臺灣博碩士論文加值系統

本文採用直接與間接邊界積分方程法結合傅立葉級數與退化核來求解含圓形邊界雙諧和方程之板與史托克流場問題。其中退化核即為分離核，係基本解中將場、源點分離所導得的級數形式，藉由退化核的內外域表示式可避免主值積分的計算。而未知的邊界密度函數則以傅立葉級數做展開。在邊界上均勻佈點，並配合邊界條件可得一線性代數方程式，其未知的傅立葉係數即可輕易求得，將之代回邊界積分方程中可得場解。本法屬於半解析法，其主要誤差來源為所截取的傅立葉項數，經由項數的增加，可收斂到正解。本文所得之數值結果將與有限元素法套裝軟體 (ABAQUS) 之結果和前人研究 (包含板問題與偏心圓磨潤學) 做一比較，以驗證本法的可行性。

In this thesis, the direct and indirect boundary integral equation methods (BIEMs) in conjunction with Fourier series and degenerate kernels are proposed to solve the biharmonic equations with circular boundaries for the multiply-connected plate and Stokes problems. The degenerate kernels in the direct BIEM and indirect BIEM are expanded by using the separation of field point and source point. The improper boundary integrals are novelly avoided since the appropriate interior and exterior expansion of degenerate kernels are used. The unknown boundary densities are expressed in terms of Fourier series. A linear algebraic system can be obtained by matching the boundary conditions and by collocating the boundary points. The unknown Fourier coefficients are determined easily and then substituted into the boundary integral equation to have the solution. The present method can be regarded as a semi-analytical approach since error only occurs in truncating the number of terms in the Fourier series. Convergence study is also addressed for the plate problems. Finally, the numerical solutions are compared with the finite element method software (ABAQUS) data and previous results (Bird and Steele, Kamal, Kelmanson and Ingham) to demonstrate the validity of the present method.

A semi-analytical approach for Stokes flow and plate problems with circular boundaries

Contents

Contents Ⅰ
Table captions Ⅲ
Figure captions Ⅳ
Notations Ⅵ
摘要 Ⅸ
Abstract Ⅹ

Chapter 1 Introduction 1
1.1 Literature review and motivation of the research 1
1.2 Organization of the thesis 4

Chapter 2 Null-field integral equation approach for plate problems with circular boundaries
6
2.1 Problems statement for a plate 6
2.2 Boundary integral equations for the domain point 7
2.3 Null-field integral equations 8
2.3.1 Expansion of Fourier series for boundary densities 9
2.3.2 Expansion of kernels 9
2.4 Adaptive observer system and vector decomposition for the normal derivative
10
2.4.1 Adaptive observer system 10
2.4.2 Vector decomposition 10
2.5 Linear algebraic system 11
2.6 Numerical results and discussions 15
2.7 Concluding remarks 17

Chapter 3 Direct and indirect boundary integral equation methods for Stokes flow problems
18
3.1 Formulation of the Stokes flow problems 18
3.2 Direct and indirect boundary integral equation methods 19
3.2.1 Direct formulation 19
3.2.2 Indirect formulation 20
3.3 Adaptive observer system and vector decomposition for the normal derivative
21
3.4 Solution procedures of the semi-analytical approaches 21
3.4.1 Direct boundary integral equation method 22
3.4.2 Indirect boundary integral equation method 31
3.5 Numerical examples 33
3.6 Concluding remarks 36

Chapter 4 Conclusions and further research 38
4.1 Conclusions 38
4.2 Further research 39

References 40
Appendix 1. Degenerate kernels 43















Table captions
Table 3-1 Comparison of analytical and numerical results of for the eccentric bearing
49






























Figure captions
Fig. 1-1 The biharmonic problems with arbitrary boundaries 50
Fig. 2-1 The Kirchhoff plate subject to the essential boundary conditions 51
Fig. 2-2 Degenerate kernel for
51
Fig. 2-3 Adaptive observer system when integrating the corresponding circular boundaries
52
Fig. 2-4 Vector decomposition (Collocation on and integration on )
52
Fig. 2-5 Collocation point and boundary contour integration in the null-field integral equation
53
Fig. 2-6 Boundary integral equation for the domain point 53
Fig. 2-7 Flowchart of the present method 54
Fig. 2-8 An annular plate subject to the essential boundary conditions 55
Fig. 2-9 The contour plot of displacement for the annular plate subject to the essential boundary conditions by using different method
55
Fig. 2-10 Error estimation of the moment and shear force on the boundaries for the concentric circular domain
56
Fig. 2-11 A circular plate containing three circular holes subject to the essential boundary conditions
57
Fig. 2-12 The contour plot of displacement for the plate containing three circular holes subject to the essential boundary conditions by using different method

57
Fig. 2-13 Parseval sum versus terms of Fourier series 59
Fig. 3-1 Collocation null-field point near in the direct formulation
60
Fig. 3-2 Boundary integral equation for the domain point 60
Fig. 3-3 Sketch of the null-field points near the inner cylinder for the centric case 61
Fig. 3-4 Sketch of the null-field points near the outer cylinder for the eccentric case
61
Fig. 3-5 Collocation method for the constraint equation 62
Fig. 3-6 (a) Biharmonic equation with the essential boundary condition 62
Fig. 3-6 (b) Biharmonic equation with the essential boundary condition 62
Fig. 3-7 Comparison of the contour plots for the biharmonic problem with the essential boundary conditions
63
Fig. 3-8 The flow between eccentric cylinders 64
Fig. 3-9 Comparison of contour plots of streamlines for
64
Fig. 3-10 Comparison of streamlines contour plots for
65
Fig. 3-11 Comparison of vorticity contour plots for
66
Fig. 3-12 Comparison of vorticity contour plots for
67
Fig. 3-13 The streamlines contour plot for by using indirect BIEM
68
Fig. 3-14 The streamlines contour plot for by using indirect BIEM
79
Fig. 3-15 The streamlines contour plot for by using indirect BIEM
70
Fig. 3-16 The streamlines contour plot for by using indirect BIEM
71
Fig. 3-17 The streamlines contour plot for by using indirect BIEM
72
Fig. 3-18 The streamlines contour plot for by using indirect BIEM
73
Fig. 3-19 The streamlines contour plot for by using indirect BIEM
74
Fig. 3-20 The streamlines contour plot for by using indirect BIEM
75
Fig. 3-21 The streamlines contour plot for by using indirect BIEM
76
Fig. 3-22 The streamlines contour plot for by using indirect BIEM
77
Fig. 3-23
Comparison of eccentricity versus degree of separation point by using different method
78

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