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A semi-analytical approach for Stokes flow and plate problems with circular boundaries

Introducing a semi-analytical method utilizing Fourier series for plate and Stokes flow problems with circular boundaries. The research includes direct and indirect boundary integral equation methods, numerical examples, and conclusions for practical engineering applications.

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A semi-analytical approach for Stokes flow and plate problems with circular boundaries

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  1. A semi-analytical approach for Stokes flow and plate problems with circular boundaries 報 告 者：蕭嘉俊 指導教授：陳正宗 博士 呂學育 博士 日 期：2005/6/16 地 點：河工二館307

  2. Outlines • Introduction • Direct boundary integral equation method • Indirect boundary integral equation method • Numerical examples • Conclusions

  3. Outlines • Introduction • Direct boundary integral equation method • Indirect boundary integral equation method • Numerical examples • Conclusions

  4. 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. 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. 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. Literature review Torsion bar with circular holes Laplace problems Steady state heat conduction of tube (Shen) Electromagnetic wave Engineering problems Helmholtz problems Membrane vibration Water wave and Acoustic problems (Chen) Plane elasticity：Airy stress function Biharmonic problems Solid mechanics：plate problem Fluid mechanics：Stokes flow (Hsiao)

  8. 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. Purpose • A semi-analytical approach in conjunction with Fourier series, degenerate kernels and adaptive observer system is extended to biharmonic problems. • Advantages： 1. Mesh free. 2. Accurate. 3. Free of CPV and HPV.

  10. Outlines • Introduction • Direct boundary integral equation method • Indirect boundary integral equation method • Numerical examples • Conclusions • Further research

  11. Problem statement Governing equation: Essential boundary condition: :slope :lateral displacement, Natural boundary condition: : moment, : shear force

  12. Direct boundary integral equations • BIEs are derived from the Rayleigh-Green identity： BIE for the domain point Null-field integral equation Interior problem

  13. Slope Displacement : Poisson ratio Moment Displacement Shear force Displacement Boundary integral equation for the domain point

  14. Slope Displacement : Poisson ratio Moment Displacement Shear force Displacement Null-field integral equation

  15. Relation among the kernels Continuous (Separable form of degenerate kernel) is the fundamental solution, which satisfies

  16. r Degenerate kernels x x s O

  17. Fourier series The boundary densities are expanded in terms of Fourier series: M: truncating terms of Fourier series

  18. Adaptive observer system : Collocation point : Radius of the jth circle : Origin of the jth circle : Boundary of the jth circle

  19. Vector decomposition for normal derivative True normal direction Tangential direction Radial direction : normal derivative : tangential derivative

  20. Linear algebraic system Null-field integral equations for and formulations H: number of circular boundaries Collocation circle index Routing circle index

  21. 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. Stokes flow problems (Eccentric case) Governing equation: Essential boundary condition: on on (Stationary) : stream function : normal derivative of stream function

  23. Linear algebraic system Given Unknown Unknown constant

  24. Constraint equation Vorticity: Constraint:

  25. Trapezoid integral Inner circle Outer circle Vector decomposition Numerical Analytical Trapezoid integral Series sum

  26. Linear algebraic augmented system Unknown

  27. Outlines • Introduction • Direct boundary integral equation method • Indirect boundary integral equation method • Numerical examples • Conclusions • Further research

  28. Indirect boundary integral equation Indirect boundary integral equation is originated from the physical concept of superposition : Vorticity : single layer fictitious densities : double layer fictitious densities

  29. Stokes flow problems (Eccentric case) Governing equation: Essential boundary condition: on on (Stationary) : stream function : normal derivative of stream function

  30. Linear algebraic system Unknown Given Unknown constant :Collocation point

  31. Constraint equation Vorticity: Constraint:

  32. Trapezoid integral Inner circle Outer circle Vector decomposition Numerical Analytical Trapezoid integral Series sum

  33. Linear algebraic augmented system Unknown

  34. Outlines • Introduction • Direct boundary integral equation method • Indirect boundary integral equation method • Numerical examples • Conclusions

  35. Numerical examples

  36. Numerical examples

  37. Plate problems (Case 1) Geometric data: and Essential boundary conditions: and on and on Exact solution：

  38. Contour plot of displacement (No. of nodes=1,920, No. of elements=3,600) FEM mesh Exact solution FEM (ABAQUS) Present method (M=10)

  39. Boundary densities for outer circle Exact solution: Exact solution:

  40. Boundary densities for inner circle Exact solution: Exact solution:

  41. Plate problems (Case 2) Geometric data: Essential boundary conditions: on and on and on and on and (Bird & Steele, 1991)

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

  43. 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)

  44. Parseval sum for convergence

  45. Parseval sum for convergence

  46. Stokes flow problems (Case 1) Boundary conditions: Exact solution: (Mills, 1977) where

  47. Contour plot of Streamline Exact solution (Mills, 1977) Present method (N=161) Analytical solution, P=100 (Wu, 2004) (Null-field + collocation) Exact solution: (Mills, 1977) (Closed-form solution) (Null-field equation solution) (Wu, 2004) Analytical solution: (Trefftz solution)

  48. Stokes flow problems (Case 2) Governing equation: Angular velocity: Boundary conditions: on and (Stationary) on and Eccentricity:

  49. Comparison of stream function n: number of boundary nodes N: number of collocation points

  50. BIE (Kelmanson) Present method Analytical solution Comparison for (160) (28) u1 (320) (640) (36) (∞) (44) DOF of BIE (Kelmanson) DOF of present method

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