Interface Problems-Fluid Structure Interaction: Description,

Application and Review

VIVEK KUMAR SRIVASTAV1, SRINIVASARAO THOTA2∗,

LATE M. KUMAR3, AMAN RAJ ANAND4

1Department of Applied Science & Humanities

Government engineering College, Siwan, Bihar, Pin–841226, INDIA

2Department of Mathematics, Amrita School of Physical Sciences,

Amrita Vishwa Vidyapeetham, Amaravati, Andhra Pradesh–522503, INDIA

3Department of Mathematics, Motilal Nehru National Institute of Technology,

Allahabad, Uttar Pradesh–211004, India. (Till April 2021).

4Department of Mechanical Engineering

Motihari College of Engineering, Motihari, Bihar, Pin- 845401, INDIA

Email: 1vivekapril@gmail.com, 2srinithota@ymail.com,

3manoj@mnnit.ac.in, 4amanraj3962@gmail.com

Abstract: - This paper presents a critical review of numerical methods for solving a wide variety of interface

problems emphasizing the immersed finite element method (IFEM). It is found in the literature that most of the

researchers considered the well-known methods with some modifications, however limited number of research

articles proposed new algorithms. Apart from the algorithm, this study highlights the wide range of applications

of interface problems specifically in biomedical, heat-transfer and turbo-machinery. Different numerical methods

for interface problems with their major finding are listed in tabulated form at the end.

Key-Words: - Interface Problems, Algorithms, Immersed Finite Element Methods, Fluid-Structure Interactions,

Numerical Models, Biomedical Engineering, Heat-Transfer, Turbo Machinery.

Received: August 25, 2023. Revised: February 11, 2024. Accepted: March 12, 2024. Published: May 23, 2024.

1 Introduction

In the last few decades, several numerical methods

are discussed to study the interface problems. Nowa-

days, Bio-medical engineering is one of the emerg-

ing fields in which medical practitioners have fo-

cused on computational results for quick diagnosis

and treatment. Most of the diseases are growing on

the surface-wall (blood flow in artery, airflow in hu-

man airways/lungs) because of interaction between

fluid (air, blood etc.,) and solid surface where in-

terface phenomena exists between the two surfaces.

The Fluid-Structure Interface (FSI) method is used for

solving interface-problem in many bio-medical ap-

plications. In general, interface issues include dif-

ferential equations, for which the solutions and input

data are discontinuous across some interfaces. Con-

struction and testing of any physical prototypes is

expensive in terms of cost and time, and even un-

able to provide sufficient results. Mathematical meth-

ods, like the Immersed Finite Element Method, are

more economical and effective for analysing intri-

cate geometries that include the interface of two or

more entities. It is found that in most of the real

life problems, To solve the entire system, a huge

number of linear systems must be used. with more

efficient method because of various realistic needs

such as curvature, random dimension, moving inter-

face. In this reference, [1], have studied the bilinear

and two-dimensional linear IFE solutions computed

from the algebraic multi-grid solver for both moving

and stationary interface problems.The study, [1], dis-

cussed several issues of immersed interface methods

including development and behavior (immersed com-

pressible and incompressible continuum). They have

also discussed how the Peskin developed immersed

boundary method for blood flow in human artery-

valves, [2], [3]. Initially, In situations when an inte-

gral equation may not be accessible, the Immersed In-

terface Method (IIM) was designed as a second order

accurate finite difference approach to solve elliptic

and parabolic partial differential equations. Further,

this method was expanded for two and three dimen-

sional problems such as parabolic, elliptic, hyperbolic

and mixed type equations, [4]. It is worth mentioning

that at the initial stage of the development of inter-

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face methods; only three famous methods (Smoothing

method, Both the Method of Harmonic Averaging and

Peskin’s Immersed Boundary Method (IBM) were ap-

plied. After that several additional boundary condi-

tions (jump conditions, stationary & moving bound-

aries, etc.) have been incorporated that escort to non-

symmetrical matrix where standard numerical solvers

fails, [5]. In [6], presented several points of Immersed

Finite Element Method in their review paper, in im-

mersed finite element method, Lagrangian dense grid

shift on the topmost of Eulerian fluid grid that is ex-

panded over the entire computational domain. In the

end, modelling of both the fluid and solid domains is

carried out, and continuity between the solid and fluid

sub-domains is ensured by force distribution utilising

the reproducing kernel particle technique (RKPM)

