1. 1 Nonuniqueness problems in numerical methods J T Chen( 陳正宗 ), Life-time Distinguished Prof. Taiwan Ocean University August 8, 11:00-12:00 2008 高海大旗津校區, 高雄 (CFD15-2008-chen.ppt) National Taiwan Ocean University MSVLAB ( 海大河工系 ) Department of Harbor and River Engineering

2. 2 Overview of numerical methods ( 中醫式的工程分析法 ) Nonuniqueness problems - review BEM failure (mathematical degeneracy) Degenerate boundary (No subdomain and no hypersingularity) Degenerate scale True and spurious eigensolution (interior prob.) Fictitious frequency (exterior acoustics) Conclusions Outline

3. 3 Overview of numerical methods PDE- variational IE DE Domain Boundary MFS,Trefftz method MLS, EFG 開刀 把脈把脈 針灸針灸 ( 中醫式的工程分析法 )

4. 4 BEM USA, China, UK, Germany, France, India, Italy, Iran, Japan, South Korea (Taiwan, No.11) Dual BEM (Made in Taiwan) UK, USA, Taiwan, China, Germany, France, Japan, Brazil, Australia, Singapore (No.3) (ISI information updated March 21, 2008) Top ten countries of BEM and dual BEM

5. 5 FEM USA, China, Germany, France, UK, Japan, India, Taiwan, Turkey, Italy (No.8) Meshless methods China, USA, Singapore, Germany, UK, Taiwan, Japan, Portugal, Slovakia, Australia (No.6) FDM China, USA, Japan, India, France, Taiwan, Canada, UK, Italy, South Korea (No.6) (ISI information updated March 21, 2008) Top ten countries of FEM, FDM and Meshless methods

6. 6 BEM Zhang C (Germany) Sapountzakis E J (Greece) Sladek J and Sladek V (Slovakia, twin) Chen J T (Taiwan, Ocean Univ.) 119 SCI papers > 545 citing Mukherjee S (USA) Tanaka M (Japan) Dual BEM (Made in Taiwan) Aliabadi M H (UK, Imperial College. London) Chen J T (Taiwan, Ocean Univ.) Chen K H(Taiwan, Ilan Univ.) Power H (UK, Univ. Nottingham) (ISI information updated March 21, 2008) Active scholars on BEM and dual BEM

7. 7 USA 劉毅軍教 授 NTOU/MSV Taiwan 海洋大學 陳正宗終身特聘教授 北京清華大學工程力學系 - 姚振漢教授 高海大造船系 - 陳義麟博士 台大土木系 - 楊德良終身特聘教授 宜蘭大學土木系陳桂鴻博士 北京清華姚振漢教授提供 Top 25 scholars on BEM/BIEM since 2001

8. 8 Number of Papers of FEM, BEM and FDM (Data form Prof. Cheng A. H. D.) 1 2 6

9. 9 March 21, 2008 Cauchy kernel Hadamard kernel BEM (no crack) Dual BEM (crack) Small scale Large scale Early Late 3.5 : 1 FMM(degenerate kernel) ? NTOU/MSV

10. 10 Advantages of BEM Discretization: dimension reduction Infinite domain (half plane) Interaction problem Local concentration Disadvantages of BEM Integral equations with singularity Full matrix (nonsymmetric) 北京清華

11. 11 BEM and FEM (1) BEM and meshless methods can be seen as a supplement of FEM. (2) BEM utilizes the discretization concept of FEM as well as the limitation. Whether the supplement is needed or not depends on its absolutely superior area than FEM. Ｃ rack & large scale problems Ó NTUCE

12. 12 Disclaimer (commercial code) The concepts, methods, and examples using our software are for illustrative and educational purposes only. Our cooperation assumes no liability or responsibility to any person or company for direct or indirect damages resulting from the use of any information contained here. inherent weakness ? misinterpretation ? User 當自強

13. 13 BEM trap ? Why engineers should learn mathematics ? Well-posed ? Existence ? Unique ? Mathematics versus Computation equivalent ? 馮康 定理 Some examples

14. 14 Nonuniqueness in numerical methods Nonlinear equation (spurious root) Finite difference method spurious eigenvalue Finite element method & meshless methods spurious mode Boundary element method spurious eigenvalues fictitious frequency Boundary element method degenerate scale

15. 15 Nonuniqueness in solving nonlinear Eq. Nonlinear equation (spurious root)

16. 16 Why spurious solution occurs 國中數學經驗 : 兩邊平方後整理 再ㄧ次兩邊平方後整理

17. 17 Nonuniqueness in FDM for ODE Finite difference method solve first-order ODE using Euler scheme (Greenberg, 1998)

18. 18 假根、浮根、溢根 (Spurious Eigenvalue) 用中間差分的方法來逼近處理 X y(x) x0x0 x5x5 x1x1 x2x2 x3x3 x4x4 0 h 2h3h4h5h

