COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Three-dimensional Mesh Generation Using the Crossed Circles Method
H. Suzuki*, Y. Ezawa
Department of Intelligent Material & Mechatronics Systems, Toyo University, Kawagoe, 850-8585 Japan
Email: [email protected]
Abstract In this study a method for generating internal mesh nodes for two and three-dimensional problems is
presented. This method is related to the Delaunay method and is called the “Crossed circles method” because crossed
circles are used for generating internal nodes. A noteworthy byproduct of this method is a criterion for removing extra
nodes. Numerical results of the crossed circles method with this criterion berify its effectiveness and show that the
crossed circles method can produce a high-quality mesh.
Key words: Mesh generation, Delaunay method, Crossed circles method
INTRODUCTION
The Delaunay method [1] is a powerful method for triangulation. However, the quality of the mesh generated by this
method depends on the way nodes are allocated. For obtaining suitable allocations, the Bubble method [2] was
developed. In the Bubble method, all nodes, including internal nodes, have to be defined in advance. We developed a
new algorithm for creating internal nodes in the process of two-dimensional mesh generation and we called this
method, which is based on the Delaunay method, the “Crossed Circles Method (CCM)” [3]. In this study we extend the
method, applying it to three-dimensional problems.
THE DELAUNAY METHOD
Initially we developed the two-dimensional Delaunay method program using the well-known algorithm as follows:
(1) Create a virtual triangle that includes the entire set of nodes.
(2) Find a triangle containing an additional node P.
(3) Divide this triangle into three triangles with a common vertex at P.
(4) Check whether the circumcircles of adjacent triangles include P.
(5) If the circumcircles of adjacent triangles include P, swap adjacent sides, as in Figure 1(d) below.
(6) Repeat steps (4) and (5) until swapping does not occur.
(7) Go to step (2) until all nodes are added.
(8) Delete triangles outside the structure.
Figure 1: The algorithm for the Delaunay method program
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Fig. 1 shows steps (2)-(5) in the Delaunay method. Figure 1(a) is the process of finding a triangle that includes a node
P. Figure 1(b) is the step of dividing a triangle into three triangles meeting at a common vertex P. Figure 1(c) is the step
of checking whether the circumcircles of the adjacent triangles include P. In Figure 1(d), the diagonal lines of
quadrilateral P-A-B-C are swapped.
Fig. 2 shows an example for studying the Delaunay method. In this example 20 nodes are allocated evenly around the
circumference of a circle of diameter 20 mm.
20 mm
Figure 2: Nodes around a circle
Using the algorithm outlined above, the following mesh is created.
Figure 3: Mesh created by the Delaunay method program
The resulting mesh is unsuitable in that the constituent triangles are far from equilateral and the mesh pattern is not
symmetric even though the original shape is symmetric. If a node is inserted at the center point of the circle, the
following mesh is created. Though the mesh in Fig. 4 is symmetric, again the triangles are far from equilateral. This
example shows that appropriate insertion of nodes is important.
Figure 4: Mesh created by the Delaunay method
THE TWO-DIMENSIONAL CROSSED CIRCLES METHOD
A method for automatic node insertion is developed. This method is related to the Delaunay method and is called the
“Crossed circles method” because crossed circles are used for inserting nodes.
The characteristic of the crossed circles method is the addition of the following step just after step (5) in the Delaunay
method.
(5-1) Calculate the positions of the centers of the swapped circumcircles that form crossed (intersecting) circles.
(5-2) Check whether both centers are included in both circumcircles.
(5-3) If so, then calculate the position of the midpoint of these centers.
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(5-4) Check whether any nodes exist in a neighborhood of this midpoint. How this neighborhood is defined is detailed
below.
(5-5) Add this midpoint to existing nodes if no such neighboring nodes exist.
The crossed circles method focuses its attention on triangles P-A-B and P-C-B in the above process (5). After
swapping diagonal lines, two circumcircles are created as shown in Figure 5(a). These two circles intersect as shown in
Figure 5(b).
A
B
P
C
(a) Circumcircles of triangles after swapping
(b) Crossed circles
Figure 5: Crossed circles
Figure 6(a) indicates the step of checking whether both centers are contained in both circumcircles. Figure 6(b) shows
the position of the midpoint of these centers in this case.
