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Transcript of 1 力學聲響振動研究室 (MSVLAB) Degenerate scale analysis for membrane and plate problems...
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1 力學聲響振動研究室 (MSVLAB)
Degenerate scale analysis for membrane and plate problems using the meshless method and
boundary element method
研究生:吳清森 指導教授:陳正宗 教授 陳義麟 博士國立台灣海洋大學河海工程學系 結構組
碩士班論文口試日期 : 2004/06/16 09:00-10:20
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2 力學聲響振動研究室 (MSVLAB)
Frame of the thesis
Chapter 1 Introduction
Free term and Jump term
Chapter 5 Free terms for plate problem
(Biharmonic problem)
Degenerate kernel
Chapter 2 Green’s function and Poisson integral formula
(Laplace problem)
Chapte3 BIEM and BEM for degenerate scale problem
(Laplace and biharmonic problem)
Chapter 4 Meshless method for degenerate scale problem
(Laplace and biharmonic problem)
Chapter 6 Conclusions and further research
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3 力學聲響振動研究室 (MSVLAB)
Literature review (Engineering background)
Engineers Year Problem Journal Treatment
He et al.(China)
1996 2-D potential and plane elasticity problem
Computer & Structures
CNMENecessary and sufficient boundary
integral formulation (NSBIE)
Zhou et al.
(China)
1999 2-D elasticity problem
CNME Boundary contour method(BCM)
He, W. J. (China)
2000 BIE---thin plate Computer & Structures
Equivalent BIE
Chen et al.
(Taiwan)
2002 Plane elasticity(Dual BIEM)
IJNME Hypersingular formulation
Chen et al.
(Taiwan)
2002 2-D Laplace equation(Degenerate kernel)
EABE Combined Helmholtz exterior integral equation formulation (CHEEF)
Chen et al.
(Taiwan)
2003 2-D Laplace and Navier problem
IJNME Addition of rigid body termHypersingular formulation
CHEEF concept
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4 力學聲響振動研究室 (MSVLAB)
Literature review (Mathematical background)
Mathematician Year Problem Journal Treatments and logarithm capacity
Soren Christiansen(Denmark)
1982 Detect non-unique solution
Applicable Analysis
Scaling methodRestriction method
Constanda(U. K.)
1995 Non-unique solution in plane elasticity problem
Quart. Appl. Math.
First kind integral equations
Martin et al.(France)
1996 Invertibility of single layer potential operator
Integr. Equat.
Oper. Th.
Logarithm capacity
Soren Christiansen(Denmark)
1998 Investigation of direct BIE for biharmonic
problem
JCAM Logarithm capacity
Soren Christiansen(Denmark)
2001 Detecting non-uniqueness of solution
through SVD
JCAM Logarithm capacity
23
21
10 ,,,
eeee
43
21
10 ,,,
eeee
21
10 ,,
eee
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5 力學聲響振動研究室 (MSVLAB)
Motivation
(1) BIEM, BEM
(2) MFS, Trefftz Method
Methods Techniques(1) Degenerate kernel
(2) Circulants
Membrane
(Laplace equation)
Plate
(biharmonic equation)
Statics
Degenerate scale problem
1-D case (Euler beam)
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6 力學聲響振動研究室 (MSVLAB)
S
x
rx (field point): variable
s (source point): fixed
Degenerate kernel
jjj
E
jjj
I
sxxBsAsxU
sxsBxAsxU
sxU),()(),(
),()(),(
),(
x
sO1
R
iU eUA
),(),(
Rsx
O2
x
sR
iUeU
B
O1
O2
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7 力學聲響振動研究室 (MSVLAB)
Alternative derivations for the Poisson integral formula
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8 力學聲響振動研究室 (MSVLAB)
G. E.: xxu ,0)(2
B. C. :
)(fu
a
Derivation of the Poisson integral formula
Traditional method
R 'R
Image source
Null-field integral equation method
Reciprocal radii method
Poisson integral formula
Image concept
Methods
Free of image concept
Searching the image point
Degenerate kernel
2
0 22
22)(
)cos(221),( df
aaa
u
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9 力學聲響振動研究室 (MSVLAB)
Null-field integral equation in conjunction with degenerate kernels
xsdBstxsUsdBsuxsTxuB
IF
B
IF ,)()(),()()(),()(2
a
Bx ),( ),( Rs
Bx
BcE
FB
EF xsdBstxsUsdBsuxsT ),()(),()()(),(0
1
0 ))sin()cos(()()(n
nn nbnaafsu Boundary densities: .))sin()cos(()(
10
n
nn nqnppst
Degenerate kernel
Unknown coefficients
dfma
dnbnaama
u
m
m
nnn
m
m
2
0 1
2
01
