True and Spurious Eigensolutions of Elliptical Membranes using Null-Field Boundary Integral Equations

1. National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering True and spurious eigensolutions of elliptical membranes by using null-field boundary integral equations Jai-Wei Lee, Jeng-Tzong Chen and Shyue-Yuh Leu Date: Dec., 10, 2009 Time: 10:30~10:50 am Place: Lunghwa University of Science and Technology

2. Outline • Introduction of NTOU/MSV group • Motivation and problem statement • Method of solution • Illustrative examples • Conclusions

3. Outline • Introduction of NTOU/MSV group • Motivation and problem statement • Method of solution • Illustrative examples • Conclusions

4. 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics Keelung

5. NTOU/MSV Group members (2009)

6. 國立台灣海洋大學力學聲響振動實驗室(NTOU/MSV Lab) 冰箱 微波爐 信封 講義 論文 書 櫃 敬啟者： 若本人不在辦公室請勞駕至 河工二館三樓三○六實驗室 分機：六一七七 歡迎蒞臨指導 陳正宗敬啟 電腦相關 論文 講義 討論桌 SEVER 大學部 蕭宇志 (國科案) 陳力豪 休 息 區 陳正宗 陳義麟 陳桂鴻 徐文信 周克勳 呂學育 李為民 書 櫃 資 料 庫 茶水 期刊論文 伙 食 期刊論文 期刊論文 期刊論文 期刊論文 工 作 台 大學部 蕭宇志 (國科案) 陳力豪 論 文 論 文 影 本 會 議 資 料 吳建鋒 蔡振鈞 紀志昌 程 式 文具櫃 高聖凱 書櫃 投 影 片 徐胤祥 解 答 沙 發 休 息 區 李文哲 書櫃 李家瑋 軟 體 負責老師：陳正宗 終身特聘教授 (海洋大學河海工程學系) 地點：河工二館 HR2306 室 　陳義麟 副教授 (高雄海洋科技大學造船學系) 聯絡電話：886-2-24622192 ext.6177 or 6140 李為民 副教授 (中華技術學院機械系) URL：http://ind.ntou.edu.tw/~msvlab 呂學育 助理教授 (中華技術學院航空機械系) E-mail：jtchen@mail.ntou.edu.tw 陳桂鴻 副教授 (國立宜蘭大學土木系) Fax：886-2-24632375 　 徐文信 助理教授 (屏東科技大學教學資源中心) 范佳銘 助理教授 (海洋大學河海工程學系) 木櫃 期刊雜誌 技術報告 博碩士論文 論文資料 MSC/NASTRAN 鐵櫃 CTEX 軟體使用手冊 圖書文具 入口

7. 陳俊賢 (J S Chen, UCLA) Jeong-Guon Ih (KAIST, Korea) (黃晉, China) 陳 鞏(USA, Texas A M) (M.Tanaka, Japan) 余德浩 中國科學院 程宏達 (Alex H.-D. Cheng, USA) 陳清祥 (C. S. Chen, USA) 姚振漢 (Yao Z H, China) 美國 中國 NTOU/MSV visitors 杜慶華 (Q. H. Du,China) 吳漢津 (H C Wu, Iowa, USA) 日 本 南 韓 吳鼎文 (T. W. Wu, USA) 祝家麟 (J. L. Zhu, China)

8. Outline • Introduction of NTOU/MSV group • Motivation and problem statement • Method of solution • Illustrative examples • Conclusions

9. Free vibration of a membrane G. E. : Eigenproblems of Laplace operator Helmholtz equation displacement wave number domain

10. Advantages and disadvantages of FEM, BEM and BIEM

11. Extension to the eigenproblems with elliptical boundaries Degenerate kernel (Elliptic coordinates) Degenerate kernel (Polar coordinates) OK Extension Key point

12. Outline • Introduction of NTOU/MSV group • Motivation and problem statement • Method of solution • Illustrative examples • Conclusions

13. Boundary integral equation and null-field boundary integral equation Interior case Exterior case Degenerate (separable) form

14. Degenerate (separable) form of fundamental solution (2D) Ellipse Extension Circle

15. Contour plots of the closed-form fundamental solution and the degenerate kernel Re Im Abs Closed-form fundamental solution Degenerate kernel

16. Relationship of kernel functions U(s, x) T(s, x) (Dual system) L(s, x) M(s, x)

17. Four degenerate kernels

18. Boundary densities Expand boundary densities by using the eigenfunction expansion is a constants along the elliptical boundary

19. Keypoint for solving the problem with elliptical boundaries The orthogonal relations are reserved

20. Outline • Introduction of NTOU/MSV group • Motivation and problem statement • Method of solution • Illustrative examples • Conclusions

21. Illustrative examples • Case 1: An elliptical membrane • Case 2: A confocal elliptical annulus

22. Case 1:An elliptical membrane G. E.: B. Cs.:

23. True and spurious eigenvalues True Spurious (11) Complex-valued kernel Dirichlet BC (11) Real-part kernel Note: the data inside parentheses denote the spurious eigenvalue.

