View
231
Download
0
Embed Size (px)
Citation preview
Spurious Eigensolutions and Fictitious Frequencies for ASpurious Eigensolutions and Fictitious Frequencies for Acoustic Problems with the Mixed-type Boundary coustic Problems with the Mixed-type Boundary
Conditions by using BEMConditions by using BEM
邊界元素法於混合型邊界條件問題之邊界元素法於混合型邊界條件問題之假根及虛擬頻率探討假根及虛擬頻率探討
國立台灣海洋大學河工二館 307 教室中華民國九十二年七月十五日
研 究 生: 林宗衛指導老師: 陳正宗教授
海洋大學力學聲響振動實驗室
OutlinesOutlines
Motivation
Boundary integral equations
Examples
Conclusions
Semi-analytical approach
Further research
02
Techniques and treatments
海洋大學力學聲響振動實驗室
OutlinesOutlines
Motivation
Boundary integral equations
Examples
Conclusions
Semi-analytical approach
Further research
03
Techniques and treatments
海洋大學力學聲響振動實驗室
MotivationMotivation
04
Dirichlet Neumann Mixed-type
spurious
Eigenvalue
fictitious
frequency
spurious
Eigenvalue
fictitious
frequency
spurious
Eigenvalue
fictitious
frequency
Complex-valued
BEM N Y N Y ? ?Real-part
BEM Y - Y - ? ?Imaginary-part
BEM Y - Y - ? ?
MRM Y - Y - ? ?
海洋大學力學聲響振動實驗室
OutlinesOutlines
05
Motivation
Boundary integral equations
Examples
Conclusions
Semi-analytical approach
Further research
05
Techniques and treatments
海洋大學力學聲響振動實驗室
Governing equationGoverning equation
Duk x ,0x 22
where
D : the domain of interest
k : the wave number
x : the domain point
u(x): the acoustic potential 06
海洋大學力學聲響振動實驗室
Types of boundary conditionsTypes of boundary conditions
Mixed-type
u=u
Neumann Dirichlet
t= =un
u
tt
uu
07
海洋大學力學聲響振動實驗室
The null-field integral formulationsThe null-field integral formulations
c
BB
c
BB
D),(dB)(t),(L)(dB)(u),(M0
D),(dB)(t),(U)(dB)(u),(T0
x ss xsss xs
x ss xsss xs
D Dc
8
DDc
海洋大學力學聲響振動實驗室
s
xs
n
),(U
x
xs
n
),(U
sx
xs
nn
),(U2
Dc: the complementary domain of D,x,s: the field and source point,u(s): the potential on the boundaryt(s): the normal derivative of potential on the boundaryU (s,x): the kernel function,
T(s,x) M(s,x)
L (s,x)9
海洋大學力學聲響振動實驗室
UU ( (ss,,xx)) kernel in different methods kernel in different methods
2
1
0krHi
U
x,s
2
0krY
U
xs,
2
0krJ
U
xs,
krJk
krYU00
]2
[ln2
)sx,(
Complex-valued BEM:
Real-part BEM:
Imaginary-part BEM:
MRM:
10
海洋大學力學聲響振動實驗室
Discretization (singular formulatioDiscretization (singular formulation)n)
11 NNNNNN uTtU
11
N
NNRL
N
NNRL u
uTT
t
tUU
ut
11
海洋大學力學聲響振動實驗室
Rearrangement of the Rearrangement of the influences matricesinfluences matrices
11
N
NNRL
N
NNRLt
uUT
u
tTU
1NNN1NNN qBpA
12
海洋大學力學聲響振動實驗室
DiscretizationDiscretization(hypersingular formulation)(hypersingular formulation)
11 NNNNNN uMtL
11 NNNNNN qDpC
13
海洋大學力學聲響振動實驗室
OutlinesOutlines
14
Motivation
Boundary integral equations
Examples
Conclusions
Semi-analytical approach
Further research
Techniques and treatments
海洋大學力學聲響振動實驗室
0
)(
)(
t
t
kC
kA
0)( pkC
0)( pkA
kt: the true eigenvalues
15
Extraction of the true Extraction of the true eigensolutions by using SVD updatieigensolutions by using SVD updating termsng terms