(a) Step (5-1)
(b) Step (5-3)
Figure 6: The algorithm for the CCM program
We now describe how the decision is made concerning the addition or otherwise of this midpoint to the set of existing
nodes.
In step (5-4), the criterion for determining whether neighboring nodes exist is important. For this purpose, we utilize
the following condition:
d2 <
r12 + r22
× R1 or
2
d 2 < (R2 × L )
2
(1)
where d is the minimum distance between the midpoint and existing nodes, and r1 and r2 are the radii of the intersecting
circumcircles. R1 and R2 are adjustable non-negative constants; L is a constant which is defined in terms of the
distances between existing nodes. If condition (1) is true, there is a neighboring node and consequently the midpoint is
not added as a new node
r1
r2
d
Figure 7: Judgment of whether a neighboring node exists
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Some numerical cases are studied. Firstly, the problem shown in Figure 1 is considered. In this problem, L is chosen as
3.14; i.e. the minimum length between existing nodes. If R1 in condition (1) is taken as 1.0, no node is inserted, so R1
should be chosen less than 1.0. Figure 8 shows a mesh created by the crossed circles method with R1 = 0.99 and R2 = 0.
This mesh is the same as the one in Figure 4.
Figure 8: Mesh created by the crossed circles method ( R1 = 0.99 , R2 = 0 )
Figure 9 shows the mesh for R1=0.95. In these meshes each triangle is approximately equilateral and the mesh pattern
is almost symmetric.
(a) R1 = 0.95 , R2 = 0
(b) R1 = 0.95 , R2 = 1.0
Figure 9: Mesh created by the crossed circles method (R1=0.95)
THE THREE-DIMENSIONAL CROSSED CIRCLES METHOD
Figure 10: The algorithm for the Delaunay method program
In three-dimensional problems, the following process is used for the Delaunay method.
(1) Create a virtual tetrahedron that includes the entire set of nodes.
(2) Find a tetrahedron that includes an additional node P, as shown in Fig. 10(a).
(3) Divide this tetrahedron into four tetrahedra meeting at a vertex P, as shown in Fig. 10(b).
(4) Determine whether circumscribing spheres of adjacent tetrahedra contain P, as shown in Fig. 10(c).
(5) If a circumscribing sphere of an adjacent tetrahedron contains P, transform two tetrahedra to three tetrahedra as
shown in Fig. 11. The number of tetrahedra increases by one.
(6) Repeat steps (4) and (5) until swapping does not occur.
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(7) Go to step (2) until all nodes are added.
(8) Delete tetrahedra outside the structure.
Figure 11: Transformation of tetrahedra
In the three-dimensional crossed circles method, the following step is inserted after step (5).
(5-1) Calculate the positions of the centers of the circumscribing spheres.
(5-2) Check whether these centers are included in all/two circumscribing spheres.
(5-3) If so, calculate the position of the centroid of the triangle whose vertices are the centers of the circumscribing
spheres.
(5-4) Check whether any nodes exist in the neighborhood of this centroid.
(5-5) Add the centroid to existing nodes if there is no neighboring node.
ｙ
z
ｘ
Figure 12: Overlapping circumscribing spheres
In step (5-4), the criterion for determining whether neighboring nodes exist is important. For this purpose, we make use
of the following condition:
d 2 < (r1 + r2 + r3 )LR
(2)
where d is the minimum distance between the centroid and existing nodes, r1, r2 and r3 are the radii of the intersecting
circumscribing spheres, L is a constant related to the structure size, and R is also a constant. If condition (2) is satisfied,
the centroid is not added; if not, the centroid becomes a new node.
r1
d
r3
r2
Figure 13: Decision about existence of neighboring nodes
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NUMERICAL EXAMPLES
First, we studied a simple example as shown in Fig. 14. Eight nodes were allocated at the vertices of a hexahedron. In
this case the constant L was set at 1.0 mm. Fig. 15 illustrates the mesh created by the Delaunay method. Fig. 16 depicts
the mesh created by the crossed circles method with R equal to 0.3. In this case, one central node was added to the eight
original nodes.
1 mm
Figure 14: Points on hexahedron
Figure 15: Mesh created by the Delaunay method
Figure 16: Mesh created by CCM
The mesh created by CCM is more symmetric than the mesh created by the Delaunay method.
