#  An Overview of Computational Fluid Structure Interaction: Methods and
Applications

Sumant R. Morab
Doctoral Student
Department of Mechanical Engineering
Indian Institute of Technology, Bombay
Email<EMAIL_ADDRESS>
Atul Sharma
Professor
Department of Mechanical Engineering
Indian Institute of Technology, Bombay
Email<EMAIL_ADDRESS>Address all correspondence related to this
author.

Over the past few decades, there has been a rapid improvement in computational
power as well as techniques to simulate the real world phenomenon which has
enabled us to understand the physics and develop new systems which outperform
the existing ones. In the domain of multi-physics problems, fluid and
structure interactions have been studied and various numerical methods are
introduced to solve them. An extensive review of most commonly employed
numerical techniques to solve fluid structure interaction(FSI) problems has
been done in this article. It also becomes utmost important to understand the
applicability of each method and hence a critical reasoning for usage of
different methods has been provided. Application domains are presented which
range from energy harvesting processes to simulation of human vocal cords and
movement of bolus(food) inside esophagus. Suitable numerical methods applied
for each application have also been discussed. Numerical methods developed so
far are classified in particular groups based on the discretization, the
manner in which coupling is performed and on the basis of meshing(division of
entire domain into small blocks inside which variation of a variable is
approximated). Challenges and instabilities posed by presently available
numerical methods are discussed and potential applications where there is a
possibility of errors due to these methods have been listed. A brief set of
application areas which have not been explored through the lens of fluid
structure interaction also have been discussed at the end of this article.
Keywords: Multi-physics problems, Fluid Structure Interaction(FSI), Bolus
transport, Discretization.

Nomenclature
e Shear strain in fluid
E Youngs Modulus of solid
f External forces
G Bulk Modulus of solid
n Normal direction to surface
p Hydrodynamic pressure
t time
v Velocity
x co-ordinates

superscripts
f fluid domain
s solid domain

subscripts
i,j directions in tensor form

Greek Symbols
$\delta~{}~{}$Kronecker delta
$\epsilon~{}~{}$lateral strain inn solid
$\lambda~{}~{}$Lame Constant
$\mu~{}~{}$Dynammic viscosity
$\nu~{}~{}$Poissons ratio
$\Omega~{}~{}$closed surface/volume
$\rho~{}~{}$denisty
$\tau~{}~{}$Shear stress
$\Gamma~{}~{}$Interface
$\sigma~{}~{}$Hydrodynamic stress

##  1 Introduction

Major research in the domain of Energy Harvesting, Transportation and Medical
science involves multi-physics and interactions between a fluid and a solid.
Modern research in disease diagnosis and treatment has helped us to take a
strong forward leap in healthcare and this medical research owes a part to
fluid structure interaction. Interaction between a fluid and a solid occur on
day to day experiences like movement of leaves due to wind, movement of flag
in air, production of sound etc. and in most complex engineering applications
like flutter of airfoil and energy harvesting.

Fluid Structure Interaction (FSI) is a class of multi-physics problem where
dependence between fluid and structural part plays an important role. The
deformation and displacement of a structure is affected by the forces applied
by the fluid flowing past through it and change in geometry or position of
solid inturn has an effect on fluid flow pattern. Thus, there is a two way
coupling that exists between fluid and solid and capturing this through
computational techniques can have humongous applications in engineering design
and medical field. Various examples of FSI found in our nature include
locomotion of aquatic animals, chirping and flying of birds, dispersal of
pollen and seeds by wind, circulation of blood through the entire body,
beating of heart etc. which has inspired humans to mimic them and build
efficient machines. Engineering applications of FSI include development of
energy harvesting devices, designing underwater vehicles to decrease drag and
improve performance, design of turbine blades for various applications, design
of high rise buildings to avoid vibrations etc. moreover in medical field, we
find applications in diagnosis of various cardiovascular diseases (CVD) like
atherosclerosis, arteriosclerosis, aneurysms, atherogenesis and prediction of
future growth prospects. In recent years, FSI has been used to simulate sound
production from vocal cords and determine the physics behind bolus transport
in esophagus. Also, there has been growing use of FSI for surgical planning
and treatment in various sectors. The above mentioned applications have been
discussed in detail in later sections of this report. Thus, it becomes very
important to develop predictive FSI techniques that provides solutions to the
major concerns faced by modern era.

For independent problems on fluid or solid, there have been some analytical
results [1] which help us to understand the physics. In case of coupling
between fluid and solid, there have been rarely any analytical solutions
except by Womersley [2]. Another possibility of experimentation has been
carried out for some applications like vortex induced vibration [3] but when
it comes to medical applications, invivo experimentation on humans seems
highly impossible. Thus, in such cases medical images obtained from computed
tomography (CT) or magnetic resonance angiography(MRA) have been used to
construct three dimensional geometry and perform computational experiments.
From the above argument we can infer that development of computational
techniques for FSI problems becomes very crucial and taking the advantage of
huge contribution by scientific community towards the development of various
numerical schemes which can efficiently handle the complexities involved in
geometry, material and fluid properties becomes crucial. The major challenge
for FSI problems is the coupling between structure and fluid beside ensuring
that boundary conditions are implemented accurately at the fluid and solid
interface. There have been different set of models developed to tackle this
issue and each one of them along with their applicability for certain kinds of
problems has been briefed in the next section.

##  2 Mathematical Formulation

Consider the whole domain of interest to be split into Solid ($\Omega$s) and
fluid ($\Omega$f) with interface between them($\Gamma$) as shown in Fig. 1.
The boundaries of fluid not in contact with fluid are assumed to be rigid and
thus a standard boundary condition may be applied on them. The representation
of the symbols presented below are provided in the nomenclature.

