1 / 70

Reporter ： Professor D. L. Young 2008/01/03

Department of Civil Engineering and Hydrotech Research Institute National Taiwan University. The Application of H ypersingular M eshless M ethod for 3D Potential and Exterior Acoustics Problems. Reporter ： Professor D. L. Young 2008/01/03.

rufin
Download Presentation

Reporter ： Professor D. L. Young 2008/01/03

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

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

  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. Brief detail of MFS

  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. Brief detail of MFS Domain method MFS

  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. Brief detail of MFS • Method of Fundamental Solutions (MFS)

  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. 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. Matrix of the singular values Orthogonal matrix

  11. Introduction Time-independent G.E. B.C.

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

  13. Introduction (Burgers’ equation) 1984Evans & Abdullah 1980 Varoglu & Finn1981Caldwell & Wanless 1982 Nguyen & Reynen2004Dogan 1990 Kakuda & Tosaka 1998 Hon 2000 Lin & Atluri 2002 Li, Hon & Chen Modified Helmhotz fundamental solution Domain-type method

  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. Introduction (2/4) • This research extends the Hypersingular Meshless Method to solve the 3D potential and exterior acoustics problems.

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

  17. Introduction (4/4)Comparison of HMM and MFS

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

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

  20. The Diagonal Coefficient of Influence Matrices

  21. Analytical derivation of diagonal coefficients • Analytical solution: where wave number number of nodes radius of sphere

  22. Case 1-1:Dirichlet boundary case:

  23. Point Distribution with Normal Vectors

  24. ( analytical solution, numerical result) ResultsCross-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)

  25. Comparison of Three MethodsNumericalvalues at x=0.5, z=0.5

  26. Absolute Error Distribution Map at x=0.5MFS

  27. Absolute Error Distribution Map at x=0.5HMM

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

  29. ( analytical solution, numerical result) ResultsCross-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)

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

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

  32. Point Distribution Comparison

  33. Sensitivity Test of Point Distribution • The distance of the nodes on the top surface is fixed at 0.0833

  34. Sensitive Test of Number of nodes

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

  36. Point Distribution with Normal Vectors 1991 nodes

  37. ResultsCross-section at z=0 MFS HMM FEM 1991 nodes 1991 nodes 1320 nodes (5000 elements)

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

  39. Point Distribution with Normal Vectors

  40. ( analytical solution, numerical result) ResultsCross-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

  41. Case 3-3:

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

  43. Case 3-4

  44. Point Distribution with Normal Vectors 2261 nodes

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

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

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

  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. Analytical derivation of diagonal coefficients • Analytical solution: where wave number number of nodes radius of sphere

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

More Related