Isogeometric Fluid-Structure Interaction Analysiswith Applications to Arterial Blood Flow

Y. Bazilevs1, V. M. Calo2, Y. Zhang2, and T. J. R. Hughes3

Institute for Computational Engineering and Sciences, TheUniversity of Texas at Austin,201 East 24th Street, 1 University Station C0200, Austin, TX78712, USA

Abstract

A NURBS (non-uniform rational B-splines)-based isogeometric fluid-structure interac-tion formulation, coupling incompressible fluids with nonlinear elastic solids, and allowingfor large structural displacements, is developed. This methodology, encompassing a verygeneral class of applications, is applied to problems of arterial blood flow modeling andsimulation. In addition, a set of procedures enabling the construction of analysis-suitableNURBS geometries directly from patient-specific imaging data is outlined. The approachis compared with representative benchmark problems, yielding very good results. Compu-tation of a patient-specific abdominal aorta is also performed, giving qualitative agreementwith computations by other researchers using similar models.

Key words: Isogeometric analysis, NURBS, fluid-structure interaction, vascularmodeling, Navier-Stokes equations, elastic arterial wall, mesh movement, blood flow

1 Graduate Research Assistant2 Postoctoral Fellow3 Professor of Aerospace Engineering and Engineering Mechanics, Computational andApplied Mathematics Chair III

Contents

1 Introduction 2

2 Formulation of the fluid-structure interaction problem 3

2.1 The fluid problem 5

2.2 The solid problem 5

2.3 Motion of the fluid subdomain problem 7

2.4 The coupled problem 8

3 Isogeometric analysis and construction of vascular models 10

3.1 A brief review of isogeometric analysis employing NURBS 10

3.2 Construction of vascular models for isogeometric analysis 13

4 Numerical examples 14

4.1 Wave propagation in an elastic tube 14

4.2 Blood flow in an idealized aneurysm 19

4.3 Patient-specific abdominal aorta 23

5 Conclusions and future work 24

1 Introduction

Isogeometric Analysis based on NURBS (non-uniform rational B-splines) was firstintroduced in [24] as an attempt to improve on and generalizethe standard finiteelement method. Further study of isogeometric analysis showed that results supe-rior to standard finite elements are obtained in the context of structural vibrations[7]. Mathematical analysis of the isogeometric approach was performed in [2]. Op-timal approximation estimates inp, the polynomial order used to define NURBSfunctions, were obtained forh-refined meshes. Stability and optimal convergencewas proved mathematically and verified numerically for problems of compressibleand incompressible elasticity, Stokes flow, and scalar advection-diffusion. In thispaper, isogeometric analysis is applied to fluid-structureinteraction (FSI) problemswith particular emphasis on arterial modeling and blood flow. It is believed that theability of NURBS to accurately represent smooth exact geometries, that are naturalfor arterial systems, but unattainable in the faceted finite-element representation,

2

and the high order of approximation of NURBS, should render fluid and structuralcomputations more physiologically realistic.

Initial attempts to simulate blood flow in arteries made use of simplified geometries.This approach had limited applicability because of its inability to represent complexflow phenomena occurring in real blood vessels. The concept of patient-specificcardiovascular modeling was first established in [39], where real-life geometrieswere used to simulate blood flow. This opened the door for designing predictivetools for vascular modeling and treatment planning. Dramatic improvements in thecomputational results were observed in [39], yet the blood vessel wall was treatedas being rigid. As was shown earlier, for example with the flexible and rigid wallcomputations [46–49], the rigid wall assumption precludespressure wave propa-gation and overestimates wall shear stress. There exists a variety of methods toinclude the effect of the moving wall in computations, the most prevalent being thearbitrary Lagrangian-Eulerian (ALE) approach. For a general discussion of ALE,the reader is referred to [8–10, 22, 29] and references therein. Applications of ALEto hemodynamics are discussed in [12, 15, 16, 31] and references therein. Someof the other techniques include the coupled momentum method[14], the immersedfinite element method [54], and the space-time finite elementmethod [40–42].

This work adopts the ALE framework. The arterial wall is treated as a nonlinearelastic solid in the Lagrangian description governed by theequations of elastody-namics. Blood is assumed to be a Newtonian viscous fluid governed by the incom-pressible Navier-Stokes equations written in the ALE form.The fluid velocity isset equal to the velocity of the solid at the fluid-solid interface. The coupled FSIproblem is written in a variational form such that the stresscompatibility condi-tion at the fluid-solid interface is enforced weakly. The ALEequations require thespecification of the fluid region motion. This motion is foundby solving an auxil-iary static linear-elastic boundary-value problem for which the fluid-solid boundarydisplacement acts as a Dirichlet boundary condition (see, e.g., [26]).

