Konference ANSYS 2009
Optimal Mapped Mesh on the Circle
doc. Ing. Jaroslav Štigler, Ph.D.
Brno University of Technology, Faculty of Mechanical Engineering, Energy Institut,
Abstract: This paper brings out some ideas and recommendations how to build mapped mesh on
the circle with high quality mesh cells.
Keywords: Mapped mesh, circle,
1. Introduction
Quality of mesh significantly influences the solution precision and convergence rate in case of
every numerical solution of fluid flow. There are several types of mesh. The best results can be
obtained with Mapped mesh. The cell’s shape in this mesh type is close to the square in case of 2d
mesh or to the cube in case of 3d mesh.
When we are solving a fluid flow in pipes with the rounded cross-section the basic questions arise
in our mind: How we can make a mapped mesh on circle? Is it possible to find an optimal solution
how to make mapped mesh on the circuit? Let us try to answer these questions in the next
chapters.
2. Different types of mapped meshes
There are two basic ways of mapped mesh creating. The easiest way is to divide the border of
circuit into four pieces. This is very easy and quick way. Bat there is a problem with the mesh
quality in this case. Quality of such mesh is so bad that the solution does not start. This type of
mesh is depicted on the fig. 01 a.
a)
b)
Fig. 01 Different types of mapped mesh on circuit.
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The next way is more difficult. Area of the circuit has to be divided into some number of sub-areas
which shape is more sufficient for the mapped meshing. The five sub-areas will be used in our
case. The circle consists of square area in the middle of the circle and four areas around this
square. This type of mapped mesh on circuit is depicted in the fig. 01b.
The first type of mapped mesh is useless because of their quality. Therefore our attention will be
focused on the second type of mapped mesh using mapped sub-areas. Dividing of circuit into
mapped sub-areas and their dimensions is depicted in the fig. 2.
a)
b)
Fig. 02 Dimensions of mapped sub-areas on circuit.
3. Optimal mapped mesh on a circle with rectangular inner square.
Basic ideas of optimal mesh creating on the circle with inner rectangular square are outlined in
detail in a technical report [1].
If we want divide the circle into mapped sub-areas, than the length of edge E has to be known.
Influence of size of the edge E is clear from the pictures in fig 03. When the E is small the mesh is
very dense in the middle square of circuit and very rough at the border of circle. This is depicted in
fig 03a. In opposite situation when the Edge E is big the cells near edge G are very deformed. The
first task is to find optimal size of edge E.
a)
b)
Fig. 03 Influence of size of square inside of the circuit on the mesh.
Konference ANSYS 2009
The square inside circle is the best area for mapped mesh. The problem lies in the four sub-areas
round the square. In case of square the opposite sides have the same length. But this is not true in
case of other sub-areas of circle.
Now the attention will be focused on one of the peripheral sub areas of the circle. This sub-area is
depicted on the fig. 02b. Now it will be useful to express the lengths of edges F, H, and G. All
these lenghts will depend on the radius R or on the length of edge E or on both of them.
π
F = R. ,
2
H=R−
E
,
2
G=R−
E
2
,
In the best mapped mesh of such area will be obtained whent the next ratios are equal 1.
G
E
= 1 and
=1
H
F
It is obvious that these conditions cannot be accomplished. Both of these conditions are in contrary
on one to other. First condition will be fulfilled in case E=0. The second condition will be fulfilled
in case E=R. π/2 . But E cannot be bigger than R . 2 . If it happens then the inner squared area
exceeds the circle. Both these conditions cannot be satisfied together. It is possible to receive the
compromise solution that both conditions will be equal.
G E
=
H F
We can add previous expressions into this condition and we receive quadratic equation for E.
 π

