2.3  Three dimensional Surfaces


In the subroutines that solve acoustic problems in three dimensions ( AIBEM3, AEBEM3 and AMBEM3) the boundaries must be represented in the form of a set of planar triangles. In order that the normal to the boundary points outward rather than inward the three nodes that define each element must be listed in the anti-clockwise direction when it is viewed from just outside the surface. The programs AIBEM3_T, AEBEM3_T and AMBEM3_T each solve acoustic problems in which the boundary under consideration is that of a sphere of unit radius. The boundary is represented by 36 planar triangles and has 20 vertices. The approximate surface is illustrated in Figure 2.4. In this example the representation of the boundary is quite a severe approximation on the original boundary and hence some corresponding loss of accuracy should be expected in the BEM solution.

Fig 2.4. Representation of the sphere by flat triangular panels.

In order to pass the description of the boundary to the subroutines it is represented by the two tables data 2.C and 2.D. Table 2.C lists the (x,y,z) coordinates of the vertices and is identified by the real array VERTEX. Table 2.D lists the index of the three vertices that define each panel and is identified by the integer array SELV.

Table 2.C: Vertices of approximate sphere boundary ( VERTEX)
Index x y z Index x y z
1 0.000 0.000 1.000 11 -0.500 -0.866 0.000
2 0.000 0.745 0.667 12 -1.000 0.000 0.000
3 0.645 0.372 0.667 13 -0.500 0.866 0.000
4 0.645 -0.372 0.667 14 0.000 0.745 -0.667
5 0.000 -0.745 0.667 15 0.645 0.372 -0.667
6 -0.645 -0.372 0.667 16 0.645 -0.372 -0.667
7 -0.645 0.372 0.667 17 0.000 -0.745 -0.667
8 0.500 0.866 0.000 18 -0.645 -0.372 -0.667
9 1.000 0.000 0.000 18 -0.645 0.372 -0.667
10 0.500 -0.866 0.000 20 0.000 0.000 1.000

Table 2.D: Panels that constitute the sphere ( SELV))
Index Vertex 1 Vertex 2 Vertex 3 Index Vertex 1 Vertex 2 Vertex 3
1 1 3 2 18 8 15 14
2 1 4 3 19 8 9 15
3 1 5 4 20 9 16 15
4 1 6 5 22 9 10 16
5 1 7 6 23 10 17 16
6 1 2 7 24 10 11 17
7 2 3 8 25 11 18 17
8 3 9 8 26 11 12 18
9 3 4 9 27 12 19 18
10 4 10 9 28 12 13 19
11 4 5 10 29 13 14 19
12 5 11 10 30 13 8 14
13 5 6 11 31 14 15 20
14 6 12 11 32 15 16 20
15 6 7 12 33 16 17 20
16 7 13 12 34 17 18 20
17 7 2 13 35 18 19 20
18 2 8 13 36 19 14 20

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