In section 1 we formulate the FSI problem mathematically at the continuous leveland give some discussion of spatial discretization and timeintegration. In section 2we give a brief review of the isogeometric analysis framework based on NURBS.In section 3 we give an overview of patient-specific vascularmodel constructionfor use in NURBS-based isogeometric analysis. In section 4 we present two bench-mark problems and flow in a patient-specific abdominal aorta.In section 5 we drawconclusions and give a summary of planned future work.

2 Formulation of the fluid-structure interaction problem

In this section we present the formulation of the fluid-structure interaction problem.We begin by introducing notation. LetΩ0 ∈ R

d, d = 2, 3, open and bounded, be the

3

initial or the reference configuration. LetΩt ∈ Rd, open and bounded, represent the

current configuration, namelyΩt is the image ofΩ0 under the motionx = x(X, t)with X ∈ Ω0, andt ∈ (0, T ), the time interval of interest. In what follows,x willbe referred to as current coordinates, andX as reference coordinates. Note thatx(X, 0) = X . The domainΩ0 admits a decomposition

Ω0 = Ωf0 ∪ Ωs

0, (1)

whereΩf0 is a subset ofΩ0 occupied by the fluid, andΩs

0 is a subset ofΩ0 occupiedby the solid. The decomposition is non-overlapping, that is

Ωf0 ∩ Ωs

0 = ∅. (2)

Likewise,

Ωt = Ωft ∪ Ωs

t , (3)

with

Ωft ∩ Ωs

t = ∅. (4)

Let Γfs0 denote the boundary between the fluid and the solid regions inthe initial

configuration, and, analogously, letΓfst be its counterpart in the current configura-

tion. The above setup is illustrated in Figure 1. It is important to emphasize thatthe motion of the fluid domain is not the particle motion of thefluid. It does, how-ever, conform to the particle motion of the arterial wall. That is, the Lagrangiandescription is adopted for the artery wall.

Ωf0

Ωs0

Γfs0

Ωft

Ωst

Γfstx(X, t)

Fig. 1. Abstract setting for the fluid-structure interaction problem. Depiction of the initialand the current configurations related through the ALE mapping. The initial configurationalso serves as the reference configuration.

4

2.1 The fluid problem

In this section we give a weak formulation of the incompressible Navier-Stokesfluid in the ALE description. LetVf = Vf (Ωf

t ) denote the trial solution space ofvelocities and pressures and letWf = Wf (Ωf

t ) denote the trial weighting space forthe momentum and continuity equations. Letv, p denote the particle velocity-pressure pair andwf , qf the weighting functions for the momentum and conti-nuity equations. Letδ denote the displacement of the fluid region with respect tothe initial configuration and∂δ

∂tbe its velocity. Note that, unless the Lagrangian

description is utilized in the fluid domain,∂δ∂t

6= v. We also assume that the fluidparticle velocity field satisfies the boundary condition,v = gf onΓf,D

t , the Dirich-let part of the fluid boundary. The variational formulation is stated as follows:

Findv, p ∈ Vf such that∀wf , qf ∈ Wf ,

Bf (wf , qf, v, p;∂δ

∂t) = F f(wf , qf) (5)

where

Bf(wf , qf, v, p;∂δ

∂t) =

(

wf , ρf ∂v

∂t

)

Ωft

+

(

wf , ρf(v −∂δ

∂t) · ∇xv

)

Ωft

(6)

+(qf ,∇x · v)Ω

ft− (∇x · w

f , p)Ω

ft

+(

∇sxw

f , 2µf∇sxv)

Ωft

,

and

F f(wf , qf) = (wf , ρff f )Ω

ft

+ (wf , hf)Γ

f,Nt

. (7)

The above equations are written over the current configuration, and(·, ·) defines thecorrespondingL2 inner product. The subscriptx on the partial derivative operatorsindicates that the derivatives are taken with respect to thecurrent coordinatesx.Γf,N

t is the Neumann part of the fluid domain boundary,hf is the boundary tractionvector,f f is the body force per unit mass, andρf andµf are the density and thedynamic viscosity of the fluid, respectively.

