1. 1 Null-field integral equations and engineering applications I. L. Chen Ph.D. Department of Naval Architecture, National Kaohsiung Marine University Mar. 11, 2010

2. 2 Research collaborators Prof. J. T. Chen Prof. J. T. Chen Dr. K. H. Chen Dr. K. H. Chen Dr. S. Y. Leu Dr. W. M. Lee Dr. S. Y. Leu Dr. W. M. Lee Dr. Y. T. Lee Dr. Y. T. Lee Mr. W. C. Shen Mr. C. T. Chen Mr. G. C. Hsiao Mr. W. C. Shen Mr. C. T. Chen Mr. G. C. Hsiao Mr. A. C. Wu Mr.P. Y. Chen Mr. A. C. Wu Mr.P. Y. Chen Mr. J. N. Ke Mr. H. Z. Liao Mr. J. N. Ke Mr. H. Z. Liao Mr. Y. J. Lin Mr. Y. J. Lin Mr. C. F. Wu Mr. J. W. Lee Mr. C. F. Wu Mr. J. W. Lee

3. 3 Introduction of NTOU/MSV group Introduction of NTOU/MSV group Outline

4. 4 URL: http://ind.ntou.edu.tw/~msvlab E-mail: jtchen@mail.ntou.edu.tw 海洋大學工學院河工所力學聲響振動實驗室 nullsystem2007.ppt` Elasticity & Crack Problem Laplace Equation Research topics of NTOU / MSV LAB on null-field BIE (2003-2010) Navier Equation Null-field BIEM Biharmonic Equation Previous research and project Current work (Plate with circulr holes ) BiHelmholtz Equation Helmholtz Equation (Potential flow) (Torsion) (Anti-plane shear) (Degenerate scale) (Inclusion) (Piezoleectricity) (Beam bending) Torsion bar (Inclusion) Imperfect interface Image method (Green function) Green function of half plane (Hole and inclusion) (Interior and exterior Acoustics) SH wave (exterior acoustics) (Inclusions) (Free vibration of plate) Indirect BIEM ASME JAM 2006 MRC,CMESEABE ASME JoM EABE CMAME 2007 SDEE JCA NUMPDE revision JSV SH wave Impinging canyons Degenerate kernel for ellipse ICOME 2006 Added mass 李應德 Water wave impinging circular cylinders Screw dislocation Green function for an annular plate SH wave Impinging hill Green function of`circular inclusion (special case:staic) Effective conductivity CMC (Stokes flow) (Free vibration of plate) Direct BIEM (Flexural wave of plate) AOR 2009

5. 5 Outlines Motivation and literature review Motivation and literature review Mathematical formulation Mathematical formulation Expansions of fundamental solution Expansions of fundamental solution and boundary density and boundary density Adaptive observer system Adaptive observer system Vector decomposition technique Vector decomposition technique Linear algebraic equation Linear algebraic equation Numerical examples Numerical examples Conclusions Conclusions

6. 6 Motivation Motivation Numerical methods for engineering problems FDM / FEM / BEM / BIEM / Meshless method BEM / BIEM (mesh required) Treatment of singularity and hypersingularity Boundary-layer effect Ill-posed model Convergence rate Mesh free for circular boundaries ?

7. 7 Motivation and literature review Fictitious BEM BEM/BIEM Null-field approach Bump contour Limit process Singular and hypersingular Regular Improper integral CPV and HPV Ill-posed Fictitious boundary Collocation point

8. 8 Present approach 1.No principal value 2. Well-posed 2. Well-posed 3. No boundary-layer effect 3. No boundary-layer effect 4. Exponetial convergence 4. Exponetial convergence 5. Meshless 5. Meshless Advantages of degenerate kernel Degenerate kernel Fundamental solution CPV and HPV No principal value

9. 9 Engineering problem with arbitrary geometries Degenerate boundary Circular boundary Straight boundary Elliptic boundary (Fourier series) (Legendre polynomial) (Chebyshev polynomial) (Mathieu function)

