immersed finite element method

Comput. Methods Appl. Mech. Engrg. 193 (2004) 2051–2067

www.elsevier.com/locate/cma

Immersed finite element method q

Lucy Zhang a, Axel Gerstenberger a, Xiaodong Wang b, Wing Kam Liu a,*

a Department of Mechanical Engineering, The Technological Institute, Northwestern University, 2145 Sheridan Road,

Evanston, IL 60208, USAb Department of Mechanical Engineering, Six MetroTech Center, Polytechnic University, Brooklyn, NY 11201, USA

Received 15 April 2003; received in revised form 10 September 2003; accepted 8 December 2003

Abstract

In this paper, the immersed finite element method (IFEM) is proposed for the solution of complex fluid and

deformable structure interaction problems encountered in many physical models. In IFEM, a Lagrangian solid mesh

moves on top of a background Eulerian fluid mesh which spans over the entire computational domain. Hence, the mesh

generation is greatly simplified. Moreover, both fluid and solid domains are modeled with the finite element methods

and the continuity between the fluid and solid sub-domains are enforced via the interpolation of the velocities and the

distribution of the forces with the reproducing kernel particle method (RKPM) delta function. In comparison with the

immersed boundary (IB) method, the higher-ordered RKPM delta function enables the fluid domain to have non-

uniform spatial meshes with arbitrary geometries and boundary conditions. The use of such kernel functions may

eventually open doors to multi-scale and multi-resolution modelings of complex fluid–structure interaction problems.

Rigid and deformable spheres dropping in channels are simulated to demonstrate the unique capabilities of the pro-

posed method. The results compare well with the experimental data. To the authors� knowledge, these are the first

solutions that deal with particulate flows with very flexible solids.

� 2004 Elsevier B.V. All rights reserved.

Keywords: Immersed finite element method; Reproducing kernel particle method; Fluid–structure interaction; Particulate flow;

Immersed boundary method

1. Introduction

For the past few decades, tremendous research efforts have been directed to the development of mod-elling and simulation approaches for fluid–structure interaction problems. Methods [11,12] developed by

Tezduyar and his coworkers are widely used in the simulations of fluid–particle (rigid) and fluid–structure

interactions. To accommodate the complicated motions of fluid–structure boundaries, we often use

qThis paper is an extended version of the paper presented in the Sixth Japan–US International Symposium on Flow Simulation and

Modeling (May 29–31, 2002, Fukuoka, Japan).* Corresponding author. Tel.: +1-847-491-7094; fax: +1-847-491-3915.

E-mail address: [email protected] (W.K. Liu).

0045-7825/$ - see front matter � 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.cma.2003.12.044

mail to: [email protected]

2052 L. Zhang et al. / Comput. Methods Appl. Mech. Engrg. 193 (2004) 2051–2067

adaptive meshing or the arbitrary Lagrangian Eulerian (ALE) techniques [6,14,17,18,20]. Recently, suchapproaches have also been adopted by Hu et al. in the modelling of fluid–particle (rigid) systems [5].

Nevertheless, mesh update or remeshing algorithms can be time consuming and expensive and a detailed

discussion on this issue is presented in Ref. [7]. In 1999, Glowinski et al. developed the distributed Lagrange

multiplier (DLM) method for the modelling of viscous flows interacting with rigid particles [4]. Their finite

element formulations enable the use of a fixed structured fluid grid along with fast and efficient solvers.

However, none of the aforementioned methods deals with the interaction between fluid and flexible solid

particles with large deformations.

In the 1970�s, Peskin developed the immersed boundary (IB) method [2] to study flow patterns aroundheart valves. The mathematical formulation of the IB method employs a mixture of Eulerian and

Lagrangian descriptions for fluid and solid domains. In particular, the entire fluid domain is represented by

a uniform background grid, which can be solved by finite difference methods with periodic boundary

conditions; whereas the submerged structure is represented by a fiber network. The interaction between

fluid and structure is accomplished by distributing the nodal forces and interpolating the velocities between

Eulerian and Lagrangian domains through a smoothed approximation of the Dirac delta function. The

advantage of the IB method is that the fluid–structure interface is tracked automatically, which removes the

costly computations due to various mesh update algorithms.Nevertheless, one major disadvantage of the IB method is the assumption of the fiber-like one-dimen-

sional immersed structure, which may carry mass, but occupies no volume in the fluid domain. This

assumption also limits accurate representations for immersed flexible solids which may occupy finite vol-

umes within the fluid domain. Furthermore, uniform fluid grids also set limitations in resolving fluid do-

mains with complex shapes and boundary conditions.

