(34.204.201.220) 您好!臺灣時間:2021/04/20 11:51
字體大小: 字級放大   字級縮小   預設字形  
回查詢結果

詳目顯示:::

我願授權國圖
: 
twitterline
研究生:林英傑
研究生(外文):Ying-Chieh Lin
論文名稱:以基本解數值法求解双諧和方程式及薄板震動問題
論文名稱(外文):Method of Fundamental Solutions for the Bi-harmonic Equations and Plate Vibration Problems
指導教授:楊德良楊德良引用關係
指導教授(外文):Der-Liang Young
學位類別:碩士
校院名稱:國立臺灣大學
系所名稱:土木工程學研究所
學門:工程學門
學類:土木工程學類
論文種類:學術論文
論文出版年:2003
畢業學年度:91
語文別:英文
論文頁數:83
中文關鍵詞:基本解法流線函數史托克斯方程式諧和方程式奇異值分解法赫姆霍滋方程式特徵值特徵模態
外文關鍵詞:Method of fundamental solutionsStream functionStokes equationsBi-Harmonic equationSingular value decompositionHelmholtz equationEigenvalueEigenmode
相關次數:
  • 被引用被引用:0
  • 點閱點閱:201
  • 評分評分:系統版面圖檔系統版面圖檔系統版面圖檔系統版面圖檔系統版面圖檔
  • 下載下載:0
  • 收藏至我的研究室書目清單書目收藏:0
本論文主要是利用基本解無網格法來模擬流線函數、史托克斯方程式、赫姆霍滋方程式以及板震動問題。而我們主要的議題是利用基本解無網格法來求解双諧和方程式,而諧和方程式被廣泛的應用在薄板理論以及緩慢的流體問題之中。在本論文之中我們對流線函數的邊界條件做了一些假設,把部分的邊界條件移除,並和有限元素法模擬結果相比較,結果仍是相當精準。同時我們藉由一些假設來利用二薄維板震動的基本解求解二維流線函數,而數值結果也證實我們的假設是正確的。
在薄板震動問題之中我們會遭遇到邊界條件為零的情形,為了求得一有意義之非零解,本文藉由非奇異值分解法來求得其特徵值與特徵向量,並且模擬二維赫姆霍滋方程式與解析解做比較作為驗證,結果顯示在相當不錯且有效率。接著應用此模式在二維板震動的問題,在固定邊界條件下可以有效的求得特徵頻率。

In this study, we adopt the method of fundamental solutions (MFS) to simulate the two-dimensional stream function, the two-dimensional Stokes equations, the two-dimensional Helmholtz equation, and two-dimensional thin plate vibration problem. Our objective is to use the MFS to solve the Bi-Harmonic equations that are widely employed in the theory of thin plate and slow flow problems. Some reasonable assumptions have been made on the boundary conditions for the stream function. Comparison of the results by MFS to the results from FEM reveals that both are very close. On the other hand, we made some asymptotic assumptions of the fundamental solutions for thin plate vibration problem to simulate the two dimensional stream function and the numerical results show that our asymptotic behaviors have been justified.
In thin plate vibration problems, we will deal with the governing equation with the homogeneous boundary conditions. In order to find out meaningful non-trivial solutions, we utilize the singular value decomposition (SVD) method to detect the eigenvalues and eigenmodes. Therefore in order to justify the correct usage of the SVD method, we have simulated two dimensional Helmholtz equation with homogeneous boundary condition, and compared with the analytical solutions. The result indicates that the MFS is very accurate and efficient as far as computational aspects are concerned. Furthermore, we utilize singular value decomposition (SVD) method to solve the thin plate vibration problem and the eigenfrequencies have been obtained efficiently by employing the clamped boundary conditions for circular thin plates.

