1 / 41 A monolithic fluid structure interaction solver Verification and Validation Application: venous valve motion Chen-Yu CHIANG O. Pironneau, T.W.H. Sheu, M. Thiriet Laboratoire Jacques-Louis Lions (LJLL), Sorbonne U. Scientific Computing and Cardiovascular Simulation (SCCS) Lab, NTU FreeFem++ days 2017
2 / 41 Outline 1 2 3 4
Formulation of coupled system A general fluid-structure coupled system General Laws of Continuum Mechanics ( ) u ρ t + u ( u) = (σ) + f (1) For Newtonian incompressible fluid : σ f = p f I + µd For hyperelastic incompressible material : σ s = p s I + F ΨF T For hyperelastic compressible material : σ s = J 1 F Ψ F T D = ( u + u ) T X: Ω 0 (0, T ) Ω t : X ( x 0, t ) : Lagrangian position at t of x 0 d = X ( x 0, t ) x 0 : displacement F i,j = x 0 j X i ; x 0 j : original position J = det F, the Jacobian of the deformation 3 / 41
4 / 41 Hyperelastic material model Mooney Rivlin model Ψ(F) = c 1 tr F T F + c 2 ( ) tr (F T F) tr 2 2 F T F F ΨF T = (2c 1 4c 2 tr B ) B + 4c 2 B 2, where B = FF T. (2) S t Venant Kirchhoff model Ψ (F) = λ s 2 tr2 E + µ s tr E 2, E = 1 ( F T F I ) 2 [ ( F ΨF T trb = λ s 2 3 ) ] µ s B + µ s B 2 (3) 2
5 / 41 Lagrangian points ( x 0, t ) to Eulerian points (x, t) F T ( = x 0X = x 0 d + x 0 ) = x 0d + I = F T d + I F = (I d) T C = I B = Dd d T d Cayley Hamilton theorem For a general n n invertible matrix A, i.e., one with nonzero determinant, A 1 can thus be written as a n th order polynomial expression in A n = 3, A 3 tr A A 2 + 1 2 (tr2 A tr A 2)A det A I = 0 n=3 B = tr B I 1 ( ) tr 2 2 B tr B 2 B 1 + J 2 B 2, B 2 = 1 [ ( ) tr 2 2 B + tr B 2 I + J 2 1 ( ) ] (4) tr 3 2 B tr B tr B 2 B 1 + tr B J 2 B 2
Stress tensor for hyperelastic models (n=3) Mooney Rivlin model F ΨF T = 2c 1 ( Dd d T d ) 2 + 2c3 ( Dd d T d ) + αi (5) c 3 := c 1 2 S t Venant Kirchhoff model ( tr 2 B tr B 2 4 ) 2c 2 F ΨF T = αi + β ( Dd d T d ) + γ ( Dd d T d ) 2 (6) { α = λ s 4 (tr B 3) ( 2tr B tr 2 B + tr B 2 + 2J 2) + µs 2 tr ( B 2trB + tr B 2 tr 2 B 2 + 2J 2) β = λs 4 (tr B 3) ( tr 2 B tr B 2 4J 2) ( + µs 2 tr 3 B tr B tr B 2 tr 2 B + tr B 2 + 2J 2 4tr B J 2) γ = λs 2 (tr B 3) J 2 + µ s (tr B 1) J 2 6 / 41
7 / 41 Variational formulation (n=3) (ρ f D tu û + µ ) 2 Du: Dû p û ˆp u Ω t f + + β 2 For simplicity, Ω t s [ ρ sd tu û + γ ( 2 Dd d d) T : Dû 2 ( ) Dd d T d ] : Dû + α û = (7) f û Ω t D td = u, (8) homogeneous boundary conditions on Γ Ω homogeneous Neumann conditions on Ω t \ Γ. So, given Ω 0 f, Ω0 s, d, and u at t = 0, we must find ( ) u, p, d, Ω t f, Ωt s with u Γ = 0 and equation (7) and (8)