01
)()](cos[)(2121
))sin()cos(()](cos[)(2121
),(
c
unknown
specified
Fundamental solution
Green’s identity
,ln),( rsxU F
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10 力學聲響振動研究室 (MSVLAB)
Degenerate scale for plate analysis using the BIEM and BEM
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11 力學聲響振動研究室 (MSVLAB)
Engineering problem governed by biharmonic equation
1. Plane elasticity:
2. Slow viscous flow (Stokes’ Flow):
3. Solid mechanics (Plate problem):
functionstressAiry:,04
functionstream:,04
ntdisplacemelateraluu :,04
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12 力學聲響振動研究室 (MSVLAB)
Problem statement
uniform pressure
a
B
w=constant
0)(,0)( xxu
: flexure rigidity
: uniform distributed load
: domain of interest
)(xu
D
)(xw
Governing equation: xDxwxu ,)()(4
Boundary condition: Bxxxu ,0)(,0)(
Splitting method
Governing equation: xxu ,0)(*4
Boundary condition: Bxxxxuxu ),()(),()(****
: deflection of the circular plate
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13 力學聲響振動研究室 (MSVLAB)
Boundary integral equations for plate
),()}(),()(),()(),()(),({)(8 sdBsuxsVsxsMsmxssvxsUxuB
),()}(),()(),()(),()(),({)(8 sdBsuxsVsxsMsmxssvxsUxB
),()}(),()(),()(),()(),({)(8 sdBsuxsVsxsMsmxssvxsUxm mmB
mm
),()}(),()(),()(),()(),({)(8 sdBsuxsVsxsMsmxssvxsUxv vvB
vv
)(, xK
)(, xmK
)(, xvK
(2) Slope
(3) Normal moment
(4) Effective shear force
(1) Displacement )(, sK)(, smK )(, svK
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14 力學聲響振動研究室 (MSVLAB)
Operators
nK
)()(
2
22 )()1()()(
nK
m
))(()1()()(22
tntnK
v
Slope
Normal moment
Effective shear force
t
n
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15 力學聲響振動研究室 (MSVLAB)
Kernel functions
)(8),(4 sxsxUx Fundamental solution:
)ln(),( 2 rrsxU
Kernel functions:)),((),( , xsUKxs s
)),((),( , xsUKxsM sm
)),((),( , xsUKxsV sv
),()}(),()(),()(),()(),({)(8 sdBsuxsVsxsMsmxssvxsUxuB
)(, sK
)(, smK )(, svK
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16 力學聲響振動研究室 (MSVLAB)
Degenerate kernels for biharmonic operator
RmRmm
mRmm
RRRRRxsU
RmR
mmm
Rmm
RRxsU
m m
m
m m
mI
m m
m
m m
mE
xsU,)](cos[
)1(1
)](cos[)1(
1)cos()ln21(ln)ln1(),(
,)](cos[)1(
1)](cos[
)1(1
)cos()ln21(ln)ln1(),(
2 21
222
2 21
222
),(
1 2 2
11
2 11 1
22
)],(cos[1
1)](cos[
)1(2
)cos()ln21()ln1(2),(
,)](cos[)1(
2)](cos[
11
)cos()ln23()ln21(),(),( m m m
m
m
mI
m m
m
m m
mE
RmRm
mRmm
mRRRxsU
RmR
mmm
mR
mR
RxsU
xsU
RmRmm
mm
RmRRR
Rxs
RmR
mm
Rmm
mRxs
m m
m
m m
mI
m m m
m
m
mE
xs,)](cos[
)1(2
)](cos[1
1)cos()ln23()ln21(),(
,)](cos[1
1)](cos[
)1(2
)cos()ln21()ln1(2),(
2 11 1
22
1 2 2
11),(
RmRm
mm
Rmm
RR
xs
RmR
mm
mR
mmR
xs
m m
m
m m
mI
m m
m
m m
mE
xs,)]([cos
12
)]([cos12
)cos()ln23(2
),(
,)](cos[12
)]([cos12
)cos()ln23(2
),(
2 1
1
1 1
1
2 1
1
1 1
1),(
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17 力學聲響振動研究室 (MSVLAB)
Mathematical analysis --- Discrete model
c
cc
ff
mv
SM2
1][
NN
c
UU
SM44
][
For the clamped circular plate (u and are specified):
)(sv)(sm
a}]{[}]{[}{ 1 mvUf c
}]{[}]{[}{ 2 mvUf c
,u formulation:
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18 力學聲響振動研究室 (MSVLAB)
Circulant
NNN
NNN
NN
N
zzzzz
zzzzzzzzzzzz
U
22012321
3201222
221012
12210
][
12,...,2,1,0,),,,()],,,([)
21
(
)21
(
NmaaaUdaaaUz
m
mmm
NNe Ni
,1,..,2,1,0,22
NNezzN
m
Nm
im
N
m
mm
U ),1(,...,2,1,0,12
0
2212
0
][
daaaU
aaaUm m
N
mN
U
2
0
12
0
][
)]0,,,()[cos(
)]0,,,([)cos(lim
a 1
23
45
2N-12N-22N-3
2N
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19 力學聲響振動研究室 (MSVLAB)
Eigenvalues of the four matrices
NNlalll
laa
laaU
l
),1(,...,3,2,)1)(1(
2
1),ln223(
0),ln21(2
3
3
3
][
NNlalll
l
laa
laa
l
),1(,...,3,2,]1
1)1(
2[
1),ln225(
0),ln1(4
2
2
2
][
NNlall
ll
laa
laaU
l
),1(,...,3,2,])1(
21
1[
1),ln225(
0),ln1(4
2
2
2
][
NNlall
ll
laa
la
l
),1(,...,3,2,]12
12[
1),ln223(
0,4][
U
U
kernel kernel
kernelkernel
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20 力學聲響振動研究室 (MSVLAB)
Determinant
1
11
11
00
00
][
U
U
U
UcSM
N
N
UU
U
UcSM)1(
][][][][ )(det]det[
NNlllla
laalaaa
),1(,...,3,2,)1)(1(
21)),ln(1(4
0],))(ln()ln(1[8
2
42
42
242
0)ln(1 aDegenerate scale