2.2 The solid problem

This section gives a weak formulation of the hyperelastic nonlinear solid in theLagrangian description. LetVs = Vs(Ωs

0) denote the trial solution space for dis-placements and letWs = Ws(Ωs

0) denote the trial weighting space for the linearmomentum equations. Letu denote the displacement of the solid body with respect

5

to the initial configuration and letws be the weighting function for the momentumequation. We also assume that the displacement satisfies theboundary condition,u = gs on Γs,D

0 , the Dirichlet part of the solid domain boundary. The variationalformulation is stated as follows:

Findu ∈ Vs such that∀ws ∈ Ws,

Bs(ws, u) = F s(ws) (8)

where

Bs(ws, u) =

(

ws, ρs0

∂2u

∂t2

)

Ωs0

+ (∇Xws, FS)Ωs

0

, (9)

and

F s(ws) = (ws, ρs0f

s)Ωs0+ (ws, hs)

Γs,N0

. (10)

The above relations are written over the reference configuration. The subscriptXon the partial derivative operators indicates that the derivatives are taken with re-spect to the material coordinatesX. Γs,N

0 is the Neumann part of the solid boundary,hs is the boundary traction vector,ρ0

s is the density of the solid in the initial con-figuration, andf s is the body force per unit mass. The displacementu is definedas

u = x − X, (11)

F is the deformation gradient

F =∂x

∂X= I +

∂u

∂X, (12)

andS is the second Piola-Kirchhoff stress tensor. We consider the St.Venant-Kirchhoffconstitutive relation:

S = C : E, (13)

where

E =1

2(F T F − I), (14)

C = λsI ⊗ I + 2µs(III −1

3I ⊗ I), (15)

Iijkl =1

2(δikδjl + δilδjk), (16)

6

E is the Green-Lagrange strain tensor,δij is the Kronecker delta, andλs andµs arethe Lame constants. Note that the fourth-order elastic tensorC is assumed constantin this model.

The St. Venant-Kirchhoff model is not without shortcomings. It exhibits a seem-ingly spurious material instability under states of strongcompression. Howeverthis is not felt to be important in the present applications.The essential point is thatit represents an objective generalization of the linear isotropic theory to the nonlin-ear case. Of course, there is no physical justification of themodel beyond the linearstrain regime.

2.3 Motion of the fluid subdomain problem

This section gives a weak formulation of the motion of the fluid subdomain. Partialdifferential equations of linear elastostatics subject toDirichlet boundary condi-tions coming from the displacements of the solid region define the ALE mappingx(X, t) on the fluid domain. This construction, which is by no means unique, im-poses sufficient regularity on the ALE mapping so as to make the fluid problem (5)well-posed. For precise conditions on the regularity of theALE map, see Nobile[31]. In the discrete setting, the fluid subdomain motion problem is referred to as“mesh moving.”

Let Vm = Vm(Ωft ) denote the trial solution space of displacements and letWm =

Wm(Ωft ) denote the weighting space for the “elastic equilibrium” equations. Letδ

denote the displacement of the fluid domain from its initial configuration and letwm be the weighting function for the linear momentum equationsgoverning themotion of the fluid subdomain. LetΩt be the configuration ofΩ0 at t < t. Wethink of this as a “nearby” configuration that in numerical computations will typi-cally represent the final configuration of the previous time step. Then, letδ be thedisplacement of the reference domain at timet. The variational formulation of theproblem is stated as follows:

Findδ ∈ Vm such that∀wm ∈ Wm,

Bm(wm, δ) = 0, (17)

subject to

δ |Γ

fst

= u x−1|Γ

fst

, (18)

and

wm |Γ

fst

= 0, (19)

7

where

Bm(wm, δ) =(

∇sxw

m, 2µm∇sx(δ − δ) + λm∇x · (δ − δ)

)

Ωft

. (20)

The above relations are written over the current configuration. The subscriptx onthe partial derivative operators indicates that the derivatives are taken with respectto the current coordinatesx. Constantsµm andλm are the Lame parameters of thelinear elastic model characterizing the motion of the fluid region. Their choice atthe continuous level should be such that the problem (17) is well-posed. In the dis-crete setting they should be selected such that the fluid meshquality is preservedfor as long as possible. In particular, mesh quality can be preserved by dividing theelastic coefficients by the Jacobian of the element mapping,effectively increasingthe stiffness of the smaller elements [43], which are typically placed at fluid-solidinterfaces. For advanced mesh moving techniques see [37, 38]. Parts of the bound-ary of the fluid region may also have motion prescribed to themindependent of themotion of the solid region. This is handled in a standard way as a Dirichlet bound-ary condition. The remainder of the fluid region boundary is subjected to a “zerostress” boundary condition.

