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1 Study on eigenproblems for Helmholtz equation with circular and spherical boundaries by using the BIEM and the multipole Trefftz method Reporter ： Shing-Kai.

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Presentation on theme: "1 Study on eigenproblems for Helmholtz equation with circular and spherical boundaries by using the BIEM and the multipole Trefftz method Reporter ： Shing-Kai."— Presentation transcript:

1 1 Study on eigenproblems for Helmholtz equation with circular and spherical boundaries by using the BIEM and the multipole Trefftz method Reporter ： Shing-Kai Kao Advisor: Dr. Jeng-Tzong Chen Date ： 2009/07/25

2 M M S S V V MSVLAB, HRE, NTOU 2 Outline Motivation Derivation of the formulation Null-field integral equation Trefftz method Addition theorem Multipole expansion Linear algebraic equation Numerical examples Eigenproblems by using the BIEM Eigenproblems by using the multipole Trefftz method Conclusions

3 M M S S V V MSVLAB, HRE, NTOU 3 Outline Motivation Derivation of the formulation Null-field integral equation Trefftz method Addition theorem Multipole expansion Linear algebraic equation Numerical examples Eigenproblems by using the BIEM Eigenproblems by using the multipole Trefftz method Conclusions

4 M M S S V V MSVLAB, HRE, NTOU 4 Motivation BEM / BIEM Treatment of singularity and hypersingularity Ill-posed model Boundary-layer effect Convergence rate Null-field integral equation approach Free of calculating the principal values Free of Boundary-layer effect Well posed Exponential convergence Spurious eigenvalue SVD updating term and SVD updating document Degenerate kernels (kernel functions) Spherical harmonics (boundary densities)

5 M M S S V V MSVLAB, HRE, NTOU 5 Motivation Trefftz method Eccentric case Multipole Trefftz method Addition theorem ExteriorInterior This thesisLiterature

6 M M S S V V MSVLAB, HRE, NTOU 6 Outline Motivation Derivation of the formulation Null-field integral equation Trefftz method Addition theorem Multipole expansion Linear algebraic equation Numerical examples Eigenproblems by using the BIEM Eigenproblems by using the multipole Trefftz method Conclusions

7 M M S S V V MSVLAB, HRE, NTOU 7 BIEM and null-field integral equation Interior problem Exterior problem Degenerate (separable) form 7

8 M M S S V V MSVLAB, HRE, NTOU 8 3D degenerate kernels for Helmholtz equation

9 M M S S V V MSVLAB, HRE, NTOU 9 3D degenerate kernels for Helmholtz equation

10 M M S S V V MSVLAB, HRE, NTOU 10 Spherical harmonics of boundary densities

11 M M S S V V MSVLAB, HRE, NTOU 11 Outline Motivation Derivation of the formulation Null-field integral equation Trefftz method Addition theorem Multipole expansion Linear algebraic equation Numerical examples Eigenproblems by using the BIEM Eigenproblems by using the multipole Trefftz method Conclusions

12 M M S S V V MSVLAB, HRE, NTOU 12 Trefftz method (interior problem)

13 M M S S V V MSVLAB, HRE, NTOU 13 Trefftz method (exterior problem)

14 M M S S V V MSVLAB, HRE, NTOU 14 Trefftz method-annular problem

15 M M S S V V MSVLAB, HRE, NTOU 15 Outline Motivation Derivation of the formulation Null-field integral equation Trefftz method Addition theorem Multipole expansion Linear algebraic equation Numerical examples Eigenproblems by using the BIEM Eigenproblems by using the multipole Trefftz method Conclusions

16 M M S S V V MSVLAB, HRE, NTOU 16 Addition theorem

17 M M S S V V MSVLAB, HRE, NTOU 17 Alternative technique Tensor transformation x1x1 y1y1 xjxj yjyj xixi yiyi x2x2 y2y2 B1B1 B2B2 BiBi BjBj osos osos osos osos Adaptive observer system Is it possible free of x

18 M M S S V V MSVLAB, HRE, NTOU 18 Outline Motivation Derivation of the formulation Null-field integral equation Trefftz method Addition theorem Multipole expansion Linear algebraic equation Numerical examples Eigenproblems by using the BIEM Eigenproblems by using the multipole Trefftz method Conclusions

