COMPUTATIONAL MECHANICS WCCM VI in conjunction with APCOM’04, Sept. 5-10, 2004, Beijing, China © 2004 Tsinghua University Press & Springer-Verlag
A Coupled Momentum Method to Model Blood Flow in Deformable Arteries Alberto Figueroa1*, Kenneth E. Jansen3, Thomas J.R. Hughes4, Charles A. Taylor1,2 1
Departments of Mechanical Engineering, 2Surgery and Pediatrics, Stanford University, E350 Clark Center, 318 Campus Drive, Stanford, CA 94305, USA 3 Department of Mechanical Engineering, Aeronautical and Engineering & Mechanics, Rensselaer Polytechnic Institute, Troy, NY 12180, USA 4 Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin, Austin, TX 78712, USA e-mail:
[email protected],
[email protected],
[email protected],
[email protected] Abstract Blood velocity and pressure fields in large arteries are greatly influenced by the deformability of the vessel. However, computational methods for simulating blood flow in three dimensional models of arteries have either considered a rigid wall assumption for the vessel or significantly simplified geometries. Computing blood flow in deformable domains using standard techniques like the ALE method remains an intractable problem for realistic anatomic and physiologic models. We have developed a new method to model blood flow in three dimensional deformable models of arteries. The method couples the equations of the deformation of the vessel wall at the variational level as a boundary condition for the fluid domain, by assuming that for a thin-walled structure the internal traction due to the fluid friction is felt uniformly through the vessel wall. We consider a strong coupling of the degrees of freedom of the fluid and the solid domains, and a linear membrane model (enhanced with through-plane stiffness) for the vessel wall. We have used the generalized-alpha method to integrate the time evolution of the resulting equations for the deformable system. We present here the mathematical formulation of the method and discuss issues related to the fluid-solid coupling, membrane formulation, time integration method, and boundary and initial conditions (including pre-stressing the membrane). Implementation issues will be discussed and initial results with simple geometries will be presented. Key words: fluid-solid interaction, blood flow, finite element, wall mechanics, time-integration. INTRODUCTION In recent years, computational techniques have been used widely by researchers seeking to simulate blood flow in three-dimensional models of arteries. Applications include disease research, medical device design and, more recently, surgical planning. With few exceptions, computational techniques applied to model blood flow in arteries have only examined the velocity field (not the pressure field) and have treated the vessel walls as rigid [1]. The rigid-wall approximation is made in large part because of the difficulty of solving the coupled blood flow / vessel deformation problem and is justified by the observation that, under normal conditions, wall deformability does not significantly alter the velocity field [2]. However, this observation was made for arteries where wall motion is small and may not be valid for arteries where deformations are larger (e.g. the thoracic aorta). Perhaps most importantly, the assumption of rigid vessel walls precludes wave propagation phenomena, fundamentally changes the character of the resultant solutions, and results in difficulties in coupling three-dimensional domains with domains described using one-dimensional wave propagation methods.
One of the better-known methods for including wall deformability is the ALE (arbitrarily Lagrangian-Eulerian) formulation for fluid-structure interaction problems [3]. This technique considers a Lagrangian description for the solid mesh and an Eulerian description for the fluid mesh. However, ALE formulations have been proven to be computationally very expensive and not very robust since they necessitate the continual updating of the geometry of the fluid and structural elements. Although significant progress has been made in recent years in solving fluid-structure interaction problems using ALE methods (see, e.g., [4]), these problems remain computational “grand challenges.” Application of fluid-structure interaction methods in simulation-based medical planning is additionally demanding in that multiple surgical interventions need to be modeled, solved, analyzed, and compared in a clinically relevant time-frame (hours). Simpler methods for incorporating wall motion are essential for these applications. In this work a new formulation called the Coupled Momentum Method for Fluid-Solid Interaction problems (CMM-FSI) is presented. The main features of this formulation can be summarized as follows: 1. The elastodynamics equations of the vessel wall are used to define a boundary condition of the fluid domain. The elastodynamics equations are coupled to the conservation of momentum equations governing blood flow by expressing the traction on the boundary of the fluid domain as a function of the equations of the motion of the vessel wall. With this approach only minimal changes in the stiffness matrices and residuals of the rigid wall model are required to incorporate the effects of the wall motion. 