In the proposed immersed finite element method (IFEM), we will eliminate the aforementioned draw-

backs of the IB method and adopt parts of the work on the extended immersed boundary method (EIBM)

recently developed by Wang and Liu [16]. With finite element formulations for both fluid and solid do-mains, the submerged structure is solved more realistically and accurately in comparison with the corre-

sponding fiber network representation. The fluid solver is based on a stabilized equal-order finite element

formulation applicable to problems involving moving boundaries [11–13]. This stabilized formulation

prevents numerical oscillations without introducing excessive numerical dissipations. Moreover, in the

proposed IFEM, the background fluid mesh does not have to follow the motion of the flexible fluid–

structure interfaces and thus it is possible to assign sufficiently refined fluid mesh within the region around

which the immersed deformable structures in general occupy and move.

Unlike the Dirac delta functions in the IB method which yield C1 continuity [3,15], the discretized deltafunction in IFEM is the Cn shape function often employed in the meshfree reproducing kernel particle

method (RKPM). Because of the higher-order smoothness in the RKPM delta function, the accuracy is

increased in the coupling procedures between fluid and solid domains [16]. Furthermore, the RKPM shape

function is also capable of handling nonuniform fluid grids.

We first give a detailed description of the IFEM formulations in Section 2. In Sections 3.1 and 3.2, we

discuss the derivations of the governing equations and their discretizations in fluid and solid domains. The

RKPM delta function for nonuniform spacing and its discretized form are presented in Section 4, together

with the fluid–structure interpolation and distribution procedures. Section 5 summarizes the discretizedgoverning equations and provides an outline of the IFEM algorithm. Several numerical examples of

particulate flows are shown in Section 6. To the authors� knowledge, these are the first solutions that deal

with particulate flows with a large number of moving flexible solids under large deformations. In Section 7,

we summarize the IFEM concept and its attributes.

L. Zhang et al. / Comput. Methods Appl. Mech. Engrg. 193 (2004) 2051–2067 2053

2. Formulations of immersed finite element method

2.1. Nomenclature and kinematics

Let us consider an incompressible three-dimensional deformable structure in Xs completely immersed in

an incompressible fluid domain Xf . Together, the fluid and the solid occupy a domain X, but they do not

intersect:

Xf [ Xs ¼ X; ð1aÞ

Xf \ Xs ¼ ;: ð1bÞIn contrast to the IB formulation, the solid domain can occupy a finite volume in the fluid domain. Since

we assume both fluid and solid to be incompressible, the union of two domains can be treated as one

incompressible continuum with a continuous velocity field. In the computation, the fluid spans over the

entire domain X, thus an Eulerian fluid mesh is adopted; whereas a Lagrangian solid mesh is constructed on

top of the Eulerian fluid mesh. The coexistence of fluid and solid in Xs requires some considerations when

developing the momentum and continuity equations.

As shown in Fig. 1, in the computational fluid domain X, the fluid grid is represented by the time-

invariant position vector x; while the material points of the structure in the initial solid domain Xs0 and the

current solid domain Xs are represented by Xs and xsðXs; tÞ, respectively. The superscript s is used in the

solid variables to distinguish the fluid and solid domains.

In the fluid calculations, the velocity v and the pressure p are the unknown fluid field variables; whereas

the solid domain involves the calculation of the nodal displacement us, which is defined as the difference of

the current and the initial coordinates: us ¼ xs � Xs. The velocity vs is the material derivative of the dis-

placement dus=dt. The definitions of the spatial coordinate, displacement, velocity, and acceleration in both

fluid and solid domains are summarized in Table 1. Since displacement and acceleration are not explicitly

used in the fluid calculations, they are not defined under the column of the fluid domain.

2.2. The equations of motion

In this section, we start by introducing the equations of motion for a continuum that contains both fluid

and solid. The governing equations of both domains are then extracted from the equations of motion along

Fig. 1. The Eulerian coordinates in the computational fluid domain X are described with the time-invariant position vector x; the solid

positions in the initial configuration Xs0 and the current configuration Xs are represented by Xs and xsðXs; tÞ, respectively. The dis-

placement of the solid is defined as us ¼ xs � Xs.

Table 1

IFEM nomenclature

Fluid domain ðXÞ Solid domain ðXsÞSpatial coordinate x xs

Displacement – us ¼ xs � Xs

Velocity v vs ¼ dus=dt ¼ _us

Acceleration – as ¼ d2us=dt2 ¼ €us


with a delta function for the coupling between fluid and solid domains. The detailed mathematical deri-vation and the proof of equivalency between strong and weak forms will be illustrated in a separate paper.