8 / 41 A first order in time consistent approximation Find Ω n+1, u n+1 H 1 0(Ω n+1 ), p n+1 L 2 0(Ω n+1 ) such that for all û n+1 H 1 0(Ω n+1 ), ˆp n+1 L 2 0(Ω n+1 ), n+1 u u n Y n+1 ρn+1 Ωn+1 δt + δt Ω n+1 s û + Ω n+1 f ( p n+1 û ˆp u µ ) 2 Dun+1 : Dû [γ n+1 ( Du n+1 u n+1 T dn dn T u n+1 ) ( D d n d n T dn ) + βn+1 ( Du n+1 u n+1 T T n+1 )] dn dn u : Dû 2 [ γ n+1 + (D d n Ω n+1 d n T ) 2 β n+1 ( dn : Dû + D d n d n T ) dn : Dû s 2 2 +α n+1 ] û = f û Ω n+1 (9) where d n stands for d n (Y n+1 ) and where d n is updated by d n+1 = d n Y n+1 + δtu n+1 (10)
Solution algorithm Fixed point iteration 1 Set ρ = ρ n, α = α n, β = β n, γ = γ n, Ω = Ω n, u = u n h, Y (x) = x uδt, 2 Solve equation (9), 3 Set u = u n+1 h, Y (x) = x uδt, Ω r = Y 1 (Ω n r ), r = s, f, update α, β, γ and ρ. 4 If not converged return to step 2. 9 / 41
10 / 41 Outline A thin elastic plate clamped into a small rigid square body immersed A thin elastic plate clamped into a rigid cylinder immersed in a flowin 1 2 A thin elastic plate clamped into a small rigid square body immersed in a flowing fluid A thin elastic plate clamped into a rigid cylinder immersed in a flowing fluid 3
A thin elastic plate clamped into a small rigid square body immersed A thin elastic plate clamped into a rigid cylinder immersed in a flowin A thin elastic plate clamped into a small rigid square body immersed in a flowing fluid Table: Material parameters [ ρ f g cm 3] [ ν f g cm 1 s 1] ρ s [g cm 3] E [g cm 1 s 2] U in [cm s 1] 1.18 10 3 1.82 10 4 2 2 10 6 31.5 u=uin v=0 v=0 u y=0 1 1 4 u x=0 0.06 v x=0 12 v=0 u y=0 6.5 14.5 11 / 41
A thin elastic plate clamped into a small rigid square body immersed A thin elastic plate clamped into a rigid cylinder immersed in a flowin A thin elastic plate clamped into a small rigid square body immersed in a flowing fluid Author present, hmin = 0.02 present, hmin = 0.015 Hubner (a) magnitude 1.85 1.94 2.0 frequency 1.2 1.2 0.8 (b) Figure: Contours of velocity magnitude at (a) t = 12.5 and (b) t = 13.0. 12 / 41
A thin elastic plate clamped into a small rigid square body immersed A thin elastic plate clamped into a rigid cylinder immersed in a flowin A thin elastic plate clamped into a rigid cylinder immersed in a flowing fluid Poiseuille profile inflow U (y) = 1.5U in y(h y) (0.5H) 2 and H = 0.41 For fluid, ρ f = 1000 and ν f = 0.001 Two steady cases, g s = 2, 4 and U in = 0 (structure ρ s = 1000 and E = 5 10 5 ) FSI1, U in = 1, ρ s = 1000 and E = 5 10 5 FSI2 U in = 2, ρ s = 10000 and E = 2 10 5 u=u(y) v=0 0.2 u=v=0 0.1 0.35 u=v=0 2.3 σ n=0 0.02 0.41 13 / 41
14 / 41 A thin elastic plate clamped into a small rigid square body immersed A thin elastic plate clamped into a rigid cylinder immersed in a flowin g s=2 g s=4 d x d y d x d y Present h min = 0.008 7.179 65.82 24.84 120.82 Dunne [?] 7.187 66.10 - - Hron and Turek [?] 7.187 66.10 - - Richter [?] 7.149 66.07 25.10 122.16 Wick [?] 7.150 64.90 25.33 122.30 (a) (b) Figure: (a) displacement on x and y with respect to time, (b) centerlines position.
