research article computational methods for coupled fluid-structure...

Research ArticleComputational Methods for CoupledFluid-Structure-Electromagnetic Interaction Models withApplications to Biomechanics

Felix Mihai, Inja Youn, Igor Griva, and Padmanabhan Seshaiyer

Department of Mathematical Sciences, College of Science, George Mason University, 4400 University Drive,MS 3F2, Fairfax, VA 22030, USA

Correspondence should be addressed to Padmanabhan Seshaiyer; [email protected]

Received 29 August 2014; Accepted 15 November 2014

Academic Editor: Kim M. Liew

Copyright © 2015 Felix Mihai et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Multiphysics problems arise naturally in several engineering and medical applications which often require the solution to coupledprocesses, which is still a challenging problem in computational sciences and engineering. Some examples include blood flowthrough an arterial wall and magnetic targeted drug delivery systems. For these, geometric changes may lead to a transient phasein which the structure, flow field, and electromagnetic field interact in a highly nonlinear fashion. In this paper, we considerthe computational modeling and simulation of a biomedical application, which concerns the fluid-structure-electromagneticinteraction in themagnetic targeted drug delivery process. Our study indicates that the strongmagnetic fields, which aid in targeteddrug delivery, can impact not only fluid (blood) circulation but also the displacement of arterial walls. A major contribution of thispaper is modeling the interactions between these three components, which previously received little to no attention in the scientificand engineering community.

1. Introduction

In the last decade, the rapid development of computationalscience has provided new methodologies to solve complexmultiphysics applications involving fluid-structure interac-tion to a variety of fields. These include solving applicationsinvolving blood flow interactions with the arterial wall tocomputational aeroelasticity of flexible wing micro-air vehi-cles to magnetohydrodynamic of liquid-metal cooled nuclearreactor to ferromagnetics with biological applications. Inthese applications, the challenge is to understand and developalgorithms that allow the structural deformation, the flowfield, and temperature variations to interact in a highlynonlinear fashion.

Coupling these multiphysics with electromagnetic effectsmakes the associated computational model too complex. Notonly is the nonlinearity in the geometry challenging butin many of these applications the material is nonlinear aswell, which makes the problem even more complex. Direct

numerical solution of the highly nonlinear equations gov-erning even the most simplified two-dimensional models ofsuch multiphysics interaction requires that all the unknownfields, such as fluid velocity, pressure, the magnetic and theelectric field, the temperature field, and the domain shape, bedetermined as part of the solution, since neither is known apriori.

The past few decades, however, have seen significantadvances in the development of finite element and domaindecomposition methods. These have provided new algo-rithms for solving such large scale multiphysics simulations.There have been several methods that have been introducedin this regard and their performance has been analyzed fora variety of problems. One such technique is the mortarfinite element method which has been shown to be stablemathematically and has been successfully applied to a varietyof applications and references therein. The basic idea isto replace the strong continuity condition at the inter-faces between the different subdomains modeling different

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 253179, 10 pageshttp://dx.doi.org/10.1155/2015/253179

2 Mathematical Problems in Engineering

multiphysics by a weaker one to solve the problem in acoupled fashion. Such novel techniques provide hope forus to develop new faster and efficient algorithms to solvecomplexmultiphysics applications. A variety ofmethods havebeen introduced including the level set methods [1], fictitiousdomain methods [2, 3], nonconforming hp finite elementmethods [4, 5], multilevel multigrid methods [6], and theimmersed boundary methods [7]. While these methods helpenhance our ability to understand complex processes, thereis still a great need for efficient computational methods thatcannot only help simulate physiologically realistic situationsqualitatively but also analyze and studymodeling of such pro-cesses quantitatively. Such multiphysics applications involvethe interaction of various components, such as fluid with thestructure, electromagnetics with the fluid, or fluid-structureinteracting completely with electromagnetics.