19. 19 假根、浮根、溢根 (Spurious Eigenvalue) h=0.05

20. 20 Nonuniqueness in FDM for eigenproblems Finite difference method solve eigenproblem (S. Zhao, 2007) spectral type nonspectral type rod, beam and membrane

21. 21 Nonuniqueness in FEM and meshless method Hour glass mode (solid mechanics) shear locking incompressible (solid propellant grain) Solid mechanics incompressible flow Fluid mechanics reduced integration Edge element-divergence free (electromagnetics)

22. 22 Solid mechanics (spurious mode) UCLA J S Chen, 2008 Physics Mathematics

23. 23 Nonuniqueness in BEM for degenerate boundary BEM with degenerate boundary 12 3 4 56 7 8 Cutoff wall crackThin airfoil

24. 24 What Is Boundary Element Method ? Ó NTUCE 12 3 4 5 6 12 geometry node the Nth constant or linear element N 西醫郎中

25. 25 Dual BEM Why hypersingular BIE is required (Two ways since 1986) Ó NTUCE 12 3 4 56 7 8 12 3 4 56 7 8 9 10 Artifical boundary introduced ! BEM Multi-domain Dual integral equations needed ! Dual BEM Single-domain Degenerate boundary

26. 26 Some researchers on Dual BEM (1012) Chen (1986) 544 citings in total Hong and Chen (1988 ） 78 citings ASCE EM Portela and Aliabadi (1992) 212 citings IJNME Mi and Aliabadi (1994) Wen and Aliabadi (1995) Chen and Chen (1995) 新竹清華 Yao (2005) 北京清華 黎在良等 --- 斷裂力學邊界數值方法 (1996) 周慎杰 (1999) Chen and Hong (1999) 88 citings ASME AMR Niu and Wang (2001) Kuhn G, Wrobel L C, Mukherjee S, Tuhkuri J, Gray L J Yu D H, Zhu J L, Chen Y Z, Tan R J … Ó NTUCE cite

27. 27 Dual Integral Equations by Hong and Chen(1984-1986) Ó NTUCE Singular integral equation Hypersingular integral equation Cauchy principal valueHadamard principal value (Mangler principal value) Boundary element methodDual boundary element method normal boundary degenerate boundary 1969 1986 2008

28. 28 Degenerate boundary geometry node the Nth constant or linear element (0,0) (-1,0.5) (-1,-0.5) (1,0.5) (1,-0.5) 12 3 4 56 7 8 N 5(+) 6(+) 5(+) 6(-) 5(+) 6(+) 5(+) 6(+) 5(+) 6(+) 5(+) 6(-) 5(+) 6(-) 5(+) 6(-) dependency Nonuniqueness

29. 29 The number of constraint equation is not enough to determine coefficients of p and q. Another constraint equation is obtained by differential operator How to get additional constraints

30. 30 Original data from Prof. Liu Y J (1984) crack BEM Cauchy kernel singular DBEM Hadamard kernel hypersingular FMM Large scale Degenerate kernel Desktop computer fauilure (2000) Integral equation 1888

31. 31 Successful experiences since 1986 (degenerate boundary)

32. 32 Solid rocket motor (Army 工蜂火箭 )

33. 33 X-ray detection ( 三溫暖測試 ) Crack initiation crack growth Stress reliever

34. 34 FEM simulation

35. 35 Stress analysis

36. 36 BEM simulation (Army)

37. 37 Shong-Fon II missile (Navy)

38. 38 V-band structure (Tien-Gen missile)

39. 39 FEM simulation

40. 40

41. 41 Seepage flow (Laplace equation) Sheet pile Cutoff wall

42. 42 Meshes of FEM and BEM

43. 43 FEM (iteration No.49) BEM(iteration No.13) Initial guess After iteration Remesh area Remesh line Free surface seepage flow using hypersingular formulation

44. 44 Incomplete partition in room acoustics (Helmholtz equation) b a e c t=0

45. 45 Water wave problem with breakwater (modified Helmholtz equation) Free water surface S x Top view O y z O x z S breakwater oblique incident water wave

46. 46 Reflection and Transmission

47. 47 Cracked torsion bar

48. 48 IEEE J MEMS Comb drive

49. 49

50. 50

51. 51 Is it possible ! No hypersingularity ! No subdomain !

