臺灣博碩士論文加值系統

無限元素法發展多年，利用其特殊的元素大小相似性使得各層元素之間的材料性質亦存在特殊比例的關係式，搭配數值方法中的疊代法，已經應用於多種穩態的工程問題，例如：結構剛性問題、溫度溼度傳遞問題等。無限元素法特別在求解奇異點的問題上具有簡易建模，計算效率優異等特性，並且易於被當作多節點的超級元素與有限元素結合，進行各項分析。
本文主旨為發展無限元素法應用於薄板元素與薄膜元素。在Mindlin-Reissner薄板理論下，其中特殊的剪力剛性矩陣，須要先經過重新組合，才能得到無限元素法中的元素相似性，方能夠使用疊代法求得多節點的剛性矩陣(combined stiffness matrix)；配合無限元素法中的封閉模式，分析圓孔的大小、位置以及形狀對於平板剛性(Sb)的影響。另一方面，應用無限元素法求解薄膜元素的特徵值問題，使用疊代法求得最後的質量與剛性矩陣方法已不敷使用。在本文中，假設無限元素層與層之間的位移關係式為冪級數(power series)，使用數值軟體求解各項係數後，代入關係式方能夠得到多節點的質量與剛性矩陣(combined mass and stiffness matrix)。使用無限元素法中的裂紋模式，分析並探討裂紋的大小、數目、位置以及角度對於薄膜的自然模態波形影響(MAC-value)。
關鍵字:無限元素法、有限元素法、Mindlin-Reissner、plate、membrane、crack、Helmholtz equation

The Infinite Element Method (IEM) is widely used for the analysis of elastostatic structures containing singularities. In the IEM method, the problem domain is partitioned into multiple element layers, where the stiffness matrix of each layer is similar to that of the other layers in the discretized domain. However, in Mindlin-Reissner plate theory, the stiffness matrix varies through the layers of the plate, and thus the conventional IEM algorithm cannot be applied. Accordingly, the present study proposes a Plate Infinite Element Method (PIEM) in which the element stiffness matrix is separated into two sub-matrices; each being similar to the equivalent sub-matrix of the element layers above and below it. The PIEM algorithm is then coupled with an FEM algorithm and used to investigate the effects of the hole size, hole position and hole profile / area on the bending strength (Sb) of plates containing through-thickness holes. Besides, this study also proposes the detailed two-dimensional Infinite Element Method (IEM) formulation with Infinite Element (IE)-Finite Element (FE) coupling scheme for analyzing the out-of-plane vibration of isotropic membranes containing one or more tip cracks. The results show that the fundamental frequency of the membrane changes in accordance with the membrane thickness, but is unaffected by the number of cracks. However, it is shown that a change in the location of the cracks may cause a shift or rotation of the wave peaks of the structural mode shape.
keywords: IEM、FEM、Mindlin-Reissner、plate、membrane、crack、Helmholtz equation

目錄
摘要 I
Abstract III
目錄 V
圖目錄 VIII
表目錄 XII
符號表 XIII
符號表 XIV
符號表 XV
第一章 緒論 1
1-1 前言 1
1-2 文獻回顧 2
1-2-1 無限元素法的發展 2
1-2-2 裂紋分析文獻回顧 3
1-2-3 薄板理論發展與應用 5
1-2-4 薄膜振動模態的發展與應用 6
1-2-5 其他數值方法 8
1-3 文獻回顧總結 9
1-4 研究目的與方法 10
1-5 論文架構 11
第二章 基本無限元素法 15
2-1 平面元素公式推導 15
2-2 元素相似性 19
2-3 疊代法 21
2-4 無限元素網格劃分法 24
第三章 薄板無限元素法 25
3-1 Mindlin–Reissner 薄板理論 25
3-2 薄板無限元素法(Plate Infinite Element Method, PIEM) 28
3-3 薄板數值分析 31
3-3-1 PIEM程式驗證 32
3-3-2 應用PIEM與FEM分析 37
第四章 薄膜振動響應理論 43
4-1 特徵值問題 (Helmholtz equation) 43
4-2 薄膜面外(out-of-plane)振動理論 44
4-3 楊氏係數(E)與張力(T)轉換 47
4-4 應用有限元素法於薄膜振動理論 49
4-5 有限元素與無限元素結合 49
第五章 薄膜無限元素法 51
5-1 無限元素法之元素相似性 51
5-2 疊代法 55
5-3 韓厚德理論 55
5-4 應隆安理論 57
5-5 驗證有限/無限元素法 60
第六章 薄膜裂紋有限/無限數值分析 67
6-1 應用韓厚德理論 67
6-2 多裂紋薄膜問題 69
6-3 薄膜厚度的影響 75
6-4 多裂纹圓形薄膜 77
6-5 裂纹角度影響 83
第七章 結論 91
參考文獻 94

[1]P. Silvester, and I. A. Cermark, “Analysis of coaxial line discontinuities by boundary relaxation,” IEEE Transactions on Microwave Theory and Techniques, vol. 17, no. 8, pp. 489-495, 1969.