15 / 41 A thin elastic plate clamped into a small rigid square body immersed A thin elastic plate clamped into a rigid cylinder immersed in a flowin
16 / 41 A thin elastic plate clamped into a small rigid square body immersed A thin elastic plate clamped into a rigid cylinder immersed in a flowin FSI2 FSI3 amplitude frequency amplitude frequency Present h min = 0.004 7.61 10 2 2.03 s 1 2.80 10 2 5.15 s 1 Present h min = 0.002 7.75 10 2 2.03 s 1 3.01 10 2 5.18 s 1 Turek [?] 8.06 10 2-3.44 10 2 - Dune [?] 8.00 10 2 1.953 s 1 3.00 10 2 5.04 s 1 Thomas [?] 8.06 10 2 1.93 s 1 - -
A thin elastic plate clamped into a small rigid square body immersed A thin elastic plate clamped into a rigid cylinder immersed in a flowin U in = 1.07 Whole solid system flaps due to instability of wake region. Front cylinder rotates at frequency of 6.38 Hz and 7.35 Hz, experimentally and numerically determined value. u=uin v=0 0.022 u=v=0 0.05 0.01 σ n=0 0.00004 0.004 0.24 Origin of coordinate is placed at center of cylinder. Measure points A and B locate at (0.082, 0) and (0.082, 0.04). 0.155 u=v=0 0.593 17 / 41
18 / 41 A thin elastic plate clamped into a small rigid square body immersed A thin elastic plate clamped into a rigid cylinder immersed in a flowin
19 / 41 A thin elastic plate clamped into a small rigid square body immersed A thin elastic plate clamped into a rigid cylinder immersed in a flowin
20 / 41 A thin elastic plate clamped into a small rigid square body immersed A thin elastic plate clamped into a rigid cylinder immersed in a flowin Relatively good agreement in displacement, frequency and velocity at measure points. 30000 elements (P1b,P1), and first order in time.
A thin elastic plate clamped into a small rigid square body immersed A thin elastic plate clamped into a rigid cylinder immersed in a flowin [ ] [ ] An elastic plate h 2, h b 2 2, b [0, L] 2 is clamped to a rectangular tube [ 5b, 16b] [ 8b, 8b] [0, 17b] h/b = 0.2 and L/b = 5 A constant inflow U in = 1 along x direction at Γ in Re = U inb ν = 1600 and ρ f = 1 ρ s = 0.678, E = 19054.9, and ν s = 0.4 z x Buoyancy force of solid f b = 0.2465 17b Γ in Γ out h L 5b 16b 21 / 41
22 / 41 A thin elastic plate clamped into a small rigid square body immersed A thin elastic plate clamped into a rigid cylinder immersed in a flowin (a) (b) Figure:. (a) Experimental results [?], (b) computational results.
23 / 41 A thin elastic plate clamped into a small rigid square body immersed A thin elastic plate clamped into a rigid cylinder immersed in a flowin C D D x /b D z /b Present 1.03 2.13 0.54 Luhar and Nepf [?] 1.15 (with 10% error) 2.14 0.59 Tian and Dai etc [?] 1.03 2.12 0.54 Table: Comparison of drag coefficient C D and deflection in x and z direction with respect to referenced data.
A thin elastic plate clamped into a small rigid square body immersed A thin elastic plate clamped into a rigid cylinder immersed in a flowin A cylindrical chamber with r 2 = 76.2 and length of 200 Two inlet pipes with r 1 = 21.0 and length of 20 A silicon filament with h = 2 thick, b = 11 wide, and l = 65 long ρ f = 1.1633 10 3 and ν f = 12.5 10 3 ρ s = 1.0583 10 3, Young s modulus E s = 216260, and Poisson ratioν s = 0.315 Gravity g = 9810 along y direction r 1 r 1 r 2 b (0,0,0) h l h y r 2 r 1 y r 1 z x 24 / 41
Initial configuration due to buoyancy force A thin elastic plate clamped into a small rigid square body immersed A thin elastic plate clamped into a rigid cylinder immersed in a flowin Figure: Initial deflection of silicone filament because of buoyancy force. 25 / 41
A thin elastic plate clamped into a small rigid square body immersed A thin elastic plate clamped into a rigid cylinder immersed in a flowin Case I : steady deformation Inlet velocity are Poiseuille profile with peak velocityof 630 and 615 for upper and lower inlet respectively. (a) (b) Figure: Computational results for phase I (a) position of center line along z direction, (b) velocity profiles at symmetric plane x = 0. 26 / 41
27 / 41 Case II : starting flow A thin elastic plate clamped into a small rigid square body immersed A thin elastic plate clamped into a rigid cylinder immersed in a flowin n 1 n 2 n 3 d 0 b 1 b 2 b 3 b 4 Î k ˆv x -11.37-28.99 7.73 1.38 0.24 3.59-3.14 1 [0, 4.07] ˆv y 14.95 11.88-2.17 2.06-2.0 4.95-3.50 1 [0, 5.51] ˆv z 367.10 363.40-62.24 1.21-0.38 3.76-3.19 1 [0, 5.27] Table: For case II, curve fitting coefficients of inlet peak velocity for ˆv k (t) = Σ 3 i=1 n it i /Σ 4 j=0 b j t j with ˆv k = 0 for t I\I k. Note, that flow in y direction is applied only in the upper inlet.