Figure 1: Computational Domain

###  2.1 Fluid Dynamics

It is assumed that the fluid is in-compressible and possesses a Newtonian
rheology. The mass conservation associated with each cell leads to Continuity
given by

$\frac{\partial\rho v_{i}^{f}}{\partial{{x}_{i}}}=0$ (1)

Momentum conservation at Control volume leads to

$\rho\frac{Dv_{i}^{f}}{Dt}=\frac{\partial\sigma_{ij}^{f}}{\partial{{x}_{j}}}-f_{i}^{f}$
(2)

Where

$\begin{multlined}\frac{D{{v}_{i}}}{Dt}=\frac{\partial{{v}_{i}}}{\partial
t}+{{c}_{j}}\frac{\partial{{v}_{i}}}{\partial{{x}_{j}}}\\\
{{\sigma}_{ij}}=-p{{\delta}_{ij}}+{{\tau}_{ij}}\\\
{{\tau}_{ij}}=2\mu({{e}_{ij}}-{{\delta}_{ij}}{{e}_{kk}}/3)\\\
{{e}_{ij}}=\frac{\partial{{v}_{j}}}{\partial{{x}_{i}}}+\frac{\partial{{v}_{i}}}{\partial{{x}_{j}}}\\\
\end{multlined}\frac{D{{v}_{i}}}{Dt}=\frac{\partial{{v}_{i}}}{\partial
t}+{{c}_{j}}\frac{\partial{{v}_{i}}}{\partial{{x}_{j}}}\\\
{{\sigma}_{ij}}=-p{{\delta}_{ij}}+{{\tau}_{ij}}\\\
{{\tau}_{ij}}=2\mu({{e}_{ij}}-{{\delta}_{ij}}{{e}_{kk}}/3)\\\
{{e}_{ij}}=\frac{\partial{{v}_{j}}}{\partial{{x}_{i}}}+\frac{\partial{{v}_{i}}}{\partial{{x}_{j}}}\\\
$

###  2.2 Structure Dynamics

Consider the solid strcuture ($\Omega$s) to be elastic, homogenous and
isotropic. According to principle of virtual work , control volumes of solid
have to satisfy

${{\rho}^{s}}\frac{Dv_{i}^{s}}{Dt}=\frac{\partial\sigma_{ij}^{s}}{\partial{{x}_{j}}}-f_{i}^{s}$
(3)

Where

$\begin{gathered}\sigma_{ij}^{s}=\lambda{{\delta}_{ij}}{{\varepsilon}_{ll}}+2G{{\varepsilon}_{ll}}\\\
{{\varepsilon}_{ij}}=\frac{1}{2}\left(\frac{\partial{{u}_{i}}}{\partial{{x}_{j}}}+\frac{\partial{{u}_{j}}}{\partial{{x}_{i}}}\right)\\\
G=\frac{E}{2(1+\nu)}\\\ \lambda=\frac{E\nu}{2(1+\nu)(1-2\nu)}\end{gathered}$

###  2.3 Coupling equations for Interface

There are mainly two conditions which have to be satisfied at the interface of
any solid and fluid.

  1. 1.

Velocity of fluid boundary at interface must be same as the velocity of solid
boundary(velocity due to its displacement). This is generally termed as
kinematic boundary condition or no slip condition.

  2. 2.

The stress exerted by fluid at the interface should be same as the external
force which acts on solid. this is termed as Dynamic Boundary Condition or
stress continuity constraint.

No Slip and stress continuity is applied at the interface($\Gamma$) as shown.

$\begin{multlined}v_{i}^{s}=v_{i}^{f}\\\
\sigma_{ij}^{s}{{n}_{i}}=\sigma_{ij}^{f}{{n}_{i}}\\\
\end{multlined}v_{i}^{s}=v_{i}^{f}\\\
\sigma_{ij}^{s}{{n}_{i}}=\sigma_{ij}^{f}{{n}_{i}}\\\ $

##  3 Computational FSI Methods

There have been various computational methods to solve equations governing the
fluid dynamics [4] (usually the Navier-Stokes equation) and structure dynamics
separately. They have been broadly called as Computational Fluid Dynamics
(CFD) and Computational Structure Dynamics (CSD). Coupling between them
started to pick up eventually when people started to think about the reason
behind unfortunate events like the collapse of the Tacoma Narrows bridge,
Vibration of deep water risers in oil rigs etc. and eventually computational
techniques started getting developed. Major classification of numerical
methods is based on the way coupling between fluid and solid is introduced,
the way grid is generated (i.e, whether it changes with fluid/solid domain)
and the way in which discretization has been performed to obtain the final
algebraic equations to be solved. These classifications are discussed in
detail below.

###  3.1 Monolithic and Partitioned Approach

This classification is made on the basis of how the mathematical framework has
been designed to solve for fluid and solid domains. In case of Monolithic
approach a unified mathematical governing equations are defined for both fluid
and solid and the whole domain is solved as one entity. The main advantage
behind using this method is that the interfacial conditions are inherently
present in the formulated mathematical framework and a single discretization
scheme can be applied. It has also been observed [5] that a better accuracy is
observed in case of well defined multi-physics problems. One of the main
issues with this approach is that the code development becomes heavily
problem-specific and a sense of generality is lost. Also, some kind of
expertise is required in the domain specific area to develop such an
algorithm. Fig. 2 shows the basic flow in these two approaches.