2. Fixed fluid domain and linearized wall kinematics. The assumption throughout the three dimensional models that the geometry of the arteries and surrounding structure are fixed but the deformations of the structure are accounted for through the use of linearized kinematics is utilized. The displacements of the structure are assumed small so that the geometry of the structure does not need to be updated throughout the cardiac cycle. Likewise, the contiguous fluid domain is held fixed but the velocity at the fluid-solid interface is non-zero – there are small velocity components in the plane of the lumen as well as the axial direction, compatible with the deformations of the artery. The in-plane component of velocity creates the possibility of flow “storage” that is impossible in a fixed geometry with no-slip boundary condition under the assumption of flow incompressibility. 3. Simple enhanced-membrane formulation for the wall. Some simplifications will be invoked in the arterial structural modeling given that the cardiac pulse has a long wavelength (meters) compared with the diameters of arteries (centimeters). Consequently, arteries tend to respond primarily in membrane mode rather than in bending mode. Therefore, a linear enhanced-membrane element is used instead of a more complex shell element accounting for bending behavior. We will adopt a strong coupling approach for the degrees of freedom of the fluid and the vessel wall. A node located on the vessel wall will be a part of the boundary of the fluid domain and also part of the membrane model for the vessel wall. No additional degrees of freedom are present with respect to a rigid wall formulation of the problem. METHODS 1. Governing equations (strong and weak forms): 1.1 Fluid mechanics problem (strong form) The strong form of the continuity and momentum equations is written in the so-called advective form ([1, 5, 6]) for a domain Ω ∈ ℜ nsd (see Fig. 1), where nsd is the number of spatial dimensions:
Fig. 1. Fluid and solid domains and boundary conditions.
K K K f : Ω× ( 0,T ) → ℜnsd , g : Γ g × ( 0,T ) → ℜnsd , h : Γ h × ( 0 ,T ) → ℜ nsd K K K K p 0 : Ω → ℜ ; find v( x,t ) and p( x,t ) ∀x ∈ Ω , ∀t ∈ [ 0,T ] such that K K ( x,t ) ∈ Ω× ( 0,T ) ∇⋅v = 0
Given
K
,
K v 0 : Ω → ℜ nsd ,
(1)
ρ v,t + ρ v ⋅∇v = −∇p + ∇ ⋅τ + f
K ( x,t ) ∈ Ω× ( 0,T )
(2)
K K v=g
K ( x,t ) ∈ Γ g × ( 0,T )
(3)
K
K
K
K K K K K ( x,t ) ∈ Γ h × ( 0 ,T ) (4) tnK = σ n = [ − pI + τ ] n = h K K K K ( x,t ) ∈ Γ s × ( 0 ,T ) (5) tnK = σ n = t f K K K K K K K v( x,0 ) = v 0 ( x ); p( x,0 ) = p 0 ( x ) (6) x ∈Ω K K where v represents the velocity, p is the pressure, ρ is the fluid density, f is the prescribed body force K per unit volume and τ is the viscous stress tensor. The initial velocity field v 0 is assumed to be K K K divergence-free. The prescribed velocity field on Γ g is g , whereas h and t f are the prescribed
tractions on Γ h and Γ s respectively. The lateral vessel wall is Γ s . While a no-slip boundary condition would be prescribed on Γ s in the case of a rigid wall approximation, this constraint is removed here to K enable non-zero wall velocity. Furthermore, at this point the magnitude t f is still unknown and is to be determined using equations coming from the solid problem. 1.2 Fluid mechanics problem (weak form) The discrete trial solution and weighting function spaces for the semi-discrete formulation are given by
{K K
K
K
K
S hk = v v( ⋅,t ) ∈ H 1( Ω )nsd ,t ∈ [ 0,T ],v xK∈Ω ∈ Pk ( Ωe )nsd ,v( ⋅,t ) = gˆ on Γ g e
{K K
K
K
K
}
Whk = w w( ⋅,t ) ∈ H 1( Ω )nsd ,t ∈ [ 0,T ],w xK∈Ω ∈ Pk ( Ωe )nsd ,w( ⋅,t ) = 0 on Γ g e
{
}
Ph k = p p( ⋅,t ) ∈ H 1( Ω ),t ∈ [ 0,T ], p xK∈Ω ∈ Pk ( Ωe ) e
(7)
}
(8) (9)
It is important to note that, because stabilized methods are used, the local approximation space Pk ( Ωe ) is the same for both the velocity and pressure variables [7]. The stabilized formulation used in the present work is based on that described by Taylor et al. [1] and Whiting and Jansen [5]. The diffusive term, pressure term and continuity equation are all integrated by parts as described by Gresho and Sani [6]. Considering the spaces defined above, the semi-discrete Galerkin finite element formulation produces the following weak form of (1)-(5): K Find v ∈ S hk and p ∈ Ph k such that K K K K K K K K K K BG ( w,q;v , p ) = w ⋅ ρ v,t + ρ v ⋅∇v − f + ∇w : ( − pI + τ ) − ∇q ⋅ v dx Ω (10) K K K Kf − w ⋅ h + qvn ds + − w ⋅ t + qvn ds + qvn ds = 0 +
∫{ ( ∫ { Γh
K for all w ∈Wh k and q ∈ Ph k .