delta function and velocity interpolation. When com-

pared to the Immersed Boundary (IB) approach, it

is discovered that the higher order replicating ker-

nel particle method performs better at solving non-

uniform spatial meshes with arbitrary geometries and

boundary constraints. It is widely known, interface

boundary value issues connected with different kinds

of partial differential equations or systems may be

used to represent a wide range of phenomena in biol-

ogy, chemistry, engineering, and physics. As a result,

the researchers have put up a number of strategies to

address various real-world issues. The use of particu-

lar method varies from problem-to-problem and with

their boundary conditions. Keeping in view, the re-

quirement of new researchers, we described crux of

the published papers with the intension to provide all

important material related to partial differential equa-

tions with one or more interfaces for new researchers

in a condense form. In the next section, a few real-life

problems have been taken to elaborate the application

of Immersed Finite Element Method.

2 Some Real Life Applications on

Immersed Finite Element Method

The various real life problems deal with Immersed Fi-

nite Element Method, few of them are listed as fol-

lows.

2.1 Immersed Finite Element Method for

Biomedical Problems

It is found that there are several real-world problems

dealing with interface-phenomena. One of the vital

use of interface problem for solving the partial differ-

ential equations in biological systems (Human lungs,

Heart, Skins etc.). Human lugs and heart (coronary

artery) are flexible in nature where the interface phe-

nomena exist on the wall. It is seen that inner injury in

human lungs (Figure 1) occurs between the wall and

air. Elastic interface problems have broad applica-

tions in continuum mechanics specially for the prob-

lems involves stress and strain, [7]. Wall shear stress

is the main parameter that is used for the interface-

problems (between wall & air, between wall & blood,

between wall & fluids) in biological system. Nowa-

days, Finite element method and computational fluid

dynamics software?s are useful for solving such inter-

face problems where large numbers of nodes/control

volumes occur.

Figure 1: Interface problems in Human Lungs, [8]

Human heart (Figure 2) is another part of human

body where researchers are trying to solve partial

differential equations for interface problems between

blood and arteries. In this context, [9], have discussed

simulation of expandable stent using immersed finite

element method, which was introduced for solving

complex fluid-structure interaction (FSI) problems.

Coronary stents is used to physically open the chan-

nel of blocked artery. They presented model and sim-

ulation of balloon elastic stent (Figure 3) cooperating

with its nearby fluid that is a class of interface prob-

lem.

Numerical methods play the major role to solve

these interfaces problems of biological system so that

the medical practitioner can exactly diagnose and fur-

ther prognosis the disease.

2.2 Immersed Finite Element Method for

Turbo Machinery Problems

Turbo-machinery is another area of application of

interface-problem. It is found that the inner wall

of water-pump becomes erosion due to high pres-

sure water/oil and that reduce the efficiency of pump.

These interface problems are also used for cavitations

in turbo-machinery. It is seen that turbo-machinery

contains two important parts namely stationary and

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Figure 2: Interface problems in human heart, [10]

Figure 3: Stent in artery wall using IFEM

moving (rotating) components. Interface is occurred

between stationary and moving components which

pay a crucial role for increasing the efficiency of

turbo-machinery (Figure 4). It is known that fluid and

solid are an important component of the rotor system

which influence each other and which creates sensa-

tions in some conditions, [11].

2.3 Immersed Finite Element Method for

Heat Problems

Heat-transfer problems are another important real ap-

plication where an interface phenomenon occurs near

the moving boundaries, [12]. The details of such

problems are discussed in the elliptic interface sec-

tion 3.1.

Figure 4: Interface problem in Turbo-Machinery

3 Classification of Interface Finite

Element Methods

Because of several variability conditions in interface

problems (fluid-solid interaction, fluid-fluid interac-

tion, solid-solid interaction etc.), it is very difficult to

used exact interface method for explaining the partial

differential equations. The following classifications

are done on the basis of their application/popularity in

different areas of sciences and engineering (Figure 5).

Figure 5: Classification of IFEM

These above methods are further classified in the

literature depending upon the problems and their

boundary conditions.

3.1 Elliptic Interface Problem

It is seen that elliptic interface phenomena comes

from the elliptic boundary value problems having dis-

continuity in their coefficients or solution around one

or more interfaces of the flow domain. In such cir-

cumstances, it is very difficult to find approximate re-

sults using standard finite difference method due to ir-

regularity or non-smooth near the interface. There are

numerous numerical methods that use Cartesian grids

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for evaluating elliptic interface problems. Finite dif-

ference method has been modified such as immersed

interface method to provide most accurate results but

still there is need to develop new methods for refin-

ing the solution or exact solution, [13]. The elliptic

interface equation of second order is

{[−∇.{β∇u}] = f(x)X∈Ω

[u(X)] = g(X)X∈∂Ω

Jump conditions

|u|Γ= 0

|β∂u

∂n|Γ= 0

where,

−→

uis the fluid velocity,

−→

X(s, t)is the Lagrangian representation of the

immersed moving boundary,

υis the constant fluid viscosity,

−→

f(s, t)is the force strength along the

interface,

ρis the fluid pressure,

Ω= open rectangular domain,

Γ= interface curve,

β(X)= positive piecewise constant.