a)(sv
)(sm
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21 力學聲響振動研究室 (MSVLAB)
,u
mu, 0lnln)ln1)(1(2)ln21)(1( 2 aaaa
vu, 0)ln2ln24()ln1( aaa
m,
v,
vm,
Degenerate scales for the clamped case Degenerate scales for the simply-supported case
0ln1 a
02ln)ln1()ln1( aaa
0ln2)ln23( aa42C
6 options
Formulation Equation of the degenerate scale in the BEM
Never zero
),()}(),()(),()(),()(),({)(8 sdBsuxsVsxsMsmxssvxsUxuB
),()}(),()(),()(),()(),({)(8 sdBsuxsVsxsMsmxssvxsUxB
),()}(),()(),()(),()(),({)(8 sdBsuxsVsxsMsmxssvxsUxm mm
Bmm
),()}(),()(),()(),()(),({)(8 sdBsuxsVsxsMsmxssvxsUxv vvB
vv
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22 力學聲響振動研究室 (MSVLAB)
Degenerate scale
0 0.2 0.4 0 .6 0.8 1 R ad ius a
1E -005
0.0001
0.001
0.01
0.1
1
10
D e t
0 .3 6 8a
1ea
0 0.2 0.4 0.6 0.8 1R ad iu s a
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
10
100
D e t
0.071
0.368
a
u formulation vu formulation 0)ln2ln24()ln1( aaa 0ln1 a3.0
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23 力學聲響振動研究室 (MSVLAB)
Degenerate scales for the free case
,u
mu,
vu,
m,
v,
vm,
Formulation Equation of the degenerate scale in the BEM
Never zero except three rigid body modes
Never zero except three rigid body modes
Never zero except three rigid body modes
Never zero except three rigid body modes
Never zero except three rigid body modes
Never zero except three rigid body modes
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24 力學聲響振動研究室 (MSVLAB)
Relationship between the Laplace problem and biharmonic problem (a) translation:
),( u
(b) rotation:
cos4
1),(
au
0 0.1 0.2 0.3 0 .4 0 .5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
sin4
1),(
au
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
constant
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25 力學聲響振動研究室 (MSVLAB)
Nontrivial modes in FEM and BEM FEM BEM
Rigid body mode Spurious mode(Hour-glass mode)
(zero-energy mode)
Rigid body modes Spurious mode
(Null-field)
Physically realizable Mathematical
realizable Physically realizable Mathematical
realizable
Q4 or Q8Q4 or Q8
0u
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26 力學聲響振動研究室 (MSVLAB)
Number of degenerate scales (Laplace problem)
Laplace problem:
0 0.25 0.5 0.75 1a
-1-0.500.5
-0.3-0.2-0.1
00.1
F,a0 0.25 0.5 0.75 1
a
-1-0.500.5
0 0.25 0.5 0.75 1a
-1-0.5
00.5
0
20
40
F,a0 0.25 0.5 0.75 1
a
-1-0.5
00.5
UT formulation:
LM formulation:
0)ln( a
No degenerate scale
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27 力學聲響振動研究室 (MSVLAB)
Number of degenerate scales (biharmonic problem)
0 0.25 0.5 0.75 1a
-1-0.5
00.5
-1
0
1
F,a0 0.25 0.5 0.75 1
a
-1-0.5
00.5
0 0.25 0.5 0.75 1a
-1-0.5
00.5
-1
0
1
2
F,a0 0.25 0.5 0.75 1
a
-1-0.5
00.5
0.2 0.4 0.6 0.8 1 1.2 1.4-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.2 0.4 0.6 0.8 1 1.2 1.4-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0 0.25 0.5 0.75 1a
-1-0.500.5
-6
-4
-2
0
F,a0 0.25 0.5 0.75 1
a
-1-0.500.5
0.2 0.4 0.6 0.8 1 1.2 1.4-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
u
mu
vu
formulation
formulation
formulation
0 0.2 0.4 0.6 0.8 1 Rad ius a
1E-005
0.0001
0.001
0.01
0.1
1
10
D et
0 .3 6 8
0 0.2 0.4 0.6 0 .8 1R ad iu s a
1E-01 3
1E-01 2
1E-01 1
1E-01 0
1E-00 9
1E-00 8
1E-00 7
1E-00 6
1E-00 5
0.0 001
0 .00 1
0.01
0.1
1
10
1 00
D et
0.071
0.368
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28 力學聲響振動研究室 (MSVLAB)
0 0.25 0.5 0.75 1a
-1-0.5
00.5
-3
-2
-1
0
F,a0 0.25 0.5 0.75 1
a
-1-0.5
00.5
0.2 0.4 0.6 0.8 1 1.2 1.4-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
Number of degenerate scales (biharmonic problem)
0 0.25 0.5 0.75 1a
-1-0.5
00.5
-2
0
2
4
F,a0 0.25 0.5 0.75 1
a
-1-0.5
00.5
0 0.25 0.5 0.75 1
a
-1-0.50
0.5
0
5
10
F,a0 0.25 0.5 0.75 1
a
0.2 0.4 0.6 0.8 1 1.2 1.4-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.2 0.4 0.6 0.8 1 1.2 1.4-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
m
v
vm
formulation
formulation
formulationNo degenerate
scale occurs
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29 力學聲響振動研究室 (MSVLAB)
Illustrative example (JFM, Mill 1977)
