九十四年電子計算機於土木水利工 程應用研討會 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

2 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 Linear algebraic equation Linear algebraic equation Numerical examples Numerical examples Conclusions Conclusions

3 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 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 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 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 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 Linear algebraic equation Linear algebraic equation Numerical examples Numerical examples 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 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 Linear algebraic equation Linear algebraic equation Numerical examples Numerical examples Conclusions Conclusions

18 Adaptive observer system collocation point

19 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 Linear algebraic equation Linear algebraic equation Numerical examples Numerical examples Conclusions Conclusions

20 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 Linear algebraic equation Linear algebraic equation Numerical examples Numerical examples Conclusions Conclusions

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

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

23 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

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

25 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 Linear algebraic equation Linear algebraic equation Numerical examples Numerical examples Conclusions Conclusions

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

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) Present method (M=10) Exact solution BIEformulation

30 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

31 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

32 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

33 Torsional rigidity (N=2,3,4) Number of holes Present method (M=10)BIEformulationFirst-ordersolution

34 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 Linear algebraic equation Linear algebraic equation Numerical examples Numerical examples Conclusions 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. 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. 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. 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 Graf’s addition theorem Complex variable Complex variable Real part If Bessel’s function