Coupling Stokes and Darcy equations: modeling and numerical methods

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1 Coupling Stokes and Darcy equations: modeling and numerical methods Marco Discacciati NM2PorousMedia 24 Dubrovnik, September 29, 24 Acknowledgment: European Union Seventh Framework Programme (FP7/27-23), Marie Curie grant

Motivation We want to model the filtration of a free fluid through a porous medium: Environmental application Biomedical simulations 5. 9.96 9.9 9.84 9.78 9.72 9.66 9.6 9.54 9.48 9.42 9.36 9.3 9 9.

2 Motivation We want to model the filtration of a free fluid through a porous medium: Environmental application Biomedical simulations [simulation by E. Miglio] [simulation by P. Zunino] Industrial applications: filters, porous foams, fuel cells... We must introduce a suitable modeling framework and set up effective numerical methods.

3 Modeling filtration (i) To model the filtration process three different approaches are generally considered: I - Microscale II - Mesoscale III - Macroscale [Chandesris and Jamet, Transp. Porous Med., 29]

4 Modeling filtration (ii) I - Microscale approach The geometry of the porous medium must be taken into account and the Navier-Stokes equations must be solved in the whole domain. [R. Hilfer, Lecture Notes in Physics 554, p (2)]

5 Modeling filtration (iii) II - Mesoscale approach A transition region is considered between the porous medium and the free fluid [Whitaker and Ochoa-Tapia (995); Jackson et al (22)]. Penalization approach [Iliev et al. (24,27); Angot (999)]: divσ(u,p)+(u )u+ nν K u+ C F K u u = f divu = in Ω f Ω d The penalization terms are set to zero in Ω f.

6 Modeling filtration (iv) III - Macroscale approach Sharp interface Γ with continuity conditions across it. Γ Ω f Ω d Free fluid Porous medium Free flow: Navier-Stokes equations divσ(u f,p f )+(u f )u f = f f divu f = in Ω f where σ(u f,p f ) = 2µ s u f p f I is the Cauchy stress tensor. Porous media flow: Darcy equation u d + K µ ( p ( ) d f d ) = K div divu d = µ ( p d f d ) = in Ω d

7 Modeling filtration (v) Continuity conditions (I) Continuity of normal velocities (due to incompressibility) u f n = u d n on Γ 2 Continuity of normal stresses n σ(u f,p f ) n = p d on Γ (pressures may be discontinuous across Γ: p f p d on Γ); 3 Beavers Joseph Saffman condition τ σ(u f,p f ) n = α d µ τ K τ u f τ [Beavers and Joseph (967); Saffman (97); Jäger and Mikelić (996,2,2) Miglio et al (22); Layton et al (23)]

8 Modeling filtration (vi) Continuity conditions (II) Continuity of normal velocities (due to incompressibility) u f n = u d n on Γ 2 Continuity of pressures p f = p d on Γ [Ene and Sanchez-Palencia (975); Levy and Sanchez-Palencia (975)]

9 Modeling filtration (vii) The microscale approach is very demanding: careful reconstruction of the geometry and solution of the Navier-Stokes equations. The macroscale approach allows using ad-hoc models in each subdomain. Once the coupling conditions are set, a global well-posed problem may be set up and non-overlapping domain decomposition methods can be used to solve the corresponding interface problem (substructuring techniques). The mesoscale approach requires solving a more expensive problem in Ω f Ω d than the macroscale approach.

10 Coupling through a sharp interface: domain decomposition methods without overlap (macroscale setting)

11 Weak form of the Stokes-Darcy problem Find u f H (Ω f ), p f L 2 (Ω f ), p d H (Ω d ): 2µ s u f : s α d µ v+ (u f ) τ v τ p f divv Ω f Γ τ K τ Ω f + p d (v n) = f f v Γ Ω f q f divu f = Ω f q d K Ω d µ p K d (u f n)q d = Γ Ω d µ f d q d

12 Algebraic form A suitable conforming finite element approximation of the coupled problem leads to the linear system: (A f +ᾱm Γ ) II (A f ) IΓ D T I (A f ) ΓI (A f ) ΓΓ D T Γ C T Γ D I D Γ C Γ A d (u f ) I (u f ) n p f = b p d with (u f ) n vector of the nodal values of u f n on Γ