a
02 u0
nu
1
nu1
0 We adopt the null-field integral equation in conjunction with degenerate kernel to derive the analytic solution.
- 1 - 0 . 8 - 0 . 6 - 0 . 4 - 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 1
B I E M ( N = 2 0 )
- 1
- 0 . 8
- 0 . 6
- 0 . 4
- 0 . 2
0
0 . 2
0 . 4
0 . 6
0 . 8
1
-1 -0 .8 -0 .6 -0 .4 -0.2 0 0.2 0.4 0.6 0.8 1
E xact so lution
-1
-0 .8
-0 .6
-0 .4
-0 .2
0
0.2
0.4
0.6
0 .8
1
2 1 1 011 1 1( , ) (1 )[ ( tan ) ( tan )]2 1 2 1 2
r ru r r tan tanr r
- 1 - 0 . 8 - 0 . 6 - 0 . 4 - 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 1- 1
- 0 . 8
- 0 . 6
- 0 . 4
- 0 . 2
0
0 . 2
0 . 4
0 . 6
0 . 8
1
Exact solution : M=20 M=50
0u
2sin]
)(8[)22(
81
)(1
1
222
m
RmR
RRxu
M
mm
m
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30 力學聲響振動研究室 (MSVLAB)
On the equivalence of the Trefftz method and MFS for Laplace and biharmonic equations
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31 力學聲響振動研究室 (MSVLAB)
Trefftz method and MFS
Method Trefftz method MFSDefinition
Base uj(x) (T-complete function) , r=|x-s|
G. E. L u(x)=0, L U(x,s)=0, (singularity at s)
Match B. C. Determine cj Determine wj
TN
jjj xucxu
1
)()(
MN
jjj sxUwxu
1
),()(
Dx
is the number of complete functions TN
MN is the number of source points in the MFS
Dx
sDu(x)
~x
r
~s
D
u(x)
~x
)(),( rsxU
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32 力學聲響振動研究室 (MSVLAB)
Statement for Laplace problem
Two-dimensional Laplace problem with a circular domain:
Dxxu ,0)(2G.E. : B.C. : Bxuxu ,)(
B
D
a aD
B
Interior : Exterior :
N
n
n
N
n
n nbnaaau11
0 )sin()cos(),( Analytical solution:
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33 力學聲響振動研究室 (MSVLAB)
By matching the boundary condition at a
,00 aa
Tnn
n Nna
aa ,...2,1,
Tnn
n Nna
bb ,...2,1
TT N
n
nn
N
n
nn
I nabnaaaau11
0 )sin()cos(),(
TT N
n
nn
N
n
nn
E nabnaaaaau11
0 )sin()cos(ln),(
Interior problem:
Exterior problem: ,
ln1
00 aa
a
Tnn
n Nnaaa ,...2,1,
Tnn
n Nnbab ,...2,1
Derivation of unknown coefficients(Trefftz method)
Field solution:Interior :
Exterior :
T-complete set functions :
)sin(),cos(,1 nn nn
)sin(),cos(,ln nn nn Interior:
Exterior:
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34 力學聲響振動研究室 (MSVLAB)
Degenerate kernel :
RnRn
RU
RnRn
RRUrRU
nne
nni
1
1
)),(cos()(1)ln(),;,(
)),(cos()(1)ln(),;,(ln),,,(
MN
jj Rca
10 )ln(
M
N
jj
njn
n NnnRn
ca M
,...2,1,)cos()1(11
M
N
jj
njn
n NnnRn
cb M
,...2,1,)sin()1(11
20],))(cos()(1
)[ln(),(1 1
MN
j nj
nj
I nRa
nRcau
20],))(cos()(1)[ln(),(1 1
MN
j nj
nj
E naR
nacau
Interior problem:
Exterior problem:
MN
jjca
a1
0)ln(
1
M
N
jj
njn
n NnnRn
caaM
,...2,1,)cos()(1
1
M
N
jj
njn
n NnnRn
cbaM
,...2,1,)sin()(1
1
Field solution: Interior :
Exterior :
Derivation of unknown coefficients(MFS)
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35 力學聲響振動研究室 (MSVLAB)
,2
2
1
1
0
N
N
ba
babaa
w
12
2
5
4
3
2
1
N
Ncc
ccccc
c
cKw Trefftz MFS
Relationship between the two methods
)sin()1(1)sin()1(1)sin()1(1)sin()1(1
)cos()1(1)cos()1(1)cos()1(1)cos()1(1
)sin(1)sin(1)sin(1)sin(1
)cos(1)cos(1)cos(1)cos(1lnlnlnln
12321
12321
12321
12321
NNNNN
NNNNN
N
N
I
NRN
NRN
NRN
NRN
NRN
NRN
NRN
NRN
RRRR
RRRR
RRRR
K
)sin()(1
)sin()(1
)sin()(1
)sin()(1
)cos()(1
)cos()(1
)cos()(1
)cos()(1
)sin()sin()sin()sin()cos()cos()cos()cos(
1111
12321
12321
12321
12321
NNNNN
NNNNN
N
N
E
NRN
NRN
NRN
NRN
NRN
NRN
NRN
NRN
RRRRRRRR
K
Interior:
Exterior:
By setting 12 TN
Trefftz method
MFS
= MN = 12 N
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36 力學聲響振動研究室 (MSVLAB)
)12()12(1221
1221
1221
1221
1221
1221
)sin()sin()sin()cos()cos()cos(
)2sin()2sin()2sin()2cos()2cos()2cos(
)sin()sin()sin()cos()cos()cos(
111
][
NNN
N
N
N
N
N
NNNNNN
T
)12()12(212
0000
212
00
02
120
0012
][][
NN
T
N
N
NN
TT
Matrix TTK R
numbernaturalNNT N
N
,02
)12(]det[21
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37 力學聲響振動研究室 (MSVLAB)
Matrix RT
NaR ill-posed
problem
Degenerate scale problem
0ln R
)12()12(
2
2
)1(10000
0)1
(1
00000
)1
(21
)1
(21
01
00
00001
0
00000)ln(
][
NN
N
N
IR
RN
RN
R
R
R
R
R
T
)12()12(
2
2
)(10000
0)(100000
)(21
)(21
00000000000001
][
NN
N
N
ER
RN
RN
R
RR
R
T
)1( R
NaR ill-posed
problem
Degenerate scale problem?