19 M M S S V V MSVLAB, HRE, NTOU 19 Multipole expansion

20 M M S S V V MSVLAB, HRE, NTOU 20 Outline Motivation and literature review Derivation of the formulation Trefftz method Addition theorem Multipole expansion Linear algebraic equation Numerical examples Eigenproblems by using the BIEM Eigenproblems by using the multipole Trefftz method Conclusions

21 M M S S V V MSVLAB, HRE, NTOU 21 Linear algebraic system

22 M M S S V V MSVLAB, HRE, NTOU 22 Linear algebraic system Don’t need to expand by using the addition theorem. The addition theorem of J(kr)

23 M M S S V V MSVLAB, HRE, NTOU 23 Flowchart of multipole Trefftz method Original problem Expansion Addition theorem Fourier series of boundary conditions Calculation of the unknown coefficient Linear algebraic system Total field Trefftz bases

24 M M S S V V MSVLAB, HRE, NTOU 24 Outline Motivation Derivation of the formulation Null-field integral equation Trefftz method Addition theorem Multipole expansion Linear algebraic equation Numerical examples Eigenproblems by using the BIEM Eigenproblems by using the multipole Trefftz method Conclusions

25 M M S S V V MSVLAB, HRE, NTOU 25 Problem statement

26 M M S S V V MSVLAB, HRE, NTOU 26 Null-field integral equation - UT

27 M M S S V V MSVLAB, HRE, NTOU 27 Dirichlet B.C. (fixed-fixed) - UT Eigenequations

28 M M S S V V MSVLAB, HRE, NTOU 28 Eigenvalue (k)-(fixed-fixed) n012345678 6.283198.9868211.526913.975916.365118.711621.025723.314125.5816 12.566415.450518.1920.834223.409825.933128.414830.862633.282 18.849621.808224.645927.39630.079332.709435.295937.84640.3649 25.132728.132431.029233.847236.602539.306341.966944.590747.1825 n012345678 6.283196.572017.111587.845048.71689.68210.707711.770812.8557 12.566412.721413.026113.471114.043714.729415.513316.380617.3173 18.849618.954419.162519.471119.845820.371820.953321.614122.3481 25.132725.211825.369225.603825.913726.296826.750227.27127.8561 It’s a special case that a =0.5 b.

29 M M S S V V MSVLAB, HRE, NTOU 29 Neumann B.C. (free-free) - UT Eigenequations

30 M M S S V V MSVLAB, HRE, NTOU 30 Eigenvalue (k )-(free-free) n012345678 1.840273.151184.389965.574546.717537.831128.9249510.0056 6.572016.911527.553628.438879.5010110.677711.916513.177814.4344 12.721412.885213.208713.68414.301415.050415.920416.900517.9775 18.954419.062119.27619.593720.011520.525321.130521.822822.5981 n012345678 6.283198.9868211.526913.975916.365118.711621.025723.314125.5816 12.566415.450518.1920.834223.409825.933128.414830.862633.282 18.849621.808224.645927.39630.079332.709435.295937.84640.3649 25.132728.132431.029233.847236.602539.306341.966944.590747.1825 Dependent on the formulation

31 M M S S V V MSVLAB, HRE, NTOU 31 The eigenvalues by using BIEM and SVD 12345678910 6.2806.5707.1107.8508.7208.9909.680 U kernel 12345678910 1.8403.1504.3905.5706.2806.5706.7206.9107.5507.830 T kernel 12345678910 4.1606.2806.5706.6807.1107.8408.7208.9909.0309.680 L kernel 12345678910 1.8403.1504.1604.3905.5706.5706.6906.7206.9107.550 M kernel Dirichlet Neumann

32 M M S S V V MSVLAB, HRE, NTOU 32 Dirichlet B.C. (fixed-fixed)-True L SVD updating terms U 0246810 The wave number ( k ) 400 410 420 430 440 450 T h e d e t e r m e n t o f t h e i n f l u e n c e m a t r i c e f o r U k e r n e l T 6.280 (6.283) T 6.570 (6.572) T 7.110 (7.111) T 7.850 (7.845) T 8.720 (8.717) T 9.680 (9.682) Determinant of the influence matrix k ( ) Analytical solution

33 M M S S V V MSVLAB, HRE, NTOU 33 Neumann B.C. (free-free)-True SVD updating terms T M Determinant of the influence matrix k ( ) Analytical solution

34 M M S S V V MSVLAB, HRE, NTOU 34 Singular formulation -Spurious SVD updating document U T Determinant of the influence matrix k ( ) Analytical solution

