Civil Engineering Reference
In-Depth Information
Table 8.24
Statistics for examples 1, 2 and 3
Example
Objects
NH
NT
NL
NZ
NR
NH*
NT*
NY*
Figures
1
Cylinder1
1200
7200
8
9
2
860
2913
209
8.139 a-g
Cylinder2
640
3840
8
468
2546
512
2
Curveplate1
2400
14,400
8
9
2
2165
3319
170
8.140 a-f
Handle
1200
7200
8
1008
2600
576
3
Curveplate1
2400
14,400
3
4
1
1454
4474
462
8.141 a-e
Curveplate2 2400 14,400 4 2208 2800 576
Note: NH = number of hexahedral elements in the mesh; NT = number of tetrahedral elements in the mesh; NL = number
of boundary intersection loops; NZ = number of zones (patches) on the boundary surface; NR = number of regions (vol-
umes) of intersection; NH* = number of hexahedral elements recovered; NT* = number of tetrahedral elements remained
in the mesh; NY* = number of pyramid elements created at interfaces.
8.7.3 Examples
In this section, three examples of merging hexahedral meshes of various characteristics and
complexity are presented. The summary of the essential features of the examples is shown
in Table 8.24 in which the number of hexahedral elements before and after merging is given.
The shape quality of the tetrahedral elements has neither been optimised nor calculated, as
this is not a main concern or restriction to the merging algorithm. As far as finite element
applications are concerned, the tetrahedral and pyramid elements could be further enhanced
in connectivity and shape by means of standard optimisation procedures. As for the hexa-
hedral elements, only those regular hexahedral elements of the original mesh are retrieved
from the tetrahedral mesh.
The intersection of two hollow cylinders is considered in example 1. The larger cylinder
is discretised into 1200 hexahedral elements, each of which is further divided into six tet-
rahedral elements, as shown in Figure 8.139a. The smaller cylinder is discretised into 640
hexahedral elements, which is further divided into 3840 tetrahedral elements, as shown
in Figure 8.139b. Eight intersection loops are found in the intersection of the cylinders,
which divide the larger cylinder into nine zones and the smaller cylinder into eight zones,
as shown in Figure 8.139c. From the surface partition of the boundary surfaces, two
regions of intersection can be identified, as shown in Figure 8.139d. A total of 468 hexa-
hedral elements (green in colour) are recovered, 512 pyramid elements (yellow in colour)
are formed and 2546 tetrahedral elements (blue in colour) remain in the intersected tetra-
hedral mesh of the smaller cylinder, as shown in Figure 8.139e; 860 hexahedral elements
are recovered, 209 pyramid elements are formed and 2913 tetrahedral elements remain
in the intersected tetrahedral mesh of the larger cylinder, as shown in Figure 8.139f. The
finite element mesh as a result of merging the two cylinders is shown in Figure 8.139g. The
intersection of a curved hollow cylinder and a curved plate is considered in example 2, as
shown in Figure 8.140a-f. The intersection of two curved plates is studied in example 3,
as shown in Figure 8.141a-e.
8.7.4 Closing remarks
In view of the success of merging arbitrary solid tetrahedral finite element meshes, the algo-
rithm is extended to merge hexahedral meshes by first dividing each hexahedral element into
five or six tetrahedral elements. Non-intersected regular hexahedral elements are readily
recovered from the intersected tetrahedral mesh as the constituent tetrahedra as a subdivi-
sion of the original hexahedral elements are intact and present in the mesh. If necessary,
