Numerical Methods of Electromagnetic Field Theory I (NFT I ... · Numerical Methods of...
33
1 Numerical Methods of Electromagnetic Field Theory I (NFT I) Numerische Methoden der Elektromagnetischen Feldtheorie I (NFT I) / 8th Lecture / 8. Vorlesung Universität Kassel Fachbereich Elektrotechnik / Informatik (FB 16) Fachgebiet Theoretische Elektrotechnik (FG TET) Wilhelmshöher Allee 71 Büro: Raum 2113 / 2115 D-34121 Kassel Dr.-Ing. René Marklein [email protected] http://www.tet.e-technik.uni-kassel.de http://www.uni-kassel.de/fb16/tet/marklein/index.html University of Kassel Dept. Electrical Engineering / Computer Science (FB 16) Electromagnetic Field Theory (FG TET) Wilhelmshöher Allee 71 Office: Room 2113 / 2115 D-34121 Kassel
Transcript of Numerical Methods of Electromagnetic Field Theory I (NFT I ... · Numerical Methods of...
1
Num
eric
al M
etho
ds o
f El
ectr
omag
netic
Fie
ld T
heor
y I (
NFT
I)N
umer
isch
e M
etho
den
der
Elek
trom
agne
tisch
en F
eldt
heor
ie I
(NFT
I) /
8th
Lect
ure
/ 8.
Vorle
sung
Univ
ersi
tät K
asse
lFa
chbe
reic
h El
ektr
otec
hnik
/ In
form
atik
(F
B 16
)Fa
chge
biet
The
oret
isch
e El
ektr
otec
hnik
(F
G T
ET)
Wilh
elm
shöh
er A
llee
71Bü
ro: R
aum
211
3 /
2115
D-3
4121
Kas
selDr.
-Ing
. Ren
éM
arkl
ein
mar
klei
n@un
i-ka
ssel
.de
http
://w
ww
.tet.e
-tec
hnik
.uni
-kas
sel.d
eht
tp:/
/ww
w.u
ni-k
asse
l.de/
fb16
/tet
/mar
klei
n/in
dex.
htm
l
Univ
ersi
ty o
f Kas
sel
Dep
t. El
ectr
ical
Eng
inee
ring
/ Co
mpu
ter
Scie
nce
(FB
16)
Elec
trom
agne
tic F
ield
The
ory
(FG
TET
)W
ilhel
msh
öher
Alle
e 71
Off
ice:
Roo
m 2
113
/ 21
15D
-341
21 K
asse
l
2
3-D
FD
TD –
Der
ivat
ion
of th
e D
iscr
ete
Equa
tions
/
3D-F
DTD
–Ab
leitu
ng d
er d
iskr
eten
Gle
ichu
ngen
The
first
two
Max
wel
l’s E
quat
ions
are
in d
iffer
entia
l for
m /
D
ie e
rste
n be
iden
Max
wel
lsch
en G
leic
hung
en la
uten
in D
iffer
entia
lform
:
m e
(,)
(,)
(
,)
(,)
(,
) (
,)
tt
tt
tt
tt∂
=−∇
−∂ ∂
=∇
−∂
BR
×ER
JR
DR
×HR
JR
In C
arte
sian
Coo
rdin
ates
we
find
for
the
Curl
oper
ator
app
lied
to E
and
H /
Im
Kar
tesi
sche
n Ko
ordi
nate
nsys
tem
find
en w
ir fü
r de
n Ro
tatio
nsop
erat
or a
ngew
ende
t auf
E u
nd H
:
(,)
