1 Part I: Direct problem Null-field integral equation approach for boundary value problems with circular boundaries J. T. Chen Ph.D. 海洋大學特聘教授 陳正宗博士 11:50 ~ 12:40, Oct. 08, 2005 NCTU, HsinChu Workshop on Inverse Problems NCTU, HsinChu, Taiwan

2 Outlines Part I: Direct problem Null-field integral equation approach for boundary value problems with circular boundaries Part I: Direct problem Null-field integral equation approach for boundary value problems with circular boundaries

3 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

4 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

5 Present approach 1. No principal value 2. Well-posed Advantages of degenerate kernel Degenerate kernel Fundamental solution CPV and HPV No principal value

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

7 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

8 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

9 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.

10 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

11 Boundary integral equation and null-field integral equation Interior case Exterior case Null-field integral equation

12 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

13 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

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

15 Separable form of fundamental solution (2D)

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

17 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

18 Adaptive observer system collocation point

19 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

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

21 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

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

23 Explicit form of each submatrix [U pk ] and vector {t k } Fourier coefficients Truncated terms of Fourier series Number of collocation points

24 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

25 Comparisons of conventional BEM and present method BoundarydensitydiscretizationAuxiliarysystemFormulationObserversystemSingularity ConventionalBEMConstant,Linear,Qurdrature…FundamentalsolutionBoundaryintegralequationFixedobserversystem CPV, RPV and HPV PresentmethodFourierseriesexpansionDegeneratekernelNull-fieldintegralequationAdaptiveobserversystemNoprincipalvalue

26 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

27 Numerical examples Laplace equation Laplace equation Eigen problem Eigen problem Exterior acoustics Exterior acoustics Biharmonic equation Biharmonic equation

28 Laplace equation Steady state heat conduction problems Steady state heat conduction problems Electrostatic potential of wires Electrostatic potential of wires Flow of an ideal fluid pass cylinders Flow of an ideal fluid pass cylinders A circular bar under torque A circular bar under torque An infinite medium under antiplane shear An infinite medium under antiplane shear Half-plane problems Half-plane problems

29 Steady state heat conduction problems Case 1 Case 2

30 Steady state heat conduction problems Case 3 Case 4

31 Case 1: Isothermal line Exact solution (Carrier and Pearson) BEM-BEPO2D(N=21) FEM-ABAQUS (1854 elements) Present method (M=10)

32 Relative error of flux on the small circle

33 Convergence test - Parseval’s sum for Fourier coefficients Parseval’s sum

34 Case 2: Isothermal line Caulk’s data (1983) IMA Journal of Applied Mathematics Present method (M=10) FEM-ABAQUS (6502 elements)

35 Case 3: Isothermal line FEM-ABAQUS (8050 elements) Present method (M=10) Caulk’s data (1983) IMA Journal of Applied Mathematics

36 Case 4: Isothermal line FEM-ABAQUS (8050 elements) Present method (M=10) Caulk’s data (1983) IMA Journal of Applied Mathematics

37 Laplace equation Steady state heat conduction problems Steady state heat conduction problems Electrostatic potential of wires Electrostatic potential of wires Flow of an ideal fluid pass cylinders Flow of an ideal fluid pass cylinders A circular bar under torque A circular bar under torque An infinite medium under antiplane shear An infinite medium under antiplane shear Half-plane problems Half-plane problems

38 Electrostatic potential of wires Hexagonal electrostatic potential Two parallel cylinders held positive and negative potentials

39 Contour plot of potential Exact solution (Lebedev et al.) Present method (M=10)

40 Contour plot of potential Onishi’s data (1991) Present method (M=10)

41 Laplace equation Steady state heat conduction problems Steady state heat conduction problems Electrostatic potential of wires Electrostatic potential of wires Flow of an ideal fluid pass cylinders Flow of an ideal fluid pass cylinders A circular bar under torque A circular bar under torque An infinite medium under antiplane shear An infinite medium under antiplane shear Half-plane problems Half-plane problems

42 Flow of an ideal fluid pass two parallel cylinders is the velocity of flow far from the cylinders is the velocity of flow far from the cylinders is the incident angle is the incident angle

43 Velocity field in different incident angle Present method (M=10)

44 Laplace equation Steady state heat conduction problems Steady state heat conduction problems Electrostatic potential of wires Electrostatic potential of wires Flow of an ideal fluid pass cylinders Flow of an ideal fluid pass cylinders A circular bar under torque A circular bar under torque An infinite medium under antiplane shear An infinite medium under antiplane shear Half-plane problems Half-plane problems

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

46 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

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

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

49 Laplace equation Steady state heat conduction problems Steady state heat conduction problems Electrostatic potential of wires Electrostatic potential of wires Flow of an ideal fluid pass cylinders Flow of an ideal fluid pass cylinders A circular bar under torque A circular bar under torque An infinite medium under antiplane shear An infinite medium under antiplane shear Half-plane problems Half-plane problems

50 Infinite medium under antiplane shear The displacement Boundary condition Total displacement on

51 Shear stress around the hole of radius a 1 (x axis) Shear stress σ zq around the hole of radius a 1 (x axis) Present method (M=20) Honein’s data (1992) Quarterly of Applied Mathematics

52 Shear stress around the hole of radius a 1 (y axis) Shear stress σ zq around the hole of radius a 1 (y axis) Present method (M=20) Honein’s data (1992) Quarterly of Applied Mathematics

53 Shear stress around the hole of radius a 1 (45 degrees) Shear stress σ zq around the hole of radius a 1 (45 degrees) Present method (M=20) Honein’s data (1992) Quarterly of Applied Mathematics

54 Shear stress around the hole of radius a 1 (Touching) Shear stress σ zq around the hole of radius a 1 (Touching) Present method discontinuous discontinuous Honein’s data (1992) Quarterly of Applied Mathematics Gibb’s phenomenon

55 Two equivalent approaches Displacement approach Stress approach Present method Bird and Steele (1992) ASME Journal of Applied Mechanics

56 Shear stress around the hole of radius a 1 Shear stress σ zq around the hole of radius a 1 Present method (M=20) Steele’s data (1992) Stress approach Displacement approach Honein’s data (1992) % 0.02% Analytical 0.06%

57 Convergence of stress σ zq at q=45 degrees versus R 0

58 Three circular holes with centers on the x axis

59 Three circular holes with centers on the y axis

60 Three circular holes with centers on the line making 45 degrees

61 Laplace equation Steady state heat conduction problems Steady state heat conduction problems Electrostatic potential of wires Electrostatic potential of wires Flow of an ideal fluid pass cylinders Flow of an ideal fluid pass cylinders A circular bar under torque A circular bar under torque An infinite medium under antiplane shear An infinite medium under antiplane shear Half-plane problems Half-plane problems

62 Half-plane problems Dirichlet boundary condition (Lebedev et al.) Mixed-type boundary condition (Lebedev et al.)

63 Dirichlet problem Exact solution (Lebedev et al.) Present method (M=10) Isothermal line

64 Mixed-type problem Exact solution (Lebedev et al.) Present method (M=10) Isothermal line

65 Numerical examples Laplace equation Laplace equation Eigen problem Eigen problem Exterior acoustics Exterior acoustics Biharmonic equation Biharmonic equation

66 Problem statement Doubly-connected domainMultiply-connected domain Simply-connected domain

67 Example 1

68 k1k1k1k1 k2k2k2k2 k3k3k3k3 k4k4k4k4 k5k5k5k5 FEM(ABAQUS) BEM (Burton & Miller) BEM(CHIEF) BEM(null-field) BEM(fictitious) Present method Analytical solution[19] The former five true eigenvalues by using different approaches

69 The former five eigenmodes by using present method, FEM and BEM

70 R=1 e=0.5 c=0.1 c c c c Example 2

71 The former five modes by using present method, BEM and FEM

72 Numerical examples Laplace equation Laplace equation Eigen problem Eigen problem Exterior acoustics Exterior acoustics Biharmonic equation Biharmonic equation

73 u= x y Sketch of the scattering problem (Dirichlet condition) for five cylinders

74 (a) Present method (M=20) (b) Multiple DtN method (N=50) The contour plot of the real-part solutions of total field for

75 The contour plot of the real-part solutions of total field for (a) Present method (M=20) (b) Multiple DtN method (N=50)

76 The real part of total field for the data for the five artificial boundaries versus for The real part of total field for the data for the five artificial boundaries versus for (a) c(0.0, 0.0), R=1.0 (b) c(1.5, 1.5), R=0.9 (c) c(-1.5, 1.5), R=1.1 (d) c(-1.5, 1.5), R=1.0 (e) c(-1.5, -1.5), R=0.8 (f) c(0.0, 0.0), R=1.0 (g) c(1.5, 1.5), R=0.9 (h) c(-1.5, 1.5), R=1.1 (i) c(-1.5, 1.5), R=1.0 (j) c(-1.5, -1.5), R=0.8 (a) ~ (e) multiple DtN method (N=50) (f) ~ (j) present method (M=20)

77 The real part of total field for the data for the five artificial boundaries versus for (a) c(0.0, 0.0), R=1.0 (b) c(1.5, 1.5), R=0.9 (c) c(-1.5, 1.5), R=1.1 (d) c(-1.5, 1.5), R=1.0 (e) c(-1.5, -1.5), R=0.8 (a) ~ (e) multiple DtN method (N=50) (f) c(0.0, 0.0), R=1.0 (g) c(1.5, 1.5), R=0.9 (h) c(-1.5, 1.5), R=1.1 (i) c(-1.5, 1.5), R=1.0 (j) c(-1.5, -1.5), R=0.8 (f) ~ (j) present method (M=20)

78 Fictitious frequencies

79 Numerical examples Laplace equation Laplace equation Eigen problem Eigen problem Exterior acoustics Exterior acoustics Biharmonic equation Biharmonic equation

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

81 Contour plot of displacement Present method (N=21) Present method (N=61) Present method (N=41) Present method (N=81)

82 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)

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

84 Comparison for DOF of BIE (Kelmanson) DOF of present method BIE (Kelmanson) Present method Analytical solution (160) (320) (640) u1u1 (28) (36) (44) (∞) Algebraic convergence Exponential convergence

85 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

86 Contour plot of vorticity for Present method (N=21) Present method (N=41) Kelmanson (n=160)

87 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

88 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.

89 Conclusions Engineering problems with circular boundaries which satisfy the Laplace Helmholtz and Biharminic problems can be solved by using the proposed approach in a more efficient and accurate manner. Engineering problems with circular boundaries which satisfy the Laplace Helmholtz and Biharminic problems can be solved by using the proposed approach in a more efficient and accurate manner.

90 The End Thanks for your kind attentions. Your comments will be highly appreciated. URL:

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