1. 1 The Application of Hypersingular Meshless Method for 3D Potential and Exterior Acoustics Problems Reporter ： Professor D. L. Young 2008/01/03 Department of Civil Engineering and Hydrotech Research Institute National Taiwan University Scientific Computing & Visualization Lab

2. 2 含超強奇異性無網格法於三維勢能及 外域聲學問題之應用 楊德良 教授 2008/01/03 Department of Civil Engineering and Hydrotech Research Institute National Taiwan University Scientific Computing & Visualization Lab

3. National Taiwan University 3 Outline: Introduction Potential problems Formulation The diagonal coefficient of influence matrices Numerical results Cube Cylinder Arbitrary shape Exterior acoustics problems Formulation The diagonal coefficient of influence matrices Numerical results Scattering by a soft sphere Scattering by a rigid sphere Scattering by a bean shape obstacle Conclusions Further researches

4. National Taiwan University 4 Brief detail of MFS

5. National Taiwan University 5 Brief detail of MFS Method of fundamental solutions （ MFS ） is involved through the combination of meshless and the concept of indirect boundary element method. The MFS considers an artificial boundary outside the computational domain, to locate the source points and some field points locate on the boundary. Using these points and boundary conditions can solve the coefficients used in the fundamental solution.

6. National Taiwan University 6 Brief detail of MFS Domain method MFS

7. National Taiwan University 7 Brief detail of MFS From the principles of method of fundamental solutions, for the given governing equation, the free space Green’s function has to be satisfied. For example of the Laplace equation as follows the free space Green’s function can be written where is the fundamental solutions is the Dirac delta function, is the position of the field point, and is the position of the source point.

8. National Taiwan University 8 Brief detail of MFS Method of Fundamental Solutions (MFS )

9. National Taiwan University 9 Brief detail of MFS Using the above expression, the approximate solution can be obtained as And the field points located on the boundary and combined with boundary condition that can solve coefficients and advance to solve any region in the solution domain.

10. National Taiwan University 10 Singular Value Decomposition SVD is the technique for dealing with sets of equations or matrices are either singular or else numerically very close to singular. Orthogonal matrix Matrix of the singular values

11. National Taiwan University 11 Introduction B.C. G.E. Time-independent

12. National Taiwan University 12 Introduction (n+1)dt y t (n)dt (n- )dt Field point Source point Time-dependent

13. National Taiwan University 13 Numerical methods for Burgers’ eq. Mesh method FDM FEM Mesh-reduction method BEM Meshless method MQ MLPG MFS-DRM 1980 Varoglu & Finn 1981 Caldwell & Wanless 1982 Nguyen & Reynen 2004 Dogan 1984 Evans & Abdullah 1990 Kakuda & Tosaka 1998 Hon 2002 Li, Hon & Chen Introduction (Burgers’ equation) 2000 Lin & Atluri Modified Helmhotz fundamental solution Domain-type method

14. National Taiwan University 14 Introduction (1/4) The Method of Fundamental Solutions proposed by Kupradze and Aleksidze, 1964. The MFS has been generally applied to solve some engineering problems. It is a kind of meshless methods, since only boundary nodes are distributed. However because of the controversial artificial boundary (off-set boundary) outside the physical domain, the MFS has not become a popular numerical method. MFS only works well in regular geometry with the Dirichlet and Neumann boundary conditions.

15. National Taiwan University 15 Introduction (2/4) This research extends the Hypersingular Meshless Method to solve the 3D potential and exterior acoustics problems. Young et al.2005 J. Comput. Phys. Potential problems in 2D. Chen et al.2006 Eng. Anal. Bound. Elem. Multiply-connected-domain Laplace problem in 2D. Young et al.2006J. Acoust. Soc. Am. Exterior acoustics problems in 2D.

