seminar ge
TRANSCRIPT
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Fluid and Deformable-Structure Interactions in Bio-Mechanical Systems and Micro-Air Vehicles
Lucy Zhang
Department of Mechanical, Aerospace, and Nuclear Engineering
Rensselaer Polytechnic InstituteTroy, NY
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log (m)
-2
-7
-6 -5
-4 -3
-8
biomaterial
Multiscale bio-mechanical systems
platelet
protein
red blood cell
vesselheart
Fluid-structure interactions at all scales
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Numerical methods for fluid-structure interactions
• Commercial softwares (ABAQUS, ANSYS, FLUENT…)• Explicit coupling technique - generate numerical instabilities (oscillations), diverged solutions
• Arbitrary Lagrangian Eulerian (ALE)• limited to small mesh deformations• requires frequent re-meshing or mesh update
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• Goals:• accurate (interpolations at the fluid-structure interface)• efficient (less/no mesh updating required)• flexible (deformable and rigid structures, boundary conditions)• extensibility (multi-phase flows, various applications)
• Immersed Boundary Method (Peskin) - flexible solid immersed in fluid• structures are modeled with elastic fibers• finite different fluid solver with uniform grid
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Immersed finite element method (IFEM): Fluid-deformable structure interaction
t=0
Assumptions:
• No-slip boundary condition at the fluid-solid interface• Solid is completely immersed in the fluid• Fluid is everywhere in the domain
solid
t = t1
solid
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Equations of motion
ext,d
dijij
i ftv
+=σρ
Principle of virtual work:0
dd ext
, =⎟⎠⎞⎜⎝
⎛ −−∫Ω ijiji
i ftvv σρδ
1 2 3
ssis
isif
iif
i dt
vvdtvvd
tvv
ssΩΩΩ
ΩΩΩ δδ
δδ
δδ ρδρδρδ ∫∫∫ +−1
ssjiji
sfjiji
fjiji ss
dvdvdv ΩΩΩΩΩΩ ∫∫∫ +− ,,, σδσδσδ2
si
si
si
fii
fi dgvdgvdgv
ssΩΩΩ
ΩΩΩρδρδρδ ∫∫∫ +− 3
Ω Ωs
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Ω
IFEM continued …
€
f iFSI ,s = ρ f − ρ s( )
dv is
dt+ σ ij, j
s −σ ij, jf + ρ s − ρ f( )giSolid: in Ωs
€
ρ f (∂v i
∂t+ v jv i, j ) = σ ij, j
f + f iFSI(x,t)
v i,i = 0
fluid: in Ω
Overlapping
Ωs
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Interpolations at the interface
∫ −=Ω
Ωd)(),(),( sFSI,FSI ss tt xxXfxf δ
€
vs(Xs,t) = v(x,t)δ(x − x s)dΩΩ∫
Force distribution
Velocity interpolation
solid nodeInfluence domainSurrounding fluid nodes
Uniform spacing
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Read solid & fluid Geometries
Apply initial conditions
Distribute F onto the fluidFFSI,s -> FFSI Update solids positions
dsolid=Vsolid*dt
Interpolate vfluid onto solids Vsolid
vfluid->Vsolid
Fluid analysis (N-S)Solve for vfluid
Structure analysis Solve for FFSI,s
IFEM Algorithm
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Validations
Flow past a cylinder
Soft disk falling in a channel
Leaflet driven by fluid flow
3 rigid spheres dropping in a channel
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Particle (elastic):
Density= 3,000 kg/m3
Young modulus: E = 1,000 N/m2
Poisson ratio: 0.3
Gravity: 9.81 m/s2
Particle mesh: 447 Nodes and 414 Elements
Fluid:
Tube diameter, D = 4d =2 cm
Tube height, H = 10 cm
Particle diameter, d = 0.5 cm
Density= 1,000 kg/m3
Fluid viscosity = 0.1 N/s.m2
Fluid initially at rest
Fluid mesh: 2121 Nodes and 2000 Elements
A soft disk falling in a viscous fluid
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Fluid recirculation around the soft disk
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Pressure distribution
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yyσ xyσ
t = 0.0 s
t = 1.1 s
t = 2.2 s
t = 3.3 s
t = 4.35 s
xxσ
Stress distribution on the soft disk
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Comparison between the soft sphere and the analytical solution of a same-sized rigid sphere
Terminal velocity of the soft disk
€
ut =ρ s − ρ( )gr2
4μln L
r ⎛ ⎝ ⎜
⎞ ⎠ ⎟− 0.9157 +1.7244 r
L ⎛ ⎝ ⎜
⎞ ⎠ ⎟2
−1.7302 rL ⎛ ⎝ ⎜
⎞ ⎠ ⎟4 ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
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3 rigid spheres dropping in a tube
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3 rigid spheres dropping in a tube
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• Why is it unique?
