指導教授:黃三益 教授 學生: M954020031 陳聖現 M954020033 王啟樵 M954020042 呂佳如.
研究生:吳清森 指導教授:陳正宗 教授 陳義麟 博士
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Transcript of 研究生:吳清森 指導教授:陳正宗 教授 陳義麟 博士
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
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
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
CNME
Necessary 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 term
Hypersingular formulation
CHEEF concept
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 method
Restriction 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
2
3
2
110 ,,,
eeee
4
3
2
110 ,,,
eeee
2
110 ,,
eee
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)
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
iUeU
A
),(
),(
Rs
x
O2
x
s
R
iUeU
B
O1
O2
7 力學聲響振動研究室 (MSVLAB)
Alternative derivations for the Poisson integral formula
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(22
1),( df
aa
au
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[)(212
1
))sin()cos(()](cos[)(212
1),(
c
unknown
specified
Fundamental solution
Green’s identity
,ln),( rsxU F
10 力學聲響振動研究室 (MSVLAB)
Degenerate scale for plate analysis using the BIEM and BEM
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
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: xD
xwxu ,
)()(4
Boundary condition: Bxxxu ,0)(,0)(
Splitting method
Governing equation: xxu ,0)(*4
Boundary condition: Bxxxxuxu ),()(),()(****
: deflection of the circular plate
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
14 力學聲響振動研究室 (MSVLAB)
Operators
nK
)(
)(
2
2
2 )()1()()(
nK
m
))(
()1()(
)(22
tntnK
v
Slope
Normal moment
Effective shear force
t
n
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
16 力學聲響振動研究室 (MSVLAB)
Degenerate kernels for biharmonic operator
RmRmm
mRmm
RRRRRxsU
RmR
mmm
R
mmRRxsU
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[
1
1)cos()ln23()ln21(),(
),( m m m
m
m
mI
m m
m
m m
mE
RmRm
mRmm
mRRRxsU
RmR
mm
mm
R
mR
RxsU
xsU
RmRmm
mm
RmRRR
Rxs
RmR
mm
R
mm
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
Rm
mR
Rxs
RmR
m
mm
R
m
mRxs
m m
m
m m
mI
m m
m
m m
mE
xs,)]([cos
1
2)]([cos
1
2)cos()ln23(
2),(
,)](cos[1
2)]([cos
1
2)cos()ln23(
2),(
2 1
1
1 1
1
2 1
1
1 1
1),(
17 力學聲響振動研究室 (MSVLAB)
Mathematical analysis --- Discrete model
c
cc
f
f
m
vSM
2
1][
NN
c
U
USM
44
][
For the clamped circular plate (u and are specified):
)(sv)(sm
a}]{[}]{[}{ 1 mvUf c
}]{[}]{[}{ 2 mvUf c
,u formulation:
18 力學聲響振動研究室 (MSVLAB)
Circulant
NNN
NNN
NN
N
zzzzz
zzzz
zzzz
zzzz
U
22012321
3201222
221012
12210
][
12,...,2,1,0,),,,()],,,([)
2
1(
)2
1(
NmaaaUdaaaUz
m
mmm
NNe Ni
,1,..,2,1,0,2
2
NNezzN
m
N
mi
m
N
m
mm
U ),1(,...,2,1,0,12
0
2
212
0
][
daaaU
aaaUm m
N
mN
U
2
0
12
0
][
)]0,,,()[cos(
)]0,,,([)cos(lim
a1
23
45
2N-12N-22N-3
2N
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),ln22
3(
0),ln21(2
3
3
3
][
NNlalll
l
laa
laa
l
),1(,...,3,2,]1
1
)1(
2[
1),ln22
5(
0),ln1(4
2
2
2
][
NNlall
l
l
laa
laaU
l
),1(,...,3,2,])1(
2
1
1[
1),ln22
5(
0),ln1(4
2
2
2
][
NNlal
l
l
l
laa
la
l
),1(,...,3,2,]1
2
1
2[
1),ln22
3(
0,4][
U
U
kernel kernel
kernelkernel
20 力學聲響振動研究室 (MSVLAB)
Determinant
1
11
11
0
0
0
0
][
U
U
U
UcSM
N
N
UU
U
UcSM)1(
][][][][ )(det]det[
NNllll
alaa
laaa
),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
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
22 力學聲響振動研究室 (MSVLAB)
Degenerate scale
0 0.2 0.4 0.6 0.8 1
R adius 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 1
R 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 et
0.071
0.368
a
u formulation vu formulation 0)ln2ln24()ln1( aaa 0ln1 a3.0
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
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
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
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
-0.10
0.1
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
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
R ad ius a
1E -005
0.0001
0.001
0.01
0.1
1
10
D e t
0 .3 6 8
0 0.2 0.4 0.6 0.8 1
R 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 et
0.071
0.368
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
29 力學聲響振動研究室 (MSVLAB)
Illustrative example (JFM, Mill 1977)
a
02 u0
n
u
1n
u1
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
Exact 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 tan
r 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(
8
1)(
11
222
m
Rm
R
RRxu
M
mm
m
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
31 力學聲響振動研究室 (MSVLAB)
Trefftz method and MFS
Method Trefftz method MFS
Definition
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
s
Du(x)
~x
r
~s
D
u(x)
~x
)(),( rsxU
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 nbnaaau
11
0 )sin()cos(),( Analytical solution:
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 nabnaaaau
11
0 )sin()cos(),(
TT N
n
nn
N
n
nn
E nabnaaaaau
11
0 )sin()cos(ln),(
Interior problem:
Exterior problem: ,