In a continuum, the inertial force of a particle is balanced with the derivative of the Cauchy stress r and

the external force fext exerted on the continuum:

qdvidt

¼ rij;j þ f exti : ð2Þ

If the solid density qs is different from the fluid density qf , i.e. q ¼ qs in Xs and q ¼ qf in Xf , we can divide

the inertial force into two components within the solid domain Xs:

qdvidt

¼qf

dvidt

; x 2 X=Xs;

qfdvidt

þ ðqs � qfÞ dvidt

; x 2 Xs:

8><>: ð3Þ

Note that to illustrate the point of addition and subtraction of sub-domains, we represent the fluid domain

Xf as the entire domain X minus the solid domain Xs, namely, X=Xs. Moreover, in this paper, the externalforce f ext

i is illustrated as the gravitational force and likewise, we obtain

f exti ¼ qfgi; x 2 X=Xs;

qfgi þ ðqs � qfÞgi; x 2 Xs:

�ð4Þ

Since the computational fluid domain is the entire domain X, we ignore the hydrostatic pressure and rewrite

Eq. (4) as

f exti ¼ 0; x 2 X=Xs;

ðqs � qfÞgi; x 2 Xs:

�ð5Þ

Furthermore, the derivative of the Cauchy stress in Eq. (2) can also be decomposed as:

rij;j ¼rfij;j; x 2 X=Xs;

rfij;j þ rs

ij;j � rfij;j; x 2 Xs:

(ð6Þ

Note that in Eq. (6) the fluid stress in the solid domain in general is much smaller than the corresponding

solid stress [8]. Let us now define the fluid–structure interaction force within the domain Xs as f FSI;si , where

FSI stands for fluid–structure interaction:

f FSI;si ¼ �ðqs � qfÞ dvi

dtþ rs

ij;j � rfij;j þ ðqs � qfÞgi; x 2 Xs: ð7Þ

Naturally, the interaction force f FSI;si in Eq. (7) is calculated with the Lagrangian description. Moreover, a

Dirac delta function d is used to distribute the interaction force from the solid domain onto the compu-tational fluid domain:


f FSIi ðx; tÞ ¼

ZXs

f FSI;si ðXs; tÞdðx� xsðXs; tÞÞdX: ð8Þ

Hence, the governing equation for the fluid can be derived by combining the fluid terms and the interaction

force as:

qf dvidt

¼ rfij;j þ f FSI

i ; x 2 X: ð9Þ

Since we consider the entire domain X to be incompressible, we only need to apply the incompressibility

constraint once in the entire domain X:

vi;i ¼ 0: ð10ÞTo delineate the Lagrangian description for the solid and the Eulerian description for the fluid, we

introduce different velocity field variables vsi and vi to represent the motions of the solid in the domain Xs

and the fluid within the entire domain X. The coupling of both velocity fields is accomplished with the Dirac

delta function:

vsi ðXs; tÞ ¼ZXviðx; tÞdðx� xsðXs; tÞÞdX: ð11Þ

Finally, with the variables in the solid domain Xs defined with the Lagrangian description and the fluid

domain X with the Eulerian description, we can summarize the derivations of the governing equations as

well as the interpolation and distribution processes as follows:

qf dvidt

¼ rfij;j þ f FSI

i ; in X; ð12aÞ

vi;i ¼ 0; in X; ð12bÞ

f FSIi ðx; tÞ ¼

ZXs

f FSI;si ðXs; tÞdðx� xsðXs; tÞÞdX; ð12cÞ

f FSI;si ¼ �ðqs � qfÞ dv

si

dtþ rs

ij;j � rfij;j þ ðqs � qfÞgi; in Xs; ð12dÞ

vsi ðXs; tÞ ¼

ZXviðx; tÞdðx� xsðXs; tÞÞdX; ð12eÞ

where the velocity in the solid domain is simply denoted as vs instead of v.

3. Governing equations and discretizations

In this section, we discuss in detail the strong and weak forms of governing equations for both fluid and

solid domains. In particular, using the kinematic and dynamic matching along the fluid–structure interface,

for brevity, we exclude the fluid–structure interface boundary integral terms in the corresponding weak

forms for both fluid and solid domains.

3.1. Fluid dynamics

This section explains in detail the 3D finite element fluid solver used in the IFEM formulation. Thederivations include the strong form, namely, the Navier–Stokes equations, the Petrov–Galerkin weak form,

and the corresponding discretization.


3.1.1. Strong form

For the Eulerian description of the fluid, the material time derivative dvi=dt is expressed as the time

derivative and the convective terms:

dvidt

¼ vi;t þ vjvi;j: ð13Þ

By substituting Eq. (13) into Eq. (12a), the governing equations for the fluid domain X are written as

qfðvi;t þ vjvi;jÞ ¼ rfij;j þ f FSI

i ; ð14aÞ

vi;i ¼ 0: ð14bÞ

Moreover, the Cauchy stress tensor rfij is also defined as

rfij ¼ �pdij þ 2lsij; ð15aÞ

sij ¼ 12ðvi;j þ vj;iÞ; ð15bÞ

where l is the dynamic viscosity and p is the pressure.