• fluid- deformable structure interactions
• two-way coupling, higher order interpolation function
• Limitations?
• time step constraint
• rigid solid case
• Possible expansions?
• compressible system
• multiphase flow
• Usefulness?
• numerous applications!
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Use numerical methods to understand and study cardiovascular diseases.
Find non-invasive means to predict physical behaviors and seek remedies for diseases Simulate the responses of blood flow (pressure and velocities) under different physiologic conditions. Compare our results (qualitatively) with published clinical data and analyze the results.
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Biomechanical applications
Red Blood Cell aggregationHeart modeling - left atrium
Deployment of angioplasty stent
Venous valves
Large deformation (flexible)
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Why heart?
Cardiovascular diseases are one of the leading causes of death in the western world.
Cardiovascular diseases (CVD) accounted for 38.0 percent of all deaths or 1 of every 2.6 deaths in the United States in 2002. It accounts for nearly 25% of the deaths in the word.
In 2005 the estimated direct and indirect cost of CVD is$393.5 billion.
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Cardiovascular system
D: The oxygen-poor blood (blue) from the superior vena cava and inferior vena cava fills the right atrium.
E: The oxygen-poor blood in the right atrium fills the right ventricle via tricuspid valve.
F: The right ventricle contracts and sends the oxygen-poor blood via pulmonary valve and pulmonary artery to the pulmonary circulation.
A: The oxygen-rich blood (red) from the pulmonary vein fills the left atrium.
B: The oxygen-rich blood in the left atrium fills the left ventricle via the mitra valve.
C: The left ventricle contracts and sends the oxygen-rich blood via aortic valve and aorta to the systemic circulation.
AF
D
E
CB
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During Atrial Fibrillation (a particular form of an irregular or abnormal heartbeat):
The left atrium does not contract effectively and is not able to empty efficiently.
Sluggish blood flow may come inside the atrium.
Blood clots may form inside the atrium.
Blood clots may break up
Result in embolism.
Result in stroke.
Atrial fibrillation and blood flow
Without blood clots
with a blood clot
Left atrial appendage
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From Schwartzman D., Lacomis J., and Wigginton W.G., Characterization of left atrium and distal pulmonary vein morphology using
multidimensional computed tomography. Journal of the American College of Cardiology, 2003.
41(8): p. 1349-1357Ernst G., et al., Morphology of the
left atrial appendage. The Anatomical Record, 1995. 242: p.
553-561. Left atrium
Left atrial appendage
Pulmonary veins
Left atrium geometry
77mm
28mm
20mm
17mm 56mm
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During diastole (relaxes, 0.06s < t < 0.43s) , no flow through the mitral valve (v=0)
During systole (contracts, 0.43s < t < 1.06s), blood flow is allowed through the mitral valve (free flow)
Blood is assumed to be Newtonian fluid, homogenous and incompressible. Maximum inlet velocity: 45 cm/sBlood density: 1055 kg/m3
Blood viscosity: 3.5X10-3 N/s.m2
Fluid mesh: 28,212Nodes, 163,662 ElementsSolid mesh: 12,292 Nodes, 36,427 Elements
Left atrium with pulmonary veins
Klein AL and Tajik AJ. Doppler assessment of pulmonary venous flow in healthy subjects and in patients with heart disease. Journal of the American Society of Echocardiography, 1991, Vol.4, pp.379-392.
