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Anisotropic Adaptive Finite Element Method for Modeling Blood Flow

J. Mullera∗, O. Sahnia, X. Lia, K. E. Jansena, M. S. Shepharda, C. A. Taylorb

aScientific Computation Research Center,Rensselaer Polytechnic Institute, 110 8th Street, Troy 12180 NY, USA

b Stanford University, Clark Center, Room E350B318 Campus Drive, Stanford, CA 94305-5431, USA

In this study we present an adaptive anisotropic finite element method (FEM) and demonstrate how computa-tional efficiency can be increased when applying the method to the simulation of blood flow in the cardiovascularsystem. We use the SUPG formulation for the transient 3D incompressible Navier-Stokes equations which arediscretized by linear finite elements for both the pressure and the velocity field.Given the pulsatile nature of the flow in blood vessels we have pursued adaptavity based on the average flow

over a cardiac cycle. Error indicators are derived to define an anisotropic mesh metric field. Mesh modificationalgorithms are used to anisotropically adapt the mesh according to the desired size field. We demonstrate theefficiency of the method by first applying it to pulsatile flow in a straight cylindrical vessel and then to a porcineaorta with a stenosis bypassed by a graft. We demonstrate that the use of an anisotropic adaptive finite elementmethod can result in an order of magnitude reduction in computing time with no loss of accuracy compared toanalyses obtained with isotropic meshes.

Keywords: Computational blood flow,

Anisotropic finite element mesh adaptivity, Wall

shear stress

1. INTRODUCTION

In recent years, computational fluid dynam-ics methods have been used to solve the three-dimensional equations governing pulsatile bloodflow in arteries, (Perktold et al., 1991; Tayloret al., 1998). Simulations in idealized modelsconstructed from computer aided design soft-ware have been displaced by analyses performedin subject-specific models generated from three-dimensional medical imaging data (Taylor et al.,1996, 1999; Moore et al., 1999; Steinman, 2002;Cebral et al., 2002).

In order to employ the finite element (FE)method to simulate hemodynamics, the geometricmodel has to be subdivided into a finite numberof elements. The obstacles that have to be over-come when discretizing a domain include gener-

∗Tel.: +1-518-276-8045; fax: +1-518-276-4886. E-mail

address: jmueller@scorec.rpi.edu (J. Muller).

ating a mesh that sufficiently represents the ge-ometry of the blood vessel and ensuring the meshresolves all the desired solution quantities, includ-ing the primary variables of velocity and pressureand derivative quantities such as the wall shearstresses. It has been proven that, for consistentmethods, increased mesh density leads to moreaccurate solutions (Franca and Frey, 1992). Un-fortunately, the exact mesh density required torepresent the solution fields to a desired level ofaccuracy is unknown a priori.

Generally, one assumes the mesh density isadequate when further refinements of the meshdo not significantly change the numerical solu-tion (“mesh independence”), (Zhao et al., 2002;Prakash and Ethier, 2001). However, for certaingeometric models that evoke singular behavior ofthe solution field and its derivatives, as is the casefor bifurcation points for instance, the numericalFE-solution will indefinitely improve however finethe mesh becomes (Muller and Korvink, 2004).In that sense, some caution is appropriate whenattempting to obtain a “mesh independent” so-lution. A much weaker claim, known as a good

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“convergence rate” of the solution as meshes be-come finer, would be more appropriate in thesecases.

To accelerate convergence behavior, meshadaptive techniques have been used for a rangeof physical problems including those in solid me-chanics, fluid mechanics, and electromagnetics, toname a few. Refining the mesh uniformly in orderto achieve a better solution would require compu-tational resources that could be orders of magni-tudes higher than those needed when employingadaptive techniques (Muller and Korvink, 2004).In view of the complexity of blood vessel models,an adaptive approach becomes highly desirablewhen simulating hemodynamics. For a summaryof different adaptive techniques, see Ainsworthand Oden (2000) or Verfurth (1996). Commonto all adaptive meshing methods is the ability toreduce computational expense while controllingthe accuracy of the solution that is obtained. Forblood flow modeling, adaptive meshing methodshave only recently been described (Prakash andEthier, 2001).