In case of partitioned approach, solid and fluid governing equations are
solved separately in their respective domains and coupling between them is
introduced by applying suitable interface conditions. Discretisation scheme
and meshing can be completely different for both the domains but a sense of
match must exist at interface location. If meshes are not in a matching
condition at the interface, a ghost patch [6] is usually introduced wherein
communication is transmitted from solid to fluid or vice versa through this
ghost patch. This ghost patch is developed in such a way that it matches solid
and fluid domains on their respective sides. This method was developed in the
beginning stages of computational FSI development and is still being applied
and developed since it has an inherent advantage of using robust fluid and
solid solution algorithms which have been developed over the past few decades.
These methods are prone to errors especially at the interfaces due to mismatch
of discretisation and mesh. Another issue with this method is that the
computational cost is high compared to that of Monolithic approach since
interface conditions needs to be applied every time and remeshing of the whole
domain has to be done either for every time step or when there is a large
displacement which could possibly disturb the skewness and other properties of
the mesh.

Figure 2: Monolithic and Partitioned Approach

###  3.2 Conforming and Non-Conforming Mesh

A classification based on the type of mesh generated to solve fluid solid
interactions leads us to conforming and non-conforming type mesh (Fig. 3). In
case of conforming type system, the interface between fluid and solid is
considered to be a physical boundary. In most of the cases, fluid particles
are tracked through Eulerian approach whereas solid movement is tracked via
Lagrangian formulation. Boundary conditions are applied at the interface for
both fluid and solid governing equations and thus the communication between
both the domains happens through these boundary conditions. Fluid and solid
domains can utilise various mesh types but needs to be updated at
every/regular time-steps. For example, if a body contracts the mesh needs to
be updated to either remove the solid cells outside contracted body or to
contract the existing mesh. This task becomes cumbersome when there are a
large number of elements in mesh and deformation is high. Tracking of the
boundary is also essential while going to the next time step. Methods under
this category include Arbitrary Lagrangian Eulerian (ALE)([7], [8]), Deformed
Spatial Domain/Stabilised space time (DSD/SST) [9] etc. which will discussed
in the upcoming topics.

In case of non-conforming mesh methods, the interface conditions are not
treated as physical boundary conditions for fluid and solid domain. They are
treated in governing equations itself so as to give the same effect of
boundary conditions (no slip and specified traction). The major advantage with
these methods is that though fluid and solid are discretized quite
differently, a single mesh is enough to solve the problem of FSI. Thus, there
won’t be any issue of re meshing in this type which would be very effective in
reducing computational cost. Efficient mathematical formulation of governing
equation for accurate interface conditions needs to be considered. Various
methods have been developed under this category which include Immersed
Boundary Method(IBM) [10], Immersed Interface Method(IIM) [11], Coupled-
Momentum method [12], Direct Forcing method etc. which will be detailed in the
next topic.

Figure 3: Non-Conforming (top) and Conforming (Bottom) mesh

###  3.3 Based on Numerical discretisation and Interface coupling schemes

After the development of some initial Fluid-Structure-Interaction techniques
based on a naive approaches, various discretisation techniques started getting
developed for specific problems to improve accuracy. Some of these methods
which are applicable for general FSI problems are discussed below.

####  3.3.1 Arbitrary Lagrangian Eulerian (ALE) methods

In the initial stages, this method was developed to obtain solutions for
interfacial and free surface flows. Hirt et al. [7] brought ALE method which
was applicable to all flow regimes. It used a Finite Difference discretisation
in which nodal points could move with fluid or be held at fixed position or
could be moved in a specific direction as specified by user. This led to the
generation of word ‘Arbitrary Lagrangian Eulerian’. Based on the value of mesh
velocity, the discretization became either fully Eulerian or fully Lagrangian
or hybrid. The computations in a particular time step have been performed in
three separate phases. These phases include explicit velocity computation,
implicit calculation of pressure and updating the velocity and rezoning using
mesh movement. The method was tested on several cases which included shock
tube flow with moderate and very large sound speed and flow around a
rectangular block. Stability analysis was also performed with this method.

Kennedy et al. [8] developed a computational method to analyse the effect of
pressurised bubble on a structure used in reactor vessels. They used Finite
Element discretization on the fluid domain and for energy equation. After
obtaining fluid solution, displacements of solid was calculated through
traction force exerted by fluid. After this step, mesh velocity and co-
ordinates were updated to perform remeshing. The numerical method was tested
on a fluid inside a rigid and flexible walls. A brief algorithm followed by
them is as follows:

  1. 1.

Calculate acceleration of fluid as

$\frac{\partial\mathbf{v}}{\partial
t}={{\mathbf{M}}^{-1}}\left({{\mathbf{f}}^{ext}}-{{\mathbf{f}}^{\operatorname{int}}}\right)$
(4)

M is found through Navier Stokes equation (2.2).

  2. 2.

Obtain fluid velocity v

${{\mathbf{v}}^{n+1/2}}={{\mathbf{v}}^{n-1/2}}+\Delta{{t}^{n}}\frac{\partial{{\mathbf{v}}^{n}}}{\partial
t}$ (5)

  3. 3.

Set Grid Velocity v G as some fraction of fluid particle velocity at interface
obtained through structure solution.

  4. 4.

Update mesh point co-ordinates as

$\begin{multlined}\mathbf{x}=\mathbf{x}+\Delta{{t}^{n+1/2}}{{\mathbf{v}}^{G,n+1}}\\\
\Delta{{t}^{n}}=\frac{1}{2}\left(\Delta{{t}^{n+1/2}}+\Delta{{t}^{n-1/2}}\right)\\\
\end{multlined}\mathbf{x}=\mathbf{x}+\Delta{{t}^{n+1/2}}{{\mathbf{v}}^{G,n+1}}\\\
\Delta{{t}^{n}}=\frac{1}{2}\left(\Delta{{t}^{n+1/2}}+\Delta{{t}^{n-1/2}}\right)\\\
$

  5. 5.

March to the next time step.

