A monolithic FEM solver for fluid structure

Transcription

1 A monolithic FEM solver for fluid structure interaction p. 1/1 A monolithic FEM solver for fluid structure interaction Stefan Turek, Jaroslav Hron jaroslav.hron@mathematik.uni-dortmund.de Department of Applied Mathematics, LS III, University of Dortmund

2 A monolithic FEM solver for fluid structure interaction p. 2/1 Fluid structure interaction large deformation structure in internal/external flow (bioengineering) pulsative flow in large blood vessel with obstacles flow through heart flaps flow in a heart ventricle... physical model parts to deal with viscous fluid flow elastic body undergoing large deformations numerical tasks involved space and time discretization nonlinear system solution solution of large linear system the interaction between the two parts

3 A monolithic FEM solver for fluid structure interaction p. 3/1 Problem description Γ 2 Γ 1 Γ3 Ω s Ω f Γ 0 χ s χ f Γ 2 t Γ 1 t Γ 0 t Γ 3 t Ω s t Ω f t reference configuration current configuration structure part fluid part χ s : Ω s [0, T ] Ω s t u s = χ s (X, t) X, v s = us t F = I + Grad u s, J = det F χ f : Ω f [0, T ] Ω f t u f = χ f (X, t) X v f : Ω f t [0, T ] Rn

4 A monolithic FEM solver for fluid structure interaction p. 4/1 Governing equations structure part fluid part v s t = div(jσ s F T ) + f in Ω s det(i + u s ) = 1 in Ω s u s = 0 on Γ 2 σ s n = 0 on Γ 3 v f t + ( v f )v f = div σ f + f in Ω f t div v f = 0 in Ω f t v f = v 0 on Γ 1 t or σ f n = 0 on Γ 1 t interface conditions v f = v s σ f n = σ s n on Γ 0 t on Γ 0 t

5 A monolithic FEM solver for fluid structure interaction p. 5/1 Arbitrary Lagrangian-Eulerian Formulation χ : Ω [0, T ] Ω t, v = χ t, F = χ X, J = det F ζ R : R [0, T ] R t, R t Ω t t [0, T ], v R = ζ R t, F R = ζ R X, J R = det F R t Z Z ϱdv + ϱ(v v R ) n Rt da = 0 R t R t t (ϱj R) + div ϱj R (v v R )F T R = 0 Lagrangian description: ζ R = χ F R = F, J R = J, v R = v t (ϱj) = 0 Eulerian description: ζ R = Id F R = I, J R = 1, v R = 0 ϱ t + div(ϱv) = 0

6 A monolithic FEM solver for fluid structure interaction p. 6/1 Governing equations structure part v s t = div(jσ s F T ) + f in Ω s det(i + u s ) = 1 in Ω s u s = 0 on Γ 2 σ s n = 0 on Γ 3 fluid part v f t + ( v f )F 1 v f = div div Jv f F T = 0 Jσ f F T + f in Ω f in Ω f v f = v 0 or σ f n = 0 on Γ 1

7 A monolithic FEM solver for fluid structure interaction p. 7/1 Coupling strategies separated, weak coupling t n fluid t n solid t n+1 fluid t n+1 solid... separated, strong coupling t n fluid t n solid t n+1 fluid t n+1 solid... monolithic t n fluid & solid t n+1 fluid & solid...

8 A monolithic FEM solver for fluid structure interaction p. 8/1 Uniform formulation Ω = Ω f Ω s, u : Ω [0, T ] R 3, v : Ω [0, T ] R 3, 8 < v in Ω s : u ( mesh deformation operator ) in Ω f 8 < div `Jσ s F T in Ω s : β( v)f 1 (v u t ) + div `Jσ f F T in Ω f 8 < J 1 in Ω s 0 = : div(jvf T ) in Ω f u t = β v t = σ f n = σ s n on Γ 0 t v = v B u = 0 σ s n = 0 on Γ 1 t on Γ 2 t on Γ 3 t