[2]應隆安, “The infinite similar element method for calculating stress intensity factors,” Scientia Sinica, vol. 11, no. 1, pp.19-43, 1978.
[3]應隆安, “無限元方法,” 北京大學出版社, 1981.
[4]L. A. Ying, “Infinite element methods,” Peking University Press, 1995.
[5]L. A. Ying, “Infinite element method for elliptic problems,” Science in China (Series A), vol. 34, no. 12, pp. 1438-1447, 1991.
[6]D. S. Liu, and D. Y. Chiou, “A coupled IEM/FEM approach for solving elastic problems with multiple cracks,” International Journal of Solids and Structures, vol. 4, no. 8, pp. 1973-1993, 2003.
[7]D. S. Liu, and D. Y. Chiou, “A hybrid 3D thermo-elastic infinite element modeling for area-array package solder joints,” Finite Elements in Analysis and Design, vol. 40, pp. 1703-1727, 2004.
[8]邱德義, “發展無限元素法及其應用於二維與三維之靜彈性問題,” 國立中正大學機械工程研究所, 博士論文, 2004.
[9]陳青揚, “以微觀力學計算模型為基礎探討構裝體中異質性接合交層結構的機溼熱相關問題,” 國立中正大學機械工程研究所, 博士論文, 2004.
[10]莊鎮瑋, “應用無限元素法分析構裝體中異性接合膠層結構的機濕熱相關問題,” 國立中正大學機械工程研究所, 博士論文, 2011.
[11]L. Demkowicz, and F. Ihlenburg, “Analysis of a coupled finite-infinite element method for exterior Helmholtz problems,” Numerische Mathematik, vol. 88, pp. 43-73, 2001.
[12]李曉龍, 王复明, 徐平, “自然元與無限元耦合方法在岩土工程粘彈性分析中的應用,” Journal of Vibration and Shock, vol. 27, no. 12, pp. 122-125, 2008.
[13]J. T. Tang, and J. Z. Gong, “A practical scheme for 3D geoelectrical forward modeling with finite-infinite element coupling method,” Progress In Electromagnetics Research Symposium Proceedings, pp. 437-440, 2010.
[14]H. Han, “The error estimates for the infinite element method for eigenvalue problems”, Analyse numerique/Numerical Analysis, vol. 16, no. 2, pp. 113-128, 1981.
[15]H. Han, “The infinite element method for eigenvalue problems,” J. Sys. Sci & Math. Scis., vol. 3, no. 3, pp. 163-171, 1983.
[16]韓厚德, 應隆安, “無限元迭代法,” 計算數學, 第一期, 1979.
[17]R. L. Wilson, and S. A. Meguid, “On the determination of mixed mode stress intensity factors of an angled crack in a disc using FEM,” Finite Elements in Analysis and Design, vol. 18, pp. 433-448, 1995.
[18]Q. Huang, C. Zou, and Z. Yin, “Study on stress intensity factors of multi-pin-hole connected structure with multi-cracks,” Journal of Mechanical Strength, vol. 27, no. 5, pp. 661-665, 2005.
[19]H. Nahvi, and M. Jabbari, “Crack detection in beams using experimental modal data and finite element model,” International Journal of Mechanical Sciences, vol. 47, pp. 1477-1497, 2005.
[20]R. D. Mindlin, “Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates,” ASME Journal of Applied Mechanics, vol. 18, pp. 31–38, 1951.