24. Mode shapes (11) Even Even Even Odd Even

25. A confocal elliptical annulus G. E.: B. Cs.: Simply-connected Multiply-connected

26. True and spurious eigenvalues (42) True Spurious UT equation Note: the data inside parentheses denote the spurious eigenvalue. Eigenvalues of an elliptical membrane (case 1) (11)

27. Mode shapes (42) Even Odd Even Odd Even

28. Outline • Introduction of NTOU/MSV group • Motivation and problem statement • Method of solution • Illustrative examples • Conclusions

29. Conclusions (Simply-connected domain) 1.Elliptic coordinates 1.Polar coordinates 2.Fourier series 2.Mathieu function 3.Bessel function 3.Modified Mathieu function Kernel Real-part Imaginary-part Kuo et al. Int. J. Numer. Meth. Engng. 2000 Present, 2009 Spurious eigenequations depend on 1. The real-part kernel used 2. The imaginary-part used

30. Conclusions (Multiply-connected domain) 1.Elliptic coordinates 1.Polar coordinates 2.Fourier series 2.Mathieu function 3.Bessel function 3.Modified Mathieu function Complex-valued kernel UT or LM Inner boundary Chen et al. Proc. R. Soc. Lond., Ser. A, 2002 & 2003 Present, 2009 Spurious eigenequations depend on 1. The geometry of inner boundary 2. The approach used (Singular or Hypersingular)

31. The end Thanks for your kind attentions Welcome to visit the web site of MSVLAB/NTOU http://msvlab.hre.ntou.edu.tw/

32. Successful experiences in 2-D eigenproblems with circular boundaries Complex-valued kernel Kernel UT or LM Real-part Inner boundary Imaginary-part Degenerate kernel (Polar coordinates) UT equation (Singular) LM equation (Hypersingular) Spurious eigenvalues Spurious eigenvalues Chen et al. Proc. R. Soc. Lond., Ser. A, 2002 & 2003 Kuo et al. Int. J. Numer. Meth. Engng. 2000 (Found and treated) Key point

33. Elliptic coordinates and Mathieu function angular coordinate radial coordinate Mathieu function Modified Mathieu function

34. Degenerate kernels Addition theorem (Morse and Feshbach’s book) Methods of Theoretical Physics, 1953, p.1421 Modified Mathieu functions of the third kind Orthogonal relations (norm) Analytical study

35. True and spurious eigenequations True UT Even (Singular) Complex-valued Odd Spurious Even UT (Singular) Odd Real-part

36. True and spurious eigenequations True Even UT (Singular) LM Odd (Hypersingular) Spurious Even UT Odd (Singular) Even LM (Hypersingular) Odd

37. Confocal elliptical annulus

38. True and spurious eigenequations

39. True and spurious eigenequations Neumann BC Dirichlet BC

40. True and spurious eigenequations True Spurious B.C. fixed-fixed Even Even Odd Odd UT Even Even Odd Odd LM

41. Successful experiences in 2-D problems with circular boundaries using the present approach Degenerate kernel Fundamental solution (Laplace) (Helmholtz) • Advantages of present approach: • No principal value • Well-posed model • Exponential convergence • Free of mesh generation The proposed approach will be extended to deal with 2-D problem with elliptic boundaries

42. Why spurious solution occurs • FDM for ODE • Real-part BEM & MRM (Simply-connected problem) • Complex-valued BEM (Multiply-connected problem)

43. Separation of variables in the elliptic coordinates Cartesian coordinates Elliptic coordinates separation of variables

44. Addition theorem Q r a b P O Addition theorem + = Subtraction theorem

45. A circular membrane with an elliptical hole Note: the data inside parentheses denote the spurious eigenvalue.

46. An elliptical membrane with a circular hole Note: the data inside parentheses denote the spurious eigenvalue.

47. Note: the data inside parentheses denote the spurious eigenvalue.

48. Nonunique solution t(a,0) Near-trapped mode (physical) Non-unique solution: Fictitious frequency (Numerical) (1) CHIEF method (Schenck, JASA , 1968) Additional constraint (CHIEF point) (2) Burton and Miller method (Burton and Miller, PRS , 1971) (3) SVD updating term technique (Chen et al., JSV, 2002)

49. SVD updating technique (去蕪[ ]存菁( )術) The same U T The same The same L M The same true mode, rigid body mode (physics) spurious mode, fictitious mode (mathematics)

50. Degenerate cases in mathematics and mechanics