1 equivalence between the trefftz method and the method of fundamental solutions for the green’s...
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1
Equivalence between the Trefftz method and the method of fundamental
solutions for the Green’s function of concentric spheres using the addition
theorem and image concept J.T. Chen
Life-time Distinguished ProfessorDepartment of Harbor and River Engineering,
National Taiwan Ocean UniversitySep. 2-4, 2009
New Forest, UK
BEM/MRM 31
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Outline
Numerical methods
Trefftz method and MFS Image method (special MFS)
Trefftz method
Equivalence of solutions derived by
Trefftz method and MFS
Conclusions
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Numerical methods
Numerical methods
Boundary Element MethodFinite Element Method Meshless Method
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Method of fundamental solutions
MN
jjj xsUcxu
1
),()(
is the fundamental solution),( xsU
Interior case Exterior case
This method was proposed by Kupradze in 1964.
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Optimal source location
Conventional MFS Alves CJS & Antunes PRS
Not good Good
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Optimal source location
Conventional MFS Alves & Antunes
GoodNot Good
?
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The simplest image method
Neumann boundary Neumann boundary conditioncondition DirichletDirichlet boundary conditio boundary conditionn
Mirror
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Conventional method to determine the image location
R
R’
O
a rr’
aOR
ORa
ORa
ORa
PR
RP
'''
2
''
OR a aOR
a OR OR
P
AB
aa
O R’RO
PPLord Kelvin(1824~1907)
Greenberg (1971)
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Image location using degenerate kernel (Chen and Wu, 2006)
a
s 's2'
'ss
R
a
aR
R
a
R
1
1ln cos ( )
s
m
sm
ma
RR
m
1
1ln cos ( )
m
m
a mm
R
a
a s2
''s
s
aR
R
a
R
R
a
1
1ln cos ( )
ms
m
a mm
R
a
1
1ln cos ( )
m
m
a
RR m
m
Rigid body term
's
u=0
u=0
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Degenerate kernel-2D (addition theorem)
1
1
),(cos)(1
ln
,)(cos)(1
lnln
m
m
m
m
RmR
m
RmRm
Rr
s( , )R q
R
r
rx( , )r f
x( , )r f
o
iU
eU
rsxU ln),(
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Addition theorem & degenerate kernel
Addition theorem Subtraction theorem
( )ik x s ikx ikse e e sxsxsx sinsincoscos)cos(
sin( ) sin cos cos sinx s x s x s
( ) /ik x s ikx ikse e e cos( ) cos cos sin sinx s x s x s sin( ) sin cos cos sinx s x s x s
Degenerate kernel for Laplace problem
1-D
2-D
RmR
m
RmRm
Rr
m
m
m
m
,)(cos)(1
ln
,)(cos)(1
lnln
1
1
sxifxs
sxifsxr
,
,
sx
3-D next page
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3-D degenerate kernel
11 0
11 0
1 ( )!cos ( ) (cos ) (cos ) ,
( )!1
1 ( )!cos ( ) (cos ) (cos ) ,
( )!
nni m m
m n n nn m
nne m m
m n n nn m
n mU m P P R
R n m R
r n m RU m P P R
n m
1, 02 , 1,2,...,m
mm
s ( , , )( , , )
