dimension equal to 2 or 3.

The variational formulation of Stokes problem consists in finding a velocity-pressure pair[v, p] such that :

Bs([v, p], [w, q]) = Ls([w, q]) (6)

where w and q are weighting functions defined in H1Γs,D

(Ωs)m and L2(Ωs) respectively.

The bilinear form Bs and the linear form Ls are defined in Stokes by :

Bs([v, p][w, q]) = 2µ < ε(v) : ε(w) > − < p,∇.w > − < q,∇.v >Ls([w, q]) = < fs,w > + < hs, q > + < σn,w > (7)

σn is the normal stress. The variational formulation of Darcy’s problem consists in finding avelocity-pressure pair [v, p] such that :

Bd([v, p], [w, q]) = Ld([w, q]) (8)

where w and q are weighting functions defined in H1Γd,D(Ωd)

m and L2(Ωd).The bilinear form Bd and the linear form Ld are defined in Darcy by :

Bd([v, p][w, q]) =µ

k< v,w > − < p,∇.w > − < q,∇.v >

Ld([w, q]) = < fd,w > + < hd, q > − < σn,w > (9)

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where σn is the normal stress.The mixed formulation of the Stokes-Darcy problem is established by considering a velocity

v on Ω and a pressure field p on Ω such as v/Ωi= vi and p/Ωi

= pi with i = s or i = d.The mixed weak formulation of Stokes-Darcy is obtained by summing up Equations 7 and 9and taking into consideration the conditions imposed on the Stokes-Darcy interface describedin Section 2. Hence, the variational formulation of the Stokes-Darcy coupled problem consistin finding [v, p] ∈ V ×Q such that :

Bc[(v, p), (w, q)] = Lc([w, q]

where [w, q] are weighting functions defined in V ×Q.

V = H1Γs,D×H1

Γd,D(div,Ωd)

Q = L2(Ωs)× L2(Ωd)

The bilinear form Bc and the linear form Lc are defined by :

Bc([v, q], [w, q]) =∫

Ω2µε(v) : ε(w)HsdΩ (10)

+∫

Ω

µ

kv.wHd dΩ −

∫Ωp divw dΩ−

∫Ωq divv dΩ +

∫Γ

αµ√kv .wHd dΓ

Lc([v, p], [w, q]) = < f ,w > + < h, q >

(f , h) are defined by (fd,hd) in Darcy and (fs,hs) in Stokes. Hi is heaviside function equal to 1in domain i and vanishing elswhere.

4 FINITE ELEMENT METHOD FOR STOKES-DARCY PROBLEMS

The whole computional domain Ω ⊂ Rd is discretized with one single unstructed mesh. Thismesh is made up of triangles K if m = 2 and of tetrahedrons K if m = 3. Let Vh and Qh bethe finite element spaces of the linear and continous elements vh and ph. The Galerkin approx-imation of both the Stokes and the Darcy problems requires the use of velocity-pressure inter-polation that satisfy the adequate inf sup condition. Different interpolations pairs are known tosatisfy this condition for each problem independently, but the key issue is to find interpolationsthat satisfy both at the same time. In this paper, we choose the use of stabilized finite elementmethods. The philosophy of the stabilized methods is to strenghthen classical variational for-mulations so that discrete approximation which would otherwise be unstable becomes stableand convergent. One of the stabilized finite element method approximation for Stokes-Darcyproblem is VMS (Variational Multiscale Method) [4], [5]. The basic idea of this method is toapproximate the effect of the component of the continous solution which cannot be captured bythe finite element solution. It consists in splitting the continous solution for velocity and pres-sure into two components, one coarse corresponding to the finite element scale [vh, ph], and afiner component corresponding to lower scale [v′, p′] for resolutions. The velocity is approxi-mated as v = vh + v′ and the pressure field is approximated as p = ph + p′. We consider asubgrid space V × Q = (Vh × Qh)

⊕(V ′ × Q′). Invoking this decomposition in the continous

problem for both the solution and test functions, we get the two scales systems:

Bc([vh, ph]), [wh, qh]) +Bc([w′, p′], [wh, qh]) = Lc([wh, qh]) (11)

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Bc([vh, ph], [w′, p′]) +Bc[v