1.1. Electromagnetic-Fluid Interaction. An important appli-cation involving interaction of electromagnetics with fluidwhich describes the behavior of electrically conducting fluidis very complex under a magnetic field, since the additionalLorentz force is caused by the interaction between velocityfield and electromagnetic field. Understanding such coupledbehavior not only helps us to create efficient algorithms butalso applies to a variety of magnetohydrodynamic (MHD)applications. Due to its multidisciplinary applications, a solidunderstanding of the MHD is required. In this regard, theHartmann flow has been studied extensively. The Hartmannflow is the steady flow of an electrically conducting fluidbetween two parallel walls, under the effect of a normalmagnetic and electric field. A thorough understanding ofsuch models for electromagnetic fluid interaction can helpus in developing new techniques for complex problems suchas magnetic drug targeting in cancer therapy. Such a modelwould involve ferrohydrodynamics of blood that helps tostudy external magnetic field and its interaction with bloodflow containing a magnetic carrier substance. The analyticmodels would involve solving Maxwell’s equations in con-junction with Navier-Stokes equations. While new models inthis area are just starting to evolve, these often consider thestructure to be fixed. There is a need to extend these modelsto include fluid-structure interaction with electromagnetics,which would be another focus of this work.

1.2. Proposed New Models. In this paper, we will developa computational infrastructure for solving coupled fluid-structure interaction with electromagnetic and temperatureeffects. The rest of the work is organized as follows. Section 2presents the models, methods, and background requiredto develop and solve the coupled multiphysics systems.In Section 3, we consider the model of a blood vessel, apermanent magnet, and surrounding tissue and air in twodimensions. We will consider both a nonmoving structureand a moving structure. The deformed structure provides anew geometry, where the Navier-Stokes equations are solvedfor the velocity andpressure fields in the bloodstream.Amag-netic vector potential generated by the permanent magnet iscalculated, which in turn creates amagnetic volume force that

Permanentmagnet

Blood flow

Blood vessel wall

Blood vessel wall

System boundary

Figure 1: Electromagnetic fluid-structure interaction model.

impacts the flow in the blood vessel. The flow field changesthe displacement of the structure, and the problem is solvedonce again for the new geometry. The proposed modelsare validated against benchmark applications numerically.Section 4 presents conclusion and a discussion of the results.Future work on the proposed problems is also presented.

A magnetically targeted drug delivery system [8] isbased on magnetic particles under the action of an externalmagnetic field. This is becoming an increasingly effectiveapproach in drug therapy. As this field has evolved in thelast decade, lots of scientific interest led to this inquiry intoefficient computational models that simulate this experimen-tal process [9]. Our study indicates that the strong magneticfields which aid in targeted drug delivery can impact not onlyfluid (blood) circulation but also the displacement of arterialwalls. Thus, it is important to have a model, which includesthe interactions between fluid, structure, and magnetic fieldin order to study and optimize drug delivery.

In this section, we will present a model that describesthe interaction between these three components, whichpreviously received little to no attention in the scientificcommunity. To develop an electromagnetic fluid-structureinteraction, we incorporate the effects of the electromagneticfield into a fluid-structure model. Gaining a thorough under-standing of such a coupled model can help us to understandthe efficacy of magnetic nanoparticle-based drug deliveryfor diseases such as cancer as has been proposed by variousresearchers [10, 11].There is significant evidence that indicatesa need for more promising models which overcome currentlimitations and improve magnetic targeting technique.

2. Mathematical Model and GoverningEquations

The model we consider is a blood vessel with a permanentmagnet near its surface, as illustrated in Figure 1. For sim-plicity of presentation, we consider a computational modelthat comprises three components. Let the computationaldomain Ω ⊂ R2 be an open set with global system boundaryΓ. Let Ω be decomposed into the four disjoint open sets,a fluid subdomain Ω

𝑓denoted by blood flow, two solid

Mathematical Problems in Engineering 3

subdomains Ω𝑖𝑠, 𝑖 = 1, 2 (blood vessel walls) with respective

boundaries Γ𝑓and Γ𝑠, and one electromagnetic domain Ω

𝑚

(permanent magnet). Let Γ𝑗𝐼, 𝑖 = 1, 2, 3, 4 be the interface

between the solid, fluid, and electromagnetic domains. Thestructural domain consists of two symmetric arterial vesselwalls denoted by Ω1

𝑠and Ω

2

𝑠. The electromagnetic domain

consists of a permanentmagnet of dimensions 10 𝜇m×40 𝜇mplaced in free space. The arterial wall describes a structuralmechanism that interacts with the flow dynamics of bloodwhich in turn is impacted by a permanent magnet, which isdescribed next.

