7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
A Full Eulerian finite difference approach for fluidstructure coupling problems
K. Sugiyama* S. Ii* S. Takeuchi* i S. Takagil and Y Matsumoto*
Department of Mechanical Engineering, The University of Tokyo, 731 Hongo, Bunkyoku,
Tokyo 1138656, JAPAN
t Department of Mechanical Engineering, Osaka University, Yamadaoka 21, Suita,
Osaka, 5650871, JAPAN
Organ and Body Scale Team, CSRP, RIKEN, 21, Hirosawa, Wako, Saitama 3510198, JAPAN
sugiyama@fel.t.utokyo.acjp
Keywords: FluidStructure Interaction, Finite Difference Method, Eulerian Formulation, Hyperelastic Material
Abstract
A full Eulerian finite difference method has been developed to facilitate solution of dynamic interaction problems
between Newtonian fluid and hyperelastic material for a given initial configuration of a multicomponent geometry
represented by voxelbased data on a fixed Cartesian mesh. The solid volume fraction, and the left CauchyGreen
deformation tensor are temporally updated on the Eulerian frame, respectively, to distinguish the fluid and solid
phases, and to describe the solid deformation. Two validation tests are performed: one is a comparison with the
available simulation of the solid motion in the liddriven flow by Zhao et al. (2008, J. Comput. Phys., 227, 3114),
in which the deformed solid motion is solved in the finite element approach; the other is an examination of the
reversibility in shape of the hyperelastic material when it is released from stress. To demonstrate the feasibility in
dealing with a system involving a large number of deformed bodies, the present Eulerian method is applied to two
and threedimensional motions of biconcave neoHookean particles in a Poiseuille flow.
1 Introduction
Numerical simulation of FluidStructure Interactions
(FSI) would make it possible to predict the effect of a
medical treatment and help decide the treatment strat
egy in clinical practice. Recently, there are growing ex
pectations for its applications along with a progress in
imaging and computational technologies. It is also ex
pected to contribute to the field of life sciences, such as
in the understanding of the very essence of life and the
demonstration of pathological mechanisms. It is of great
importance to develop numerical techniques suitable for
the characteristics of body tissues, which are flexible
and complicated in shape, when attempting to rational
ize and generalize the fluidstructure coupled analyses.
As in the case of industrial products, if a given
blueprint can provide precise information on coordi
nates, the generation of meshes can be easily auto
mated in many cases and accurate computations can
be performed. However, a blueprint does not exist for
the human body, and it therefore requires the acquisi
tion of the multicomponent geometric data converted
from CT/MRI medical images before the mesh genera
tion. The basic idea of the medical imagebased sim
ulation is found in Taylor et al. (1998) and Torii et al.
(2001) dealing with blood flows in arteries. To predict
the motion of moving/deforming objects, a finite ele
ment method based on an interfacetracking approach
using a bodyfit Lagrangian mesh, which includes Ar
bitrary Lagrangian Eulerian (Hirt et al. (1974); Huges et
al. (1981)) and DeformingSpatialDomain/SpaceTime
(Tezduyar et al. (1992a,b)) methods, attains a highly ac
curate computation. An EulerianLagrangian approach
such as immersed boundary (Peskin (1972, 2002)) and
immersed finite element (Zhang et al. (2i" '1)) methods,
in which the fluid and solid phases are separately formu
lated on the fixed Eulerian and Lagrangian grids, respec
tively, is also feasible and effective for the purpose.
In the full or semi Lagrangian approach, one must
convert the medical image into voxel data and further
into the finite element mesh before starting the compu
tation, and reconstruct the mesh with time advancement.
As pointed out in Matsunaga et al. (2002), Yokoi et al.
(2005) and Noda et al. (2006), technical knowledge and
expertise are required for the mesh generation and re
construction. In general, as the shape of a target be
comes more complicated or the size of the system used
in computation becomes larger, the automation of mesh
production will become more difficult. To extend the
applicability of the FSI simulations to certain additional
classes of problems in the medical field, an alternative
simulation technique should be accessible even without
a specialist or expertise on mesh generation.
We have been developing a FSI simulation tech
nique based on a full Eulerian method (Sugiyama et al.
