BMT 06.03 Steven F. Petit Validation of Mesh Machine ... · Validation of Mesh Machine Tetrahedron...

1

BMT 06.03

Steven F. Petit

ID 0489685

Validation of Mesh Machine Tetrahedron

Meshes for Flow Simulations in Sepran

Suresnes, France

31th

December 2005

Supervisors:

Franck Laffargue MediSys PMS, Paris

Sander de Putter Tue, Philips HIT, Best

Prof. dr. ir. Frans van de Vosse Tue

2

Chapter 1: Introduction ...................................................... .......................................4

Chapter 2: Theoretical Background ................................... .......................................6 2.1 The Navier Stokes equations and the Finite Element method ...............................................6

2.2 Influence of the MM mesh .....................................................................................................9

2.3 Quality and Accuracy of the meshes....................................................................................10

2.4 Quality of mesh elements in Sepran.....................................................................................11

Chapter 3: Mesh Generation and Simulation ..................... .....................................12 3.1 Mesh generation by Sepran ..................................................................................................12

3.2 Mesh generation by Sepran in combination with the method of Berent Wolters ................12

3.3 Mesh generation by MM ......................................................................................................12

3.4 Simulations...........................................................................................................................13

Chapter 4: Descriptions and analysis of the simulations ... .....................................14 4.1 Stationary flow in a axisymmetric cylindrical tube..............................................................14

4.1.1 Theoretical Section........................................................................................................14

4.1.2 Simulations....................................................................................................................16

4.1.3 Sepran Brick meshes .....................................................................................................16

4.1.4 Mesh Machine tetrahedral meshes ................................................................................20

4.1.5 Mesh Accuracy depending on Reynolds numbers ........................................................22

4.2. Oscillating flow in a axis symmetric cylindrical tube.........................................................27


4.2.2 Geometry and meshes ...................................................................................................28

4.2.3 Simulations....................................................................................................................29

4.2.4 Results of the simulations .............................................................................................29

4.2.5 Discussion and Conclusion ...........................................................................................30

4.3 Stationary flow in a diverging tube ......................................................................................31


4.3.2 Geometry and Meshes ...................................................................................................33

4.3.3 Simulations....................................................................................................................35



4.4 Oscillating flow in a Diverging Tube...................................................................................46

4.4.1 Geometry and Meshes ...................................................................................................46

4.4.2 Simulations....................................................................................................................46



4.5 Stationary flow in a curved tube. .........................................................................................50

4.5.1 Theoretical Overview....................................................................................................50

4.5.2 Geometry and meshes ...................................................................................................53

4.5.3 Simulations....................................................................................................................54


4.5.5 Discussion and Conclusions..........................................................................................62

4.6 Stationary flow in a Bifurcating Tube..................................................................................64


4.6.2 Geometry and mesh.......................................................................................................65

4.6.3 Simulations....................................................................................................................66


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Chapter 5: Discussion......................................................... .....................................76

Chapter 6: Conclusion ........................................................ .....................................79

Future Prospectives ............................................................ .....................................79

APPENDIX I ...................................................................... .....................................81

APPENDIX II..................................................................... .....................................82

4

Chapter 1: Introduction

The goal of the HemoDyn project is “the improvement of the diagnostis and treatment of

cardiovascular diseases by means of patient-specific computation fluid dynamic (CFD)

simulation of the blood flow and of the short and long term reaction of the cardiovascular system

to this flow.” In the HemoDyn project the CFD simulations are performed by means of a finite

element method (FEM). The geometrical information is obtained from segmentation of medical

images in the form of surfaces meshes. Based on the surface meshes, volume meshes are

generated. The volume meshes of the vessels are used for finite element analysis, resulting in

information regarding flow velocity, pressure and stresses. This information is used to perform

solid dynamic simulation on the walls of the vessels. The results of these simulations give insight

in short and long term reaction of the vascular system to flow.

A critical step in this process is the generation of the volume meshes based on the surface

meshes. In three dimensions, finite element meshes are usually composed of hexahedral (brick)

elements or tetrahedral elements. Associated with these types, there are two mesh models,

structured and unstructured. The structured meshes consist of a set of points with regular

connections (constant adjacency number) in each point, such that these connections can be stored

in a matrix. The unstructured meshes consist of a set of points with irregular connections. The

connections in each point should be explicitly defined and stored. The main advantage of

structured meshes with the FEM is the computation time, that can be significantly shorter than

with unstructured meshes. In addition structured meshes may result in more accurate calculations

compared to unstructured meshes with a comparable number of nodal points. The main

disadvantage is, that the connectivity constraints limit the possibility to automatically generate

meshes. Unstructured meshes, on the other hand, are much more suitable for this purpose.

For the HemoDyn project, a specially designed in house mesh generation package was used. This

package is referred to as Mesh Machine (MM). MM generates volume meshes based on

triangulated surface meshes and uses tetrahedral elements.

The combination of MM meshes and the Sepran finite element package has led to some problems

in the past. The usage of MM meshes several times resulted in Sepran simulation errors,

presumably caused by mesh quality problems. At this moment little information is available

about the accuracy of flow simulations based on the MM meshes. The goal of the present study is

to test and validate the tetrahedron meshes that are generated by MM for flow simulations in

Sepran. The methodology consists of two parts. With the first part the requirements of Sepran

regarding mesh quality are investigated and compared with the quality measures implemented in

MM. The second part consists of an evaluation of simulation results. The general methodology is

to start with a simple tubular mesh and gradually make this mesh more complex to make it

resemble an abdominal aortic aneurysm geometry. Four different geometries are used: a perfect

cylinder, a diverging tube, a curved tube and a bifurcating tube. Of all geometries both MM

meshes and brick meshes are generated. The results of the simulation with the different meshes

are compared. Simulations with stationary and with instationary flow are performed. The flow

simulations in a perfect cylinder are compared with the corresponding analytical solution. The

flow simulations with the curved tube and the bifurcating tube are compared with flow

measurement data from the PhD-thesis of Gijsen [Gijsen, 1998]. It is expected that in the

situations for which no reference data is available, brick meshes can be considered as a reference

to the test the MM meshes. This assumption is validated for the situations, for which reference

data is available

5

The following chapter describes how flow is modeled with the FEM, what the influence of mesh

quality on the FEM is and how Sepran controls mesh quality. Chapter 3 is devoted to the

different methods to generate volume meshes and describes the Sepran input for the simulations.

Chapter 4 gives an outline of the performed simulations and an analysis of the results. Chapter 5

offers a general discussion followed by the conclusion.

6

Chapter 2: Theoretical Background

In this study, it was assumed that blood in an incompressible viscous fluid that follows a purely

Newtonian law and hence can be described by the Navier – Stokes equations given by

Dpuut

uηρ 2⋅∇+∇−=

∇⋅+

∂

∂ rrrrrr

(1)

with ρ the density of the fluid, ur

the velocity of the fluid, p the pressure on the fluid, η the

constant dynamic viscosity and D the rate of strain tensor. Gravity forces are not taken into

account.

( )( )TuuD rrrr ∇+∇=2

1 (2)

The Navier – Stokes equation in 3D is thus system of 3 coupled non-linear partial differential

equations. The equations can only be solved analytically when they describe flow in a simple

geometry, like a cylindrical tube with a constant radius. In order to solve the Navier-Stokes

equation in more complicated geometries, computational methods like the Finite Element Method

(FEM) must be used. In the present study the FEM was used to solve the Navier-Stokes equations

of flow in a number of different geometries. The next section describes how the Navier-Stokes

equation is transformed into a set of discrete equations that can be solved numerically.

2.1 The Navier Stokes equations and the Finite Element method

Chapter 1 and chapter 7 of [Baaijens] were used as the basis for the information presented in this

section.

The objective of the FEM is to find approximate solutions to boundary values problems, which

are governed by partial differential equations. These problems often can not be solved

analytically. The goal is to transform the differential equations into a set of discrete equations,

that can be solved numerically. The FEM proceeds along three steps:

1 Transformation of the original set of differential equations into an integral equation by means

of the weighted residuals principle [Baaijens].

2 Discretization of the solution by interpolation. When the solution is known in a finite number of

points an approximation of the continuous solution can be found by interpolation.

3 With the discretization, the integral equation is transformed into a linear set of equations, from

which the point-values can be solved.