)ln()ln(
aa
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38 力學聲響振動研究室 (MSVLAB)
Circulants
12212
222
1210][
NNNNN CaCaCaIaU
where
0000110000
0010000010
12
NC
dkUmkmUN
mNk )cos()0,(1)cos()0,(lim 2
0
12
0
2
1
1
212 1)(
ln2)2(]det[
N
NIRa
RRa
NUInterior:
Degenerate scale problem (R=1)
a
R=1fail
2
1
1
212 1)(ln2)2(]det[
N
NE
aR
aaRNUExterior
:
Nonunique problem (a=1)
a=1R
fail
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39 力學聲響振動研究室 (MSVLAB)
Numerical Examples
0)(:.. xuEG
DB
a
x
y)3cos()( xu
)3cos(),( 3 rru
)3cos(1ln),( 3 r
rcru
Simply-connected problem Multiply-connected problem
0u 1u
5.2lnln),( u
5.22 a
11 a
0u 1u
X
Y
}cos816cos8116{
2ln21),( 2
2
u
11 a
5.22 a
D
Ba
)3cos()( xu
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40 力學聲響振動研究室 (MSVLAB)
Numerical Example 1
Trefftz method for the simply-connected problem
Interior problem Exterior problem
Exact solution Numerical solution Exact solution Numerical solution5 Points: B.C. aliasing base deficiency
9 Points:
a=1 5 Points: B.C. aliasing Failure ( )9 Points:
Failure ( )
a=2 5 Points: B.C. aliasing
9 Points:-1 .0 -0 .8 -0 .6 -0 .4 -0 .2 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0
E x act so lu tion
-1 .0
-0 .8
-0 .6
-0 .4
-0 .2
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
-1 .0 -0 .8 -0 .6 -0 .4 -0 .2 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0
T refftz m eth o d (5 sets)
-1 .0
-0 .8
-0 .6
-0 .4
-0 .2
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
-1 .0 -0 .8 -0 .6 -0 .4 -0 .2 0.0 0 .2 0.4 0 .6 0 .8 1 .0
T re fftz m eth o d (9 sets)
-1 .0
-0 .8
-0 .6
-0 .4
-0 .2
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
-5 .0 -4 .0 -3 .0 -2 .0 -1 .0 0 .0 1 .0 2.0 3. 0 4.0 5. 0
T re f ftz so lu t ion (9 se ts)
-5 .0
-4 .0
-3 .0
-2 .0
-1 .0
0 .0
1 .0
2 .0
3 .0
4 .0
5 .0
-5 .0 -4.0 -3 .0 -2 .0 -1 .0 0 .0 1 .0 2.0 3 .0 4 .0 5 .0
T r e fftz so lu ti on (5 sets )
-5 .0
-4 .0
-3 .0
-2 .0
-1 .0
0 .0
1 .0
2 .0
3 .0
4 .0
5 .0
ln
ln- 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5
- 5
- 4
- 3
- 2
- 1
0
1
2
3
4
5
- 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5- 5
- 4
- 3
- 2
- 1
0
1
2
3
4
5
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41 力學聲響振動研究室 (MSVLAB)
MFS for the simply-connected problemInterior problem Exterior problem
Exact solution Numerical solution Exact solution Numerical solution
5 Points: B.C. aliasing
9 Points:
20 Points:
a=1: 5 Points: B.C. aliasing
Failure ( ln a)9 Points:
Failure ( ln a)
a=2: 5 Points: B.C. Aliasing
9 Points:
55 Points:
Numerical Example 2
-1 .0 -0 .8 -0 .6 -0 .4 -0 .2 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0
E xact so lu tion
-1 .0
-0 .8
-0 .6
-0 .4
-0 .2
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
-1 .0 -0 .8 -0 .6 -0 .4 -0 .2 0.0 0 .2 0.4 0 .6 0 .8 1.0
M F S (5 se ts)
-1 .0
-0 .8
-0 .6
-0 .4
-0 .2
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
-1 .0 -0 .8 -0 .6 -0 .4 -0 .2 0 .0 0 .2 0 .4 0 .6 0.8 1 .0
M F S (9 sets)
-1 .0
-0 .8
-0 .6
-0 .4
-0 .2
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
-1 .0 -0 .8 -0 .6 -0 .4 -0 .2 0.0 0 .2 0 .4 0 .6 0.8 1 .0
M F S (55 sets)
-1 .0
-0 .8
-0 .6
-0 .4
-0 .2
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
-5 .0 -4 .0 -3 .0 -2 .0 -1 .0 0.0 1.0 2.0 3 .0 4 .0 5 .0
E xac t so lut ion (r= 1 )
-5 .0
-4 .0
-3 .0
-2 .0
-1 .0
0 .0
1 .0
2 .0
3 .0
4 .0
5 .0
-5 .0 -4 .0 -3 .0 -2 .0 -1 .0 0 .0 1 .0 2 .0 3 .0 4 .0 5 .0
E x a ct so lu tio n (r= 2 )
-5 .0
-4 .0
-3 .0
-2 .0
-1 .0
0 .0
1 .0
2 .0
3 .0
4 .0
5 .0
-5 .0 -4 .0 -3 .0 -2 .0 -1 .0 0 .0 1 .0 2 .0 3 .0 4 .0 5 .0
M F S (5 se ts )
-5 .0
-4 .0
-3 .0
-2 .0
-1 .0
0 .0
1 .0
2 .0
3 .0
4 .0
5 .0
-5.0 -4.0 -3 .0 -2 .0 -1 .0 0 .0 1 .0 2 .0 3 .0 4 .0 5 .0
M F S (9 se ts )
-5 .0
-4 .0
-3 .0
-2 .0
-1 .0
0 .0
1 .0
2 .0
3 .0
4 .0
5 .0
-5 .0 -4.0 -3 .0 -2 .0 -1.0 0.0 1 .0 2.0 3 .0 4 .0 5 .0
M F S (65 se ts )
-5 .0
-4 .0
-3 .0
-2 .0
-1 .0
0 .0
1 .0
2 .0
3 .0
4 .0
5 .0
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42 力學聲響振動研究室 (MSVLAB)
Numerical Example 3
Trefftz method for the multiply-connected problem
Concentric circle Eccentric circle
Exact solution Numerical solution Exact solution Numerical solution
