研究生：陳佳聰 指導教授：陳正宗 教授 陳義麟 副教授

1. Null-field integral equation approach for Helmholtz (interior and exterior acoustic) problems with circular boundaries 研究生：陳佳聰 指導教授：陳正宗　教授 　　　　　 陳義麟　副教授 國立台灣海洋大學河海工程學系 結構組 碩士班論文口試 日期: 2005/06/16 13:30-15:00 MSVLAB

2. Outlines • Motivation • Literature review • Present method • Scattering problems in full and half planes • Alternative derivation of the semi-circular canyon subject to the SH waves • Numerical examples-membrane vibration • Numerical examples-exterior acoustics • Special case for Laplace problem • Conclusions MSVLAB

3. Outlines • Motivation • Literature review • Present method • Scattering problems in full and half planes • Alternative derivation of the semi-circular canyon subject to the SH waves • Numerical examples-membrane vibration • Numerical examples-exterior acoustics • Special case for Laplace problem • Conclusions MSVLAB

4. Motivation Potential formulation Regular formulation Singularity and hypersingularity ‧ 1 1 Fictitious boundary (indirect BEM) 2 2 Null-field integral equation Conjunction with degenerate kernel Bump contour Null-field integral equation Collocation point C.P.V. and H.P.V. ill-posed ??? Regularization techniques MSVLAB

5. Outlines • Motivation • Literature review • Present method • The technique for solving scattering problems in full and half planes • Alternative derivation of the Trifunac’s solution • Numerical examples-membrane vibration • Numerical examples-exterior acoustic • Special case for Laplace problem • Conclusions MSVLAB

6. Spurious characteristic equation of BEM (abroad researcher) • Tai and Shaw 1974, De Mey 1976 (real-part or imaginary-part kernel) • Niwa et al. 1982 (Only complex-value kernel satisfies Green’s function) • Hutchinson 1985 (real-part kernel) • Kitahara 1985 (Spurious eigenvalues were found) • Chen and Hong1988 (Dual BEM) • Golub & Van 1989 (SVD updating terms) • Partridge 1992 (Dual reciprocity boundary element method) • Kamiya & Andoh 1993, Itagaki & Brebbia 1993, Nowak & Neves 1994 (Multiple reciprocity method) MSVLAB

7. Spurious characteristic equation of BEM (MSV LAB) • Huang 1999 (Dual multiple reciprocity method using the singular value decomposition technique, simply-connected domain) • Lin 2000 (BEM using Burton & Miller approach, multiply-connected domain) • Chang 2001 (MFS, simply-connected domain in three dimension) • Chen and Lin 2002 (BEM using CHEEF concept, multiply-connected domain) • Liu 2002 (BEM using SVD updating technique, multiply-connected domain, analytical solution for annular case) • Lee 2004 (MFS, multiply-connected domain ) • Highly precision of prediction in spurious eigenvalue?? MSVLAB

8. Successful experiences of spuriouseigenvalue (BEM) (Membrane) Simply-connected problem Multiply-connected problem (Membrane) MSVLAB

9. Fictitious frequency (abroad researcher) • Seybert and Rengarajan 1968 (CHIEF method) • Ohmatsu 1983 (Combined integral equation method (CIEM)) • Achenbach et al.1988 (Off-boundary approach to BEM) • Fancis 1989 (SVD technique to solve the electromangetic resonance problem) • Wu and Seybert 1991 (CHIEF-block method using the weighted residual formulation) • Lee and Wu 1993 (Enhanced CHIEF method) • Juhl 1994 and Poulin 1997 (combined the CHIEF method and SVD technique) • Dokumaci and Sarigül 1995 (Surface Helmholtz integral equation (SHIE) and CHIEF method for radiation problem of two spheres) MSVLAB

10. Fictitious frequency (MSV LAB) • Chen 2000 (Dual BEM conjunction with Burton & Miller method, simply-connected domain) • Chen 2002 (Dual BEM conjunction with CHIEF method, simply-connected domain) • Chen 2004 (Dual BEM conjunction with Fast Multipole expansion method(FMM), simply-connected domain) • Multiply-connected domain ???? MSVLAB

11. Outlines • Motivation • Literature review • Present method • scattering problems in full and half planes • Alternative derivation of the Trifunac’s solution • Numerical examples-membrane vibration • Numerical examples-exterior acoustic • Special case for Laplace problem • Conclusions MSVLAB

12. Governing equation Helmholtz equation D u : acoustic potential k : wave number, : angular frequency c : sound speed D : domain of interest :Laplacian operator D MSVLAB

13. Integral representation D D Dual integral equation formulation for domain point: Null-field integral equation formulation: singular formulation hyper-singular formulation u : acoustic potential t: the normal derivation of u U, T, L, M: kernel function D : domain of interest x x x x MSVLAB