This procedure is explained for the Navier-Stokes equations. Before steps 1,2 and 3 can be

applied, first a θ – scheme is used to perform the temporal discretization. Secondly the equations

are rewritten in order to solve them by a Newton – iteration process.

The Navier – Stokes equations are given by (1). The fluid is assumed to be incompressible,

therefore the continuity equation reduces to

0=⋅∇ uvr

(3)

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Equation (1) and (3) are descritized in time by means of the θ – scheme. Many possible schemes

are available, but only the θ – scheme is presented here. Application of the θ – scheme yields

substitution of t

uu

t

u nn

∆

−=

∂

∂ +1 and of u = θun+1 + (1- θ)un, which leads to

( )

0

2)1()1(2

1

=⋅∇

⋅∇−+∇−−⋅∇+∇−

=

∇⋅−+∇⋅+

∆

−

u

DpDp

uuuut

uu

nn

nn

n

vr

rrrr

vvvvvvvv

ηθθηθθ

θθρ

(4)

where the notations 1+= nuuvv

and 1+= npp are used. All the t=tn quantities are moved to the left

side and all t = tn+1 to the right side. The following formulas are obtained

0

2)1()1()1(

2

=⋅∇

⋅∇−+∇−−

∇⋅−−

∆

=⋅∇−∇+

∇⋅+

∆

u

Dpuut

u

Dpuut

u

nnnn

n

rv

vvrrrr

vvrrrr

ηθθθρ

ηθθθρ

(5)

Because of the non-linearity of the uurrr

∇⋅ term the equation needs to be solved in an iterative way. For this purpose the Newton-iteration process is used. This is applied as follows.

Let f(x) = 0 be the equation we want to solve and let xreal be the exact solution and xi a first

approximation of the solution. When higher order term are neglected a Taylor series expansion

around this estimate yields

( )0)( =+ i

i

i xdx

xdfxf δ (6)

with δxi an estimate of the error in xi. This estimate can be found by solving equation (6). The

new estimation of xreal, xi+1 = xi+ δxi is used for the next iteration. This process is repeated until

it converges, i.e. it is stopped when for example one of the following terms is satisfied.

δxi < εabs (7)

or

rel

i

i

x

xε

δ<

+1

(8)

with εabs and εrel prescribed constants.

Now let iur

denote an estimate of the solution iur

, and iuv

δ the estimation of the error, both at the

ith

iteration. If this is implemented in equation (6) with θ equal to 1 (Euler implicit scheme) the

following equations are obtained (an extension for other values of θ is straightforward).

)()(2 iiiiiiii uruDpuuuut

u rrvvvrrrrrr

=⋅∇−∇+

∇⋅+∇⋅+

∆δηδδδ

δρ (9)

ii uu ⋅∇−=⋅∇vrv

δ (10)

with

8

)(2)( iiiini

i uDpuut

uuur

vvvvvvvv

vvηρ ⋅∇+∇−

∇⋅+

∆

−−= (11)

Step 1: Transformation of the differential equation into a integral equation.

At this stage step 1 is applied. The method of weighted residuals is used to transform the

differential equation into a integral equation. The method of weighted residuals states that if a

given function g(x) is equal to zero on a given domain bxa ≤≤ this is equivalent to

0)()( =∫ dxxgxvb

a

for all v.

The function v(x) is called the weighting function and can be any function that is continuous on

the integration domain. The goal of the weighted residuals method is to transform the

requirement that a function must be equal to zero on a given domain at an infinite number of

points, into a single number, the integral, that must be equal to zero. Using this method, equation

(9) is transformed into an integral equation (with omitting of the subscript i).

Ω⋅=Ω

⋅∇−∇+

∇⋅+∇⋅+

∆⋅ ∫∫

ΩΩ

durvduDpuuuut

uv )()(2

rrvvvvrrrrrr

vδηδδδ

δρ (12)

This integral contains second order derivatives of the function u. This makes it difficult to find

appropriate interpolation functions. Fortunately these second order derivates can be removed by

integration by parts leading to

Ω⋅=Ω⋅∇−Ω+Ω

∇⋅+∇⋅+

∆⋅ ∫∫ ∫∫

ΩΩ ΩΩ

drvdvpDdDduuuut

uv v

rvvvrrrrrr

vηδδ

δρ 2: (13)

and

∫∫ΩΩ

Ω⋅∇−=Ω⋅∇ duqduqvvvv

δ (14)

with D the rate of strain tensor of u and Dv the rate of strain tensor of v.

Step 2: Spatial discretization of the solution u. Suppose the values of a function u are known in the points xi and u(xi) = ui. A polynomial

approximation of degree n-1 of u, uh is given by

uh = a0 + a1x + a2 x2 + … + an-1x

n-1

if u is known in n points. The coefficients ai of uh can be expressed by ui and can be found by

solving

=

−

−

−

nnn

nnn

n

n

u

u

u

a

a

a

xxx

xxx

xxx

MM

L

MMMMM

L

L

2

1

1

0

12

1

1

2

22

1

1

2

11

1

1

1

(15)

The values ai are linearly dependent on ui, therefore the polynomial may be written as

9

∑=

=n

i

iih uxNu1

)( (16)

with the shape function, Ni(x), being linear functions of ai and polynomial expressions of the

order n-1 in terms of the coordinate x. We chose to use the Galerkin method, i.e. the same shape

functions are used for the unknown u and the weighting functions v.

The differentiation of uh is straightforward since ui are not dependent on x and Ni(x) are simple

functions of x. The shape of the shape functions is dependent on the type of element used and on

the interpolation method.

Using (16) equation (13) can be transformed to

e

Tee

Tpu

Teeuue

Tee

C

eTee

C

eTe

ee

Te

fvpKvuKvuKvuKvt

uMv =++++

∆δδδ

δ 21 (17)

which is the discrete counterpart of (13) for one element with ve and δue vectors containing the

information of the nodal points in element e. The matrices Me, KeC1

and KeC2

, Kuue and Kpu are

matrices composed of integrations of the shape functions and there derivatives. KeC2

and KeC1

are

functions of u and v as well. fe is the term associated with the residual, r, of equation (13). For a

detailed description of the matrices, we would like to refer to chapter 7 of [Baaijens].

In a similar manner equation (14) is transformed yielding

eT

eepu

T

e gvuKv =δ (18)

with ge a vector associated with the incompressibility term.

Step 3: Formation of the final set of equations

After the assembly process where the matrices and the vectors describing the solution in the

elements are assembled to large matrices and vectors for the entire mesh, the resulting set of

equations is given by

=

+++

∆~

~

~

~

21

~

~

000

01

g

f

p

u

K

KKKK

p

uM

t pu

up

CC

uuδδ

(19)

2.2 Influence of the MM mesh

The matrices M, KC1

, KC2

, Kuu, Kup and Kpu in equation (19) are dictated by the mesh on which

the problem is solved. The equations of the shape functions and the number of shape functions

used per element is directly dependent on the type of elements the mesh is composed of. The

manner in which Me, KeC1

, KeC2

, Kuue, Kupe and Kpue are assembled to form M, K

C1, K

C2 , Kuu, Kup

and Kpu respectively is dependent on the connections between the different elements. The

accuracy of the interpolation is dependent on the number of elements in the mesh and the

conditioning of the mesh is important for calculation time and the accuracy. The mesh is thus a

key factor in determining the outcome of the FEM. The question that now arises is: what is a

good mesh for a particular application with the FEM? In this study meshes that are composed of

hexahedrons with 27 points are compared with meshes that are composed of tetrahedrons with

10

15 points. An advantage of the usage of tetrahedron elements is that it is possible to mesh more

complex 3D domains than is possible with hexahedrons. The usage of hexahedrons on the other

hand is known to result in more accurate calculations. In this study the differences between

tetrahedron and hexahedron meshes for our application are investigated.

2.3 Quality and Accuracy of the meshes

Optimal sizing and shaping of the elements is very important for accurate and fast calculations

with the FEM. Just a few badly shaped elements can compromise accuracy and speed of some

applications. Unfortunately our understanding of the relationship between mesh geometry,

numerical accuracy and stiffness matrix conditioning remains incomplete. Roughly spoken

equilateral tetrahedral elements perform usually well and skewed tetrahedral elements are usually

bad. Brick elements perform best when their shapes are close to cubic.