}
)
}
∫ { Γs
}
∫
Γg
The integral in equation (10) given by K K − w ⋅ t f ds
∫
(11)
Γs
is not known yet. We will determine it using information coming from the elastodynamics problem. Before describing the treatment of the solid domain, we must recall that the standard Galerkin method is unstable for advection-dominated flows and in the diffusion dominated limit for equal order interpolation of velocity and pressure. A stabilized method is utilized to address these deficiencies of Galerkin’s method. The formulation becomes: K Find v ∈ S hk and p ∈ Ph k such that K K K K B( w,q;v , p ) = BG ( w,q;v , p ) nel
+
K ∑ ∫ {∇w : (τ e =1 nel
+
}
K K K K ⎫ K ⎧ K ⎛ ∆K K ⎞ K K w ⋅ ⎜ ρ v ⋅∇v ⎟ + ∇w : (τ L ( v , p ) ⋅∇v ) ⊗ L ( v , p ) ⎬ dx ⎨ Ωe ⎠ ⎩ ⎝ ⎭ τ K K K ∇q ⋅ M L ( v , p ) dx = 0
∑∫ e =1 nel
+
Ωe
K K ( v , p ) ⊗ vK ) + ∇ ⋅ wKτ C ∇ ⋅ vK dxK
ML
∑∫ e =1
Ωe
(12)
ρ
K K K for all w ∈Wh k and q ∈ Ph k . L ( v , p ) represents the residual vector of the momentum equation K K K K K K L ( v , p ) = ρ v,t + ρ v ⋅∇v + ∇p − ∇ ⋅τ − f
(13)
1.3 Solid mechanics problem (strong form) The classic elastodynamics equations are used to describe the motion of the vessel wall in a domain s Ω ∈ ℜ nsd (see Fig. 1 ). Therefore, the strong form of this problem can be described as follows: K K K K Given b s : Ω s × ( 0,T ) → ℜnsd , g s : Γ sg × ( 0 ,T ) → ℜnsd , h s : Γ sh × ( 0 ,T ) → ℜnsd , u 0 : Ω s → ℜ nsd and K K K K u,t0 : Ω s → ℜnsd ; find u( x,t ) ∀x ∈ Ω s , ∀t ∈ [ 0,T ] such that K K K ( x,t ) ∈ Ω s × ( 0,T ) ρ s u,tt = ∇ ⋅ σ s + b s (14) K K K u = gs (15) ( x,t ) ∈ Γ sg × ( 0 ,T ) K K K tnK = σ s n = h s K K K K u( x,0 ) = u 0 ( x ) K K K K u,t ( x,0 ) = u,t0 ( x )
( xi ,t ) ∈ Γ sh × ( 0,T )
(16)
K x ∈ Ωs
(17)
K K where u is the displacement field, ρ s is the density of the vessel wall, b s is the prescribed body force per K K K K unit volume, σ s is the vessel wall stress tensor, and u 0 ( x ) and u,t0 ( x ) are the given initial conditions for displacement and velocity, respectively. Similarly, Γ sg and Γ hs represent the parts of the boundary of Ωs K K where the essential ( g s ) and natural ( h s ) boundary conditions are prescribed.