(2010a,b)), in which the process of mesh generation is
not required. The moving boundary kinematics and the
dynamics in interaction between Newtonian fluid and
hyperelastic material are incorporated. In consideration
that the voxel data contain the volume fractions of fluid
and solid, a VolumeOfFluid (VOF) function (Hirt &
Nichols (1981)) is introduced to describe the multi
component geometry. Further, the left CauchyGreen
deformation tensor, which quantifies a level of solid de
formation, is introduced on each grid point and tem
porally updated to describe a nonlinear MooneyRivlin
law.
The paper is organized as follows. In 2, we outline
the basic equations of the system consisting of Newto
nian fluid and hyperelastic material, and how to imple
ment the given voxelbased geometry into the code. In
3, we explore the validity of the advocated numerical
method. In 4, we demonstrate the motions of bicon
cave neoHookean particles in a Poiseuille flow. In 5,
we draw conclusion and some perspectives.
2 Basic equations
2.1 Governing equations
The fluid and solid are assumed to be incompressible
and to possess the same density. We use one set of gov
erning equations for the whole flow field, which is one
of the standard ways in multiphase flow simulations, and
referred to as a onefluid formulation (Tryggvason et al.
(2007)). The mass and momentum conservation are
Vv 0, (1)
( v .v = Vp + V r, (2)
where p denotes the density, v the velocity vector, p the
pressure deviation from the driving pressure, and r the
deviatoric Cauchy stress to be modeled in the subsequent
subsection.
2.2 Constitutive equations
The constitutive equations are described on an Eulerian
frame adjusted to the voxelbased geometry. To distin
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
guish the fluid and solid phases, the volume fraction p~
of solid obeys the transport equation
i9ts + v VOs = 0.
(Hereafter, the subscripts s and f stand for the solid and
fluid phases, respectively.)
We assume that fluid is Newtonian, and the solid
stress has both the elastic and viscous components. The
deviatoric Cauchy stress is written in a mixture form
7 =2(1 0,)pfD'
+ s {2cB' + 2c2(tr(B)B' (B B)')
+4c3(tr(B) 3)B' + 2pD',
where p and denotes the viscosity, c1, c2 and c3 the mod
uli of elasticity, I the unit tensor, D(= (Vv + VVT))
the strain rate, and B the left CauchyGreen deforma
tion tensor (Bonet & Wood (2008)). The prime on
the secondorder tensor stands for the deviatoric tensor,
namely T' T jtr(T)I for a tensor T. Note that in
(4), the nonlinearity in the MooneyRivlin law (Mooney
(1940); Rivlin (1948)) up to O(B2) is considered.
To avoid the numerical instability stemming from a
rather rough distribution of B in the fluid region (Liu &
Walkington (2001)), we introduce a modified deforma
tion tensor B to be zero in the fluid region
B= /2B for Ps >min
S 0 for s < Omin.
The second term in the righthandside of (4) immedi
ately reads
21/2c B' + 2c2(tr(B)B' (B B)')
+ 4c3(tr(B) 30/2)B' + 2 pDD'.
The modified left CauchyGreen tensor obeys the zero
Oldroyd rate, namely
tB + v VB L B B LT = 0,
where L(= Vv) denotes the velocity gradient tensor.
2.3 General descriptions of numerical
methods
The basic equations are directly solved by means of the
finite difference method using uniform square or cubic
mesh. All the spatial derivatives are approximated by
the secondorder central difference scheme on a stag
gered grid (Harlow & Welch (1965)), except the ad
vection terms in (3) and (6) solved by the fifthorder
WENO method (Jiang & Shu (1996); Osher & Fedkiw
(2003)). To integrate the equations in time, we employ
the secondorder AdamsBashforth and CrankNicolson
schemes. To complete the time marching in the mo
mentum equation (2) and simultaneously satisfying the
solenoidal condition (1) of the velocity field, we employ
the SMAC procedure (Amsden & Harlow (1970)) by
solving a Poisson equation for the pressure.
This paper concerns the applicability of the voxel
based geometry, which is directly implemented into the
FSI solver. The solid volume fraction p~o at the start
ing point is given as a set of artificial voxel data. As a
preprocessing, we numerically compute the ratio of the
occupied solid to each control volume from the initial
configuration of the system geometry, and construct the
distribution of 0so.