β(X) = {[β−X∈Ω−

β+X∈Ω+

Figure 6: The domain Ωhas divided into two sub-

domain and representation of Grid generation

Numerical methods such as finite element meth-

ods (FEM) and finite difference methods (FDM) can

be used to tackle the interface problem mentioned

above. Finding the precise answer to the curved-

interface problem is a challenging endeavor, [14].

There are many types of interrelated problems de-

pending on geographic area and environment (Fig-

ure 6). In this context, [4], discussed four types of

elliptic interface problems in their review article, [4].

These are: (i) The coefficients of the differential

equation are constant (ii) The coefficients of the dif-

ferential equation are not constant. (iii) The interface

is fixed (iv) the interface is movable, [4]. Seyidmame-

dovy and Özbilgey discussed the transportation prob-

lems of the crisis in the state in the media. In this

study, two different types of jumps of the propagation

method are presented, and the local error for each case

is estimated on the non-uniform grid, especially us-

ing the standard deviation, [15]. Mathematical meth-

ods are described in this reference by [3]. It includes

different types of problems including: discussed the

computational problems of the immersed finite ele-

ment method.

In [16], author is discussed three types of computa-

tional schemes involving non-conforming rotated Im-

mersed Finite Element functions for evaluating the el-

liptic interface problems, [16].

(I). Finite Element Galerkin Immersed Methods.

(II). Partially Penalized Galerkin (PPG) Immersed Fi-

nite Element Methods.

(III). Interior Penalty Discontinuous Galerkin (IPDG)

Schemes.

An and Chen (2014) studied a partially penalty im-

mersed interface finite element method for solving the

anisotropic elliptic interface problems. The proposed

method is based on linear immersed interface finite

elements and used not-continuous Galerkin formula-

tion nearby the boundary, [17]. Further, higher degree

immersed finite element methods for second order el-

liptic interface problem was discussed by [18].

[16], has mentioned several real life applications

of elliptic interface problems such as plasma particle

simulations in ion thruster optics. It is known that

ion thruster is an electronic propulsion device that re-

leases a high-energy ion beam to drive a spacecraft.

It is also used in projection methods to solve Navier-

Stokes problems in multiphase-flow. Another, ap-

plication of elliptic interface problem is used for the

topology-optimization in heat-conduction problems.

Several methods are discussed by the researchers

in which immersed interface method is very popular

because of method efficiency. In immersed interface

method, jump conditions are included in standard fi-

nite difference formula across the interface to obtain

minimum error. It is found that discontinuity in gen-

eral solution and its derivative occurs because of dis-

continuity in the coefficient of differential equation.

After that Immersed Interface method was further im-

plemented by the researchers and given decomposed

immersed interface method for solving elliptic equa-

tions that provides non-smooth and discontinuous so-

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lution. Later on, fourth order accurate finite differ-

ence was given for solving elliptic equations having

embedded interface of discontinuity, [4].

3.2 Galerkin Method of Matching

Interfaces and Bounds (MIB)

It is well known that interfaces exist in many real-

world models and devices that help solve relevant

problems and support the development of new algo-

rithms. Due to the different properties of various nat-

ural materials and models, the part of the equation

with constant coefficients and positive terms needs

to be solved. A partial equation is difficult to solve

due to interactions and interactions between two or

more interacting objects. The integrated and bound-

ary layer approach was adopted by [19], Solving

multicomponent interface problems. This method in-

volves extending interface-based solutions from var-

ious perspectives on both sides of the interface, [20].

Matching interfaces and boundary Galerkin tech-

niques were introduced by [21], in a specific study

solved two-dimensional partial equations with com-

plex interfaces, geometric singularities, and rare so-

lutions. Since higher order methods are important

for many problems involving high-frequency waves,

they proposed further development of three- and four-

dimensional matching interface and boundary meth-

ods . Furthermore, for three-dimensional elliptic in-

terface issues, [22], establish the Galerkin formula-

tion of the MIB technique. Boundary Galerkin tech-

nique and the suggested three-dimensional Matched

Interface are verified for correctness and stability

across three different types of elliptic interfaces prob-

lems. The first type of interface is analytically de-

fined by level set functions that admits geometrical

singularities. The second types of Protein surfaces de-

fine interfaces. and last one is generated from multi-

protein complexes.