16. National Taiwan University 16 Introduction (3/4) Source point location MFS HMM

17. National Taiwan University 17 Introduction (4/4) Comparison of HMM and MFS HMMMFS Meshless features Yes Source point locationReal boundaryFiction boundary Accuracy AcceptableBetter Potential Double layerSingle layer Kernel functions for 3D potential problems Kernel functions for 3D exterior acoustics

18. 18 Potential Problems Department of Civil Engineering and Hydrotech Research Institute National Taiwan University

19. 19 Formulation Governing equation:, the representation of the solution for interior problem can be approximated as: Kernel functions:

20. National Taiwan University 20 The Diagonal Coefficient of Influence Matrices

21. National Taiwan University 21 Analytical derivation of diagonal coefficients Analytical solution: wherewave number number of nodes radius of sphere

22. National Taiwan University 22 Case 1-1: Dirichlet boundary case:

23. National Taiwan University 23 Point Distribution with Normal Vectors

24. National Taiwan University 24 Results Cross-section at x=0.5 MFS HMM FEM RMSE: 6.26E-5 RMSE: 2.01E-3 RMSE: 4.10E-3 1350 NODES 1350 NODES 3375 NODES (13720 Elements) ( analytical solution, numerical result)

25. National Taiwan University 25 Comparison of Three Methods Numerical values at x=0.5, z=0.5

26. National Taiwan University 26 Absolute Error Distribution Map at x=0.5 MFS

27. National Taiwan University 27 Absolute Error Distribution Map at x=0.5 HMM

28. National Taiwan University 28 Case 1-2: Dirichlet and Neumann mixed boundary case:

29. National Taiwan University 29 Results Cross-section at x=0.5 MFS HMM FEM RMSE: 8.23E-5 RMSE: 2.02E-3 RMSE: 4.10E-3 1350 NODES 1350 NODES 3375 NODES (13720 Elements) ( analytical solution, numerical result)

30. National Taiwan University 30 Case 2: Analytical solution: : Bessel function : The root of

31. National Taiwan University 31 Results Cross-section at z=0.5 MFS HMM 1800 nodes 1800 nodes RMSE: 3.80E-3 RMSE: 2.41E-2 ( analytical solution, numerical result)

32. National Taiwan University 32 Point Distribution Comparison RMSE : 2.41E-2RMSE : 0.1523 RMSE : 8.89E-2RMSE : 0.1738

33. National Taiwan University 33 Sensitivity Test of Point Distribution The distance of the nodes on the top surface is fixed at 0.0833 Number of nodes on Z axis Distance of the nodes on Z axis RMSE 60.16660.3764 70.14280.2871 80.12500.2131 90.11110.1510 100.10009.84E-2 110.09095.44E-2 120.08332.41E-2 130.07693.39E-2 140.07146.10E-2 150.06668.73E-2

34. National Taiwan University 34 Sensitive Test of Number of nodes NodesRMSE 2346.870E-2 4203.070E-2 6842.726E-2 10502.481E-2 15182.507E-2 18002.412E-2 21062.319E-2 28422.299E-2 37202.253E-2 42402.172E-2

35. National Taiwan University 35 Case 3-1: Inside radius: 1 Outside radius: 2 Height: 1

36. National Taiwan University 36 Point Distribution with Normal Vectors 1991 nodes

37. National Taiwan University 37 Results Cross-section at z=0 MFS HMM FEM 1991 nodes 1991 nodes 1320 nodes (5000 elements)

38. 38 Case 3-2: BC: Analytical solution:

39. 39 Point Distribution with Normal Vectors

40. 40 Results Cross-section at x=0 MFS (d=0.5) MFS (d=1) HMM 2826 nodes 2826 nodes 2826 nodes RMSE: 9.63E22 RMSE: 1.26E-4 RMSE: 4.12E-2 ( analytical solution, numerical result)

41. 41 Case 3-3:

42. 42 Results Cross-section at x=0 2826 nodes 2981 nodes 2826 nodes (a) MFS (d=1) (b) LDQ (c) HMM

43. 43 Case 3-4

44. 44 Point Distribution with Normal Vectors 2261 nodes

45. 45 Results Cross-section at z=0 2826 nodes 2981 nodes 2826 nodes (a) MFS (d=2) (b) LDQ (c) HMM