Most commonly used discretization scheme for solids has been finite elements
because of its robustness to handle material and geometric nonlinearities.
There have been some attempts [13] to discretize solid through Finite Volumes
(FVM) and compare the accuracy with that of Finite Element Method(FEM). The
main motivation behind using such a discretization is to allow efficient and
smooth coupling at the interface of solid and fluid (which is mostly
discretized through Finite Volumes in an Eulerian frame). Tsui et al. [14]
adopted Finite Volume discretization and partitioned approach for both fluid
and solid domains. Their method had a contrasting uniqueness where fluid
variables were placed at grid centroids whereas solid variables were at the
nodes of cell. They successfully validated results for flow past a cylindrical
bluff body with plate fixed on the rear side.

Hubner et al.[5] developed a monolithic approach for FSI problems where-in
simultaneous solution was achieved for fluid and solid. Remeshing was
performed before marching to the next time step. Discretization was performed
through space time finite elements [15]. The simultaneous solution was
possible by formulatining fluid and solid discretized equation into a single
equation. The solid was modelled with a complete Lagrangian fashion where
displacement and Piola-Kirchhoff stresses were the variables to be solved for.
Fluid domain was modelled in an Arbitrary-Lagrangian Eulerian representation
where boundaries of fluid domain intersecting with solid were in Lagrangian
mode. The method was validated by solving flow over a building roof, and
through vortex excited elastic plate.

Though these methods are found to be accurate and easier(as they use existing
solvers of fluid and solid domain), remeshing has to be carried out at
successive interval time which increases the computational cost by a huge
margin. These methods encounter issues for interface matching due to which
significant accuracy at interfaces is lost. When we deal with moving bodies
(eg. object falling under gravity) or with large deformation problems
(expansion of air inside balloon etc.,), mesh generated at different intervals
may be of lesser order or lead to entanglement (when solid rotates). Thus, in
these cases, ALE is found to be not very suitable.

####  3.3.2 Immersed Boundary Methods (IBM)

The IBM is a Non-Conforming methods that uses a Cartesian grid on whole domain
and boundary is approximated either as a curve(for 2D flow) or a surface(3D
flow). Structure is solved separately to update the coordinates of curve.
Further, the governing equations in IBM are modified so as to incorporate the
boundary conditions of no-slip by introducing source terms. This method was
introduced by Peskin [16] where blood flow patterns in a model heart was
analysed. The method used Eulerian frame for fluid (blood) that is discretized
by finite differences and applied Chorin’s projection method for obtaining
fluid solution. Solid was modelled in Lagrangian frame where forces generated
were modelled through combination of springs.

Immersed Boundary Method is broadly classified in two types based on the
manner in which forcing term is applied (refer Fig. 4). In the first method,
which is popularly known as ‘Continuous Forcing’, force term developed through
solid at the interface is applied on the fluid momentum equation (Eqn. (2))
and then a suitable discretization is performed. The following equations may
help to visualise the flow of solution scheme with this approach—

$[L]({{v}_{i}})=\frac{\partial{{v}_{i}}}{\partial
t}+{{v}_{j}}\frac{\partial{{v}_{i}}}{\partial{{x}_{j}}}+\frac{1}{\rho}\frac{\partial
p}{\partial{{x}_{i}}}-\frac{\mu}{\rho}\left(\frac{{{\partial}^{2}}{{v}_{i}}}{\partial
x_{j}^{2}}\right)$ (6)

A forcing term is introduced as follows

$[L]\\{{{v}_{i}}\\}=\\{{{f}_{b}}\\}$ (7)

The forcing terms are interpolated from solid domain using dirac delta
functions as follows (k referes to the discrete point on interface curve).

${{\mathbf{f}}_{m}}(\mathbf{x},t)=\sum\limits_{k}{{{\mathbf{F}}_{k}}(t)\delta(|\mathbf{x}-\mathbf{X}|)}$
(8)

Where ‘x’ denotes co-ordinates in fluid domain(eulerian) and ‘X’ denotes co-
ordinates in solid field(Lagrangian). Various algebraic expressions are
presented for ‘$\delta$’ by Peskin [10]. Velocity at the Lagrangian points is
obtained through

$\frac{\partial{{\mathbf{X}}_{k}}}{\partial
t}=\mathbf{v}({{\mathbf{X}}_{k}},t)$ (9)

It was observed by Mittal and Iaccarino [17] that continuous forcing approach
was suitable for immersed elastic bodies and moving boundaries immersed in a
fluid but produced errors when simulating rigid bodies and bodies with complex
physical boundaries as it softens the interface.

Another class of IBM are the discrete forcing approach based models where a
normal fluid governing expression without forcing terms are discretized first
and then a modified forcing term is applied to discretized equations via dirac
delta kernels (so that force applies only on boundary and near boundary
nodes). This method further has classifications like indirect boundary
condition imposition where fluid velocity is predicted without forcing terms
and then corrected through suitable usage of kernel functions for force and
direct boundary condition imposition methods which prevent smoothing of
boundaries. This indirect boundary condition imposition is specially used for
boundary layer problems like flow through vented cylinders, pectoral fin
locomotion analysis. The broad classifications of IBM have been provided
below.

Figure 4: Flowchart of Immersed Boundary Method

####  3.3.3 Immersed Interface Method (IIM)

After the introduction of Immersed Boundary methods, it was observed that
Pressure and viscous stress were discontinuous at the interface and IBM
regularized this discontinuity. The main reason behind this was the usage of
discrete dirac delta functions for singular force term calculation at the
interfaces. Thus, it was found to have first-order accuracy at interfaces
[11]. Though sharp interface IBM was developed [10], full second order
accuracy was not possible. The initial Immersed Interface Method (IIM) was
developed by Li [18] but was restricted to simpler elliptic PDEs applied to
phase change problems and stokes equations. Li and Lai [19] developed IIM for
Navier-Stokes equation which was able to achieve second-order accuracy at
interface. The major modification with respect to IBM was that the jump
conditions of variables calculated through Taylor series were applied at the
interface instead of discrete delta functions. The derivation for jump theorem
and calculation of jump are reported in Li and LeVeque [20]. The method was
successfully applied on moving interface problems that involved phase changes.