13 An equivalent interface equation The Stokes-Darcy problem can be formulated in terms of the solution λ (normal velocity across Γ) of the interface problem S f fluid operator: S f : λ (normal velocities on Γ) S f λ+s d λ = χ on Γ solve n σ(u f,p f ) n (normal stresses on Γ). Stokes S d porous medium operator: S d : λ (normal velocities on Γ) solve Darcy p d Γ (pressure p d on Γ). Discrete interface equation for the normal velocity: (Σ f +Σ d )(u f ) n = χ s +χ d [MD, Quarteroni (23)]

14 Dimensional analysis of the Stokes-Darcy problem The Stokes-Darcy problem can be characterized in terms of three dimensionless numbers: the Reynolds number Re = U fx f ρ µ the Euler number Eu = P f ρu 2 f the Darcy number [Bear (99)] Da = K X 2 f [MD (in preparation)]

15 Dimensionless Schur complement system The Schur complement system with respect to (u f ) n becomes: (Σ s +Σ d )(u f ) n = χ s +χ d with Σ f = (ReEu) ˆΣf Σ d = (ReEuDa) ˆΣd and ˆΣ f = (A f ) ΓΓ (A f ) ΓI (A f +αda /2 M Γ ) II (A f IΓ ( ) + (A f ) ΓI (A f +αda /2 M Γ ) II D T I D T Γ ( ) D I (A f +αda /2 M Γ ) II D T I ( ) D I (A f +αda /2 M Γ ) II (A f ) IΓ D Γ ˆΣ d = C T Γ A d C Γ

16 If we multiply by (ReEuDa), we obtain: (Da ˆΣ f + ˆΣ d )(u f ) n = ˆχ f + ˆχ d Remark that we have independence of Re. We can prove that cond((da ˆΣ f + ˆΣ d )) c k + k Dak +h Dak + + k k+ h h Re = 2 Da: h = / Re = h = / h = / e+3.289e+3 h = / e e+3 h = / h = / e+3.289e+3 h = / h = / e e+3

17 Dirichlet-Neumann type preconditioners (i) It can be proved that cond((da ˆΣ f ) (ˆΣ f + ˆΣ d )) C + k Da + k h 2 k + k + Da from which we can see that if Da h 2, then cond((da ˆΣ f ) (ˆΣ f + ˆΣ d )) if Da h 2, then cond((da ˆΣ f ) (ˆΣ f + ˆΣ d )) h 2 PCG iterations, tol =, Re = Da: h = / h = / h = / h = / no prec. Da: h = / h = / h = / h = / P = (Da ˆΣ f )

18 Dirichlet-Neumann type preconditioners (ii) It can be proved that cond(ˆσ d (ˆΣ f + ˆΣ + k + k + Da k d )) C h 2 +k Da from which we can see that if Da h 2, then cond(ˆσ d (ˆΣ f + ˆΣ d )) h 2 if Da h 2, then cond(ˆσ d (ˆΣ f + ˆΣ d )) PCG iterations, tol =, Re = Da: h = / h = / h = / h = / no prec. Da: h = / h = / h = / h = / P = (ˆΣ d )

19 Comments Using (ˆΣ f ) requires solving a Stokes problem: y = ˆΣ f w y = u f n on Γ where Stokes(u f,p f ) = in Ω f n σ(u f,p f ) n = w on Γ Using (ˆΣ d ) requires solving a Darcy problem: y = ˆΣ d w y = K p d µ n div( K µ p d on Γ where ) = in Ω d p d = w on Γ

.6.4.2.8.6.4.2 Numerical results (i) We consider the following setting proposed in [Hanspal et al, 29]:.2.5.5.5 2 2.5 3 3.5 4 - Ω f =.5.5 m, Ω d =.5.25 m - Inflow with max horizontal velocity.