′, p′], [w′, q′]) = Lc([w′, q′]) (12)

for all [wh, qh] ∈Vh×Qh and [w′, q′] ∈ V ′×Q′. After approximating (12) with an algebraic for-mulation, by introducing the operator of projection P ′ onto V ′, the approximated fields [v′, p′]are taken into account in the finite element problem (11), we get the stabilized forms of thebilinear and linear forms in Stokes, Darcy, and Stokes-Darcy coupled problem. The stabilizedproblem in Stokes can be written as follow:Find [vh, ph] ∈Vh ×Qh such as :

Bs,stable([vh, ph], [wh, qh]) = Ls,stable([wh, qh])(13)

The bilinear stabilized form Bs,stable and the linear stabilized form Ls,stable are defined by :

Bs,stable([vh, ph], [wh, qh]) = Bs([vh, ph], [wh, qh]) (14)+τq

∑K

< P ′h,p(∇.vh),∇.wh > + τv∑K

< P ′h,u(−µ∆vh +∇ph), µ∆wh −∇qh >

Ls,stable([wh, qh]) = Ls([wh, qh])

+τq∑K

< P ′h,p(hs),∇.wh > + τv∑K

< P ′h,p(fs), µ∆wh −∇qh >

where Bs([vh, ph], [wh, qh]) and Ls([wh, qh]) are defined in (7). τq,τv are the stabilization pa-rameters (obtained by Fourier transform) that we compute as:

τq = c1 µ

τv = c1 µh2k (15)

c1 is an algorithmic constant and hk is the size of mesh. P ′h,u is the broken L2 projection onto V ′

and P ′h,p is the broken L2 projection onto Q′. The simplest approach is to take P ′h as the identityoperator when acting on the FE residual. Assuming this, we obtain the stabilized method thatwe called Algebraic Subgrid Scale (ASGS) method. Invoking, this expression of P ′h, we get thefollowing stabilized finite form Bs and Ls in Stokes:

Bs,stable([vh, ph], [wh, qh]) = Bs([vh, ph], [wh, qh]) (16)+τq

∑K

< ∇.vh,∇.wh) > + τv∑K

< −µ∆vh +∇ph, µ∆wh −∇qh >

Ls,stable([wh, qh]) = Ls([wh, qh])

+τq∑K

< h,∇.wh > + τv∑K

< f, µ∆wh −∇qh >

By using ASGS method, the stabilized problem in Darcy can be written as follow :Find [vh, ph] ∈ Vh ×Qh such as:

6


Bd,stable([vh, ph][wh, qh]) = Ld,stable([wh, qh]) (17)

Bd,stable([vh, ph], [wh, qh]) = Bd([vh, ph], [wh, qh])

+τp∑K

< ∇.vh,∇.wh) > + τu∑K

<µ

kvh +∇ph,−

µ

kwh −∇qh >

Ld,stable([wh, qh]) = Ld([wh, qh])

+τp∑K

< hd,∇.wh > + τu∑K

< fd,−µ

kwh −∇qh > (18)

where Bd([vh, ph], [wh, qh]) and Ld([wh, qh]) are defined in 9. τp, τu are the stabilization pa-rameters, that we compute as

τp = cpµ

kl2p

τu = (cuµ

klu)−1h2

k (19)

with cp and cu algorithmic constants. lu and lp are length scales which we choose to take(L0 hk)

2, L0 is a characteristic length of domain and hk is the size of mesh.For Stokes and Darcy flow coupled through the interfaces, the stabilized problem with ASGS

can be written as follow:

Find [vh, ph] ∈ Vh ×Qh such as :

Bc,stable([vh, ph][wh, qh]) = Lc,stable([wh, qh]) (20)

Bc,stable([vh, ph], [wh, qh]) = 2µ∫

ΩHsε(vh) : ε(wh) +

µ

k

∫ΩHdwhvh

−∫

Ω∇.wh ph −

∫Ω∇.vh qh + τp,c

∫Ω∇vh∇wh

+∫

Γαµ√k

(vh.τ)(wh.τ) + τu,c

∫Ω< −∆vh +

µ

kvh +∇ph, µ∆wh −

µ

kwh −∇qh >

Lc,stable([wh, qh]) = < f,wh > + < h, qh >

+τp,c < hc,∇.wh > + τu,c < fc, µ∆wh −µ

kwh −∇qh > (21)

τp,c, τu,c are the stabilization parameters, that we compute as

τp,c = cpµ

kl2p + c1µ

τu,c = (c1µ+ cuµ

klu)−1h2

k (22)

where Hi is a heaviside function equal to 1 in domain i and vanishing elsewhere. The surfaceintegral

∫Γ α

µk(vh.τ)(wh.τ) is turned into a volume integral for calculation simplification.