3 Validation tests
3.1 Comparison with independently
conducted FSI analysis
We make a comparison with wellvalidated FSI analy
sis. We perform full Eulerian simulations of deformable
solid motion in a liddriven cavity with the same setup
and conditions as in Zhao et al. (2008), who employed
a mixed Lagrangian and Eulerian approach. The initial
setup is schematically illustrated in figure l(a). The size
of the cavity is L, x L = 1 x 1. Initially, the system is
at rest. The unstressed solid shape is circular with a ra
dius of 0.2, and centered at (0.6, 0.5). At t = 0, to drive
the fluid and solid motions, the top wall starts to move at
a speed of Vw = 1 in x direction. The noslip condition
is imposed on the walls. The solid component is neo
Hookean material. The material properties are p 1,
Pf y s 10 2, cl 0.05 and 2 c3s 0.
Figure 1 visualizes the particle deformation and the
flow field for eight consecutive time instants. The
dashed curve in figure 1 represents the outline of the par
ticle obtained by Zhao et al. (2008), who computed the
solid deformation on the Lagrangian mesh. The solid
lines represent the instantaneous particle shapes, corre
sponding to the isoline at ps = 1/2, obtained by the
present full Eulerian simulation. The dotted material
points are tracked just to transfer images of the parti
cle deformation, but we did not use these material points
for computing solid stress and strain. The particle moves
and deforms driven by the fluid flow, and exhibits highly
deformed shape when the particle approaches the wall.
It should be noticed that no special artifact for avoiding
a particlewall overlap is implemented into the present
method because the particlewall hydrodynamic repul
sion is likely to be brought due to the geometry change
via the particle deformation. The solid shapes obtained
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
by the present Eulerian simulation are in excellent agree
ment with the wellvalidated result by Zhao et al. (2008).
10 .
06
04
06
04
( 3a) 4
I 2,34
g) (h)
00 t7,03 t 203 ....
00 02 04 06 08 1000 02 04 06 08 10
x x
Figure 1: The solid deformation in the liddriven flow.
The dashed outline represents the result of Zhao et al.
(2008), in which the Lagrangian tracking approach was
employed to describe the solid deformation. The solid
outline, the dotted material points and the streamlines
correspond to the present simulation results based on the
full Eulerian approach with a mesh 1024 x 1024.
3.2 Reversibility in shape of
hyperelastic material
The hyperelastic material generally exhibits reversibil
ity in shape when it is released from stress. In the to
tal Lagrangian method using the finite element mesh,
since the tracked material points link to both the refer
ence and current configurations, the reversibility can be
captured with little difficulty. By contract, the Eulerian
fixed grid point retains no information on the reference
configuration. Therefore, one may raise a shortcoming
that the Eulerian approach is likely to lose the informa
tion about the original shape once the material is stressed
to deform. We demonstrate a reversibility examination
(Sugiyama et al. (2010b)).
1.5
1.5
Figure 2: Snapshots of the velocity (arrows) and
vorticity (color) fields involving a circular particle in
the imposingreleasing shear flow between two parallel
plates. The number of grid points is 1024 x 256. The up
per and lower plates move at speed of 1 and 1 within
a period of t E [0, 4], and then stop after t = 4. The
solid obeys an incompressible Saint VenantKirchhoff
law with p = 0, Aam, 6 and / a 4.
We deal with a shear flow between two plane plates
involving a hyperelastic particle. The distance between
the plates is Ly = 2. The computational extent in x di
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
reaction is set to L, = 8. The upper and lower plates
are located at y = 1 and y = 1, respectively. Ini
tially, the system is at rest. An unstressed solid particle
is initially circular with a radius of 0.75, and centered at
the middle position (0, 0) between the plates as depicted
in figure 2(a). The noslip condition is imposed on the
plates, whereas the periodic condition is applied in x di
rection. The material properties are p 1, f = 1,
p 0, cl 4, 2 2 and c3 1.75. Note that
the solid phase for these moduli follows an incompress
ible Saint VenantKirchhoff law (Giiltop (2003)) with
Lam6's constants AX = 6 and L = 4. Within a pe
riod of 0 < t < 4, the upper and lower plates move at
speeds of VP"r = 1 and V wer 1 in x direction,
respectively, to drive the fluid and solid motions. After
t =4, the moving plates stop (i.e. Vper' V wer
0) to release the particle from the shearing force.