A recent study by Griffith and Patankar [21] showed that there were some
discrepancies in the solution obtained for flow past cylinders with IBM. They
stated that this problem of leakage of mass flux inside cylinder could be
easily overcome by the usage of IIM. When a channel flow was tried to be
solved through IBM [22], it was observed that velocity diffused outwards of
channel walls. They introduced a novel formulation for solving Fluid-Structure
Interaction problems through Immersed Interface Method by calculating jump
conditions from projections. Performing simulations with discrete IB
formulation reveals that results near the interface have lower order accuracy
and thus produce erroneous results. In case of Vortex Induced Vibration
problem, it was observed to have some internal flow which is not desirable. In
case of channel flow, deviation from analytical results were observed in flow
rate with the use of classical IB formulation. They have projected the jump
conditions on the subspace containing interface and then derived the jump
conditions. This adjustment has been done to avoid the discontinuity in the
interface normal directions. Ray-Casting Algorithm has been used to find the
intersection between finite difference stencil and the interface. They have
used IBAMR software to incorporate their new methodology and carry-out
simulations. Many fundamental flow problems like flow in 2D and 3D channel
(straight and inclined), flow over stationary and rotating cylinder have been
solved with the proposed as well as classical IB method. Better results (in
terms of sharpening of boundary and avoiding internal flow) were observed with
IIM (with projected jump conditions applied on all variables). It was observed
that solution was globally second order accurate for velocity and traction
variables whereas the jump conditions with least order were considered using
discretisation. This was achieved due to the projection of jump conditions
across the interface. Thekkethil and Sharma [23] developed Level-Set based IIM
formulation to solve FSI problems involving flexible bodies. Boundary
conditions at interface were applied by the Level Set(LS) function which was
calculated based on the geometry of solid.

####  3.3.4 Coupled Momentum Method

Although there have been some Non-Conforming techniques like IBM and IIM which
can cut the computational cost and make programming clearer, coupling accuracy
especially while using different discretizations can be error prone at
interfaces. A new method which is monolithic in nature and avoids errors at
interfaces was developed by Figueroa et al. [12] and is popularly called as
coupled momentum method. Using the thin wall approximation, the lateral body
of solid surface (Eqn. (10) ) is projected onto the fluid surface using
quantitative wall thickness. This process is approximated as

$\int\limits_{{{\Omega}^{s}}}{(\ldots)dx=\zeta}\int\limits_{{{\Gamma}^{s}}}{(\ldots)ds}$
(10)

The Nominal governing equation for fluid is written in integral form of Finite
Element discretisation approach. Traction boundary condition tf is brought
into the governing equations for fluid. It was assumed that surface traction
tf acting on the lateral surface is equal and opposite to vessel wall surface
traction i.e, tf = - ts . This inturn was incorporated in body force term of
solid governing equation and an expression for tf was obtained. This force was
substituted in fluid-governing equation. Thus, a common governing equation for
fluid and solid was thus obtained. Simulations were performed on idealized
model of common carotid artery and deformation model was compared with that of
rigid model. Resistance, Impedence and Pressure type boundary conditions were
used and displacement of artery wall along with flow rates were measured.
Recently a verification study for this method was published by Filonova et al.
[24]. Some challenges faced by this method include incorporating spatially
varying material properties and development of method for varying thickness
models.

##  4 Computational FSI Applications

In this section, an attempt has been made to cover some important applications
involving interactions between fluid and solid. The research gap which exists
in individual application domains have been tried to be addressed so that it
would pave way for working on those areas to provide solutions. The
applications are presented below in separate subsections for VIV induced
energy harvesting, nature inspired vehicles and bio-medical applications.

###  4.1 Vortex Induced Vibrations

After the incident of Tacoma Narrows Bridge in 1940, scientific community
started to study the reason behind the collapse of reason. One theory
suggested by community suggested the shedding frequency of swirls behind the
bridge were actually matching the natural frequency of vibration of bridge
because of which it started to resonate. The study of vortices and the effect
it has on a structure thus began which led to the development of study related
to interaction between fluid and solid. Study of VIV has been applied in
various fields which include prevention of vibrations of vertical structures
like chimney stack, offshore risers and harvesting energy (called vortex
power).

Vortex induced vibrations were initially studied analytically by Facchinetti
et al. [25] where they considered a lower order equation for structural
vibration and wake vibration (van der Pol equation). Authors considered three
different models for coupling wake and structure oscillations through
velocity, displacement and inertia terms. The analytical results from these
three models were compared with experimental results for various parameters.
It was observed that inertia model based on acceleration coupling yielded the
best results. A major disadvantage of these models was the ignorance of some
terms during derivation (like non-linear variables) and approximation of some
parameters based on experiments. Hence, a computational approach applicable to
all set of problems becomes necessary.

Herjfold et al. [26] developed a cost effective computational methodology for
analyzing the VIV phenomenon on deepwater risers placed in offshore plants so
as to test the efficiency of riser design before actually developing it.
Authors considered finite number of two dimensional fluid planes along the
length of riser on which incompressible Navier-Stokes equations were solved
separately. Fractional step algorithm was used using NAVSIM program to solve
for velocity and pressure. The values of surface force obtained on fluid
simulation were interpolated on surface of structure. Displacement field for
solid was computed based on the interpolated surface forces using algorithm of
USFOS (Ultimate strength of framed offshore structures). Validation of the
developed model was performed by comparing structure response at various
sections with the one obtained from experiments. A small amount of under
prediction was observed at bottom of the riser. This might have happened due
to ignorance of ground effects.