2.4 The coupled problem

In this section we present the coupled FSI problem, which is based on the indi-vidual subproblems introduced in previous sections of thispaper. The variationalformulation for the coupled problem is stated as:

Find v, p ∈ Vf , u ∈ Vs, andδ ∈ Vm such that∀wf , qf ∈ Wf , ∀ws ∈ Ws,and∀wm ∈ Wm,

Bf(wf , qf, v, p;∂δ

∂t) − F f(wf , qf) +

Bs(ws, u) − F s(ws) + Bm(wm, δ) = 0. (21)

The following auxiliary relations hold in the sense of traces:

v|Γ

fst

=∂u

∂t x−1|

Γfst

, (22)

δ |Γ

fst

= u x−1|Γ

fst

, (23)

wf |Γ

fst

= ws x−1|Γ

fst

, (24)

8

and

wm |Γ

fst

= 0. (25)

Relationship (22), the kinematic constraint, equates the fluid velocity with that ofthe solid at the fluid-solid boundary. Equation (24) leads tothe compatibility ofstresses at the fluid-solid interface. Relations (23) and (25) indicate that the fluidregion motion is driven by the displacement of the fluid-solid boundary acting as aDirichlet boundary condition for the fluid motion subdomainproblem.

Spatial discretization of the coupled problem (21) makes use of isogeometric anal-ysis based on NURBS, an alternative approach to standard finite elements, pro-posed in Hughes, Cottrell, and Bazilevs [24]. Isogeometricanalysis is briefly re-viewed in the next section. Galerkin’s method is employed for the structural andthe fluid subdomain motion parts of the formulation, while a residual-based multi-scale method, given in Calo [5], and Hughes, Calo, and Scovazzi [23], is used forthe fluid equations. The class of residual-based multiscalemethods may be thoughtof as stabilized methods, such as SUPG (see Brooks and Hughes[4]), extended tothe nonlinear realm.

The resultant semi-discrete equations are advanced in timeusing the generalized-α algorithm proposed by Chung and Hulbert [6] for the equations of structuralmechanics and extended to the equations of fluid mechanics byJansen, Whiting,and Hulbert [25]. The generalized-α method embodies a family of second-ordertime integrators with strict control over high frequency dissipation. In the contextof fluid-structure interaction, the generalized-α method was applied to the couplingof the linearized Euler equations with a nonlinear structure in one spatial dimensionby Kuhl, Hulshoff, and de Borst [28].

The kinematic constraint (22) is enforced strongly by requiring basis functions to beC0-continuous across the fluid-solid interface. Alternatively, this constraint may beenforced weakly by appropriately modifying the discrete formulation (see Bazilevsand Hughes [3]).

The coupled nonlinear system is solved monolithically, that is, the fluid, the struc-tural, and the mesh solution increments in the Newton iteration are obtained si-multaneously. The effect of the structural and the mesh motion on the fluid equa-tions is included in the left-hand-side matrix for robustness. The coupled systemis solved iteratively by the GMRES procedure [34] with simple diagonal scaling.More sophisticated preconditioning strategies are under current investigation. Fora description of different solution strategies for FSI equation systems see [44].

9

3 Isogeometric analysis and construction of vascular models

3.1 A brief review of isogeometric analysis employing NURBS

This section gives a very brief overview of isogeometric analysis based on NURBS.A more detailed description of the isogeometric approach may be found in [7, 24].For an introductory text on NURBS, see Rogers [33], while a more detailed treat-ment is given in the book of Piegl and Tiller [32]. Mathematical theory of isogeo-metric analysis forh-refined meshes may be found in the recent work of Bazilevset al. [2].

3.1.1 One-dimensional B-splines

A B-spline basis is comprised of piece-wise polynomials joined with prescribedcontinuity. In order to define a B-spline basis of polynomialorderp in one dimen-sion one needs the notion of aknot vector. A knot vector in one dimension is a setof coordinates in the parametric space, written asΞ = ξ1, ξ2, ..., ξn+p+1, wherei is the knot index,i = 1, 2, ..., n + p + 1, ξi ∈ [0, 1] is theith knot, andn is thetotal number of basis functions. A knot vector is said to beopenif its first and lastknots are repeatedp + 1 times. Basis functions formed from an open knot vectorare interpolatory at the end points of the parametric interval, but they are not, ingeneral, interpolatory at the interior knots.