[21]M. Biljana, B. Zoran, M. Marina, and D. Petar, “Application of Mindlin’s theory for analysis of footing plate bending based on experimental research,” Architecture and Civil Engineering, vol. 8, no. 2, pp. 211-223, 2010.
[22]C. M. Wang, G. T. Lim, J. N. Reddy, and K. H. Lee, “Relationships between bending solutions of Reissner and Mindlin plate theories,” Engineering Structures, vol. 23, pp. 838-849, 2001.
[23]P. Yu, W. Guo, C. She, and J. Zhao, “The influence of Poisson’s ratio on thickness-dependent stress concentration at elliptic holes in elastic plates,” International Journal of Fatigue, vol. 30, pp. 165-171, 2008.
[24]Z. Yang, C. B. Kim, H. G. Beom, and C. Cho, “The stress and strain concentrations of out-of-plane bending plate containing a circular hole,”International Journal of Mechanical Sciences, vol. 52, pp. 836-846, 2010.
[25]Z. Yang, C. B. Kim, C. Cho, and H. G. Beom, “The concentration of stress and strain in finite thickness elastic plate containing a circular hole,” International Journal of Solids and Structures, vol. 45, pp. 713-731, 2008.
[26]M. A. Komur, and M. Sonmez, “Elastic buckling of rectangular plates under linearly varying in-plane normal load with a circular cutout,” Mechanics Research Communications, vol. 35, pp. 361-371, 2008.
[27]M. Mohammadi, J. R. Dryden, and L. Jiang, “Stress concentration around a hole in a radially inhomogeneous plate,” International Journal of Solids and Structures, vol. 48, pp. 483-491, 2011.
[28]E. Maiorana, C. Pellegrino, and C. Modena, “Elastic stability of plates with circular and rectangular holes subjected to axial compression and bending moment,” Thin-Walled Structures, vol. 47, pp. 241-255, 2009.
[29]A. H. Sheikh, and A. Chakrabarti, “A new plate bending element based on high-order shear deformation theory for the analysis of composite plates,” Finite Element in Analysis and Design, vol. 39, pp. 883-903, 2003.
[30]X. Y. Cui, G. R. Liu, and G. Y. Li, “Analysis of Mindlin-Reissner plates using cell-based smoothed radial point interpolation method,” International Journal of Applied Mechanics, vol. 2, no. 3, pp. 653-680, 2010.
[31]H. N. Xuan, T. Rabczuk, S. Bordas, and J. F. Debongnie, “A smoothed finite element method for plate analysis,” Computer Methods in Applied Mechanics and Engineering, vol. 197, pp. 1184-1203, 2008.
[32]C. Chinosi, C. Lovadina, and L. D. Marini, “Nonconforming locking-free finite elements for Reissner-Mindlin plates,” Computer Methods in Applied Mechanics and Engineering, vol. 195, pp. 3448-3460, 2006.
[33]M. H. Lee, and J. M. Park, “Flexural stiffness of selected corrugated structures,” Packaging Technology and Science, vol. 17, pp. 275-286, 2004.
[34]P. Feraboli, and K. T. Kedward, “Four-point bend interlaminar shear testing of uni- and multi-directional carbon/epoxy composite systems,” Composites Part A: Applied Science and Manufacturing, vol. 34, pp. 1265-1271, 2003.
[35]M. Huurman, R. Gelpke, and M. J. Jacobs, “A theoretical investigation into the 4 point bending test,” 7th RILEM International Conference on Cracking in Pavements, pp. 475-486, 2012.
[36]M. Tanner, and F. Smith, “MEMS reliability in a vibration environment,”Proceedings. 38th Annual International Reliability Physics Symposium, pp. 139-145, 2000.
[37]L. M. Zhang, B. Culshaw, and P. Dobson, “Measurement of Young’s modulus and internal stress in silicon microresonators using a resonant frequency technique,” Measurement Science and Technology, vol. 1, pp. 1343-1346, 1990.
[38]M. Ebert, R. Gerbach, and S. Michael, “Numerical identification of geometric from dynamic measurement of grinded membranes on wafer level,” 7th. International Conference on Thermal, Mechanical and Multiphysics Simulation and Experiments in Micro-Electronics and Micro-Systems, 2006.