Key features of adaptive meshing methods in-clude the following.

• An a posteriori error estimation/indication:estimating or obtaining an indication of thediscretization error since the solution andtherefore the error are unknown a priori.

• An adaptive strategy by which elementsare modified in order to reduce the over-all error. This can be achieved by equidis-tribution of local error indicators over themesh. The most straightforward way wouldbe to refine (or split) elements with higherror and to coarsen elements with low er-ror. An alternate approach, described inthis paper, is to incorporate directional er-ror information. In this case, error reduc-tion is achieved by collapsing elements in di-rections where low error values prevail andby splitting elements in the direction of higherrors.

• Geometric mesh modification algorithms: anumber of methods have been presented

in the literature ranging from conform-ing, splitting and coarsening of simplicialmeshes in 2D and 3D, see Rivara (1984)and Bansch (1991b) to complex mesh op-erations that were developed only recentlyand which enable directional alignment inaccordance with pre-defined mesh metricfields in 2D (Almeida et al., 2000) and 3D(Li et al., 2003b,a; Pain et al., 2001; Freyand Alauzet, 2003).

A posteriori error estimation is well understoodfor elliptic problems and to a lesser extent forparabolic problems. However, error estimationmethods for the time dependent incompressibleNavier-Stokes equations are lacking. Early resultscan be found in Bansch (1991a) and a more re-cent survey on possible approaches is presented inBarth and Herman. (2003). Attempts to improvesimulation efficiency of hemodynamics by meansof mesh adaptivity up to now are restricted tosteady flow (Prakash and Ethier, 2001).

While most of the work on error estimation fortransient phenomena also try to incorporate thetime discretization error, it seems less advisableto do so for flow that is pulsatile and thereforeperiodic in time. We propose a method in whichhigh errors are identified by averaging the flowfield over a cardiac cycle and then base the erroranalysis on that averaged field.

In order to be able to extract directional in-formation of the error which in turn can be con-verted into a mesh metric field we use the Hes-sian strategy (Borouchaki et al., 1997; Remacleet al., 2002), a method where the field’s secondderivatives are used to obtain information on theerror distribution. When linear finite elementsare used, the interpolation error is equivalent tosecond derivatives and it has been shown that alarge proportion of the discretization error is cov-ered by this error indicator, (Fortin, 2000). Higheigenvalues of the field’s Hessian matrix defined ata particular node imply high errors and thereforethe size of the elements surrounding that nodein the direction of the corresponding eigenvectorshould be decreased. Conversely, low eigenvaluesin a particular eigendirection suggest the elementscan be increased in size in that direction.

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In most hemodynamic studies, a primary quan-tity of physiological interest is the wall shearstress which contains derivatives of the velocityfield. It is widely believed that low wall shearstress affects the localization of atheroscleroticplaques in regions of complex flow (Caro et al.,1971). As is the case for most discretization tech-niques, derivative field quantities are not as wellresolved as the primary field. The result is simplya lower convergence rate in its appropriate norm.

Investigating wall shear stresses therefore notonly poses an important challenge from a physi-ological viewpoint but also the nature of the nu-merical method at hand has to be properly en-hanced by an efficient adaptive method to accu-rately resolve the fields where otherwise unrea-sonable values would tempt the analyst to drawfalse conclusions.

This paper is organized as follows. We pro-vide a brief introduction to the numerical methodthat we use to solve hemodynamic flows. Then,an overview of the error indication employedtogether with the basic concepts of the meshmodification techniques are given. The thirdsection encompasses an in depth study of wallshear stress computation employing our adaptivemethod. We first demonstrate the efficiency ofour method by applying it to a steady flow casein a straight cylindrical vessel. Here, we com-pare the computed wall shear stress with the an-alytical values. We then increase the complexityof the problem by extending it to pulsatile flowfor the same geometric configuration. Finally, weapply the method to a model of blood flow in aporcine aorta bypass model for which experimen-tal validation of the finite element method hasbeen obtained (Ku et al., 2002). In each of thesecases, we compare the adaptive method to a non-adaptive method. For this method, we attempt toimprove the solution quality by using a series ofdifferent uniform meshes, meaning that elementsizes (in terms of edge lengths) for a given meshare roughly the same.

