7  Results for the Eigenvalue Problem

In this section results from the application of the method of section 6 to the test problem are presented. The computed eigenvalues from the application boundary and shell element methods are given for 8+1, 16+2 and 32+4 boundary + shell elements (h = [1/2], [1/4] and [1/8]) with intervals [kA, kB] of length 2.0 and 1.0 and interpolating polynomials of second and fourth degree (m=2,4). The first three computed eigenvalues from the direct method are given in table V, the results from the indirect method are given in table VI.

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Table V: Computed Eigenvalues via the direct BSEM
h kB-kA m 1st eigenvalue 2nd eigenvalue 3rd eigenvalue
[1/2] 2.0 2 1.6134 + i0.0073 3.2156 - i0.0138 4.7904 - i0.0118
[1/2] 2.0 4 1.6007 + i0.0080 3.2048 - i0.0031 4.8051 - i0.0067
[1/2] 1.0 2 1.6032 + i0.0025 3.2029 - i0.0022 4.8069 - i0.0066
[1/2] 1.0 4 1.6011 + i0.0022 3.2046 - i0.0029 4.8053 - i0.0066
[1/4] 2.0 2 1.5952 + i0.0067 3.2689 - i0.0181 4.7228 - i0.0101
[1/4] 2.0 4 1.5804 + i0.0057 3.2550 - i0.0047 4.7403 - i0.0027
[1/4] 1.0 2 1.5825 + i0.0013 3.2520 - i0.0035 4.7432 - i0.0026
[1/4] 1.0 4 1.5807 + i0.0011 3.2548 - i0.0044 4.7405 - i0.0026
[1/8] 2.0 2 1.5892 + i0.0062 3.3026 - i0.0157 4.7027 - i0.0085
[1/8] 2.0 4 1.5737 + i0.0047 3.2868 - i0.0013 4.7208 - i0.0044
[1/8] 1.0 2 1.5757 + i0.0007 3.2835 - i0.0001 4.7239 - i0.0002
[1/8] 1.0 4 1.5740 + i0.0004 3.2866 - i0.0010 4.7210 - i0.0003


Table VI: Computed Eigenvalues via the indirect BSEM
h kB-kA m 1st eigenvalue 2nd eigenvalue 3rd eigenvalue
[1/2] 2.0 2 1.6398 - i0.0722 3.2100 - i0.1047 4.7591 - i0.0961
[1/2] 2.0 4 1.6368 - i0.0771 3.2031 - i0.0908 4.7738 - i0.0865
[1/2] 1.0 2 1.6362 - i0.0859 3.2010 - i0.0878 4.7772 - i0.0852
[1/2] 1.0 4 1.6331 - i0.0851 3.2030 - i0.0905 4.7741 - i0.0863
[1/4] 2.0 2 1.6187 - i0.0403 3.2967 - i0.0756 4.7293 - i0.0515
[1/4] 2.0 4 1.6108 - i0.0435 3.2835 - i0.0599 4.7455 - i0.0429
[1/4] 1.0 2 1.6110 - i0.0503 3.2802 - i0.0584 4.7486 - i0.0424
[1/4] 1.0 4 1.6085 - i0.0497 3.2833 - i0.0596 4.7458 - i0.0428
[1/8] 2.0 2 1.6061 - i0.0223 3.3233 - i0.0525 4.7128 - i0.0316
[1/8] 2.0 4 1.5950 - i0.0247 3.3081 - i0.0369 4.7301 - i0.0232
[1/8] 1.0 2 1.5958 - i0.0304 3.3048 - i0.0357 4.7332 - i0.0229
[1/8] 1.0 4 1.5936 - i0.0301 3.3079 - i0.0366 4.7303 - i0.0231