Shewchuck [Shewchuck, 2002] analysed tetrahedron element quality for FEM. He showed that

errors in the interpolation can be reduced by using smaller elements. However, to avoid large

differences between the gradient of the true function and the gradient of the approximation, using

smaller elements does not suffice. In addition elements with angles approaching 180o should be

excluded from the mesh. Large conditioning numbers of the stiffness matrix mean that iterative

solvers will run slowly and direct methods may incur excessive roundoff errors. The conditioning

number K is defined as minmax / λλ=K with maxλ the largest and minλ de smallest eigenvalue of

the element stiffness matrices of the elements. minλ is hardly dependent on the shape of the

elements and becomes smaller as the elements decrease in size. Shewchuck shows that the largest

eigenvalue can grow arbritary large when an angle in an element approaches 0o or 180

o. Taking

all the information into account, it can be concluded that in general a good mesh contains

elements with shapes close to perfect tetrahedrons, that are small enough to results in low

interpolation errors, but that are not too small in order to avoid long calculation times. In order to

compare the shape of an element to a perfect tetrahedron a number of different quality measures

can be found in the literature. Shewchuck [Shewchuck, 2002] himself proposes a number of

different quality measures related to interpolation error or stiffness matrix conditioning for a

single element. According to Seveno [Seveno, 1997] the most commonly used quality measures

are the inner radius over the longest edge ratio, q1, and the inner-radius over circum-radius ratio,

q2. Both are normalized between [0,1]. They are defined as follows

(20)

Rq

ρα=2 (21)

with

S

V3

32

=

=

ρ

α

hq

ρα=1

11

and V the volume of the tetrahedral element, S the surface of the faces, h the longest edge of the

tetrahedron and R the radius of the circumsphere of the tetrahedron. Both quality measures

respond approximately in the same manner, but q1 manages to discriminate better between good

and bad elements [Laffargue, 2005]. We chose to use the distribution of elements with q1 values

to express mesh quality of tetrahedral meshes.

For hexahedron elements less information was available in literature regarding element quality. A

minimal condition is that the determinant of the Jacobian matrix of the vertices of the elements

should be positive, corresponding to a positive volume. For a vertex of an 8-point hexahedron the

Jacobian matrix is formed as follows. Let x in R3 be the position of this vertex and xi in R

3 for i

=1, 2 ,3 be the position of the neighbours in a fixed order. Using edge vectors ei = xi – x with i =

1, 2,3 the Jacobian matrix is then A = [e1, e2, e3]. The determinant of the Jacobian is usually

called the Jacobian. The fixed order is obtained in the following manner. First a top and a bottom

face are defined. Then, starting at vertex x, an imaginary quarter of a circle between the two other

vertices in the face that contains vertex x is drawn (thus the top or bottom face), with vertex x as

center. When one looks from the opposite plane to this plane and imagines that the quarter of a

circle is drawn counterclockwise, the node 1 is the starting point of the quarter of the circle, node

2 is the end point of the quarter of the circle and node 3 is the point that is not in the plane.

The Jacobian of an arbitrary tetrahedral element consisting of four vertices vi with i = 0,1,2,3 with

coordinates xi in R3 and edge vectors ek,n = xk –xn with k ≠ n can be found as follows [4]. Vertex

vn has three attached edge vectors, en+1,n, en+2,n and en+3,n. The Jacobian matrix, An, of vertex vn is

given by

An = (-1)n ( en+1,n en+2,n en+3,n ) (22)

The Jacobian is defined as the determinant of An. A right-handed rule is assumed for the edge

ordering, so that a positive volume of a tetrahedron corresponds with positive Jacobians of the

vertices.

The Jacobians of 27 points hexahedrons and 15 points tetrahedrons can be found in the same

manners as described above, when only the corner vertices are analyzed.

2.4 Quality of mesh elements in Sepran

Sepran uses only one criteria to test whether a mesh suffices. Namely the Jacobians of all

elements. Sepran checks whether they do not become too small and do not have changing signs.

Unfortunately it is not known if Sepran uses a threshold value to test the elements and what value

a possible threshold would have. The Jacobian of an element is dependent on the volume. It is not

known how Sepran takes this into account when working with different meshes with dimensions

of different order of magnitude.

12

Chapter 3: Mesh Generation and Simulation

This chapter describes the different methods that are used to create meshes based on the analysed

geometries. The first section describes the user defined mesh generation of hexahedral meshes in

Sepran. The second section describes how the hexahedral mesh of the curved tube is constructed

by the method of Berent Wolters and the third section describes the mesh generation by MM.

3.1 Mesh generation by Sepran

Sepran allows the user to manually define the nodal points of the mesh, the curves that connect

the nodal points, the different surfaces and the different volumes. This is a relatively easy yet

very tedious way to create a desired hexahedron or tetrahedron mesh of a geometry. However the

method is limited by the available surface generators that for instance can not create the surface

of a curved cylinder. The hexahedron meshes of the cylindrical tube and the diverging tube are

constructed by the Sepran mesh generator. To construct a mesh of a curved tube a program

provided by Wolters was used.

3.2 Mesh generation by Sepran in combination with the method of Berent Wolters

First Sepran is used to create a mesh of a cylindrical tube. The user specifies the centreline and

the radius of the desired curved tube. Based on this information contours are create that describe

the surface of the desired mesh. A Fourier fit is made on the data points in the contours to make

the data more smooth. With the Fourier components of the contours the simple cylinder is formed

into the desired curved tube.

3.3 Mesh generation by MM

MM was designed to make tetrahedral meshes of deformable model based segmentations of

human organs based on imaging data. For our purpose MM was adapted to make tetrahedral

meshes of predefined geometries. The MM software is based on the Constrained Delaunay

Tetrahedrization algorithm. A tetrahedron is Delaunay valid if and only if it circumsphere defined

by 4 vertices encloses no other vertex of the mesh. The entire mesh is Delaunay valid when all its

elements are. Based on this criteria an iterative method is designed to add new vertices to a mesh

that is Delaunay valid. The method works as follows. A vertex is added to the mesh, and all the

tetrahedrons of which the circumspheres overlap the new vertex are located and removed from

the mesh (not the vertices). This defines an enclosing cavity. New tetrahedrons are created in the

cavity by connecting the new vertex to the cavity faces.

In order to generate a tetrahedral mesh of a geometry MM needs the corresponding surface mesh

as input. In practise the procedure is as follows. The tetrahedral mesh is initialised by a bounding

box containing 5 tetrahedrons. Next the vertices of the input triangle mesh are inserted one by

one into the newly created bounding mesh. The tetrahedrons of which the circumsphere contains

the new nodal point are located and new tetrahedrons are formed based on the Delaunay

13

principle. At the end the result is a Delaunay valid tetrahedral box mesh containing all the

vertices of the input triangle mesh together with the 8 vertices of the box. Based on the initial

triangulation in the volume, the tetrahedrons that are located outside the input object are removed

and possibly additional tetrahedrons are removed or reconfigured to recreate the initial

triangulation of the surface of the object.

At this stage the quality of the tetrahedral mesh is extremely poor and the mesh needs to be

refined. The goal of the refinement process is to generate a mesh containing only elements with

shapes close to the shape of a perfect tetrahedron and without large differences in element size.

To ensure that both requirements will be fulfilled the refinement process is guided by a quality

function (MM uses q1 (20)) and a size function that calculates the optimal size of a tetrahedron

based on the sizes of the triangles at the surface. After the refinement process, the mesh is

optimized by relocating the mesh vertices that do not lie on the surface without altering the

tetrahedrization. For this purpose a constrained version of the Laplacian smoothing is

implemented, that will only relocate the vertices if this increases the mesh quality.

3.4 Simulations

A laptop PC with a RAM memory of 320 MB and a speed of 650 MHz was used in combination

with the Sepran package to perform the FEM calculations. The available memory allowed

calculations with meshes with a maximum number of nodal points around 22 000. To limit

calculation time, in general meshes are generated with around 10 000 nodal points.

No slip conditions served as boundary conditions at the wall. At the plane(s) of outflow stress

free boundary conditions were applied for most simulations. At the plane of inflow the velocity

was prescribed. For the simulations with time-dependent inflow a Euler implicit method was used

to descritize the equations in time. 64 steps per period were used. Before the first period a linear

increase in inflow velocity was applied with a length equal to one period. 4 complete periods

were computed. The convective terms in the equations were linearized with the Newton scheme.

