FSI with Application in Hemodynamics Analysis and Simulation

Transcription

1 FSI with Application in Hemodynamics Analysis and Simulation Mária Lukáčová Institute of Mathematics, University of Mainz A. Hundertmark (Uni-Mainz) Š. Nečasová (Academy of Sciences, Prague) G. Rusnáková (Uni-Košice/Uni-Mainz)

2 II. Numerical Modelling

3 Numerical schemes for the FSI problems strong coupled schemes (inner subiterations or fully composite schemes) weakly coupled schemes (no subiterations needed)

4 Numerical schemes for the FSI problems strong coupled schemes (inner subiterations or fully composite schemes) weakly coupled schemes (no subiterations needed) - for aerodynamical problems: weakly coupled schemes: OK - for biomechanical probelms: kind of stronger coupling, subiterations are typically needed

5 Numerical schemes for the FSI problems strong coupled schemes (inner subiterations or fully composite schemes) weakly coupled schemes (no subiterations needed) - for aerodynamical problems: weakly coupled schemes: OK - for biomechanical probelms: kind of stronger coupling, subiterations are typically needed - many numerical approaches for the FSI: immersed boundary, Lagrangian, Arbitrary Eulerian-Lagrangian (ALE)

6 FSI for hemodynamical applications 1 Quarteroni, Formaggia, Veneziani, Nobile et al. ( 01-07)... inner subiterations

7 FSI for hemodynamical applications 1 Quarteroni, Formaggia, Veneziani, Nobile et al. ( 01-07)... inner subiterations 2 Hron, Turek et al. ( 09) monolitic scheme (strong coupling)

8 FSI for hemodynamical applications 1 Quarteroni, Formaggia, Veneziani, Nobile et al. ( 01-07)... inner subiterations 2 Hron, Turek et al. ( 09) monolitic scheme (strong coupling) 3 Guidoboni, Glowinski, Cavallini, Čani`c ( 09): weak coupling: kinematical coupling

9 FSI for hemodynamical applications 1 Quarteroni, Formaggia, Veneziani, Nobile et al. ( 01-07)... inner subiterations 2 Hron, Turek et al. ( 09) monolitic scheme (strong coupling) 3 Guidoboni, Glowinski, Cavallini, Čani`c ( 09): weak coupling: kinematical coupling 4 Janela, Moura, Sequeira ( 10): simulations for non-newtonian FSI

10 FSI for hemodynamical applications 1 Quarteroni, Formaggia, Veneziani, Nobile et al. ( 01-07)... inner subiterations 2 Hron, Turek et al. ( 09) monolitic scheme (strong coupling) 3 Guidoboni, Glowinski, Cavallini, Čani`c ( 09): weak coupling: kinematical coupling 4 Janela, Moura, Sequeira ( 10): simulations for non-newtonian FSI 5 Hundertmark, Lukáčová, Rusnáková ( 08, 10, 11): analysis and simulations for non-newtonian FSI schemes (global and kinematical coupling)

11 Outline Numerical methods Discretization FSI Stability Energy estimates Energy estimate for the A operator Energy estimates for B operator Numerical experiments Inflow data and parameters Experiment I: stenotic vessel Experiment II: bifurcation vessel Convergence study

12 Discretization method Fluid equations UG software toolbox

13 Discretization method Fluid equations UG software toolbox FV-type schemes

14 Discretization method Fluid equations UG software toolbox FV-type schemes pseudo compressibility stabilization

15 Discretization method Fluid equations UG software toolbox FV-type schemes pseudo compressibility stabilization Newton/fixed point linearization of convective term

16 Discretization method Fluid equations UG software toolbox FV-type schemes pseudo compressibility stabilization Newton/fixed point linearization of convective term fixed point iterations of non-linear viscosity µ( D( u) )D( u) µ( D( u old ) )D( u)

17 Discretization method Fluid equations UG software toolbox FV-type schemes pseudo compressibility stabilization Newton/fixed point linearization of convective term fixed point iterations of non-linear viscosity µ( D( u) )D( u) µ( D( u old ) )D( u) Structure equation 2 η t 2 a 2 η x1 2 + b(1/η)η c 3 η x1 2 t = RHS(u, p, R 0) - finite difference method in time (Newmark scheme) - finite difference method in space (central approximations)... second order

