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Linton and Evans’ approach Multi-pole Trefftz method Reporter ： Shing-Kai Ka o Advisor: Jeng-Tzong Ch en Date ： 2009/04/01
Linton and Evans’ approachMulti-pole Trefftz method
Reporter ： Shing-Kai Kao Advisor: Jeng-Tzong Chen
Date ： 2009/04/01
MMSS VV
MSVLAB, HRE, NTOU
Outline Trefftz method Addition theorem Problem statement Multi-pole Trefftz method
MMSS VV
MSVLAB, HRE, NTOU
Outline Trefftz method Addition theorem Problem statement Multi-pole Trefftz method
MMSS VV
MSVLAB, HRE, NTOU
Trefftz method-interior problem
1R
1
1 1 1 1
1
( ) 0 0 0
0 ( ) 0 0
0 0 0 ( )
p p p
p p p
p p p
J kR c a
J kR c a
J kR c a
( ) ( ) inn n
n
u x c J kr e
2 2 ( ) 0k u x
( ) ,inn
n
u x a e x B
MMSS VV
MSVLAB, HRE, NTOU
Trefftz method-exterior problem
2
1 2 1 1
2
( ) 0 0 0
0 ( ) 0 0
0 0 0 ( )
p p p
p p p
p p p
H kR d b
H kR d b
H kR d b
2 2 ( ) 0k u x
2R
(1)( ) ( ) inn n
n
u x d H kr e
( ) ,inn
n
u x b e x B
MMSS VV
MSVLAB, HRE, NTOU
Trefftz method-annular problem
2R1R
(1)1 1
(1)1 1
(1)2 2
(1)2 2
( ) 0 0 0 0
0 0 0 0
0 0 ( ) 0 0
( ) 0 0 0 0
0 0 0 0
0 0 ( ) 0 0
p pp p
p pp p
p pk p
p pp p
c aJ kR H kR
c aJ kR H kR
d bJ kR H kR
d bJ kR H kR
(1)( ) ( ) ( )in inn n n n
n n
u x c J kr e d H kr e
2 2 ( ) 0k u x
1( ) ,inn
n
u x a e x B
2( ) ,inn
n
u x b e x B
MMSS VV
MSVLAB, HRE, NTOU
Outline Trefftz method Addition theorem Problem statement Multi-pole Trefftz method
MMSS VV
MSVLAB, HRE, NTOU
Addition theorem
2 1 r r b
2
12r
b
1r
1
2
1
( )1 1
2( )
1 1
( ) ( ) ,
( )
( ) ( ) ,
ini m nm n n
nimm
ini m nm n n
n
H kb e J kr e r b
H kr e
J kb e H kr e r b
2 1
1
( )2 1
( )1
( ) ( ) ( )
( ) ( )
im ini m nm m n n
n
i m n inm n n
n
J kr e J kb e J kr e
J kr e J kb e
MMSS VV
MSVLAB, HRE, NTOU
Outline Trefftz method Addition theorem Problem statement Multi-pole Trefftz method
MMSS VV
MSVLAB, HRE, NTOU
Problem statement
2R1R
2 2 ( ) 0k u x
1( ) 0,u x x B
2( ) 0,u x x B
1
2
2
0.5
0.5
R
R
b
b
MMSS VV
MSVLAB, HRE, NTOU
Outline Trefftz method Addition theorem Problem statement Multi-pole Trefftz method
MMSS VV
MSVLAB, HRE, NTOU
Multi-pole Trefftz method
2R1R
1 2(1)1 2( ) ( ) ( )in in
n n n nn n
u x c J kr e d H kr e
2 2 ( ) 0k u x
11( ) ,in
nn
u x a e x B
22( ) ,in
nn
u x b e x B
1 2 1(1)1 2( ) ( )in ih in
n n h h nn h n
c J kR e d H kr e a e
1 1 1( ) (1)1 1( ) in im ini n m
n n n n m m nn h m n
c J kR e d J kb e H kR e a e
1 1 1(1) ( )1 1( ) in in ini m n
n n n m m n nn n m n
c J kR e H kR e d J kb e a e
MMSS VV
MSVLAB, HRE, NTOU
Multi-pole Trefftz method
(1) ( ) (1) ( )1 1 1
(1) ( ) (1) ( )1 1 1
( )( ) ( )( ) (1)2 2 2
2
( ) 0 0
0 0 0
0 0 ( )
( ) ( ) 0 0
0 0 0
( )
i p p i p pp p p p p p p
i p p i p pp p p p p p p
i p p i p pp p p p p p p
p
J kR H kR J kb e H kR J kb e
J kR H kR J kb e H kR J kb e
J kR J kb e J kR J kb e H R
J kR J
( )( ) ( )( ) (1)2 2( ) 0 0
k k
k k
k k
i p p i p pk kp p p p p p
c a
c a
d b
d bkb e J kR J kb e H R
1 1 1(1) ( )1 1( ) in in ini m n
n n n m m n nm
c J R e H kR e d J kb e a e
(1) ( )1 1
( )( ) (1)2 2
( )
( )
i m nn n n m m n n
m
i m nn m m n n n n
m
c J R H kR d J kb e a
J kR c J kb e d H R b
MMSS VV
MSVLAB, HRE, NTOU
2D Problem
3.85
2.1
0.41
MMSS VV
MSVLAB, HRE, NTOU
3D problemInterior problem Exterior problem
1r
1o
2o
o
2o
o
r2r
1o
radiation
scattering
concentric sphere
eccentric sphere
rr
2r
2r
1r
r
2r
1r
1r
MMSS VV
MSVLAB, HRE, NTOU
~Thanks for your kind attentions~
