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 Professor Department of Harbor and River Enginee ring, National Taiwan Ocean University Sep. 2-4, 2009 New Forest, UK BEM/MRM 31
32
Embed
1 Equivalence between the Trefftz method and the method of fundamental solutions for the Green’s function of concentric spheres using the addition theorem.
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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
2
Outline
Numerical methods
Trefftz method and MFS Image method (special MFS)
Trefftz method
Equivalence of solutions derived by
Trefftz method and MFS
Conclusions
3
Numerical methods
Numerical methods
Boundary Element MethodFinite Element Method Meshless Method
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
26
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
27
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
28
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
29
Equivalence of Trefftz method and MFS
3-D
Trefftz method MFS (image method)
30
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.
31
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.
32
Thanks for your kind attentionsYou can get more information from our website