ln
100 a
aa
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:
34 力學聲響振動研究室 (MSVLAB)
Degenerate kernel :
RnR
nRU
RnRn
RRUrRU
n
ne
n
ni
1
1
)),(cos()(1
)ln(),;,(
)),(cos()(1
)ln(),;,(ln),,,(
MN
jj Rca
1
0 )ln(
M
N
jj
njn
nNnn
Rnc
a M
,...2,1,)cos()1
(1
1
M
N
jj
njn
nNnn
Rnc
b M
,...2,1,)sin()1
(1
1
20],))(cos()(1
)[ln(),(
1 1
MN
j n
jn
jI n
R
a
nRcau
20],))(cos()(1
)[ln(),(
1 1
MN
j n
jn
jE n
a
R
nacau
Interior problem:
Exterior problem:
MN
j
jcaa
1
0)ln(
1
M
N
j
jn
jnn NnnR
ncaa
M
,...2,1,)cos()(1
1
M
N
j
jn
jnn NnnR
ncba
M
,...2,1,)sin()(1
1
Field solution: Interior :
Exterior :
Derivation of unknown coefficients(MFS)
35 力學聲響振動研究室 (MSVLAB)
,2
2
1
1
0
N
N
b
a
b
a
b
a
a
w
12
2
5
4
3
2
1
N
N
c
c
c
c
c
c
c
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(1
lnlnlnln
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
RRRR
RRRR
K
Interior:
Exterior:
By setting 12 TN
Trefftz method
MFS
= MN = 12 N
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
NNN
NNN
T
)12()12(2
12000
02
1200
02
120
0012
][][
NN
T
N
N
NN
TT
Matrix TTK R
numbernaturalNN
TN
N
,02
)12(]det[
2
1
37 力學聲響振動研究室 (MSVLAB)
Matrix RT
N
aR ill-posed problem
Degenerate scale problem
0ln R
)12()12(
2
2
)1
(1
0000
0)1
(1
000
00
)1
(2
1
)1
(2
1
01
00
00001
0
00000)ln(
][
NN
N
N
IR
RN
RN
R
R
R
R
R
T
)12()12(
2
2
)(1
0000
0)(1
000
00
)(2
1
)(2
1000
00000
000001
][
NN
N
N
ER
RN
RN
R
R
R
R
T
)1( R
N
aR ill-posed problem
Degenerate scale problem?
)ln(
)ln(
a
a
38 力學聲響振動研究室 (MSVLAB)
Circulants
12212
222
1210][
NNNNN CaCaCaIaU
where
00001
10000
00100
00010
12
NC
dkUmkmU
N
mNk )cos()0,(
1)cos()0,(lim
2
0
12
0
2
1
1
212 1)(
ln2)2(]det[
N
NI
R
a
R
RaNUInterior
:
Degenerate scale problem (R=1)
a
R=1fail
2
1
1
212 1)(
ln2)2(]det[
N
NE
a
R
a
aRNUExterior
:
Nonunique problem (a=1)
a=1R
fail
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(1
ln),(3
r
rcru
Simply-connected problem Multiply-connected problem
0u 1u
5.2ln
ln),(
u
5.22 a
11 a
0u 1u
X
Y
}cos816
cos8116{
2ln2
1),(
2
2
u
11 a
5.22 a
D
Ba
)3cos()( xu
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 solution
5 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 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
T refftz m eth od (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 refftz m eth od (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 refftz so lu tion (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 refftz so lu tio n (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
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
41 力學聲響振動研究室 (MSVLAB)
MFS for the simply-connected problem
Interior 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 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 (9 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 (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 x a c t so lu tio n (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 c t 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 (6 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
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 a ct so lu tio n
-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 eth o d (tt= 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 a ct so lu tio n
-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 a ct so lu tio n
-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 eth o d (tt= 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 eth 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
5.2ln
ln),(
u
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=2.6
Inner 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
5.2ln
ln),(
u
- 3 - 2 - 1 0 1 2 3
- 3
- 2
- 1
0
1
2
3
}cos816
cos8116{
2ln2
1),(
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
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
j
jjx
xhn
xux
1
* )()(
)(
Trefftz method:
MFS:
,),()(1
MN
jjj sxUvxu
MM N
jjj
N
j x
jj
x
sxvn
sxUvx
n
xu
11
),(),(
)()(
Field solution :
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(
2
1)sin(
2
1
)cos(2
1)cos(
2
1)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
v
v
v
v
v
v
NRNN
NRNN
NRNN
NRNN
NRNN
NRNN
RRR
RRR
RRR
NRNN
NRNN
NRNN
NRNN
NRNN
NRNN
RRRRRR
RRRRRR
RRRRRR
d
c
d
c
c
b
a
b
a
a
M
M
M
M
M
M
M
M
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
sinsinsinsin
coscoscoscos
sinsinsinsin
coscoscoscos
1111
sinsinsinsin
coscoscoscos
sinsinsinsin
coscoscoscos
1111
][
NNNN
NN
NN
NN
NN
NN
NN
NN
NNNN
NNNN
NNNN
NNNN
T
)24()24(
2
2
2
)1(
1)1(
1
2
12
1ln1
)1(
1)1(
1
)ln21(
)ln21(
ln
][
NNN
N
N
N
R
NNR
NNR
R
R
RNNR
NNR
RR
RR
RR
T
1
2
3
12 N
4
0)12(2]det[ 12 NNT
47 力學聲響振動研究室 (MSVLAB)
)24()24(
2
2
2
)1(
1)1(
1
2
12
1ln1
)1(
1)1(
1
)ln21(
)ln21(
ln
][
NNN
N
N
N
R
NNR
NNR
R
R
RNNR
NNR
RR
RR
RR
T
Diagonal matrix TR
Existence of the degenerate scales
12
10 ,,
eeeRNonuniqueness
(in numerical aspect)
Rln
Rln21
Rln1 Degenerate scale problem
0
0 1)24()24()24(1)24( }{][}{ NjNNNj vKg O. K.!