1.4 Solid mechanics problem (weak form) In order to make the solid problem stated in equations (14)-(17) compatible with the fluid problem Γ s needs to be mapped on Ωs . Assuming a thin-walled structure, integrals over Γ hs and Ωs can be related with integrals over the lateral boundary of the fluid domain Γ s according to the following expressions
∫
Ωs
( • ) dx = a ∫Γ ( • ) ds
∫
Γ hs
s
( • ) ds = a ∫∂Γ ( • ) dl
(18)
s
K The surface traction t f on the fluid lateral wall due to the interaction with the solid is equal and K K K opposite to the surface traction t s on the vessel wall due to the fluid motion ( t f = −t s ). Using again a K K thin wall approximation, this surface traction t s can be considered as a body force b s for the solid domain by assuming that the internal surface traction is felt uniformly through the wall thickness a . Therefore, it follows that Kf K (19) bs = −t a
and thus it is possible to relate the new integral term given by equation (11) with the weak form for the solid mechanics problem that will be introduced next. This approach is analogous to Womersley’s [8] derivation of an analytical solution for pulsatile blood flow in an elastic vessel. There, Womersley considered just two components of the stress due to the fluid motion being applied to the vessel wall: a longitudinal component (related to the viscous stress) and a radial component (related to the pressure). Here, the full stress vector ti f = −tis is taken into account. The weak counterpart of equations (14)-(17) is to be obtained next. Considering the spaces S hk and Wh k defined before, the semi-discrete Galerkin finite element formulation produces the following weak form: K Find v ∈ S hk such that K K K K K K K K K K (20) ρ s w ⋅ v,t dx + s ∇w : σ s dx = s w ⋅ b s dx + s w ⋅ h s ds s Ω Ω Ω Γh K for all w ∈Wh k . Now, equation (18) is utilized to rewrite equation (20) in terms of the new boundary decomposition for the fluid problem given in Fig. 1 as follows: K K K K K K K (21) a ρ s w ⋅ v,t ds + a ∇w : σ s ds = a w ⋅ b s ds +a w ⋅ h s dl Γs Γs Γs ∂Γ s K Considering the expression for the body force b s given by equation (19), the final expression of the weak form for the solid domain is K K K K K K K (22) − w ⋅ t f ds = a ρ s w ⋅ v,t ds + a ∇w : σ s ds − a w ⋅ h s dl Γs Γs Γs ∂Γ s
∫
∫
∫
∫
∫
∫
∫
∫
∫
∫
∫
∫
1.5 Combined problem: the CMM-FSI weak form Equation (22) provides an expression for the unknown term in equation (10) as a function of the solid internal stresses and inertial forces. If these two equations are combined together, it results in the following weak Galerkin form for the Coupled Momentum Method for Fluid-Solid Interaction problems: K K K K K K K K K K BG ( w,q;v , p ) = w ⋅ ρ v,t + ρ v ⋅∇v − f + ∇w : ( − pI + τ ) − ∇q ⋅ v dx Ω K K − w ⋅ hds + qvn ds Γh Γh (23) K K K K K K +a w ⋅ ρ s v,t + ∇w : σ s ( u ) ds − a w ⋅ h s dl + qvn ds Γs ∂Γ s Γs
∫{ ( ∫ ∫ { + ∫ qv ds Γg
n
}
)
∫
}
∫
∫
The red integrals in equation (23) define the new terms added to the rigid wall theory. In order to complete the formulation of the problem, we need to discuss the details of the mechanical model for the vessel wall, as well as the linearization and time integration schemes used to discretize equation (23). 2. Finite element model for the vessel wall The need for a simple model for the vessel wall mechanics has already been justified. Since we consider a strong coupling of the degrees of freedom of the lateral boundary of the fluid domain and the vessel wall, a membrane element with translational degrees of freedom only represents the simplest choice. A coordinate transformation between the global reference system and the local plane of the triangular element must be defined in order to represent the membrane behavior (see Fig. 2). Furthermore, since the geometry of the fluid domain is kept fixed, the choice of the infinitesimal elasticity theory for the constitutive model of the wall is reasonable. The mesh of the vessel wall will be defined in most cases by linear triangles, since the internal fluid mesh is generated by linear tetrahedra. However, it is well known that linear constant strain triangles (CST) representing membrane modes only are inappropriate when used in three dimensional geometries with loads applied perpendicularly to their plane (see [9], [10]). We have decided to augment the stiffness of the linear membrane element with a through-plane shear as illustrated in Fig. 2 (although the effects of this transversal shear are expected to be small in the applications envisaged). This allows for an improved representation of the wall mechanics using only the translational degrees of freedom {u1 , u2 , u3 } .