10. 10 Motivation and literature review Analytical methods for solving Laplace problems with circular holes Conformal mapping Bipolar coordinate Special solution Limited to doubly connected domain Lebedev, Skalskaya and Uyand, 1979, “Work problem in applied mathematics”, Dover Publications Chen and Weng, 2001, “Torsion of a circular compound bar with imperfect interface”, ASME Journal of Applied Mechanics Honein, Honein and Hermann, 1992, “On two circular inclusions in harmonic problem”, Quarterly of Applied Mathematics

11. 11 Fourier series approximation Ling (1943) - torsion of a circular tube Ling (1943) - torsion of a circular tube Caulk et al. (1983) - steady heat conduction with circular holes Caulk et al. (1983) - steady heat conduction with circular holes Bird and Steele (1992) - harmonic and biharmonic problems with circular holes Bird and Steele (1992) - harmonic and biharmonic problems with circular holes Mogilevskaya et al. (2002) - elasticity problems with circular boundaries Mogilevskaya et al. (2002) - elasticity problems with circular boundaries

12. 12 Contribution and goal However, they didn’t employ the null-field integral equation and degenerate kernels to fully capture the circular boundary, although they all employed Fourier series expansion. However, they didn’t employ the null-field integral equation and degenerate kernels to fully capture the circular boundary, although they all employed Fourier series expansion. To develop a systematic approach for solving Laplace problems with multiple holes is our goal. To develop a systematic approach for solving Laplace problems with multiple holes is our goal.

13. 13 Outlines (Direct problem) Motivation and literature review Motivation and literature review Mathematical formulation Mathematical formulation Expansions of fundamental solution Expansions of fundamental solution and boundary density and boundary density Adaptive observer system Adaptive observer system Vector decomposition technique Vector decomposition technique Linear algebraic equation Linear algebraic equation Numerical examples Numerical examples Conclusions Conclusions

14. 14 Boundary integral equation and null-field integral equation Interior case Exterior case Degenerate (separate) form

15. 15 Outlines (Direct problem) Motivation and literature review Motivation and literature review Mathematical formulation Mathematical formulation Expansions of fundamental solution Expansions of fundamental solution and boundary density and boundary density Adaptive observer system Adaptive observer system Vector decomposition technique Vector decomposition technique Linear algebraic equation Linear algebraic equation Numerical examples Numerical examples Degenerate scale Degenerate scale Conclusions Conclusions

16. 16 Gain of introducing the degenerate kernel Degenerate kernel Fundamental solution CPV and HPV No principal value? interior exterior

17. 17 How to separate the region

18. 18 Expansions of fundamental solution and boundary density Degenerate kernel - fundamental solution Degenerate kernel - fundamental solution Fourier series expansions - boundary density Fourier series expansions - boundary density

19. 19 Separable form of fundamental solution (1D) Separable property continuous discontinuous

20. 20 Separable form of fundamental solution (2D)

21. 21 Boundary density discretization Fourier series Ex. constant element Present method Conventional BEM

22. 22 Outlines Motivation and literature review Motivation and literature review Mathematical formulation Mathematical formulation Expansions of fundamental solution Expansions of fundamental solution and boundary density and boundary density Adaptive observer system Adaptive observer system Vector decomposition technique Vector decomposition technique Linear algebraic equation Linear algebraic equation Numerical examples Numerical examples Conclusions Conclusions

23. 23 Adaptive observer system collocation point

24. 24 Outlines Motivation and literature review Motivation and literature review Mathematical formulation Mathematical formulation Expansions of fundamental solution Expansions of fundamental solution and boundary density and boundary density Adaptive observer system Adaptive observer system Vector decomposition technique Vector decomposition technique Linear algebraic equation Linear algebraic equation Numerical examples Numerical examples Conclusions Conclusions

25. 25 Vector decomposition technique for potential gradient Special case (concentric case) : Non-concentric case: True normal direction

26. 26 Outlines Motivation and literature review Motivation and literature review Mathematical formulation Mathematical formulation Expansions of fundamental solution Expansions of fundamental solution and boundary density and boundary density Adaptive observer system Adaptive observer system Vector decomposition technique Vector decomposition technique Linear algebraic equation Linear algebraic equation Numerical examples Numerical examples Conclusions Conclusions