Thus, Numerous issues in the real world may

be solved with the renowned Matched Interface and

Boundary (MIB) Galerkin approach.

3.3 Immersed Boundary Method

The Immersed Boundary (IB) method was developed

by [2], to solve the problem of interaction between

blood and the vessel wall. The main advantage of

this method is that it has a network update algo-

rithm between the liquid and the walls. This method

is based on the difference method by dividing line,

and the submergence is represented by the fiber net-

work, [3], [6]. Fluid interaction is achieved by di-

viding the node forces and interacting between the

Eulerian and Lagrangian fields using the Dirac delta

function. However, one of the limitations of this

method is that it provides an immersed fibrous struc-

ture that has mass but does not retain volume in the

liquid, [6]. In [3], authors proposed the continuous

expansion method for boundary conditions (EIBM).

Instead of using small particles in the elastic barrier,

they use material in the elastic material that holds

the small volume inside the liquid. In the proposed

method, the meshless Regenerative Kernel Particle

Technique (RKPM) kernel function is used by the

discretized delta function. They achieved better re-

sults with these modifications, [23], [6] It briefly in-

troduces the immersed finite element method for solv-

ing complex fluid and solid problems. Lagrangian ob-

jects and Eulerian fluid networks control all calcula-

tions in the immersed finite element method. In solv-

ing non-uniform spatial networks, the Kernel Parti-

cle Propagation Method (RKPM) outperforms the im-

mersed boundary method. They eliminated the disad-

vantages of the submerged boundary layer and incor-

porated the extended finite element method (EFEM)

concept of [23], into their design. The computational

fluid dynamics (CFD) fails to solve many problems

such as elastic boundary moves fluid and eventually

fluid reverse back against it. These issues lead to the

interface problem with a solitary source and discon-

tinuous coefficient, [4].

The Stoke’s equation:

∇p=ν∆−→

u+−→

F(−→

x , t)

∇.−→

u= 0

where, −→

F(−→

x , t)is boundary force and it can be de-

fined as:

−→

F(−→

x , t)=∫Γ(s,t)

−→

f(s, t)δ2(−→

x−−→

X(s, t))ds.

The discrete delta function was used in the im-

mersed boundary approach to distribute the singu-

lar source to the closest grid, and a collection of la-

grangian points discretizes the immersed boundary.

Additionally, a discrete sum is used in place of the in-

tegral function, while a discrete approximation with

length support is used in place of the delta function.

This approach is popular because it can be imple-

mented easily and therefore, used in many real life

application, [4].

3.4 Reproducing Kernel Particle Method

(RKPM)

[6], suggested an immersed finite element approach

for solving the fluid-structure interaction issue. To

ensure continuity between the fluid and solid sub-

domains, they employed the replicating kernel parti-

cle technique (RKPM). They have also incorporated

the concept of the extended immersed boundary tech-

nique (EIBM). The fluid-solid interaction is used to

model the flow vibration in vocal folds during phona-

tion. [24], described a linked sharp-interface im-

mersed boundary-finite element approach for inter-

action problems involving human phonation. They

investigated the dynamics of the glottal jet in a three-

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dimensional model, as well as the two-dimensional

laryngeal model. [25], developed an immersed fi-

nite element approach with integral equation correc-

tion. They portrayed the physical domain border as

a domain to aid in the mapping of finite element and

integral equation solutions. [4], discussed numerous

numerical approaches for handling elliptic interface

issues. In their survey piece, they focused on newly

discovered numerical approaches and their benefits

and drawbacks. They determined that the Finite Dif-

ference Method (FDM) is easier to adopt than the Fi-

nite Element Method (FEM), which has less regular-

ity. It is observed that numerical techniques using

grids are more costly than meshless methods. [26], in-

troduced immersed finite element approaches for el-

liptic interface issues with non-homogeneous jumps.

This approach was proposed to solve second-order

elliptic problems with discontinuous jump condi-

tions. [27], addressed the Immersed smoothed finite

element method for simulating fluid-structure inter-

actions in aortic valves. They addressed the solid

domain using a smoothed finite element approach,

which is appropriate for handling hyperelastic ma-

terials with substantial deformation. [28], reported

an immersed finite volume approach for simulating

unsteady three-dimensional heat transfers and turbu-

lent flows in an industrial furnace with three conduct-

ing solid bodies. Heltaia and Costanzo (2012) pro-

posed the generalised finite element immersed bound-

ary technique (FEIBM), in which an incompressible

Newtonian fluid interacts with a generic hyperelas-

tic solid. The suggested technique is based on inde-

pendent discretization for both the fluid and solid do-

mains.The fluid and solid domains are modelled by

the Eulerian and Lagrangian models, [29]. [30], inves-

tigated the immersed smooth finite element approach

for two-dimensional fluid-structure interaction issues.