A thin elastic plate clamped into a small rigid square body immersed A thin elastic plate clamped into a rigid cylinder immersed in a flowin (a) (b) (c) (d) (e) (f) Figure: Deflection of the silicone filament in phase II (a) t = 0.073, (b) t = 0.721, (c) t = 1.153, (d) t = 1.585, (e) t = 2.017, (f) t = 4.781. 28 / 41
29 / 41 A thin elastic plate clamped into a small rigid square body immersed A thin elastic plate clamped into a rigid cylinder immersed in a flowin (a) (b) (c) (d) Figure: At midplane x = 0 mm, flow velocity component u z at (a) z 0.20l, (b) z 0.51l, (c) z 0.82l, (d) z 1.12l ( l = 65 mm, length of the silicon filament ).
30 / 41 Variational inequality/constrain for contact Valved veins Outline 1 2 3 Variational inequality/constrain for contact Valved veins 4
31 / 41 Variational inequality/constrain for contact Valved veins Variational inequality/constrain for contact A (u, v) + Λ (u, v) = b (v) where Λ is a variational constrain for contact { λ (x, t) 0, x, Λ (u, v) = λn v, Ω t d S t (x) λ (x, t) = 0, S t i i To each simply onnected disjointed part Si t, i = 1 ns is associated a signed distance function x d S t measuring distance of x to S t i i. d S t i d S t i d S t i < 0, structure region = 0, contact point/line/surface > 0, fluid region for some x Ω/S t i
32 / 41 Variational inequality/constrain for contact Valved veins Semi Smooth Newton Method In k th iteration for each time step n + 1 ( λ n+1,k+1 min {0, λ n+1,k + c 0 d S x + δtu n+1,k)}. x S ti n+1 i Original equations can be recast into G ( u n+1,k+1, v ) = ns A ( u n+1,k+1, v ) + τ ck G ( u n+1,k+1, v ) = b (v) i=1 S i n+1 {x : λ n+1,k +c 0 d S i n+1 (x+δtu n+1,k ) } g ( u n+1,k+1, v ) where function f is associated to rebound velocity and variant with different model.
Variational inequality/constrain for contact Valved veins Rebound velocity For example g ( u n+1,k+1) = u n+1,k+1 f ( u n+1,k n ) n f(u n )=-βu n, 0<β f(u n )=ε closer away 33 / 41
34 / 41 Variational inequality/constrain for contact Valved veins Free falling ball hit a slope
35 / 41 Variational inequality/constrain for contact Valved veins d h w=d/20 h T=0.92d w=0.625d h B=0.58d L v 0.8d h v=0.01d (a) (b) Figure: Shape and size of one valved vein
36 / 41 Variational inequality/constrain for contact Valved veins
37 / 41 Variational inequality/constrain for contact Valved veins
38 / 41 Outline 1 2 3 4
Modeling In/out-let boundary conditions of blood vessels Anisotropic hyperelastic material model for vascular walls and valves Model for agregation and deformation of red blood capsules Simulation Parallize whole computational process with domain decomposition method Vascular model and simulation in 3D Comparison with experimental data 39 / 41
40 / 41 Thanks for your listening.
41 / 41 Flow rate and volume (a) (b)