Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 16.6.2 - A Full Eulerian finite difference approach for fluid-structure coupling problems
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 Material Information
Title: 16.6.2 - A Full Eulerian finite difference approach for fluid-structure coupling problems Fluid Structure Interactions
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Sugiyama, K.
Ii, S.
Takeuchi, S.
Takagi, S.
Matsumoto, Y.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: fluid-structure interaction
finite difference method
Eulerian formulation
hyperelastic material
 Notes
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 multi-component geometry represented by voxel-based data on a fixed Cartesian mesh. The solid volume fraction, and the left Cauchy-Green 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 lid-driven 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 twoand three-dimensional motions of biconcave neo-Hookean particles in a Poiseuille flow.
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
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7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


A Full Eulerian finite difference approach for fluid-structure coupling problems


K. Sugiyama* S. Ii* S. Takeuchi* i S. Takagil and Y Matsumoto*

Department of Mechanical Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku,
Tokyo 113-8656, JAPAN
t Department of Mechanical Engineering, Osaka University, Yamadaoka 2-1, Suita,
Osaka, 565-0871, JAPAN
Organ and Body Scale Team, CSRP, RIKEN, 2-1, Hirosawa, Wako, Saitama 351-0198, JAPAN
sugiyama@fel.t.u-tokyo.acjp
Keywords: Fluid-Structure 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 multi-component geometry
represented by voxel-based data on a fixed Cartesian mesh. The solid volume fraction, and the left Cauchy-Green
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 lid-driven 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 three-dimensional motions of biconcave neo-Hookean particles in a Poiseuille flow.


1 Introduction

Numerical simulation of Fluid-Structure 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 fluid-structure 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 multi-component geometric data converted
from CT/MRI medical images before the mesh genera-


tion. The basic idea of the medical image-based 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 interface-tracking approach
using a body-fit Lagrangian mesh, which includes Ar-
bitrary Lagrangian Eulerian (Hirt et al. (1974); Huges et
al. (1981)) and Deforming-Spatial-Domain/Space-Time
(Tezduyar et al. (1992a,b)) methods, attains a highly ac-
curate computation. An Eulerian-Lagrangian 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 Volume-Of-Fluid (VOF) function (Hirt &
Nichols (1981)) is introduced to describe the multi-
component geometry. Further, the left Cauchy-Green
deformation tensor, which quantifies a level of solid de-
formation, is introduced on each grid point and tem-
porally updated to describe a nonlinear Mooney-Rivlin
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 voxel-based geometry into the code. In
3, we explore the validity of the advocated numerical
method. In 4, we demonstrate the motions of bicon-
cave neo-Hookean 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 one-fluid formulation (Tryggvason et al.
(2007)). The mass and momentum conservation are

V-v 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 voxel-based 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 Cauchy-Green deforma-
tion tensor (Bonet & Wood (2008)). The prime on
the second-order tensor stands for the deviatoric tensor,
namely T' T jtr(T)I for a tensor T. Note that in
(4), the nonlinearity in the Mooney-Rivlin 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 right-hand-side 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 Cauchy-Green 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 second-order central difference scheme on a stag-
gered grid (Harlow & Welch (1965)), except the ad-
vection terms in (3) and (6) solved by the fifth-order











WENO method (Jiang & Shu (1996); Osher & Fedkiw
(2003)). To integrate the equations in time, we employ
the second-order Adams-Bashforth and Crank-Nicolson
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 well-validated FSI analy-
sis. We perform full Eulerian simulations of deformable
solid motion in a lid-driven 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 no-slip 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 particle-wall overlap is implemented into the present
method because the particle-wall 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 well-validated 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 lid-driven 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 imposing-releasing 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 Venant-Kirchhoff
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 no-slip 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 Venant-Kirchhoff 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
fluid-structure 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 tank-treading 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 non-zero vor-
ticity forms near the fluid-structure 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 tank-treading 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 shear-imposing 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. Venant-Kirchhoff
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 n-th 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 first-order accuracy in capturing unstressed
shape. It should be noticed that in the much simpler
system consisting of the fluid-structure layers bounded
with the oscillating top and bottom plates, the present
fluid-structure couping method involves the first-order
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 imposing-releasing 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 three-dimensional motions
of neo-Hookean biconcave particles in Poiseuille flows
(Sugiyama et al. (2010a)) are demonstrated below.


4.1 Two-dimensional motions with
large number of particles

We here address a two-dimensional 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)t-1O (d) t=-20
-12 12 -12 12


Figure 5: The snapshots of 112 biconcave particles and
16 elliptic particles of the neo-Hookean material in a
two-dimensional 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 neo-Hookean 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) t-50


Figure 6: The snapshots of 16 discoid biconcave parti-
cles of the neo-Hookean material in a three-dimensional
Poiseuille flow. The left and right panels show the parti-
cle interfaces at (a) t 0 and (b) t = 50, respectively.


4.2 Three-dimensional motions

Extending the two-dimensional system, we apply the
full Eulerian approach to the three-dimensional system
containing 16 discoid biconcave particles in a circular
pipe. The no-slip 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
multi-component 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 well-validated
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 Venant-Kirchhoff material, and the
present fluid-structure coupling scheme turned out to
be of first-order 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 three-dimensional motions of bicon-
cave neo-Hookean 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 large-scale 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 fluid-structure 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 interface-capturing 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 Next-Generation Integrated Simulation of
Living Matter, a part of the Development and Use of the
Next-Generation Supercomputer Project of the Ministry
of Education, Culture, Sports, Science and Technology
(MEXT), and by the Grant-in-Aid for Young Scientist
(B) (No.21760120) of MEXT.


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