For this, we use Maxwell’s equation for the magnetostaticcase (the field quantities do not vary with time) that relatesthemagnetic field intensityH and the electric current densityJ [12]:

∇ ×H = J,

∇ ⋅ J = 0.(1)

The constitutive relations between B and H depend on thedomain [12, 13]:

B =

{{

{{

{

𝜇0𝜇𝑟,magH + Brem for the permanent magnet

𝜇0(H +M

𝑓𝑓(H)) for the blood stream

𝜇0H for the tissue and air,

(2)

where 𝜇0

is the magnetic permeability of vacuum(V⋅s/(A⋅m)), 𝜇

𝑟,mag is the relative magnetic permeability ofthe permanent magnet (dimensionless),Brem is the remanentmagnetic flux (A/m), andM

𝑓𝑓is the magnetization vector in

the blood stream (A/m), which is a function of the magneticfield,H. By defining a magnetic vector potential A such that

B = ∇ × A, with ∇ ⋅ A = 0, (3)

we get

∇ × (1

𝜇∇ × A −M) = J. (4)

Assuming no perpendicular currents, we can simplify to a 2Dproblem and reduce this equation to

∇ × (1

𝜇0

∇ × A −M) = 0. (5)

This assumes that themagnetic vector potential has a nonzerocomponent only perpendicularly to the plane, which isA = (0, 0, 𝐴

𝑧). The induced magnetization M

𝑓𝑓(𝑥, 𝑦) =

(𝑀𝑓𝑓𝑥,𝑀𝑓𝑓𝑦) is characterized by [14–17]

M𝑥= 𝛼 arctan(

𝛽

𝜇0

𝜕𝐴𝑧

𝜕𝑦) ,

M𝑦= 𝛼 arctan(

𝛽

𝜇0

𝜕𝐴𝑧

𝜕𝑥) .

(6)

To capture the magnetic fields of interest we can linearizethese expressions to obtain

M𝑥=𝜒

𝜇0

𝜕𝐴𝑧

𝜕𝑦, M

𝑦=𝜒

𝜇0

𝜕𝐴𝑧

𝜕𝑥, (7)

where 𝜒 = 𝛼𝛽 is the magnetic susceptibility. This magneticfield induces a body force on the fluid. With the assumptionthat the magnetic nanoparticles in the fluid do not interact,the magnetic force F = (𝐹

𝑥, 𝐹𝑦) on the ferrofluid for relatively

weak fields is given by [16]

F = |M| ∇ |H| . (8)

Substituting (2) and (3) in (8) leads to the expression

𝐹𝑥= 𝑘𝑓𝑓

𝜒

𝜇0𝜇2𝑟

(𝜕𝐴𝑧

𝜕𝑥

𝜕2𝐴𝑧

𝜕𝑥2+𝜕𝐴𝑧

𝜕𝑦

𝜕2𝐴𝑧

𝜕𝑥𝜕𝑦) ,

𝐹𝑦= 𝑘𝑓𝑓

𝜒

𝜇0𝜇2𝑟

(𝜕𝐴𝑧

𝜕𝑥

𝜕2𝐴𝑧

𝜕𝑥𝜕𝑦+𝜕𝐴𝑧

𝜕𝑦

𝜕2𝐴𝑧

𝜕𝑦2) ,

(9)

where 𝑘𝑓𝑓

is the fraction of the fluid which is ferrofluid. Thevector 𝐹

𝑓= (𝐹𝑥, 𝐹𝑦) is the volume force, which is input for

the Navier-Stokes equations in the next subsection.

2.1. Modeling the Unsteady Blood Flow. We model the fluiddomain for the blood flow via the unsteady Navier-Stokesequations for an incompressible, isothermal fluid flowwrittenin nonconservative form as

𝜌𝑓

𝜕𝑢𝑓

𝜕𝑡+ 𝜌𝑓(𝑢𝑓⋅ ∇) 𝑢𝑓+ ∇𝑝 = ∇ ⋅ 𝜏

𝑓+ 𝐹𝑓,

𝜌𝑓∇ ⋅ 𝑢𝑓= 0,

(10)

where 𝑢𝑓is the velocity, 𝜌

𝑓is the density, 𝑝 is the pressure,

and 𝐹𝑓

is the body forces. The viscous stress tensor is𝜏(𝑢𝑓) = 2𝜂𝐷(𝑢

𝑓), where 𝜂 is the dynamic viscosity and the

deformation tensor is

𝐷(𝑢𝑓) = 𝜇𝑠(

∇𝑢𝑓+ (∇𝑢

𝑓)𝑇

2) . (11)

