1. National Taiwan Ocean University
MSVLAB
Department of Harbor and River Engineering
Dual BEM since 1986 with emphasis on vibrations and acoustics J T Chen (陳正宗特聘教授)
Department of Harbor and River Engineering
National Taiwan Ocean University, Keelung, Taiwan
Jan. 08, 15:10-15:40, 2007 台日聲學研討會
(Tai-Jap-aco2007.ppt)

2. Outlines Overview of BEM and dual BEM Mathematical formulation
Hypersingular BIE
Nonuniqueness and its treatments
Degenerate scale
True and spurious eigensolution (interior prob.)
Fictitious frequency (exterior acoustics)
Conclusions

3. Top ten countries of BEM and dual BEM
USA (3423) , China (1288) , UK, Japan, Germany, France, Taiwan (551), Canada, South Korea, Italy (No.7)
Dual BEM (Made in Taiwan)
UK (119), USA (90), Taiwan (69), China (46), Germany, France, Japan, Australia, Brazil, Slovenia (No.3)
(ISI information Nov.06, 2006)
台灣加油 FEM Taiwan (No.9/1311)

4. Top ten countries of FEM, FDM and Meshless methods
USA, China, Japan, France, Germany, England, South Korea, Canada, Taiwan, Italy
Meshless methods
USA, China, Singapore, Japan, Spain, Germany, Slovakia, England, France, Taiwan
FDM
USA, Japan, China, England, France, Germany, Canada, Taiwan, South Korea, Italy
(ISI information Nov.06, 2006)

5. Top three scholars on BEM and dual BEM
Aliabadi M H (UK, Queen Mary College)
Mukherjee S (USA, Cornell Univ.)
Chen J T (Taiwan, Ocean Univ.) 56 篇
Tanaka M (Japan, Shinshu Univ.)
Dual BEM (Made in Taiwan)
Aliabadi M H (UK, Queen Mary Univ. London)
Chen J T (Taiwan, Ocean Univ.) 43 篇
Power H (UK, Univ Nottingham)
(ISI information Nov.06, 2006)
NTOU/MSV 加油

6. Top 25 scholars on BEM/BIEM since 2001
NTOU/MSV
Taiwan
北京清華姚振漢教授提供(Nov., 2006)

7. Overview of numerical methods
Domain
Boundary
MFS
Trefftz method
MLS, EFG DE
PDE-
variational IE 7

8. Number of Papers of FEM, BEM and FDM6 2 1
(Data form Prof. Cheng A. H. D.)

9. Growth of BEM/BIEM papers
(data from Prof. Cheng A.H.D.)

10. Advantages of BEM Disadvantages of BEM 北京清華
Discretization dimension reduction
Infinite domain (half plane)
Interaction problem
Local concentration
Disadvantages of BEM
北京清華
Integral equations with singularity
Full matrix (nonsymmetric)

11. BEM and meshless methods can be seen as a supplement of FEM.
BEM and FEM
BEM and meshless methods can be seen as a supplement of FEM.
(2) BEM utilizes the discretization concept of FEM as well as the limitation. Whether the supplement is needed or not depends on its absolutely superior area than FEM.
Ｃrack & large scale problems Ó
NTUCE

12. What Is Boundary Element Method ?
Finite element method Boundary element method
What Is Boundary Element Method ? 5 4 1 2 6 3 1 2
geometry node
the Nth constant
or linear element N Ó
NTUCE

13. Artifical boundary introduced ! Dual integral equations needed !
Dual BEM
Why hypersingular BIE is required
(potential theory)
Degenerate
boundary 7 4 7 4 6 5 6 5 8 3 8 3 10 9 1 2 1 2
Artifical boundary introduced !
BEM
Dual integral equations needed !
Dual BEM Ó
NTUCE

14. Some researchers on Dual BEM
Chen(1986) 412 citings in total
Hong and Chen (1988） 71 citings ASCE
Portela and Aliabadi (1992) 188 citings IJNME
Mi and Aliabadi (1994)
Wen and Aliabadi (1995)
Chen and Chen (1995) 新竹清華
黎在良等---斷裂力學邊界數值方法(1996)
周慎杰(1999)
Chen and Hong (1999) 76 citings ASME
Niu and Wang (2001)
Yu D H, Zhu J L, Chen Y Z, Tan R J …
cite Ó
NTUCE

15. 1969 2006 1986 Dual Integral Equations by Hong and Chen(1984-1986)
Singular integral equation
Hypersingular integral equation
Cauchy principal value
Hadamard principal value
Boundary element method
Dual boundary element method
1969
2006
1986
normal
boundary
degenerate
boundary Ó
NTUCE

16. Degenerate boundary geometry node the Nth constant or linear element 7
(-1,0.5)
(1,0.5) 7 4
the Nth constant
or linear element N 6 5 8 3
(0,0) 1 2
(-1,-0.5)
(1,-0.5)
5(+) 6(+)
5(+) 6(-)
5(+)
6(+)
5(+)
6(+)
5(+) 6(+)
5(+) 6(-)
5(+)
6(-)
5(+)
6(-)

17. How to get additional constraints
The constraint equation is not enough to determine coefficients of p and q,
Another constraint equation is required

18. Original data from Prof. Liu Y J
BEM
Cauchy
kernel
singular
DBEM
Hadamard
kernel
hypersingular
crack
1888
Integral equation
(1984)
(2000)
FMM
Large scale
Degenerate kernel
Original data from Prof. Liu Y J
Desktop computer fauilure

19. Fundamental solution Field response due to source (space)
Green’s function
Casual effect (time)
K(x,s;t,τ)

20. Green’s function, influence line and moment diagram
Force
Force s x s
s=1/2
x=1/4
G(x,s)
G(x,s) x s
Moment diagram
s:fixed
x:observer
Influence line
s:moving
x:observer(instrument)

21. Two systems u and U U(x,s) Domain(D) u(x) s source Boundary (B)
Infinite domain

22. Dual integral equations for a domain point (Green’s third identity for two systems, u and U)
Primary field
Secondary field
where U(s,x)=ln(r) is the fundamental solution.

