臺灣博碩士論文加值系統

本文探討使用邊界元素法/邊界積分方程法在解決二維彈力問題時，由於問題的幾何外型在某些特定尺度的時候會使得影響矩陣產生秩降問題，這不同於物理問題上的尺度效應，而是數學方法本身所造成的。針對退化尺度發生的機制，
在離散系統我們透過兩個數值技巧來求得，其一為透過奇異值分解法，逐一搜尋影響係數矩陣之最小奇異值為零時所對應的尺度;其二為將二維彈力的退化尺度問題化簡為二乘二矩陣的特徵值問題。而在連續系統首先回顧Kelvin解在極座標系統下的退化核形式，以及推導在橢圓座標系統下的退化核形式。其次將基本解以分離核形式呈現，邊界密度及邊界條件使用特徵方程表示，透過積分和比較係數後，圓形與橢圓形的退化核尺度即可以被解析的推導而得。此外，我們亦透過分離核解析探討積分方程對唯一解的規範是否充分且必要以及積分運算元值域缺損的問題。針對給定不同類型的邊界條件，我們透過模態參與係數的觀念來探討對應奇異值為零的模態是否容易被激發出來。為了處理解不唯一的問題，藉由處理在二維拉普拉斯因退化尺度所產生的病態問題之成功經驗，我們根據Firchra 法，藉由增加自由常數以及對應束制條件來提升影響係數矩陣的秩數，解決退化尺度問題中解不唯一的情況。此外亦使用Firchra法處理病態矩陣的想法，透過奇異值分解和增邊矩陣，藉由增加自由度及相對應的束制條件，我們推導出秩降矩陣自我正則化方法，並透過自我正則化方法成功解決自由結構之奇異柔度/勁度矩陣。透過增加的自由常數是否為零也可做為判斷結構之給定外力是否滿足力平衡的有解依據。

For solving two-dimensional (2-D) elasticity problems by using the boundary element method (BEM), the special size of geometry results in a rank-deficient matrix, which is different from the size effect in physics. Direct searching technique (singular value decomposition) and the only two-trials technique ( matrix eigenvalue problem) are used to find the numerical solution of degenerate scales, respectively. By using the boundary integral equation method (BIEM), the range deficiency also exists in the degenerate-scale problem. The degenerate kernel for the 2-D elasticity Kelvin solution in the polar coordinates is reviewed first and the degenerate kernel in the elliptical coordinates is derived. By using the indirect BIEM in conjunction with the degenerate kernel, the analytical solutions of degenerate scales for a circle and an ellipse are derived. Besides, the sufficient and necessary condition of ensure unique solution of the boundary integral equation (BIE) is also addressed. The ill-conditioned system is derived from the incomplete integral equation. By using the degenerate kernel, the space-deficiency of solution in the degenerate scale problems for 2-D elasticity is analytically studied. The modal participation factor is used to numerically examine the excited mode corresponding to different types of boundary conditions. According to the experience of the degenerate-scale problems for 2-D Laplace equation, we extend Fichera’s idea to solve the degenerate scale problems for 2-D elasticity by adding a constant and the corresponding constraint. In addition, the rigid body mode results in a rank-deficient matrice also exist in the finite element method. Motivated by Fichera’s idea for regularizing the rank-deficiency model, we derive the self-regularization technique by using the bordered matrix and the singular value decomposition. The self-regularization technique is adopted to solve the free-free structure problems by adding n slack variables and corresponding constraints. The value of the extra degree of freedom can judge no solution (nonzero value) and infinite solution (zero value) with respect to the loading vector.

Contents I
Table captions IV
Figure captions V
Notations IX
Abstract XII
摘要 XIII

Chapter 1 Introduction .................................1
1.1 Motivation of the research and literature review .................................................1
1.2 Organization of the present thesis .........3

Chapter 2 Derivation of the degenerate scales for two-dimensional elasticity problem by using the BEM/BIEM ...7
Summary ........................................7
2.1 Introduction ...............................7
2.2 Problem statement and formulation ..........9
2.2.1 Problem statement ........................9
2.2.2 Boundary integral equation formulation ...9
2.2.3 Expansions of fundamental solution and boundary information ..................................10
2.2.3.1 Review of the Kelvin solution by using the degenerate kernel in terms of the polar coordinates .......................................................10
2.2.3.2 Derivation of the degenerate kernel for the Kelvin solution in terms of the elliptic coordinates .......................................................12
2.2.3.3 Fourier series expansion for boundary information in the circular domain ....................14
2.2.3.4 Eigenfunction expansion for boundary information in the elliptical domain ..................15
2.3 Derivation of analytical degenerate scales by using the degenerate kernels ..........................15
2.3.1 Degenerate scales on the circular domain for a 2-D elasticity problem ..........................15
2.3.2 Degenerate scales on the circular domain for a 2-D elasticity problem ..........................16
2.4 Numerical examination of degenerate scales by using the direct searching technique and only two ordinary trials .......................................18
2.4.1 Direct searching technique to find the degenerate scales ............ 18
2.4.2 Only two-trials to find degenerate scales .......................................................18
2.5 Illustrative examples .....................21
2.6 Concluding remarks ........................23

Chapter 3 A necessary and sufficient BEM/BIEM for two-dimensional elasticity problems ...................36
Summary .......................................36
3.1 Introduction ..............................36
3.2 Problem statement and formulation .........38
3.2.1 Problem statement .......................38
3.2.2 Derivation of the analytical solution for a circular case .........................................39
3.2.3 Derivation of the analytical solution for an elliptical case ....................................40
3.2.4 Regularized method for the degenerate-scale problems ..............................................43
3.2.5 Modal participation factor to examine the numerical instability .................................44
3.3 Illustrative examples and discussions .......................................................45
3.4 Concluding remarks ........................48

Chapter 4 A self-regularization approach for solving the free-free structure problems ..........................59
Summary .......................................59
4.1 Introduction ..............................59
4.2 Derivation of the formulation .............62
4.2.1 Firchera’s method for the boundary element method ................................................62
4.2.2 Self-regularization approach for the linear algebraic equation ....................................62
4.2.3 Linkage of the slack variables corresponding to the bordered matrix to the Fredholm alternative theorem ...................................65
4.3 Illustrative examples .....................66
4.4 Concluding remarks ........................77

Chapter 5 Conclusions and further researches ..........91
5.1 Conclusions ...............................91
5.2 Future researches .........................92

References ............................................93
Appendix A ...........................................101
Appendix B ...........................................105

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