Figure 2 visualizes the particle deformation and the
flow field for six consecutive time instants. As the shear
flow is induced by the moving plates, the shearing force
is imposed on the solid particle, and causes the particle
elongation toward the extensional direction. In the tran
sient state during the development of the deformation,
it is observed in figure 2(b)(c) that the transverse elas
tic waves travel inside the solid, and are reflected by the
fluidstructure interface. The wave amplitude is damped
through the repetitious reflections with time as shown in
figure 2(d)(e). As shown in figure 2(e), the vorticity in
side the particle at t = 4 is negative, indicating that the
particle experiences a tanktreading like motion. After
the shearing force is released by setting the wall veloc
ities to be zero at t = 4, the fluid flow rapidly decays
and the deformed particle gradually recovers the circu
lar shape. At t = 6 as shown in figure 2(f), the vorticity
in the bulk fluid is almost zero, while the nonzero vor
ticity forms near the fluidstructure interface, indicating
the particle shape is under recovery.
To directly demonstrate whether the reversibility can
be captured, the distributions of the tracers for four con
secutive time instants are shown in figure 3. As depicted
in figure 3(a), the tracers are initially seeded on the con
centric circles inside the solid to see the local displace
ments inside the solid. Figure 3(b) shows the tracer dis
tribution at the most deformed instant t = 4 when the
particle is under the tanktreading like motion. After the
wall velocities is set to be zero at t = 4, the tracer par
ticles gradually move back toward the initial concentric
circles with time. It should be noted that because the
degree of freedom corresponding to the rigid rotation is
allowed, the tracer distributions in figure 3(c)(d) turn in
the clockwise directions about 80 degrees with respect
to the initial distribution in figure 3(a). At the instant
t = 8, when the same period as the shearimposing stage
(four unit time) has elapsed after the walls stop, the dis
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
crepancy between the tracer location and the concentric
circle is clearly shown in figure 3(c), indicating that the
recovery in the particle shape is still underway. After a
sufficiently long time (t = 200), the tracers are found
to be back in the concentric circles as shown in figure
3(d). The present Eulerian approach can capture the re
versibility in shape under certain right circumstances.
St. VenantKirchhoff
o (a)' (b)' t=4
1 0
05
10 05 00 05 10 10 05 00
x x
05 10
Figure 3: Material point distribution in the imposing
releasing shear flow involving a circular particle be
tween two parallel plates with the 1024 x 256 mesh. The
conditions are same as those of figure 2. The colored
filled circles are distributed to demonstrate the rotation.
To reveal the grid convergence behavior in the parti
cle deformation, the deformation modes I R, ( ((R, +
R ,)1/2) at t = 200, indicating the level of the spuri
ous residual deformation, are shown in figure 4. Here,
R,, and R,, denote the nth order deformation modes,
which satisfy
Ro = Ro + (Rc,, cos n + R,, sin nO), (7)
n=l
where R(O) is the distance from the solid centroid to
the interface. The deformation modes are determined
via the orthogonality in sine and cosine functions. The
residual amplitudes are nearly proportional to N 1, in
dicating the firstorder accuracy in capturing unstressed
shape. It should be noticed that in the much simpler
system consisting of the fluidstructure layers bounded
with the oscillating top and bottom plates, the present
fluidstructure couping method involves the firstorder
accuracy of the velocity with respect to the grid size
(Sugiyama et al. (2010b)), which is reflected on the grid
convergence of the reversibility in shape.
o 102
0'
N
10 
64 128 256 512 1024
Figure 4: Residual modal amplitudes IRnI of the par
ticle deformation at t = 200 versus the number N, of
grid points in the imposingreleasing shear flow. The
conditions are same as those of figure 2.
4 Biconcave particles in a Poiseuille
flow
One of the strong advantage of the present simulation
method is characterized by no process of generation and
reconstruction of the unstructured meshes. If the La
grangian method is used for the analysis of a large sys
tem involving complicated shape, in which the boundary
between fluid and solid changes with time, it requires not
only a extensive computational cost but also a great ef
fort for the mesh generation and reconstruction to ensure
accuracy and stability. On the other hand, the use of the
Eulerian method makes it easily possible to perform a
FSI simulation even on a target with a complicated ge
ometry, if the distribution of the solid volume fraction
at the starting point is provided. As examples of the
Eulerian analyses, two and threedimensional motions
of neoHookean biconcave particles in Poiseuille flows
(Sugiyama et al. (2010a)) are demonstrated below.
