| Summary: | 碩士 === 國立海洋大學 === 河海工程學系 === 90 === We provide a perspective on the degenerate problems, including degenerate scale,
degenerate boundary,
spurious eigensolution and fictitious frequency,
in the boundary integral formulation.
All the degenerate problems originate from the rank deficiency
in the influence matrix.
Both the Fredholm alternative theorem and singular value decomposition (SVD) technique
are employed to study the degenerate problems.
Updating terms and updating documents of the SVD technique are utilized.
The roles of right and left unitary vectors of the influence matrices in BEM and their relations
to true, spurious and fictitious modes are examined by using the Fredholm alternative theorem.
A unified method for dealing with the degenerate problem
in BEM is proposed.
For the degenerate scale problem,
three regularization techniques, hypersingular formulation,
method of adding a rigid body mode and CHEEF concept,
are employed to deal with the rank-deficiency problem.
Instead of direct searching for the degenerate scale by trial and error,
a more efficient technique is proposed to directly obtain the singular case
since only one normal scale needs to be computed.
The existence of degenerate scale is proved for the two-dimensional
Laplace problem using the integral formulation. The addition of a rigid body term, $c$, in the
fundamental solution can
shift the original degenerate scale
to a new degenerate scale by a factor $e^{-c}$.
Instead of using either the multi-domain BEM or the dual BEM
for degenerate-boundary problems,
the eigensolutions for membranes with stringers
are obtained in a single domain by using
the conventional BEM in conjunction with the SVD technique.
The occuring mechanism of both the spurious and fictitious
eigensolutions are unified by using the Fredholm alternative
theorem and SVD technique.
The criterion to check the validity of CHIEF and CHEEF points
is also addressed.
Several examples are demonstrated to check the validity of the proposed method.
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