0q
海洋大學力學聲響振動實驗室
Detection of the spurious eigensolDetection of the spurious eigensolutions by using utions by using SVD updating documentsSVD updating documents
by using the Fredhohm’s alternative theorem
16
0)(
)(
T
s
T
s
kB
kA
0)(
0)(
T
T
kB
kA
T: transposeks: the spurious eigenvalues
海洋大學力學聲響振動實驗室
Burton & Miller methodBurton & Miller method
[ikU(s, x)+L(s, x)]u(s) = [ikT(s, x)+M(s, x)]t(s)
1NNN1NNN uMtL
1NNN1NNN uTtU
+)
ik
17
海洋大學力學聲響振動實驗室
CHIEF methodCHIEF method
11 NNNNNN uTtU
1N
N)aN(c
Na
BNN
1N
N)aN(c
Na
BNN u
T
Tt
U
U
18
海洋大學力學聲響振動實驗室
OutlinesOutlines
19
Motivation
Boundary integral equations
Examples
Conclusions
Semi-analytical approach
Further research
Techniques and treatments
海洋大學力學聲響振動實驗室
Interior problemsInterior problems
20
海洋大學力學聲響振動實驗室
A finite rodA finite rod
s0=0x x
0 1
s1=1
Type I : u(0)=0; t(1)=0Type II: t(0)=0; u(1)=0
21
B.C.:
海洋大學力學聲響振動實驗室
Comparison of the eigenfuctionsComparison of the eigenfuctions
Interior problem
Dirichlet
u(0)=0, u(1)=0
Neumann
t(0)=0, t(1)=0
Mixed-type
(u(0)=0, t(1)=0)
True Spurious True Spurious True Spurious
Complex –valued BEM
UT Sink - Sink - Cosk -
LM Sink - Sink - Cosk -
Real-part
BEM
UT Sink Sink Sink Sink Cosk Sink
LM Sink Sink Sink Sink Cosk Sink
Imaginary-part BEM
UT Sink Sink Sink Sink Cosk Sink
LM Sink Sink Sink Sink Cosk Sink
22
海洋大學力學聲響振動實驗室 23
Real-part BEM
Imaginary-part BEM
Complex-valued BEM
0 4 8 12
k
1E-006
1E-005
0.0001
0.001
0.01
0.1
1
S ingular form ula tion
H ypersingu lar form ulationS ingular form ula tion
H ypersingu lar fo rm ulation
0 4 8 12
k
1E-007
1E-006
1E-005
0.0001
0.001
0.01
0.1
1
Singular form ula tion
H ypersingular form ulation
0 4 8 12
k
1E-008
1E-007
1E-006
1E-005
0.0001
0.001
0.01
0.1
1
T
T
T
TTT T
T T
T
T
T
S
S
SSS S
海洋大學力學聲響振動實驗室
A circular cavityA circular cavity
u2=0 t1=0R=1 m
24
Elements: N=30
海洋大學力學聲響振動實驗室
Detection the eigenvalues using thDetection the eigenvalues using the complex-valued BEMe complex-valued BEM
0 1 2 3 4 5
k
0.0001
0.001
0.01
0.1
1
0 1 2 3 4 5
k
0.01
0.1
1
10
C
A
0 1 2 3 4 5
k
0.0001
0.001
0.01
0.1
1
0 1 2 3 4 5
k
0.01
0.1
1
10
B
D
25
海洋大學力學聲響振動實驗室
True
Spurious
kYn
of Zeros kYn
of Zeros
0 1 2 3 4 5
k
1E-005
0.0001
0.001
0.01
0.1
1
NNA
0 1 2 3 4 5
k
0.01
0.1
1
10
NN2C
A
Updating term
0 1 2 3 4 5
k
0.0001
0.001
0.01
0.1
1
10
N2NBA
Updating document
Detection of true and spurious Detection of true and spurious eigenvalues using eigenvalues using the real-part BEMthe real-part BEM
26
海洋大學力學聲響振動實驗室
Detection of true and spurious Detection of true and spurious eigenvalues using eigenvalues using the real-part BEMthe real-part BEM
0 1 2 3 4 5
k
0.01
0.1
1
10
NN2C
A
0 1 2 3 4 5
k
0 . 1
1
1 0
N2NDC
0 1 2 3 4 5
k
0.0001
0.001
0.01
0.1
1
10
NNC
True
Spurious
kYn
' of Zeros kYn
' of Zeros Updating term
Updating document
27
海洋大學力學聲響振動實驗室
The comparison of the spurious The comparison of the spurious eigenvalues with different B.C.eigenvalues with different B.C.
