A Semi-Analytical Approach for Boundary Value Problems with Circular Boundaries


In this paper, a semi-analytical approach is developed to deal with problems including multiple circular boundaries. The boundary integral approach is utilized in conjunction with degenerate kernel and Fourier series. To fully utilize the circular geometry, the fundamental solutions and the boundary densities are expanded by using degenerate kernels and Fourier series, respectively. Both direct and indirect formulations are proposed. This approach is a semi-analytical approach, since the error stems from the truncation of Fourier series in the implementation. The unknown Fourier coefficients are easily determined by solving a linear algebraic system after using the collocation method and matching the boundary conditions. Five goals: (1) free of calculating principal value, (2) exponential convergence, (3) well-posed algebraic system, (4) elimination of boundary-layer effect and (5) meshless, of the formulation are achieved. The proposed approach is extended to deal with the problems containing multiple circular inclusions. Finally, the general-purpose program in a unified manner is developed for BVPs with circular boundaries. Several examples including the torsion bar, water wave and plate vibration problems are given to demonstrate the validity of the present approach.


Boundary Element Method Circular Plate Linear Algebraic System Circular Boundary Degenerate Kernel 
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  1. Bates RHT, Wall DJN (1977) Null field approach to scalar diffraction. I. General method. Philos. Trans. R. Soc. Lond. 287: 45–78CrossRefMathSciNetGoogle Scholar
  2. Boström A (1982) Time-dependent scattering by a bounded obstacle in three dimensions. J. Math. Phys. 23: 1444–1450MATHCrossRefMathSciNetGoogle Scholar
  3. Chen JT, Hong H-K (1999) Review of dual boundary element methods with emphasis on hypersingular integrals and divergent series. ASME Appl. Mech. Rev. 52: 17–33CrossRefGoogle Scholar
  4. Chen JT, Shen WC, Wu AC (2005) Null-field integral equations for stress field around circular holes under anti-plane shear. Eng. Anal. Bound. Elem. 30: 205–217CrossRefGoogle Scholar
  5. Chen JT, Hsiao CC, Leu SY (2006) Null-field integral equation approach for plate problems with circular boundaries. ASME J. Appl. Mech. 73: 679–693MATHCrossRefGoogle Scholar
  6. Chen JT, Chen CT, Chen PY, Chen IL (2007a) A semi-analytical approach for radiation and scattering problems with circular boundaries. Comput. Meth. Appl. Mech. Eng. 196: 2751–2764CrossRefMATHGoogle Scholar
  7. Chen JT, Wu CS, Lee YT, Chen KH (2007b) On the equivalence of the Trefftz method and method of fundamental solution for Laplace and biharmonic equations. Comput. Math. Appl. 53: 851–879MATHCrossRefMathSciNetGoogle Scholar
  8. Khurasia HB, Rawtani S (1978) Vibration analysis of circular plates with eccentric hole. ASME J. Appl. Mech. 45: 215–217Google Scholar
  9. Kress R (1989) Linear integral equations. Springer, BerlinMATHGoogle Scholar
  10. Lee WM, Chen JT, Lee YT (2007a) Free vibration analysis of circular plates with multiple circular holes using indirect BIEMs. J. Sound Vib. 304: 811–830CrossRefGoogle Scholar
  11. Lee YT, Chen JT, Chou KS (2007b) Revisit of two classical elasticity problems using the null-field BIE. in APCOM’07 in conjunction with EPMESC XI, Kyoto, JapanGoogle Scholar
  12. Linton CM, Evans DV (1990) The interaction of waves with arrays of vertical circular cylinders. J. Fluid Mech. 215: 549–569MATHCrossRefMathSciNetGoogle Scholar
  13. Martin PA (1981) On the null-field equations for water-wave radiation problems. J. Fluid Mech. 113: 315–332MATHCrossRefMathSciNetGoogle Scholar
  14. Muskhelishvili NI (1953) Some basic problems of the mathematical theory of elasticity. Noordhoff, GroningenMATHGoogle Scholar
  15. Perrey-Debain E, Trevelyan J, Bettess P (2003) Plane wave interpolation in direct collocation boundary element method for radiation and wave scattering: numerical aspects and applications. J. Sound Vib. 261: 839–858CrossRefMathSciNetGoogle Scholar
  16. Schaback R (2007) Adaptive Numerical Solution of MFS Systems, in ICCES Special Symposium on Meshless Methods, Patras, GreeceGoogle Scholar
  17. Tang RJ (1996) Torsion theory of the crack cylinder. Shanghai Jiao Tong University Publisher, Shanghai (in Chinese)Google Scholar
  18. Waterman PC (1965) Matrix formulation of electromagnetic scattering. Proc. IEEE. 53: 805–812CrossRefGoogle Scholar
  19. Waterman PC (1976) Matrix theory of elastic wave scattering. J. Acoust. Soc. Am. 60: 567–580MATHCrossRefMathSciNetGoogle Scholar

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© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of Mechanical and Mechatronic Engineering, Department of Harbor and River EngineeringNational Taiwan Ocean UniversityKeelungTaiwan
  2. 2.Department of Harbor and River EngineeringNational Taiwan Ocean UniversityKeelungTaiwan
  3. 3.Department of Mechanical EngineeringChina Institute of TechnologyTaipeiTaiwan

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