23. Dual integral equations for a boundary point (x push to boundary)
Singular integral equation
Hypersingular integral equation
where U(s,x) is the fundamental solution.

24. Potential theory Single layer potential (U) Double layer potential (T)
Normal derivative of single layer potential (L)
Normal derivative of double layer potential (M)

25. Physical examples for potentials
Force
Moment
U:moment diagram
T:moment diagram
L:shear diagram
M:shear diagram

26. Order of pseudo-differential operators
Single layer potential (U) (-1)
Double layer potential (T) --- (0)
Normal derivative of single layer potential (L) --- (0)
Normal derivative of double layer potential (M) --- (1)
Pseudo differential operator
Real differential operator

27. Calderon projector

28. How engineers avoid singularity
BEM / BIEM
Improper integral
Singularity & hypersingularity
Regularity
Fictitious BEM
Bump contour
Limit process
Fictitious boundary
Achenbach et al. (1988)
Null-field approach
Guiggiani (1995)
Gray and Manne (1993)
Collocation point
CPV and HPV
Ill-posed
Waterman (1965)

29. Definitions of R.P.V., C.P.V. and H.P.V. using bump approach
R.P.V. (Riemann principal value)
C.P.V.(Cauchy principal value)
H.P.V.(Hadamard principal value)
Definitions of R.P.V., C.P.V. and H.P.V. using bump approach Ó
NTUCE

30. Principal value in who’s sense
Common sense
Riemann sense
Lebesgue sense
Cauchy sense
Hadamard sense (elasticity)
Mangler sense (aerodynamics)
Liggett and Liu’s sense
The singularity that occur when the base point and field point coincide are not integrable. (1983)

31. Two approaches to understand HPV
Differential first and then trace operator
(Limit and integral operator can not be commuted)
Trace first and then differential operator
(Leibnitz rule should be considered)

32. Bump contribution (2-D)s T s x x L s M s x x

33. Bump contribution (3-D)0` s s x x s s x x

34. Successful experiences since 1986

35. Solid rocket motor (Army工蜂火箭)

36. X-ray detection (三溫暖測試)
Crack initiation
crack growth
Stress reliever

37. FEM simulation

38. Stress analysis

39. BEM simulation

40. Shong-Fon II missile (Navy)

41. IDF (Air Force)

42. Flow field

43. V-band structure (Tien-Gen missile)

44. FEM simulation

46. Seepage flow

47. FEM (iteration No.49) BEM(iteration No.13)
Free surface seepage flow
using hypersingular formulation
FEM (iteration No.49) BEM(iteration No.13)
Initial guess
Initial guess
After iteration
After iteration
Remesh area
Remesh line

48. Meshes of FEM and BEM

49. Incomplete partition in room acousticsb a e c
t=0

50. Water wave problem with breakwater
Free water surface S x
Top view O y z
breakwater
oblique incident water wave

51. Reflection and Transmission

52. Cracked torsion bar

53. IEEE J MEMS

56. Radiation and scattering problems
Nonuniform radiaton scattering 2

57. Adaptive Mesh
BEM
FEM
DtN interface 5

58. Strategy of adaptive BEM
Mesh
Singular
Equation
Hypersingular
Equation
Error
estimator
Error
estimator
Solution
u,t
u,t 21

59. Nonuniform radiation： Dirichlet problem
Numerical solution: BEM
Numerical solution: FEM
64 ELEMENTS
2791 ELEMENTS
Nonuniform radiation： Dirichlet problem 9

60. Is it possible !
No hypersingularity !
No subdomain !

61. Degenerate boundary problems
Multi-domain BEM
u=0
r=1
interface
Subdomain 1
Subdomain 2
Dual BEM

62. Conventional BEM in conjunction with SVD
Singular Value Decomposition
Rank deficiency originates from two sources:
(1). Degenerate boundary
(2). Nontrivial eigensolution
Nd=5
Nd=4
Nd=5

63. UT BEM + SVD (Present method) Sub domain Dual BEM versus k
Determinant versus k
Dual BEM
Determinant versus k

64. k=3.14
k=3.82
k=4.48
UT BEM+SVD
k=3.09
k=3.84
k=4.50
FEM (ABAQUS)

65. BEM trap ? Why engineers should learn mathematics ?
Well-posed ?
Existence ?
Unique ?
Mathematics versus Computation
equivalent ?
Some examples

66. Numerical and physical resonance
Numerical resonance
radiation
incident
wave

67. Numerical phenomena (Fictitious frequency)
t(a,0)
A story of NTU Ph.D. students

68. Numerical phenomena (Spurious eigensolution)

69. Treatments SVD updating term Burton & Miller method CHIEF method
Mathematical analysis and numerical study for free vibration of plate using BEM-69

70. Conclusions Introduction of dual BEM
The role of hypersingular BIE was examined
Successful experiences in the engineering applications using BEM were demonstrated
The traps of BIEM and BEM were shown

71. Thanks for your kind attention
The End
Thanks for your kind attention

72. E-mail: jtchen@mail.ntou.edu.tw
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