2. SIMULATION METHODS

2.1. Flow Solver

The governing equations for blood flow, assum-ing Newtonian constitutive behavior and rigidblood vessel walls, are the incompressible 3Dtime-dependent Navier-Stokes equations. Theweak form of the Navier-Stokes equations is dis-cretized by spatially subdividing the domain intoa finite number of elements and by defining dis-crete subspaces of the spaces in which the weakform is defined. The domain boundary is decom-posed into portions with natural boundary con-ditions and essential boundary conditions.

In the present work we use a stabilized finite el-ement (SUPG) formulation that is based on thatdescribed by Taylor et al. (1998). The stabiliza-tion allows us to choose the same local approx-imation space, for both the velocity and pres-sure variables (typically piecewise polynomials;all calculations described herein were performedwith a linear basis). To develop a discrete sys-tem of equations, the weight functions, the so-lution variables, velocity ui, and pressure p, andtheir time derivatives are expanded in terms ofthe finite element basis functions. Since we havea non-linear, time dependent system of equations,Gauss quadrature of the spatial integrals resultsin a system of first-order, non-linear ordinary dif-ferential equations.

Finally this system of non-linear ordinary dif-ferential equations is discretized in time via ageneralized-α time integrator (Jansen et al., 1999)resulting in a non-linear system of algebraic equa-tions. This system is in turn linearized with New-ton’s method which yields a linear algebraic sys-tem of equations which in turn is solved (at eachtime step) and the solution is updated for each ofthe Newton iterations. The linear algebra solverof Shakib (1999) is used to solve the linear systemof equations.

2.2. Error indication

The numerical solution that we obtain throughthe method described above is subject to twokind of errors: first, insufficiency in the approxi-mation of the geometric model, and second, thediscretization error itself. To render the second

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type more precise, consider a model that can beperfectly represented by any given mesh, e.g. astraight sided quadrilateral. Then, even in thiscase, numerical FE solutions generally carry er-rors that exclusively depend upon the coarsenessof the mesh.

To further illustrate the origin of the discretiza-tion error, we recall that a function which is suf-ficiently smooth can be developed into a Taylorseries. When trying to interpolate that functionwith a piecewise linear function the error will havea lowest order error term (second order) propor-tional to the second derivative of the function.Therefore, if we assume that the “true” finite el-ement solution (meaning the one that is definedthrough the weak form) is smooth enough andassuming that our finite element solution is a lin-ear interpolation to that (which is not always thecase, however) a good error measure is related tothe second derivatives of the discrete solution.

The discrete solution is piecewise linear on eachelement, thus the first and the second derivativesfor each node i have to be reconstructed. The re-construction that we employ uses the derivativesof the patch Si of all elements T surrounding nodei. For example, the recovered gradient of a givenfield component u on node i can be computed as

gradr(u)i =1

Vol(Si)

∑

T∈Si

Vol(T ) ∇u|T . (1)

This is equivalent to a lumped-mass approxima-tion to a least squares reconstruction of a gra-dient for linear elements. The same procedureapplied to the second field derivatives yields theHessian matrix. We should note that values thatare reconstructed on the mesh boundary in thisway usually do not very well represent the “real”Hessians, therefore an extrapolation technique isused to project interior values onto the nodes thatlie on the boundary. A mesh metric field is thenconstructed by calculating a scaled eigenspace ofthe Hessian matrix. A high eigenvalue in a par-ticular eigendirection for a given node indicatesthat the lengths of element edges in the vicinityof that node (aligned with that direction) shouldbe smaller than in directions with lower eigenval-ues.

