7  Application of the method to test problems

In this section some results from the application of the boundary element method outlined in the previous section to the test problems of a sphere and a cylinder and the solutions are assumed to be axisymmetric. The boundary functions are approximated by a constant on each element. The wavenumber range [0,kmax] is divided into equal subintervals in each test. The numerical integrations required in the discretization of the integral operators were carried out with sufficient accuracy to ensure that the significant error in the method arises through the application of collocation and the approximation (39).

7.1  Cylinder

The cylinder is of radius 1.0 and height 1.0 with the z axis being the axis of the cylinder, the cylinder lying between the z=0 and z=1 planes. The boundary is exactly represented. The Neumann eigenvalues can be found through first writing the solution in the form
j(r, z) = jl(r(k2 - j2 p2)[1/2]) cos(j pz)
for j=0,1,... where r is the radial coordinate and jl is the Bessel function of integer order l. This representation satisfies the boundary condition at z=0 and z=1. In order that the boundary condition at r = 1 is satisfied we must have
j
r
 = (k2 - j2 p2)jl¢(r(k2 - j2 p2)[1/2]) cos(j pz) ûr = 1 = 0 .
This implies that jl¢((k2 - j2 p2)[1/2]) = 0. Thus the eigenvalues are given by
k* = ( alm2 + j2 p2 )[1/2] .
where the alm, m=1,2,... are the zeros of jl¢.
The generator of the cylinder is divided into n elements of equal length. In table 1 the real part of the approximation to the first five Neumann eigenvalues are shown. Results are given for each combination n = 24,48 and kB - kA = 1.0, 0.5. The computed results are expressed to four decimal places.


Table 1: Computed Eigenfrequencies of the cylinder
Exact values ® 3.1416 3.8317 4.9550 6.2832 7.0156
n kB - kA Computed solutions
24 1.0 3.1449 3.8359 4.9477 6.2880 7.0421
24 0.5 3.1441 3.8349 5.9576 6.2910 7.0234
48 1.0 3.1143 3.8151 4.9486 6.2869 7.0397
48 0.5 3.1405 3.8337 4.9547 6.2852 7.0199

7.2  Sphere

The boundary in this test problem is a sphere is of radius 1.0. Writing the Helmholtz equation in spherical polar coordinates reduces it to the Bessel differential equation. The Neumann eigenvalues are the zeros of Jl¢(k) where Jl(k) is the spherical Bessel function of integer order l.
For the application of the boundary element method, the surface is represented by a set of n axisymmetric conical elements so that the edges of each element sweep through an equal angle about the centre. Thus, in this example, the boundary is not exactly represented.
In table 2 the real part of the approximation to the first six Neumann eigenvalues are shown. Results are given for each combination n = 16,32,64 and kB - kA = 1.0, 0.5. The computed results are expressed to four decimal places. Note that the third and fourth eigenvalues are very close and are listed together.


Table 2: Computed Eigenfrequencies of the sphere
Exact values ® 2.0816 3.3420 4.4934, 4.5131 5.6467 5.9404
n kB - kA Computed solutions
16 1.0 2.1026 3.3618 4.5087,4.5567 5.7086 5.9950
16 0.5 2.0942 3.3687 4.5117,4.5556 5.7041 5.9595
32 1.0 2.0945 3.3416 4.4968,4.5252 5.6650 5.9318
32 0.5 2.0845 3.3492 4.4929,4.5260 5.6605 5.9477
64 1.0 2.0925 3.3366 4.4938,4.5173 5.6541 5.9286
64 0.5 2.0821 3.3444 4.4904,4.5188 5.6496 5.9411

7.3  Discussion

The results in tables 1 and 2 generally show improvements in the accuracy of the results as the number of elements and the range of the quadratic interpolation is reduced.
For the sphere test problem the results for the approximated sphere in table 2 are less accurate than those obtained through applying a similar method to the exact sphere, as given in Kirkup and Amini [18]. A significant loss of accuracy can occur in the boundary element method because of boundary approximation. The interested reader is referred to Kirkup and Henwood [40] wherein the effect of approximating a circle by a regular polygon on the results from the boundary element method is experimentally explored.