The fluid equations are subject to the boundary conditions:

𝑢𝑓= 𝑢wall, 𝑥 ∈ Γ

𝑗

𝐼, 𝑗 = 2, 3

𝜏𝑓⋅ 𝑛 = 𝑡 ⋅ 𝑛, 𝑥 ∈ Γ

𝑁,

𝑢𝑓=𝜕𝑑𝑠

𝜕𝑡𝑥 ∈ Γ𝑗

𝐼, 𝑗 = 2, 3,

(12)

where 𝑡 = −𝑝𝐼 + 2𝐷 (𝑢𝑓) is the prescribed tractions on the

Neumann part of the boundary with 𝑛 being the outwardunit normal vector to the boundary surface of the fluid.Conditions of displacement compatibility and force equilib-rium along the structure-fluid interface are enforced. In orderto solve a fluid-structure interaction problem in a coupledfashion we employ an arbitrary Lagrangian-Eulerian (ALE)formulationwhere the characterizing velocity is no longer thematerial velocity 𝑢

𝑓, but a grid velocity ��

𝑓. This allows us to

replace the material velocity 𝑢𝑓in (10) with the convective


velocity 𝑐 = 𝑢𝑓− ��𝑓[5]. The weak variational formulation of

the fluid problem then becomes

∫Ω𝑓

𝜏𝑓⋅ ∇𝜙 𝑑Ω

𝑓

= ∫Ω𝑓

𝐹 ⋅ 𝜙 𝑑Ω𝑓+ ∫Γ𝑓

𝑡 ⋅ 𝜙 𝑑Γ

+ ∫Ω𝑓

𝜌𝑓

𝜕𝑢

𝜕𝑡⋅ 𝜙 𝑑Ω

𝑓+ ∫Ω𝑓

𝜌𝑓(𝑐 ⋅ ∇) 𝑢

𝑓⋅ 𝜙 𝑑Ω

𝑓,

∫Ω𝑓

𝑞∇ ⋅ 𝑢 𝑑Ω𝑓= 0.

(13)

2.2. Modeling the Structure Equations. The structuraldomains for the blood vessel walls consist of the arterialvessel walls denoted by Ω1

𝑠, Ω2𝑠. They are modeled via the

following equation:

𝜌𝑠

𝜕2𝑑𝑠

𝜕𝑡2= ∇ ⋅ 𝜏

𝑠+ 𝐹𝑠, (14)

where 𝑑𝑠is the structure displacement, 𝜌

𝑠is the structure

density, 𝜏𝑠is the solid stress tensor, and 𝜕2𝑑

𝑠/𝜕𝑡2 is the local

acceleration of the structure.This is solvedwith the boundaryconditions:

𝑑𝑠= 𝑑𝐷

𝑠𝑥 ∈ Γ𝐷

𝑆,

𝜏𝑠⋅ 𝑛𝑠= 𝑡𝑠

𝑥 ∈ Γ𝑁

𝑆,

𝜏𝑠⋅ 𝑛𝑠= −𝜏 ⋅ 𝑛 + 𝑡

𝐼

𝑆𝑥 ∈ Γ𝑗

𝐼𝑗 = 2, 3.

(15)

Here Γ𝐷𝑆

and Γ𝑁

𝑆are the respective parts of the structural

boundary where the Dirichlet and Neumann boundaryconditions are prescribed. Also, 𝑡

𝑆are the applied tractions

on Γ𝑁

𝑆and 𝑡

𝐼

𝑆are the externally applied tractions to the

interface boundaries Γ𝑗𝐼, 𝑗 = 1, 2, 3, 4. The unit outward

normal vector to the boundary surface of the structure is𝑛𝑠. The stresses are computed using the constitutive relation

described next. Equations (15) enforce the equilibrium ofthe traction between the fluid and the structure on therespective fluid-structure interfaces. The total strain tensorfor a typical geometrically nonlinearmodel iswritten in termsof displacement gradients:

𝜀 =1

2(∇𝑑𝑠+ ∇𝑑𝑇

𝑠+ ∇𝑑𝑠∇𝑑𝑇

𝑠) . (16)

For small deformations, the last term on the right hand sideis omitted to obtain a geometrically linear model. Since theobjective of this section is to investigate the influence ofelectromagnetic effects on fluid-structure interactionmodels,we will consider a geometrically linear model combined witha linear constitutive law. The solid stress tensor 𝜏

𝑠is given in

terms of the second Piola-Kirchoff stress 𝑆:

𝜏𝑠= (𝑆 ⋅ (𝐼 + ∇𝑑

𝑠)) . (17)

20𝜇m

20𝜇m

40𝜇m

100𝜇m

10𝜇m

300𝜇m

∙ (150, 105)

∙ (150, 95)

∙ (150, 50)

∙ (150, −5)

Figure 2: Domain and points of interest.