Truncation values hmin and hmax are specifiedto scale the eigenvalues to be the desired meshsizes. The modified eigenvalues of the Hessianmatrix then read

λi = min(max(c |λi|,1

h2max

),1

h2min

) (2)

where c is a scale factor, determined in terms ofthe leading element interpolation error aiming toequilibrate the distribution of the error. At eachvertex the mesh metric field M then is obtainedby multiplying the matrix of the modified eigen-values with the matrix R of eigenvectors

M = RΛR−1, Λ =

λ1 0 0

0 λ2 0

0 0 λ3

. (3)

The goal then is for the length of each edge e

of the mesh, to assume unit value in the spacewhere distances are measured by the metric M.Expressed in terms of a quadratic form, this isequivalent to

(e, Mee) = 1 (4)

where Me is the mesh metric field of the nodesadjacent to the given edge. We also refer to thisoperation as the measuring of sizes in the trans-formed space.

2.3. Mesh modifications

Given the mesh metric field M defined overthe domain, the ideal goal is to apply local meshmodification operators to yield a mesh with alledges satisfying equation (4). However, the factthat we cannot pack unit equilateral tetrahedrato satisfy a constant unit mesh metric field in-dicates that a perfect match to equation (4) isimpossible. Therefore, we relax that criteria andconsider a mesh satisfying the metric field if, inthe transformed space, (i) all its measuring edgelengths fall into an interval close to one, referredto as [Llow, Lup] , and (ii) no sliver element ex-ists. A restriction, Llow ≤ 0.5Lup, in selecting theinterval is applied. This is to ensure that the mea-suring size of two new edges created by a bisectionwill not be shorter than Llow so that oscillationbetween subdivision and coarsening is prevented.

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The overall mesh modification procedure consistsof (Li et al., 2003a):

• Incrementally reduce the maximal measur-ing edge length in the transformed space un-til below Lup by marking a set of longestmesh edges and subdividing all elementsbounding these edges in each iteration;

• Collapse as many mesh edges shorter thanLlow (in the transformed space) as possibleto coarsen or fix the mesh;

• Eliminate sliver elements with respect tothe mesh size field to improve element qual-ity.

The subsections that follow illustrate the con-cepts of the major mesh modification proceduresin two dimensions. Similar rules hold for theslightly more complex 3D case, described in detailin de Cougny and Shephard (1999).

Subdivision of elements

The first mesh modification operator is the el-ement subdivision. Given a set of edges to berefined, it consists in modifying a cavity C (orcavities) composed of elements neighboring theseedges by replacing these elements with nested andsubdivided ones, see Figure 1a.

Edge collapsing

The second mesh modification operator is edgecollapsing. If an edge is too short (< Llow) wedecide to remove it. The operator modifies a cav-ity C composed of all the triangles (tetrahedra in3D) neighboring one end vertex of the short edge,see Figure 1b. It can be seen as pulling one endvertex of the edge to another.

Edge swapping

This operation consists of modifying a cavityC composed of two neighboring elements e1 ande2 by swapping the edge and replacing e1 and e2by e3 and e4 as represented in Figure 1c.

Projecting to the model boundary

Once the mesh size field requires that a newnode be introduced, by subdividing an elementthat is on the mesh boundary, the algorithm en-sures that the new node lies on the model bound-

Tn Tn+1

C C'

Tn Tn+1

C C'

Tn Tn+1

e1 e2e3

e4

C C'

Tn Tn+1

C C'

Tn Tn+1

u1

u2

f(u1,u2)

C C'

a)

b)

c)

d)

e)

Figure 1. Mesh modification in 2D.

ary as well (Li et al., 2003b). At the same time weavoid nodes being removed from the mesh whenthey lie on the boundary, see Figure 1d. Hereby,each time a modification procedure is invoked, weassure that the approximation of the model by themesh only can improve.

Easy solution transfer

We use a callback mechanism which is invokedeach time a mesh modification takes place. Thatis, once an element is split by dividing its edges alist of elements of the old mesh as well as elementsof the new mesh are handed in to the callbackfunction which then accesses the solution valuesattached to the old mesh and transfers these val-ues according to a user defined interpolation ruleonto the new mesh, see Figure 1 e. This is ofparticular importance when the mesh is modified

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during a time stepping sequence within the solu-tion of transient problems.

