FSI with Application in Hemodynamics Analysis and Simulation

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)

II. Numerical Modelling

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

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

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

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)

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)

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

Discretization method Fluid equations UG software toolbox

Discretization method Fluid equations UG software toolbox FV-type schemes

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

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

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

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, 3... - 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

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)

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

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

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 )

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 )

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

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

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

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

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 )) + 1 2 ξn+ 1 2 2 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+ 1 2 2 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

Energy of the solution Energy of A-operator: ξ n+ 1 2 2 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+1 1 2 L 2 (0,L) + b ηn+1 2 L 2 (0,L) + ξn+1 2 L 2 (0,L) = a ηx 0 1 2 L 2 (0,L) + b η0 2 L 2 (0,L) + ξ0 2 L 2 (0,L) N ( ) + ξ n+ 1 2 2 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)

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 )

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)

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 1 0.8 0.6 0.4 0.2 0.5 1 1.5 2 2.5 time

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 30 20 10 0.2 0.4 0.6 0.8-10 3 Q(t) = -20 u inflow dx 2 = f (t) = Q(t) Γ in 4VR(t)

Some hemodynamical quantities domain deformation: stenosed and straight channel 0.06 0.04 0.02-4 -2 2 4-4 -2 2 4-0.02 cm cm ---Carreau,q=-0.322-0.04...NS Yeleswarapu -0.06 _._.Carr.,q=-10-0.08 time 0.36 s time 0.96 s cm cm cm 0.06 0.05 0.04 0.03 0.02 0.01-4 -2 2 4-0.01 cm 0.02 0.01 time 0.36 s time 0.96 s -4-2 -0.01 2 4-0.02-0.03-0.04 NS -0.05 ---Carreau, q=-0.322-0.06 cm cm

Some hemodynamical quantities WSS-stenosed channel dyn.cm^-2 150 125 100 dyn.cm^-2...ns ---Carreau,q=-0.32-6 Yeleswarapu _._.Carreau,q=-10-4 -2 2 4-10 cm 75 50 25 time 0.36 s -4-2 2 4 cm -12-14 -16-18 -20

Some hemodynamical quantities OSI OSI 0.175 OSI 0.15 0.15 0.125 Carreau 0.1 0.075 0.05 0.125 0.1 0.075 0.05 Carreau NS 0.025 NS 0.025-4 -2 2 4 cm -4-2 2 4 cm M.L. & A. Hundertmark [Comp.& Math.Appl. ( 10)]

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