26 Points 26 Points 6 Points
14 Points
26 Points
-1 .5 -1 .0 -0 .5 0.0 0.5 1 .0 1 .5 2 .0 2.5 3.0 3.5
E x act so lu tion
-2.5
-2.0
-1.5
-1.0
-0.5
0 .0
0 .5
1 .0
1 .5
2 .0
2 .5
-1 .5 -1 .0 -0 .5 0.0 0 .5 1 .0 1 .5 2 .0 2.5 3.0 3.5
T re fftz m eth od (t t= 1 )
-2.5
-2.0
-1.5
-1.0
-0.5
0 .0
0 .5
1 .0
1 .5
2 .0
2 .5
-1 .5 -1 .0 -0 .5 0.0 0.5 1 .0 1 .5 2 .0 2.5 3.0 3.5
E x act so lu tion
-2.5
-2.0
-1.5
-1.0
-0.5
0 .0
0 .5
1 .0
1 .5
2 .0
2 .5
-1 .5 -1 .0 -0 .5 0 .0 0 .5 1 .0 1.5 2.0 2.5 3.0 3.5
E xact so lu t ion
-2 .5
-2 .0
-1 .5
-1 .0
-0 .5
0 .0
0 .5
1 .0
1 .5
2 .0
2 .5
-1 .5 -1 .0 -0 .5 0.0 0 .5 1 .0 1.5 2 .0 2.5 3.0 3.5
T ref ftz m eth o d (t t= 3 )
-2 .5
-2 .0
-1 .5
-1 .0
-0 .5
0 .0
0 .5
1 .0
1 .5
2 .0
2 .5
-1.5 -1 .0 -0.5 0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0 3.5
T refftz m e th o d ( tt=6)
-2 .5
-2 .0
-1 .5
-1 .0
-0 .5
0 .0
0 .5
1 .0
1 .5
2 .0
2 .5-2 .5 -2 -1 .5 -1 -0 .5 0 0 .5 1 1.5 2 2.5-2 .5
-2
-1 .5
-1
-0 .5
0
0.5
1
1.5
2
2.5
-2 .5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
5.2lnln
),(
u
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43 力學聲響振動研究室 (MSVLAB)
Numerical Example 4
MFS for the multiply-connected problem
Concentric circle Eccentric circle
Exact solution Numerical solution Exact solution Numerical solution
Inner circle: 20 points outer circle: 60points
Inner circle: a1=0.9 outer circle :a2=2.6Inner circle: 20 points outer circle: 60points
Inner circle: 20 points outer circle: 60points
Inner: 20points; outer: 60points; inner a1=0.9
outer a2=2.6 outer a2 =3.0
outer a2 =4.0 outer a2 =10.0
Inner: 20points; outer: 60points; outer a22.6
inner a1=0.5 inner a1 =0.3- 3 - 2 - 1 0 1 2 3
- 3
- 2
- 1
0
1
2
3
- 3 - 2 - 1 0 1 2 3
- 3
- 2
- 1
0
1
2
3
5.2lnln
),(
u
- 3 - 2 - 1 0 1 2 3
- 3
- 2
- 1
0
1
2
3
}cos816cos8116{
2ln21
),(
2
2
u
- 3 - 2 - 1 0 1 2 3
- 3
- 2
- 1
0
1
2
3
- 3 - 2 - 1 0 1 2 3
- 3
- 2
- 1
0
1
2
3
- 3 - 2 - 1 0 1 2 3
- 3
- 2
- 1
0
1
2
3
- 3 - 2 - 1 0 1 2 3
- 3
- 2
- 1
0
1
2
3
- 3 - 2 - 1 0 1 2 3
- 3
- 2
- 1
0
1
2
3
- 3 - 2 - 1 0 1 2 3
- 3
- 2
- 1
0
1
2
3
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44 力學聲響振動研究室 (MSVLAB)
Trefftz method and MFS for biharmonic equation
)(sin)(cos)()(sin)(cos),(1
2
1
220
110
* mdmccmbmaauTTTT N
m
mm
N
m
mm
N
m
mm
N
m
mm
Analytical solution:
TN
j
jj xugxu1
* )()(Field solution :
T-complete functions: )sin(),cos(,),sin(),cos(,1 222 mmmm mmmm
TN
jjj
xxh
nxu
x1
* )()(
)(
Trefftz method:
MFS:,),()(
1
MN
jjj sxUvxu
MM N
jjj
N
j x
jj
xsxv
n
sxUvx
nxu
11
),(),(
)()(
Field solution :
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45 力學聲響振動研究室 (MSVLAB)
Relationship between the Trefftz method and MFS
Coefficients of the
Trefftz method Coefficients of the
MFS Mapping matrix
[K]
1)24(24
14
4
3
2
1
)24()24(21
21
21
21
22
21
2
22
21
2
21
21
222
1)24(
1
1
0
1
1
0
)sin(1
)1(1
)sin(1
)1(1
)sin(1
)1(1
)cos(1)1(
1)cos(1)1(
1)cos(1)1(
1
)sin(2
1)sin(
21
)sin(2
1
)cos(2
1)cos(
21
)cos(2
1ln1ln1ln1
)sin()1
()1(
1)sin()
1(
)1(1
)sin()1
()1(
1
)cos()1
()1(
1)cos()
1(
)1(1
)cos()1
()1(
1
)sin()ln1()sin()ln1()sin()ln1()cos()ln1()cos()ln1()cos()ln1(
lnlnln
NN
N
NNNNNN
NNNN
N
N
NNNN
NNNN
N
N
NN
N
N
N
vv
vvvv
NRNN
NRNN
NRNN
NRNN
NRNN
NRNN
RRR
RRR
RRR
NRNN
NRNN
NRNN
NRNN
NRNN
NRNN
RRRRRRRRRRRR
RRRRRR
dc
dccba
baa
M
M
M
M
M
M
M
M
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46 力學聲響振動研究室 (MSVLAB)
Decomposition of the K matrix
][][][ TTK R
)24()24(241421
241421
241421
241421
241421
241421
241421
241421
sinsinsinsincoscoscoscos
sinsinsinsincoscoscoscos
1111sinsinsinsincoscoscoscos
sinsinsinsincoscoscoscos
1111
][
NNNN
NN
NN
NN
NN
NN
NN
NN
NNNNNNNN
NNNNNNNN
T
)24()24(
2
2
2
)1(1
)1(1
21
21
ln1)1(
1)1(
1
)ln21()ln21(
ln
][
NNN
N
N
N
R
NNR
NNR
R
R
RNNR
NNR
RRRR
RR
T
1
2
3
12 N
4
0)12(2]det[ 12 NNT
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47 力學聲響振動研究室 (MSVLAB)
)24()24(
2
2
2
)1(1
)1(1
21
21
ln1)1(
1)1(
1
)ln21()ln21(
ln
][
NNN
N
N
N
R
NNR
NNR
R
R
RNNR
NNR
RRRR
RR
T
Diagonal matrix TR
Existence of the degenerate scales
121
0 ,,
eeeRNonuniqueness
(in numerical aspect)
RlnRln21Rln1 Degenerate scale
problem0
0 1)24()24()24(1)24( }{][}{ NjNNNj vKg O. K.!