For the linear material model, we employ the followingconstitute law relating the stress tensor to the strain tensor:

𝑆 = 𝑆0+ 𝐶 : 𝜀, (18)

where 𝐶 is the 4th order elasticity tensor and “:” stands forthe double-dot tensor product. 𝑆

0and 𝜀0are initial stresses

and strains, respectively. The weak variational form of thestructural equations then becomes the following: find thestructure displacement 𝑢

𝑠such that

∫Ω𝑓

𝜏𝑠⋅ 𝜀𝑠𝑑Ω𝑠

= ∫Ω𝑓

𝐹𝑠⋅ 𝜙𝑠𝑑Ω𝑠+ ∫Γ𝑁𝑆

𝑡𝑠⋅ 𝜙𝑠𝑑Γ

− ∫Ω𝑓

𝜌𝑠

𝜕2𝑑𝑠

𝜕𝑡2⋅ 𝜙𝑠𝑑Ω𝑠− ∫Γ𝐼

(𝑡𝐼

𝑆− 𝜏𝑓⋅ 𝑛) ⋅ 𝜙

𝑠𝑑Γ.

(19)

3. Numerical Results

In this section, we present the numerical results for theelectromagnetic-fluid-structure interaction model problempresented in this section. To understand the effects of thecoupling between electromagnetic field and fluid-structureinteraction models better, we first consider the interactionwith a rigid structure, which is often employed in themost research problems that are only interested in study-ing the electromagnetic-fluid interaction.The computationaldomain (see Figure 2) represents a blood vessel that is 300micrometers long and 100 micrometers in diameter, withwalls 20micrometers in thickness. All the results presentedare for three magnetic fields: 0T (no magnetic field), 0.5T,and 1T. The structure model we consider is linear (MLGL),which was introduced in Section 2.

3.1. Coupled Interaction with Rigid Structure. Figures 3(a),4(a), and 5(a) illustrate the influence of the magnetic fieldon the interaction. These figures show the surface von Mises


400

350

300

250

200

150

100

50

0

−50

350300250200150100500−50

−100

−150

−2000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 0

0

00.1032

(a) Nonmoving structure

400

350

300

250

200

150

100

50

0

−50

350300250200150100500−50

−100

−150

−2000

0.01

0.02

0.03

0.04

0.05

0.06

0.2

0.4

0.6

0.8

1

1.2

0.07

0.08

0.09

0.1

0

0.103

75.734

1.3904 × 104

×104

(b) Moving structure

Figure 3: Surface von Mises stress with streamlines of spatial velocity field and magnetic field for Brem = 0T at 𝑡 = 0.215. Time = 0.215,surface: von Mises stress (N/m2), surface: velocity magnitude (m/s), and streamline: velocity field (spatial).

400

450

350

300

250

200

150

100

50

0

−50

350300250200150100500−50

−100

−150

−200

−250

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 0

0

00.1032

33.992

30.414

26.836

23.258

19.681

16.103

12.525

8.9471

5.3692

1.7914

−1.7865

−5.3643

−8.9421

−12.52

−16.098

−19.676

−23.253

−26.831

−30.409

−33.987

−3.3987 × 10−6

3.3992 × 10−6

×10−7


400

450

350

300

250

200

150

100

50

0

−50

350300250200150100500−50

−100

−150

−200

−250

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0

0.103

33.992

30.414

26.836

23.258

19.681

16.103

12.525

8.9471

5.3692

1.7914

−1.7865

−5.3643

−8.9421

−12.52

−16.098

−19.676

−23.253

−26.831

−30.409

−33.987

−3.3987 × 10−6

3.3992 × 10−6

×10−7

0.2

0.4

0.6

0.8

1

1.2

75.713

1.389 × 104

×104


Figure 4: Surface von Mises stress with streamlines of spatial velocity field and magnetic field for Brem = 0.5T at 𝑡 = 0.215. Time =0.215, surface: von Mises stress (N/m2), surface: velocity magnitude (m/s), contour: magnetic vector potential, 𝑧 component (Wb/m), andstreamline: velocity field (spatial).

stress along with streamlines of spatial velocity field and the𝑧-component of the magnetic vector potential. While thereis no significant impact of increasing the magnetic field onthe velocity profile in each of the graphs in Figures 3(a), 4(a),and 5(a), the impact on the magnetic vector potential is as

expected. As it can be seen, the 𝑧-component of the magneticpotential doubles when magnetic field doubles.

