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研究生:林書睿
論文名稱:邊界元素法中退化問題之探討
論文名稱(外文):Study on the degenerate problems in BEM
指導教授:陳正宗陳正宗引用關係
學位類別:碩士
校院名稱:國立海洋大學
系所名稱:河海工程學系
學門:工程學門
學類:河海工程學類
論文種類:學術論文
論文出版年:2002
畢業學年度:90
語文別:英文
論文頁數:137
中文關鍵詞:邊界元素法退化問題Fredholm二擇一定理奇異值分解法
外文關鍵詞:boundary element methoddegenerate problemFredholm alternative theoremsingular value decomposition
相關次數:
  • 被引用被引用:0
  • 點閱點閱:320
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  • 收藏至我的研究室書目清單書目收藏:0
本文提出了邊界積分方程中退化問題的統一觀點,
此退化問題包含了退化尺度、退化邊界、內域真假特徵值和外域虛擬頻率。
而所有的退化問題均源自於影響係數矩陣的秩降現象。
針對這些退化問題,
可應用Fredholm二擇一定理、奇異值分解法(SVD),
以及奇異值分解法中的補充行、補充列的技巧來加以探討。
邊界元素法中的影響係數矩陣經由奇異值分解所得到
的左酉及右酉單位向量矩陣,利用Fredholm二擇一定理
與奇異值補充行、補充列的技巧
來檢視它們與真假與虛擬特徵模態的關係。
本文針對此退化問題提出一個統一的推導。
在退化尺度的問題,採用三種正規化的方法;超強奇異積分式、
加剛體運動項法、CHEEF法來
處理秩數不足的問題。此外,一個
更有效率的方法為只需一個正規尺度即可找出退化尺度而不需試誤法。
同時,在二維Laplace問題證明出退化尺度的存在。
我們亦證明出基本解加剛體運動項法在解退化尺度問題時,新的退化尺度為
原本退化尺度的e^{-c}倍。
含束制條的振動薄膜退化邊界問題,將採用
傳統邊界元素法伴隨奇異值分解法來解決,
這種方法取代了過去以對偶邊界元素法或多領域邊界元素法解退化邊界問題。
內域真假特徵值與外域虛擬頻率出現的機制
將一貫採用
Fredholm二擇一定理以及奇異值分解法來探討。
在本文並提出CHIEF與CHEEF點數目和位置的判定準則。
數值結果均驗證本方法的有效性。

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.