20 Numerical results (i) We consider the following setting proposed in [Hanspal et al, 29]: Ω f =.5.5 m, Ω d =.5.25 m - Inflow with max horizontal velocity. m/s, Re = 5 K: m 2 2 m 2 4 m 2 m 2 2 m 2 4 m 2 Da: h = / h = / h = / >25 >25 no prec. P = (Da ˆΣ f ) m 2 2 m 2 4 m P = (ˆΣ d )

21 Numerical results (ii) Following [Zunino (22)] we consider an artery with - radius =.3 cm - wall thickness =.3 cm - blood density ρ b =.4 g/cm 3 - blood (kinematic) viscosity ν b =.33 cm 2 /s - wall permeability K = 2 4 cm 2 This corresponds to - Re = Da = Dofs (u f + p f + p d ) h No prec. PCG with ˆΣ d / / / /

22 Numerical results (iii) Ω f = (,) (,2), Ω d = (,) (,) Re = Variable permeability 8 K 6 m 2 (Da = 3.3 ) Iterations Relative residual (PCG with ˆΣ d ) (tol = ) Velocity field Vertical velocity Hydrodynamic pressure x

23 Coupling through a transition region: interface control domain decomposition methods (mesoscale setting) Joint work with: Paola Gervasio (Università di Brescia) Alfio Quarteroni (EPF Lausanne)

24 The ICDD method for the Stokes-Darcy problem (i) Similarly to the mesoscale approach, we introduce a transition region overlapping Ω f and Ω d : Γ 2 Ω f Ωfd Γ Stokes divσ(u f,p f ) = f f in Ω f div u f = in Ω f Φ f (u f,p f ) = λ on Γ Ω d Darcy ( ) K div µ ( p d f d ) = in Ω d Φ d (p d ) = λ 2 on Γ 2 λ, λ 2 control interface functions

25 The ICDD method for the Stokes-Darcy problem (ii) Consider controls corresponding to the traces of the Stokes velocity and the Darcy pressure on the interfaces: u f = λ on Γ and p d = λ 2 on Γ 2. We set the minimization problem: inf λ,λ 2 J(λ,λ 2 ) with J(λ,λ 2 ) = 2 u f u d 2 L 2 (Γ ) + 2 p f p d 2 L 2 (Γ 2 ) [MD, Gervasio, Quarteroni (submitted 24)]

26 The ICDD method for the Stokes-Darcy problem (iii) The corresponding optimality system becomes: state problems: divσ(u f,p f ) = f f in Ω ( ) f K div u f = in Ω div f µ ( p d f d ) = in Ω d u f = λ on Γ p d = λ 2 on Γ 2 pseudo-adjoint problems: divσ(w f,q f ) = in Ω ( ) f K div w f = in Ω div f µ ψ d = in Ω d w f = u f u d on Γ ψ d = p d p f on Γ 2 pseudo-euler-lagrange equations: (u f u d )+ψ d = on Γ (p d p f )+q f = on Γ 2.

27 Numerical results (i) BiCGStab iterations: Dependence on K K iter inf J 9.6e-2 6.8e e-25.3e-25 7.e e-3.5e-24 u f u d L 2 (Ω fd ) 3.9e-6.7e-6.9e-6.9e-6.9e-6.9e-6.9e-6 p f p d L 2 (Ω fd ) 4.5e- 2.7e-9 4.7e-9 5.2e-9 5.3e-9 5.3e-9 5.3e-9 Dependence on δ δ iter (K = 8 ) 4 4 iter (K = 2 ) 2 2 Physical considerations lead to choose δ K. Dependence on h h iter (K = 8 ) iter (K = 2 ) 2 2 2

28 Comparison with the sharp-interface solution (K = 7 m 2 ) ICDD: Sharp interface (Beavers-Joseph-Saffman): Horizontal velocity Vertical velocity Hydrodynamic pressure

29 Comparison with the sharp-interface solution (K = 2 m 2 ) ICDD: Sharp interface (Beavers-Joseph-Saffman): Horizontal velocity Vertical velocity Hydrodynamic pressure

30 Comparison with the sharp-interface solution Stokes domain K = 7 m y 2 2 x y 2 x y K = 2 m 2 x y 2 2 x y 2 x y u ICDD u BJS u BJS v ICDD v BJS v BJS p ICDD p BJS p BJS x 2 3

31 Numerical results (ii) K =, K 2 = 4, K 3 = 6 m 2 δ =.5%, convergence in 4 iterations (tol = 9 ) ICDD: Velocity field Hydrodynamic pressure Sharp interface (Beavers-Joseph-Saffman):

Numerical results (iii) K = 7, K 2 = m 2, Re

Stokes Hydrodynamic pressure Darcy This case

32 Numerical results (iii) K = 7, K 2 = m 2, Re =. δ =.5%, convergence in 2 iterations (tol = 9 ) ICDD: Velocity field Hydrodynamic pressure Stokes Hydrodynamic pressure Darcy This case is not meaningful using Beavers-Joseph-Saffman: ICDD can treat more general fluid regimes.