Figures 6(a), 7(a), and 8(a) compare the effect of varyingthe magnetic field on the surface pressure. Unlike the impacton the velocity profile, these figures suggest that the surface


400

350

300

250

200

150

100

50

0

−50

300250200150100500

−100

−150

−200

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 0

0

00.1032

67.984

60.828

53.672

46.517

39.361

32.205

25.05

17.894

10.738

3.5828

−3.5729

−10.729

−17.884

−25.04

−32.196

−39.351

−46.507

−53.663

−60.818

−67.974

−6.7974 × 10−6

6.7984 × 10−6

×10−7


400

450

350

300

250

200

150

100

50

0

−50

350300250200150100500−50

−100

−150

−200

−250

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0

0.103

0.2

0.4

0.6

0.8

1

1.2

1.4

75.6

1.4126 × 104

×104

67.984

60.828

53.672

46.517

39.361

32.205

25.05

17.894

10.738

3.5828

−3.5729

−10.729

−17.884

−25.04

−32.196

−39.351

−46.507

−53.663

−60.818

−67.974

−6.7974 × 10−6

6.7984 × 10−6

×10−7


Figure 5: Surface vonMises stress with streamlines of spatial velocity field andmagnetic field forBrem = 1T at 𝑡 = 0.215. Time= 0.215, surface:von Mises stress (N/m2), surface: velocity magnitude (m/s), contour: magnetic vector potential, 𝑧 component (Wb/m), and streamline:velocity field (spatial).

0

0

5

10

15

20

25

30

35

40

45

46.609

350

300

250

200

150

100

50

0

−50

350300250200150100500−50

−100

−150


0

0

5

10

15

20

25

30

35

40

40.587

350

300

250

200

150

100

50

0

−50

350300250200150100500−50

−100

−150


Figure 6: Pressure for Brem = 0T shown at 𝑡 = 4. Time = 4; surface: pressure (Pa).

pressure is impacted by increasing the magnetic field and thedoubling effect is also seen as expected.

3.2. Coupled Interaction with Moving Structure. Next, weconsider the benchmark problem presented with the struc-ture moving. For this, we employ the ALE formulation forthe fluid-structure interaction as described in Section 2. Wenotice from Figures 3(b), 4(b), and 5(b) that, at 𝑡 = 0.215

(when the fluid velocity has maximum value), the structureand the flow pattern are not very much impacted by the mag-net. For the maximum studied magnetic field of 1T, the arte-rial wall is slightly bent towards the magnet. For even largermagnetic fields not shown in the picture (the order of mag-netic field of 5T), the magnet intersects with the arterial wall.

Even though we have not seen a big difference instructural deformation and fluid flow for our study case, the


0

0

5

10

15

20

25

30

35

40

45

47.129

350

300

250

200

150

100

50

0

−50

350300250200150100500−50

−100

−150


0

40.336

350

300

250

200

150

100

50

0

−50

350300250200150100500−50

−100

−1500

5

10

15

20

25

30

35

40


Figure 7: Pressure for Brem = 0.5T shown at 𝑡 = 4. Time = 4; surface: pressure (Pa).

0

10

20

30

40

50

60

70

80

90

100

100.78

350

300

250

200

150

100

50

0

−50

350300250200150100500−50

−100

−150

−0.0369


0

0

10

20

30

40

50

60

70

80

90

98.84

350

300

250

200

150

100

50

0

−50

350300250200150100500−50

−100

−150


Figure 8: Pressure for Brem = 1T shown at 𝑡 = 4. Time = 4; surface: pressure (Pa).

fluid pressure is entirely different between two consideredmagnetic fields (see Figures 6(b), 7(b), and 8(b)). If forBrem =

0T the pressure is completely symmetric with respect tothe 𝑥-axis, the pressure around the magnet increases whenmagnetization is 0.5T and becomes more than double themaximum pressure in the rest of the fluid when Brem = 1T.