3. SIMULATION RESULTS

To demonstrate the efficiency of our methodwe first apply it to stationary flow in a cylindri-cal straight blood vessel. This example providesan opportunity to explicitly quantify the majorimprovements achievable with our method. Thesymmetry of the geometry and therefore the flowprofile allow easy measurement of the computa-tional savings while comparing numerical solu-tions obtained on different meshes to an analyt-ical value. The quantity of interest is the wallshear stress, defined on any point on the domainboundary. The wall shear stress can be definedvia the surface traction vector t whose compo-nents are given as

ti = (−pδij + τij) nj (5)

p denoting the pressure, τij are the componentsof the stress tensor and nj are the components ofthe normal n to the surface. The WSS magnitudethen is defined, on each point on the surface, as

tw = |tw| = |t− (t · n) · n|, (6)

that is, the magnitude of the traction vector’scomponent that is in the plane of the surface.

3.1. Straight cylindrical vessel

We compute the flow in a straight cylindricalblood vessel, as depicted in Fig.2. The vessel hasa radius of 1 cm and is 20 cm long.

3.1.1. Stationary flow

We apply a parabolic inflow profile while theouter wall is assumed to have no-slip (zero ve-locity) boundary condition and the outflow hasa zero natural pressure condition. The velocityfield only has a single non-zero component (thez-component) and this component is constant inz direction, only varying radially. Therefore ,the wall shear stress assumes a constant valueof tw = |c ν| over the entire cylinder wall. Inour example, we have set the velocity inflow pro-file such that c = 2.0 and the viscosity is set toν = 0.01dyn s/cm2.

z

Figure 2. Model of straight cylindrical vessel.The axis is aligned with the z coordinate axis.

Two different meshing strategies are comparedin what follows. First, we obtain results ona series of successively refined uniform meshes,hereby reflecting the fact that the values shouldbecome more accurate as the mesh becomes finer.Second, we try to more efficiently calculate theWSS by applying error estimation/indication andanisotropic mesh adaption. Therefore we com-pute the flow field on the coarsest uniform mesh(3021 nodes), then reconstruct the second deriva-tives of the speed field, which in turn is trans-formed into a mesh metric field according to (3).Then, the metric field is input into the mesh mod-ification tool box which, after a number of iter-ation cycles produces a highly anisotropic mesh.Fig. 3 shows the surfaces of three uniform meshesand an anisotropically adapted mesh according tothe method described in Section 2.

Table 1 shows the mean values and the stan-dard deviation for the WSS which is interpolatedon the surface mesh along a cut plane perpendic-ular to the cylinder’s z-axis, as indicated in Fig.2.

The mean values together with the standarddeviations σ cover the analytically expected valuein all four cases. Fluctuations of the WSS ob-tained on the uniform meshes are smaller thefiner the mesh is. We observe that we achievehigher accuracy in terms of both the mean andthe standard deviation when using an anisotropi-cally adapted mesh of 1258 vertices as comparedto a uniform mesh of 18184 vertices. Figure 4shows the spatial distribution of the WSS along

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3021 Nodes 6939 Nodes

18184 Nodes 1258 Nodes

Figure 3. Three successively refined uniform meshes and one anisotropically adapted mesh.

Table 1WSS mean values and standard deviation σ forstationary flow in the cylindrical vessel.

Mesh type # nodes mean WSS σ

uniform 3021 2.062e-2 1.477e-3

uniform 6939 1.977e-2 0.531e-3

uniform 18184 1.985e-2 0.388e-3

anisotropic 1258 1.991e-2 0.316e-3

the vessel wall circumference for a constant z-value.

To further quantify the amount of savingsgained through the adaptive method, we notethat, by considering the mean value along withthe standard deviation as a measure, a factorlarger than 15 reduction in the number of de-grees of freedom provides a comparable qualitysolution.

We should note that this gain factor has to bereduced when taking into account that first an ini-tial solution has to be calculated on the coarsestuniform mesh. However, the gain factors given

0 50 100 150 200 250 300 3500.015

0.017

0.019

0.021

0.023

0.025Uniform : 3021N Uniform : 6939N Uniform : 18184NAniso : 1258N

dyne

s/cm

2

Angle

Figure 4. Wall shear stress along the circumfer-ence of the straight cylindrical blood vessel forstationary flow.

above hold for the best case scenario where theflow solver scales linearly with the number of de-grees of freedom. In practice, the computer timeversus the number of nodes increases by a higherorder than linear and, therefore, our gain factor

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becomes even larger.