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5 NUMERICAL VALIDATION

In this section, the accuracy, robustness and convergence of the proposed method is inves-tigated in two situations. The first one is the case of a flow perpendicular to the Stokes-Darcyinterface, while the second one proposes to study a flow parallel to this interface. The per-pendicular flow case has been found to be more severe than the parallel one. In particular, themethod developped in [3] shows consistency errors and spurious oscillations of the velocity inthe vicinity of the interface in this situation. That is why we investigate the convergence of ourapproach in the perpendicular case, and additionally, we compare our results to those obtainedin [3].

5.1 Perpendicular flow

To validate continuity of normal velocity on Stokes-Darcy interface, we study a test case ofa flow perpendicular to the interface of normal y in the global frame. Let us consider a domainΩ = [0; 5]×[0; 2] m2 composed of two sub-domains: a pure fluid domain Ωs = [0; 5]×[1; 2] m2

and porous medium Ωd = [0; 5]× [0; 1] m2.A normal stress σn of 1 bar is applied on the top and a pressure of 0 bar is applied on thebottom. Other boundary conditions are zero normal velocity on the left and right hand sides ofthe geometry (Figure 3). For this flow, the constants in τu,c and τp,c are choosen as:c1 = 1, cp = 10 , cu = 0.5 and L0 = 1.

Figure 3: Boundary conditions for a flow perpendicular to the interface.

Figure 4: Pressure field p for permeability k = 10−14m2.

We present results for pressure field in Figure 4 for k = 10−14m2. The small values ofthe permeabilites (down to 10−14m2) are realistic permeabilities met in LCM processes [6] .The pressure distribution doesn’t depend on the permeability. Permeability tests for k down to10−15m2 are conducted. For example, in the perpendicular flow test (Figure 5), we compare

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the velocity vy obtained by using ASGS method with the one obtained by using P1+P1/HVMmethod [3]. Tests are realized on structured and unstructured meshes For permeability k =10−14m2, normal velocity must be equal to 10−9m/s. Figure 5 shows oscillations of velocityand consistency error in Stokes for the P1+/P1-HVM method contrary to ASGS method wherenormal velocity is continous and does not let show any oscillations at the interface.

Figure 5: comparison between ASGS method and P1+/P1-HVM method in perpendicular flow test

5.1.1 SOLUTION CONVERGENCE

Convergence of the solution, as well as relative errors, have been considered for the case ofthe perpendicular flow. We compare the obtained numerical results for ASGS and HVM/P1+P1methods with analytical ones, where

vs,x = vd,x = 0,vs,y = vd,y = − k

µ∇p = −10−9m/s,

(23)

ASGS HVM/P1+P1 [3]h, [m] ‖vy,error‖stokes, [%] ‖vy,error‖darcy, [%] ‖vy,error‖stokes, [%] ‖vy,error‖darcy, [%]0.08 0.9 % 0.7 % 10.41 % 9.37 %0.04 0.406 % 0.242 % 6.3 % 5.9 %0.02 0.3 % 0.05 % 3.37 % 3.38 %

Table 1: Relative errors for normal velocities in Stokes and Darcy regions, ASGS method and P1+/P1-HVM,k = 10−14m2, perpendicular flow.

A large decrease in the relative error between ASGS method and P1+/P1-HVM methodyields improvements to the importance of Variational Multiscale Methods. The errors are com-puted: ‖verror‖ =

‖va−vh‖L2

‖va‖L2, where va is the analytical solution and vh is the obtained numerical

solution, ‖v‖L2 = (∫Ω v

2dΩ)1/2 - norm L2. We obtain results for specified above perpendicular

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flow for different sizes of the mesh, when µ = 1 Pa.s, k = 10−14 m2 presented in Table 1 forboth ASGS and P1+P/HVM method. Orders of convergence are also studied using manufac-tured solutions. We present manufactured solutions in another paper, but the method confirmsthe theoritical order of convergence in both L2 and H1 norms.

5.1.2 3D simulation.

For the case of perpendicular flow, 3D simulations were conducted. Geometry and boundaryconditions are the same as in 2D case, extruded along axis z. We use unstructed mesh inthis case. We present results for pressure and velocity fields in Figure 6. These results are inaccordance with the results obtained for the 2D case. Normal velocity is continous, but forunstructed meshes, some oscillations appear around the Stokes-Darcy interface. However, theirintensity is limited and their consistency effect is limited to few nodes. Furthermore, they donot affect the consistency and the convergence of the method.

a) b)Figure 6: Pressure field (a) and velocity field (b) for permeability k = 10−14m2, perpendicular flow.