Another experiment we perform is to measure the veloc-ity profile and displacement of two specific points. FromFigures 9 and 10, we notice that, as expected, the velocityand pressure decrease at the center and increase around the

boundaries when the structure is moving, mainly becauseof the dilatation of the structure. While the pressure inthe center is not affected much by the presence or absenceof magnetic field, near the magnet the pressure is steadilyincreasing with the time.

For the measured displacement, we notice in Figure 11that the wall towards the magnet is getting closer to themagnet because of the increasing pressure, while the otherwall is virtually unaffected by the presence of the magneticfield.


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

15

20

25

30

35

40

45

50

t (s)

Pres

sure

(Pa)

Moving mesh, B = 0TMoving mesh, B = 0.5 T

Nonmoving mesh, B = 0TNonmoving mesh, B = 0.5 T

(a) Velocity profile for coordinates (150, 50)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

t (s)

Velo

city

mag

nitu

de (m

/s)



(b) Velocity profile for coordinates (150, 95)

Figure 9: Velocity for a center and edge point inside the fluid.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

15

20

25

30

35

40

45

50

t (s)

Pres

sure

(Pa)



(a) Pressure profile for coordinates (150, 50)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

15

20

25

30

35

40

45

50

t (s)

Pres

sure

(Pa)



(b) Pressure profile for coordinates (150, 95)

Figure 10: Pressure for a center and edge point inside the fluid.

4. Conclusion

In this work, we presented the computational modelingand simulation of coupled multiphysics applications. Theseincluded a variety of processes such as fluid dynamics,structural mechanics, and electromagnetic interaction thatimpacted the behavior of the physical system in a coupledway. Specifically, this work considered the research questionof “how does incorporating electromagnetic field into fluid-structure interaction models influence the fluid flow andstructural deformation?” In answering this question, this

work led to the development of a two-dimensional multi-physics problem involving electromagnetics coupled withfluid-structure interaction.

In order to answer this research question, we first pre-sented the mathematical background and simulation of theinteraction between fluid, structure, and magnetic field. Themotivation of this came from researchingmodels for targeteddrug delivery for delivering drugs in human body, to increasethe concentration of the drug in the target area. For example,the chemotherapy drug dosage is limited by the negativeimpact on the drugs on the healthy cells. By delivering the


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1

0

1

2

3

4

5

t (s)Moving mesh, B = 0TMoving mesh, B = 0.5 T


Disp

lace

men

t fiel

d Y

com

pone

nt (𝜇

m)

(a) 𝑦-displacement for coordinates (150, 105)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−5

−4

−3

−2

−1

0

1

t (s)Moving mesh, B = 0TMoving mesh, B = 0.5 T


Disp

lace

men

t fiel

d Y

com

pone

nt (𝜇

m)

(b) 𝑦-displacement for coordinates (150, −5)

Figure 11: 𝑦-displacement of two points.

drugs with high accuracy and maximum concentration tospecific areas of the body, it is possible to increase local dosageof the drug on the tumor, with lower concentration in therest of the body. The drug effectiveness is increased while theside effects are reduced. Other examples of the applicationsof magnetic drug targeting are treatment of cardiovascularconditions, such as stenosis and thrombosis. Thus, it isimportant to model not only the blood circulation but alsothe deformation of the blood vessel, in order to improvethe accuracy which is the focus of the second problem inthis thesis. In particular, it is important to have an accuratemodel of the interaction between the three components foroptimizing the shape, size, and magnetic power, in orderto deliver the drugs efficiently in the desired place andminimize the side effects. Our results from this work clearlyindicate the importance of the magnetic field to be coupledwith a fluid-structure interaction model. More importantly,the results suggest the importance of using moving wallsversus nonmoving walls in this coupled electromagneticfluid-structure interaction.