0 1 2 3 4 5
k
0.0001
0.001
0.01
0.1
1
10
0 1 2 3 4 5
k
0 . 1
1
1 0
0 1 2 3 4 5
k
0.001
0.01
0.1
1
10
0 1 2 3 4 5
k
0.1
1
10
kYn
of Zeros
kYn
' of Zeros
Neumann or Dirichlet Mixed-typed
28
海洋大學力學聲響振動實驗室
The spurious eigenvalues using diThe spurious eigenvalues using different methodsfferent methods
Complex-valued
BEM
Real-part
BEM
Imaginary-part
BEMMRM
Singular
formulation -
Hyperingular
formulation -'
nY'
nY '
nJ
nYnJ
nY
.5772.0krJ]2
k[ln
2krYY nnn
and where
29
海洋大學力學聲響振動實驗室
The comparison for the former fivThe comparison for the former five eigenmodese eigenmodes
Real –part BEM
FEM
30
海洋大學力學聲響振動實驗室
The comparison for the former five The comparison for the former five eigenvalueseigenvalues
k1 k2 k3 k4 k5
Real –part BEM 1.222 2.544 2.954 3.802 4.231
FEM(ABAQUS)
1.254 2.593 2.934 3.842 4.194
31
海洋大學力學聲響振動實驗室
Exterior problemsExterior problems
32
海洋大學力學聲響振動實驗室
A semi-infinite rodA semi-infinite rod
s0=1x
0 1
s1=
33
B.C.: mt(1)+u(1)=n
海洋大學力學聲響振動實驗室
Comparison of Comparison of the fictitious frequenciesthe fictitious frequencies
Interior problem
Dirichlet
u(1)=0
Neumann
t (1)=0
Mixed-type
mt(1)+u(1)=n
Fictitious eigenfunctions
T(s,x0)=0UT cosk cosk cosk
LM sink sink sink
U(s,x0)=0UT sink sink sink
LM cosk cosk cosk
34
海洋大學力學聲響振動實驗室T(s,x)=0
0 4 8 12
k
-8000
-4000
0
4000
Pot
entia
l
Singular form ulation
H ypersingular form ulation
Exect so lu tion: u(k)=1
0 4 8 12
k
-8000
-4000
0
4000
Pot
entia
lExect so lu tion: u(k)=1
Singular form ulation
H ypersingular form ulation
U(s,x)=035
The fictitious frequency of 1-D The fictitious frequency of 1-D problemproblem
海洋大學力學聲響振動實驗室
A circular radiatorA circular radiator
θ4)k(H
)kR(H)(u
)1(4
)1(4
2 cos
θ4)R
4-
)(kρH
(kR)H(k)(t
)1(4
)1(5
1 cos
36
R=1 m
Elements: N=30
海洋大學力學聲響振動實驗室0 2 4 6 8
k
- 2
0
2
4
u(1,
0)
Singular form ulation
Hypersingular form ulation
Analytica l solution: u(k)=1
J'12
J41J13J02
J'51J'31J'21
CHIEF m ethod (X)
Burton & M iller m ethod
Analytical solution: u(k)=1
0 2 4 6 8
k
- 2
0
2
4
u(1,
0)
The positions of irregular for The positions of irregular for u(1,0) u(1,0) and their treatmentsand their treatments
x
37
海洋大學力學聲響振動實驗室0 2 4 6 8
k
- 2
0
2
4
t(1,
)
Singular form ulation
Hypersingular form ulation
Analytica l solution: t(k)=1
J'12
J'51
J'31
J'21
The positions of irregular for The positions of irregular for t(1,t(1,) ) and their treatmentsand their treatments
CHIEF m ethod (X)
Burton & M iller m ethod
Analytical solution: t(k)=1
0 2 4 6 8
k
- 2
0
2
4
t(1,
)x
38
海洋大學力學聲響振動實驗室
Comparison of the BEM and exact Comparison of the BEM and exact solutions for radiation problemsolutions for radiation problem
BEM solution Analytical solution
39
海洋大學力學聲響振動實驗室
OutlinesOutlines
40
Motivation
Boundary integral equations
Examples
Conclusions