A second example is shown in Fig. 17. Twenty six nodes were allocated on the surface of a hexahedron. Fig. 18 shows
the mesh created by CCM when L was set to 1.0 mm and R was set to 0.1. In this case also a central node was added.
Figure 17: Nodes on the surface of a hexahedron
Figure 18: Mesh ceated by CCM
About 1 mm
Figure 19: A Fullerene-like Structure
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For a third example we considered a more complex structure that had 60 points on a “fullerene”-like sphere as shown
in Fig.19. In this case, the diameter was about 1.0 mm. So the value L was taken to be 1.0 mm.
Figure 20 illustrates the mesh created by the Delaunay method. Figure 21 shows the mesh created by the crossed
circles method with R set at 1.0. In this case only one node was added at the center of the structure.
Figure 20: Mesh created by the Delaunay method
Figure 21: Mesh created by CCM (R=1.0)
With R taken as 0.8, the mesh created is shown in Figure 22.
Figure 22: Mesh created by CCM (R=0.8)
In this case, twelve nodes were added.
In order to evaluate the quality of the mesh, we defined the following parameter M.
A smaller value for M indicates a better quality of mesh.
6
M =
∑ (l
i =1
m
− li ) 2
(3)
6l m2
6
where l m =
∑l
i =1
6
i
, and l i (i = 1, L ,6) are the lengths of the six edges in a tetrahedron.
Figure 23 indicates the quality of the mesh shown in Figure 20. In this graph, the quality parameter M defined by
equation (3) is the variable along the horizontal axis and the vertical axis indicates the number of elements for each M.
It can be seen from Figure 23 that there are lots of distorted elements in the mesh of Figure 20.
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Number of elements
30
25
20
15
10
60
55
57.5
50
52.5
45
47.5
40
42.5
35
37.5
30
32.5
25
27.5
20
22.5
15
17.5
10
12.5
7.5
2.5
0
5
5
Ｍ
Figure 23: The quality of mesh resulting from the Delaunay method
Figure 24 shows the quality of mesh resulting from CCM (with R=1.0), indicating fewer distorted elements.
Number of elements
30
25
20
15
10
60
57.5
55
50
52.5
47.5
45
42.5
40
37.5
35
32.5
30
27.5
25
22.5
20
17.5
15
12.5
10
5
2.5
0
7.5
5
M
Figure 24: The quality of mesh resulting from CCM (with R=1.0)
Figure 25 shows the quality of mesh resulting from CCM (with R=0.8), indicating even fewer distorted elements.
Number of elements
70
60
50
40
30
20
10
60
55
57.5
50
52.5
47.5
45
40
42.5
37.5
35
32.5
30
27.5
25
22.5
20
17.5
15
12.5
10
7.5
5
2.5
0
M
Figure 25: The quality of mesh resulting from CCM (with R=0.8)
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Figure 26 shows the relation between the parameter R and the mean value of M. The broken line indicates the mesh
created using the Delaunay method and the solid line indicates the mesh created using CCM. The number in each
square is the number of additional nodes. This figure indicates that the quality of the mesh is best when R is 0.8. When
R is greater than 0.8, the quality becomes worse because the number of additional nodes is too large. Therefore,
determination of the parameter M is important for establishing an appropriate value for R, and consequently an
appropriate number of additional nodes.
Mean value of M
18
16
Delaunay
14
12
10
8
1
44
CCM
1
6
4
12
18
2
0
1.1
1
0.9
0.8
0.7
0.6
0.5
Parameter R
Figure 26: Relation between the parameter R and the quality of mesh
CONCLUSIONS
We have presented a method for generating internal nodes in mesh generation for two and three-dimensional
problems. The feature of this method is that internal nodes are created automatically. Numerical results of the crossed
circles method verified its effectiveness and show that the crossed circles method can result in a high-quality mesh.
REFERENCES
1. George PL. Automatic mesh generation – application to finite element methods –. Wiley, 1991.
2. Shimada K, Gossard DC. Bubble mesh generation: automated 2D/3D triangulation via physically-based node
placement and modified Delaunay tessellation. WCCM III, Vol. I, Tokyo, August, 1994, pp. 900-902.
3. Ezawa Y. Triangulation with internal points generated by crossed circles. WCCM VI in conjunction with
APCOM’04, Sept. 5-10, 2004, Beijing, China, 2004.
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