xs R
x
exterior
x
interior
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Trefftz method and MFS
Method Trefftz method MFS
Definition
Figure sketch
Base , (T-complete function) , r=|x-s|
G. E.
Match B. C. Determine cj Determine wj
( , ) lnU x s r
1( ) ( , )
N
j jj
u x w U x s
( )2 0u xÑ = ( )2 0u xÑ =
D
u(x)
~x
s
Du(x)
~x
r
~s
is the number of complete functions MN is the number of source points in the MFS
1( )
M
j jj
u x c
j
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Derivation of 3-D Green’s function by using the image method
Interior problem
Exterior problem
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15
Weightings of the image source in the 3-D problem
y
z
1a
x
y
1 a
x
z
),,(2
sR
as ),,( sRs
),,( sRs
),,(2
sR
as
Interior problem Exterior problem
1sR
a
1sR
a
True source
Image source
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Weighting and locations of succesive images
22 15 91
1 5 9 4 32 3
2 32 1
2 6 10 4 22
2 32 13 7 11
3 7 11 4 12 3
2
4 8 2
, ( ), ( ) ( )
, ( ), ( ) ( )
, ( ), ( ) ( )
, (
nn
nn
nn
aR a RR b b b b b b bw w w w
b R b R a b R a R aa a a a a a a a a
w w w wR bR R b b R R b R baR a R a Rb b b b b b b
w w w wbR a b R a a b R a a a aa a a a
w wb b b
3
2 112 43
), ( ) ( )nn
a a a a aw w
b b b b b b
2 2 2 2 21
1 5 4 32 2
2 2 2 2 21
2 6 4 22 2
2 2 2 2 21
3 7 4 12 2 2 2 2
2 2 2 2 21
4 8 42 2 2 2 2
, ........ ( )
, ....... ( )
, ... ( )
, ... ( )
nn
nn
nn
nn
b b b b bR R R
R R a R a
a a a a aR R R
R R b R b
b R b R b b R bR R R
a a a a a
a R a R a a R aR R R
b b b b b
Weighting of successive images
Location of successive images
s
Ä
1s
Ä
2s
e
3s
e
4s
e
5s
e
6s
Ä
7s
ÄK
8s
ÄK
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Derivation of analytical solution using interpolation functions
a
b
( , ) ( , ) 0a bG x s G x s ( ) ( )
( , ) ( , ) ( , ) ( , ),( ) ( )m m b m a
b a a bG x s G x s G x s G x s a b
b a b a
4 3 4 2 4 1 4
1 4 3 4 2 4 1 4
1 1( , ) lim
4
( ) 1
( ) ( )
Ni i i i
Ni i i i i
N Ns s
s s
w w w wG x s
x s x s x s x s x s
R a a b Ra a
b R b a b R b a
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4 3 4 2 4 1 4
1 4 3 4 2 4 1 4
1 1 ( )( , ) lim ( )
4
Ni i i i
Ni i i i i
w w w w d NG x s c N
x s x s x s x s x s
Derivation of analytical solution using complementary solutions
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Numerical approach by collocation BCs
a
b
4 3 4 2 4 1 4
1 4 3 4 2 4 1 4
4 3 4 2 4 1 4
1 4 3 4 2 4 1 4
1 1 ( )( , ) ( ) 0
4
1 1 ( )( , ) ( ) 0
4
Ni i i i
aia a i a i a i a i
Ni i i i
bib b i b i b i b i
w w w w d NG x s c N
x s x s x s x s x s a
w w w w d NG x s c N
x s x s x s x s x s b
0
0
)(
)(1
1
11
1
1
14
4
14
14
24
24
34
34
14
4
14
14
24
24
34
34
Nd
Nc
b
a
sx
w
sx
w
sx
w
sx
w
sx
sx
w
sx
w
sx
w
sx
w
sx
iib
i
ib
i
ib
i
ib
i
b
iia
i
ia
i
ia
i
ia
i
a
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Numerical and analytic ways to determine c(N) and d(N)
0 2 4 6 8 10
N
0
0.02
0.04
0.06
0.08
0.1c (N ) a n d d (N )
A n a ly tica l c (N )N u m er ica l c (N )A n a ly tica l d (N )N u m er ica l d (N )
Coefficients
c(N)
d(N)
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Derivation of 3-D Green’s function by using the Trefftz Method
1G
2G
11 GG
22 GG
PART 1 PART 2
PART 1
11 0
11 0
1 1 ( )!cos ( ) (cos ) (cos ) ,
4 ( )!( , )
1 1 ( )!cos ( ) (cos ) (cos ) ,
4 ( )!
nnm m
m n n snn ms s
Fnn
m msm n n sn
n m
n mm P P R
R n m RG x s
Rn mm P P R
n m
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Boundary value problem
1( , )
TN
T j jj
G x s c
11 GG
22 GG
Interior:
)(cos)sin(),(cos)cos(,1 m
n
nm
n
n PmPm
Exterior:
)(cos)sin(),(cos)cos(,1 )1()1(
m
n
nm
n
n PmPm
( 1)0000
1 0
( 1)
( , ) [ (cos )cos( ) (cos )cos( )
(cos )sin( ) (cos )sin( )]
nn m n m
T nm n nm nn m
n m n m
nm n nm n
BG x s A A P m B P m
C P m D P m
00
00
4( )
4
s
s
s
s
R a
R b aAB a b R
R b a
2 1 2 1
1 2 1 2 1
2 1 2 1 2 1
1 2 1 2 1
( )!(cos )cos( )
4 ( )!
( )!(cos )cos( )
4 ( )!
n nmm s
nn n nsnm
n n nnm s mm
nn n ns
R an mP m
n m R b aA
B a b Rn mP m
n m R b a
2 1 2 1
1 2 1 2 1
2 1 2 1 2 1
1 2 1 2 1
( )!(cos )sin( )
4 ( )!
