Méthode de quasi-newton en interaction fluide-structure. J.-F. Gerbeau & M. Vidrascu

Transcription

1 Méthode de quasi-newton en interaction fluide-structure J.-F. Gerbeau & M. Vidrascu

2 Motivations médicales Athérosclérose :

3 Sistole Diastole La formation des plaques est liée, entre autre, la dynamique du sang et aux contraintes exercées sur la paroi. S S Recirculation Zone Ce que pourrait permettre la simulation : mieux comprendre le phénomène prévoir les conséquences d opérations chirurgicales améliorer les prothèses

4 1. Problem statement Purpose : simulate the mechanical interaction between the wall and the blood in a portion of large vessels

5 1.1. General equations Fluid models : Navier-Stokes 2D or 3D (u, p) in Arbitrary Lagrange Euler formulation (w) Structure models : Structure 1D (d = d r e r ) : cylindrical geometry, linear elasticity ρ w h 2 d r t 2 kgh 2 d r z 2 + Eh 1 ν 2 d r R 2 0 γ 3 d r z 2 t = f Σ Structure 2D (d). Shell model in large displacements (MITC) Coupling Fluid-Structure interface Σ t (reference configuration ˆΣ) : < f Σ, ϕ >= u Σt = d t ˆΣ Σ t (pn 2µɛ(u) n) ϕ dσ

6 1.2. Variational formulation Let ˆv and ϕ test fonctions independent of t on ˆΩ, with v = 0 on Σ t. Ω F (t) d dt ρ F u v div w dx Ω F (t) ρ F u v dx + Ω F (t) p div v dx + Ω F (t) ρ F (u w) u v dx Ω F (t) 2µɛ(u) ɛ(v) dx = 0 Ω F (t) Ω F (t) q div u = 0 w v = 0 ˆΩ S ρ S 2 d t 2 ϕ dˆx + a(d, ϕ) = < f Σ t, ϕ >

7 1.3. Time discretization. Fluid. (implicit Euler) Ω n+1 F 1 δt ρ F u n+1 v + Ω n+1 F ρ F u n+1 v div w n+1 ρ F (u n+1 w n+1 ) u n+1 v Ω n+1 F p div v + Ω n+1 F 2µɛ(u n+1 ) ɛ(v) = 1 δt Ω n+1 F Ω n+1 F q div u n+1 = 0 ρ F u n v Ω n F u n+1 Σ n+1 = dn+1 d n Ω n+1 w n+1 v = 0 δt w n+1 Σ n+1 = u n+1 Σ n+1 f Σ n+1 = F(d n+1 )

8 1.4. Time discretization. Structure. (mid-point) v ρ S ˆΩS n+1 S v n S δt ϕ dˆx a(dn, ϕ) a(dn+1, ϕ) = < f Σ n+1, ϕ > d n+1 d n δt = vn+1 S + v n S 2 where < f Σ n+1, ϕ > is the residual of the fluid problem d n+1 = S(f Σ n+1)

9 1.5. Fluid/structure coupling. Two main approaches : (i) Staggered schemes (Piperno, Fahrat, Larrouturou,...) At each time step : basically one resolution of F and one resolution of S. this seems to be unsuitable for blood flows (ii) Implicit schemes (Le Tallec & Mouro) At each time step a nonlinear problem to be solved : d n+1 = S F(d n+1 ) Advantages : stability (energy balance) Drawback : expensive!

10 2.1. Fixed point algorithms Naive fixed point algorithm (i) d 0 = d n (ii) d k+1 = S F(d k ) does not converge! 2. Numerical algorithms

11 Relaxed fixed point algorithm with prediction (i) d 0 = d n + 3δt 2 vn S δt 2 vn 1 S (ii) d k+1 = S F(d k ) (iii) d k+1 = ω k dk+1 + (1 ω k )d k

12 How to choose ω k? : (i) by hand : ω k = ω (ii) Domain decomposition (Le Tallec - Mouro) : reformulate the fixed point algorithm as a preconditioned gradient method applied to an interface problem (iii) Aitken-like acceleration formula (Mok - Wall - Ramm) : heuristic generalization in n dimensions, of the formula that gives the exact solution for affine functions in 1D : ω k = (d k d k 1 ) (d k S F(d k ) d k 1 + S F(d k 1 )) d k S F(d k ) d k 1 + S F(d k 1 ) 2 Mok & co. concluded that (iii) gave slightly better results compared to (ii).