3.1.2. Pulsatile flow

As blood flow is unsteady in arteries, it is im-portant to understand the effect of pulsatile flowin this straight cylindrical blood vessel. Here,we use the same model that we have used forthe stationary case. The inflow boundary con-dition is assumed to be a Womersley profile withWomersley number α = 5.6 and a time periodof 5s, (Womersley, 1955), for which the flowrateis depicted in Fig. 5. As in the previous exam-ple, we apply zero (no-slip) velocity conditionson the cylinder wall and zero natural pressureon the outlet. The density and the viscosity areρ = 1g/cm3, ν = 0.04dyn s/cm2, respectively.

0 0.2 0.4 0.6 0.8 10

20

40

60

80

t/tp

Flow Vs. Time

Flow

rate

(cm

/s)

3

Figure 5. Flow rate profile at the inlet of thestraight cylindrical vessel.

Similarly to the stationary case, the only non-zero velocity component is the z-component.This component varies periodically in time ateach location within the domain, reflecting thepulsatile nature of the flow. We analyze the WSSboth at a given instant during a time cycle andtemporally averaged over the period. The latteris also known as the wall shear stress magnitude,

defined as

tmag =1

T

∫ T

0

|tw|dt (7)

where T is the time period of the cycle. tmag isdiscretely approximated in time by the sum overthe values at each time step.

As in the stationary case we compare resultsobtained by a series of uniformly refined meshes(as shown in Figure 3) to the results obtainedon an anisotropically adapted mesh. Here, incontrast to the time-independent flow, it is notadvisable to use the speed field at an arbitraryinstant during the cycle to compute the secondfield derivatives. Instead, we use the time aver-aged speed field to compute the Hessian values.An equally valid approach of considering the Hes-sian at several steps in the cycle and construct-ing a mesh satisfying the “worst case” needs willbe pursued in subsequent studies. As can be ex-pected, the mesh after adaptation looks similarto the one depicted in Fig. 3. Table 2 lists themean values, along the wall circumference (seeFig. 2), of the time averaged tmag at a constantz location. In this case we can use the analyti-cal Womersley solution to compute the wall shearstress and its magnitude and compare it to thenumerical value. Employing the adaptive tech-

Table 2WSS magnitude (tmag) mean values and standarddeviation σ for pulsatile flow in the cylindricalvessel.

Mesh type # nodes mean tmag σ

uniform 3021 2.8782 0.7638

uniform 6939 2.8328 0.2630

uniform 18184 2.6569 0.1612

uniform 103409 2.5969 0.0451

anisotropic 4238 2.6045 0.0504

analytic 2.5900

nique suggests a gain factor of around an order

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of magnitude again, when considering the meanvalues along with the standard deviations of theWSS magnitude.

We further illustrate this by presenting thetime dependent behavior of the WSS magnitudeat a specific point on the wall in Fig. 6. While theshape of the curves follow the same pattern withina cycle, the magnitude is too high for the coarsermeshes. When comparing to the analytical so-lution we can state that the anisotropic mesh iscapable of adequately resolving the time depen-dent behavior of the fields.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8Uniform : 3021N Uniform : 6973N Uniform : 18184N Uniform : 103409NAniso : 4238N Exact

t/tp

dyne

s/cm

2

Figure 6. WSS magnitude during a cycle in thestraight cylindrical vessel at a surface point fordifferent meshes.

3.2. Porcine aorta

As the last but practically most relevant appli-cation we study the performance of our methodby applying it to the simulation of pulsatile flowin a porcine aorta with a stenosis and a bypassgraft. The dimensions of the model are approxi-mately 10cm in length while the vessels diameterat the inflow is approximately 1.6cm. Blood wasmodeled as an incompressible Newtonian fluidwith a constant density of 1.06g/cm3 and a con-stant viscosity of 0.04dyn s/cm2. The geometric

model is shown in Fig. 7.

P

Figure 7. Model of porcine aorta with bypass.