GivenΞ andp, B-spline basis functions are defined recursively startingwith piece-wise constants(p = 0) :

Bi,0(ξ) =

1 if ξi ≤ ξ < ξi+1,

0 otherwise.(26)

Forp = 1, 2, 3, ..., they are defined by

Bi,p(ξ) =ξ − ξi

ξi+p − ξi

Bi,p−1(ξ) +ξi+p+1 − ξ

ξi+p+1 − ξi+1

Bi+1,p−1(ξ). (27)

Basis functions of orderp havep − 1 continuous derivatives at knots. If a knot isrepeatedk times, then the number of continuous derivatives decreasesby k. Whenthe multiplicity of a knot is exactlyp, the basis function is interpolatory. Basisfunctions form a partition of unity, each one is compactly supported on the interval[ξi, ξi+p+1], and they are point-wise non-negative. These properties areimportantand make these functions attractive for use in analysis.

10

3.1.2 Multi-dimensional B-splines and geometrical objects

Let α be a positive integer such that1 ≤ α ≤ d, whered is the number of spacedimensions. Givenα knot vectorsΞγ,pγ

, γ = 1, . . . , α, multi-dimensional B-splinesare constructed by taking tensor products of their one-dimensional counterparts:

Bi1,...,iα(ξ1, . . . , ξα) = ⊗αγ=1Biγ ,pγ

(ξγ). (28)

B-spline functions are defined on a parametric domain(0, 1)α, andpγ is the poly-nomial order in the parametric directionγ.

Objects of B-spline geometry can be most generally characterized as unions ofpatchesin R

d. A patch is an image under a mapping of the parametric interval(0, 1)α, expressed as a linear combination of spline basis functions (28) and pointsin R

d, that is,

Ω = F(ξ) | ξ ∈ (0, 1)α, (29)

F(ξ) =∑

i∈I

CiBi(ξ),

whereI is the index set

I = i = (i1, . . . , iα) ∈ Nα | 1 ≤ iγ ≤ nγ + pγ + 1, (30)

Ω represents an object and theC’s are the so-calledcontrol points. Various geomet-rical objects may be obtained by varyingα, namely: the case ofα = 1 correspondsto aB-spline curve, α = 2 generates aB-spline surface, and aB-spline solidis ob-tained by settingα = 3. A piece-wise linear interpolation of the control points forcurves, and a piece-wise multi-linear interpolation of thecontrol points for surfacesand solids is called acontrol mesh.

3.1.3 NURBS functions and geometry

The geometric framework based on B-splines is limited in that basic elements ofengineering design, such as exact circles, ellipses, and other conic sections cannotbe obtained by using B-spline functions alone. NURBS were devised to overcomethis shortcoming. NURBS geometrical objects inR

d are preciselyprojective trans-

11

formationsof B-spline geometrical objects inRd+1 (see Farin [11]), that is,

Ω = F(ξ) | ξ ∈ (0, 1)α,

F(ξ) = P(∑

i∈I

Ci, wiBi(ξ)) =

∑

i∈I

(Ci

wi

)(wiBi(ξ)

∑

j∈I wjBj(ξ)) =

∑

i∈I

Ci

wiBi(ξ)

w(ξ)=∑

i∈I

CiRi(ξ). (31)

In the aboveCi,Ci are inRd, wi ∈ R, strictly positive, is theweight, andw(ξ) =

∑

i∈I wiBi(ξ) is theweighting function. The weighting function is defined on thegeometry and stays unchanged throughout the refinement process. The last line of(31) defines NURBS basis functionsRii∈I :

Ri(ξ) =wiBi(ξ)

w(ξ). (32)

It is important to note that while a B-spline basis depends only on the structureof the knot vector, construction of a NURBS basis requires information about thegeometry. Multi-dimensional NURBS basis functions are no longer tensor productsof one-dimensional entities, in contrast with B-splines. Properties such as partitionof unity, positivity, and compact support are retained for NURBS bases. Continuityof NURBS functions is also the same as that of B-splines.

3.1.4 Analysis framework based on NURBS

Isogeometric analysis framework consists of the followingitems and features:

(1) A physical domain consists of a union of patches, each defined as an image ofthe parametric space as

Ω = F(ξ) | ξ ∈ (0, 1)α (33)

(2) A mesh for a NURBS patch is defined by the union of NURBS elements,denoted byK, each one being an image under a NURBS map of a knot spanin the parametric space

K = F(ξ) | ξ ∈ Q = ⊗αγ=1(ξiγ ,γ, ξiγ+miγ ,γ). (34)

wheremiγ is the multiplicity of knotξiγ ,γ.(3) The basis for the solution space in the physical domain isdefined through a

composite mapping as follows:

Ni = Ri F−1, i ∈ I. (35)

12

This definition gives the isoparametric construction [21],that is, the fieldsin question (e.g., displacement, velocity, temperature, etc.) are represented interms of the same basis functions as the geometry. Coefficients of the basisfunctions, orcontrol variables, are the degrees-of-freedom of the discrete sys-tem. The isoparametric approach is most convenient for applications involv-ing Lagrangian and ALE descriptions of continuous media where geometry isconstantly updated as the physical system evolves in time.