Fig. 2. Coordinate transformations and stresses for the membrane patch with transverse shear. 3. Newton-Raphson linearization and time integration scheme We apply Newton’s Method in two variables (one vector, one scalar) to the weak form defined by equation (23). Linearization is done with respect to the time derivatives of the velocity and pressure at time tn +1 as described by [5]. By doing this, we obtain the following system: ⎡ K ijab ⎢ ⎢⎣ D ab j
(
)
b ⎧ ⎫ a Giab ⎤ ⎪ ∆v j,t ⎪ ⎪⎧( Ri )m ⎪⎫ + 1 n ⎥⎨ ⎬ = −⎨ a ⎬ C ab ⎥⎦ ⎪ ( ∆p )b ⎪ ⎪⎩ Rc ⎪⎭ ,t n +1 ⎭ ⎩
(24)
where { Ri }m and Rc are the terms of equation (23) that represent the momentum and continuity equations ab residuals. The matrices K ijab , Giab , D ab j and C are defined by
∂ ( Ri )m a
Kijab
=
( )
∂ v j,t
b n +1
∂ ( Ri )m a
,
Giab
=
∂ ( p,t )n+1 b
,
D ab j
=
∂ ( Rc )
( )
∂ v j,t
a
b n +1
, C
ab
=
∂ ( Rc )
a
∂ ( p,t )n +1 b
(25)
This semi-discrete system is advanced in time using the generalized-alpha method as described by [11] and [12].
The method as presented here has been coded into a stabilized finite element solver of the Navier-Stokes equations. So far, the integrals in red in equation (23) have only been included into the right-hand-side vector of the system of algebraic equations defined by equation (24), resulting in an explicit implementation of the method. With this explicit approach, the left-hand-side matrices K ijab , Giab , D ab j and C ab are the same as in the standard rigid wall formulation. Further work needs to be done to add these integrals to the left-hand-side matrices and thus make the method implicit. However, some interesting results can be observed with this explicit implementation of the method as shown in the next section. NUMERICAL RESULTS
We present the results obtained for a simple flow in a rectangular channel with lateral deformable walls. The geometry, mesh, and boundary conditions are described in Fig. 3 below.
Fig. 3. Domain, mesh and boundary conditions for the rectangular channel flow The material parameters of the model are all given in cgs units. Their values are shown in Table 1. Table 1. Material parameters and constants.
ρ
µ 3
gr/cm 1.06
ρs
E
dynes/cm ⋅ s 0.04 2
2
dynes/cm 5320
3
gr/cm 1.09
ν
k
a
vmax
pout
0.5
0.833
cm 0.1
cm/s 12.5
dynes/cm2 2660
E and ν are the Young modulus and Poisson ratio of the vessel wall, respectively; k is the transverse shear correction parameter; a is the thickness of the vessel wall and µ is the fluid viscosity. The values of the material parameters are all physiologically relevant, expect for the Young modulus, whose value is softer than in real vessels. We have to do this since thus far we have coded only the explicit implementation of the method. Under these conditions, the larger the Young modulus, the harder it is to get a convergent solution without adding the solid contributions to the left-hand-side matrices defined in equations (24)-(25). However, interesting conclusions can be made from the results presented below. We are going to present the time history of the displacement and velocity of the nodes located on the deformable walls of the model –i.e. nodes 11, 12, 13 and 14. These nodes are free to move in the longitudinal –z direction (due to effect of the fluid shear) and in the transversal –x direction (due to the effect of the pressure). The initial conditions for the velocity and pressure are as follows:
⎧ 0 ⎪⎪ K K v( x, 0 ) = ⎨ 0 ⎪ 1 − x2 ⎩⎪
(
)
⎫ ⎪⎪ ⎬ ⎪ ⎭⎪
K p( x, 0 ) = pout
(26)
Two cases are presented here. In the first one, the walls are completely unloaded at the beginning of the simulation, and therefore the initial displacement field for the wall nodes is zero. In the second case, the walls are pre-stressed so that they are in equilibrium with the internal traction of the fluid domain. Case 1. Walls without pre-stress Nodal displacement - z direction
Nodal displacement - x direction 15
5
Node 11 Node 12
0 0
2
4
6
8
10
Node 13 Node 14
-5