5.2 Parallel flow

This simulation is for a flow parallel to the interface (horizontal) to validate the Stokes-Darcycoupling presented in this paper, and more particularly the enforcement of the BJS condition.

Let Ω be the computational domain divided into a purely fluid domain and a porous medium.The boundary conditions considered on ∂Ω in velocity and in pressure are shown in Figure7. The physical parameters for this simulation are µ = 1 Pa.s, α = 1 , p = 105pa andk = 10−14m2. Results for pressure and velocity fields are presented in Figure 8. We obtain thisresults by using structured mesh. For this flow, the constants in τu,c and τp,c are choosen as:c1 = 1, cp = 10 , cu = 0.5 and L0 = 1. Computed results coincide with analytical ones. Inthe purely fluid domain, the velocity normal to the interface (vy ) is equal to zero, while thetangential velocity (vx), solution of the Stokes equations, and verifying the BJS condition onthe interface Γ and the condition v = 0 on the upper side (i.e. for y = H), can be analyticallycalculated:

vx = −K2η

(λ2 + 2αλ

1 + αλ

)dpdx

(1 +

α√Ky)+

1

2η

(y2 + 2αy

√K)dpdx

(24)

vy = 0

10


Figure 7: Boundary conditions for a flow parallel to the interface.

with λ = H√K

. In the porous medium, the computed velocity is equal to the theorical values :

vx = −Kη∇p (25)

vy = 0

a) b)

Figure 8: Velocity field (a) and pressure field (b) for permeability k = 10−14m2, parallel flow

5.2.1 3D simulation.

For the case of parallel flow, 3D simulations were conducted. The physical parameters forthis simulation are µ = 1Pa.s, α = 1, k = 10−14m2 and p = 105Pa. Geometry and boundaryconditions are the same as in 2D case extruded along axis z. We present results for pressure andvelocity fields in Figure 9. We use unstructed mesh in this case. These results are in accordancewith the results obtained for the 2D case.

6 CONCLUSIONS

The monolithic approach has been developped to solve Stokes-Darcy coupled problem. Theoriginality of this method is the use of one single mesh and the use of level set to describe theStokes-Darcy interface which passes through the mesh. This approach is now perfectly robust

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a)

b)

Figure 9: Velocity field (a) and pressure field (b) for permeability k = 10−14m2, parallel flow.

due to the introduction of the ASGS sub-grid scale stabilization, for permeabilities down to10−15m2. Convergence and robustness of the method is validated. At the end of the paper,2D and 3D simulations of parallel and perpendicular flow were presented. Flows on complexshapes were performed. The future work and developments will be dedicated to couple the resinflow and the thermo-physico-chemistry of the resin.

REFERENCES

[1] P. Celle, S. Drapier, and J-M. Bergheau. Numerical modelling of liquid infusion intofibrous media undergoing compaction. European Journal of Mechanics, A/Solids(27)(2008), 647–661.

[2] A. Masud and T.J. R. Hughes. A stabilized mixed finite element method for Darcy flow.Computer Methods in Applied Mechanics and Engineering, 191(39-40) (2002), 4341-4370.

[3] G. Pacquaut, J. Bruchon, N. Moulin, and S. Drapier. Combining a level set method andmixed stabilized p1/p1 formulation for coupling stokes-darcy flows. International Journalfor Numerical Methods in Fluids, doi: 69,459-480,2012.

[4] S.Badia and R.Codina. Stabilized Continuous and discontinuous Galerkin techniquesfor Darcy flow. Computer Methods in Applied Mechanics and Engineering,199,654-1667,2010.

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[5] S.Badia and R.Codina. Unified stabilized finite element formulation for the Stokes and theDarcy problems. SIAM journal on Numerical Analysis,47,1971-2000,2008.

[6] P. Wang, S. Drapier, J. Molimard, A. Vautrin and J-C Minni, Numerical and experimentalanalyses of resin infusion manufacturing processes of composite materials. Journal ofComposite Materials, 46(13): 1617-1631, 2012.

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MONOLITHIC APPROACH OF STOKES-DARCY COUPLING FOR …bruchon/conferences/Abouorm_ECCOMAS2012.pdf · structures with organic matrix, especially for large pieces in aeronautics. Resin