4.1 Twodimensional motions with
large number of particles
We here address a twodimensional Poiseuille flow con
taining 112 biconcave particles and 16 elliptic particles.
As shown in figure 5(a) for the solid volume fraction
Ps field at t = 0, the initial particle position and ori
entation are randomly given. The system is bounded
by the bottom (y 0) and top (y L,) plates, and
periodic in x direction The computational domain is
L, x Ly = 28.8 x 28.8 with the number of grids
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
0 I 12 12
'SI..g^ go W
(c)t1O (d) t=20
12 12 12 12
Figure 5: The snapshots of 112 biconcave particles and
16 elliptic particles of the neoHookean material in a
twodimensional Poiseuille flow. The upper left panel
(a) shows the solid volume fraction field at t = 0. The
others show the velocity (arrows) and vorticity (back
ground color) fields for various instants t 5 (b), t 10
(c), and t 20 (d). The colors inside the particles are
given for convenience in displaying the temporal evolu
tion of each particle position and orientation.
of 1024 x 1024. The material properties are p 1,
pf = ps 1, cl 50, and c2 cs 0. To pump the
fluid and solid, the uniform pressure gradient AP/L,
is applied to the system for the time t > 0. The system
is supposed in stationary equilibrium before the driving
pressure is imposed. The temporal evolution of the par
ticle position and orientation is shown in figure 5(b), (c),
and (d). The biconcave particles tend to be more mixed
as the time goes on, whereas the small elliptic particles
remain in the vicinity of the walls with some fluctuating
motions.
It should be noticed that the Eulerian analyses of dy
namics of a pair of neoHookean biconcave particles
were demonstrated in Sugiyama et al. (2010a). It was
shown that similar to the numerical study on the red
blood cell motion in a circular pipe in Gong et al. (2009),
in which Skalak's constitutive laws of the membrane
were considered, the deformation, the relative position
and orientation of the particles are strongly dependent
on the initial configuration.
(a) t=u
(b) t50
Figure 6: The snapshots of 16 discoid biconcave parti
cles of the neoHookean material in a threedimensional
Poiseuille flow. The left and right panels show the parti
cle interfaces at (a) t 0 and (b) t = 50, respectively.
4.2 Threedimensional motions
Extending the twodimensional system, we apply the
full Eulerian approach to the threedimensional system
containing 16 discoid biconcave particles in a circular
pipe. The noslip boundary condition is imposed on the
pipe wall surface. The system is periodic in x direction.
The pipe has a radius of 7.2 and a length of 7.2. The
computational domain is Lx x Ly x L, 7.2 x 7.2 x 7.2
with the number of grids of 128 x 128 x 128. Same as
in 4.1, the uniform pressure gradient AP/L = 1 is
applied for t > 0. The material properties are p 1,
p 1, f p=s 1, cl 25, and c2 cs 0. Fig
ure 6(a) shows the initial particle position. Figure 6(b)
shows the deformed particles at t = 50 in the flow.
It should be noted that within a framework of the finite
difference method using the rectangular grid, the algo
rithm in Yokoi et al. (2005), where the level set method
and ghost fluid method are used to approximate the com
plex geometry, would be helpful in applying the present
approach to more practical system. Simulation of a flow
field bounded by deformable walls is the ongoing sub
ject (Nagano et al. (2010)).
5 Conclusion and perspectives
A full Eulerian simulation method for solving Fluid
Structure Interaction (FSI) problems has been devel
oped. It is suitable for the voxel data, which are con
verted from the medical CT/MRI image and describe
multicomponent geometry. It releases the coupling
simulation from the mesh generation procedure. The
present study demonstrated the validity of the simulation
method, which consistently captured the wellvalidated
simulation data for the deformed solid motion in the lid
driven cavity (Zhao et al. (2008)). Furthermore, the level
of the reversibility in shape was estimated for the in
compressible Saint VenantKirchhoff material, and the
present fluidstructure coupling scheme turned out to
be of firstorder accuracy for the reversal with respect
to the grid resolution. To demonstrate the feasibility
in dealing with a system involving a large number of
deformed bodies, the present Eulerian method was ap
plied to two and threedimensional motions of bicon
cave neoHookean particles in the Poiseuille flow.