They employed a characteristic-based split technique

to compute fluid-structure interaction forces. In this

approach, a fictional fluid mesh is employed to cal-

culate the fluid-structure interaction force acting on

the solid. [31], has used adaptive interface finite ele-

ment method for elastic interface. He has established

a residual based posteriori error estimate constant for

refining the finite element mesh. This posteriori er-

ror was further tested and is found competitive na-

ture of adaptive method. [32], explored the immersed

molecular finite element technique (IMFEM). It was

discovered that the interaction between submerged

objects and the fluctuating fluid is caused by hydrody-

namic forces, which gave rise to the fluid-structure in-

teraction term. It has been stated that a molecular type

force field paired with a coulomb potential between

submerged objects enables the immersed molecular

finite element method to provide a thorough descrip-

tion of nanoscale structure.

3.5 Algorithm Focused on Solid-Surface

The literature on related topics seems to be limited

to researchers focusing solely on the material, so this

chapter introduces many aspects of the material. [33],

studied the modified immersed finite element method

for fluid structure assembly. The proposed system fo-

cuses on material dynamics rather than fluid dynam-

ics. This approach is especially important in situa-

tions where strong forces play an important role in

fluid interactions. Captures material behavior better

than standard interface finite element method algo-

rithms, especially at high Reynolds numbers.

[34], discussed the use of the immersed finite

element method to describe tissues interacting with

fluids. They found three biological applications us-

ing the interfacial finite element method. A special

method for combining finite element technical formu-

las is required for stable and accurate calculations of

liquids and solids. [35], applied the solution method

to fluid interface problems. They found that the ap-

plicability of the method is general and independent

of the specific choice of finite field in waste and con-

tinuous water.

[36], describes the immersed finite element

method to solve symmetric and similar problems. The

proposed method uses a non-uniform finite element

method, which is symmetric and follows the second

truth. [37], discussed the regular Galerkin immersed

finite volume element method for anisotropic flow

model in porous media. This method was developed

by mapping the orbital function space into the im-

mersed finite element space. [38], proposed a nonlin-

ear finite element method for nonlinear material me-

chanics. Therefore, the authors used this method to

provide the most important solution for the competi-

tion.

4 Conclusions

Many variables and boundary conditions must be con-

sidered while developing an appropriate numerical

approach for fluid-structure interaction issues. It is

seen that a lot of work has been done on the fluid-

solid interaction problem; still it is far away to predict

the accurate result. Use of correct numerical method

is one of the challenging task as well as important

parameter which have been discussed in the present

study. Based on the review of recent literature as dis-

cussed above, the point wise conclusions and sugges-

tions are furnished herewith:

(i) Immersed boundary method is one of the popular

methods that have been used/implemented by several

researchers because of finite difference based method

with uniform grid distribution.

(ii) Most of the methods used finite difference method

for the development of new-methods such as im-

mersed finite element method.

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(iii) It is seen that immersed boundary (IB) method

is based on finite difference with uniform grid distri-

bution and having mesh updating algorithms between

fluid and solid wall.

(iv) The reproducing kernel particle approach is ef-

fective for addressing non-uniform spatial meshes.

This approach used a discretized delta function in

a mesh-free distribution, and greater order precision

improved the interaction between fluid and solid.

(v) In the present paper, real-life problems have been

presented to make out the application of interface fi-

nite element method.

(vi) The different types of numerical methods pre-

sented in this study can be applied in several real life

problems depending upon the interface conditions.

Acknowledgment:

The authors thank the reviewers, editor, and editing

department for giving valuable inputs and

suggestions to get the current form of the manuscript.

References:

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WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE

DOI: 10.37394/23208.2024.21.22

Vivek Kumar Srivastav, Srinivasarao Thota,

Late M. Kumar, Aman Raj Anand

E-ISSN: 2224-2902

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

Dr. V.K. Srivastav is involved in the formation and

derivation of the mathematical calculations. Dr. S.

Thota, M. Kumar, and A.R. Anand are involved in

suggestions, revisions, and verification of the mathe-

matical calculations. All authors read and approved

the final manuscript.