Zhu and Peng [27] developed a computational model to analyse the effect of
heaving motion of foil for the purpose of efficient vortex control. The
mathematical formulation of Navier-Stokes equation was performed in XY-
coordinate system which was fixed in space and a xy-coordinate system which
was fixed with foil. Discretization was performed based on finite difference.
In order to incorporate interaction between fluid and structure, an iterative
coupling model was used. This model followed some steps wherein initial
excitation was provided to the foil and which the deflection (heave) of foil
was computed based on the lift force acting on the surface of foil at
different sections. They came to a conclusion that partial recovery of energy
of flapping could be recovered when leading edge vortex is made to be formed
as far as possible from the pitching axis.
Williamson and Govardhan [28] discussed the overall development in terms of
computational and experimental methods to study the VIV phenomenon so as to
apply the findings in industrial applications. Review of methods was performed
based on their effectiveness to satisfy the Griffin-plot and other parameters.
It has been observed that though sufficient computational techniques have been
developed to address VIV, much efforts on high Reynolds number VIV with
inclusion of turbulence model through Direct Numerical Solution or other
methods is the need of the hour to appreciate the real world flows. This might
lead us to efficiently develop Industrial products with more confidence.

###  4.2 Aquatic Locomotion and Insect Flight

Requirements for Enhancing the lift and reducing drag in mechanical systems
for locomotion has led the scientific community to find inspiration from
aquatic and flying species on Earth. An aerodynamic model was proposed by
Weis-Fogh [29] through which small insects like wasps were found to enhance
lift during their flight. He proposed that insects used clap and fling motion
during their flight to optimise performance. Miller and Peskin [30] used a
‘target boundary’ version of Immersed Boundary Method to simulate clap and
fling in a tiny wasp (called Encarsaria formosa). They performed simulation at
various flow conditions (8¡=Re¡=128) to analyse the lift enhancement
mechanism. Displacement and rotation of the wings were specified to understand
the interaction between fluid and wing structure. Lift and Drag coefficients
were obtained as the outputs which were validated with experimental results.
It was found that leading and trailing edge vortices existed for single wing
case throughout the cycle whereas only leading edge vortex (LEV) was present
in two wing case during initial fling motion. Vortical asymmetry during
translation and dominance of leading edge vortex were found to be the reason
for lift enhancement. Results obtained for one wing and two wings at various
reynolds number showed that clap and fling mechanism was most suitable for
insects flying at lower speeds.

Efficient propulsion found in aquatic animals is attributed to the optimal
vortex control done by them through movement of body and tail fin. In order to
understand the mechanism behind effective propulsion system of fish and derive
the optimal parameters for body motion, Triantafyllou [31] performed a
systematic review of experimental observations and computational methods. It
was shown that when an oncoming vortex moves over a foil (which has generated
its own vortex), it can either interact constructively to form a large
trailing edge vortex or interact destructively to weaken the vortex or form
pairs to broaden the wake region. The experiments carried out on oscillating
and thrust producing foils showed that there was an optimum St (around 0.25 -
0.35) where maximum propulsive efficiency was noted.

Bhalla et al. [32] developed a computational Fluid-Structure Interaction model
using cartesian based Immersed Boundary Method and applied Adaptive Mesh
Refinement to capture thin Boundary Layer at Fluid surface Interface. They
validated their model by applying it on a 3D model of black ghost knifefish.
The fluid was modelled in an Eulerian framework whereas solid was modelled
through Lagrangian framework. Specified motion is provided to the body and
force term fc is incorporated in governing equations to account for that
whereas a term fb is used to account for deformations of solid body. Reverse
karman vortex street is observed in eel locomotion close to trailing edge
which suggests the self propelling nature observed in eels. Forward and
Backward motion of black ghost knifefish was simulated to understand its
ability to swim in the reverse direction.

Recently Thekkethil et al.[33] performed a unified hydrodynamics study based
on various types of motions (pitch, heave and undulation) through an in-house
code based on Level Set Immersed boundary formulation. Apart from solving
regular Navier stokes equation, two separate equations based on advection
velocity at the interface and level-set function were used to classify the
domains. A non dimensional number based on wavelength of undulation was used
for unifying various motions. Simulations were performed at several St and
propulsive efficiency was derived. The role of reverse Karman street in
enhancement of propulsion was described by authors. A sensing mechanism used
by predator fish was also proposed after visualising the vortex in the
downstream direction. Performance characteristics of self propulsive motion of
NACA airfoil (which represents fish cross-section) has been investigated by
Thekkethil et al. [34] using the in-house developed Level Set (LS) based
Immersed Interface Method. It was observed that pitching type fishes have
higher propulsive efficiency and stability during the initial phases of
propulsion. They extended the study to 3D batoid type fish motion [35] with an
added parameter of 3D aspect ratio. Horshoe type vortices with multiple vortex
rings were observed for pitching type batoids whereas the propulsive
efficiency was found to be high for intermediate wavelength numbers and larger
aspect ratios.