18 Outline Numerical methods Discretization FSI Stability Energy estimates Energy estimate for the A operator Energy estimates for B operator Numerical experiments Inflow data and parameters Experiment I: stenotic vessel Experiment II: bifurcation vessel Convergence study

19 Fluid-Structure Coupling 1 I. Strong coupling - global iteration method k=1,2,... - solve fluid eq. & structure eq. on the domain Ω(η k ) for t n, n = 1, 2, homogeneous Dirichlet or Neumann boundary conditions

20 Fluid-Structure Coupling 1 I. Strong coupling - global iteration method k=1,2,... - solve fluid eq. & structure eq. on the domain Ω(η k ) for t n, n = 1, 2, homogeneous Dirichlet or Neumann boundary conditions (ξ = t η) ξ n+1 ξ n t aα 2 η n+1 x 2 1 = RHS(p n, u n, R 0 ) + (1 α) η n+1 η n t + bαη n+1 cα 2 ξ n+1 ( a 2 η n x 2 1 x 2 1 = αξ n+1 + (1 α)ξ n, α = 0.5 ) bη n + c 2 ξ n x 2 1

21 Global iteration with respect to the domain k=0, h=η k initial condition, solve NS and deformation problem t1,. t2,.... tn 1.deformat. problem, use 2.flow problem, use η (x,t2) k u(x,t1) in Ω(h) k+1. values η (x,t1) k+1 u (x,t1) k+1 η (x,t2) k+1 u (x,t2) k+1.. η (x,tn) k+1 u (x,tn) k+1 if convergence STOP if not update domain geometry Ω(h), k+1 h= η (x,t)

22 Fluid-Structure Coupling 2 II. Kinematical coupling [Guidoboni et al. ( 09)] - solve Fluid eq. + Structure eq. on Ω(η n ) for t [t n, t n+1 ] - split structure eq. into 2 parts - kinematical lateral B.C. u 2 y=h = η = ξ t

23 Fluid-Structure Coupling 2 II. Kinematical coupling [Guidoboni et al. ( 09)] - solve Fluid eq. + Structure eq. on Ω(η n ) for t [t n, t n+1 ] - split structure eq. into 2 parts - kinematical lateral B.C. u 2 y=h = η = ξ t 1. hydrodynamic part fluid eq. + ξ = u 2 y=h 2. elastic part ξ t = c 2 ξ + RHS(u, p) x 2 1 η = ξ t ξ t = a 2 η x 2 b(1/η)η + RHS(R 0) 1

24 Fluid-Structure Coupling 3 Kinematical coupling / Discretization A operator: hydrodynamic part Fluid solver + ξ n+1/2 ξ n t cα 2 ξ n+1/2 x 2 1 = RHS(p n, u n ) + c(1 α) 2 ξ n, x 2 1 B operator: elastic part η n+1 η n = βξ n+1 + (1 β)ξ n+1/2, β = 0.5 t ξ n+1 ξ n+1/2 aα 2 η n+1 t x1 2 + b(1/η n )αη n+1 = RHS(R 0 ) + a(1 α) 2 η n b(1/η n )(1 α)η n α = 1, 0.5 x 2 1

25 Types of splitting kinematical splitting scheme of the Marchuk-Yanenko type 1st order accurate A t B t in the (n + 1) th time-step on Ω(t n )

26 Types of splitting kinematical splitting scheme of the Marchuk-Yanenko type 1st order accurate A t B t in the (n + 1) th time-step on Ω(t n ) kinematical splitting scheme of the Strang-type kinematical splitting 2nd order accurate B t/2 A t B t/2 in the (n + 1) th time-step on Ω(t n )

27 Outline Numerical methods Discretization FSI Stability Energy estimates Energy estimate for the A operator Energy estimates for B operator Numerical experiments Inflow data and parameters Experiment I: stenotic vessel Experiment II: bifurcation vessel Convergence study

28 ALE - mapping Arbitrary Lagragian-Eulerian Mapping A t : Ω 0 Ω t A t (X) = x Ω t f (t, x) f (t, A 1 t x) f (t, X) D A u Dt ALE derivative w := x t grid velocity" = D A u(t, x) ũ(t, X) := Dt t u(t, x) + w u(t, x) t