13 Domain decomposition method (formal) A F II A F IΣ 0 A F ΣI A F ΣΣ + AS ΣΣ AS ΣI 0 A S IΣ A S II df I d Σ d S I = bf I b Σ b S I (d F I = δt u F I ) (i) Fluid solver with Dirichlet boundary conditions : A F II df I,k+1 = bf I AF IΣ d Σ,k (ii) Residual (force) computation : F Σ,k+1 = A F ΣI df I,k+1 + AF ΣΣ d Σ,k (iii) Structure solver with Neumann boundary conditions : [ A S ΣΣ A S ΣI A S IΣ A S II ] [ dσ,k+1 d S I,k+1 ] = [ bσ F Σ,k+1 b S I ] (iv) d Σ,k+1 = ω k dσ,k+1 + (1 ω k )d Σ,k

14 Eliminate d F I,k+1 in (i) and (ii) (SF = A F ΣΣ AF ΣI (AF II ) 1 A F IΣ ) F Σ,k+1 = S F d F Σ,k + A F ΣI(A F II) 1 b F I Eliminate d S I,k+1 in (iii) (SS = A S ΣΣ AS ΣI (A II) 1 A S IΣ ) S S dσ,k+1 = b Σ A S ΣI(A S II) 1 b S I F Σ,k+1 S S dσ,k+1 = S F d Σ,k + b Σ A S ΣI(A S II) 1 b S I A F ΣI(A F II) 1 b F I d Σ,k+1 = (S S ) 1 ( b S F d Σ,k )

15 Relaxation : d Σ,k+1 = ω k dσ,k+1 + (1 ω k )d Σ,k = = d Σ,k + ω k ([ d Σ,k+1 d Σ,k ) ] d Σ,k + ω k (S S ) 1 ( b S F d Σ,k ) + d Σ,k = d Σ,k + ω k (S S ) 1 [ b (S F + S S )d Σ,k ] We recognize a gradient method for the solution of (S F + S S )d Σ = b preconditionned by S S. this gives a way to design ω k (e.g. steepest descent)

16 Benchmark test ( Mok, Wall, Ramm in 2D) Fluid : ρ F = 1, µ = 0.01 Structure : ρ S = 500, E = 250, h = u(t) = 1 cos ( ω t ) rigid wall rigid wall flexible bottom

17 MODULEF : vidrascu 03/09/02 tracou.data NOMBRE DE COURBES : 3 EXTREMA EN X : E+03 EXTREMA EN Y : -0.68E : B : A : C TRACE DE COURBES

18 Pressure wave in artery Fluid blood Structure artery Po P in Pout t = 0 : u = 0, P = P 0 = 0. { 0 < t < 5 ms : Pin = 10 mmhg t > 5 ms : P in = 0

19 160 Aitken constant relaxation number of iterations time

20 2.2. Quasi-Newton algorithms (i) d 0 = d n + 3δt 2 vn S δt 2 vn 1 S (ii) R (d k )δd k = R(d k ) (iii) d k+1 = d k + λ k δd k R is a suitable approximation of R R = I S F Rd = 0 λ k is determined via a linesearch strategy.

21 First possibility : automatic differentiation... Second possibility : (Brown & Saad) The linear system at step (ii) is solved approximatively with a few iterations of GMRES. The matrix/vector products R (d k )z are evaluated by computing : R(d k + hz) R(d k ) h or (better) R(d k + hz) R(d k hz) 2h It works, but it s expensive and it s difficult to choose h. Third possibility : evaluate the continuous Jacobian of the fluid structure problem (Fernandez & Moubachir). To be tested! Fourth possibility : use a reduced model.

