Fluid-structure interaction. + large deformation of a structure in internal/external
flow. I biomechanics: blood flow - mostly laminar, incompressible, isothermal.
Monolithic solver for fluid-structure interaction problems Jaroslav Hron, Josef M´ alek, Martin M´ adl´ık Neˇ cas Center for Mathematical Modeling & Mathematical Institute
AIME@CZ 2011
Fluid-structure interaction
+ large deformation of a structure in internal/external flow I
biomechanics: blood flow - mostly laminar, incompressible, isothermal
I
viscoelastic behaviour, mechanical interaction with surroundings, chemical reactions
mathematical models I
viscous fluid flow
I
elastic body under large deformations
I
interaction between the two parts
numerical tasks involved I
space and time discretization
I
nonlinear system
I
solution of large linear system
testing and validation I
benchmarking
I
accuracy, efficiency, robustnes
I
error estimation, adaptivity
AIME@CZ 2011
Governing equations - continuum mechanics
I
balance equations %
I
@vv + div (%vv ⌦ v ) = rp + div @t div v = 0
constitutive equations T=
I
+ %ff
pII +
=
C, D , ...) pII + F (C
boundary conditions v = vB n =g ( pII + )n v · n = 0,
n ·t ↵vv · t = ( pII + )n
AIME@CZ 2011
Constitutive equations - fluid part I
incompressible Newtonian fluid T=
I
D pII + 2µD
1 (rvv + rvv T ) 2
generalized Newtonian fluid T=
I
D=
D| , c, ...)D D pII + 2µ(|D
general non-Newtonian simple viscous fluid, implicit constitutive law T, D , c, ...) = 0 G(T
I
rate type models, visco-elastic models, ... T, G(T
T D dT dD , ..., D , , ..., c, ...) = 0 dt dt
Blood: shear thinning property (power law, Casson law, Careau law,...) ✓ ◆ µ0 µ 1 µ = 2 µ1 + D|)a (1 + |D Questions: Existence and qualitative properties of the solution. M´ alek: Mathematical properties of flows of incompressible power-law-like fluids that are described by implicit constitutive relations, 2008.
AIME@CZ 2011
Fluid-structure interaction
+ Constitutive relation for solid part: I
hyperelastic material, anisotropy s T =
F) =↵(IC (F F) =↵1 (IC (F
F pII + 2F
@ T F F @F
3) 3) + ↵2 (IIC
Fe | 3) + ↵3 (|F
1)
2
Holzapfel, Ogden: Constitutive modelling of arteries, 2010.
+ Coupled formulation for fluid and structure: I I
separated strong or weak coupling: problem dependent stability, smaller problems for solvers monolithic ALE formulation: fully coupled, good stability, mesh deformation
AIME@CZ 2011
Problem description
2
⌦s
1
⌦f
reference configuration
structure part
s
X , t) (X
~f
0 t
1 t
⌦ft
current configuration
fluid part
: ⌦s ⇥ [0, T ] 7! ⌦st
us =
⌦st
2 t 0
s
3 t
~s
3
X,
F = I + Grad u s ,
vs =
us @u
@t J = det F
v f : ⌦ft ⇥ [0, T ] 7! Rn f f
: ⌦f ⇥ [0, T ] 7! ⌦ft
u =
f
X , t) (X
X
AIME@CZ 2011
Governing equations
structure part us @u = vs @t @vv s Ts F = div(JT @t det F = 1
fluid part
T
)+f
in ⌦s in ⌦s
us = 0
on
2
Ts n = 0
on
3
@vv f + (rvv f )vv f = div T f + f @t div v f = 0
in ⌦ft in ⌦ft
vf = v0
on
1 t
f
on
1 t
T n =0
interface conditions vf = vs
on
0 t
Tf n = Ts n
on
0 t
AIME@CZ 2011
Governing equations - ALE structure part us @u s =v @t s @vv Ts F = div(JT @t F) = 1 det(F
T
)+f
in ⌦
s
in ⌦
s s
in ⌦
s u =0
on
2
s
on
3
T n =0
fluid part f u =0
f
in ⌦ , f ⇣ u @vv f @u f F 1 (vv f Tf F + (r v )F ) = div JT @t @t ⇣ ⌘ f T div Jvv F =0
f s u =u ⌘ T +f
on
0
in ⌦
f
in ⌦
f
f v = v0
on
1
f
on
1
T n =0 u f |s , F = I + ru
J = det F
AIME@CZ 2011
Uniform formulation
⌦ = ⌦ f [ ⌦s ,
u : ⌦ ⇥ [0, T ] ! R3 ,
v : ⌦ ⇥ [0, T ] ! R3 ,
8 u < @u =v in ⌦s @t : u = 0 (“mesh deformation operator”) in ⌦f 8 ⇣ ⌘ @vv > < %s Ts F T = div JT @t ⇣ ⌘ u @u > %f @vv = %f ( v )F 1 : Tf F T ) + div JT r F (vv @t @t ⇢ J=1 in ⌦s div(Jvv F T ) = 0 in ⌦f Tf F JT
Ts F JT
T
T
Ts F N = JT
T
N on
0
v = vB
on
1
u =0
on
2
N =0
on
3
in ⌦s in ⌦f
AIME@CZ 2011
Mesh deformation examples
I I I
Laplace operator ) problem on non-convex domain