(4) Boundaries of NURBS geometrical objects are themselveslower dimensionsNURBS objects (e.g., a NURBS solid is bounded by NURBS surfaces, which,in turn, are bounded by NURBS curves). As a result, the easiest way to setDirichlet boundary conditions on a patch face is to constrain control vari-ables that correspond to that face. Interpolation or projection needs to be em-ployed in cases when the prescribed function is not in the discrete space. Thisamounts to “strong” satisfaction of the boundary conditions. An alternativeformulation of Dirichlet conditions can be based on “weak” satisfaction, astandard feature of the discontinuous Galerkin method. Neumann boundaryconditions are satisfied as in the standard finite element method.

(5) Mesh refinement strategies are developed from a combination of knot inser-tion andorder elevationtechniques. These enable analogues of classicalh-refinement andp-refinement methods, and the new possibility ofk-refinement.Details of refinement algorithms may be found in [24].

3.2 Construction of vascular models for isogeometric analysis

Fig. 2. Stages 3 and 4 of the patient-specific cardiovascularmodel construction for isogeo-metric analysis. Left: depiction of a cross-section surface template. Cross-section surface isbounded by a closed quadratic NURBS curve defined in terms of the control polygon con-sisting of 16 points. Middle: arterial path identified by skeletonization. Right: solid NURBSgeometry, ready for refinement and analysis.

Construction of patient-specific models for isogeometric analysis is a process in-

13

volving four stages, described below.

(1) In scanned Computed Tomography (CT), or Magnetic Resonance Imaging(MRI), of patient-specific data, intensity contrast may notbe sufficiently sharp,images are often “noisy”, and the luminal surface is frequently blurred. As aresult, preprocessing of of the CT/MRI data is necessary to improve its qual-ity. Techniques such as contrast enhancement [53], filtering [1], classification[45], and segmentation [52] are employed for this purpose.

(2) The blood vessel surface model can then be constructed from preprocessedimaging data using isocontouring methods. The two main isocontouring meth-ods, that make use of imaging data, are: primal contouring ormarching cubes[30], and dual contouring [27]. The latter is chosen in this work for isosurfaceextraction, as it tends to generate control meshes possessing better aspect ra-tios. In some cases geometric editing is required to remove unnecessary com-ponents and features. Once the luminal surface is identified, skeletonizationtechnique [17] is employed in order to find paths.

(3) We have developed a skeleton-based sweeping method to construct hexahe-dral NURBS control meshes for blood vessels [55]. The template faceted con-trol polygon of a circle, projected onto the true surface, isswept along thearterial path to create a quadrilateral surface control mesh for a given arterialbranch. Arterial branches are also arranged in a hierarchy,ranging from thelargest to the smallest. Different cross-section templates are applied to dif-ferent branches in the hierarchy. Templates for various branch intersections,such as bifurcations and trifurcations, are also worked outand applied on acase-by-case basis. See [55] for details.

(4) Finally, solid NURBS meshes are constructed by filling inthe volumes radi-ally from the outer surface inward. Arterial wall meshes areobtained by ex-tending the outer surface in the normal direction by a user-prescribed amount.

Stages 3 and 4 of the process are demonstrated on a simple example in Figure 2.

4 Numerical examples

In all the examples, the wall is modeled by two elements and four C1-continuoussecond order basis functions through the thickness. See [24] for further details re-garding modeling of shell-like structures as solids.

4.1 Wave propagation in an elastic tube

Our first test case, taken from Greenshields and Weller [18],deals with wave prop-agation in a fluid-filled elastic tube. In this example the tube length isL = 10 cm,

14

Fig. 3. Wave propagation in a fluid-filled elastic tube problem setup.H(t) is the heavisidefunction.

its inner radius isRi = 1 cm, and its outer radius isRo = 1.2 cm. The solid regionis enclosed betweenRi andRo while the fluid occupies the rest of the tube. Theproblem setup and boundary conditions are illustrated in Figure 3. Material prop-erties representative of blood flow in arteries are defined asfollows: the density ofthe solid isρs = 1 g

cm3 , and Young’s modulus and Poisson’s ratio areE = 107 dyncm2

andν = 0.3, respectively. The fluid density is alsoρf = 1 gcm3 , and its viscosity is

µf = 0.04 gcm s.