27. 27 Linear algebraic equation where Column vector of Fourier coefficients (Nth routing circle) Index of collocation circle Index of routing circle

28. 28 Flowchart of present method Potential of domain point Analytical Numerical Adaptive observer system Degenerate kernel Fourier series Linear algebraic equation Collocation point and matching B.C. Fourier coefficients Vector decomposition Potential gradient

29. 29 Comparisons of conventional BEM and present method BoundarydensitydiscretizationAuxiliarysystemFormulationObserversystemSingularityConvergenceBoundarylayereffect ConventionalBEMConstant,linear,quadratic…elementsFundamentalsolutionBoundaryintegralequationFixedobserversystem CPV, RPV and HPV LinearAppear PresentmethodFourierseriesexpansionDegeneratekernelNull-fieldintegralequationAdaptiveobserversystemDisappearExponentialEliminate

30. 30 Outlines Motivation and literature review Motivation and literature review Mathematical formulation Mathematical formulation Expansions of fundamental solution Expansions of fundamental solution and boundary density and boundary density Adaptive observer system Adaptive observer system Vector decomposition technique Vector decomposition technique Linear algebraic equation Linear algebraic equation Numerical examples Numerical examples Conclusions Conclusions

31. 31 Numerical examples Laplace equation (EABE 2005, EABE 2007) Laplace equation (EABE 2005, EABE 2007) (CMES 2005, ASME 2007, JoM2007) (CMES 2005, ASME 2007, JoM2007) (MRC 2007, NUMPDE 2010) (MRC 2007, NUMPDE 2010) Biharmonic equation (JAM, ASME 2006 ) Biharmonic equation (JAM, ASME 2006 ) Plate eigenproblem (JSV ) Plate eigenproblem (JSV ) Membrane eigenproblem (JCA) Membrane eigenproblem (JCA) Exterior acoustics (CMAME, SDEE ) Exterior acoustics (CMAME, SDEE ) Water wave (AOR 2009) Water wave (AOR 2009)

32. 32 Laplace equation A circular bar under torque A circular bar under torque (free of mesh generation) (free of mesh generation)

33. 33 Torsion bar with circular holes removed The warping function Boundary condition where on Torque

34. 34 Axial displacement with two circular holes Present method (M=10) Caulk’s data (1983) ASME Journal of Applied Mechanics Dashed line: exact solution Solid line: first-order solution

35. 35 Torsional rigidity ?

36. 36 Numerical examples Biharmonic equation Biharmonic equation (exponential convergence) (exponential convergence)

37. 37 Plate problems Geometric data: and on Essential boundary conditions: (Bird & Steele, 1991)

38. 38 Contour plot of displacement Present method (N=101)Bird and Steele (1991) FEM (ABAQUS) FEM mesh (No. of nodes=3,462, No. of elements=6,606)

39. 39 Stokes flow problem Governing equation: Boundary conditions: and on and on Eccentricity: Angular velocity: (Stationary)

40. 40 Contour plot of Streamline for Present method (N=81) Kelmanson (Q=0.0740, n=160) Kamal (Q=0.0738) e Q/2 Q Q/5 Q/20 -Q/90 -Q/30 0 Q/2 Q Q/5 Q/20 -Q/90 -Q/30 0

41. 41 An infinite plate with two inclusions

42. 42 Distribution of dynamic moment concentration factors by using the present method and FEM( L/a = 2.1)

43. 43 Membrane eigenproblem ( ) Membrane eigenproblem ( Nonuniqueness problems ) A confocal elliptical annulus

44. 44 A confocal elliptical annulus G. E.: B. Cs.:

45. 45 True and spurious eigenvalues Note: the data inside parentheses denote the spurious eigenvalue. (42) (11) Eigenvalues of an elliptical membrane UT equation Spurious BEM mesh FEM mesh True

46. 46 (42) Mode shapes EvenOddEven Odd

47. 47 Water wave ( ) Water wave ( Nonuniqueness problems ) Interaction of water waves with vertical cylinders

48. 48 Trapped mode （ nonuniqueness in physics ） M.S. Longuet-higgins JFM, 1967. A.E.H. Love 1966. Williams & Li OE, 2000.