The dynamic viscosity of the fluid was equal to 3.5 10-6

Ns/mm [www.cvphysiology.com] and

the density was equal to 1.05 kg/mm3 [www.phsysic.nist.gov]. These values are based on the

characteristic parameters of blood, although blood is a non-newtonian fluid with varying

viscosity.

The equations of the simulations with the Newtonian fluid were solved using the integrated

method with elements of type 902. An explanation of this type of element can be found in [Segal,

2003] at paragraph 7.1 and page 23. Renumbering of the unknown pressure and velocity

quantities was performed to avoid zero diagonal elements in the assembled matrix. A

BiCGSTAB solver was applied and the iteration process was stopped when the following criteria

was satisfied

3

010, −=≤ εε

res

resk

(23)

with resk and res

0 the residual at iteration k and 0 respectively.

14

Chapter 4: Descriptions and analysis of the simulations

This chapter contains descriptions and the results of the different tests that are performed. Each

test is described in a single paragraph. The paragraphs describe the goals of the tests, the used

geometries and meshes, the simulations and the analysis of the results. The first paragraph

contains tests of flow in a cylindrical tube with stationary inflow. The second paragraph describes

a test with a cylindrical tube and instationary inflow. The third and fourth paragraph contain

simulations with a diverging tube with stationary and instationary inflow respectively. The fifth

paragraph describes a test with a curved tube with stationary inflow and the sixth paragraph is

devoted to the simulations with a bifurcating tube with stationary inflow. The simulations with

the curved tube and the bifurcating tube are compared with measurement data by Gijsen [Gijsen,

1998]. The dimensions of the meshes are based on geometrical characteristics of the aorta. In

general the meshes have a radius of 10 mm and a length of 100 mm.

4.1 Stationary flow in a axisymmetric cylindrical tube

In the first situation steady, fully developed flow in a cylindrical tube with constant radius is

modeled. The cylindrical tube is the basis of the other geometries. For this situation the analytical

solution of the Navier – Stokes equations is available. For these reasons it offers appropriate

conditions to test basic problems.

As mentioned in the introduction it is expected that the calculations with the brick meshes can act

as references for the calculations with MM meshes. Since the quality of the meshes is dependent

on the element distributions, the first tests are performed to find the brick element distribution in

a cylindrical tube that leads to the highest accuracy of the solutions. This reference element

distribution (or variations of it) is used in the rest of the study.

The second test regards the quality of the MM meshes. As mentioned in the introduction one of

the goals of the present study was to get more insight in the quality of MM meshes regarding

accuracy of the solution and stability of the calculations. With the second test the quality of the

MM meshes is lowered and the effect on accuracy of the solution and stability of the calculations

is studied.

With the third test simulations are performed to test the influence of the Reynolds number on the

accuracy of the numerical solution with the brick mesh and the MM mesh.

This paragraph starts with a theoretical section about the Navier –Stokes equations of fully

developed, stationary flow in a cylindrical tube, a section that presents information of the

simulation, and three different sections for the performed tests.

4.1.1 Theoretical Section

The information presented in this section comes from [Heijst, 2002].

Dimensionless Navier- Stokes equation

Often it is convenient to express the Navier-Stokes equation in dimensionless quantities. The

equations can be made dimensionless by means of characteristic scales of the length, time and

velocity. A typical velocity, length and frequency are defined as L, V, ω-1

respectively. The

velocities are scaled by dividing by V, the dimensions are scaled by dividing by L, the time is

scaled by multiplying with ω and ∇ is scaled by multiplying with L. De pressure is made

15

dimensionless by dividing by ρV2. The Navier - Stokes equations (1) with the scaled quantities

becomes

( ) '2'''''''

'2

22

DL

Vp

L

Vuu

L

V

t

uV η

νϖ ⋅∇+∇−=∇⋅+

∂

∂ rrrrrr

(24)

The terms with the primes are dimensionless, the terms without primes are not. The equations are

made dimensionless by dividing by V2/L. This yields

( ) '2'''''''

'D

VLpuu

t

u

V

Lη

νϖ⋅∇+∇−=∇⋅+

∂

∂ rrrrrr

(25)

with

1/Re =VL

ν (26)

the inverse of the Reynolds number, (Re), and

Sr = V

Lϖ (27)

The Strouhal number. The Re is a measure of how important the convective terms are in relation

to the viscous terms and the Sr gives the relation between the importance of the instationary

terms and the convective terms.

Analytical solution of fully developed, stationary flow in a cylindarical tube.

In this paragraph simulations with fully developed flow with stationary boundary conditions are

analysed. The flow in the cylindrical tube is presumed to be axis symmetrical, fully developed

and only in the axial direction. If we use a cylindrical coordinate system (Appendix I) with the

axial axis in the z-direction, it follows from these presumptions that

)(,0,0,0 rvvvvz

uzzr ====

∂

∂=

∂

∂θ

θ

When this is applied to the Navier – Stokes equations with the following boundary conditions

vz(a) = 0 and at r =0 0=dr

dvz

the result is a Poiseille profile:

−=

2

2

12)(a

rurv mz (28)

with vz(r) the axial velocity in the cylinder as function of the radial distance to the central axis, um

the mean axial velocity, r, the radial distance to the axis and a the inner radius of the tube. The

velocity in the directions perpendicular to the radial axis are in this situation equal to 0. The full

description of the obtainment of this results can be found in [Heijst, 2002].

Inflow length of stationary flow in a cylindrical tube.

Because of viscosity effects at the border of the wall of a tube and the fluid, a boundary layer will

develop between the wall of the tube and the core of the flow. This is a layer of fluid that is

moving with a lower velocity relative to the main flow. More about boundary layers can be found

in the theoretical section of paragraph 4.3. The flow in a tube is called fully developed when the

boundary layer contains the complete cross section. When a flat velocity profile is prescribed on

16

the inflow boundary of a cylinder, the flow needs an inflow length to become fully developed.

The inflow length for stationary, laminar flow in a cylindrical tube is according to [Schlichting,

1960] given by

Le = 0.112Re a (29)

4.1.2 Simulations

The properties of the simulations are exactly as described in paragraph 3.4.

To compare the axial velocity in the nodal points calculated by the FEM and with the analytical

solution, the nodal points are divided into 50 categories dependent on their radial distance to the

central axis. For each category the difference between the calculated solution and the analytical

solution, eabs, in the nodal points is calculated with

∑=

−=N

i

iexactiFEMabs vvN

e1

,,

1 (30)

VFEM,i is the velocity in the axial direction in nodal point i calculated with the FEM and Vexact,i is

the analytically calculated velocity in nodal point i, which is solely dependent on the radial

distance to the central axis. N is the number of nodal points in a category. The relative error is

expressed by

∑=

−=

N

i iexact

iexactiFEM

relv

vv

Ne

1 ,

,,1 (31)

4.1.3 Sepran Brick meshes

This section describes the different simulations that are performed to find the element distribution

in a Sepran brick mesh that leads to the most accurate solution. A cylindrical tube with a radius of

10 mm and a length of 100 mm was used. The tube was aligned in the negative y-direction, i.e.

the plane of inflow has a y-coordinate of –100. Figure 1 shows the cross section of a brick mesh

on a cylinder. Different brick meshes are used to investigate the effect of the element distribution

on the solution. The number of elements along curves c, s, r and l are varied. Where c, s, r and l

are, respectively, the number of elements along a quarter of the circumference of the cross

section, the number of elements along the sides of the inner square, the number of elements along

the lines connecting the corners of the inner square with the circumference and the number of

elements along the length of the cylinder. Table 1 sums up the different geometries that are used

(e.g. 55510 means c = 5, s = 5, r =5 and l = 10). Simulations are performed with an inflow

velocity of 5 mm/s. This corresponds to a Re equal to 30 and a Le of 34 mm. The Re of 30 is low

in relation to flow in AAA’s. But to analyze fully developed flow in the cylinder a cylindrical

tube with a length higher than Le should be used. Higher Re’s lead proportionally to higher Le’s

and in order to be able to analyze fully developed flow, longer geometries must be used. It would

have been possible to prescribe a Poiseuille profile at the plane of inflow to simulate fully

developed flow, but since the results of the simulations are compared with this profile, little

difference between the accuracy of the simulations with different meshes is expected.

Since the other meshes of the present study are based on the dimensions mentioned in the

introduction of this chapter, and since the goal of this test is to find a good element distribution in

brick meshes that can be used for the other geometries, we stick with a mesh with a length of 100

mm. This is a limitation for the maximal Re. The results of these tests are shown in Figure 2.

17

Figure 1: Element distribution in the planes perpendicular to the central axis.

Table 1. Element distributions of the brick meshes that were tested.

Mesh configuration Number of elements Number of nodes Description

222100 (i.e. 100 elements

in the length)

2000 17889 A large number of elements in

the direction of the flow, few

elements on the cross section

perpendicular to the flow.

33335 (i.e. 35 elements in

the length)

1575 13703 This leads to elements that are

more or less cubic, i.e. the

length of the elements is equal

to the square root of the

average surface of one element

in the planes perpendicular to

the flow direction.


the length)

1600 13817 Less elements in the length

more elements in the planes

perpendicular to the flow

direction.


the length)

1250 10941 Less elements in the length

more elements in the planes

perpendicular to the flow

direction

18

Figure 2: The absolute difference between VFEM and Vexact as function of the radial distance to the center of

the tube. The curves shown are the averages of all data points with -50 mm < r < –20. The errorbars represent

the standard deviations. From left to right and top to bottom, the first figure represents Brick Mesh 222100,

the second Brick Mesh 33335, the third Brick Mesh 44420, and the fourth Brick Mesh 55510.

Figure 2 shows the absolute value of the difference between the velocity in the direction of the

central axis calculated with the FEM method and with the analytical solution. The error bars

represent the standard deviation. According to (29) the inflow length in this situation is equal to

33.6 mm. To study only fully developed flow and to minimize outflow effects only nodes from

the second half of the tube (-50 < y < -20) are taken into account. The figures suggest that a

higher resolution of elements in the plane perpendicular to the central axis results in a lower

absolute error of the solution in the nodal points.

With equation (31) the average relative error, erel, is calculated in each category. Figure 3 shows

it as a function of the radius.

19

Figure 3: The relative difference between VFEM and Vexact as function r. Only the nodal points with -50 < r 8 mm, there exists no strong relation between the

location of the nodes and the erel, which is more or less constant. The low standard deviations

indicate that the solution of the FEM method is solely dependent on the radial distance to the

central axis of the tube. This corresponds well to the analytical solution. For nodes with r > 8 mm

the erel fluctuates more strongly, but it can be concluded that this is the effect of dividing the error

in this region by low (exact) velocity values. Figure 3 shows clearly that the 55510 conformation

leads to the lowest error. Table 2 shows the total calculation time and the calculation time per

nodal point per iteration of the simulations with the different meshes. It seems remarkable that

the mesh with the smallest number of nodes (55510) results in the highest calculation time.

However, this mesh contains the most elongated elements and Sepran fails to construct the

preconditioning matrix, which resulted in higher calculation times. Based on the results, it is

suggested that brick elements that are elongated in the direction of the flow do not necessarily

influence the accuracy of the solution.

Table 2: calculation time, the number of iterations needed before convergence is reached and the relative

calculation time per iteration, per node.

Mesh Calculation time [s] Number of iterations

needed

Calculation time per node per

iteration [10-4

s]

222100 132 45 1.64

33335 136 61 1.63

44420 176 88 1.45

55510 427 351 1.09

20

Discussion and Conclusion

For the four meshes the solution in the nodes is almost exclusively dependent on the radial

distance of the nodes to the central axis and the relative error is almost constant as function of the

radial distance. The FEM method of Sepran performs, for a situation with a cylindrical tube with

stationary inflow, best for the 55510 configuration even though this mesh has the smallest

number of elements and nodes. However there is little difference with the 44420 configuration,

although is has more elements and nodes. Both methods result in average relative errors of 4% to

5%. It is remarkable that the 33335 mesh with elements with close to cubic shapes responds

significantly less accurate than the 44420 and 55510 meshes, since the latter two contain more

elongated elements. It is suggested that, because of the elements are aligned in the direction of

the flow, and the gradients are small in this direction, elongating the elements is possible without

the loss in accuracy. For more complicated flow the effect of elongated elements on the accuracy

of the calculations is usually larger. Analysis of the calculation time showed that although the

55510 mesh contains the lowest number of nodes, the calculation time was significant longer than

with the other simulations. This is due to the fact that Sepran does not manage to construct a

preconditioning matrix with this mesh, resulting in a large number of iterations that is needed.

4.1.4 Mesh Machine tetrahedral meshes

MM optimizes the distribution and shape of tetrahedron elements in its meshes. In this section the

effect of lowering mesh quality on calculation time and on the accuracy of the solution is

investigated for MM meshes. As a starting point a simulation is performed with an optimized

MM mesh of a cylindrical tube with a radius of 10 mm and a length of 100 mm. The mesh

contains 10773 nodal points and 2236 tetrahedron elements. At the inflow boundary a flat

velocity profile is prescribed of 5 mm/s corresponding to a Re of 30. The nodal points that lie

between y = -60 and y = - 20 are analyzed to minimize inflow and outflow effects. The graph on

the top of Figure 4 show erel.

An extension of the MM program was generated that randomly moves the vertices of the

elements of the optimized mesh to produce a mesh with at least one element with a user defined

worse quality. 5 Meshes are created this way with a lowest q1 values around 0.005 (on a scale of

0 to 1). These meshes are thus all based on the same cylindrical tube and contain the same

number of elements.

Results

The first simulation resulted in a Sepran error, stating that the Jacobian of element 391 was

almost equal to 0. This element had a q1 value of 0.0061. This was not the element with the

lowest q1 value in the mesh. The other 4 simulations resulted in converging solutions. The results

are shown in Figure 4. Table 2 shows the q1 values of the elements with the lowest q1 values in

the 5 meshes and the calculation time of the simulations. The calculation time of a similar

simulation with an optimized mesh with lowest q1 of 0.27 was 59 s.

21

Table 3: The minimal q1 values of the elements in the different meshes and the calculation time needed.

simulation Minimal q1 Calculation time [s]

Mesh 1 0.0056 -

Mesh 2 0.0041 57

Mesh 3 0.0048 60

Mesh 4 0.0040 58

Mesh 5 0.0043 56

Optimized mesh 0.27 59

Figure 4: The graphs show erel as a function of r. The top figure represents an optimized mesh, the other

figures represent 4 different meshes with minimal q1 values around 0.005.

The difference in the relative error as function of the radial distance to the central axis is

practically independent of the used mesh. All 5 meshes result in an almost identical solution,

including the simulation with the optimized mesh. Since the minimal q1 values of the four

different meshes are all lower than the q1 value of the elements that caused the error, the results

suggest that for this particular situation, the minimal q1 value of the mesh is not the only or most

important factor in determining simulation outcome.

Extra simulations are performed to investigate whether the positions of the badly shaped elements

are of importance , but these simulations revealed no results of interest.

The result of the simulations with the meshes with a few badly shaped elements, suggest that one

skewed element in a mesh can cause Sepran to stop, but that it is not necessarily the case. In

addition a number of simulations are performed with meshes with a certain percentage of all

elements having q1 values between 0 and 0.1. The first 3 simulations are performed with meshes

22

that contain 9.5% of these elements. All simulations resulted in errors, stating that the Jacobian of

a certain element was too small of had changing sign. This was also the case with three

performed simulations with meshes with 5% of the elements having q1 values between 0 and 0.1.

Two out of three simulations with meshes with 1% of the elements having q1 values between 0

and 0.1 resulted in converging solutions, which were equal to the solution with the optimized

mesh. The third simulation resulted in a Sepran error. The highest value of the Jacobian of one of

the elements that caused the error was equal to 0.978. Regarding q1 the highest value of one of

these elements was equal to 0.0168.


Based on all performed simulations, Sepran errors were not caused by elements with q1 values

higher than 0.0168. The results suggest that the presence of bad elements does not necessarily

influence the accuracy of the solution, nor did it influence calculation time. This is in contrast to a

number of published articles. The fact that a relative simple geometry is used could be of

influence as well as the low Re of the simulations. It is possible that in situations with more

complicated flow one badly shaped element always leads to an error in Sepran. The exact

mechanism with which Sepran uses the Jacobian to judge elements remains unclear. Nor is it

understood whether the location of the skewed elements is of influence on simulation outcome.