( )!(cos )sin( )
4 ( )!
n nmm s
nn n n
nm
n n nnm s mm
nn n ns
R an mP m
n m R b aC
D a b Rn mP m
n m R b a
PART 2
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PART 1 + PART 2 :
1G2G11 GG
22 GG
( , ) ( , ) ( , )F TG x s G x s G x s
11 0
11 0
1 1 ( )!cos ( ) (cos ) (cos ) ,
4 ( )!( )
1 1 ( )!cos ( ) (cos ) (cos ) ,
4 ( )!
nnm m
m n n snn ms s
Fnn
m msm n n sn
n m
n mm P P R
R n m RG x
Rn mm P P R
n m
( 1)0000
1 0
( 1)
( , ) [ (cos )cos( ) (cos )cos( )
(cos )sin( ) (cos )sin( )]
nn m n m
T nm n nm nn m
n m n m
nm n nm n
BG x s A A P m B P m
C P m D P m
( 1)0000
1 0
( 1)
1( , ) [ (cos )cos( ) (cos )cos( )
4
(cos )sin( ) (cos )sin( )],
nn m n m
nm n nm nn m
n m n m
nm n nm n
BG x s A A P m B P m
x s
C P m D P m
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Results
-10 -8 -6 -4 -2 0 2 4 6 8 10-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10-10
-8
-6
-4
-2
0
2
4
6
8
10
Trefftz method (x-y plane) Image method (x-y plane)
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Outline
Motivation and literature reviewDerivation of 2-D Green’s function
by using the image methodTrefftz method and MFS
Image method (special MFS)Trefftz method
Equivalence of solutions derived by Trefftz method and MFS
Boundary value problem without sourcesConclusions
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Trefftz solution
( 1)0000
1 0
( 1)
1( , ) [ (cos )cos( ) (cos )cos( )
4
(cos )sin( ) (cos )sin( )],
nn m n m
nm n nm nn m
n m n m
nm n nm n
BG x s A A P m B P m
x s
C P m D P m
2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1
1 1 2 1 2 11 0
( )1 1( , ) + +
4 ( ) ( )
( )! + cos[ ( )] (cos )
4 ( )! ( )
s s
s s
n n n n n n n nnmm s s
nn n n nn m s
R a a b RG x s
x s R b a R b a
R a a b a Rn mm P
n m R b a
Without loss of generality
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Mathematical equivalence the Trefftz method
and MFS Trefftz method series expansion
2 1 2 1 2 1 2 1
2 1 2 1 1 2 1 2 1 1 1 2 1 2 1 1 2 1 2 1(cos )
( ) ( ) ( )
n n n n n n n nm
nn n n n n n n n n n n n
R a a b a RP
b a R b a R b a b a
Image method series expansion
1212
12
12
12
24
222
121
9
91
5
51
1
1
1
nn
nn
n
n
n
nn
n
nnn
n
nn
n
n
n
n
n
n
ab
R
ba
bR
b
Ra
aR
b
b
R
R
b
Rw
Rw
Rw
s s1s2s4 s3 s5 s9s7
)(1121211
1212
12
12
11
12
12
42
1
2
110
1016
612
2
nnnn
nn
n
n
nn
n
nnn
n
nn
n
n
n
n
n
n
n
abR
ba
baR
a
Rb
a
bR
a
R
a
R
aRw
Rw
Rw
s s1 s3s2s4s6s8s10
s s1s2s4 s3 s5 s9s7
)(1)()(
12121
12
12
12
112
12
144
44
2
2
122
22
1
7
71
3
3
nnn
nn
n
n
nn
nn
nn
nn
nn
nn
n
n
n
n
abR
a
baRb
a
Rb
a
a
b
Rb
a
a
b
Rw
Rw
s s1 s3s2s4s6s8s10
)(1)()(
12121
12
12
12
112
12
14
4
2
2
12
2
1
881
44
nnn
nn
n
n
nn
nn
nn
nn
nn
nn
n
n
n
n
ab
Ra
ba
bRa
b
Ra
b
a
b
Ra
b
aRw
Rw
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Equivalence of solutions derived by Trefftz method and image method (special MFS)
Trefftz method MFS (image method)
1, cos( ) (cos ),
sin( ) (cos )
, , 0,1,2,3, , ,
1,2, , ,
n m
n
n m
n
m P
m P
m
n
r f q
r f q
= ¥
= ¥
K L
L
1,
j
j Nx s
-Î
-
Equivalence
Addition theorem
Linkage
3-D
True source
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Equivalence of Trefftz method and MFS
3-D
Trefftz method MFS (image method)
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Conclusions
The analytical solutions derived by using the Trefftz method and MFS were proved to be mathematically equivalent for Green’s functions of the concentric sphere.
In the concentric sphere case, we can find final two frozen image points (one at origin and one at infinity). Their singularity strength can be determined numerically and analytically in a consistent manner.
It is found that final image points terminate at the two focuses of the bipolar (bispherical) coordinates for all the cases.
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References
J. T. Chen, Y. T. Lee, S. R. Yu and S. C. Shieh, 2009, Equivalence between Trefftz method and method of fundamental solution for the annular Green’s function using the addition theorem and image concept, Engineering Analysis with Boundary
Elements, Vol.33, pp.678-688.
J. T. Chen and C. S. Wu, 2006, Alternative derivations for the Poisson integral formula, Int. J. Math. Edu. Sci. Tech, Vol.37, No.2, pp.165-185.
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