46. 46 Exterior Acoustics Problems Department of Civil Engineering and Hydrotech Research Institute National Taiwan University

47. 47 Formulation Governing equation: Sommerfeld radiation condition: the representation of the solution for exterior problem can be approximated as: Kernel functions:

48. 48 The Diagonal Coefficient of Influence Matrices The kernel function will be approximated by: The diagonal coefficients for the exterior problem can be extracted out as:

49. 49 Analytical derivation of diagonal coefficients Analytical solution: wherewave number number of nodes radius of sphere

50. 50 Scattering of a Plane Wave by a Soft Sphere Governing equation: Plane wave incidence: y z a x Analytical solution of total field: J. J. Bowman,T. B. A. Senior, P. L. E.Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes, Hemisphere publishing Corp., 1987. Analytical solution of the scattered field:

51. 51 Point Distribution and Normal Vectors 1866 nodes

52. 52 Results Values on the y=0, z=0 line Real part Imaginary part

53. 53 Results Cross-section of y=0 plan for real part RMSE: 8.73E-5 RMSE: 4.73E-3 1866 nodes 1866 nodes MFS HMM Exact solution Numerical solution

54. 54 Results Cross-section of y=0 plan for imaginary part RMSE: 1.26E-5 RMSE: 9.69E-3 1866 nodes 1866 nodes MFS HMM Exact solution Numerical solution

55. 55 Scattering of a Plane Wave by a Rigid Sphere Governing equation: Sommerfeld radiation condition: Neumann boundary condition: y z a x Associated Lengendre polynomial

56. 56 Analytical Solution where K. Gerdes, L. Demkowicz, Solution of 3D-Laplace and Helmholtz equations in exterior domains using hp-infinite elements. Comput. Meth. Appl. Mech. Eng. 137 1996 239–273. Spherical Hankel function of the first kind

57. 57 Results Real part Imaginary part

58. 58 Results Cross-section of RMSE: 1.34E-4 RMSE: 1.47E-2 1866 nodes 1866 nodes MFS HMM Exact solution Numerical solution

59. 59 Results Cross-section of RMSE: 5.98E-3 RMSE: 5.76E-2 1866 nodes 1866 nodes MFS HMM Exact solution Numerical solution

60. 60 Scattering of a Plane Wave by a Soft Bean Shape Obstacle Governing equation: Sommerfeld radiation condition: Plane wave incidence: M. Ganesh and I.G. Graham, A high-order algorithm for obstacle scattering in three dimensions, J. Comput. Phys. 198 2004 211–242. Radius parameter R:

61. 61 Mesh of the Bean Shape Obstacle 2600 elements

62. 62 Point Distribution and Normal Vectors 2500 nodes

63. 63 Results real part imaginary part

64. 64 Results of Real Part by MFS d=-0.1d=-0.2 d=-0.15d=-0.25

65. 65 Results of Imaginary Part by MFS d=-0.1d=-0.2 d=-0.15d=-0.25

66. National Taiwan University 66 Conclusions The controversy of the artificial (off-set) boundary outside the physical domain by using the MFS no longer exists. From the series cases of the complex irregular shape, MFS required a lot of time to adjust the distance of source points, HMM has figured out acceptable answers immediately. From the sensitivity test of point distribution, we can know that to obtain the high accuracy of HMM, improving the point seeding is necessary. HMM required the uniform point distribution to obtain the good results.

67. National Taiwan University 67 Further Researches For the next step, to solve the Helmholtz problem in vector field which relate to the electromagnetic problem in three dimensions is we are going to do. The combination of other numerical methods such as method of particular solutions (MPS) or domain decomposition method (DDM) and HMM to solve Poisson, Helmholtz, modified Helmholtz equation would be interesting topics to research.

68. 68 Department of Civil Engineering and Hydrotech Research Institute National Taiwan University Thank You Scientific Computing & Visualization Lab

69. 69 The Diagonal Coefficient of Influence Matrices for BEM

70. 70 The Diagonal Coefficient of Influence Matrices for HMM The kernel function will be approximated by: The diagonal coefficients for the exterior problem can be extracted out as: