bmt 06.03 steven f. petit validation of mesh machine ... · validation of mesh machine tetrahedron...
TRANSCRIPT
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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
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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
4.2.1 Theoretical Section........................................................................................................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
4.3.1 Theoretical Section........................................................................................................31
4.3.2 Geometry and Meshes ...................................................................................................33
4.3.3 Simulations....................................................................................................................35
4.3.4 Results of the simulations .............................................................................................35
4.3.5 Discussion and Conclusion ...........................................................................................45
4.4 Oscillating flow in a Diverging Tube...................................................................................46
4.4.1 Geometry and Meshes ...................................................................................................46
4.4.2 Simulations....................................................................................................................46
4.4.3 Results of the simulations .............................................................................................47
4.4.4 Discussion and Conclusion ...........................................................................................49
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
4.5.4 Results of the simulations .............................................................................................55
4.5.5 Discussion and Conclusions..........................................................................................62
4.6 Stationary flow in a Bifurcating Tube..................................................................................64
4.6.1 Theoretical Section........................................................................................................64
4.6.2 Geometry and mesh.......................................................................................................65
4.6.3 Simulations....................................................................................................................66
4.6.4 Results of the simulations .............................................................................................67
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4.6.5 Discussion and Conclusion ...........................................................................................74
Chapter 5: Discussion......................................................... .....................................76
Chapter 6: Conclusion ........................................................ .....................................79
Future Prospectives ............................................................ .....................................79
APPENDIX I ...................................................................... .....................................81
APPENDIX II..................................................................... .....................................82
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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
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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.
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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
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)(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
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∑=
=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
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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
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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.
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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
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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.
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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.
44420 (i.e. 20 elements in
the length)
1600 13817 Less elements in the length
more elements in the planes
perpendicular to the flow
direction.
55510 (i.e. 10 elements in
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.
Discussion and Conclusion
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).
Discussion and Conclusion
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.
4.2.1 Theoretical Section
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.
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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
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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.
4.3.1 Theoretical Section
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.
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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
>−
=−
=
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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.
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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
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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
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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