The inlet flow is prescribed as pulsatile Wom-ersley flow profile that is based upon PC-MRIthrough-plane flow rate data, see Ku et al. (2002).Figure 8 shows the velocity at an instant duringthe cardiac cycle and the inset depicts the veloc-ity profile at a point near the center of the inflowplane for one cardiac cycle (together with the in-stant the velocity snapshot is taken at).

80

60

40

Velocity Profile (Pig)

t/tp

Vel

ocity

(cm

/s)

0.2 0.6 0.80.4

50 100 150 200 Speed (cm/s)

Figure 8. Flow profile during a cycle (inset) andspeed field at an instant. Bright shades indicatehigh speed.

Similar to the straight vessel cases we obtainsimulation results on five different meshes, four ofthem being uniform and one being an anisotrop-ically adapted mesh.

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Again, the latter was obtained by first comput-ing the solution on the coarsest uniform mesh,then calculating the interpolation error in termsof the Hessians of the averaged speed field. TheHessians then are converted into a mesh metricfield that in turn is handed over to the meshmodification tool box. Figure 9 shows four of themeshes used in this simulation together with theirtotal number of vertices.

A clip plane parallel to the flow direction of themain vessel further illustrates the effect of themesh modification procedure, see Fig. 12. Weobserve that slender and elongated elements werecreated in regions where flow speed and also sec-ond derivatives are highest, such as near the ves-sel walls (boundary layer effect). This becomesmore obvious when comparing Fig. 12 and theflow field at an instant, depicted in Fig. 8. Otherregions where major mesh adaption takes place(see the zooms in Fig. 12) are the stenosis area,re-entrant corners of the model and in the mainartery where the re-directed flow impacts on thevessel wall. Typically, element sizes are predomi-nately small but isotropic in the re-entrant corner,reflecting the well known fact that solution behav-ior near that region tends to be nearly singular.Small isotropic elements also dominate the steno-sis area, accounting for strongly varying flow inthat region; for example, back-flow occurs withina particular interval during the cycle.

We analyze the transient behavior of the WSSin two different ways. First, we compare the WSSdistribution for different meshes at the same in-stant during the cardiac cycle. Figs. 10 and 11show the WSS at t/tp = 0.2 and t/tp = 0.8, re-spectively. Secondly, in Fig. 13 we compare thetime dependent behavior of the WSS for the dif-ferent meshes at one particular point on the vesselwall, located near the downstream vessel branch-ing and labelled as P (depicted in Fig. 7). TheWSS are computed on a series of uniform meshesranging from 3456 nodes to over 857K nodes andon an adapted mesh consisting of 42196 nodes.

While the WSS values near the stenosis aremuch higher than at the other locations, we ob-serve that in both cases the values heavily dependon the number of degrees of freedom that are usedfor their computation, see Figs. 10, 11 and 13. On

the coarser meshes the calculated WSS generallyis too high, whereas the values tend to convergefor the finer meshes. Even though the conver-gence is not uniform during the cardiac cycle wedo observe convergence patterns that are similarto that of the pulsatile flow in the straight vesselcase.

For location P, see Fig. 13, assuming thatthe finest uniform mesh of over 857K nodes (cor-responding to over 4.7 million tetrahedra) suf-ficiently resolves all the flow features includingthe derivative quantities, we observe that theanisotropically adapted mesh follows the patternof the finest uniform mesh well in the first half ofthe cycle but is of slightly lower magnitude afterthe peak in the cycle’s second half.

The convergence behavior can be regarded suf-ficient for the uniform meshes, i.e. the meshconsisting of around 217K nodes and the meshconsisting of 857K nodes yield roughly the sameresults all over the cycle. The values of theanisotropically adapted mesh with 42K nodes fol-low those of the finest uniform mesh for the mostpart. The anisotropic mesh even seems to capturethe WSS pattern better than the 217K mesh atthe beginning of the cycle. Minor inconsistenciesoccur in the second half of the cycle. Consideringthe complex flow pattern resulting in a conver-gence behavior which cannot be fully classifiedas spatially and temporally uniform and that isfurther exacerbated by our attempt to reduce er-rors in WSS in each point on the surface, we stillare able to claim considerable gains in computa-tional time. Even though stringent margins for again factor do not seem appropriate for this spe-cific case we do observe that results are far betterthan those obtained on uniform meshes of twicethe number of nodes. And, they are quite close tothose obtained on meshes of either 217K or 857Knodes.