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Pressure distribution at the center of the atrium during a diastole and systole cycle
Transmitral velocity during diastole
Left atrium with appendage
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Rigid wall
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Pressure distribution at the
center of the atrium during
one cardiac cycle
Transmitral velocity during
one cardiac cycle
Kuecherer H.F., Muhiudeen I.A., Kusumoto F.M., Lee E., Moulinier L.E., Cahalan M.K. and Schiller N.B., Estimation of
mean left atrial pressure from transesophageal pulsed Doppler echocardiography of pulmonary venous flow
Circulation, 1990, Vol 82, 1127-1139
E
A
Left atrium (comparison with clinical data)
5
Pressure (mm hg)
2 Time (s)
1.510
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Transmitral velocity during one cardiac cycle (with and without the appendage)
Velocity inside the appendage during one cardiac cycle
Influence of the appendage
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Left atrium geometry
Courtesy of Dr. A. CRISTOFORETTI,[email protected]
University of Trento, Italia
G. Nollo, A. Cristoforetti, L. Faes, A. Centonze, M. Del Greco, R. Antolini, F. Ravelli: 'Registration and Fusion of Segmented Left Atrium CT Images with CARTO Electrical Maps for the Ablative Treatment of Atrial Fibrillation', Computers in Cardiology 2004, volume 31, 345-348;
Pulmonary veins
Pulmonary veins
Left atrium
Left atrial appendage
Pulmonary veins
Pulmonary veins
Mitral valveLeft
atriumBlood clots
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Red blood cells and blood
RBC FEM RBC model
From Dennis Kunkel at http://www.denniskunkel.com/
10ìm2ìm
Property of membrane•Thickness of RBC membrane: 7.5 to 10 nm•Density of blood in 45% of hematocrit: 1.07 g/ml•Dilation modulus: 500 dyn/cm•Shear modulus for RBC membrane: 4.2*10-3dyn/cm•Bending modulus: 1.8*10-12 dyn/cm.
Property of inner cytoplasm •Incompressible Newtonian fluid
empirical function
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The shear rate dependence of normal human blood viscoelasticity at 2 Hz and 22 °C (reproduced from http://www.vilastic.com/tech10.html)
Bulk aggregates Discrete cells Cell layers
Red blood cells and blood
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Shear of a RBCs Aggregate
The shear of 4 RBCs at low shear rate
The RBCs rotates as a bulk
The shear of 4 RBCs at high shear rate
The RBCs are totally separated and arranged at parallel layers
The shear of 4 RBCs at medium shear rate
The RBCs are partially separated
RBC-RBC protein dynamic force is coupled with IFEM (NS Solver)C-Cf
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Micro-air vehicles
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http://www.fas.org/irp/program/collect/docs/image1.gif
three types of MAVs:1. airplane-like fixed wing model, 2. helicopter-like rotating wing model, 3. bird-or insect-like flapping wing model.
potential military and surveillance use
10-4
1
0-3
10-2
1
0-1
1
10
1
02
10
3
10
4
105
1
06
Gro
ss W
eigh
t (Lb
s)
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MAVs
Features:• improved efficiency, • more lift, • high maneuverability,• reduced noise.
Loitering wings: high span and a large surface areaFast wings: a low wing span and a small areaFlying efficiently at high speed: small, perhaps, swept wings Flying at slow speed for long periods: long narrow wings
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Bio-inspired flapping wings
muscle contraction
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Acknowledgement
•Graduate students: • Mickael Gay, Yili Gu
• Collaborators: • Dr. Holger Salazar (Cardiology Department, Tulane University)• Dr. A. Cristoforetti (University of Trento, Italy)
• Funding agencies: NSF, NIH, Louisiana BOR
• Computing resources: •Center for Computational Sciences (CCS) - Tulane• SCOREC (RPI)
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What can you do?