In addition to these equations of motion, we stipulate the following boundary conditions:

vi ¼ gi; x 2 Cgi ; ð16aÞ

rfijnj ¼ hi; x 2 Chi ; ð16bÞ

where gi and hi are given functions.

For each spatial degree of freedom i, we assume that the fluid boundary C can be partitioned as

C ¼ Cgi [ Chi , with Cgi \ Chi ¼ ; (i.e. the Dirichlet and Neumann boundaries span the entire fluid surface

but do not intersect).

3.1.2. Petrov–Galerkin weak form

For the fluid domain, transforming the strong form of the conservation equations into the weak formrequires two test functions: the velocity test function dvi and the pressure test function dp for the

momentum equation and the continuity equation, respectively. To reduce or eliminate the numerical

oscillations in the velocity and pressure solutions from the standard Galerkin method for incompressible

materials, we employ the test functions d~vi and d~p along with the stabilization parameters sm and sc. Thealgorithm follows the stabilized equal-order finite element formulation developed in Refs. [11–13]:

d~vi ¼ dvi þ smvkdvi;k þ scdp;i; ð17aÞ

d~p ¼ dp þ scdvi;i; ð17bÞwhere sm and sc are the scalar stabilization parameters dependent of the computational grid, time step size,

and flow variables.The weak form of the continuity equation (14b) is obtained by multiplying the pressure test function d~p

and integrating over X,ZXðdp þ scdvi;iÞvj;j dX ¼ 0: ð18Þ

Similarly, the weak form of the momentum equation (14a) is achieved by multiplying the velocity testfunction d~vi,


ZXðdvi þ smvkdvi;k þ scdp;iÞ qfðvi;t

�þ vjvi;jÞ � f FSI

i

�dX

�ZXdvirf

ij;j dX�Xe

ZXe

ðsmvkdvi;k þ scdp;iÞrfij;j dX ¼ 0: ð19Þ

Applying integration by parts and the divergence theorem, we can rewrite Eq. (19) asZXðdvi þ smvkdvi;k þ scdp;iÞ qfðvi;t


i

�dX�

ZChi

dvihi dC

þZXdvi;jrf

ij dX�Xe

ZXe

ðsmvkdvi;k þ scdp;iÞrfij;j dX ¼ 0; ð20Þ

where individual finite elements are denoted with the subscript e and the stabilization supplementary termsare evaluated at the element interiors.

The detailed illustrations of the stabilized finite element formulations are available in Refs. [11–13]. Let

us assume that there is no traction applied on the fluid boundary, i.e.RChi

dvihi dC ¼ 0, the final weak form

yieldsZXðdvi þ smvkdvi;k þ scdp;iÞ qfðvi;t


i

�dXþ

ZXdvi;jrf

ij dX

�Xe

ZXe

ðsmvkdvi;k þ scdp;iÞrfij;j dXþ

ZXðdp þ scdvi;iÞvj;j dX ¼ 0: ð21Þ

The nonlinear systems are solved with the Newton–Raphson method. Moreover, to improve the compu-

tation efficiency, we also employ the GMRES iterative algorithm and compute the residuals based on

matrix-free techniques [9,21].

3.1.3. Finite element discretization

The discretization process using finite elements follows the procedures in Ref. [10]. The velocity andpressure functions, vi and p, along with the test functions dvi and dp, are interpolated as:

vhi ¼XI

NIdiI ; ð22aÞ

dvhi ¼XI

NIciI ; ð22bÞ

ph ¼XI

NIpI ; ð22cÞ

dph ¼XI

NIqI ; ð22dÞ

where NI is the nodal shape function at node I .Substituting the interpolations into Eq. (21), we derive

XI

ZX

ciINI

�þ smvhkciINI ;k þ scqINI ;i

�qfðvhi;th

þ vhj vhi;jÞ � f FSI;h

i

idXþ

XI

ZXciINI;jr

f ;hij dX

�XI

Xe

ZXe

smvhkciINI ;k

�þ scqINI ;i

�rf;hij;j dXþ

XI

ZXðqINI þ scciINI ;iÞvhj;j dX ¼ 0: ð23Þ


Combining the test function vectors ci and q, we have four set of equations with respect to four unknowns

at each node I :

cTi rwi ¼ 0; ð24aÞ

qT rq ¼ 0; ð24bÞ

where the residual vectors rwiI and rqI are given as follows:

rwiI ¼ZXðNI þ smvhkNI ;kÞ qfðvhi;t

hþ vhj v

hi;jÞ � f FSI;h

i

idXþ

ZXNI ;jr

f;hij dX

�Xe

ZXe

smvhkNI ;krf;hij;j dXþ

ZXscNI ;ivhj;j dX; ð25Þ

rqI ¼ZXscNI;i q

fðvhi;th

þ vhj vhi;jÞ � f FSI;h

i

idX�

Xe

ZXe

scNI;irf;hij;j dXþ

ZXNIvhj;j dX: ð26Þ

The pressure and velocity unknowns are solved by minimizing the residual vectors in Eqs. (25) and (26)

iteratively in each time step.