Table of Contents
摘要 Ⅰ
Abstract Ⅱ
Table of Contents Ⅲ
Figure List Ⅴ
Table List Ⅷ
Chapter 1 Introduction
1.1 Motivations and Objectives 1
1.2 Literature Review 2
1.3 Outline of the Thesis 4
Chapter 2 Method of Fundamental Solutions and Radial Basis Functions
2.1 Introduction to Method of Fundamental Solutions 5
2.2 Radial Basis Functions 8
Chapter 3 Stream Function and the Stokes Equations
3.1 Stream Function 12
3.1.1 Governing Equation 12
3.1.2 Numerical Results 14
3.2 Stokes Equations and their Fundamental Solutions 41
3.2.1 Governing Equations and Fundamental Solutions 41
3.2.2 Numerical Results 43
Chapter 4 Helmholtz Equations and Free Vibration Analysis of Plate
4.1 Helmholtz Equations 50
4.1.1 Governing Equation 50
4.1.2 Numerical Results 52
4.2 Free Vibration Analysis of Plate 64
4.2.1 Governing Equation 64
4.2.2 Numerical Results 65
Chapter 5 Conclusions and Future Studies
5.1 Conclusions 78
5.2 Future Studies 79
References 80

References
[1] Karageorghis, A. and Fairweather, G., “The method of fundamental solutions for the numerical solution of the biharmonic equation”, Journal of Computational Physics, pp. 434-459, 1986.
[2] Karageorghis, A. and Fairweather, G., “The almansi method of fundamental solution for solving biharmonic problems”, international Journal for Numerical Methods in Engineering, Vol. 26, pp. 1665-1682, 1988.
[3] Karageorghis, A. and Fairweather, G.,”The simple layer potential method of fundamental solutions for certain biharmonic problems”, International Journal for Numerical Methods in Fluids, Vol. 9, pp. 1221-1234, 1989.
[4] Karageorghis, A., “Modified methods of fundamental solutions for harmonic and biharmonic problems with boundary singularities”, Numerical Methods for Partial Differential Equations, vol. 8, pp. 1-19, 1992.
[5] Lin, T. W. and Lin, C. H., “Numerical methods and programs.”, Graphics and Literature Boos Publisher. Taipri, Taiwan, 1997. (In Chinese)
[6] Chiu, C. L., “Non-singular boundary integral equation for the analysis of the electromagnetic problems.”, MS Thesis, National Taiwan University, Taipei, Taiwan, 2002.
[7] Tasi, C. C., “Meshless Numerical Methods and their Engineering Applications.”, Ph D Dissertation, National Taiwan University, Taipei, Taiwan, 2002.
[8] Lee, Y.T., Chen, I.L., Chen, K.H., and Chen, J.T., “A new meshless method for free vibration analysis of plates using radial basis function”, The 26th National Conference on Theoretical and Applied Mechanics, 2002.
[9] Nardini, D. and Breebia, C.A., “A new approach to free vibration analysis using boundary elements”, Boundary Element Methods in Engineering, Springer-Verlag, pp. 312-326, 1982.
[10] Cheng, A.H.-D, Young, D.L., and Tsai, C.C., “Solution of Poisson’s equation by iterative DRBEM using compactly supported, positive definite radial basis function”, Engineering Analysis with Boundary Elements, 24, pp. 549-557, 2000.
[11] Young, D.L., Tsai, C.C., Eldho, T.I., and Cheng, A.H.-D., “Solution of Stokes flow using an iterative DRBEM based on compactly-supported, positive-definite radial basis function”, Computers and Mathematics with Applications, 43, pp. 607-619, 2002.
[12] Tsai, C.C., Young, D.L., and Cheng, A.H.-D., “An iterative DRBEM for three-dimensional Poisson’s Equation”, Boundary Element Technology, XIV, WIT Press, pp. 323-332, 2000.
[13] Trefftz, “E. Ein Gegenstuck zum Ritzchen Verharen”, Proceeding of 2nd Int. Cong. Appl. Mech., Zurich, pp. 131-137, 1926.
[14] Golberg, M.A., “The method of fundamental solutions for Poisson’s equation”, Engineering Analysis with Boundary Elements, vol. 16, pp. 205-213, 1995.
[15] Li, J., ”Mathematical justification for RBF-MFS”, Engineering Analysis with Boundary Elements, vol. 25, pp. 897-901, 2001.
[16],Tsai, C.C., Young, D.L., and Cheng, A.H.-D, “Meshless BEM for steady three-dimensional Stokes Flows”, Proceedings of International Conference on Computational Engineering & Sciences, Tech. Science Press , 2001.