3.3. Discussion

We have demonstrated how computer time canbe reduced and at the same time numerical ac-curacy increased using adaptive meshing tech-niques. We have compared the quality of the so-lution that is obtained when employing our adap-tive method to the solutions obtained on a series

11

3456 Nodes 41971 Nodes

217757 Nodes 42196 Nodes

Figure 9. Three successively refined uniform meshes and one anisotropically adapted mesh (The 857Knodes mesh is not shown in this figure).

Figure 12. Clip plane through anisotropicallyadapted mesh of pig aorta.

of uniformly refined meshes. For a straight ves-sel, we first observed how the computed WSS con-verges pointwise to the value that can be expectedfrom the analytical theory. Then, we quantifiedthe convergence behavior for the same field andthe same geometry when pulsatile inlet boundary

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

Uniform : 3456N Uniform : 41971N Uniform : 217757NUniform : 857164NAniso : 42196N

dyne

s/cm

2

t/tp

Figure 13. WSS during a cycle at a point P.

conditions are imposed. Here, we also investi-gate the transient behavior of the WSS at arbi-trary locations on the domain boundary. In bothcases, our method proves to be highly efficientand a gain factor in terms of degrees of freedomof approximately one order of magnitude can be

12

>50

30

10

dynes/cm2

Uniform 41971 Nodes Uniform 217757 Nodes

Uniform 857164 Nodes Anisotropic Adaptive 42196 Nodes

Figure 10. Wall shear stress at t/tp = 0.2.

achieved.Finally, we demonstrated our anisotropic adap-

tive meshing methods on a numerical simulationof blood flow in a porcine pig aorta with a by-passed stenosis. We demonstrated pointwise con-vergence of the WSS at two particular instantsduring the cardiac cycle. We further detailed theconvergence behavior by analyzing the WSS at anarbitrarily chosen location on the domain bound-ary, located near the vessel branching. For thislocation we computed the transient behavior ofthe WSS. The complex nature of the flow in thismodel reqires more flexibility when it comes tothe quantification of the gain factors that we areable to achieve. We observed significant savingswhen we employed anisotropic adaptive methods.

4. Summary

We have presented a method by which hemody-namic finite element simulations can be enhanced.The method is based on mesh adaptivity and useserror indication together with mesh modificationtechniques. The error indicator uses interpolationestimates that yield information on the directionin which the mesh should be refined and in which

direction it can be coarsened in order to obtaina more accurate solution at considerably reducedcomputational costs.

This approach is novel in computational hemo-dynamics where it is generally assumed that thesolution field the analyst is interested in is ac-curate enough when it does not change signifi-cantly if the underlying computational mesh isuniformly refined. This traditional approach isfar too costly in terms of computer resources anddisregards the effect of possible singular valuesof the solution field and particularly its deriva-tives, on the convergence behavior of the solu-tion. Singularities of that kind are rooted in thefinite element formulation for a given model, es-pecially for complex geometries such as branchingvessels. Moreover, the non-adaptive approach ne-glects anisotropic behavior of the solution fieldssuch that a fine mesh density is used where acoarse density would be sufficient.

Future work will incorporate a more stringenterror analysis to further accelerate the efficiencyof the adaptive method. We will also extend thesimulation to larger and therefore more realistichemodynamic systems.

13

>50

30

10

2

Uniform 41971 Nodes Uniform 217757 Nodes

Uniform 857164 Nodes Anisotropic Adaptive 42196 Nodes

dynes/cm

Figure 11. Wall shear stress at t/tp = 0.8.

ACKNOWLEDGEMENTS

We gratefully acknowledge the support of thiswork by NSF grant ACI-0205741. This workwas facilitated through an allocation of advancedcomputing resources by NPACI (NRAC program)through the support of the National ScienceFoundation.

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