3.2. Structural mechanics

In this section, we start with the governing equation of the solid presented in Section 2. The weak form

and its discretization are then derived.

3.2.1. Strong form

The solid domain Xs is solved using the Lagrangian description. The material time derivative dvsi=dt cantherefore be directly introduced as €usi and Eq. (7) yields

ðqs � qfÞ€usi ¼ rsij;j � rf

ij;j þ ðqs � qfÞgi � f FSI;si : ð27Þ

The force balance equation in Eq. (27) is the linear momentum equation in the updated Lagrangian

description. The interaction force fFSI;s can be treated as an additional body force in the solid continuum.Obviously, if the densities are the same for both fluid and solid, and if the solid behaves as the fluid, the

entire computational domain would behave as the fluid with f FSI;si ¼ 0.

3.2.2. Weak form

To develop the weak form, we multiply the momentum equation, Eq. (27), by a test function dui andintegrate over the current configuration Xs:Z

Xs

dui ðqs

�� qfÞ€usi �

orsij

oxj� ðqs � qfÞgi þ f FSI;s

i

�dXs ¼ 0: ð28Þ

Note that for brevity we ignore the fluid stress within the solid domain. The transformation of the weak

form from the updated Lagrangian to the total Lagrangian description is to change the integration domain

from Xs to Xs0. Since we consider incompressible fluid and solid, and the Jacobian determinant is 1 in the

solid domain, the transformation of the weak form to total Lagrangian description yieldsZXs0

dui ðqs

�� qfÞ€usi �

oPijoXj

� ðqs � qfÞgi þ f FSI;si

�dXs

0 ¼ 0; ð29Þ


where the first Piola–Kirchhoff stress Pij is defined as Pij ¼ JF �1ik rs

kj and the deformation gradient Fij asFij ¼ oxi=oXj.Using integration by parts and the divergence theorem, we can rewrite Eq. (29) asZ

Xs0

duiðqs � qfÞ€usi dXs0 þ

ZXs0

dui;jPij dXs0 �

ZXs0

duiðqs � qfÞgi dXs0 þ

ZXs0

duifFSI;si dXs

0 ¼ 0: ð30Þ

Note again that the boundary integral terms on the fluid–structure interface for both fluid and solid do-

mains will cancel each other and for brevity are not included in the corresponding weak forms.

3.2.3. Finite element discretization

The discretization of the equilibrium equation (30) is done by introducing the finite element interpola-

tions of the test and trial functions,

duhi ¼XI

NIduiI ; ð31aÞ

uhi ¼XJ

NJuiJ : ð31bÞ

By substituting the interpolation functions into Eq. (30), we can obtain the discretized form:XI

XJ

ZXs0

NIduiIðqs � qfÞNJ€usiJ dXs0 þ

XI

ZXs0

NI;jduiIPij dXs0 �

XI

ZXs0

NIduiIðqs � qfÞgi dXs0

þXI

ZXs0

NIduiIfFSI;si dXs

0 ¼ 0: ð32Þ

Using the arbitrariness of duiI , we deriveXJ

ZXs0

NIðqs � qfÞNJ€usiJ dXs0 þ

ZXs0

NI;jPij dXs0 �

ZXs0

NIðqs � qfÞgi dXs0 þ

ZXs0

NIfFSI;si dXs

0 ¼ 0: ð33Þ

Therefore, at a discrete submerged point I represented with the position vector xsIðX

sI ; tÞ, we have the

following force equilibrium:

f inertiI þ f int

iI � f extiI þ f FSI;s

iI ¼ 0; ð34Þ

where f inertiI , f int

iI , f extiI , and f FSI;s

iI stand for the inertial, internal, external, and fluid–structure interactionforces at node I , respectively.

In the current IFEM implementation, we consider a three-dimensional almost incompressible hyper-

elastic material model with the Mooney–Rivlin material description [16], in which the elastic energy po-

tential W is given as:

W ¼ C1ðJ1 � 3Þ þ C2ðJ2 � 3Þ þ j2ðJ3 � 1Þ2; ð35Þ

where C1, C2, and j are the material constants and J1, J2, and J3 are functions of the invariants of theCauchy–Green deformation tensor C defined as Cij ¼ FmiFmj.