52. 52 Dual BEM Degenerate boundary problems u=0 r=1 u=0 r=1 u=0 r=1 interface Subdomain 1 Subdomain 2 Subdomain 1 Subdomain 2 Multi-domain BEM

53. 53 Rank deficiency due to degenerate boundary and rigid body mode (SVD) PhysicsMathematics Left unitary matrix Right unitary matrix U Spurious True L T M Rigid body mode spurious mode(fictitious mode) (mathematics) true mode, rigid body mode (physics)

54. 54 SVD Technique (Google searching) [C] SVD decomposition [U] and [V} left and right unitary vectors

55. 55 Physical meaning of SVD 變形前 變形後 假根真根 Chen et al., 2002, Int. J. Comp. Numer. Anal. Appl. 先拉再轉 先轉再拉

56. 56 Conventional BEM in conjunction with SVD Singular Value Decomposition Rank deficiency originates from two sources: (1). Degenerate boundary (2). Nontrivial eigensolution N d= 5 N d= 4

57. 57 Dual BEM UT BEM + SVD (Present method) versus k Determinant versus k Sub domain

58. 58 k=3.14 k=3.82 k=4.48 UT BEM+SVD k=3.09 k=3.84k=4.50 FEM (ABAQUS)

59. 59 Nonuniqueness in BEM for exterior acoustics BEM for exterior acoustics Numerical and physical resonance incident wave radiation Physical resonance Numerical resonance

60. 60 Radiation and scattering problems Nonuniform radiaton scattering 2

61. 61 Mesh Mesh Error estimator Error estimator Solution Strategy of adaptive BEM Singular Equation Hypersingular Equation u,t 21

62. 62 BEM FEM Adaptive Mesh 5 DtN interface

63. 63 Numerical solution: BEMNumerical solution: FEM 64 ELEMENTS2791 ELEMENTS Nonuniform radiation ： Dirichlet problem 9

64. 64 Numerical phenomena (Fictitious frequency) t(a,0) A story of Ph.D. students

65. 65 Nonuniqueness in BEM for degenerate scale BEM with degenerate scale Gamma contour Critical value Logarithmic capacity Solvability of integral equations Invertibility of integral operator

66. 66 Numerical phenomena (Degenerate scale) Error (%) of torsional rigidity a 0 5 125 Previous approach : Try and error on a Present approach : Only one trial Commercial ode output ? Stokes Flow biharmonic Torsion Laplace

67. 67 Nonuniqueness in BEM for multiply connected domain problem Spurious eigensolution

68. 68 Numerical phenomena (2-D) (Spurious eigensolution)

69. 69 Numerical phenomena (3-D) (Spurious eigensolution) x y z a 0.5a 0.4a BIEMExperimentInner (spurious)ABAQUS 111.0113 85.357 201.2204 186.49 （ 2 ） 277.1279 209.85 364.9364 247.75 （ 2 ） 438.5441 426.3274.5 642.1640 610.0 305.05 （ 2 ） 782.6784782.0 333.3 849.2854852.5 350.02 912.6907 374.71 （ 2 ） 931.3933 399.3 （ 2 ） 996.1990 ……….. 1040.61033 ……….. ? ? 呂學育博士 林羿州 Fillipi, JSV Spurious eigenvalue

70. 70 Treatments SVD updating term Burton & Miller method CHIEF method Mathematical analysis and numerical study for free vibration of plate using BEM- 70

71. 71 SVD structure for four influence matrices UT LM spurious mode, fictitious mode (mathematics) true mode, rigid body mode (physics) The same

72. 72 SVD updating technique ( 去蕪 [ ] 存精 ( ) 術 ) UT LM spurious mode, fictitious mode (mathematics) true mode, rigid body mode (physics) The same [ ] ()

73. 73 BEM trap ? Why engineers should learn mathematics ? Well-posed ? Existence ? Unique ? Mathematics versus Computation equivalent ? 馮康 定理 Some examples

74. 74 馮康 定理 同一個物理或工程問題，可以有各種不同 的數學模式來模擬。 這些數學模式在物理上是等效的 (Equivalent) 。但在離散後，數值的實踐 (Implementation) 就不見得等效。 唯有在不同的數值方法中，儘量保有問題 的基本特徵，方是做計算數學與計算力學 的最高指導原則

75. 75 Conclusions Review of numerical methods ( 中醫式的工程分析法 ) Review of nonuniqueness problems in numerical methods Successful experiences in the engineering applications with degenerate boundaries were demonstrated Nonuniquness due to degenerate boundary, degenerate scale, spurious eigenvalue, fictitious frequency is shown SVD structures for the nonuniqueness are examined.

76. 76 Acknowledgement 台大土木系 洪宏基 終身特聘教授 中山科學院 王政盛博士 高海大造船系 陳義麟博士 宜大土木系 陳桂鴻 博士 中華技術學院機械系 李為民 博士 中華技術學院機械系 呂學育 博士 中山科學院 全湘偉博士 NTOU/MSV group members

77. 77 The End Thanks for your kind attention

78. 78 歡迎參觀海洋大學力學聲響振動實驗室 烘焙雞及捎來伊妹兒 http://ind.ntou.edu.tw/~msvlab/ http://140.121.146.149/ E-mail: jtchen@mail.ntou.edu.tw

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