1. 1 The Method of Fundamental solutions for exterior acoustic problems 報告人：陳義麟 國立高雄海洋科技大學造船系 高雄 2006/3/19 NSC-92-2611-E-022-004

2. Outline Introduction Method of fundamental solutions Governing equation Mathematical analysis Numerical examples Conclusions 2

3. Introduction Method of Fundamental Solutions Governing Equation Mathematical analysis Numerical examples Conclusions 3

4. Numerical method Finite Difference Method Finite Element Method Boundary Element Method Numerical method Meshless Method Mesh Singular integral XXX X 4

5. Introduction Method of Fundamental Solutions Governing Equation Mathematical analysis Numerical examples Conclusions 5

6. Method of fundamental solutions Field representation: xixi sjsj r ij s j+1 Interior problemExterior problem xixi sjsj r ij s j+1 Source point Observation point 6

7. Acoustic problem Governing Equation: is the acoustic pressure is the wave number is the Laplacian operator is the domain of the interest 7

8. Fundamental solution The fundamental solution satisfy 8 is the Dirac delta function

9. Kernel functions Fundamental solution 9

10. Acoustic pressure and flux Single-layer potential approach 10 Double-layer potential approach

11. Using MFS to solve the acoustic field 11 For the Dirichlet B.C., single-layer potential approach

12. Using MFS to solve the acoustic field 12 For the Neumann B.C., single-layer potential approach

13. Using MFS to solve the acoustic field 13 For the Dirichlet B.C., double-layer potential approach

14. Using MFS to solve the acoustic field 14 For the Neumann B.C., double-layer potential approach

15. Introduction Method of Fundamental Solutions Governing Equation Mathematical analysis Numerical examples Conclusions 15

16. Degenerate kernels for circular case (0,0) X1X1 Y1Y1 R1R1 Y2Y2 X2X2 R2R2 I U E U 16

17. Circulant Discritization into 2N nodes on the circular boundary 17

18. Circulant : eigenvalue of C 2N 18

19. Eigenvalue of U kernel Riemann sum reduces to integral: 19

20. Eigenvalues of L, T and M influence matrices 20

21. Fictitious frequency (exterior problem) ka u(a,0) 21

22. Radiation problem with Neumann B. C., 22 single-layer potential approach

23. Radiation problem with Dirichlet B. C., 23 single-layer potential approach

24. Radiation problem with Neumann B. C., 24 double-layer potential approach

25. Radiation problem with Dirichlet B. C., 25 double-layer potential approach

26. Introduction Method of Fundamental Solutions Governing Equation Mathematical analysis Numerical examples Conclusions 26

27. Radiation problem (Dirichlet type) for a cylinder Radiator a 27

28. Collocation point Source point R x s R a The located position of source and collocation points 28

29. The contour plot for the real-part solutions 29

30. BEM MFS k t t(a,0) versus k for by using the MFS and BEM. 30

31. The nonuniform radiation problem for a cylinder. 31

32. Numerical solution for the nonuniform radiation problem 32

33. UL method TM method Burton & Miller Analytical solution k u u versus k using the MFS for the nonuniform radiation by a circular cylinder. 33

34. Introduction Method of Fundamental Solutions Governing Equation Mathematical analysis Numerical examples Conclusions 34

35. Conclusions We have verified that the fictitious frquency depends on the location of the source point. For the MFS, the sources can be distributed on the real boundary without any difficulty. However, the sources must be distributed outside the domain to avoid the singularity when the MFS are utilized. Fictitious frequencies for the exterior acoustic problems were analytically derived by using the degenerate kernels and circulants. 1. 2. 3. 35

36. The End Thanks for your attention 36