MONOLITHIC APPROACH OF STOKES-DARCY COUPLING FOR THE SIMULATION OF LIQUID INFUSION PROCESS

Vienna, Austria, September 10-14, 2012

MONOLITHIC APPROACH OF STOKES-DARCY COUPLING FOR

THE SIMULATION OF LIQUID INFUSION PROCESS

Lara Abouorm1_{, Sylvain Drapier}1_{, Julien Bruchon}1_{, Nicolas Moulin}1 1_{Centre SMS - UMR CNRS 5146}

´

Ecole Nationale Superieure de Mines de Saint-Etienne Saint Etienne, France

e-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 with the finite element method, composite manufacturing processes. This model allows to represent the flow of a liquid resin into fibrous preforms undergoing large deformations. The numerical model is based on the coupling between the resin flow within a porous medium (Darcy) and a purely domain (Stokes). The weak formulation is obtained by summing up the variational forms of the Stokes and the Darcy equations over the whole domain. In both the purely fluid domain and 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 originality of the model consists in using one single unstructed mesh to represent the Stokes and Darcy subdomains (monolithic approach). A level set context is used to represent the Stokes-Darcy interface and to capture the moving flow front. The monolithic approach, stabilized with ASGS, can now be used to solve 2D and 3D problems.

1 INTRODUCTION

Manufacturing processes by resin infusion are competitive routes to elaborate composite structures 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 their plane. The resin is brought into a flow and infused through the preforms due to the pressure applied on the top of the mold (Figure 1). The physical and mechanical properties of the final part (the final thickness and the final fiber volume fraction) are hardly predictable. To help to the 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 (the fibrous preforms). This paper proposes a robust finite element solution for coupling flows in both purely fluid region, ruled by Stokes equations, and fibrous preforms region governed by Darcy flow. The Stokes-Darcy coupled problem has been studied by many ressearchers in many field 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 using two 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 mixed finite elements stabilized with hierarchical-based bubble function, i.e P1+/P1 finite element in Stokes domain and HVM (Hughes Variational Multiscale) for stabilization in Darcy [2, 3]. In this paper we use a robust approach which yields improvements compared to this previous ap-proach, [3]. The robustness of the apap-proach, which is assesed in this paper, is ensured by using ASGS method (Algebraic Subgrid Scale) [4, 5] to stabilize velocity and pressure approximated by linear and continous elements in Stokes and Darcy domains. Signed functions are used to represent the Stokes-Darcy interface and to capture the moving flow front. In this paper, we focus on Stokes-Darcy coupling without taking into consideration either the evolution of the front 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 domainΩformed by two non overlapping subdomains Ωs and Ωd separated by a surfaceΓ = ∂Ωs ∩∂Ωd. Index s is used to denote everything that concerns the purely fluid part (Stokes’ domain) and indexdfor modelling porous medium (Darcy’s domain). Ωsis the region occupied by the fluid, the motion of which is described by the Stokes equations. The incompressible newtonian fluid flow inΩd, the porous medium with low permeability, is governed by a Darcy law (Figure 2).

onΩssolution of

−div(2µε˙(vs)) +∇ps =fs in Ωs −divvs =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 andt is the stress vector to be pre-scribed onΓs,N,vsandpsare the velocity and pressure fields,µdenotes dynamic viscosity,ε˙is the Eulerian strain rate tensor, andσsis the Cauchy stress tensor. If the fluid is incompressible

thenhs = 0.

Figure 2: Computational domain.

The Darcy equations are expressed as: find the velocity vd and the pressurepd defined on

Ωd, solution of µ kvd+∇pd =fd in Ωd −divvd =hd in Ωd (2) vd.nd = 0 on Γd,D pd =pexton Γd,N

wherekis the permeability tensor reduced to a scalar in the isotropic case considered here,pext is a pressure to be prescribed onΓd,N andndis the outward unit vector normal to the boundary

ofΩd.