9 A monolithic FEM solver for fluid structure interaction p. 9/1 Constitutive equations incompressible Newtonian fluid σ f = pi + ν( v + v T ) ν = ν( D ), D = v + v T hyperelastic material σ s = pi + 2F Ψ F FT where C = FF T and I C = tr C, I C = 1 2 neo-hookean compressible material Ψ(F) =α(i C 3) Ψ(F) =α 1 (I C 3) + α 2 (I C 3) + α 3 ( Fe 1) 2 `tr C 2 (tr C) 2 neo-hookean incompressible material σ s = p s I + µ(ff T I) p s =λ(det F det F 1 ) σ s = p s I + µ(ff T I) det F =I (λ )

10 A monolithic FEM solver for fluid structure interaction p. 10/1 Energy estimate for the system Z c T 2 v(t ) 2 L 2 (Ω T ) + 0 µ v 2 L 2 (Ω f t ) dt + a u(t ) 2 L 2 (Ω s ) b L 1 (Ω s ) v 0 2 L 2 (Ω f ) + β 2 v 0 2 L 2 (Ω s ) U = {u L (I, [W 1,2 (Ω)] 3 ), u = 0 on Γ 2 } V = {v L 2 (I, [W 1,2 (Ω t )] 3 ) L (I, [L 2 (Ω t )] 3 ), v = 0 on Γ 1 } P = {p L 2 (I, L 2 (Ω))}

11 A monolithic FEM solver for fluid structure interaction p. 11/1 Discretization in space and time Discretization in space: FEM Q 2 /Q 2 /P disc 1 y x v h, u h p h, p h x, p h y U h = {u h [C(Ω h )] 2, u h T [Q 2 (T )] 2 T T h, u h = 0 on Γ 1 }, V h = {v h [C(Ω h )] 2, v h T [Q 2 (T )] 2 T T h, v h = 0 on Γ 2 }, P h = {p h L 2 (Ω h ), p h T P 1 (T ) T T h }. Discretization in time: Crank-Nicholson scheme with adaptive time-step selection

12 A monolithic FEM solver for fluid structure interaction p. 12/1 Discrete nonlinear system R(X) =0, X = (u h, v h, p h ) U h V h P h Mu h k 2 (M s v h + L f u h ) = rhs(u n h, vn h ) (M f + βm s )v h + k 2 N 1(v h, v h ) N 2(v h, u h ) + k 2 (Ss (u h ) + S f (v h )) kbp h = rhs(u n h, vn h, pn h ) R X (X) = N 2 u h + k 2 C(u h ) + B f T vh = 1 M k 2 Lf k 2 M 1 s 0 (N 1 +S s +S f ) u + k B h u p h M s + βm f + 1 N 2 h 2 v + k (N 1 +S 2 f ) h 2 v kbc h A B st + Bf T u v h B f T 0 h

13 A monolithic FEM solver for fluid structure interaction p. 12/1 Discrete nonlinear system R(X) =0, X = (u h, v h, p h ) U h V h P h Mu h k 2 (M s v h + L f u h ) = rhs(u n h, vn h ) (M f + βm s )v h + k 2 N 1(v h, v h ) N 2(v h, u h ) + k 2 (Ss (u h ) + S f (v h )) kbp h = rhs(u n h, vn h, pn h ) C(u h ) + B f T vh = S uu S uv 0 S vu S vv kb c u B T s c v B T f u v p = Typical discrete saddle-point problem f u f v f p 3 7 5

14 A monolithic FEM solver for fluid structure interaction p. 13/1 Solution of the nonlinear problem compute the Jacobian matrix (analytic, automatic differentiation or divided differences)» R (X n ) [R] i(x n + εe j ) [R] i (X n εe j ), X 2ε ij solve for δx» R X (Xn ) δx = R(X n ) adaptive line search strategy X n+1 =X n + ωδx ω such that f(ω) = R(X + ωδx) X MG, BiCGStab or GMRes(m) with ILU(k) preconditioner to solve the linear problems