E 2 − E.R.
+ 2  + R 2 .π = 0
 2

This equation has two solutions in form.
E1, 2


π 
 ±
= R 1 +

2. 2 

2


π 
1 +
 − π 

2. 2 


We can express this solution as
E 1, 2 = R.a 1, 2
Where
a 1 = 3,25682254 ,
a 2 = 0,96462 ≅ 0,96
Correct solution is α2, because its value lays in a interval of acceptable values a ∈ 0, 2 . The
basic geometry of sub-areas is determined. Now we need to know a number of elements on edges
F,E,G and H to be able to create mapped mesh on circle. Numbers of elements on edges will be
signed nF, nE, nG, nH. In case of mapped mesh the numbers of opposite edges have to be same. It is
assumed that successive ratio is 1 for all edges. Than the numbers of elements will be determined
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form condition that lengths of elements are as equal as it is possible. Lengths of edges elements are
signed LF, LE, LG, LH. These lengths can be calculated
LF =
F
,
nF
LE =
E
,
nE
LG =
G
,
nG
LH =
H
nH
It is impossible for all edge elements to be equal. But we can found some limits of their lengths.
These limits are.
L E ≤ LG ≤ LF
or
G
G
nF ≤ nG ≤ nE
F
E
LE ≤ LH ≤ LF
or
H
H
nF ≤ nH ≤ nE
F
E
In case of mapped mesh nE=nF and nG=nH. When all above relation are taken into consideration we
receive.
1 1 
2
a 
1 −
n F ≤ n G ≤  −
n F , or
π 
2
2
a
2 a
1 1
1 − n F ≤ n G ≤  −  n F
π 2
a 2
Now it is possible to write that
n G = β.n F
The value of coefficient b is on the intersection of above intervals.
1
2 a
1 

1 −  ≤ β ≤  −
π 2
2
a
Number of elements nG lies between these limit values
n G min = βmin .n F =
2 a
1 − .n F ,
π 2
1 1 
.n F
n G max = βmax .n F =  −
2
a
The number of elements nF is chosen and number of elements nG can be calculated from above
terms. The best way is to take average value of β. We can calculate nG for three different values of
a an example.
A
0,83
0,964619
1,0
β min
0,3724225
0,329572
0,3183099
β max
0,4977125
0,329572
0,2928932
nGmin
7
7
6
nGmax
10
7
6
nGav
10
7
6
Tab. 01 Calculating of nG for nF=20 and for different value of a.
In the case of ratio aopt = 0,964619 – optimal value, this interval of β is only one value. When the
ration a is bigger than aopt the interval for β is empty. When the ratio a is smaller than aopt than the
intersection interval is not empty. The best value is the average valule of βmin and βmax.
Konference ANSYS 2009
Evaluation of meshes is listed in table 02. The main criterion CEAS and CEVS is constant for all type
of these meshes with its value 0,5. So these meshes have to be sorted by different criterion for
example criterion CAR. Using this criterion the best meshes are for nGav or close to it. Best mesh is
mesh for a=0,964 and nG=7 for nF=20 in a global view.
Mesh
nG
5
7
8
0,83
9
10
14
4
0,964619 7
8
4
6
1,00
7
10
E/R
N
800
960
1040
1120
1200
1520
720
960
1040
720
880
960
1200
Evaluation of mesh quality by different criteria.
CAR
CDR
CER
CEAS CEVS CMAS
CS
0,5
0,56 0,54
2,580 1,0 2,800 0,5
0,5
0,59 0,40
1,882 1,0 2,000 0,5
0,5
0,60 0,34
1,665 1,0 1,760 0,5
0,5
0,61 0,29
1,594 1,0 1,704 0,5
0,5
0,61 0,33
1,776 1,0 1,900 0,5
0,5
0,63 0,49
2,500 1,0 2,660 0,5
0,5
0,55 0,52
2,500 1,0 2,700 0,5
0,5
0,59 0,27
1,564 1,0 1,704 0,5
0,5
0,60 0,34
1,792 1,0 1,950 0,5
0,5
0,55 0,48
2,32 1,0 2,500 0,5
1,0
0,5
0,5
0,58 0,30
1,582
1,665
0,5
0,59 0,32
1,704 1,0 1,873 0,5
0,5
0,61 0,49
2,46 1,0 2,680 0,5
Tab.02. Evaluation of mesh quality for nF=20 and different value of a
CAR
-Aspect Ratio
CEVS
-EquiSize Skew
CDR
-Diagonal Ratio
CMAS
-MidAngle Skew
CER
-Edge Ratio
CS
-Stretch
CEAS
-EquiAngle Skew
4. Optimal mapped mesh on a circle with bent inner square.
The quality of mesh can be increased by using not rectangular inner square. The edges will not be
straight but they will be bent with radius r. This type of sub-areas is depicted in fig 04.
The expression for the length of edge H has to be modified and the expression of length J has to be
added.
E
1 1

− K = E. − − b 
2
a
2






2
2
2
4.K + E
E.K
b
 = E. 4.b + 1 .arctg
J=
.arctg 
2
 0,25 − b 2
 E 

4.K
4
.
b

  − K2 
  2 

H=R−
(
)