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IFEM: Governing Equations
€
ρ s ∂2us
∂t 2
⎛ ⎝ ⎜
⎞ ⎠ ⎟−∇ ⋅σ s = f sur + ρ sg on Ωs
f = f sur (X)δ(x − X)dVΩ s
∫
ρ f ∂v∂t
+ v ⋅∇v ⎛ ⎝ ⎜
⎞ ⎠ ⎟=∇ ⋅? + f on Ω
∇ ⋅v = 0
dX /dt = v(x)δ(x − X)dVΩ∫
Navier-Stokes equation for incompressible fluid
Governing equation of structure
Force distribution
Velocity interpolation
ΩsΩ
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IFEM: Solid Force Calculationextsurint fffMa +=+
gf )(ext ρρ −= σVol
External Forces: External forces can be arbitrary forces from diverse force fields (e.g. gravity, buoyancy force, electro-magnetic fields).
g – acceleration due to gravity
VSs I
pqpqI dint ∫Ω ∂∂
=X
fε
Internal Forces: hyperelastic material description (Mooney-Rivlin material).
S – 2nd Piola Kirchhoff stress tensor
ε - Green Lagrangian strain tensorTotal Lagrangian Formulation
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€
ρ ∂v∂t
+ ρv ⋅∇v =∇ ⋅σ f + f FSI
∇ ⋅v = 0
Solve for velocity using the Navier-Stokes equation Eq. (III)
The interaction force fFSI,s is distributed to the fluid domain via RKPM delta function.
The fluid velocity is interpolated onto the solid domain via RKPM delta function
ΩsΩ
∫Ω Ω−= d)(),(),( sss tt xxxvXv φ
∫Ω Ω−= d)(),(),( sFSI,FSI ss tt xxXfxf φ
€
−Δρ ∂vs
∂t+ (∂σ s
∂x− ∂σ f
∂x) + Δρg
sΩin Ωin
),(FSI txf),(sFSI, tsXf
tt
ss
∂∂
≡∂∂ vu
2
2
The interaction force is calculated with Eq. (I)
sFSI,f
I.
IV.
III.
II.
),( txvP and v unknowns are solved
by minimizing residual vectors (derived from their
weak forms)
Distribution of interaction force
Insert this inhomogeneous fluid force field into the N-S eqn.
Update solid displacement with
solid velocity
IFEM Governing Equations
=sFSI,f
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Red blood cell modelRBC
From Dennis Kunkel at http://www.denniskunkel.com/
2ìm
Shear rate dependence of normal human blood viscoelasticity at 2 Hz and 22 °C (reproduced from http://www.vilastic.com/tech10.html)
Bulk aggregates Discrete cells Cell layers
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Venous Valve
Courtesy of H.F. Janssen, Texas Tech University.
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• Site of deep venous thrombosis formation• Prevents retrograde venous flow (reflux) • Site of sluggish blood flow• Decreased fibrinolytic activity• Muscle contraction prevents venous stasis:
– Increases venous flow velocity– Compresses veins
• Immobilization promotes venous stasis
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Venous Valve Simulation
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Venous Valve
Comparison between experiment and simulation at 4 different time steps
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Multi-resolution analysis
⎟⎟⎠⎞
⎜⎜⎝⎛
Δ−
Δ=−
j
j
jja xa
xxxa
xx φφ 1)(
• Window function with a dilation parameter:
∑=
Δ−Φ==NNP
jjjaja
Ra xxxxxuxuPxu1
);()()()(
• Projection operator for the scale a
a: dilation parameter
• Wavelet function:);();();( 22 jajaja xxxxxxxxx −Φ−−Φ=−ψ
• Complementary projection operator:
∑=
Δ−=NNP
jjjaja xxxxxuxuQ
122 );()()( ψ
)()()( 22 xuQxuPxuP aaa +=low scale + high scale