###  4.3 Hemodynamic Analysis of pulsatile blood flow

Cardiovascular ailments have become a major reason for deaths in the modern
world mainly due to the sedentary lifestyle and food habits. It was estimated
by World Health Organisation (WHO) that around 7.3 million deaths are reported
annually around the globe due to coronary artery abnormalities [36]. Many
numerical and experimental works have shown there exists a strong correlation
between Hemodynamic parameters like Wall Shear Stress(WSS), Oscillatory Shear
Index(OSI) , Relative Residence Time (RRT) etc. on the growth and progress of
diseases. Cardiac-Surgeons are usually referred to as Cardio-Vascular Fluid
mechanicians who brings the blood flow to its normal condition either by
inserting a mechanical device which enlarges artery(called stent) or by making
some other way for blood flow(Grafting). Since in-vivo measurements are
dangerous and very difficult, it becomes utmost build a most accurate
computational model which can simulate blood flow through arteries so as to
predict potential sites for disease progression, help surgeons in surgery
planning and study the causes for disease development. Major challenges for
computational development include using patient specific models, using proper
rheology of blood, modelling blood constituents and considering proper
compliance of arteries through accurate artery material properties.
Inclusion of compliance leads us to requirement of coupling fluid and
structure interaction. Crosetto et al.[37] used ALE (Section 3.1) to calculate
the variation of mean velocity and pressure at different section with respect
to the cardiac cycle. Patient specific arteries were modelled using ITK Snap
software. Simulations were performed through LifeV multi-physics software. A
hybrid Boundary condition based on pressure and traction was formulation to
obtain realistic conditions. Blood was assumed to have a Newtonian rheology as
the arteries considered were of sufficiently larger cross section. It was
observed that when the inlet pressure was specified as boundary condition,
there was a diastolic backward flow which is not usually observed in fixed
wall simulation. This phenomenon was actually reported in human blood
circulation cycle which led to closing of aortic valve. It was also shown that
WSS was overestimated by rigid wall model. Thus, it can be argued that rigid
wall models underestimates the disease growth (as CAD is associated with low
and oscillating shear stress zone).

Bathe and Kamm [38] Studied the effects of aortic stenosis severity,
asymmetric nature and input pressure conditions on the compressive and tensile
stresses, blood flow patterns and plaque rupture possibilities. Computational
method developed on Finite element and finite difference scheme was validated
with experimental results. Poly Vinyl Alcohol(PVA) - hydrogel was selected as
the artery material for experiments. Pressure measurements were obtained
through manometric techniques whereas flow rates were measured through
ultrasound flow meter. Three dimensional Mooney-Rivlin model was used to model
arteries computationally. Fluid domain was solved through finite difference
scheme in a curvilinear coordinates. FSI at interfaces was implemented through
incremental boundary iteration. Maximum compressive stress was observed near
the inner wall at stenosis throat for asymmetric artery whereas point of
maximum compression was observed to be in the near post-stenotic region.
Turbulence and Non-Newtonian rheology were ignored which have effects on
hemodynamic parameters as suggested by Mahalingam et al. [39].

Based on some of the fine literature discussed above, it can be observed that
there have been minimal efforts to consider all parameters into account to
simulate realistic blood flow in compliant arteries. A major facet ignored in
currently available studies include the effect of shear induced migration of
red blood particles [40] in complaint stenotic arteries in the carotid region.

###  4.4 Phonation Study

Phonation refers to the production of sound due to fluid induced vibration of
the vocal cords in the larynx. Structure of human larynx consists of three
different layers [18] as shown in figure. Vocal cord paresis occurs due to
damage on the upper epithelium and leads to disability in speaking. This
problem is usually found during injury or due to old age. In the process of
phonation, vocal cord is made to vibrate by the alternate vortex shedding in
the wake of the vocal cords. Medialization laryngoplasty is a surgical
treatment for vocal cord paresis where surgeon tries to insert an implant made
of a small strip which can make up of the damaged mass of epithelium thus
enabling normal sound production. This process is iterative in nature and
requires a huge experience of surgeon to handle it. Simulation of vocal cords
with particular implant can help the surgeon to plan the surgery and remove
the iterative procedure.

In the initial stages, fully coupled FSI approach was used by Ishizaka and
Flanagan [41] where airflow was considered in one dimensional and two lumped
masses for vocal cords were used. Mittal et al. [17] developed a complete two-
dimensional FSI model sharp interface immersed boundary formulation.
Incompressible NS equations for air discretized through sharp interface
immersed boundary method was coupled with cartesian BM formulation performed
on structure. Appropriate properties based on experiments were used to model
vocal folds. Triangular elements were used to discretize solid and second
order finite difference scheme was applied for temporal and spatial
discretisation. An equation for displacement eigen mode was formulated and
frequency was calculated. Normal modes found through experiments were compared
with the eigen modes obtained through simulation. It was observed that second
and third eigen modes obtained in simulation were in reversed order from
normal modes. Thus, a modification in numerical method to handle solid is
needed for accurately solving this issue.

###  4.5 Bolus transport through Esophagus

Esophagus is a thick multi-layered cylindrical tube which helps to transfer
food (called bolus in medical terms) from pharynx to stomach through pulsatile
wall motions controlled by neural stimulus. Esophagus is composed of mucosal,
Longitudinal muscle (LM) and Circular muscle (CM) layers which shorten and
contract simultaneously to produce the desired wall motion [18]. It was found
that around 5-8 percentage of the population above 50 years of age have
swallowing difficulty which is associated with esophageal dysphagia [42].
Understanding the role of muscle contraction and shortening in esophageal
transport can be achieved through numerical simulations. Quantification of
disease pathologies and planning treatment strategies are the next steps in
this technology which have to be explored.

Kou et al. [43] developed a computational FSI model to simulate peristaltic
wall motions in esophagus using Immersed Boundary approach based on continuous
formulation (refer 3.3.2). In this model, bolus was considered as a viscous
fluid and esophagus as a thick tube containing fibers at different
orientation. This whole setup of fluid inside a solid was then submerged
inside another fluid where equations were not solved. Numerical method was
validated with analytical solution for tube dilation problem. The Immersed
Boundary method developed contains momentum and incompressibility of coupled
fluid and structure in an Eulerian framework whereas structural forces are
modelled through Lagrangian framework.The three different fibers of esophagus
are modelled as springs and beams whose whose stiffness parameter was derived
through material properties (Young’s modulus and poisson’s ratio). Finite
element formulation of wall structure using triangular elements has been
performed based on the shape function calculated through local coordinates.
Muscle activation kinematics is used for wall motion through which
displacement is specified at the inner layers of Esophagus. Numerical
challenges due to a wide range of dimensional quantities in wall diameter and
esophagus length were observed due to which some amount of fluid leakage was
observed. An adaptive mesh technique may prevent this issue and keep
computational cost in check. Kou et al.[44] validated their numerical pressure
contours with clinical manometry measurements.