The computational mesh, consisting of 6,080 quadratic NURBS elements, is shownin Figure 4. Att = 0 a step change in pressure is applied at the fluid inflow bound-ary to the system that is initially at rest (all initial fieldsare zero). The pulse causesa pressure wave to propagate down the tube. Figure 5 presentssnapshots of fluidpressure at various times. Four radial cuts are shown on eachof the plots to demon-strate that the computed solution is pointwise axisymmetric. For visualization pur-poses, pressure in the solid region is set to zero to create a sharp contrast at thefluid-solid interface. As a result, radial wall displacement, which is on the order of5%, is visible in the figure.

Figure 6a shows the outer wall displacement, while Figure 6bshows the centerlinefluid pressure at various times. Isogeometric results are compared with referencecomputations of Greenshields and Weller [18], who employeda small-strain, small-displacement formulation of the solid. Discrepancies between results are assumedattributable to the fully nonlinear model used in the present study versus the linearmodel utilized in [18]. Nevertheless, results are in fairlygood agreement with thereference computations, as well as with the Joukowsky solution (see Greenshieldsand Weller [18] for details). These observations provide tentative confirmation that

15

Fig. 4. Wave propagation in a fluid-filled elastic tube mesh consisting of 6,080 NURBSelements.

the coupled momentum method for hemodynamics, proposed by Figueroaet al.[14], in which the fluid and the structure exhibit strong coupling, but the geometrystays fixed at a reference configuration, is an adequate description for blood flowcalculations.

16

t = 1.6ms t = 2.8ms

t = 4.0ms t = 5.2ms

t = 6.4ms t = 7.6ms

Fig. 5. Wave propagation in a fluid-filled elastic tube. Contours of fluid pressure at variousradial slices. Solution remains pointwise axisymmetric

17

0 1 2 3 4 5 6 7 8 9 10−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Distance along tube (cm)

Rad

ial d

ispl

acem

ent (

cm)

Isogeometric

Greenshields&

Weller

t=1.6ms

t=3.2ms

t=6.4ms

t=8.0ms

t=4.8ms

(a)

0 1 2 3 4 5 6 7 8 9 10−1

0

1

2

3

4

5

6

7x 10

4

Distance along tube (cm)

Cen

terli

ne p

ress

ure

(dyn

/cm2 )

Isogeometric

Greenshields&

Weller

t=1.6ms

t=3.2ms

t=4.8ms

t=6.4ms

t=8.0ms

(b)

Fig. 6. Wave propagation in a fluid-filled elastic tube. (a) Outer wall radial displacement.(b) Centerline pressure. Computational results of Greenshields and Weller [18] are plottedfor comparison.

18

4.2 Blood flow in an idealized aneurysm

Fig. 7. Idealized aneurysm problem setup.

In this test case, taken from [13, 35], we examine pulsatile flow in an idealizedaneurysm. The problem setup is shown in Figure 7. A time-periodic velocity wave-form, specified at the inflow plane, is parabolically distributed over the circularsurface. The period of the wave,T , is 0.84s. The domains proximal and distal tothe aneurysm region are assumed to have rigid walls, while the aneurysm wallis elastic. The density of the solid, its Young’s modulus, and Poisson’s ratio areρs = 1.2 g

cm3 , E = 6×106 dyncm2 , andν = 0.3, respectively. The fluid density and dy-

namic viscosity areρf = 1.012 gcm3 andµf = 0.035 g

cm s, respectively. A resistanceboundary condition is applied at the outflow. The value of theresistance constantis CR = 300 dyn s

cm5 . For implementation of boundary conditions employing variouspressure-flow relationships, see Heywoodet al. [19] and Vignonet al. [50]. Themesh, consisting of 14,630 quadratic NURBS elements, is shown in Figure 8.

Figure 9 shows velocity vectors superimposed on the axial velocity contours in theaneurysm region at different times. A135 “pie” slice was cut out of the domain inorder to exhibit the flow features. Distensibility of the wall contributes significantlyto the unsteadiness of the flow. Nevertheless, the flow remains axisymmetric, asmay be discerned from symmetry of the velocity vectors. As inthe previous exam-ple, no axisymmetry in the solution is assumed at the outset.It should be noted thatthe peak Reynolds number, estimated to be about 1,400-1,500based on the largest

19

Fig. 8. Idealized aneurysm mesh consisting of 14,630 NURBS elements.

diameter, is close to the transitional value for circular pipe flow (see, e.g., White[51]). Thus, relatively small perturbations in the geometry and/or flow conditionsmay lead to much more complex, unsteady solutions. Figure 10shows the inflowand outflow waveforms. Note the outflow lags the inflow due to the distensibility ofthe aneurysm wall. This well-known phenomenon was also observed in the compu-tations of Figueroaet al.[14]. Figure 10 also shows reference results from [13, 35].The agreement is excellent despite the differences in the wall models (a nonlinearshell was used in [13, 35]).