These results show that the chance on an error in Sepran increases as the number of elements

with low q1 increases. More experiments need to be carried out to investigate what the exact

criteria of Sepran are. In addition simulations with more complicated geometries should be

performed. Due to time restrictions these additional simulations are not included in the present

study.

4.1.5 Mesh Accuracy depending on Reynolds numbers

The influence of the value of the stationary inflow velocity on the mesh accuracy was

investigated. Simulations were performed with Reynolds numbers of 30, 150 and 300

respectively. Likewise simulations were performed to find the maximum inflow velocity and

corresponding Reynolds number for which the solution of the FEM converges. Equation (29) was

used to calculate the inflow lengths, Le. For the different inflow velocities, the entrance lengths in

a tube with a equal to 10 mm are 33.6, 168 and 336 mm respectively. In order to compare the

numerical solution with the analytical solution for fully developed stationary flow, meshes of

cylindrical tubes must be used with lengths longer than Le. MM tries to generate meshes with

tetrahedrons with shapes as close as possible to perfect tetrahedrons. This means that lengthening

a tube and fitting a new mesh (with a constrained number of points), leads to a lower spatial

resolution of elements in the directions perpendicular to the axial direction compared to the mesh

of the original tube. The element distribution in the original mesh is thus not a scaled version of

the element distribution in the new mesh. There is thus a trade-off between the maximum Re that

can be analyzed and the spatial resolution of the elements in a plane perpendicular to the flow, for

a constrained number of nodal points. In section 4.1.3 it is explained why a Poiseuille profile was

not used to simulate fully developed flow.

23

Figure 5: On the left side the element distribution on the plane of inflow of the tube with a length of 500 mm.

On the right side the element distribution on the tube with a length of 100 mm.

For the simulation with Re equal to 30 a tube with a length of 100 mm and a radius of 10 mm is

used. This is too short for the simulations with Re equal to 150 and 300. For these numbers a tube

with a length of 500 mm and a radius of 10 mm was used. It was chosen to use meshes with

approximately 10 000 nodal points for the first tube and with approximately 22 000 for the

second tube. This last number is close to the maximum number of nodal points in a mesh that is

allowed by the available amount of memory of the PC. Figure 5 shows the two different MM

meshes. Table 4 shows the number of nodes and information regarding the q1 values of the

elements of the different meshes.

Table 4 : The number of nodal point of the 4 different meshes.

Mesh Length tube

[mm]

Number

of nodal

points

Minimal

q1

Average

q1

Brick 100 10941 - -

MM 100 12037 0.35 0.70

Brick 500 19277 - -

MM 500 23257 0.47 0.71

The extended brick mesh is basically a scaled version of the original brick mesh; the element

distribution perpendicular to the central axis is an exact copy. In the direction of the flow 18

elements are used, instead of 10. These elements are 2.7 times as long as the elements in the

original mesh.

The numerical solution in the nodal points between Le and the length of the tube is compared

with the analytical solution of fully developed stationary flow in a tube. The results are shown in

Figure 6, Figure 7 and Figure 8.

24

Figure 6: erel and eabs as function of r. The graphs on the right side represent the Sepran brick mesh and the

graphs on the left side represent the MM mesh. The inflow velocity is equal to 5 mm/s corresponding to a Re

of 30.

Figure 6 shows that the absolute and relative errors were more than twice as low with the MM

mesh as with the brick mesh (0.016 compared to 0.042). The graphs show that the relative error is

almost independent on the radial distance of the nodal points to the centerline for both

simulations. The low standard deviation indicates that the flow is almost perfectly axis

symmetrical. The calculation time with the MM mesh was 114 s, whereas it was 427 s with the

Sepran brick mesh. The results of the simulation with a Re of 150 with the MM mesh are

presented in Figure 7. The absolute error varied between 0 mm/s at the wall and 1.3 mm/s near

the central axis, resulting in a relative error between 0.025 and 0.03. The calculation time was

equal to 468 s. The simulation with the brick mesh was manually shut down after 12 hours. For

this reason no results are presented.

25

Figure 7: erel and eabs as function of r. The graphs represent the MM mesh. The inflow velocity is equal to 25

mm/s corresponing to a Re of 150. The calculation time was equal to 468 s. The corresponding simulation

with the brick mesh was manually shut down after 12 hours.

Figure 8: erel and eabs as function of r. The graphs on the right side represent the Sepran brick mesh and the

graphs on the left side represent the MM mesh. The inflow velocity is equal to 50 mm/s corresponding to a Re

number of 300. The calculation time of the simulation with the MM mesh was 663 s, whereas the calculation

time with the brick mesh was 38 275 s (10.6 h).

Figure 8 shows the results of the simulations with a Re of 300. The Sepran Brick mesh results in

a relative error between 0.04 and 0.05. It decreases slightly with increasing r. The absolute error

26

is parabolic dependent on r. The MM mesh results in an average relative error between 0.022 and

0.03. The corresponding absolute error is parabolic dependent on r. The shear stresses are of the

order of magnitude 103 higher with the MM mesh than with the brick mesh. With both meshes

these stresses are constant on the wall except near the inflow boundary. The pressure on the wall

of the cylinder shows similar results with both meshes. The pressure is highest at the plane of

inflow and decreases to reach a minimum at the plane of outflow. Table 3 shows the minimal and

maximal pressure values for both meshes. The calculation time of the simulation with the brick

mesh was 57 times as high as with the MM mesh.

Table 5: maximum and minimum pressure values in units and dimensionless.

Mesh Max pressure [10-3

mPa]

Max Pressure

[-]

Min pressure [10-5

mPa]

Min Pressure

[-]

Brick 10 190 -7.65 -1.4

MM 8 152 -3.95 -0.7

Additional simulations are performed to investigate what the maximal Re is for which the

solution converges. The experiments are performed with a tube with a radius of 10 mm and a

length equal to 100 mm. The tube is thus too short to result in fully developed flow. The 55510

Sepran brick mesh is used and the optimized MM mesh. Simulations with the Sepran brick mesh

resulted in a maximum Re higher than 1200, the same is true for the MM mesh. The calculation

time of the simulation with the MM mesh was almost twice as short as with the Sepran brick

mesh (1098 s compared to 1986s).


The simulations show that for Re equal to 30 the MM meshes result in more accurate solutions of

the axial velocity than the Sepran Brick meshes. This Re corresponds with a velocity of 5 mm/s.

This velocity is very low considering that velocity values up to 900 mm/s are observed in the

aorta. The calculations with the brick mesh took 4 times as much time as with the MM mesh. The

simulation Re = 150 with the MM mesh took 468 s and the corresponding simulation with the

brick mesh did not results in a converging solution after 12 hours of calculated time. The

simulations with Re equal to 300 resulted in lower errors (absolute and relative) of the velocity in

the nodal points close to the centerline, with the MM mesh than with the Sepran brick mesh. The

pressure at the wall of the cylinder agreed with the brick mesh and with the MM mesh. The

calculation time was 57 times shorter with the MM mesh than with the brick mesh, suggesting

that the former mesh results in a more stable system of equations.

Basically the comparison of the brick mesh with the MM mesh is not only a comparison of the

response to different Re, since the brick mesh and the MM mesh are adapted differently to the

different Re. The brick mesh is adapted to maintain a radial resolution, whereas the MM mesh is

adapted to maintain high quality elements. This resulted in a dramatic increase in calculation time

with the brick mesh. Simulations are performed with meshes of a cylindrical tube of 100 mm to

find the maximum Re numbers which result in a converging solution. The flow in these situations

was thus not fully developed. With both meshes the Re were higher than 1200. The simulations

with Re equal to 1200 were almost twice as fast with the MM mesh as with the brick mesh. More

simulations must be done to find the absolute maximum Re for which the meshes result in

converging solutions.

27

In conclusion, in the analyzed situations with stationary flow in a cylindrical tube the MM mesh

resulted in better approximations of the solution than the brick mesh. The simulations with the

latter were much slower than with former.