Contents \dotfill I \ Table captions \dotfill V\ Figure captions \dotfill VI\ Notations \dotfill X\ 中文摘要 \dotfill XIII\ Abstract \dotfill XIV\\ }
\begin{enumerate}
\itemsep=-1pt
\item[\bf Chapter 1] {\bf Introduction} \dotfill 1
\item[1-1] Degenerate problems in BEM \dotfill 1
\item[1-2] Degenerate scale for 2-D Laplace and Navier problems \dotfill 2
\item[1-3] Degenerate boundary in boundary value problems \dotfill 3
\item[1-4] Spurious eigensolutions for interior eigenproblems \dotfill 5
\item[1-5] Fictitious frequency in exterior acoustics \dotfill
\item[1-6] Scope of the thesis \dotfill 6
\ \item[\bf Chapter 2] {\bf Degenerate scale for torsion bar problems with
arbitrary cross sections using the dual BEM} \dotfill 9
\item[2-1] Introduction \dotfill 9
\item[2-2] Dual boundary integral formulation and dual BEM for torsion problems \dotfill 12
\item[2-3] Proof of the existence for the degenerate scale
of the two-dimensional Laplace problem using the integral formulation \dotfill 15
\item[2-4] Proof of the expansion ratio of {\bf $e^{-c}$} \dotfill 17
\item[2-5] Mathematical analysis of the degenerate scale
for an elliptical bar under torsion \dotfill 18
\item[2-6] Special case - circular bar with radius $R$ \dotfill 20
\item[2-7] Detection of degenerate scales and determination of spurious modes
by using the SVD updating documents and the Fredholm alternative theorem\dotfill 23
\item[2-8] Three regularization techniques to deal with
degenerate scale problems in BEM \dotfill 25
\begin{enumerate}
\itemsep=-1.2pt
\item[2-8-1] Method of adding a rigid body mode \dotfill 25
\item[2-8-2] Hypersingular formulation \dotfill 25
\item[2-8-3] CHEEF method \dotfill 25
\end{enumerate}
\item[2-9] Numerical examples \dotfill 26
\begin{enumerate}
\itemsep=-1.2pt
\item[2-9-1] Elliptical bar \dotfill 26
\item[2-9-2] Square bar \dotfill 27
\item[2-9-3] Triangular bar \dotfill 28
\item[2-9-4] Circular bar with keyway \dotfill 29
\end{enumerate}
\item[2-10] Conclusions \dotfill 30
\item[\bf Chapter 3] {\bf Eigenanalysis for membranes
with stringers using BEM in conjunction with SVD technique}\dotfill 32
\item[3-1] Introduction \dotfill 32
\item[3-2] Integral formulation and
boundary element implementation for the membrane eigenproblem with stringers \dotfill 33
\item[3-3] Review of the multi-domain BEM and the dual BEM
for the eigenproblem with a degenerate boundary \dotfill 35
\begin{enumerate}
\itemsep=-1.2pt
\item[3-3-1] Multi-domain BEM \dotfill 35
\item[3-3-2] Dual BEM \dotfill 36
\end{enumerate}
\item[3-4] Direct-searching scheme by using determinant
and singular value in BEM \dotfill 38
\begin{enumerate}
\itemsep=-1.2pt
\item[3-4-1] Multi-domain BEM \dotfill 38
\item[3-4-2] Dual BEM \dotfill 39
\item[3-4-3] $UT$ BEM+SVD \dotfill 39
\end{enumerate}
\item[3-5] Numerical examples \dotfill 40
\item[3-6] Conclusions \dotfill 42
\item[\bf Chapter 4] {\bf On the true and spurious eigensolutions for eigenproblems
using the Fredholm alternative theorem and SVD} \dotfill 44
\item[4-1] Introduction \dotfill 44
\item[4-2] Problem statement and the methods of solution \dotfill 45
\begin{enumerate}
\itemsep=-1.2pt
\item[4-2-1] True eigensolutions by using the complex-valued BEM \dotfill 46
\item[4-2-2] True and spurious eigensolutions by using the real-part BEM \dotfill 47
\item[4-2-3] True and spurious eigensolutions by using the imaginary-part BEM \dotfill 47
\item[4-2-4] True and spurious eigensolutions by using the MRM \dotfill 48
\end{enumerate}
\item[4-3] Extraction of the spurious eigensolutions by using
the Fredholm alternative theorem and SVD updating techniques \dotfill 49
\item[4-4] Extraction of the true eigensolutions by
using the Fredholm alternative theorem and SVD updating techniques \dotfill 50
\item[4-5] Numerical examples \dotfill 51
\item[4-6] Conclusions \dotfill 52
\item[\bf Chapter 5] {\bf Fictitious frequency revisited} \dotfill 53
\item[5-1] Introduction \dotfill 53
\item[5-2] Problem statement and review of the CHIEF method \dotfill 54
\item[5-3] Detection of the fictitious frequency
and ficitious mode for exterior acoustics using the
Fredholm alternative theorem and SVD technique \dotfill 55
\item[5-4] Mathemetical structure for the updating matrix \dotfill 56
\item[5-5] Source of numerical instability - zero division by zero \dotfill 58
\item[5-6] A criterion to check the validity of CHIEF points \dotfill 59
\item[5-7] Numerical examples \dotfill 61
\item[5-8] Conclusions \dotfill 63
\item[\bf Chapter 6] {\bf Conclusions and further research} \dotfill 64
\item[6-1] Conclusions \dotfill 65
\item[6-2] Further research \dotfill 66

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