20

t ≈ 0T t ≈ 1

5T

t ≈ 2

5T t ≈ 3

4T

Fig. 9. Idealized aneurysm. Velocity vectors superimposedon axial velocity contours atvarious times. Top right and bottom left correspond to the systolic and diastolic phases,respectively. Note that the flow is axisymmetric.

21

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9−40

−20

0

20

40

60

80

100

120

Time (s)

Flo

wra

te (

cm3 /s

)

Inflow

Outflow, Fernandez et al.

Outflow, Isogeometric

Fig. 10. Idealized aneurysm. Inflow and outflow waveforms. Notice the time lag attributableto the distensibility of the wall.

22

4.3 Patient-specific abdominal aorta

Fig. 11. Patient-specific abdominal aorta geometry.

We present fluid-structure interaction calculations of a patient-specific abdominalaorta for a healthy over-55 volunteer obtained from 64-slice CT angiography. Thegeometrical model, which contains most major branches of a typical abdominalaorta, is shown in Figure 11. The inferior mesenteric arterywas not clearly capturedin the imaging and was omitted in the geometrical model. The fluid properties are:ρf = 1.06 g

cm3 , µf = 0.04 gcm s. The solid is characterized by the densityρs = 1 g

cm3 ,Young’s modulus,E = 4.144 × 106 dyn

cm2 , and Poisson’s ratio,ν = 0.4. A periodicflow waveform, with periodT = 1.05s, is applied at the inlet of the aorta, while re-sistance boundary conditions are applied at all outlets. The solid is fixed at the inletand at all outlets. Material and flow rate data, as well as resistance values are takenfrom Figueroaet al. [14], with the following exception. Poisson’s ratio is taken tobe 0.4, not 0.5 as in [14], as the latter is not allowed in the pure displacement for-mulation of an elastic solid. Wall thickness for this model is taken to be 15% of thenominal radius of each cross-section of the fluid domain model. The computationalmesh, consisting of 52,420 quadratic NURBS elements, is shown in Figure 12.

Figure 13 shows velocity isosurfaces plotted on the currentconfiguration of thegeometry at various times during the cardiac cycle. The flow appears to be fullythree-dimensional and unsteady, with most of the unsteadiness occurring in latediastole. Figure 14 shows the distribution of flow among the branches. As in theprevious example, the outflow lags the inflow due to the distensibility of the ar-terial wall. Although perfect matching with [14] cannot be expected because thegeometry and analysis models are different, the overall flowdistribution and the

23

Fig. 12. Patient-specific abdominal aorta mesh consisting of 52,420 quadratic NURBS ele-ments.

time lag are in qualitative agreement.

5 Conclusions and future work

We have developed a NURBS-based isogeometric fluid-structure interaction capa-bility coupling incompressible fluids with nonlinear elastic solids and allowing forlarge structural displacements, and applied it to problemsof arterial blood flow. Wehave also developed a set of procedures allowing us to construct analysis-suitableNURBS geometries directly from patient-specific imaging data. The new approachis evaluated on two benchmark problems and applied to the fluid-structure inter-action of a patient-specific abdominal aorta. Very good results are obtained for thebenchmark computations and the results for our patient-specific model are in quali-tative agreement with the results of Figueroaet al.[14] for a patient-specific model.

Future developments will address extensions to hyperelastic materials with anisotropyand viscoelasticity (see, e.g., [20, 36]), which are capable of representing morephysically realistic behavior of the arterial wall. Solid incompressibility and near-

24

t ≈ 1/5T t ≈ 2/5T

t ≈ 3/5T t ≈ 4/5T

Fig. 13. Patient-specific abdominal aorta. Isosurfaces of the velocity magnitude plotted onthe deformed geometry at various times.

incompressibility will be dealt with by means of a mixed formulation employingdisplacement and pressure. Comparisons with standard finite elements are alsoplanned.

25

Fig. 14. Patient-specific abdominal aorta. Inlet and outletflow waveforms. Flowrates(cm3/s) versus time (s).

26

Acknowledgments

Y. Zhang was partially supported by the J.T. Oden ICES Postdoctoral Fellowship atthe Institute for Computational Engineering and Sciences.This support is gratefullyacknowledged. We would also like to thank Miguel Fernandez of INRIA for pro-viding us with data for the idealized aneurysm as well as for his valuable commentsand suggestions.

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