The 32 nd National Conference on Theoretical and Applied Mechanics

1. The 32nd National Conference on Theoretical and Applied Mechanics Scattering of sound from axisymetric sources by multiple circular cylinders using addition theorem and superposition technique Reporter : Yi-Jhou Lin National Taiwan Ocean University Department of Harbor and River Engineering Authors :Yi-Jhou Lin, Ying-Te Lee , I-Lin Chen and Jeng-Tzong Chen Date: November 28-29, 2008 Place: National Chung Cheng University, Chia-Yi

2. Outlines • Introduction • Problem statement • Method of solution • Mathematical Equivalence • Mathematical equivalence between the solution of Green’s third identity and that of superposition technique • Numerical examples • Concluding remarks Introduction

3. Motivation Numerical methods for engineering problems FDM / FEM / BEM / BIEM / Meshless method BEM / BIEM Treatment of singularity and hypersingularity Boundary-layer effect Convergence rate Ill-posed model

4. Motivation BEM / BIEM Improper integral Singularity & hypersingularity Regularity Fictitious BEM Bump contour Limit process Fictitious boundary Achenbach et al. (1988) Null-field approach Guiggiani (1995) Gray and Manne (1993) Collocation point CPV and HPV Ill-posed Waterman (1965)

5. Present approach Degenerate kernel Fundamental solution No principal value CPV and HPV • Advantages of present approach • mesh-free generation • well-posed model • principal value free • elimination of boundary-layer effect • exponential convergence

6. Green’s third identity BIE for Green’s function

7. Outlines • Introduction • Problem statement • Method of solution • Mathematical Equivalence • Mathematical equivalence between the solution of Green’s third identity and that of superposition technique • Numerical examples • Concluding remarks Problem statement

8. Problem statement Free field Original Problem Radiation field(typical BVP) (soft)

9. Original problem Decompose two parts Free field Radiation field Expansion Degenerate kemel For fundamental solution Fourier series of boundary densities Collocate of the real boundary Linear algebraic system Calculation of the unknown Fourier BIE for the domain point Superposing the solution of two parts Total field Flowchart

10. Outlines • Introduction • Problem statement • Method of solution • Mathematical Equivalence • Mathematical equivalence between the solution of Green’s third identity and that of superposition technique • Numerical examples • Concluding remarks Method of solution

11. Method of solution Boundary integral equation and null-field integral equation Interior case Exterior case Degenerate (separate) form

12. cosnθ, sinnθ boundary distributions kth circular boundary Degenerate kernel and Fourier series x Expand fundamental solution by using degenerate kernel s O x Expand boundary densities by using Fourier series

13. U(s,x) T(s,x) L(s,x) M(s,x) Degenerate kernels

14. Degenerate kernels

15. Adaptive observer system Source point Collocation point

16. Linear algebraic system y x

17. Outlines • Introduction • Problem statement • Method of solution • Mathematical Equivalence • Mathematical equivalence between the solution of Green’s third identity and that of superposition technique • Numerical examples • Concluding remarks Mathematical Equivalence

18. Mathematical equivalence between the solution of Green’s third identity and that of superposition technique = + Green’s third identity Superposition technique

19. Outlines • Introduction • Problem statement • Method of solution • Mathematical Equivalence • Mathematical equivalence between the solution of Green’s third identity and that of superposition technique • Numerical examples • Concluding remarks Numerical examples

20. An infinite plane with two equal circular cylinders subject to a point sound source. Governing equation: Dirichlet Boundary condition: (soft) Fundamental solution:

21. Distribution potential on the artificial boundaries in the free field

22. Case 1 parameter use size and cylinder (soft) B1 b B2 b y (soft) Probe

23. Distribution potential on the artificial boundaries in the free field versus polar angle.

24. Probe Probe Relative amplitude of total field versus the probe location y (M=20). Total field (soft) B1 b B2 b (soft) Versus (soft) Free field B1 b B2 b (soft)

25. Probe Probe Relative amplitude of total field versus the probe location (M=20). (soft) Total field B1 b B2 b (soft) Versus (soft) Free field B1 b B2 b (soft)

26. Probe Probe Relative amplitude of total field versus (M=20). (soft) Total field B1 b B2 b (soft) Versus (soft) Free field B1 b B2 b (soft)

27. Convergence test of Parseval’s sum for (real part).

28. Convergence test of Parseval’s sum for (imaginary part).

29. Case 2 parameter usecylinder center-to-center (soft) B1 b Probe B2 b (soft)

30. Relative amplitude of total field versus (M=20). (soft) B1 b Probe b B2 (soft)

31. Relative amplitude of total field versus (M=20). (soft) B1 b Probe B2 b (soft)

32. Relative amplitude of total field versus (M=20). (soft) B1 b Probe B2 b (soft)

33. Relative amplitude of total field versus (M=20). (soft) B1 b Probe b B2 (soft)

34. Outlines • Introduction • Problem statement • Method of solution • Mathematical Equivalence • Mathematical equivalence between the solution of Green’s third identity and that of superposition technique • Numerical examples • Concluding remarks Concluding remarks

35. Concludingremarks • A general-purpose program for solving the problems with arbitrary number, size and various locations of circular cavities was developed. • We have proposed a BIEM formulation by using degenerate kernels, null-field integral equation and Fourier series in companion with adaptive observer system.

36. The end Thanks for your kind attention Welcome to visit the web site of MSVLAB http://ind.ntou.edu.tw/~msvlab