If the distribution of the solid volume fraction for the
initial geometry is given, the present approach facilitates
to solve the FSI problem. It provides a significant boost
of the geometrical flexibility, and thus encourages one to
tackle a target with a complicated geometry. A charac
teristic of the Eulerian analysis is to make it easily pos
sible to perform a coupled analysis using general com
putational algorithms for incompressible fluid. In con
sideration of a fact that a largescale computation is es
sential when a realistic system is analyzed, the expertise
for parallelization that has been cultivated in the field of
the computational fluid dynamics can be utilized, which
would be a large advantage in the realization of mas
sively parallel computation. The expansion of the scale
and models of the computations in figure 5 and in figure
6 allows us to analyze a series of phenomena starting
from the adsorption, under a condition in which many
red blood cells are present, and would gain insight into
the dynamics in thrombosis.
To improve the accuracy in the fluidstructure cou
pling, the ideas of the immersed interface treatment
(LeVeque & Li (1994); Li & Ito (2006)) and the local
ized strain formulation (Okada & Atluri (1995)) would
be effective. It is a challenging task to overcome the
multiphysics difficulty particularly associated with the
difference in constitutive laws for fluid and solid. The
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
sharp interfacecapturing and the robust time advance
ment are the ongoing subject (Ii et al. (2009)).
Acknowledgments
This research was supported by Research and Develop
ment of the NextGeneration Integrated Simulation of
Living Matter, a part of the Development and Use of the
NextGeneration Supercomputer Project of the Ministry
of Education, Culture, Sports, Science and Technology
(MEXT), and by the GrantinAid for Young Scientist
(B) (No.21760120) of MEXT.
References
Taylor, C.A., Hughes, T.J.R. and Zarins, C.K., Finite ele
ment modeling of blood flow in arteries, Comput. Meth
ods. Appl. Mech. Eng., Vol. 158, pp. 155196, 1998.
Torii, R., Oshima, M., Kobayashi, T and Takagi, K., Nu
merical simulation system for blood flow in the cerebral
artery using CT imaging data, JSME Int. J. Ser. C, Vol.
44,pp. 982989,2001.
Hirt, C.W., Amsden, A.A. and Cook, J.L., An ar
bitrary LagrangianEulerian computing method for all
flow speeds, J. Comput. Phys., Vol. 14, pp. 227253,
1974.
Hughes, T.J.R., Liu, W.K. and Zimmermann, T.K.,
LagrangianEulerian finite element formulation for in
compressible viscous flows, Comput. Methods Appl.
Mech. Eng., Vol. 29, pp. 329349, 1981.
Tezduyar, T.E., Behr, M. and Liou, J., A new strat
egy for finite element computations involving mov
ing boundaries and interfaces the deformingspatial
domain/spacetime procedure: I. The concept and the
preliminary numerical tests, Comput. Methods Appl.
Mech. Eng., Vol. 94, pp. 339351, 1992.
Tezduyar, T.E., Behr, M., Mittal, S. and Liou, J., A new
strategy for finite element computations involving mov
ing boundaries and interfaces the deformingspatial
domain/spacetime procedure: II. Computation of free
surface flows, twoliquid flows, and flows with drifting
cylinders, Comput. Methods Appl. Mech. Eng., Vol. 94,
pp.353371, 1992.
Peskin, C.S., Flow patterns around heart valves: a nu
merical method, J. Comput. Phys., Vol. 10, pp. 252271,
1972.
Peskin, C.S., The immersed boundary method, Acta Nu
merica., Vol. 11, pp. 479517, 2002.
Zhang, L., Gerstenbetger, A., Wang, X. and Liu, W.K.,
Immersed finite element method, Comput. Methods
Appl. Mech. Eng., Vol. 193, pp. 20512067, 2004.
Matsunaga, N., Liu, H. and Himeno, R., An immersed
based computational fluid dynamics method for haemo
dynamic simulation, JSME Int. J. Ser. C, Vol. 45, pp.
989996,2002.
Yokoi, K., Xiao, F., Lui, H. and Fukasaku, K., Three
dimensional numerical simulation of flows with com
plex geometries in a regular Cartesian grid and its ap
plication to blood flow in cerebral artery with multiple
aneurysms, J. Comput. Phys., Vol. 202, pp. 119, 2005.