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b)
a)
Fig. 04 Dimensions of mapped sub-areas on circuit with bounded inner sub-area.
Where
b=
E
.
R
Now we will take value of the aopt=0,964619. Then we will try to find the optimal ratio b. This
optimum value will be searched on the interval b ∈ <0,
(
2 −1
>. For the case b=bmin=0 it is
2
)
obtained rectangular square inner area. In the case b=bmax= 2 − 1 2 it is obtained inner area as
circuit. This optimum was found by evaluating of set meshes with constant ratio a and with
different values of b on the interval <bmin, bmax>. Results are listed in table 3. The number of
elements on edge G (nG) is calculated similar way as in the case of straight edge E only the length
of edge J is taken into consideration instead of edge E.
The interval for coefficient β is β∈<βbmin, βbmax>. Where
β b min =
2 a

1 − − b.a  ,
π 2

β b max

a 
1 −

2
2
=
π a.M
Where
M=
(4.b
)

+1
b
.arctg
2.b.π
0
,
25
− b2

2

 .

There are dependences of βbmin and βbmax on the ratio b in the fig. 05.
Results of evaluating of meshes created under previous conditions are listed in the table 03.
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b
0,0
0,020711
0,041421
0,062132
0,066274
0,0675
0,070416
0,074558
0,078701
0,082843
0,103553
0,124264
0,144975
0,165685
0,186396
0,207107
Mesh
nGav
10
10
9
9
9
9
9
9
9
9
9
9
8
8
8
7
N
2 100
2 100
1 980
1 980
1 980
1 980
1 980
1 980
1 980
1 980
1 980
1 980
1 860
1 860
1 860
1 740
CAR
1,644
1,644
1,637
1,564
1,546
1,540
1,534
1,516
1,498
1,485
1,480
1,480
1,392
1,316
1,316
1,270
Evaluation of mesh quality by different criteria.
CDR
CER
CEAS
CEVS
CMAS
2,100 1,644
0,50
0,50
0,63
1,920 1,644
0,44
0,44
0,57
1,744 1,637
0,39
0,39
0,51
1,616 1,564
0,34
0,34
0,44
1,665 1,546
0,33
0,33
0,46
1,679 1,540
0,33
0,33
0,47
1,720 1,534
0,34
0,34
0,49
1,784 1,516
0,36
0,36
0,52
1,846 1,498
0,38
0,38
0,54
1,910 1,485
0,40
0,40
0,57
2,300 1,480
0,50
0,50
0,68
2,860 1,480
0,59
0,59
0,78
3,700 1,392
0,69
0,69
0,86
5,100 1,316
0,78
0,78
0,92
7,930 1,316
0,87
0,87
0,96
17,000 1,270
0,96
0,96
0,99
CS
0,29
0,26
0,29
0,26
0,26
0,25
0,25
0,24
0,24
0,23
0,23
0,25
0,27
0,28
0,28
0,29
Tab.03. Evaluation of mesh quality for nF=30 and value of a=0,964619
From both table 03 and fig. 05 is clear, that number nG is decreasing with ratio b increasing. This is
true only in case that ratio a is a constant. It is possible to find two optimal values of ratio b. First
is for value b= 0,062132. In this case it is an optimum for two criterion CDR a CMAS. The second
optimum is for value b=0,0675 and in this case it is optimum for criterion CEAS. What optimum
will be taken into consideration in practice it depends what criterion we prefer. The dependences
of these three criterions are depicted on the fir 06. The optimum is an intersection of two different
curves in all three cases. It is obvious, because one of the curve represents quality of elements in
the outer sub-area and the second one represents quality of elements of the inner sub-area. In case
of criterion CEAS it seems to be an intersection of two straight lines. When we increase bending of
edges of inner square we are getting worse the elements quatily of inner sub-area and getting better
elements quality of outer sub-area.
Fig. 05 Coefficients for nG setting in dependence on ratio b
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Fig. 06 Coefficients for nG setting in dependence on ratio b
5. References
Štigler, J., 2004. Jak vytvořit mapovanou síť na kruhu. Technická zpráva VUT-EU 13303-QR-0305, VUT FSI v Brně, Brno, pp 29.
6.
Appendix
Author tried to find some common principles, how to create an optimal mapped mesh on the circle
in this paper. Currently it was found that best mesh can be made for these values of ratio
a=0,964619, ratio b=0,675, in case that we prefer criterion CEAS (EquiAngle Skew), and with value
of coefficient β which is an average of βbmin and βbmax, which is important for nG calculating. These
informations can be useful for first design of mapped mesh on the circle and can be useful for
automatic mesh generation.
7. Acknowledgement
Author is grateful to Grant Agency of Czech Republic for funding of this research within scope of
project with number GA 101/09/1539 with title Mathematical and Numerical Modeling of Flow
in Pipe Junction and its Comparison with Experiment.