For structures with large displacements and deformations, the second Piola–Kirchhoff stress Sij and the

Green–Lagrangian strain Eij are used in the total Lagrangian formulation:

Sij ¼oWoEij

and Eij ¼1

2ðCij � dijÞ: ð36Þ


The first Piola–Kirchhoff stress Pij can then be obtained from the second Piola–Kirchhoff stress Sij:

Pij ¼ SikFjk; ð37Þwhere the definition of Pij is employed to derive the internal force f int

iI .

Note that if the solid material model is purely incompressible as constrained in Eqs. (11) and (10), we

should have the invariant I3 ¼ 1 and its function J3 ¼ 1 based on the definition of detðCÞ ¼ 1 for incom-

pressible materials. Therefore, the third term in the definition of W should be 0. However, the divergence

free property is not strictly satisfied in the process of interpolating the velocities, which induces accumulated

numerical errors in the fluid and solid calculations and creates possible artificial stress due to the volumechange of the solid. Therefore, in the implementation, we assign the bulk modulus j to be sufficiently small

so that the last term in Eq. (35) has nearly no contribution in the energy potential. A more elaborate study

on this subject will be discussed in a separate paper.

4. Construction of delta function

The coupling between the fluid and the structure consists of two parts: the interpolation of the fluid gridvelocity vðx; tÞ onto the structural nodal velocity vsðXs; tÞ, and the distribution of the solid domain force

fFSI;sðXs; tÞ onto the equivalent fluid domain force fFSIðx; tÞ. Note again that the fluid variables are depicted

in the Eulerian coordinate x; whereas the solid variables are represented in the Lagrangian coordinate Xs.

In this section we describe the construction of the function d that is used in the interactions. Both Dirac

delta functions in the IB method [3] and EIBM [16] are implemented for uniform fluid meshes. However,

the RKPM delta function employed in IFEM can handle both uniform and nonuniform grids [19]. With

this improvement, we are able to deal with fluid domains with arbitrary geometries.

As illustrated in Ref. [19], both wavelet and smooth particle hydrodynamics (SPH) methods belong to aclass of reproducing kernel methods where the reproduced function uRðxÞ is derived as

uRðxÞ ¼Z þ1

�1uðyÞ/ðx� yÞdy; ð38Þ

with a projection operator or a window function /ðxÞ.The reproducing condition requires that up to nth order polynomial can be reproduced, i.e.,

xn ¼Z þ1

�1yn/ðx� yÞdy: ð39Þ

To satisfy the consistency condition, a correction function Cðx; x� yÞ is introduced in the finite domain

of influence or support, so that the window function yieldsZXCðx; x� yÞ 1

r/

x� yr

� dX ¼ 1; ð40Þ

where r is the dilation parameter or refinement of the window function. The additional constant, r�1, scales

the window function so that the integral over the domain of support equals one.The discretized reconstruction of the delta function for nonuniform spacing can be written as

/IðxÞ ¼ Cðx; x� xIÞ1

r/

x� xIr

� DxI : ð41Þ

The window function is used as the delta function for the interpolation of the nodal velocities and the

distribution of the nodal forces between the fluid and solid domains. To ensure the conservation of energy,

the delta functions in both distribution and interpolation procedures must be the same at each time step.


In particular, for each solid point I at its current position xsIðtÞ, we construct an influence domain X/I

that covers a subset of the fluid grid nodes J positioned at xJ . To define the influence domain for each solid

point I , a search algorithm is needed at each time step during the calculation of the RKPM shape functions.

When nonuniform fluid mesh is used, the size of the influence domain for each solid point is dependent of

the definition of the RKPM shape function in Eq. (41) and fluid mesh density.

Therefore, in the interpolation process from the fluid onto the solid grid, the discretized form of Eq. (11)

can be written as

vsiI ¼XJ

viJ ðtÞ/J ðxJ � xsIÞ; xJ 2 X/I : ð42Þ

Here, the solid velocity vsI at node I can be calculated by gathering the velocities at fluid nodes within the

influence domain X/I . A dual procedure takes place in the distribution process from the solid onto the fluid

grid. The discretized form of Eq. (8) is expressed as

f FSIiJ ¼

XI

f FSI;siI ðXs; tÞ/IðxJ � xs

IÞ; xsI 2 X/J : ð43Þ

By interpolating the fluid velocities onto the solid particles in Eq. (42), the fluid within the solid domain

is bounded to solid material points. This ensures not only the no-slip boundary condition on the surface ofthe solid, but also stops automatically the fluid from penetrating the solid, provided the solid mesh is at

least two times denser than the surrounding fluid mesh. This heuristic criterion is based on the numerical

evidence and needs further investigation.