29 Weak formulation of coupled problem test functions: v V, where { } V v W 1,p (Ω(t)) 2 : v 1 Γwall = 0, v 2 x2 =0 = 0 D A u Dt ALE derivative w := x t D A u Dt := u t + w u(x, t) grid velocity

30 Weak formulation of coupled problem test functions: v V, where { } V v W 1,p (Ω(t)) 2 : v 1 Γwall = 0, v 2 x2 =0 = 0 D A u Dt ALE derivative w := x t D A u Dt := u t + w u(x, t) grid velocity D A u v dx + ((u w) u)vdx Ω(t) Dt Ω(t) p div v dx + 1 2µ( D(u) )D(u) : D(v) dx + Ω(t) ρ Ω(t) L ( 2 ) η L ( t 2 + bη v 2 Γwall dx 1 + a η ) + c 2 η (v2 Γwall x 1 t x 1 x 1 0 R0 = 0 P in v 1 x1 =0dx 2, v V, a.e.t I. 0

31 Energy estimates with G. Rusnáková weak formulation of the semi-discrete coupled problem p 2 test with u n+1 kinematic pressure boundary conditions on infow / outflow symmetry boundary conditions x 2 = 0

32 Outline Numerical methods Discretization FSI Stability Energy estimates Energy estimate for the A operator Energy estimates for B operator Numerical experiments Inflow data and parameters Experiment I: stenotic vessel Experiment II: bifurcation vessel Convergence study

33 Energy estimate for ξ n+1/2 := u n+1 2 Γwall 1 2 un+1 2 L 2 (Ω(t n+1 )) + c 2ρ t un+1 2 W 1,2 (Ω(t n+1/2 )) ξn L 2 (0,L) + 1 c t ξn+ 2 x 2 L 2 (0,L) 1 2 un 2 L 2 (Ω(t + 1 n)) 2 ξn 2 L 2 (0,L) + c t Pn+1 in 2 - we obtain the following estimate for the A-operator ξ n L 2 (0,L) + un+1 2 L 2 (Ω(t n+1 )) ξ n 2 L 2 (0,L) + un 2 L 2 (Ω(t + n)) 2c t Pn+1 in 2

34 Outline Numerical methods Discretization FSI Stability Energy estimates Energy estimate for the A operator Energy estimates for B operator Numerical experiments Inflow data and parameters Experiment I: stenotic vessel Experiment II: bifurcation vessel Convergence study

35 Energy of the solution Energy of A-operator: ξ n L 2 (0,L) ξn 2 L 2 (0,L) + un 2 L 2 (Ω(t n)) un+1 2 L 2 (Ω(t n+1 ) + 2c t P n+1 in 2 Energy of B-operator: a ηx N L 2 (0,L) + b ηn+1 2 L 2 (0,L) + ξn+1 2 L 2 (0,L) = a ηx L 2 (0,L) + b η0 2 L 2 (0,L) + ξ0 2 L 2 (0,L) N ( ) + ξ n L 2 (0,L) ξn 2 L 2 (0,L) n=0

36 Energy of the solution Energy of A-operator: ξ n L 2 (0,L) ξn 2 L 2 (0,L) + un 2 L 2 (Ω(t n)) un+1 2 L 2 (Ω(t n+1 ) + 2c t P n+1 in 2 Energy of B-operator: a ηx N L 2 (0,L) + b ηn+1 2 L 2 (0,L) + ξn+1 2 L 2 (0,L) = a ηx L 2 (0,L) + b η0 2 L 2 (0,L) + ξ0 2 L 2 (0,L) N ( ) + ξ n L 2 (0,L) ξn 2 L 2 (0,L) n=0 Energy estimate of the solution: N E N+1 E 0 + 2c t P n+1 n=0 in 2 E k = u k 2 L 2 (Ω(t k )) +a ηk x 1 2 L 2 (0,L) +b ηk 2 L 2 (0,L) + ξk 2 L 2 (0,L)

37 Outline Numerical methods Discretization FSI Stability Energy estimates Energy estimate for the A operator Energy estimates for B operator Numerical experiments Inflow data and parameters Experiment I: stenotic vessel Experiment II: bifurcation vessel Convergence study