4.2. Oscillating flow in a axis symmetric cylindrical tube

This second paragraph describes the tests that are performed with a cylindrical tube with

oscillating inflow. The goal of this test is to investigate how well the approximations of the

solutions are with a Sepran brick mesh and a MM mesh in a situation with a simple tube with a

fluctuating inflow. The solutions of the simulations are compared with the available analytical

solution. This paragraphs starts with a theoretical section, which analyses the Navier-Stokes

equation that describes this situation. A section describing the geometry and the meshes follows.

The third section gives information regarding the simulations. The last section contains a

presentation and discussion of the results.


In this section the solution of the Navier–Stokes equations of instationary, fully developed, axis

symmetrical flow in a cylindrical tube will be presented. The information is based on [van de

Vosse, 1998]. The axis symmetry leads to a velocity in the circumferential direction equal to 0

( 0=φv ). All derivates in the φ -direction and the momentum equation are omitted. For fully

developed flow the derivates in axial direction z∂

∂ and the velocity component in the radial

direction vr are equal to zero as well. The Navier Stokes equations simplify to

∂

∂

∂

∂+

∂

∂−=

∂

∂

r

vr

rrz

p

t

v zz ν

ρ

1 (32)

The velocity can be scaled by vz*

= vz/V , the coordinates by dividing by the radius of the tube, a,

r* = r/a and z* =z/a. The pressure can be scaled as p* = p/ρV2 and the time by t* = ωt. Omitting

the asterix gives

∂

∂

∂

∂+

∂

∂−=

∂

∂

r

vr

rrz

p

t

v zz 1Re2α (33)

with α the Womersley parameter defined as

ν

ωα a= (34)

α is a measure of the relative importance of the instationary terms. In the human aorta α is

approximately equal to 10, meaning that instationary terms are more important than viscous

terms. This section presents how the flow in a cylindrical tube responds to an oscillating pressure

gradient. The pressure gradient is given by

iwte

z

p

z

p

∂

∂=

∂

∂ ˆ (35)

and the solution of the Navier – Stokes equation, the velocity in axial direction, is given by iwt

zz ervv )(ˆ= (36)

28

The solution can then be constructed by superposition of its harmonics. This is allowed since (34)

is linear in vz.

The final result is given by

( )( )

−

∂

∂=

α

α

ρω 2/30

2/3

0 /1ˆ

)(ˆiJ

ariJ

z

pirvz (37)

with J0 the Bessel function of the first kind.

In [van de Vosse 1998] it can be found that the entrance length, Le, in a tube with oscillating

inflow is of the order

= Re

2α

aOLe (38)

4.2.2 Geometry and meshes

The simulations are performed with a cylindrical tube with a length of 100 mm and a radius, a, of

10 mm. The tube is positioned in the y-direction. The plane of inflow is the plane with y = -100

mm, the plane of outflow has a y-coordinate of 0. To reduce calculation time meshes are used

with a reduced number of nodal points. It was aimed to generate meshes with approximately

5000 nodal points. The brick mesh has a 33310 conformation (see section 4.1.3) and contains

4053 nodal points and 450 hexahedral elements. The MM mesh contains 5255 nodal points and

1069 tetrahedral elements. The minimal q1 was equal to 0.415 and the average q1 of all elements

was equal to 0.69. The mesh is shown in Figure 9.

Figure 9: The MM mesh of the cylinder that was used for the simulation with instationary flow. The figure on

the right side shows the element distribution on a cross section parallel to the centerline

29

4.2.3 Simulations

The simulations are performed in the way described in paragraph 3.4. Additional information

regarding the simulation is given is this section. At the plane of inflow a spatially dependent

inflow velocity was prescribed given by )2cos( tVV a π= , with Va the amplitude, which was equal

to 5 mm/s (Re = 30). The inflow velocity was independent of the radial position. The period of

oscillation was thus 1 second. From t = 0 s to t = 1 s a linear increase in velocity from 0 to the Va

was prescribed and from t = 1 s to t = 5 s 4 periods of a cosine were prescribed. The velocity of

the fluid was analyzed between t =4 s and t =5 s in order to analyze fully developed flow. The

Womersley number was equal to 13. The entrance length, Le, is of the order of magnitude 2 mm.

The velocity in the axial direction is analyzed in the nodal points with y-coordinates between –60

mm and –20 mm at four different points in time (A, B, C and D shown in Figure 10). The

calculated maximal pressure difference between the plane of outflow and the plane of inflow is

used as the amplitude of the oscillating pressure gradient to calculated the Womersley profile.

Figure 10: Inflow velocity as function of time between t =4 and t =5 s.

4.2.4 Results of the simulations

Figure 11 shows the axial velocity of the analyzed nodal points at the points in time-points A, B,

C and D. The black curve shows the analytical solution based on the calculated pressure

difference between the plane of inflow and outflow of the brick mesh. The pressure difference

followed a perfect oscillation in time with the same period as the prescribed inflow velocity. This

difference was approximately equal for the simulations with the different meshes. The maximum

pressure with the MM mesh was 6% higher than with the brick mesh, which should result

according to (37) in a velocity profile that is 6% higher. At point A en C the data points of the

simulation with the Sepran brick mesh correspond very well to the calculated Womersley profile.

At points B and D there is an average difference of about 0.5 mm/s between the two. The spread

in the axial velocity in the data points of the simulation with the MM mesh is larger and the

maximum velocity is higher than with the simulations with the Sepran Brick mesh and than the

Womersley profile. The axial velocity does not correspond as well to the Womersley profile as

the simulations with the Sepran Brick mesh. This is not just a matter of the expected difference in

velocity of 6%, but the peak in velocity is shifted 2 mm towards the central axis. In addition the

the velocity in a number of the MM nodes has an unexpected difference in phase with the other

nodes of the MM and the Sepran Brick mesh. Table 6 shows the differences between the

calculated axial velocity in the nodal points and the corresponding exact velocity. It shows clearly

that the brick mesh results in lower errors and lower standard deviations.

30

Figure 11: The axial velocity as function of the radial distance of the nodal points to the centerline for the

Sepran brick mesh and the MM mesh with -60 < y < -20, at the points in time A,B, C and D. The little blue

dots represent the MM mesh, the red dot represent the Sepran brick mesh and the black curve is the

calculated Womersley profile.

Table 6: The mean and standard deviation of the difference between the calculated axial velocity in the nodal

points and the corresponding exact velocity (calculated with the Womersley profiles of the simulations with

the brick and the MM mesh), in different time-points.

A B C D

Brick: mean error 0.0778 0.4087 0.0588 0.4157

MM: mean error 0.3526 0.6355 0.3475 0.6344

Brick: STD error 0.0665 0.1607 0.0665 0.1648

MM: STD error 0.3848 0.4069 0.3900 0.4039

4.2.5 Discussion and Conclusion

Analytical solutions of flow in a cylindrical tube were available in the form of Womersley

profiles, which describe the velocity in the axial direction as function of the oscillating pressure

gradient. Unfortunately, to my knowledge, it was not possible to prescribe a pressure gradient on

a tetrahedron mesh in Sepran. By prescribing stress tensor components (natural boundary

conditions) the pressure can be prescribed implicitly, however this method is only implemented

in Sepran for hexahedron meshes. Prescribing the mass flux means implicitly prescribing the

pressure difference. But in order to prescribe the mass flux, special elements need to be defined

in the mesh. This is no problem with a Sepran mesh but it requires complicated manual

manipulation of the meshoutputfile of the MM method. Due to time restrictions it was not

31

possible to further investigate the possibilities. Instead we chose to do simulations with an

oscillating inflow velocity. The resulting pressure of the nodal points in the plane of inflow and

outflow was analyzed in time and space. It appeared to be independent of r. In time the pressure

difference between the plane of inflow and the plane of outflow followed a perfect oscillation.

The amplitude of this oscillation was extracted and used to calculate the Womersley profile. The

numerical velocities were compared with the calculated Womersley profile. This situation is not

ideal since the FEM solves the velocity and the pressure in a coupled way and it is thus

straightforward that there exists a clear relation between the calculated velocity and pressure. Still

a comparison between the numerical velocity and the calculated Womersley profile can provide

information about the correctness of the solution. The clear correspondence in shape between the

calculated Womersely profile and the numerical velocities for instance, indicate that the

simulations result in good solutions. The spread in the axial velocity in nodal points with the

same r, was larger with the MM mesh than with the Sepran brick mesh. The axial velocity in a

number of nodal point of the MM mesh was out of phase compared to the other nodal points. The

solution of the simulations with the Sepran brick mesh resulted in a better approximation of the

Womersley profile than the solution with the MM mesh.