5. Computational algorithm

The governing equations of IFEM in discretized form (except the Navier–Stokes equations, for con-venience) are summarized as follows:

f FSI;siI ¼ �f inert

iI � f intiI þ f ext

iI ; in Xs; ð44aÞ

f FSIiJ ¼

XI

f FSI;siI ðXs; tÞ/IðxJ � xs

IÞ; xsI 2 X/J ; ð44bÞ

qfðvi;t þ vjvi;jÞ ¼ rij;j þ qgi þ f FSIi ; in X; ð44cÞ

vj;j ¼ 0; in X; ð44dÞ

vsiI ¼XJ

viJ ðtÞ/J ðxJ � xsIÞ; xJ 2 X/I : ð44eÞ

An outline of the IFEM algorithm with a semi-explicit time integration is illustrated as follows:

(1) Given the structure configuration xs;n and the fluid velocity vn at time step n,(2) Evaluate the nodal interaction forces fFSI;s;n for solid material points, using Eq. (44a),

(3) Distribute the material nodal force onto the fluid lattice, from fFSI;s;n to fFSI;n, using the delta function as

in Eq. (44b),

(4) Solve for the fluid velocities vnþ1 and the pressure pnþ1 implicitly using Eqs. (44c) and (44d), described in

Section 3.1,

(5) Interpolate the velocities in the fluid domain onto the material points, i.e., from vnþ1 to vs;nþ1 as in Eq.(44e), and

(6) Update the positions of the structure using us;nþ1 ¼ vs;nþ1Dt and go back to step 1.


Note that even though the fluid is solved fully implicitly, the coupling between fluid and solid is explicit.If we rewrite the fluid momentum equation (for clarity only discretized in time), we have

qf vmþ1i � vmi

Dt

�þ vmþ1

j vmþ1i;j

�¼ rf ;mþ1

ij;j þ f FSI;mi : ð45Þ

It is clear that the interaction force is not updated during the iteration, i.e., the solid equations are cal-

culated with values from the previous time step. For a fully implicit coupling, this force must be a function

of the current fluid velocity and the term f FSI;mþ1i should be included into the linearization of the fluid

equations.

6. Numerical examples

In this section, we present the simulations on particulate flows with both rigid and deformable spheres

using the proposed IFEM method. The algorithm requires no mesh update or remeshing and the structures

are not restricted to only rigid cases as explained in Section 2. The demonstrated problems are: a rigid

sphere and a soft sphere dropping between two planes and multiple soft spheres dropping in a square

channel. The sphere flexibility is controlled by adjusting the material properties. Naturally, as the modulus

increases, the sphere behaves more like a rigid body.

6.1. A rigid sphere dropping between two planes

In this section, we discuss the case with a rigid sphere dropping in a channel. Since a rigid structure is

considered, the material modulus is set to be high. The channel has a size of 2 · 2 · 8 cm. The diameter of

the sphere is 0.4 cm. The properties for both fluid and solid as well as the discretization for each mesh are

summarized in Table 2.

As illustrated in Fig. 2, the use of the nonuniform fluid mesh provides the unique ability and the

advantage of IFEM over the IB method as well as EIBM.The interaction force fFSI is calculated using Eq. (44a). As shown in Fig. 3, the pressure and velocities in

the fluid domain are influenced by the movement of the sphere and the vortices and recirculation are clearly

captured near the sphere.

To verify the accuracy of IFEM algorithm, we compared the solution of terminal velocity with the

experimental data. It is explained in Ref. [1] that for a particle moving with steady terminal velocity UT in a

gravitational field, the drag force FD balances the difference between the weight and buoyancy:

FD ¼ gðqs � qfÞðp=6ÞD3 ð46Þand as a consequence the drag coefficient CD is calculated as

Table 2

Properties used for one rigid sphere dropping in a channel

Fluid 4987 nodes qf ¼ 1 g/cm3

26 030 elements l ¼ 0:01 g cm/s

Solid 997 nodes qs ¼ 2:02 g/cm3 C1 ¼ 2:93� 105 g/(cm s2)

864 elements D ¼ 0:4 cm C2 ¼ 1:77� 105 g/(cm s2)

g ¼ 980 cm/s2

Fig. 2. A gradient mesh is constructed with respect to the distance from the submerged sphere.

Fig. 3. Velocity vectors and pressure contours in the fluid domain at different time steps. (a) t ¼ 0:01 s; (b) t ¼ 0:1 s; (c) t ¼ 0:2 s;

(d) t ¼ 0:3 s; (e) t ¼ 0:4 s.


CD ¼ 4ðqs � qfÞgD=3qfU 2T ¼ 4qfðqs � qfÞgD3=3l2Re2T; ð47Þ

where ReT is the Reynolds number at the terminal velocity.

