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Thanks to the fundamental solution, both BIEs and BEM are effective approaches for solving boundary value problems. But it may result in rank deficiency of the influence matrix in some situations such as fictitious frequency, spurious eigenvalue and degenerate scale. First, the nonequivalence between direct and indirect method is analytically studied by using the degenerate kernel and examined by using the linear algebraic system. The influence of contaminated boundary density on the field response is also discussed. It's well known that the CHIEF method and the Burton and Miller approach can solve the unique solution for exterior acoustics for any wave number. In this paper, we extend a similar idea to avoid the degenerate scale for the interior two-dimensional Laplace problem. One is the external source similar to the null-field BIE in the CHIEF method. The other is the Burton and Miller approach. Two analytical examples, circle and ellipse, were analytically studied. Numerical tests for general cases were also done. It is found that both two approaches can yield an unique solution for any size.
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Table 1. Comparison of four BIEs
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Table 2. Study on the degenerate scale by using different approaches
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Table 3. Degenerate scales appearing in the conventional BEM, the approach of adding rigid body mode, the Burton and Miller approach and the method of introducing the fictitious source point
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Table 4. Comparison of the four kernel functions
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Row space, column space, null space and range of
Numerical evidences for the degenerate scale of four cases in the BEM