displacement (cm)
displacement (cm)
10
-10 -15
0.4 0.3 0.2 0.1 0 -0.1 0 -0.2 -0.3 -0.4 -0.5 -0.6
Nodal velocity - x direction
4
6
8
10
Node 12 Node 13 Node 14
Nodal velocity - z direction
100
15
80
10
40
Node 11
20
Node 12
0
Node 13 2
4
6
-40
8
10
Node 14
velocity (cm/s)
60 velocity (cm/s)
2
time (scs)
time (scs)
-20 0
Node 11
5
Nodel 11 Node 12
0 0
5
10
Node 13 Node 14
-5 -10
-60 -80
-15 time (scs)
time (scs)
Fig. 4. Displacement and velocity history of the nodes located on the walls (no pre-stressing) We can observe that the lateral walls undergo a transient due to the sudden loading with the stresses coming from the interaction with the fluid motion. As expected, the system eventually reaches an equilibrium state, defined by a finite displacement and a zero velocity for the nodes under study. The final displacement in the transversal –x direction is considerably larger than the final displacement in the longitudinal –z direction. This is consistent with the ratio between the magnitudes of the stresses in the z-direction ( the resulting shear is on the order of 1 dyne/cm2) and in the x-direction (the normal pressure is on the order of 2600 dynes/cm2). For this example, the simulation has only an additional expense of a larger number of nonlinear iterations per time step than in the rigid case. In the deformable case, we have used seven nonlinear iterations, whereas in the rigid case, two were used. The time step size used here is on the same order of magnitude as in the rigid case model ( ∆t = 0.01 sec) with the same boundary conditions.
Case 2. Walls with pre-stress
Pre-stressing the walls is a very important aspect from both a mechanical and physiologic standpoint. On the one hand, providing the right level of initial stress to the deformable walls so that they are in equilibrium with the internal fluid stress is a very desirable ‘initial condition’ which reduces the initial transients and defines a better-posed problem. Furthermore, from a physiologic perspective, the vessel walls are well known to be pre-stressed both in the circumferential and longitudinal directions (see [13]). We present the results obtained for the previous problem when the deformable nodes are pre-stressed with the equilibrium displacements shown in Fig. 4. Nodal displacement - z direction
Nodal displacement - x direction
0.0008
15
displacement (cm)
Node 11
5
Node 12 Node 13
0 0
0.5
1
1.5
2
Node 14
displacement (cm)
0.0007 10
0.0006
Node 11
0.0005
Node 12
0.0004
Node 13
0.0003
Node 14
0.0002 0.0001
-5
0 0
-10
0.5
0.03
0.006
0.02
0.004
0.01
Node 11 Node 12
0 0.5
1
1.5
2
Node 13 Node 14
0.002
Node 11
0 -0.002
Node 12 0
0.5
1
-0.006
-0.03
-0.008
1.5
2
Node 13 Node 14
-0.004
-0.02
time (scs)
2
Nodal velocity - z direction
velocity (cm/s)
velocity (cm/s)
Nodal velocity - x direction
0
1.5
time (scs)
time (scs)
-0.01
1
time (scs)
Fig. 5. Displacement and velocity history of the nodes located on the walls (with pre-stressing) The effects of the pre-stressing are very dramatic, as it can be seen in Fig. 5. The transient in the displacement and velocity fields is almost completely eliminated. Results are shown for just two seconds of physical time, since after that the variations in the solution are negligible. This confirms the importance of providing the right initial stresses for the structure, and motivates further examination of the effects of stretching the structures to make them stiffer, analogous to real blood vessels. It is important to remark that these computations have been done using a value of ρ ∞ (the spectral radius for an infinite time step, see [11]) of zero. This results in the annihilation of the highest frequencies of simulation in one time step and is a desirable property since the walls are perfectly elastic, and thus the only dissipation would come from the small fluid viscosity. Considering a viscoelastic model for the vessel walls will probably reduce this constraint in the value adopted for ρ ∞ .