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
1e2
1
e
Mathematics: rank-deficiency problem
(nonuniqueness problem)
Numerical failure
Degenerate scale problem
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(2
1)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…..
50 力學聲響振動研究室 (MSVLAB)
Free terms for the biharmonic equation using the dual boundary integral equation
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
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
t
n
t
n
)(8),(4 sxsxUx sxrrrsxU ),ln(),( 2
Explicit forms for the sixteen kernel functions
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
u
ss
u
ss
u
s
u
ss
u
s
u
s
u
s
u
s
uxusu
sn
sus
)(
)(
2
22 )(
)1()()(s
sn
sususm
ssss
s
tn
su
tn
susv
)()1(
)()(
22
Boundary
Domain
)0,0(x
B)sin,cos( s
vector component:
2,1, isxy iii
54 力學聲響振動研究室 (MSVLAB)
)(xu
)(2
xt
)(
2)(
2xuxt
Free terms of dual BIE for Laplace problem 2-D problem:
0
3-D problem:)(2 xu
)(3
2xt
)(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 ...
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 boundary2
1
Sixteen improper integrals
Density functions are expanded by the Taylor series
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
1
O
1
O
1
O
1
O
2
1
O
2
1
O
2
1
O
3
1
O
3
1
O
4
1
O
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(
xux
)()1(2
xm
)()1( xm
)(
)3
51()1(4
)(
x
xm
)(0)(
3
)3()1(8
)()3(2
2xu
x
xm
)()3)(1(2
xv
)()76(
3
4
)(2
2 xm
xv
2
)(16)5(3
3
)1(
)(
3
)7()1(8
)()2)(1(
x
xm
xv
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
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
59 力學聲響振動研究室 (MSVLAB)
Conclusions
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
61 力學聲響振動研究室 (MSVLAB)
Thanks for your kind attention
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
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
R
msxU
RmRm
RsxUsxrsxU
RmR
msx
m
m
,)](cos[)(
1lnln
1
RmRm
Rsxm
m
,)](cos[)(1
lnln1
R
a
RR
R
R 22
),( x
),( Rs
)','(' Rs
.lnln||ln||ln
lnln||ln||ln
lnln||ln||ln),;(
'
2'
'
Rasxsx
R
aasxsx
RasxsxssxU G
a
Image point
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
R
Rma
R
RasxsxssxU
m
mm
G
.,)](cos[)]()[(1
)ln(
lnln||ln||ln),;(
12
'
aRma
RR
ma
RasxsxssxU
m
mm
G
a
a
),( x
),( x
),( Rs
),( Rs
)','(' Rs
)','(' Rs
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:
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
)(
2
1
)(),;,;,(2
1),(
2
0 22
22
2
0
2
aRdf
aRRa
Ra
dafR
aRTRu G
Poisson integral formula
20,0,)())(cos(212
1),(
2
01
aRdfma
RRu
mm
m
Series-form:
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
Clamped
Simply-supported
Mathematically realizable
Physically realizable
0KDet
Flowchart of the nontrivial modes
68 力學聲響振動研究室 (MSVLAB)
By setting 12 TN
12
10 )ln(
N
jj Rca
12
1
12,...2,1),cos()1
(1N
jj
njn Nnn
Rnca
12,...2,1,)sin()1
(112
1
Nnn
Rncb
N
jj
njn
12
1
0
N
j
jca
12
1
12,...2,1),cos()(1
N
j
jn
jn NnnRn
ca
12,...2,1,)sin()(1
12
1
NnnRn
cb
N
j
jn
jn
Interior problem: Exterior problem:
Trefftz method
MFS
= MN = 12 N
Connection between the Trefftz method and MFS