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48 力學聲響振動研究室 (MSVLAB)
Special size:
: position of the source points
The occurrence of the degenerate scales using the MFS
a
0e
1e21
eMathematics: rank-deficiency problem
(nonuniqueness problem)
Numerical failure
Degenerate scale problem
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49 力學聲響振動研究室 (MSVLAB)
On the complete set of the Trefftz method and the MFS using the degenerate kernel
T-complete functions of the Trefftz method:
)sin(),cos(,),sin(),cos(,1 222 mmmm mmmm
Degenerate kernel of the MFS:
))(cos()1(
1))(cos()1(
)cos(21)cos()ln21(ln1)ln1(),(
22
2
2
322*
mRmm
mRmm
RRRRRRu
mm
m
mm
m
m=0m=0m=1m=1
m=2, 3….. m=2, 3…..
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50 力學聲響振動研究室 (MSVLAB)
Free terms for the biharmonic equation using the dual boundary integral equation
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51 力學聲響振動研究室 (MSVLAB)
History of free terms in the dual BEM
2-D and 3-D Laplace problem
2-D and 3-D elasticity problem
W. C. Chen thesis
2-D biharmonic problem
(1) Bump-contour method
(2) Taylor series expansion
Free terms
Dual boundary integral equation
Improper integrals
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52 力學聲響振動研究室 (MSVLAB)
Bump-contour method
For a smooth boundary:
B
xy
B+ B-
B’ B’
B’
D
Singular point
4 ( ) 0u x
)sin,cos( s
)0,0(x
)(su
)0,1(),1,0(
)cos,sin(),sin,(cos
x
x
s
s
tntn
)(8),(4 sxsxUx sxrrrsxU ),ln(),( 2
Explicit forms for the sixteen kernel functions
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53 力學聲響振動研究室 (MSVLAB)
Taylor expansion for boundary density functions
)(]sinsincossincoscos[!3
1
]sincossincos[!2
1]sincos[)()(
4333
2
32
212
1
32
221
33
31
3
2
21
22
22
22
21
2
21
Os
uss
ussu
su
ssu
su
su
su
su
xusu
snsu
s
)(
)(
2
22 )(
)1()()(s
sn
sususm
ssss
s
tnsu
tnsu
sv)(
)1()(
)(22
Boundary
Domain
)0,0(x
B)sin,cos( s
vector component: 2,1, isxy iii
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54 力學聲響振動研究室 (MSVLAB)
)(xu
)(2
xt
)(2
)(2
xuxt
Free terms of dual BIE for Laplace problem 2-D problem:
0
3-D problem:)(2 xu
)(3
2 xt
)(2
)(3
4xuxt
BxsdBsuxsMVPHsdBstxsLVPCxt
BxsdBsuxsTVPCsdBstxsUVPRxu
BB
BB
,)()(),(...)()(),(...)(
,)()(),(...)()(),(...)(
)(])(),()(),([)(2 sdBstxsUsuxsTxuB
)(])(),()(),([)(2 sdBstxsLsuxsMxtB
)(])(),()(),([)(4 sdBstxsUsuxsTxuB
)(])(),()(),([)(4 sdBstxsLsuxsMxtB
0
BxsdBsuxsMVPHsdBstxsLVPCxt
BxsdBsuxsTVPCsdBstxsUVPRxu
BB
BB
,)()(),(...)()(),(...)(2
,)()(),(...)()(),(...)(2
Half
Half Singular point
B
partVPC ...
partVPC ...
partVPC ...
partVPC ...
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55 力學聲響振動研究室 (MSVLAB)
Dual boundary integral equations
BxsdBsuxsVPFsdBsxsMPFsdBsmxsPFsdBsvxsUPFxuBBBB
,)()(),(..)()(),(..)()(),(..)()(),(..)(8
BxsdBsuxsVPFsdBsxsMPFsdBsmxsPFsdBsvxsUPFxBBBB
,)()(),(..)()(),(..)()(),(..)()(),(..)(8
BxsdBsuxsVPFsdBsxsMPFsdBsmxsPFsdBsvxsUPFxmB
mB
mB
mB
m ,)()(),(..)()(),(..)()(),(..)()(),(..)(8
BxsdBsuxsVPFsdBsxsMPFsdBsmxsPFsdBsvxsUPFxvB
vB
vB
vB
v ,)()(),(..)()(),(..)()(),(..)()(),(..)(8
F.P. denotes the finite part
for a smooth boundary21
Sixteen improper integrals
Density functions are expanded by the Taylor series
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56 力學聲響振動研究室 (MSVLAB)
Singular behavior of the sixteen kernels
),( xsU ),( xs ),( xsM ),( xsV
),( xsU v
),( xsU m
),( xsU
),( xsv
),( xsm
),( xs
),( xsM v
),( xsM m
),( xsM
),( xsVv
),( xsVm
),( xsV
ln2O lnO
lnO
lnO
lnO
lnO
1O
1O
1O
1O
2
1
O
2
1
O
2
1
O
3
1
O
3
1
O
4
1
O
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57 力學聲響振動研究室 (MSVLAB)
),( xsU ),( xs ),( xsM ),( xsV
),( xsU v
),( xsU m
),( xsU
),( xsv
),( xsm
),( xs
),( xsM v
),( xsM m
),( xsM
),( xsVv
),( xsVm
),( xsV
0 0 0
0 0
)(4 xu
)()1( x
)(
)3(4)()3(xu
x
)()1(2
xm
)()1( xm
)(
)35
1()1(4
)(x
xm
)(0)(
3)3()1(8
)()3(2
2 xux
xm
)()3)(1(2
xv
)()76(
34
)(2
2 xm
xv
2)(
16)5(33
)1(
)(3
)7()1(8)()2)(1(
x
xmxv
3
2
2
)()1(8
)()3(16)5(3
3)1(
)()1)(1(4)()3)(1(
2
xu
x
xmxv
Free terms due to the bump integral for the biharmonic equation
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58 力學聲響振動研究室 (MSVLAB)
Dual boundary integral equations for the biharmonic problem
After deriving the sixteen improper integrals, we have
BxsdBsuxsVPFsdBsxsMVPRsdBsmxsVPRsdBsvxsUVPRxuBBBB
,)()(),(..)()(),(...)()(),(...)()(),(...)(4
BxsdBsuxsVPFsdBsxsMPFsdBsmxsPFsdBsvxsUPFxBBBB
,)()(),(..)()(),(..)()(),(..)()(),(..)(4
BxsdBsuxsVPFsdBsxsMPFsdBsmxsVPRsdBsvxsUVPRxmB
mB
mB
mB
m ,)()(),(..)()(),(..)()(),(...)()(),(...)(4
BxsdBsuxsVPFsdBsxsMPFsdBsmxsPFsdBsvxsUPFxvB
vB
vB
vB
v ,)()(),(..)()(),(..)()(),(..)()(),(..)(4
48 for a smooth boundary
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59 力學聲響振動研究室 (MSVLAB)
Conclusions
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60 力學聲響振動研究室 (MSVLAB)
1. New methods to derive the Poisson integral formula by using the degenerate kernels and the null-field integral equations.