bi-harmonic operator, inverted Laplacian ) expansive to solve “Pseudo-elastic material” div with ⌫ < 0.0 and E (xx )
⇣
⌘ (div u )II + µ(r u + r u T ) = 0
AIME@CZ 2011
Development of numerical methods
+ space and time discretization I I
in time Crank-Nicholson scheme or fractional ✓ scheme with adaptive time-step selection in space mixed FEM stable pair (Q2 /P1disc , P2+ /P1disc ) or equal order stabilized formulation (local projection, GLS, internal penalty)
+ solving the discrete nonlinear system I I
Large scale Newton or quasi-Newton method. Jacobian computation: analytical, automatic di↵erentiation, finite di↵erences approximation
+ solving large linear system I I I
direct sparse methods, iterative Krylov space based methods, multigrid methods problem dependent smoothing operators, preconditioners e↵ective parallel implementation to use full current hardware potential
+ error evaluation I I I
error caused by modeling discretization error numerical error in solving ! adaptivity
AIME@CZ 2011
Discretization
I
In time: Crank-Nicholson scheme (2nd order) or fractional ✓ scheme (2nd order, better stability) with adaptive time-step selection
I
In 2D space: FEM Q2 /P1disc on quadrilaterals 2 2 Vh = {vv h 2 [C (⌦h )] , v h |T 2 [Q2 (T )] 8T 2 T h }, 2
Ph = {ph 2 L (⌦h ), ph |T 2 P1 (T )8T 2 T h }. I
In 3D space: FEM P2+ /P1disc (Crouzeix-Raviart) on tetrahedrons 0 3 + 3 V h ={vv h 2 [C (⌦h )] : v h dKi 2 [P2 (Ki )] ; 8Ki 2 T h } 2
Ph ={pi 2 L0 (⌦h ) : pi dKi 2 P1 (Ki )
+ P2
=P2
span{
1
...
4}
span{
8Ki 2 T h } i
j
k}
AIME@CZ 2011
Discrete nonlinear system
X ) =0 0, F (X
u h , v h , p h ) 2 Uh ⇥ V h ⇥ P h X = (u s
M uh f
f
s
s
(% M + % M )vh +
k s f n n (M vh + L uh ) = rhs(uh , vh ) 2
k 1 k s f N1 (vh , vh ) + N2 (vh , uh ) + (S (uh ) + S (vh )) 2 2 2
C (uh ) + B 2
n
n
n
kBph = rhs(uh , vh , ph ) fT
vh = 1
+
Suu 4 Sv u cu BsT
Suv Sv v cv BfT
32 3 2 3 0 u fu kB 5 4 v 5 = 4 fv 5 p fp 0
Typical discrete saddle-point problem
AIME@CZ 2011
Solution of the nonlinear problem
I
compute the Jacobian matrix (analytic, automatic di↵erentiation, divided di↵erences) F ]i (X X n + "ee j ) [F F ]i (X X n "ee j ) F [F @F X n) ⇡ (X , X ij @X 2"
I
˜ (BiCGStab or GMRes(m)/ILU(k), MG, direct solver) solve the linear system for X F @F ˜ = F (X X n) X X n) (X X @X
I
adaptive line search strategy ˜ X n + !X X n+1 =X
I
˜) ·X & X + !X ! 2 [ 1, 0) such that f (!) = F (X
continuation methods
AIME@CZ 2011
Jacobian approximation
"/TOL
F @F X @X
10
ij
X n) ⇡ (X
8
F ]i (X X n + "Xjne j ) [F
10
4
2"
F ]i (X Xn [F
10
"Xjne j )
2
,
10
1
10
8
7 /107.57 [21.52]
12 /57.08 [26.52]
12 /47.00 [23.75]
17 /33.06 [27.38]
10
4
7 /108.71 [24.57]
8 /62.75 [17.77]
10 /42.20 [18.95]
18 /31.33 [29.05]
10
2
16 /109.75 [51.65]
20 /47.35 [38.28]
25 /29.80 [38.58]
56 /16.98 [73.83]
10
1
44 /116.11 [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))
AIME@CZ 2011
Linear system solvers
I
direct sparse solver (umfpack, superLU)
I
Krylov space based iterative solver with preconditioning (general ILU(k), special preconditioners) multigrid
I
. standard geometric multigrid approach . smoother by local MPSC-Ansatz (Vanka-like smoother) 2 l+1 3 2 l 3 2 3 Av v |⌦i Av q|⌦i kB|⌦i + Cv p|⌦i v u X 4q l+1 5 = 4q l 5 ! 4Aqvv |⌦i 5 Aqq|⌦i Cqp|⌦i T B|⌦ 0 0 Patch ⌦i p l+1 pl i
1
2
3 defvl 4def lq 5 def lp
. full inverse of the local dense problems by standard LAPACK . full Q2 and P1disc prolongation P by interpolation, restriction defined by R = P T
AIME@CZ 2011
Multigrid solver
+ 1 timestep of fully developed solution + streamline di↵usion stabilization + shown: number of nonlinear steps/avg. number of linear steps [CPU time] + timestep 10 2
Level 1 2 3 4
ndof 12760 50144 198784 791552
MG(2) 2/8 [66] 2/8 [190] 2/9 [744] 2/13 [3803]
MG(4) 2/8 [92] 2/5 [198] 2/6 [852] 2/7 [3924]
BiCGStab(ILU(1)) 2/51 [32] 2/120 [200] 2/311 [1646] MEM.