35 M M S S V V MSVLAB, HRE, NTOU 35 Hypersingular formulation -Spurious SVD updating document L M Determinant of the influence matrix k ( ) Analytical solution

36 M M S S V V MSVLAB, HRE, NTOU 36 Outline Motivation Derivation of the formulation Null-field integral equation Trefftz method Addition theorem Multipole expansion Linear algebraic equation Numerical examples Eigenproblems by using the BIEM Eigenproblems by using the multipole Trefftz method Conclusions

37 M M S S V V MSVLAB, HRE, NTOU 37 Problem statement

38 M M S S V V MSVLAB, HRE, NTOU 38 Multipole Trefftz method

39 M M S S V V MSVLAB, HRE, NTOU 39 Multipole Trefftz method

40 M M S S V V MSVLAB, HRE, NTOU 40 Eigenvalues of an eccentric case Multipole Trefftz methodBEM [Chen et al. 2001] 1.741.75 2.132.14 2.462.47 2.772.78 2.962.98

41 M M S S V V MSVLAB, HRE, NTOU 41 Eigenmodes of an eccentric case Mode 1Mode 2Mode 3Mode 4

42 M M S S V V MSVLAB, HRE, NTOU 42 Eigenvalues of a concentric case Multipole Trefftz method BEM [Chen et al. 2001] Analytical solution 2.052.062.04884 2.222.232.22375 2.222.232.22375 2.662.672.65993 2.662.672.65993

43 M M S S V V MSVLAB, HRE, NTOU 43 Eigenmodes of a concentric case Mode 1Mode 2Mode 3Mode 4

44 M M S S V V MSVLAB, HRE, NTOU 44 Eigenmodes of an multiply-connected case Multipole Trefftz methodBEM [Chen et al. 2004] 4.4994.47 5.3695.37 5.3695.37 5.5495.54 5.9495.95

45 M M S S V V MSVLAB, HRE, NTOU 45 Eigenmodes of an multiply-connected case Mode 1Mode 2Mode 3Mode 4

46 M M S S V V MSVLAB, HRE, NTOU 46 Outline Motivation Derivation of the formulation Null-field integral equation Trefftz method Addition theorem Multipole expansion Linear algebraic equation Numerical examples Eigenproblems by using the BIEM Eigenproblems by using the multipole Trefftz method Conclusions

47 M M S S V V MSVLAB, HRE, NTOU 47 Conclusions (BIEM) There are still spurious eigenvalues by using BIEM to deal with concentric sphere problems. True eigenvalues are dependent on problems and spurious eigenvalues are dependent on methods. U

48 M M S S V V MSVLAB, HRE, NTOU 48 Conclusions (BIEM) Spurious eigenvalues are dependent on the inner boundary.

49 M M S S V V MSVLAB, HRE, NTOU 49 Conclusions (multipole Trefftz method) We extend the conventional Trefftz method to the multipole Trefftz method by introducing the multipole expansion (addition theorem).

50 M M S S V V MSVLAB, HRE, NTOU 50 Conclusions (multipole Trefftz method) The multipole Trefftz method has successively provided an analytical model for solving eigenvalues and eigenmodes of a circular membrane containing multiple circular holes. The eigenvalues can be found by employing the multipole Trefftz method free of pollution of spurious eigenvalues. It’s found that a small number of terms is sufficient for extraction of eigenvalues

51 M M S S V V MSVLAB, HRE, NTOU 51 ~Thanks for your kind attentions~ Welcome to the web site of MSVLAB: http://ind.ntou.edu.tw/~msvlab

52 M M S S V V MSVLAB, HRE, NTOU 52 The comparison of the null-field BIEM and the multipole Trefftz method Null-field BIEMMulti-pole Trefftz method No. of terms (O)No. of terms (X) Analytical (O) (X) Approach the boundary Membrane & Plate k=5.37 k=5.369

53 M M S S V V MSVLAB, HRE, NTOU 53 Mode 1 of eccentric cases e=1 k=1.49 e=1.4 k=1.36 e=1.495 k=1.35 e=1.499999 k=1.34 Multipole Trefftz method

54 M M S S V V MSVLAB, HRE, NTOU 54 SVD updating term (Dirichlet)

55 M M S S V V MSVLAB, HRE, NTOU 55 SVD updating document (UT)

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