49. 49 Trapped and near-trapped modes Near-trapped modeTrapped mode

50. 50 Numerical and physical resonance Physical resonanceFictitious frequency (BEM/BIEM) t(a,0) Present

51. 51 Water wave interaction with surface-piercing cylinders Governing equation: Separation variable : Seabed boundary conditions :

52. 52 Problem statement,. Dispersion relationship: Dynamic pressure: Force: Original problem Governing equation: 1 2 3 j

53. 53 Sketch of four cylinders

54. 54 Physical phenomenon and fictitious frequency Near-trapped mode Fictitious frequency Near-trapped mode Fictitious frequency Near-trapped mode Fictitious frequency Near-trapped mode 2.4 3.8 5.1 5.5 6.4 ka 4.08482

55. 55 Mechanism of fictitious frequency i =1 i = 2 N=12.40425.5201 N=23.83177.0156 N=35.13568.4172 N=46.38029.7610 N=57.588311.0647 N=68.771512.3386 a N ikNikNi

56. 56 Near-trapped mode for the four cylinders at ka=4.08482 (a/d=0.8) (a) Contour by the present method (M=20)

57. 57 Near-trapped mode for the four cylinders at ka=4.08482 (a/d=0.8) (c) Horizontal force on the four cylinders against wavenumber (b) Free-surface elevations by the present method (M=20) 54 1 2 3 4

58. 58 ka =4.08482

59. 59 By perturbing the radius of one cylinder (a1/d≠0.8) to destroy the periodical setup a 1 /d=0.82 Evans and Porter, JEM,1999.Present method

60. 60 Sketch of four cylinders 11

61. 61 i=2,3,4 Cylinder 1 Cylinder 3 ForceForce 0.860.81.150.25 0.840.81.200.25 0.820.81.300.27 0.80.854.154.1 0.780.81.020.34 0.760.81.130.30 0.740.81.190.30 i=2,3,4 Cylinder 1 Cylinder 3 ForceForce 0.860.81.150.29 0.840.81.200.28 0.820.81.270.27 0.80.854.154.1 0.780.81.120.27 0.760.81.170.26 0.740.81.160.26 Changing radiusMoving the center of one cylinder ka=4.08482 Disorder of the periodical pattern

62. 62 Outlines Motivation and literature review Motivation and literature review Mathematical formulation Mathematical formulation Expansions of fundamental solution Expansions of fundamental solution and boundary density and boundary density Adaptive observer system Adaptive observer system Vector decomposition technique Vector decomposition technique Linear algebraic equation Linear algebraic equation Numerical examples Numerical examples Conclusions Conclusions

63. 63 Conclusions A systematic approach using degenerate kernels, Fourier series and null-field integral equation has been successfully proposed to solve Laplace Helmholtz and Biharminic problems with circular boundaries. A systematic approach using degenerate kernels, Fourier series and null-field integral equation has been successfully proposed to solve Laplace Helmholtz and Biharminic problems with circular boundaries. Numerical results agree well with available exact solutions, Caulk’s data, Onishi’s data and FEM (ABAQUS) for only few terms of Fourier series. Numerical results agree well with available exact solutions, Caulk’s data, Onishi’s data and FEM (ABAQUS) for only few terms of Fourier series.

64. 64 Conclusions Physical phenomena of near-trapped mode as well as the numerical instability due to fictitious frequency in BIEM were both observed. Physical phenomena of near-trapped mode as well as the numerical instability due to fictitious frequency in BIEM were both observed. Fictitious frequency appears and is suppressed in sacrifice of higher number of Fourier terms. Fictitious frequency appears and is suppressed in sacrifice of higher number of Fourier terms. The effect of incident angle and disorder on the near-trapped mode was examined. The effect of incident angle and disorder on the near-trapped mode was examined.

65. 65 Conclusions Free of boundary-layer effect Free of boundary-layer effect Free of singular integrals Free of singular integrals Well posed Well posed Exponetial convergence Exponetial convergence Mesh-free approach Mesh-free approach

66. 66 The End Thanks for your kind attentions. Your comments will be highly appreciated. URL: http://msvlab.hre.ntou.edu.tw/ http://msvlab.hre.ntou.edu.tw/