In Stokes-Darcy coupled problem, boudary conditions must be considered on the interfaceΓ

of normaln=ns. These conditions are:

Continuity of normal velocity:

vs.ns+vd.nd = 0 on Γ (3)

Continuity of the fluid normal stress:

Beaver-Joseph-Saffman condition:

2n.ε˙(vs).τj =

α

√

k(vs.τj) (5)

whereαis a dimensionless parameter, so-called slip coefficient andτjare the tangentiel vectors

on 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 forfor-mulation of Stokes and Darcy separatly. The weak formulation of the coupled problem is obtained by summing up the weak formulation of Stokes and Darcy taking into consideration interface conditions described in Section 2. For the sake of simplicity, we choose to write theL2 _{inner product inΩ}

d,sas<, >. The spaces of velocity, pressures and test functions are defined by :

L2(Ωi) = {q,

Z

Ωi

q2dΩ<∞}

H1(Ωi)m ={q ∈L2(Ωi)m,∇q∈L2(Ωi)m×m}

H_Γi,D1 (Ωi)m ={q∈H1(Ωi)m |q = 0surΓi,D} withi=sori=dand 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) wherewandqare weighting functions defined inH1

Γs,D(Ωs)m andL2(Ωs)respectively. The bilinear formBsand the linear formLsare 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 a

velocity-pressure pair[v, p]such that :

Bd([v, p],[w, q]) = Ld([w, q]) (8) wherewandqare weighting functions defined inH1

Γd,D(Ωd)mandL2(Ωd). The bilinear formBdand the linear formLdare defined in Darcy by :

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

µ

k <v,w>−< p,∇.w>−< q,∇.v>

whereσnis 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 9 and taking into consideration the conditions imposed on the Stokes-Darcy interface described in Section 2. Hence, the variational formulation of the Stokes-Darcy coupled problem consist in finding[v, p] ∈V ×Qsuch that :

Bc[(v, p),(w, q)] = Lc([w, q] where[w, q]are weighting functions defined inV ×Q.

V =H_Γs,D1 ×H_Γd,D1 (div,Ωd)

Q=L2(Ωs)×L2(Ωd) The bilinear formBcand the linear formLcare defined by :

Bc([v, q],[w, q]) = Z Ω 2µε˙(v) : ˙ε(w)HsdΩ (10) + Z Ω µ k v.wHddΩ − Z Ω p divw dΩ− Z Ω q divvdΩ + Z Γ αµ √ kv.wHddΓ 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 1 in domainiand vanishing elswhere.

4 FINITE ELEMENT METHOD FOR STOKES-DARCY PROBLEMS

The whole computional domainΩ⊂Rd_{is discretized with one single unstructed mesh. This}

mesh is made up of triangles K if m = 2and of tetrahedrons K if m = 3. LetVh andQh be the finite element spaces of the linear and continous elementsvh andph. 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 interinter-polations pairs are known to satisfy this condition for each problem independently, but the key issue is to find interpolations that satisfy both at the same time. In this paper, we choose the use of stabilized finite element methods. The philosophy of the stabilized methods is to strenghthen classical variational for-mulations so that discrete approximation which would otherwise be unstable becomes stable and convergent. One of the stabilized finite element method approximation for Stokes-Darcy problem is VMS (Variational Multiscale Method) [4], [5]. The basic idea of this method is to approximate the effect of the component of the continous solution which cannot be captured by the 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 a finer component corresponding to lower scale[v0, p0] for resolutions. The velocity is approxi-mated as v = vh+v0 and the pressure field is approximated as p = ph +p0. We consider a subgrid spaceV ×Q= (Vh×Qh)L(V0×Q0).Invoking this decomposition in the continous problem for both the solution and test functions, we get the two scales systems:

Bc([vh, ph],[w0, p0]) +Bc[v0, p0],[w0, q0]) =Lc([w0, q0]) (12) for all[wh, qh]∈Vh×Qhand[w0, q0]∈V0×Q0. After approximating (12) with an algebraic for-mulation, by introducing the operator of projectionP0 ontoV0, the approximated fields[v0, p0]

are taken into account in the finite element problem (11), we get the stabilized forms of the bilinear and linear forms in Stokes, Darcy, and Stokes-Darcy coupled problem. The stabilized problem in Stokes can be written as follow:

Find[vh, ph]∈Vh×Qhsuch as :

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

(13)

The bilinear stabilized formBs,stableand the linear stabilized formLs,stableare defined by :

Bs,stable([vh, ph],[wh, qh]) = Bs([vh, ph],[wh, qh]) (14) +τq X K < P_h,p0 (∇.vh),∇.wh > + τv X K < P_h,u0 (−µ∆vh+∇ph), µ∆wh− ∇qh > Ls,stable([wh, qh]) = Ls([wh, qh]) +τq X K < P_h,p0 (hs),∇.wh > + τv X K < P_h,p0 (fs), µ∆wh− ∇qh >

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

τq = c1µ

τv = c1µ h2k (15)

c1is an algorithmic constant andhkis the size of mesh.Ph,u0 is the brokenL2projection ontoV

0

andP_h,p0 is the brokenL2_{projection ontoQ}0_{. The simplest approach is to takeP}0

has the identity operator when acting on the FE residual. Assuming this, we obtain the stabilized method that we called Algebraic Subgrid Scale (ASGS) method. Invoking, this expression ofP_h0, we get the following stabilized finite formBsandLs in Stokes:

Bs,stable([vh, ph],[wh, qh]) = Bs([vh, ph],[wh, qh]) (16) +τq X K <∇.vh,∇.wh)> + τv X K <−µ∆vh+∇ph, µ∆wh− ∇qh > Ls,stable([wh, qh]) = Ls([wh, qh]) +τq X K < h,∇.wh > + τv X 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:

Bd,stable([vh, ph][wh, qh]) = Ld,stable([wh, qh]) (17) Bd,stable([vh, ph],[wh, qh]) = Bd([vh, ph],[wh, qh]) +τp X K <∇.vh,∇.wh)> + τu X K < µ kvh+∇ph,− µ kwh− ∇qh > Ld,stable([wh, qh]) = Ld([wh, qh]) +τp X K < hd,∇.wh > + τu X K <fd,− µ kwh− ∇qh > (18)

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

τp = cp µ kl 2 p τu = (cu µ klu) −1 h2_k (19)

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

(L0hk)2,L0 is a characteristic length of domain andhkis 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×Qhsuch as : Bc,stable([vh, ph][wh, qh]) = Lc,stable([wh, qh]) (20) Bc,stable([vh, ph],[wh, qh]) = 2µ Z Ω Hsε˙(vh) : ˙ε(wh) + µ k Z Ω Hdwhvh − Z Ω ∇.whph − Z Ω ∇.vhqh+τp,c Z Ω ∇vh∇wh + Z Γ α√µ k(vh.τ)(wh.τ) + τu,c Z 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 = cp µ kl 2 p+c1µ τu,c = (c1µ+cu µ klu) −1_h2 k (22)

whereHi is a heaviside function equal to 1 in domainiand vanishing elsewhere. The surface integralR

Γα

µ

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-Darcy interface, 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, the method developped in [3] shows consistency errors and spurious oscillations of the velocity in the vicinity of the interface in this situation. That is why we investigate the convergence of our approach in the perpendicular case, and additionally, we compare our results to those obtained in [3].

5.1 Perpendicular flow

To validate continuity of normal velocity on Stokes-Darcy interface, we study a test case of a flow perpendicular to the interface of normalyin the global frame. Let us consider a domain

Ω = [0; 5]×[0; 2]m2_{composed of two sub-domains: a pure fluid domainΩ}

s= [0; 5]×[1; 2]m2 and porous mediumΩd= [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 the

bottom. Other boundary conditions are zero normal velocity on the left and right hand sides of the geometry (Figure 3). For this flow, the constants inτu,c andτp,c are choosen as:

c1 = 1,cp = 10,cu = 0.5andL0 = 1.

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

Figure 4: Pressure fieldpfor permeabilityk = 10−14_m2_.