14. U(s,x) T(s,x) L(s,x) M(s,x) The relation about the kernels MSVLAB

15. x s x s R O1 x O2 Degenerate kernel x (field point): variable s (source point): fixed r S O1 R O2 MSVLAB

16. Degenerate kernels Fundamental solution: Degenerate kernels: MSVLAB

17. Fourier series for boundary densities Fourier series: MSVLAB

18. Adaptive origin of observer x: collocation point . MSVLAB

19. Decomposition of gradient vector Angle derivative direction True normal direction Radial derivative direction x MSVLAB

20. 2M+1 terms Collocation points By choosing M D.O.F. of Fourier series, we select 2M+1 collocation points on the circle. MSVLAB

21. Linear algebraic equation (Membrane vibration) fixed Routing boundary index Collocation circle index Routing boundary index MSVLAB

22. Linear algebraic equation (Membrane vibration) ● ● ● ● ● ● ● ● MSVLAB

23. Degenerate kernel Fourier series Potential Null-field equation Fourier Coefficients Analytical Algebraic equation AX=0 or AX=B Numerical The flowchart of present formulation Adaptive observer system, decomposition of gradient vector Collocation point method Integral equation for domain point SVD Inverse MSVLAB

24. Outlines • Motivation • Literature review • Present method • The technique for solving scattering problems in full and half planes • Alternative derivation of the Trifunac’s solution • Numerical examples-membrane vibration • Numerical examples-exterior acoustic • Special case for Laplace problem • Conclusions MSVLAB

25. Decomposition of scattering problem into incident wave field and radiation problem Incident SH-wave = (a) Incident wave field + Incident SH-wave (b) Radiation field MSVLAB

26. Image concept for solving the half-plane wave Image incident-SH wave Incident-SH wave take free body Incident-SH wave Incident-SH wave Incident-SH wave (a) Real problem (c) Transformed problem of full plane (b) Extended problem with artificial boundaries ( ) MSVLAB

27. Outlines • Motivation • Literature review • Present method • The technique for solving scattering problems in full and half planes • Alternative derivation of the Trifunac’s solution • Numerical examples-membrane vibration • Numerical examples-exterior acoustic • Special case for Laplace problem • Conclusions MSVLAB

28. Alternative derivation of the Trifunac solution by using the present method Reflected SH wave Incident SH wave MSVLAB

29. Alternative derivation of the Trifunac solution by using the present method Incident wave field: Expansion formula of Abramowitz and Stegan: Incident wave field : where Otherwise, The normal derivative of incident wave field: MSVLAB

30. Alternative derivation of the Trifunac solution by using the present method The radiation boundary condition: Fourier coefficients for : Substituting into null-field integral equation: Substituting into integral equation for domain point: The total displacement for the full plane: MSVLAB

31. The difference between present method and BEM MSVLAB

32. Outlines • Motivation • Literature review • Present method • The technique for solving scattering problems in full and half planes • Alternative derivation of the Trifunac’s solution • Numerical examples-membrane vibration • Numerical examples-exterior acoustic • Special case for Laplace problem • Conclusions MSVLAB

33. Problem statement Simply-connected domain Doubly-connected domain Multiply-connected domain MSVLAB

34. Mesh Collocation point Element (a) Present method (b) BEM (c) FEM MSVLAB

35. Example 1 MSVLAB

36. The eigenfrequencies by using singular equation M=6, 26 collocation points Contaminated by spurious eigenvalues MSVLAB

37. The eigenfrequencies by using hyper-singular equation M=6, 26 collocation points Contaminated by spurious eigenvalues MSVLAB

38. The spurious eigenvalues are filtered by Burton & Miller method Only true eigenvalues appear M=6, 26 collocation points MSVLAB

39. The former five true eigenvalues by using different approaches MSVLAB

40. The former five eigenmodes by using present method, FEM and BEM MSVLAB

41. Example 2 MSVLAB

42. The eigenfrequencies by using singular equation M=6, 26 collocation points Contaminated by spurious eigenvalues MSVLAB

43. The eigenfrequencies by using hyper-singular equation M=6, 26 collocation points Contaminated by spurious eigenvalues MSVLAB

44. The spurious eigenvalues are filtered by Burton & Miller method Only true eigenvalues appear M=6, 26 collocation points MSVLAB

45. The former five eigenvalues by using different approaches MSVLAB

46. The former five eigenmodes by using present method, FEM and BEM MSVLAB

47. Parserval sum (b) inner circle (real part) (a) outer circle (real part) Parserval sum Parserval sum M M MSVLAB

48. Example 3 R=1 c2=0.4 c1=0.3 e=0.5 MSVLAB

49. Extraction of the spurious eigenvalues by using SVD updating document M=3 MSVLAB

50. The former five eigenvalues by using different approaches MSVLAB