38 hemodynamical parameters Wall Shear Stress WSS := τ w = T f n τ, (1) negative values of the WSS - indicate existence of large recirculation zones Oscillatory Shear Index ( OSI := 1 T 0 1 τ ) w dt 2 τ, (2) w dt measures temporal oscillations of the shear stress pointwisely - indicates areas with large stenotic plug danger Reynolds number RE 0 = ρvl µ 0 or RE = ρvl µ or RE = T 0 ρvl 1 2 (µ 0 + µ )

39 hemodynamical parameters Wall Shear Stress WSS := τ w = T f n τ, (1) negative values of the WSS - indicate existence of large recirculation zones Oscillatory Shear Index ( OSI := 1 T 0 1 τ ) w dt 2 τ, (2) w dt measures temporal oscillations of the shear stress pointwisely - indicates areas with large stenotic plug danger Reynolds number RE 0 = ρvl µ 0 or RE = ρvl µ or RE = T 0 ρvl 1 2 (µ 0 + µ )

40 hemodynamical parameters Wall Shear Stress WSS := τ w = T f n τ, (1) negative values of the WSS - indicate existence of large recirculation zones Oscillatory Shear Index ( OSI := 1 T 0 1 τ ) w dt 2 τ, (2) w dt measures temporal oscillations of the shear stress pointwisely - indicates areas with large stenotic plug danger Reynolds number RE 0 = ρvl µ 0 or RE = ρvl µ or RE = T 0 ρvl 1 2 (µ 0 + µ )

41 Inflow data inflow velocity pulsatile parabolic profile u inflow ( L, x 2 ) = V (1 x 2 )(1 + x 2 ) sin 2 (πt/ω) on Γ in, or some f (t), e.g.,illiac artery measurements g(x,t)

42 Inflow data inflow velocity pulsatile parabolic profile u inflow ( L, x 2 ) = V (1 x 2 )(1 + x 2 ) sin 2 (πt/ω) on Γ in, or some f (t), e.g.,illiac artery measurements periodic_function time

43 Inflow data inflow velocity pulsatile parabolic profile u inflow ( L, x 2 ) = V (1 x 2 )(1 + x 2 ) sin 2 (πt/ω) on Γ in, or some f (t), e.g.,illiac artery measurements 40 Inflow rate, illiac artery Q(t) = -20 u inflow dx 2 = f (t) = Q(t) Γ in 4VR(t)

44 Some hemodynamical quantities domain deformation: stenosed and straight channel cm cm ---Carreau,q= NS Yeleswarapu _._.Carr.,q= time 0.36 s time 0.96 s cm cm cm cm time 0.36 s time 0.96 s NS Carreau, q= cm cm

45 Some hemodynamical quantities WSS-stenosed channel dyn.cm^ dyn.cm^-2...ns ---Carreau,q= Yeleswarapu _._.Carreau,q= cm time 0.36 s cm

46 Some hemodynamical quantities OSI OSI OSI Carreau Carreau NS NS cm cm M.L. & A. Hundertmark [Comp.& Math.Appl. ( 10)]

47 Outline Numerical methods Discretization FSI Stability Energy estimates Energy estimate for the A operator Energy estimates for B operator Numerical experiments Inflow data and parameters Experiment I: stenotic vessel Experiment II: bifurcation vessel Convergence study

48 Summary 1 Effects due to complex geometry: wall deformation, extrema of WSS, OSI in stenosed / bifurcation regions 2 Effects due to the Non-Newtonian rheology: WSS: larger negative absolute value = recirculation zones, OSI: larger extremes = more sensible prediction of stenotic plug 3 Difference between Carreau and Yeleswarapu model: negligible

49 Summary 1 Effects due to complex geometry: wall deformation, extrema of WSS, OSI in stenosed / bifurcation regions 2 Effects due to the Non-Newtonian rheology: WSS: larger negative absolute value = recirculation zones, OSI: larger extremes = more sensible prediction of stenotic plug 3 Difference between Carreau and Yeleswarapu model: negligible 4 FSI & kinematical coupling: efficient, stable FSI algorithm