[17],Tsai, C.C., Young, D.L., and Cheng, A.H.-D, “Meshless BEM for three-dimensional Stokes Flows”, Computer Modeling in Engineering & Sciences, 3(1), pp. 117-128, 2002.
[18] Jaswon, M. A., Maiti, M. and Symm, “Numerical Biharmonic analysis and some applications.”, G. T., International Journal of Solids and Structures, vol. 3, pp. 309, 1967.
[19] Jaswon, M. A. and Maiti, M., “An integral equation formulation of plate bending problems.”, Journal of Engineering Mathematics, vol. 2, pp. 83-89 , 1968.
[20] Maiti, M. and Chakrabarti, S. K., International Journal of Engineering Science, vol. 12, pp. 793, 1974.
[21] Leissa, A.W., ”Vibration of Plates”, NASA SP-160, 1969.
[22] Murashima, S., Nonaka, Y., and Nieda, H., in Boundary Elements, Proceeding of the Fifth International Confererence, Hiroshima, Japan, 1983, edited by C. A. Brebbia, T. Futagami, and M. Tanaka (Springer-Verlag, New York), pp. 75 , 1983.
[23] Burgess G. and Mahajerin, E., “Rotational Fluid Flow Using a Least Squares Colloctaion Technique.”, Computers & Fluids , vol. 12, pp. 311, 1984.
[24] Lo D. J., “Two-dimensional Velocity-vorticity Formulation for Incompressible Flows with Free Surfaces by the Finite Element Method.”, MS Thesis, National Taiwan University, Taipei, Taiwan, 2000.
[25] Burggraf, O.R , “Analytic and Numerical Studies of Structure of Steady Separated Flow”, J. Fluid Mech. ,vol. 24, pp. 113-151, 1966.
[26] Hwu, T.Y., Young, D. L., and Chen, Y. Y., “Chaotic Advections for Stokes Flows in Circular Cavity.”, Journal of Engineering Mechanics, pp. 774-882, 1997.
[27] Chen, C.S., Golberg, M.A., Ganesh, M., and Cheng, A.H.-D., “Multilevel compact radial functions based computational schemes for some elliptic problems”, Journal of Computers and Mathematics with Application, 43, pp. 359-378, 2002.
[28] Li, X., Ho, C.H., and Chen, C.S., “Computational test of approximation of functions and their derivatives by radial basis functions”, to appear in Neural, Parallel and Scientific Computations, 2002.
[29] Nardini, D., and Breebia, C.A., “A new approach to free vibration analysis using boundary elements”, Boundary Element Methods in Engineering, Springer-Verlag, pp. 312-326, 1982
[30] Balakrishnan, K., and Ramachandrn, P.A., “A particular solution Trefftz method for non-linear Poisson problems in heat and mass transfer”, Journal of Computational Physics, vol. 150, pp. 239-267, 1999.
[31] Berbbia, C. A., Topics in Boundary Element Research, Vol. 6, Electromagnetic Applications, Springer-Verlag, 1989.
[32] Press, W. H., Teukolsky, S. A., and Vetterling, W. T., and Flannery, B. P., Numerical recipes in Fortran, ed., Cambridge, 1992.
[33] Chen, J.T., Chen, I.L., Chen, K.H., Lee, Y.T., and Yeh, Y.T., ”A meshless method for free vibration analysis of circular and rectangular clamped plates using radial basis function.”, Engineering Analysis with Boundary Elements, 5 April 2003.
[34] Kitahara, M., “Boundary Integral Equation Methods in Eigenvalue problems of Elastodynamics and Thin Plates”, Amsterdam, Elsevier, 1985.
[35] Pozrikidis, C., “Boundary integral and singularity methods for linearized viscous flow.”, Cambridge university press, 1992.
[36] Kythe, P. K. “Fundamental solution for differential operators and applications.”, Boston: Birkhauser, 1996.
[37] Wong, G.K.K. and Hutchinson, J.R. ; An improved boundary element method for plate vibrations, Boundary Element Methods, Ed. Brebbia, C.A., Springer-verlag, Berlin, pp. 272-289, 1981 (Proc. 3rd International Seminar, Irvine, California, 1981).

QRCODE
 
 
 
 
 
                                                                                                                                                                                                                                                                                                                                                                                                               
第一頁 上一頁 下一頁 最後一頁 top
系統版面圖檔 系統版面圖檔