Semi-analytical approach
Further research
Techniques and treatments
海洋大學力學聲響振動實驗室
The null-field integral formulationsThe null-field integral formulations
c
BB
c
BB
D),(dB)(t),(L)(dB)(u),(M0
D),(dB)(t),(U)(dB)(u),(T0
xss xsss xs
xss xsss xs
41
海洋大學力學聲響振動實驗室
Semi-analytical approachSemi-analytical approach
42
12n42n42n4 vW0
12n4H
海洋大學力學聲響振動實驗室 43
0 1 2 3 4 5
k
1E-024
1E-023
1E-022
1E-021
1E-020
1E-019
1E-018
1E-017
1E-016
1E-015
1E-014
1E-013
1E-012
1E-011
1E-010
1E-009
1E-008
1E-007
1E-006
1E-005
0.0001
0.001
0.01
n=5m =20
S
T
T TT
T
0 1 2 3 4 5
k
1E-016
1E-015
1E-014
1E-013
1E-012
1E-011
1E-010
1E-009
1E-008
1E-007
1E-006
1E-005
0.0001
0.001
0.01
0.1
1
n=5m =20
TT T
T T
Real-part BEM
Imaginary-part BEM
Complex-valued BEM
Interior problemInterior problem
0 1 2 3 4 5
k
100000
1000000
10000000
100000000
1000000000
10000000000
n=5m =20
S
SST
TT
T TS
海洋大學力學聲響振動實驗室 44
J01
J12
J21
J02J31
J13J41
0 2 4 6 8
k
0.001
0.01
0.1
1
n=5m =20
n=5m =20
0 2 4 6 8
k
1E-012
1E-011
1E-010
1E-009
1E-008
1E-007
1E-006
1E-005
0.0001
0.001
0.01
0.1
1
10
J '11 J '21 J'02
J'31J'41 J'51
J'22J'03
J'61
Exterior problemExterior problem
Singular Formulation
Hypersingular Formulation
海洋大學力學聲響振動實驗室
OutlinesOutlines
45
Motivation
Boundary integral equations
Examples
Conclusions
Semi-analytical approach
Further research
Techniques and treatments
海洋大學力學聲響振動實驗室
ConclusionsConclusions
The results in the numerical experiment match well with those in our semi-analytical results.
The eigensolutions and fictitious frequencies
were solved successfully by using semi-analytical approach and BEMs
46
海洋大學力學聲響振動實驗室
ConclusionsConclusions
The spurious eigenvalues and fictitious frequencies depend on the representation no matter what the given types of B.C. for the problem are specified.
47
海洋大學力學聲響振動實驗室
OutlinesOutlines
48
Motivation
Boundary integral equations
Examples
Conclusions
Semi-analytical approach
Further research
Techniques and treatments
海洋大學力學聲響振動實驗室
Further researchFurther research
49
It will be interesting to know that if spurious eigenvalues and fictitious frequencies locate at the zeros of the Mathieu functions in the shape of ellipse.
The quantity of the fictitious frequencies depend on the modal participation factor.
海洋大學力學聲響振動實驗室
Further researchFurther research
50
To derive the fictitious frequencies analytically by using the complex-valued BEM for mixed-type boundary condition problem.
The extension to multiple radiators & scatters and half-plane problems using the similar algorithm can be conducted to examine the occurrence of the fictitious frequencies.
海洋大學力學聲響振動實驗室
Further researchFurther research
51
The extension to 3-D problems deserves further study to reconfirm the conclusion proposed in the thesis.
海洋大學力學聲響振動實驗室
報告完畢
敬請指教52