15 A monolithic FEM solver for fluid structure interaction p. 14/1 Jacobian approximation» R X ij (X n ) [R] i(x n + εe j ) [R] i (X n εe j ) 2ε, ε/t OL / [21.52] 12 /57.08 [26.52] 12 /47.00 [23.75] 17 /33.06 [27.38] / [24.57] 8 /62.75 [17.77] 10 /42.20 [18.95] 18 /31.33 [29.05] / [51.65] 20 /47.35 [38.28] 25 /29.80 [38.58] 56 /16.98 [73.83] / [141.30] 48 /35.79 [81.72] 49 /17.92 [65.77] nonlinear solver it. / avg. linear solver it. [CPU time] for BiCGStab(ILU(0))

16 A monolithic FEM solver for fluid structure interaction p. 15/1 Multigrid solver standard geometric multigrid approach smoother by local MPSC-Ansatz (Vanka-like smoother) u l+1 u l 6 4v l = 6 4v l 7 5 ω X S uu Ωi S uv Ωi S vu Ωi S vv Ωi kb 7 Ωi 5 Patch Ω i c u Bs Ω T c v B T 0 i f Ω i p l+1 p l def l u 6 4def l 7 v 5 def l p full inverse of the local problems by standard LAPACK (39 39 systems) alternatives: simplified local problems (3 3 systems) or ILU(k) combination with GMRES/BiCGStab methods possible full Q 2 and P disc 1 prolongation P, restriction by R = P T

17 A monolithic FEM solver for fluid structure interaction p. 16/1 Boundary and initial conditions v f = 0 inflow Γ 0 t outflow v f = 0 inflow parabolic velocity profile is prescribed at the left end of the channel v f (0, y) = 1.5 y(h y) 2 = y(0.41 y), H outflow condition can be chosen by the user (stress free or do nothing) interface condition on Γ 0 t is vf = v s and σ f n = σ s n otherwise the no-slip condition is prescribed for the fluid on the other boundary parts. i.e. top and bottom wall and cylinder

18 A monolithic FEM solver for fluid structure interaction p. 17/1 Examples β = 1 α = ν P = 0.5 ν = (Re = 200) A, y coord A, y coord t t

19 A monolithic FEM solver for fluid structure interaction p. 18/1 Multigrid solver 1 timestep started with fully developed solution standard streamline diffusion, CN time step, each linear step solved to relative prec shown: number of nonlinear steps/avg. number of linear steps [CPU time] timestep 10 2 Level ndof MG(2) MG(4) MG(8) BiCGStab(ILU(1)) GMRES(ILU(1),200) /8 [66] 2/8 [92] 2/7 [112] 2/51 [32] 2/50 [27] /8 [190] 2/5 [198] 2/4 [302] 2/120 [200] 2/117 [151] /9 [744] 2/6 [852] 2/4 [1185] 2/311 [1646] 2/358 [1432] /13 [3803] 2/7 [3924] 2/6 [6241] MEM. MEM. timestep 10 0 Level ndof MG(2) MG(4) MG(8) BiCGStab(ILU(1)) GMRES(ILU(1),200) /12 [118] 4/11 [177] 4/10 [262] 20/160 [631] 20/801 [1579] /12 [466] 4/7 [470] 4/5 [681] 2/800 [] diverg. 13/801 [] diverg /13 [1898] 4/7 [2057] 4/5 [2874] 2/800 [] diverg. 4/801 [] diverg /15 [8678] 4/8 [9069] 4/6 [13808] MEM. MEM. robust and efficient Newton-MG scheme

20 A monolithic FEM solver for fluid structure interaction p. 19/1 Summary monolithic, fully coupled FEM (Q 2 /P 1 ) for viscous incompressible fluid and incompressible hyperelastic structure fully implicit 2nd order discretization in time (Crank-Nicholson) Newton-like method for the coupled system (Jacobian matrix via divided differences) preconditioned Krylov space linear solver (ILU(k)/GMRES(m)) adaptive time step control a priori space-adapted mesh