##  5 Conclusions and Future Scope

###  5.1 Conclusions

A brief overview of different numerical methods used to solve the multi-
physics problems of fluid and structure have been discussed with some
important application domains where computational methods become very
essential. The scope and limitations of the various methods classified based
on coupling manner and discretization have been discussed in detail. Methods
suitable for each application domain and issues which persist even today with
these models have been listed. A list of standard numerical methods, their
developers and major application domains which use these methods has been
provided in Table 1 .

Table 1: Summary of Literature for various numerical methods and applications Numerical |  |
---|---|---
Method | Significant Developers | Application Domains
ALE | Hirt et al.[7],Kennedy et al.[8] | Free surface flows, Sloshing dynamics, vibration of risers,
|  | Flapping wing aerodynamics.
DSD/SST | Tezduyar et al. [9] | Modelling patient specific arterial blood flow, Flow through
|  | Windsock.
IBM | Peskin [16] | Blood flow through heart valves, Vortex induced vibration,
|  | Phonation and Esophagal transport.
IIM | Li and Lai [19] | Hydrodynamics of aquatic locomotion, Intra Vena Cava (IVC)
|  | flow, Solidification and crystallization study.
CMM | Figueroa et al. [12] | Deformable artery flow.

###  5.2 Future Scope

There has been very significant progress in the development of computational
techniques and application of the developed methods to design useful
mechanical devices which can harvest energy, decrease the vibration of
deepwater risers, build better infrastructures and help us to understand the
way birds, insects and aquatic animals locomote. Recently, there has been a
huge advancement in the development of diagnosis techniques, surgical planning
in medical domain which have some contribution from Fluid Structure
Interaction based computational models. Here we present scope for future work
on computational FSI methods and applications in separate subsections below.

####  5.2.1 Computational FSI Methods

  1. 1.

Recently Tsui et al. (2013) have used Finite Volume discretisation for solids
in FSI problem and validated the numerical method with flow past a circular
cylinder with flexible plate attached on its lee side. These methods have an
inherent advantage of obtaining better accuracy at interface since solid and
fluid has a common coupling scheme at the interface. These methods are not
explored completely in the domain of FSI and especially in the domain of
cardio-vascular FSI. Using Eulerian framework for solids and discretizing
through FVM remains an unanswered question which can have huge implications in
accuracy and computational costs.

  2. 2.

It has been observed in most of the literature that FSI model are used to
solve problems where fluid motion is assumed to be laminar. Hence, there are
no relevant computational model found which use turbulence model for fluid and
then applying two way coupling with solid. If we consider the case of
pulsatile blood flow in small arteries like those found in cerebrum and in
deep veins, a small constriction may cause the flow to become turbulent. Thus
turbulence modelling for these kind of biological FSI problems becomes
necessary.

  3. 3.

It has been observed in Immersed Boundary method that though second order
accuracy is obtained for smooth interfaces, calculations on sharp and
especially discontinuous interfaces (usually found on walls with protrusions)
cannot achieve this accuracy. Though there are some other methods like
Immersed interface method, blob-projection method which can avoid this issue,
a core Immersed Boundary method which can obtain translation invariance and
second order accuracy needs to be developed.

####  5.2.2 Computational FSI Applications

  1. 1.

Phonoangiography is a non-invasive diagnosis technique in which arterial
murmurs (called bruits) are analysed to detect abnormalities. Though there was
substantial research for qualitative phonoangiography [45], quantitative
methods which help to localise the source and quantify the abnormalities have
been developed in the recent past through computational techniques [46]. A
major assumption made by them was ignorance of fluid induced vibration of
structure which can lead to significant change in sound spectrum. They had
considered coronary arteries which are well protected by the rib cage which
may substantially reduce the resonance peak created by structure vibration. If
we consider the case of carotid arteries, this assumption might not hold true
as it is located very close to the epidermal surface where measurements are
made. Thus, analysis of bruits on carotid arteries with Fluid Structure model
coupled with acoustics can take us close to device a Guided User
Interface(GUI) which can help medical practitioners to quantify abnormalities
quantitatively.

  2. 2.

Considering Shear Induced Migration (SIM) of Red Blood Cells (RBC) in small
complaint arteries and calculating Hemodynamic parameters to predict growth of
diseases can help surgeons in medical planning. This problems requires one to
couple FSI model for blood-artery with the mixture model which governs
transport of RBC to obtain final solution. This becomes very important in
small arteries where SIM is enhanced and may change the platelet activation
mechanisms which are the main reasons for plaque deposits.

  3. 3.

Development of new energy harvesting devices like tapered vented cylinders as
vortex generators need a very efficient FSI model which can handle geometric
complexities. It was proposed by Kumar et al. [47] that usage of such systems
can improve energy harvesting quantitatively but FSI model was not
incorporated by authors due to geometric complexity.

  4. 4.

Performance of new underwater vehicles operating on the strategy of aquatic
locomotion needs to be verified before actual design and manufacturing. These
include the use of flexible plate on the lee side of cylinder as a fin for
these vehicles in order to reduce drag. Though simulation has been performed
on part scales, a full fledged simulation using FSI models is very essential
for moving ahead with next procedures.

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