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Solution of biharmonic problems with circular boundaries using null-field integral equations

Solution of biharmonic problems with circular boundaries using null-field integral equations. Name: Chia-Chun Hsiao Date: 2005/9/2 Place: NCKU. Outlines. Introduction Formulation Numerical examples Conclusions. Outlines. Introduction Formulation Numerical examples Conclusions.

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Solution of biharmonic problems with circular boundaries using null-field integral equations

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  1. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Solution of biharmonic problems with circular boundaries using null-field integral equations Name: Chia-Chun Hsiao Date: 2005/9/2 Place: NCKU

  2. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Outlines • Introduction • Formulation • Numerical examples • Conclusions

  3. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Outlines • Introduction • Formulation • Numerical examples • Conclusions

  4. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Engineering problems with arbitrary boundaries Degenerate boundary (Chebyshev polynomials) Elliptic boundary (Mathieu function) Circular boundary (Fourier series) Straight boundary (Legendre polynomials) Degenerate boundary Circular boundary

  5. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Motivation BEM/BIEM Improper integral Desingular (Regular) Singular Indirect (Interior) Direct Null-field approach Fictitious boundary method Contour Limiting process : collocation point Fictitious boundary

  6. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Degenerate kernel Field point Present approach Motivation BEM/BIEM Improper integral Desingular (Regular) Singular Indirect (Exterior) Direct Null-field Contour Limiting process Fictitious boundary method : collocation point Fictitious boundary ill-posed CPV & HPV

  7. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Literature review Torsion bar with circular holes Laplace problems Steady state heat conduction of tube Electromagnetic wave Engineering problems Helmholtz problems Membrane vibration Water wave and Acoustic problems Plane elasticity：Airy stress function Biharmonic problems Solid mechanics：plate problem Fluid mechanics：Stokes flow

  8. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Literature review • Plane elasticity: Jeffery (1921), Howland and Knight (1939), Green (1940) and Ling (1948) 2.Solid mechanics (Plate problem): Bird and Steele (1991) 3.Viscous flow (Stokes Flow): Kamal (1966), DiPrima and Stuart (1972), Mills (1977) and Ingham and Kelmanson (1984)

  9. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Purpose • A semi-analytical approach in conjunction with Fourier series, degenerate kernels and adaptive observer system is applied to biharmonic problems. • Advantages： 1. Mesh free. 2. Accurate. 3. Free of CPV and HPV.

  10. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Outlines • Introduction • Formulation • Numerical examples • Conclusions

  11. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Problem statement Governing equation: Essential boundary condition: :slope :lateral displacement, Natural boundary condition: : moment, : shear force

  12. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Boundary integral equations • BIEs are derived from the Rayleigh-Green identity： BIE for the domain point Null-field integral equation Interior problem

  13. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Slope Displacement : Poisson ratio Moment Displacement Shear force Displacement Boundary integral equation for the domain point

  14. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Slope Displacement : Poisson ratio Moment Displacement Shear force Displacement Null-field integral equation

  15. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Relation among the kernels Continuous (Separable form of degenerate kernel) is the fundamental solution, which satisfies

  16. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 r Degenerate kernels x x s O

  17. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Fourier series The boundary densities are expanded in terms of Fourier series: M: truncating terms of Fourier series

  18. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Adaptive observer system : Collocation point : Radius of the jth circle : Origin of the jth circle : Boundary of the jth circle

  19. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Vector decomposition for normal derivative True normal direction Tangential direction Radial direction : normal derivative : tangential derivative

  20. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Linear algebraic system Null-field integral equations for and formulations H: number of circular boundaries Collocation circle index Routing circle index

  21. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Fourier series Degenerate kernels Flowchart of the present method Analytical Collocation method Matching B.C. Adaptive observer system Numerical Linear algebraic system Potential Fourier coefficients BIE for domain point

  22. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Stokes flow problems (Eccentric case) Governing equation: Essential boundary condition: on on (Stationary) : stream function : normal derivative of stream function

  23. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Linear algebraic system Given Unknown Unknown constant

  24. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Constraint equation Vorticity: Constraint:

  25. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Trapezoid integral Inner circle Outer circle Vector decomposition Numerical Analytical Trapezoid integral Series sum

  26. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Linear algebraic augmented system Unknown

  27. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Outlines • Introduction • Formulation • Numerical examples • Conclusions

  28. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Plate problems Geometric data: Essential boundary conditions: on and on and on and on and (Bird & Steele, 1991)

  29. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Contour plot of displacement Present method (N=41) Present method (N=21) Present method (N=61) Present method (N=81)

  30. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Contour plot of displacement Present method (N=101) Bird and Steele (1991) (No. of nodes=3,462, No. of elements=6,606) FEM mesh FEM (ABAQUS)

  31. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Parseval sum for convergence ( )

  32. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Parseval sum for convergence

  33. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Stokes flow problems Governing equation: Angular velocity: Boundary conditions: on and (Stationary) on and Eccentricity:

  34. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Comparison of stream function N: number of collocation points n: number of boundary nodes

  35. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 BIE (Kelmanson) Present method Analytical solution Comparison for (160) (28) u1 (320) (640) (36) (∞) (44) DOF of BIE (Kelmanson) DOF of present method

  36. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Contour plot of Streamline for 0 -Q/90 Q/20 Q/5 -Q/30 Q/2 Q Present method (N=81) 0 -Q/90 Q/20 Q/5 -Q/30 Q/2 Kelmanson (Q=0.0740, n=160) Q e Kamal (Q=0.0738)

  37. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Contour plot of Streamline for Present method (N=21) Present method (N=41) Kelmanson (Q=0.0740, n=160) Present method (N=81)

  38. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Contour plot of vorticity for Present method (N=41) Present method (N=21) Kelmanson (n=160)

  39. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Contour plot of vorticity for Present method (N=41) Present method (N=21) Kelmanson (n=160)

  40. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Outlines • Introduction • Formulation • Numerical examples • Conclusions

  41. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Conclusions • Successful applied to biharmonic problems with circular boundaries by using the present method. • Good agreement was obtained after compared with previous results, exact solution and ABAQUS data. • Stream function and vorticity were found to be independent of Poisson ratio as we predicted. • Once engineering problems satisfy the biharmonic equation with circular boundaries, our present method can be used.

  42. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Thank you for your kind attention! The end

  43. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Conclusions Null-field integral equation available? No ! Null-field !

  44. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Further research

  45. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 Further research

  46. 九十四年電子計算機於土木水利工程應用研討會九十四年電子計算機於土木水利工程應用研討會 w=constant uniform pressure a Splitting method B Governing equation: Boundary condition: Governing equation: Boundary condition: : deflection of the circular plate General form : flexure rigidity Governing equation: : uniform distributed load : domain of interest Boundary condition:

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