22 3.1. Classical 1D model : 3. Reduced FSI models cylinder axial symmetry no radial velocity z ex ez e y z z=z - A = S (area of S) - Q = - p = 1 A(z) S u z dσ (flux through S) ou u = Q A S p dσ (mean pressure moyenne)

23 mass conservation : div u = 0 = A t + Q z = 0 momentum along z : u z t +div (u zu)+ 1 ρ F p z ν u z = 0 = Q t + z (α Q2 A ) + A p ρ F z + K Q R A = 0 wall law : relation between p and A e.g. : Φ(A) = p 0 + πh0 E (1 ν 2 )A 0 ( A A0 ) = p = β 0 + β A Main difficulties : geometry : cylinder mean value variables : 3D 1D 3D

24 3.2. A new reduced model An important feature in fluid structure interaction problems in blood flow is the so-called mass added effect. This effect can be captured with a non-viscous linear fluid model : ρ F t u + p = 0 on Ω F div u = 0 on Ω F u n = t d n on Σ or, p = 0 on Ω F p n = ρ u F t n on Σ

25 3.3. The special case of the generalized wave equation Notation : p = 0 on Ω F p n = g on Σ p = p d on Γ Generalized wave equation for the structure : p Σ = M A g ρ S 2 d r t 2 a 2 d r z 2 + bd r c 3 d r z 2 t = p Σ u Simplified fluid model : p Σ = ρ F M A t n = ρ 2 d r F M A t 2 Simplified fluid-structure problem : (ρ S + ρ F M A ) 2 d r t 2 a 2 d r z 2 + bd r c 3 d r z 2 t = 0

26 3.4. Three possible applications of the reduced model With 1D model structure (wave equation) : (i) A good candidate for studying fluid-structure algorithms (ii) A possible alternative for classical FSI 1D models With general structure models : (iii) A good candidate for approximating the jacobian in Quasi-Newton algorithms

27 (i) A good candidate for studying fluid-structure algorithms it is rather simple (linear) it shares two important features with the real model : lots of sub-structuring iterations are needed to converge analogous property of propagation

28 35 Simplified model Complete model number of iterations time

29 Why staggered algorithm does not work in blood flows? Implicit algorithm : (ρ S + ρ F M A ) vn+1 S v n S δt Staggered algorithm : u n+1 n = v n+1 S a 2 d n+1/2 r z 2 n + bd n+1/2 r c 3 d n+1/2 r z 2 t = 0 u n+1 n = v n S n v n+1 S v n S ρ S δt v n S + ρ F M vn 1 S A δt a 2 d n+1/2 r z 2 + bd n+1/2 r c 3 d n+1/2 r z 2 t = 0

30 D n+1 V n+1 W n+1 = P D n V n W n Spectral property of the iteration matrix P : For ρ F /ρ S > (roughly) : λ k >> 1 λ k increases when δt decreases

31 (ii) A possible alternative for classical FSI 1D models versus [ ] A t Q + Q Q z 2 A + β = 0 3 A3/2 K Q A (ρ S + ρ F M A ) 2 d t 2 a 2 d z 2 + bd c 3 d z 2 t = 0 What has been lost? nonlinearity viscous effect What has been gained? same variables as the complete model (3D-3D) realistic structure response

32 (iii) Approximate the tangent operator of the real problem (i) Prediction : d 0 = d n + 3δt 2 vn S δt 2 vn 1 S (ii) Resolution of R (d k )δd k = R(d k ) evaluate R(d k ) (with R = I S F) Solve the linear system with GMRES (tolerance 10 3, about 8 iterations). The matrix/vector products being evaluated by : R (d k )z = z S (F(d k )) F (d k )z one resolution of a scalar Poisson problem (instead of Navier-Stokes) one resolution of the linearized structure (already computed and factorized during the structure solution) (iii) d k+1 = d k + λ k δd k In practice the linesearch procedure is never activated (λ k = 1) : the approximated jacobian is not so bad!

33 35 Quasi-Newton Aitken number of iterations time

34 50 45 Quasi-Newton Aitken number of iterations time

35 Mean number of fluid-structure evaluations : Computational time : Aitken Quasi-Newton Speed up 2D D Aitken Quasi-Newton Speed up 2D 43 min 16 min 2.7 3D 12 h 16 5 h