| Summary: | 碩士 === 國立臺灣海洋大學 === 機械與機電工程學系 === 100 === Degenerate scales for elliptical, regular N-gon and half-disc domains are studied by using the boundary element method (BEM) and conformal mapping. For solving two-dimensional Laplace problems by using the BEM, the special size of geometry results in a rank-deficient matrix, and is different from the size effect problem in physics. Degenerate scale stems from either the non-uniqueness of BIE using the logarithmic kernel or the conformal mapping of unit logarithmic capacity in the complex variables. The degenerate scale can be analytically derived by using the conformal mapping as well as numerical detection by using the BEM in this thesis. Analytical formula of an ellipse for the degenerate scale can be derived not only from the conformal mapping in conjunction with unit logarithmic capacity, but also can be derived by using the degenerate kernel. Eigenvalues and eigenfunctions for the weakly singular integral operator in the elliptical domain are both derived by using the degenerate kernel. It is found that a zero eigenvalue results in a degenerate scale. By using the conformal mapping technique, it is interesting to find that the absolute value of the logarithmic capacity equals to 1 in the case of degenerate scale. Based on the singular value decomposition, the rank-deficiency (mathematical) mode due to the degenerate scale (mathematics) is imbedded in the left singular vector for the influence matrices of weakly singular (U kernel) and strongly singular (T kernel) integral operators. On the other hand, we obtain the common right singular vector in the dual integral formulation corresponding to a rigid body mode (physics) in the influence matrices of strongly singular (T kernel) and hypersingular (M kernel) operators. To deal with the problem of non-uniqueness solution, four regularization techniques, the hypersingular BIE, the constraint of boundary flux equilibrium, addition of a rigid body term in the fundamental solution and the Combined Helmholtz Exterior integral Equation Formulation (CHEEF) approach, are employed to promote the rank of influence matrices to be full rank. Null field for the exterior domain and interior nonzero field are analytically derived and numerically verified for the ordinary scale while the null field for the interior domain and nonzero exterior field are obtained for the homogeneous Dirichlet problem in the case of the degenerate scale. It is also found that the contour of nonzero exterior field for the degenerate scale using the BEM matches well with that of the conformal mapping. Besides, no failure CHEEF point outside the domain can be found due to the nonzero field of the complementary domain in the case of degenerate scale. Only one trial in the BEM is required to determine the degenerate scale. Both analytical and numerical results agree well in the demonstrative examples.
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