2. The occurring mechanism of degenerate scales depends on the formulation instead of the boundary conditions.
3. It is interesting to find that the T-complete set in the Trefftz method is imbedded in the degenerate kernels of MFS.
4. We adopt the bump-contour method to derive the free terms surrounding the singularity. For a smooth boundary, the sum of free terms are half.
Conclusions
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61 力學聲響振動研究室 (MSVLAB)
Thanks for your kind attention
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62 力學聲響振動研究室 (MSVLAB)
Image method
xxdBxtsxUxdBxusxTsuB
FB
F ,)()(),()()(),()(2
known unknown
Image method
BxxdBxussxTsuB
G ),()(),;()(2
B
G xdBxtssxU 0)()(),;( 0|),;( BxG ssxU
B
),( Rs )','(' Rs
),( x
0|),;( BxG ssxUClosed-form Green’s function ?)',;( ssxU G
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63 力學聲響振動研究室 (MSVLAB)
Closed-form Green’s function (Interior problem)
,)],(cos[)(1
ln),(
,)],(cos[)(1
ln),(ln)ln(),(
1
1
m
mEF
m
mIF
FRm
Rm
sxU
RmRm
RsxUsxrsxU
RmRm
sxm
m
,)](cos[)(1lnln
1
RmRm
Rsxm
m
,)](cos[)(1lnln
1
Ra
RR
RR 22
),( x),( Rs
)','(' Rs
.lnln||ln||ln
lnln||ln||ln
lnln||ln||ln),;(
'
2'
'
RasxsxR
aasxsx
RasxsxssxU G
a
Image point
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64 力學聲響振動研究室 (MSVLAB)
Series-form Green’s function (degenerate kernels)
.0,)](cos[])()[(1)ln(
lnln||ln||ln),;(
12
'
Rma
RRma
R
RasxsxssxU
m
mm
G
.,)](cos[)]()[(1
)ln(
lnln||ln||ln),;(
12
'
aRma
RRma
RasxsxssxU
m
mm
G
a
a
),( x
),( x
),( Rs
),( Rs
)','(' Rs
)','(' Rs
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65 力學聲響振動研究室 (MSVLAB)
Closed-form and series-form Green’s functions for interior and exterior problems
Closed-form Series-form
0,25.1R,8.0R 1,a - 1 - 0 . 8 - 0 . 6 - 0 . 4 - 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 1
- 1
- 0 . 8
- 0 . 6
- 0 . 4
- 0 . 2
0
0 . 2
0 . 4
0 . 6
0 . 8
1
s
- 1 - 0 . 8 - 0 . 6 - 0 . 4 - 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 1- 1
- 0 . 8
- 0 . 6
- 0 . 4
- 0 . 2
0
0 . 2
0 . 4
0 . 6
0 . 8
1
s
050,M,25.1R0.8,R 1,a
- 2 - 1 . 5 - 1 - 0 . 5 0 0 . 5 1 1 . 5 2 2 . 5 3
- 2
- 1 . 5
- 1
- 0 . 5
0
0 . 5
1
1 . 5
2
s
- 2 - 1 . 5 - 1 - 0 . 5 0 0 . 5 1 1 . 5 2 2 . 5 3
- 2
- 1 . 5
- 1
- 0 . 5
0
0 . 5
1
1 . 5
2
s
0,8.0R,25.1R 1,a 050,M,8.0R,25.1R 1,a
Interior problem:
Exterior problem:
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66 力學聲響振動研究室 (MSVLAB)
x
GG n
ssxUssxT
),;(),;(
BxxdBxussxTsuB
G ),()(),;()(2
Poisson integral formula
Cosine theorem
,20,0,)()cos(2
)(21
)(),;,;,(21),(
2
0 22
22
2
0
2
aRdf
aRRaRa
dafR
aRTRu G
Poisson integral formula
20,0,)())(cos(2121),(
2
01
aRdfma
RRum
m
m
Series-form:
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67 力學聲響振動研究室 (MSVLAB)
Degenerate scale Rigid body mode
Solve u
u is a null field u is solved to be the rigid body solution
Discriminant
Laplace problems
Biharmonic problems
Dirichlet NeumannFree
ClampedSimply-supported
Mathematically realizable Physically
realizable
0KDet
Flowchart of the nontrivial modes
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68 力學聲響振動研究室 (MSVLAB)
By setting 12 TN
12
10 )ln(
N
jj Rca
12
112,...2,1),cos()1(1N
jj
njn Nnn
Rnca
12,...2,1,)sin()1
(112
1
Nnn
Rncb
N
jj
njn
12
10
N
jjca
12
1
12,...2,1),cos()(1N
jj
njn NnnR
nca
12,...2,1,)sin()(1
12
1
NnnRn
cbN
jj
njn
Interior problem: Exterior problem:
Trefftz method
MFS
= MN = 12 N
Connection between the Trefftz method and MFS