Boundary integral formulation and boundary element analysis for multiply-connected Helmholtz problems 地點: 河工二館會議室 時間: 2002. 05. 21. 口試生: 劉立偉 同學 指導教授: 陳正宗 教授 1

Outlines Motivation Methods of solution Singular & hypersingular BIEs (BEMs) Null-field BIEs (BEMs) Fictitious BIEs (BEMs) CHIEF method SVD updating techniques Numerical experiments Conclusions & further research 2

Motivation Kitahara (1985) Point out that spurious eigenvalues always appear in BEM Chang (1999) Numerical verification Lin (2001) 1. Discrete system 2. Burton & Miller method Liu (2002) 1. Continuous system 2. CHIEF method 3. SVD updating techniques 4. Truly multiply-connected problem 3

Singular and hypersingular BIEs (direct BIEs) Dual boundary integral equations Singular BIE (UT formulation) Hypersingular BIE (LM formulation) Degenerate kernel functions R<ρ ρ<R ρ R θ Aleph s =(R,) x=(,) 4

Eigenproblem of an annular domain s =(R,) x=(,) Eigenequation: True eigenequation Spurious eigenequation by using singular BIE Eigenequation: by using hypersingular BIE 5

Singular and hypersingular BEMs (direct BEMs) 1 r1=0.5 m r2=2.0 m k 6

Eigenequations for annular domains with different boundary conditions Spurious eigenvalues depend on formulations (UT or LM) True eigenvalues depend on boundary conditions (Dirichlet, Neumann or Robin ) 7

Null-field BIEs (direct BIEs) UT formulation Spurious eigenequation True eigenequation LM formulation  s =(R,) x=(,) In order to avoid the singularity, the null-field BIEs are used. The collocation points are located in the out of the domain instead of boundary. The results of null-field BIEs are different in spurious eigenequations. Spurious eigenequation True eigenequation 8

Null-field BEMs (direct BEMs) 1 1=0.45 m 2=2.2 m s s s k 9

Fictitious BIEs (indirect BIEs) single-layer potential approach double-layer potential approach s =(R,) x=(,) 10

Fictitious BEMs (indirect BEMs) r1=0.5 m r2=2.0 m    1=0.45 m 2=2.2 m s s s r’1=0.4 m r’2=2.2 m 11

CHIEF method CHEEF point (Chen et al. 2001 JASA)  CHIEF point (Chen et al. 2001 JSV) trivial (identity) 12

CHIEF or CHEEF valid ? CHIEF valid CHEEF invalid 1 s k 13

Two CHIEF points for spurious eigenvalues of multiplicity two 1 s k 14

CHIEF flowchart Filtering technique for spurious eigenvalue ks Failure Jn(k1)=0 sin n(1- 3)=0 Add a CHIEF point (r1 , 1) No ( exterior point) r1< r1 Yes (interior point) Failure Yes No Overcome the spurious eigenvalue of multiplicity one Add another interior point (r3 , 3) Jn(k3)=0 Success (overcome the spurious eigenvalues of multiplicity two) 15

SVD updating techniques: Direct BEMs SVD updating term for true eigenvalue True eigenvalues 1 s s s k 16

SVD updating techniques: Direct BEMs SVD updating documents for spurious eigenvalue Spurious eigenvalues 1 s s k 17

SVD updating techniques: Indirect BEMs SVD updating document for true eigenvalues True eigenvalues 1 k 18

SVD updating techniques: Indirect BEMs SVD updating term for spurious eigenvalues Spurious eigenvalues s s 19

Numerical experiments A circular domain with two unequal holes 1 [U] -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 - . 8 7 6 5 4 3 2 1 + BEM FEM mode 1 mode 2 mode 3 mode 4 mode 5 k=4.82 k=4.790 k=4.801 k=6.619 k=6.72 k=7.82 k=6.634 k=7.797 k1 k3 k2 s k4 k5 k 2 [U] k Each mode shape of BEM can be obtained by superimposing the other two mode shapes of FEM. The spurious eigenvalues which depends on the right inner boundary will appears first. 20

Spurious boundary mode F1 ei1q ei0q ei0q ei1q 21

Numerical experiments A circular domain with two equal holes -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 - . 8 7 6 5 4 3 2 1 + BEM FEM mode 1 mode 2 mode 3 mode 4 mode 5 k=4.50 k=6.37 k=7.16 k=4.453 k=4.512 k=6.267 k=6.269 k=6.930 Nagaya & Poltorak 1989 22

Spurious boundary modes F1 and F2 ei1q ei0q ei1q ei0q 23

Conclusions & further research Spurious eigenvalues depend on the inner boundaries, inner collocation points and inner fictitious boundaries by using the singular and hypersingular BIEs (BEMs), the null-field BIEs (BEMs) and fictitious BIEs (BEMs), respectively. The criterion of CHIEF method were discussed. SVD updating techniques were successful employed to deal with the spurious eigenvalues in the direct and indirect BEMs. SVD updating terms (true eigenvalues) SVD updating documents (spurious eigenvalues) Direct BEMs: SVD updating documents (true eigenvalues) SVD updating terms (spurious eigenvalues) Indirect BEMs: 24

Further research: The ill-posed problem arises from the singularity free approaches. Hence, it is deserves further study to overcome the ill-posed problem in depth. The extension to exterior acoustics for multiple radiators and scatters using the similar algorithm can be conducted to examine the occurrence of the fictitious frequency. Whether the spurious modes in the UT and LM formulations are the same or not needs further investigation. 25