[39]S. Michael, S. Hering, G. Holzer, T. Polster, M. Hoffmann, and A. Albrecht, “Parameter identification on wafer level of membrane structures,” 8th. International Conference on Thermal, Mechanical and Multiphysics Simulation and Experiments in Micro-Electronics and Micro-Systems, 2007.
[40]R. Gerbach, M. Ebert, and G. Brokmann, “Identification of mechanical defects in MEMS using dynamic measurements for application in production monitoring,” Microsyst Technol, vol. 16, pp. 1251-1257, 2010.
[41]N. Pugno, B. Peng, and H. D. Espinosa, “Predictions of strength in MEMS components with defects–a novel experimental–theoretical approach,” International Journal of Solids and Structures, vol. 42, pp. 647-661, 2004.
[42]J. Ricart, J. Pons-Nin, and E. Blokhina, “Control of MEMS vibration modes with pulsed digital oscillators-Part II: Simulation and experimental results,” IEEE Transactions on Circuits and Systems, vol. 57, no. 8, pp. 1879-1890, 2010.
[43]M. Ebert, R. Gerbach, and F. Naumann, “Measurement of dynamic properties of MEMS and the possibilities of parameter identification by simulation,” 8th. International Conference on Thermal, Mechanical and Multi-Physics Simulation Experiments in Micro-electronics and Micro-Systems, 2007.
[44]C. Y. Dong, S. H. Lo, and Y. K. Cheung, “Numerical solution for elastic half-plane inclusion problems by different integral equation approaches,” Engineering Analysis with Boundary Elements, vol. 28, pp. 123-140, 2004.
[45]W. M. Lee, J. T. Chen, and Y. T. Lee, “Free vibration analysis of circular plates with multiple circular holes using indirect BIEMs,” Journal of Sound and Vibration, vol. 304, pp. 811-830, 2007.
[46]E. S. Ventsel, “An indirect boundary element method for plate bending analysis,” International Journal for Numerical Methods in Engineering, vol. 40, pp. 1597-1610, 1997.
[47]J. T. Chen, C. C. Hsiao, and S. Y. Leu, “Null-field integral equation approach for plate problems with circular boundaries,” ASME Journal of Applied Mechanics, vol. 73, no.4, pp. 679-693, 2006.
[48]J. T, Chen, W. C. Shen, and A. C. Wu, “Null-field integral equations for stress field around circle holes under antiplane shear,” Engineering Analysis with Boundary Elements, vol. 30, pp. 205-217, 2006.
[49]J. T. Chen, W. C. Shen, and P. Y. Chen, “Analysis of circular torsion bar with circular holes using null-field approach,” CMES: Computer Modeling in Engineering & Sciences, vol. 12, no. 2, pp. 109-119, 2006.
[50]B. T. Kee, G. R. Liu, and C. Lu, “A least-square radial point collocation method for adaptive analysis in linear elasticity,” Engineering Analysis with Boundary Elements, vol. 32, pp. 440-460, 2008.
[51]G. R. Liu, and X. L. Chen, “A mesh-free method for static and free vibration analyses of thin plates of complicated shape,” Journal of Sound and Vibration, vol. 241, no. 5, pp. 839-855, 2001.
[52]K. Y. Liu, S. Y. Long, and G. Y. Li, “A Meshless Local Petrov-GalerkinMethod for the Analysis of Cracks in the Isotropic Functionally Graded Material,” CMC: Computers, Materials &Continua, vol. 7, no. 1, pp. 43-57, 2008.
[53]H. David, “Fundamentals of finite element analysis,” Mc Graw Hill Higher Education, Ch. 9, pp. 347-352, 2004.
[54]E. Mfoumou, C. Hedberg, and S. Kao-Walter, “Static versus low frequency dynamic elastic modulus measurement of thin films,” Electronic Journal Technical Acoustics, vol. 17, 2006.
[55]L. Meirovitch, “Principles and techniques of vibrations,” Prentice Hall International Editions, 1997.
[56]Y. W. Kwon, and H. Bang, “The finite element method using Matlab,” CRC Press, New York, 2000.
[57]Hibbitt, Karlsson, and Sorensen, “Abaqus user’s manual,” Version 6.2-1, 2001.
[58]吳家龍, “彈性力學,” 同濟大學出版社, pp. 416-418, 1992.