Recent development of BEM/BIEM in vibration and acoustics

1. Recent development of BEM/BIEM in vibration and acoustics 陳正宗 海洋大學 特聘教授 河海工程學系 Nov. 19, 2004, NSYSU, 14:10~16:00

2. Outlines • Introduction • Exterior acoustics - adaptive BEM • Interior acoustics - multiply-connected eigenproblems • Conclusions

3. Growth of BEM papers

4. Introduction • Finite difference method (FDM) • Finite element method (FEM) • Boundary element method (BEM) • Meshless method (MM) • Boundary integral equation method (BIEM) MM BEM BIEM FDM FEM No mesh No discretization for circular boundaries Domain discretization Boundary discretization No mesh No node

5. Adaptive BEM for exterior radiation and scattering problems

6. Problem statement Non-uniform radiator problemScattering problem

7. Solver Error indicator Adaptive scheme Singular formulation Hypersingular formulation R.P.V. is Riemann Principal Value C.P.V. is Cauchy Principal Value H.P.V. is Hadamard Principal Value

8. Adaptive mesh Uniform mesh Adaptive mesh

9. Refinement scheme h-version p-version r-version 1 2 3 1. Element number increasing 2. Interpolation function order increasing 3. Optimum nodal collocation

10. Mesh BEMFEM(DtN) Taiwan, NTOU US Navy. Stanford Univ.

11. Non-uniform radiation： Dirichlet problem Numerical solution: FEM(DtN) Numerical solution: BEM 64 ELEMENTS 2791 ELEMENTS Taiwan, NTOU US Navy. Stanford Univ.

12. Non-uniform radiation ：Dirichlet problem Analytical solution: n=20

13. ∥ + Superposition principle

14. Scattering ： Neumann problem Numerical solution: FEM(DtN) Numerical solution: BEM 63 ELEMENTS 7816 ELEMENTS Taiwan, NTOU US Navy. Stanford Univ.

15. Scattering ：Neumann problem Analytical solution: n=20

16. u(a,0) ka Fictitious frequency： Non-uniform radiation problem

17. • t(a, 0) ka Fictitious frequency ： The scattering Dirichlet problem

18. Summary • Fictitious frequency depends on the formulation (singular or hypersingular) instead of B.C. (Dirichlet or Neumann). • Burton & Miller method and CHIEEF method can overcome the problem of fictitious frequency. • Fictitious frequency happens to be the true eigenvalues of the interior problem (SingularDirichlet, HypersingularNeumann).

19. Spurious eigenvalues for multiply-connected problems

20. Problem domain Simply-connected domain Doubly-connected domain Multiply-connected domain

21. . . . . . . . . . . . 0 0 BEM&BIEM Fourier series Boundary discretization . u(θ) or t(θ) u(θ) or t(θ) BEM BIEM θ θ

22. Solve the fundamental solution Solve boundary data Potential BIE for domain point Linear algebraic system BIE for boundary point The flowchart to determine the eigenvalues and mode shape by BEM Given G.E. and B.C. SVD Moving to the boundary Boundary element discretization

23. Degenerate kernel Fourier series Potential Null-field equation Algebraic equation Fourier Coefficients Analytical Numerical The flowchart to determine the eigenvalues and mode shape by BIEM SVD

24. Integral Formulation Null-field integral equations:

25. s r x Degenerate kernels s R x O x：source point；s：field point

26. Degenerate kernels Degenerate kernels:

27. Degenerate kernels

28. Fourier series for boundary densities Fourier series:

29. 2M+1 terms Collocation points By choosing M terms of Fourier series, we select 2M+1 collocation points on the circle.

30. Integral representation Integral equation formulation:

31. Numerical examples Example 1

32. The eigenfrequenies by using singular equation More accurate ( ) Numerical [ ] exact Contaminated by spurious eigenvalues

33. Relation of spurious eigenvalue and true eigenvalue 0.5 True Spurious eigenvalue using singular formulation happens to be the true eigenvalue of the associated interior Dirichlet problem.

34. More accurate More accurate The eigenfrequenies by using hyper-singular equation ( ) Numerical [ ] exact Contaminated by spurious eigenvalues

35. 0.5 True Relation of spurious eigenvalue and true eigenvalue Spurious eigenvalue using hypersingular formulation happens to be the true eigenvalue of the associated interior Neumann problem.

36. The spurious eigenvalues are filtered by Burton&Miller method Only true eigenvalues appear

37. The former five eigenvalues of Helmholtz eigenproblem with an eccentric domain

38. The former five eigenmodes for eccentric case using present method, FEM and BEM

39. R=1 c2=0.4 c1=0.3 e=0.5 Numerical examples Example 2

40. Extraction of the spurious eigenvalues by using SVD updating document More accurate More accurate

41. The former five eigenvalues for a multiply-connected problem with two unequal holes using different approaches

42. The former five modes for a circle domain with two unequal holes using present method, BEM and FEM

43. Summary • Spurious eigenvalues depend on formulation (singular or hyper-singular). • Spurious eigenvalues are independent of B.C. (Dirichlet or Neumann). • Spurious eigenvalues happens to be the true eigenvalues of the interior problem (Dirichletsingular, Neumannhypersingular). • To overcome the spurious eigenvalues  Burton&Miller, SVD updating term, SVD updating document…….

44. Conclusions • Exterior acoustic problems (radiation and scattering) were solved by using adaptive BEM. • Good accuracy and efficiency of the present method were obtained in comparison with those with FEM. • Spurious eigenvalues embedded in the BIEM/BEM were examined and filtered out in this study. • Both the fictitious frequency and spurious eigenvalue depend on the formulation instead of B.C. .

45. 歡迎參觀海洋大學力學聲響振動實驗室 烘培雞及捎來伊妹兒 URL: http://ind.ntou.edu.tw/~msvlab/ Email: jtchen@mail.ntou.edu.tw The end Thanks for your kind attention