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Analysis of circular torsion bar with circular holes using null-field approach

Analysis of circular torsion bar with circular holes using null-field approach. 研究生：沈文成 指導教授：陳正宗 教授 時間： 15:10 ~ 15:25 地點： Room 47450-3. Outlines. Motivation and literature review Mathematical formulation Expansions of fundamental solution and boundary density

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Analysis of circular torsion bar with circular holes using null-field approach

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  1. Analysis of circular torsion bar with circular holes using null-field approach 研究生：沈文成 指導教授：陳正宗 教授 時間：15:10 ~ 15:25 地點：Room 47450-3 九十四年電子計算機於土木水利工程應用研討會

  2. Outlines • Motivation and literature review • Mathematical formulation • Expansions of fundamental solution and boundary density • Adaptive observer system • Linear algebraic equation • Numerical examples • Conclusions

  3. Outlines • Motivation and literature review • Mathematical formulation • Expansions of fundamental solution and boundary density • Adaptive observer system • Linear algebraic equation • Numerical examples • Conclusions

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

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

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

  7. Motivation and literature review Analytical methods for solving Laplace problems with circular holes Special solution Bipolar coordinate Conformal mapping 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 Lebedev, Skalskaya and Uyand, 1979, “Work problem in applied mathematics”, Dover Publications Limited to doubly connected domain

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

  9. Contribution and goal • However, they didn’t employ the null-field integral equationand 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.

  10. Outlines • Motivation and literature review • Mathematical formulation • Expansions of fundamental solution and boundary density • Adaptive observer system • Linear algebraic equation • Numerical examples • Conclusions

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

  12. Outlines • Motivation and literature review • Mathematical formulation • Expansions of fundamental solution and boundary density • Adaptive observer system • Linear algebraic equation • Numerical examples • Conclusions

  13. Expansions of fundamental solution and boundary density • Degenerate kernel - fundamental solution • 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 • Motivation and literature review • Mathematical formulation • Expansions of fundamental solution and boundary density • Adaptive observer system • Linear algebraic equation • Numerical examples • Conclusions

  18. collocation point Adaptive observer system

  19. Outlines • Motivation and literature review • Mathematical formulation • Expansions of fundamental solution and boundary density • Adaptive observer system • Linear algebraic equation • Numerical examples • Conclusions

  20. Outlines • Motivation and literature review • Mathematical formulation • Expansions of fundamental solution and boundary density • Adaptive observer system • Linear algebraic equation • Numerical examples • Conclusions

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

  22. Explicit form of each submatrix [Upk] and vector {tk} Truncated terms of Fourier series Number of collocation points Fourier coefficients

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

  24. Comparisons of conventional BEM and present method

  25. Outlines • Motivation and literature review • Mathematical formulation • Expansions of fundamental solution and boundary density • Adaptive observer system • Linear algebraic equation • Numerical examples • Conclusions

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

  27. Torsion problems Torque Torque Eccentric case Two holes (N=2)

  28. Torsion problems Torque Torque Three holes (N=3) Four holes (N=4)

  29. Torsional rigidity (Eccentric case)

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

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

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

  33. Torsional rigidity (N=2,3,4)

  34. Outlines • Motivation and literature review • Mathematical formulation • Expansions of fundamental solution and boundary density • Adaptive observer system • Linear algebraic equation • Numerical examples • Conclusions

  35. Conclusions • A systematic approach using degenerate kernels, Fourier series and null-field integral equation has been successfully proposed to solve torsion problems with circular boundaries. • Numerical results agree well with available exact solutions and Caulk’s data for only few terms of Fourier series.

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

  37. The end Thanks for your kind attentions. Your comments will be highly appreciated.

  38. Derivation of degenerate kernel • Graf’s addition theorem • Complex variable Bessel’s function Real part If Real part

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