GMRES(ILU(1),200) 2/50 [27] 2/117 [151] 2/358 [1432] MEM.
MG(2) 4/12 [118] 4/12 [466] 4/13 [1898] 4/15 [8678]
MG(4) 4/11 [177] 4/7 [470] 4/7 [2057] 4/8 [9069]
BiCGStab(ILU(1)) 20/160 [631] 2/800 [] diverg. 2/800 [] diverg. MEM.
GMRES(ILU(1),200) 20/801 [1579] 13/801 [] diverg. 4/801 [] diverg. MEM.
+ timestep 10
0
Level 1 2 3 4
ndof 12760 50144 198784 791552
) robust and efficient Newton-MG combination
AIME@CZ 2011
Numerical fluid-structure interaction benchmarking
+ based on the DFG flow around cylinder ( + realistic materials I I
)
Turek, Sch¨ afer , 1996
incompressible Newtonian fluid, laminar flow regime elastic solid, large deformations
+ setup with simple periodic oscillations + reasonable deformations + computable configuration ) laminar flow, reasonable aspect ratios + results collection, see http://fsw.informatik.tu-muenchen.de/intern/wiki/index.php/Benchmark
AIME@CZ 2011
Benchmarking of the experimental data Flustruc experiment, http://fsw.informatik.tu-muenchen.de/intern/wiki/index.php/Experiment I rotational degree of freedom of the cylinder I small beam thickness I rear mass with corners
Flustruc experiment, Erlangen
computation
AIME@CZ 2011
Example: 2D FSI with power-law fluid
I
(with M. Razzaq, TU-Dortmund)
2D-simplified model of aneurysm with stents, power-law fluid, interaction with elastic structure
[aneurysm]
[aneurysm]
AIME@CZ 2011
Example: 3D FSI with oscilating flow full fluid structure interaction, power law viscosity
#cores 2 4 8 16 32
assembly [s] 297.36 155.59 84.42 48.66 29.98
solver [s] #PCs 2249.93 2 1600.70 2 1090.41 2 586.39 2 479.18 4 MUMPS/PETSC for problem of 649016 degrees of freedom problem. AIME@CZ 2011
Example: aneurysm - real data
+ CT scan, voxel data of the blood vesel segmented out of the full image
+ Spatial discretization: I I
Fictious boundary method - easy to set up for complicated geometry, not optimal approximation of boundary data Classical body fitted mesh or Isogeometric representation - more involved to set up, better approximation
+ Material parameters: viscosity, wall sti↵ness + Boundary conditions: inflow/outflow location?, multiple inflow/outflows?, velocity/pressure values?...
AIME@CZ 2011
Example: aneurysm - real data - FBM I
(with R. Muenster, TU-Dortmund)
3D Navier-Stokes equations, Fictious boundary method
AIME@CZ 2011
Summary and
+outlook
I
monolithic, fully coupled FEM (Q2 /P1disc , P2+ /P1disc ) for viscous incompressible fluid and elastic solid with wide range of constituitve models
I
direct steady calculation, fully implicit 2nd order discretization in time, adaptive timestep selection
I
Newton-like method for the coupled system (Jacobian matrix via divided di↵erences)
I
combination with GMRES/BiCGStab/multigrid and direct methods
+ complete understanding of each step - from model equations, trough analysis and numerical solution + ability to estimate total error from all steps for real world data
AIME@CZ 2011