We present results for pressure field in Figure 4 for k = 10−14m2. The small values of the permeabilites (down to 10−14_m2_{) are realistic permeabilities met in LCM processes [6] .}

The pressure distribution doesn’t depend on the permeability. Permeability tests forkdown to

the velocityvy obtained by using ASGS method with the one obtained by using P1+P1/HVM method [3]. Tests are realized on structured and unstructured meshes For permeability k = 10−14m2, normal velocity must be equal to10−9m/s. Figure 5 shows oscillations of velocity and consistency error in Stokes for the P1+/P1-HVM method contrary to ASGS method where normal 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 of the perpendicular flow. We compare the obtained numerical results for ASGS and HVM/P1+P1 methods with analytical ones, where

vs,x=vd,x = 0,

vs,y =vd,y =−_µk∇p=−10−9m/s,

(23)

ASGS HVM/P1+P1 [3]

h,[m] kvy,errork_stokes, [%] kvy,errork_darcy, [%] kvy,errork_stokes, [%] kvy,errork_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 = 10−14_m2_{, perpendicular flow.}

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

kva−vhkL2

kvak_L2 , wherevais the analytical solution andvhis the obtained numerical

solution,kvk_L2 = ( R

Ωv2dΩ) 1/2

flow for different sizes of the mesh, whenµ= 1 P a.s,k = 10−14 _m2 _{presented in Table 1 for}

both ASGS and P1+P/HVM method. Orders of convergence are also studied using manufac-tured solutions. We present manufacmanufac-tured solutions in another paper, but the method confirms the theoritical order of convergence in bothL2_andH1_norms.

5.1.2 3D simulation.

For the case of perpendicular flow, 3D simulations were conducted. Geometry and boundary conditions are the same as in 2D case, extruded along axis z. We use unstructed mesh in this case. We present results for pressure and velocity fields in Figure 6. These results are in accordance with the results obtained for the 2D case. Normal velocity is continous, but for unstructed meshes, some oscillations appear around the Stokes-Darcy interface. However, their intensity is limited and their consistency effect is limited to few nodes. Furthermore, they do not affect the consistency and the convergence of the method.

a) _b)

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

5.2 Parallel flow

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

LetΩbe 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 Figure 7. The physical parameters for this simulation are µ = 1 Pa.s, α = 1 , p = 105pa and

k = 10−14m2. Results for pressure and velocity fields are presented in Figure 8. We obtain this results by using structured mesh. For this flow, the constants inτu,candτp,c are choosen as:

c1 = 1, cp = 10, cu = 0.5andL0 = 1. Computed results coincide with analytical ones. In

the purely fluid domain, the velocity normal to the interface (vy ) is equal to zero, while the tangential velocity (vx), solution of the Stokes equations, and verifying the BJS condition on the interfaceΓand the condition v = 0on the upper side (i.e. fory =H), can be analytically calculated: vx = − K 2η λ2+ 2αλ 1 +αλ dp dx 1 + √α Ky + 1 2η y2+ 2αy√Kdp dx (24) vy = 0

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

withλ = √H

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

vx = −

K

η∇p (25)

vy = 0

a) b)

Figure 8: Velocity field (a) and pressure field (b) for permeabilityk = 10−14_m2_{, parallel flow}

5.2.1 3D simulation.

For the case of parallel flow, 3D simulations were conducted. The physical parameters for this simulation areµ= 1P a.s, α = 1,k = 10−14_m2 _{andp = 10}5_{P a. Geometry and boundary}

conditions are the same as in 2D case extruded along axisz. We present results for pressure and velocity fields in Figure 9. We use unstructed mesh in this case. These results are in accordance with the results obtained for the 2D case.

6 CONCLUSIONS

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

a)

b)

Figure 9: Velocity field (a) and pressure field (b) for permeabilityk = 10−14_m2_{, parallel flow.}

due to the introduction of the ASGS sub-grid scale stabilization, for permeabilities down to

10−15m2. 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 complex shapes were performed. The future work and developments will be dedicated to couple the resin flow and the thermo-physico-chemistry of the resin.

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