The amplitude of the oscillating inflow velocity was equal to 5 mm/s, which is low compared to

the velocity in the aorta. Based on the approximation of the inflow length is it possible to

prescribe an oscillating inflow velocity with an amplitude, which is at least 10 times higher than

the one used, while using the same geometry. More simulations should be performed with higher

amplitudes. Further, it should be investigated whether it is possible to prescribe a pressure

gradient on a MM mesh. Extra simulations should be performed to find the maximum velocity

amplitude.

In summary, it can be concluded that the Sepran brick mesh resulted in more accurate

approximations of the real solution than the MM mesh. It must be mentioned, however that this is

based on just one situation with a relatively low inflow. Although, it is not expected that for

higher Re numbers and more complicated geometries the MM meshes do better, it would be

useful to analyze more different situations.

4.3 Stationary flow in a diverging tube

The second geometry that is used to test the Sepran brick meshes and MM tetrahedral meshes is a

diverging tube. The goal is to test if the meshes can be used to accurately simulate flow in an

unstable situation. No reference solution was available. The chapter starts with a theoretical

section, followed by section describing the experiments, a section describing the results and the

final sections offers a discussion and conclusion.


Because of viscosity effects at the border of the wall of a tube and a fluid, the fluid near the wall

is delayed with regard to the main flow (Figure 12). This layer of fluid is referred to as the

boundary layer. When the flow in a tube travels in the direction of an increasing pressure

gradient, there exists a risk that the boundary layer lets go off the wall, i.e. in the boundary layer a

flow develops in the opposite direction of the main flow in the tube [Heijst, 2002]. If the

conservation of mass (39) and the Bernoulli equation (40) in the direction of the main flow in a

diverging tube (Figure 13) are applied, the following relations are found.

32

(39)

2

22

2

112

1

2

1VpVp ρρ +=+ (40)

(41)

(42)

Thus the flow is in the direction of increasing pressure and thus there is a risk that the boundary

layer lets go. The described tests in this paragraphs aims at evaluating the response of the two

types of meshes to this kind of flow instabilities.

y

x

Figure 12: Schematic representation of backflow in a boundary layer with a delayed main flow.

The dashed line represent the frontier between the boundary layer and the main flow. The flow is

travelling in the direction of an increasing pressure gradient. In the boundary layer the flow does

not have enough impulse to defeat the pressure gradient, resulting in a flow in the direction

opposite to the main flow.

2211 AVAV =

( )0

2

2

1

2

2

2

112

112

>−

=−

=

33

4.3.2 Geometry and Meshes

The simulations are performed with two different geometries. Both geometries consist of a

normal cylindrical tube with a diverging tube connected to it. The tube is positioned in the y–

direction and the center of the plane that connects the cylindrical tube with the diverging tube lies

in the origin of the coordinate system. The cylindrical tube has a radius of 10 mm, a length of 30

mm and lies in the negative y-direction. The diverging tube has a length of 70 mm. Figure 14

shows a schematic representation. Two different geometries are used. The first has an outflow

radius of 20 mm (geometry 1) and the second of 40 mm (geometry 2). Table 7 gives information

about the different geometries and the different meshes.

The Sepran brick meshes of both geometries have matching element distribution and thus exactly

an equal number of nodal points and elements. The element distributions are regular. Figure 15

shows on the left side the Sepran Brick mesh of geometry 1 and on the right side the element

distribution on the cross section in the plane y=0. This distribution is identical in the diverging

part of the tube, though it is enlarged.

The MM meshes have irregular element distributions. Figure 16 shows the MM mesh of

geometry 1 and the element distribution along the length of the tube in the plane x = 0. The

meshes of geometry 2 are not shown.

Table 7: Information about the two different geometries and the four different meshes. The

divergingsteepness is defined as the steepness of the wall of the diverging tube and can be calculated by

Rout /70.

Mesh Geometry Outflow

Radius [mm]

Diverging

steepness

Number of

nodes

Number of

elements

Average

q1

Minimal

q1

Brick 1 20 0.13 10941 1250 - -

MM 1 20 0.13 12899 2706 0.7 0.42

Brick 2 40 0.43 10941 1250 - -

MM 2 40 0.43 13419 2819 0.69 0.41

V2

p2 A2

V1 p1 A1

Boundary layers

Figure 13: Schematic representation of flow and the boundary layer in a diverging tube. V, p and A represent

the average velocity, the average pressure and the cross section of the tube respectively. The subscript 1

denotes inflow and 2 outflow. The increase in cross section leads to a decrease in velocity and an increase in

pressure.

34

Figure 14: Schematic representation of the diverging tube. The values are in mm. Rout is the radius of the

plane of outflow and is equal to 20 mm with geometry 1 and 40 mm with geometry 2.

Figure 15 : Sepran Brick mesh of the diverging tube with geometry 1. On the right side the element

distribution on the cross section is shown

Figure 16 : MM mesh of the diverging tube of geometry 1. On the left side the element distribution on the

plane with x = 0 is shown. The apparent unsmoothness of the cylindrical tube of the mesh is the result of the

35

visualization tool, which draws the elements based on the vertices on the corners and does not include the

vertices in the middle of the edges.

4.3.3 Simulations

The simulations are performed in the way described in paragraph 3.4. Additional information

regarding the simulation is given is this section.

At the plane of inflow a Poiseuille profile was prescribed to simulate a fully developed stationary

inflow. Simulations are performed for multiple different mean inflow velocities, Vmean.

4.3.4 Results of the simulations

In the next section velocity numbers and Re numbers are used to refer to the different

simulations. In order to avoid confusion table 8 shows the corresponding Re numbers of a

number of Vmean values. The Re numbers are calculated based on the cylindrical tube that is

attached to the diverging tube and on the used density and viscosity values. References to

locations in a mesh are expressed in the coordinates of the mesh. The values are in mm.

For the analysis of the axial velocity, nodal points are analyzed in two planes. The first plane has

y = 0 and the second y = 51. All nodal points with a maximum axial distance to the plane of 1

mm are analyzed. This will lead to a spread in axial velocity, since the mean velocity as a

function of the axial coordinate differs in a diverging tube. Based on equation (41) the difference

in mean axial velocity can be calculated at two axial positions in the diverging tube that are

positioned perpendicular to the flow and have a mutual distance of 2 mm. For geometry 1 this

leads to differences in axial velocity of 5.5% (y=0) and of 3% (y=51). For geometry 2 this is 15%

(y=0) and 9% (y=51) respectively.

Table 8: Vmean and corresponding Re for the used geometry and fluid, Vmean (mm/s) Re

5 30

10 60

15 90

20 120

25 150

30 180

40 240

50 300

60 360

70 420

Maximum inflow velocity for which the solution converges

Table 9 summarizes the maximum inflow velocities and inflow Reynolds numbers for which the

solution converges. The brick mesh of geometry 1 can handle higher Re’s than the corresponding

MM mesh. For geometry two there is no difference.

Table 9: the maximum Re which resulted in a converging solution with the four different meshes. Geometry 1 Geometry 2

Sepran brick Mesh 480 < Re < 540 300 < Re < 360

MM Mesh 420 < Re < 450 300 < Re < 360

36

Calculation time and the number of iterations needed.

Figure 17 shows the calculation time and the number of iterations needed for the different

simulations with the different meshes. In general the calculation time of the simulations with the

brick meshes was longer than with the MM meshes. To take into account the different number of

nodes of the meshes table 10 shows the average calculation time per iteration per node for the

four meshes. It can be observed that the calculation time with the brick mesh is approximately

twice as high as the calculation time with the MM mesh. There is little difference in calculation

time between the simulations with the different geometries. This hold for the brick as well as for

the MM mesh.

Table 10: summary of the average calculation time in seconds per node per iteration for the four situations.

Number of nodes Average calculation time per

node per iteration [10-5

s].

Brick Mesh geometry 1 10941 10.3

MM mesh geometry 1 12899 5.49

Brick Mesh geometry 2 10941 10.1

MM mesh geometry 2 13419 5.1

Brick Mesh (both geometries) 10.2

MM Mesh (both geometries) 5.32

Figure 17: The two figures on top show from left to right the calculatio

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