CONCLUSION
We have successfully developed and implemented a simple method for simulating blood flow in three dimensional deformable domains. The method couples the equations of the deformation of the wall with the equations governing the fluid motion using the basic assumption of a thin-walled structure. Results have been shown for the explicit implementation of the method in an idealized channel flow problem. Work in progress includes the implicit implementation of the method, the study of the effects of pre-stressing the vessel walls, as well as the consideration of a more physiologically relevant viscoelastic constitutive model for the wall. The method could represent a computationally efficient alternative to ALE formulations for modeling blood flow in deformable domains, and would enable the coupling with lower-order downstream deformable models that accommodate wave propagation phenomena (see [14]). Acknowledgements This material is based upon work supported by the National Science Foundation under Grant No. 0205741. REFERENCES [1] C. A. Taylor, T. J. R. Hughes and C. K. Zarins, Finite Element Modeling of Blood Flow in Arteries, Computer Methods in Applied Mechanics and Engineering, 158,(1998),155-196. [2] K. Perktold and G. Rappitsch, Computer Simulation of Local Blood Flow and Vessel Mechanics in a Compliant Carotid Artery Bifurcation Model, Journal of Biomechanics, 28(7),(1995),845-856. [3] T. J. R. Hughes, W. K. Liu and T. K. Zimmermann, Lagrangian-Eulerian Finite Element Formulation for Incompressible Viscous Flows, Computer Methods in Applied Mechanics and Engineering, 29,(1981),329-349. [4] S. M. Rifai, Z. Johan, W. P. Wang, P. P. Grisval, T. J. R. Hughes and R. M. Ferencz, Multiphysics Simulation of Flow-Induced Vibrations and Aeroelasticity on Parallel Computing Platforms, Computer Methods in Applied Mechanics and Engineering, 174,(1999),393-417. [5] C. H. Whiting and K. E. Jansen, A Stabilized Finite Element Method for the Incompressible Navier-Stokes Equations Using a Hierarchical Basis, International Journal for Numerical Methods in Fluids, 35,(2001),93-116. [6] P. M. Gresho and R. L. Sani, Incompressible Flow and the Finite Element Method. 1998, Wiley: New York. [7] T. J. R. Hughes, L. P. Franca and M. Balestra, A New Finite Element Formulation for Computational Fluid Dynamics: V. A Stable Petrov-Galerkin Formulation of the Stokes Problem Accomodating Equal-Order Interpolations, Computer Methods in Applied Mechanics and Engineering, 59,(1896),85-99. [8] J. R. Womersley, Oscillatory Motion of a Viscous Liquid in a Thin Walled Elastic Tube-I: The Linear Approximation for Long Waves, The Philosophical Magazine, 7,(1955),199-221. [9] T. J. R. Hughes, The Finite Element Method. Linear Static and Dynamic Finite Element Analysis. 2000, Dover: New York. [10] R. D. Cook, D. S. Malkus, M. E. Plesha and R. J. Witt, Concepts and Applications of Finite Element Analysis. 2002, John Wiley & Sons: New York. [11] K. E. Jansen, C. H. Whiting and G. M. Hulbert, Generalized-{Alpha} Method for Integrating the Filtered Navier-Stokes Equations with a Stabilized Finite Element Method, Computer Methods in Applied Mechanics and Engineering, 190(3-4),(2000),305-319. [12] J. Chung and G. M. Hulbert, A Time Integration Algorithm for Structural Dynamics with Improved Numerical Dissipation: The Generalized-{Alpha} Method, Journal of Applied Mechanics, 60,(1993),371-375. [13] Y. C. Fung and S. Q. Liu, Change of Residual Strains in Arteries Due to Hypertrophy Caused by Aortic Constriction, Circulation Research, 65,(1989),1340-1349. [14] I. Vignon and C. A. Taylor, Outflow Boundary Conditions for One-Dimensional Finite Element Modeling of Blood Flow and Pressure Waves in Arteries, Wave Motion, 39(4),(2004), 361-374.