Noda, S., Fukasaku, K. and Himeno, R., Blood flow sim
ulator using medical images without mesh generation,
IFMBE Proc. of World Cong. on Medical Physics and
Biomedical Engineering 2006, Seoul, Korea, pp. 3640,
2006.
Hirt, C.W. and Nichols, B.D., Volume of fluid (VOF)
method for the dynamics of free boundaries, J. Comput.
Phys., Vol. 39, pp. 201225, 1981.
Sugiyama, K., Ii, S., Takeuchi, S., Takagi, S. and Mat
sumoto, Y, Full Eulerian simulations of biconcave neo
Hookean particles in a Poiseuille flow, Comput. Mech.,
accepted, 2010.
Sugiyama, K., Ii, S., Takeuchi, S., Takagi, S. and Mat
sumoto, Y, A full Eulerian finite difference approach
for solving fluidstructure coupling problems, submit
ted, 2010.
Tryggvason, G., Sussman, M. and Hussaini, M.Y, Im
mersed boundary methods for fluid interfaces, in Pros
peretti, A. and Tryggvason, G. (Eds.), 'Computational
Methods for Multiphase Flow,' Chap. 3, Cambridge
University Press, Cambridge, 2007.
Mooney, M., A theory of large elastic deformation, J.
Appl. Phys., Vol. 11, pp. 582592, 1940.
Rivlin, R.S., Large elastic deformations of isotropic ma
terials IV, Further development of general theory, Phil.
Trans. R. Soc. A, Vol. 241, pp. 379397, 1948.
Bonet, J. and Wood, R.D., 'Nonlinear Continuum Me
chanics for Finite Element Analysis,' Chap. 4, second
edition, Cambridge University Press, Cambridge, 2008.
Liu, C. and Walkington, N.J., An Eulerian description of
fluids containing viscoelastic particles, Arch. Rational
Mech. Anal., Vol. 159, pp. 229252, 2001.
Harlow, F.H. and Welch, J.E., Numerical calculation
of timedependent viscous incompressible flow of fluid
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
with free surface, Phys. Fluids, Vol. 8, pp. 21822189,
1965.
Jiang, G.S. and Shu, C.W., Efficient implementation of
weighted ENO scheme, J. Comput. Phys. Vol. 126, pp.
202228, 1996.
Osher, S. and Fedkiw, R., 'Level Set Methods and Dy
namic Implicit Surfaces,' Chap. 3, Springer, New York,
2003.
Amsden, A.A. and Harlow, F.H. A simplified mac tech
nique for incompressible fluid flow calculation, J. Com
put. Phys., Vol. 6, pp. 322325, 1970.
Zhao, H., Freund, J.B. and Moser, R.D., A fixedmesh
method for incompressible flowstructure systems with
finite solid deformation, J. Comput. Phys., Vol. 227, pp.
31143140,2008.
Giiltop, T, On the propagation of acceleration waves in
incompressible hyperelastic solids, J. Sound and Vibra
tion, Vol. 264, pp. 377389, 2003.
Gong, X., Sugiyama, K., Takagi, S. and Matsumoto, S.,
The deformation behavior of multiple red blood cells in
a capillary vessel, J. Biomech. Eng., Vol. 131, 074504,
2009.
Nagano, N., Sugiyama, K., Takeuchi, S., Ii, S., Takagi,
S. and Matsumoto, Y, Full Eulerian fluidstructure cou
pling analysis for hyperelastic wavy channel flow, Proc.
of WCCM/APCOM 2010, Sydney, Austratia, to appear,
2010.
LeVeque, R.J. and Li, Z., The immersed interface
method for elliptic equations with discontinuous coef
ficients and singular sources, SIAM J. Numerical Anal
ysis, Vol. 31, pp. 10191044, 1994.
Li, Z. and Ito, K., 'The Immersed Interface Method,'
SIAM, Philadelphia, 2006.
Okada, H. and Atluri, S.N., "Embedded localized strain
zone constitutive model in finite strain and finite rota
tion," Proc. of Int. Conf. on Computational Engineering
Science, Vol. 2, pp. 21545159, 1995.
Ii, S., Sugiyama, K., Takeuchi, S., Takagi, S. and Mat
sumoto, Y, Development of accurate numerical model
for the fluidstructure interaction problem based on Eu
lerian framework, Proc. of 10th US National Cong. on
Comput. Mech., Ohio, USA, No. 158294, 2009.
