1. Jia-Wei Lee, Chi-Feng Nien & Jeng-Tzong Chen
 Symposium of the International Association for Boundary Element Methods 2018
Combination of the CHIEF and the self-regularization technique for solving 2D exterior Helmholtz equations with fictitious frequencies in the indirect BEM and MFS
Jia-Wei Lee, Chi-Feng Nien & Jeng-Tzong Chen
Date: June 26, 2018
Time: 09:35~10:00
Place: University Paris 6
2018/06/28

2. Outline Introduction Problem statement Present approach
Numerical results
Conclusions
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3. Outline Introduction Problem statement Present approach
Numerical results
Conclusions
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4. Engineering problems Platform (Offshore structure) Exterior problem
Ringing, Rodney Eatock Taylor
BHP Billiton's giant Atlantis oilfield in the deepwater Gulf of Mexico
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5. Motivation Exterior problem FEM BEM Artificial boundary condition
Only boundary element
DtN boundary condition
Nonreflection boundary condition
(NCTS, 2013, Summer Course)
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6. Unique solution->Paradise No solution->Parasite
well-posed
ill-posed
Unique solution->Paradise
No solution->Parasite
Infinite domain
Paradise
Model
Parasite
Black-box
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7. Physical and numerical resonance
Physical resonance
Fictitious frequency
(BEM/BIEM)
t(a,0)
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8. Rank-deficency problems
Influence matrix
Spurious eigenvalue (Interior)
Fictitious frequency (Exterior)
t(a,0)
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9. Alleviation for rank-deficency problems
Spurious eigenvalue (Interior)
Fictitious frequency (Exterior)
t(a,0)
(1) CHIEF method (Schenck, JASA , 1968)
Additional constraint (CHIEF point)
(2) Burton and Miller method (Burton and Miller, PRS , 1971)
(3) SVD updating term technique (Chen et al., JSV, 2002)
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10. Mixed potential method
Four regularization techniques
Direct BEM
indirect BEM
(MFS)
Burton & Miller method or
Mixed potential method
CHIEF method
Present approach No
null-field
equation
Null-field equation
Self-regularization method +
CHIEF
Extra point
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11. Outline Introduction Problem statement Present approach
Numerical results
Conclusions
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12. Problem statement Exact solution: Domain Governing equation:
Boundary condition:
Exact solution:
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13. Indirect BEM Indirect BIE Domain Fundamental solution Boundary
Matching the boundary condition
Discretization
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14. Spectrum of minimum singular value
single root
double root
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15. Rank deficiency Domain base Range base Operator How to represent for
Domain base
Range base
Operator
How to represent
for of ??
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16. Outline Introduction Problem statement Present approach
Numerical results
Conclusions
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17. Self-regularization technique
Fichera’s method for the problem of degenerate scale
constraint equation
When the size is at the degenerate scale, the Fichera’s method can obtain the unique solution
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18. Self-regularization technique
SVD
has no
component If
Self-regularization technique by using the SVD
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19. Spectrum of U by using the self-regularization method
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20. A new point of view of CHIEF
Exterior problem governed by the Helmholtz equation
Direct BEM
Fictitious frequency
Null-field equation is
a CHIEF equation
Relation between CHIEF and self-reg. method
Risk of failure points
Domain
Indirect BEM
Fictitious frequency
No null-field equation
How to find CHIEF equation ?
Constraint equation ?
Failure points ?
G. E.
B. Cs. or
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A self-regularization method

21. Self-regularization technique
Range deficiency
Discrete system
How to determine the field
Self-regularization
???
Boundary data
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22. Self-regularization+CHIEF idea
Range deficiency
Continuous system
Extra source point
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23. Analytical derivation of the present indirect BIE
Double root
Degenerate kernel
Fourier series
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24. Analytical derivation of the present indirect BIE
Double root
The failure positions are the same with the CHIEF method in the direct BEM
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25. Present idea for MFS Constraint equations Ordinary MFS: Present MFS:
MFS source point
Extra source point
Constraint equations
Single root
Double root
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26. Outline Introduction Problem statement Present approach
Numerical results
Conclusions
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27. Numerical examples
Case 1
Case 2
Case 3
Case 4
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28. Bar chart of unitary vectors
CHIEF
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29. Contour plot
Case 1
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30. Contour plot
Case 1
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31. Contour plot
Case 2
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32. Contour plot
Case 2
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33. Bar chart of unitary vectors
CHIEF
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34. Contour plot
Case 3
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35. Contour plot
Case 4
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36. Contour plot
Case 4
Failure points
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2017/07/11

37. Numerical examples Case 4 Exact solution Ordinary MFS
The present MFS with failure points
The present MFS with suitable points
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38. Outline Introduction Problem statement Present approach
Numerical results
Conclusions
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39. Conclusions
The bordered matrix obtained by the self-regularization technique is invertible for all value of wavenumber.
The problem of fictitious frequencies can be alleviated by adding extra fundamental solutions in the direct BEM and MFS.
The present approach can fill in the gap that there is no CHIEF method in the indirect BEM and MFS.
The criterion of failure points of the present idea is the same with the CHIEF method.
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40. The end
Thanks for your kind attentions
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41. Welcome to visit the web site of MSVLAB/NTOU
The end
Thanks for your kind attentions
勞請各位老師與前輩多多指教
Welcome to visit the web site of MSVLAB/NTOU
2018/06/28

42. Contour plot
Case 3
Failure point
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2017/07/11

43. 目前需要克服的問題 COLY(NELM+1)=0.9*DCOS(Pi/(14.))
COLX(NELM+1)=0.9*DSIN(Pi/(14.))
COLX(NELM+2)=0.9
COLY(NELM+2)=0
2017/07/11
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44. Physical and numerical resonance
Physical resonance
Fictitious frequency
(BEM/BIEM)
Present
t(a,0)
2017/07/11
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45. Self-regularization+CHIEF method
怎麼進場 自救
無法表現
可表現
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46. Analytical derivation of 魯蛋版 indirect BIE
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