Since both the drag coefficient and Reynolds number are dependent of the terminal velocity UT, an

iterative procedure is needed. It is more convenient to express Re as a function of the so-called best number

ND, which is defined as

ND ¼ CDRe2T ¼ 4qfðqs � qfÞgD3=3l2: ð48ÞFinally, the Reynolds number can be expressed as a function of ND. In this case, for 5806ND 6

1:55� 107, the correlation for Re as a function of ND is expressed as

log10 Re ¼ �1:81391þ 1:34671W � 0:12427W 2 þ 0:006344W 3; ð49Þwith W ¼ log10 ND.

Fig. 4. Comparison between the experimental data and the IFEM solution.


Using the parameters shown in the above example, the experimental terminal velocity is found to be

2.203 cm/s with the terminal Reynolds number 88:11. The calculated terminal velocity from the IFEM

solution, as shown in Fig. 4, is found to be 2.24412 cm/s, which deviates only 1.87% from the experimental

value.

6.2. A soft sphere dropping in a channel

This example shows a soft sphere dropping in a viscous fluid channel. The sphere flexibility is controlledby the parameters C1 and C2 that are defined in Section 3.2 and listed in Table 3.

The deformation of the sphere as well as the stress distribution at different time steps are shown in Fig. 5.

As expected from the simulation, the sphere experiences a higher pressure at the bottom and a lower

pressure on the top as it tries to immerse deeper into the fluid. The deformation can be clearly observed as it

accelerates inside the fluid domain, shown in Fig. 5.

6.3. 20 Soft spheres dropping in a channel

This example solves 20 soft spheres dropping in a channel. These same-sized spheres are placed ran-

domly near the top of the channel. The spheres experience the gravitational and internal forces as well as

the interaction forces with the surrounding fluid. The properties used in this problem can be found in

Table 4.

Table 3

Properties used for one soft sphere dropping in a channel


22 557 elements l ¼ 10 g cm/s

Solid 997 nodes qs ¼ 2 g/cm3 C1 ¼ 2:93� 102 g/(cm s2)

864 elements D ¼ 0:5 cm C2 ¼ 1:77� 102 g/(cm s2)

g ¼ 980 cm/s2

Fig. 5. Von Mises stress of the soft sphere at different time steps as it drops in a channel. (a) t ¼ 0 s; (b) t ¼ 7:50 s; (c) t ¼ 11:25 s;

(d) t ¼ 15:00 s; (e) t ¼ 18:75 s.

Table 4

Properties for 20 soft spheres dropping in a channel


22 557 elements l ¼ 10:0 g cm/s

Solid 20 · 997 nodes qs ¼ 2 g/cm3 C1 ¼ 2:93� 101 g/(cm s2)

20 · 864 elements D ¼ 0:5 cm C2 ¼ 1:77� 101 g/(cm s2)

g ¼ 980 cm/s2

Fig. 6. The movement of 20 spheres dropping in a channel at different time steps. (a) t ¼ 0 s; (b) t ¼ 1:0 s; (c) t ¼ 2:0 s; (d) t ¼ 3:0 s;

(e) t ¼ 4:0 s.


Fig. 7. The Von Mises stress distribution and deformation of 20 spheres dropping in a channel at different time steps. (a) t ¼ 0:0 s;

(b) t ¼ 1:0 s; (c) t ¼ 2:0 s; (d) t ¼ 3:0 s; (e) t ¼ 4:0 s.


The movements of the spheres at different time steps are shown in Fig. 6. To clearly illustrate the

deformations of the spheres, we present in Fig. 7 the enlarged images of the spheres and the pressure

distributions at different time steps.

7. Conclusions

In this paper, we proposed the IFEM to solve complex fluid–structure interaction problems. In com-

parison with the IB method, we demonstrate the following advantages of IFEM.

The structural models in IFEM are not restricted to one-dimensional volumeless structures such as fi-

bers; instead, they may occupy finite volumes in the fluid. Finite elements are used to discretize the sub-

merged solid and as a consequence the calculated stress are more realistic and accurate. In addition, inIFEM, different material laws can be assigned to the submerged solid.

The higher-ordered RKPM delta function has the ability to handle nonuniform grids and is particularly

useful for fluid domains with arbitrary geometries and boundaries. Finally, unlike the periodic boundary

conditions implemented in most IB applications, essential and natural boundary conditions in the context

of finite element methods are relatively easy to implement in IFEM.

Acknowledgements

We would like to thank Prof. C.S. Peskin in the Courant Institute of Mathematical Sciences at New

York University and Prof. N.A. Patankar at Northwestern University for the helpful discussions and

suggestions. The authors are sponsored by NASA Langley.

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immersed finite element method

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