33 Conclusions Sharp-interface approach: An interface problem and ad-hoc preconditioners for the Schur complement system (Steklov-Poincaré equation). Transition-region approach: A minimization problem which takes into account the physical properties of the problem without relying on any specific interface conditions.

35 Optimized Schwarz (Robin-Robin type) preconditioner (i) How to deal with intermediate values of Da? Considering the Schur complement of the linear system associated to the Stokes-Darcy problem with respect to both (u f ) n and p d Γ, we obtain the following augmented interface system: where Σ p Σ d. ( Σs C T )( ) Γ (uf ) n C Γ Σ p p d Γ ( ) χs = χ p [MD (23)]

36 Optimized Schwarz (Robin-Robin type) preconditioner (ii) Consider the Robin-Robin method: Solve the Stokes problem in Ω f with interface conditions on Γ: τ σ(u (k) f,p (k) f ) n = α dµ τ K τ u (k) f τ n σ(u (k) f,p (k) f ) n α f u (k) f n = p (k ) d α f u (k ) d n Solve the Darcy problem in Ω d with interface condition on Γ: p (k) d +α p u (k) d n = n σ(u (k) f,p (k) f ) n+α p u (k) f n where the parameters α f and α p must be chosen in a suitable way to ensure (optimal) convergence.

37 Optimized Schwarz (Robin-Robin type) preconditioner (iii) The Robin-Robin method can be proved to be equivalent to a Gauss-Seidel scheme for the augmented interface Schur complement system multiplied by the preconditioner: P α p I OS = ( I ) αf I I which can be used in the framework of Krylov methods like GMRES or BiCGStab. The theory of Optimized Schwarz methods allows to characterize α f and α p to maximize the convergence rate of the method. [MD, Gerardo-Giorda (in preparation)]

38 Numerical results (i) Domain Ω f = (,) (,2) and Ω d = (,) (,). Solution u f = ((y ) 2 +(y )+ Da,x(x )) p f = 2 (x +y ) ReEu ( p d = (x( x)(y )+(y ReEu Da )3 /3)+2x ) We set Re = 3 and Da = 6, 8,, 2 m 2. Using optimized Schwarz methods we can estimate α f = , α p =.528. Da: h = / h = / h = / BiCGStab iterations with tol =

39 .5.5 Stokes velocity field Numerical results (iii) Setting as in [Hanspal et al, 29] but with discontinuous permeability (reduced by one order of magnitude) and inflow from the left (no-slip conditions on the remaining fluid boundary): Darcy pressure Here Re = 5 and Da between.34 5 and Optimal coefficients: α f = 3.24 and α p =.24. h: 3/ 3/2 3/4 elements iterations BiCGStab iterations with tol =

40 The ICDD method ICDD is inspired to both the Virtual Control Method by J.L. Lions, O. Pironneau (998) and the Least Squares Method by R. Glowinski, Q. Dinh, J. Periaux (983). VCM/LSM ICDD J(λ) = 2 u u 2 L 2 (Ω 2 ) J(λ) = 2 u u 2 L 2 (Γ Γ 2 ) distributed observation interface observation classical Optimality System non-standard OS (integration by parts) Interface observation makes ICDD more suitable for heterogeneous problems and it leads to more efficient algorithms than distributed observation.

Iterative Methods for Stokes/Darcy Coupling

Iterative Methods for Stokes/Darcy Coupling Iterative Methods or Stokes/Darcy Coupling Marco Discacciati Ecole Polytechnique Fédérale de Lausanne Institut d Analyse et Calcul Scientiique Chair o Modelling and Scientiic Computing marco.discacciati@epl.ch