While this work provided a lot of insight into the impor-tance of electromagnetic effects in fluid-structure interaction,there is scope to enhance this work by considering effectsof non-Newtonian rheological properties incorporated alongwith the extension to materially and geometrically nonlinearmodels. In the last two decades, collagenous soft tissueshave been found to exhibit viscoelastic behavior, whichincludes time-dependent creep and stress relaxation, rate-dependence, and hysteresis in a loading cycle. As suggestedin [18], this hysteresis is less sensitive than the stiffness tothe loading rate, and this phenomenon is generally found insoft tissues and elastomers [18]. One of the future directionswould be to extend the structural mechanics module toincorporate viscoelasticity and then study the influence ofthis on our models. The computational models in this

work included two-dimensional models for simplicity, butour models can be naturally extended to three dimensions.With increasing the size of the problem comes the needfor more computational resources. There is intensive workthat is evolving in the area of domain decomposition thathelps to address how to solve coupled multiphysics problemsefficiently. So as the problem dimension becomes bigger, onemust also resort to domain decomposition type approacheswhich can then open up more venues on parallelization ofthe algorithms that have been developed.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgment

The authors wish to thank the STEM Accelerator Programin the College of Science at George Mason University forsupporting this project in part to help promote scientificdevelopment for high school students and undergraduate andgraduate students to learn aboutmultidisciplinary research inSTEM.

References

[1] Y. C. Chang, T. Y. Hou, B. Merriman, and S. Osher, “A levelset formulation of Eulerian interface capturing methods forincompressible fluid flows,” Journal of Computational Physics,vol. 124, no. 2, pp. 449–464, 1996.

[2] F. P. Baaijens, “A fictitious domain/mortar element method forfluid-structure interaction,” International Journal for NumericalMethods in Fluids, vol. 35, no. 7, pp. 743–761, 2001.


[3] R. Glowinski, T. W. Pan, T. I. Hesla, D. D. Joseph, and J.Periaux, “A fictitious domain approach to the direct numericalsimulation of incompressible viscous flow past moving rigidbodies: application to particulate flow,” Journal of Computa-tional Physics, vol. 169, no. 2, pp. 363–426, 2001.

[4] P. Seshaiyer, “Stability and convergence of nonconformingℎ𝑝 finite-element methods,” Computers & Mathematics withApplications, vol. 46, no. 1, pp. 165–182, 2003.

[5] E. W. Swim and P. Seshaiyer, “A nonconforming finite elementmethod for fluid-structure interaction problems,” ComputerMethods in Applied Mechanics and Engineering, vol. 195, no. 17,pp. 2088–2099, 2006.

[6] E. Aulisa, A. Cervone, S. Manservisi, and P. Seshaiyer, “A mul-tilevel domain decomposition approach for studying coupledflow applications,” Communications in Computational Physics,vol. 6, no. 2, pp. 319–341, 2009.

[7] C. S. Peskin, “Numerical analysis of blood flow in the heart,”Journal of Computational Physics, vol. 25, no. 3, pp. 220–252,1977.

[8] A. O. Mulyar, Magnetically Targeted Drug Delivery System,GRIN Verlag, 2010.

[9] A. D. Grief and G. Richardson, “Mathematical modelling ofmagnetically targeted drug delivery,” Journal of Magnetism andMagnetic Materials, vol. 293, no. 1, pp. 455–463, 2005.

[10] J. Dobson, “Magnetic nanoparticles for drug delivery,” DrugDevelopment Research, vol. 67, no. 1, pp. 55–60, 2006.

[11] S.-I. Takeda, F.Mishima, S. Fujimoto, Y. Izumi, and S.Nishijima,“Development of magnetically targeted drug delivery systemusing superconducting magnet,” Journal of Magnetism andMagnetic Materials, vol. 311, no. 1, pp. 367–371, 2007.

[12] J. M. Jin, The Finite Element Method in Electromagnetics, JohnWiley & Sons, 2002.

[13] J. D. Jackson, Classical Electrodynamics, John Wiley & Sons,1998.

[14] D. Strauss, Magnetic Drug Targeting in Cancer Therapy, 2005,http://www.comsol.com/showroom.

[15] R. E. Rosensweig, “An introduction to ferrohydrodynamics,”Chemical Engineering Communications, vol. 67, no. 1, pp. 1–18,1988.

[16] R. E. Rosensweig, Ferrohydrodynamics, Dover, 1997.[17] P. A. Voltairas, D. I. Fotiadis, and L. K.Michalis, “Hydrodynam-

ics of magnetic drug targeting,” Journal of Biomechanics, vol. 35,no. 6, pp. 813–821, 2002.

[18] Y. C. Fung, Biomechanics: Mechanical Properties of LivingTissues, Springer, 1993.

Submit your manuscripts athttp://www.hindawi.com

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttp://www.hindawi.com

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com

Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Stochastic AnalysisInternational Journal of

research article computational methods for coupled fluid-structure...

Documents