(,
)(
,)(
,)
(,)
(,
)(
,)(
,)
(,)
(,)
(,)
(,
)(
,)
(,
)
(,)
(,
)
xy
y
xy
z
yy
xx
zz
xy
z
xy
y
xy
z
yz
tx
yz
Et
Et
Et
Et
Et
Et
Et
Et
Et
yz
zx
xy
tx
yz
Ht
Ht
Ht
Ht
Ht
y
∂∂
∂∇
=∂
∂∂
∂∂
∂∂
∂∂
=−
+−
+−
∂∂
∂∂
∂∂
∂∂
∂∇
=∂
∂∂
∂∂
=−
∂ee
e
×ER
RR
R
RR
RR
RR
ee
e
ee
e
×HR
RR
R
RR
(,)
(,)
(,)
(,)
yx
xz
xy
zH
tH
tH
tH
tz
zx
xy
∂
∂
∂∂
+−
+−
∂∂
∂∂
∂
RR
RR
ee
e
3
3-D
FD
TD –
Der
ivat
ion
of th
e D
iscr
ete
Equa
tions
/
3D-F
DTD
–Ab
leitu
ng d
er d
iskr
eten
Gle
ichu
ngen
If w
e in
sert
the
last
exp
ress
ions
into
the
first
two
Max
wel
l’s e
quat
ions
are
in d
iffer
entia
l for
m r
ead
/ W
enn
wir
die
letz
ten
Ausd
rück
e in
the
erst
en b
eide
n M
axw
ells
chen
Gle
ichu
ngen
in D
iffer
entia
lform
ein
setz
en,
erha
lten
wir:
m(
,)
(,
) (
,)
(,
)(
,)
(,
)
(,
)(
,)
(,
)(
,)
(,
)(
,)
xy
zx
yz
yy
xx
zz
xy
z
tt
tt
Bt
Bt
Bt
tE
tE
tE
tE
tE
tE
ty
zz
xx
y
∂=−∇
−∂ ∂
+
+
∂
∂
∂
∂∂
∂∂
=−
−+
−+
−
∂
∂∂
∂∂
∂
BR
×ER
JR
Re
Re
Re
RR
RR
RR
ee
e
mm
m
e
(
,)
(,
)(
,)
(,
)
(
,)
(,
)
(,
)(
,)
(,
)
(,
)(
,)
(,
)
xy
zx
yz
xy
zx
yz
yx
zx
Jt
Jt
Jt
tt
tt
Dt
Dt
Dt
tH
tH
tH
ty
zz
−+
+
∂=
∇−
∂ ∂
+
+
∂
∂
∂
∂=
−+
∂∂
∂
Re
Re
Re
DR
×HR
JR
Re
Re
Re
RR
Re
ee
e
(,
)(
,)
(,
)
(,
)(
,)
(,
)
yx
zy
z
xy
zx
yz
Ht
Ht
Ht
xx
y
Jt
Jt
Jt
∂
∂∂
−+
−
∂
∂∂
−+
+
RR
Re
e
Re
Re
Re
Six
deco
uple
d sc
alar
equ
atio
ns! /
Se
chs
entk
oppe
lte s
kala
re G
leic
hung
en!
4
3-D
FD
TD –
Der
ivat
ion
of th
e D
iscr
ete
Equa
tions
/
3D-F
DTD
–Ab
leitu
ng d
er d
iskr
eten
Gle
ichu
ngen
If w
e in
sert
the
last
exp
ress
ions
into
the
first
two
Max
wel
l’s e
quat
ions
are
in d
iffer
entia
l for
m w
e re
ad /
W
enn
wir
die
letz
ten
Ausd
rück
e in
die
ers
ten
beid
en M
axw
ells
chen
Gle
ichu
ngen
in D
iffer
entia
lform
ei
nset
zen,
erh
alte
n w
ir:
m m m e
(,
)(
,)
(,
)(
,)
(,
)(
,)
(,
)(
,)
(,
)(
,)
(,
)(
,)
(,
)(
,)
(,
)
(,
)
(
yz
xx
xz
yy
yx
zz
yz
xx
y
Et
Et
Bt
Jt
ty
z
Et
Et
Bt
Jt
tz
xE
tE
tB
tJ
tt
xy
Ht
Ht
Dt
Jt
ty
z
D t
∂
∂
∂=−
−−
∂∂
∂
∂
∂∂
=−
−−
∂∂
∂
∂
∂
∂=−
−−
∂∂
∂
∂
∂
∂=
−−
∂∂
∂
∂ ∂
RR
RR
RR
RR
RR
RR
RR
RR
Re e
(,
)(
,)
,)
(
,)
(,
)(
,)
(,
)
(,
)
xz
y
yx
zz
Ht
Ht
tJ
tz
xH
tH
tD
tJ
tt
xy
∂∂
=−
−
∂
∂
∂
∂
∂=
−−
∂∂
∂
RR
R
RR
RR
5
3-D
FD
TD –
Der
ivat
ion
of th
e D
iscr
ete
Equa
tions
/
3D-F
DTD
–Ab
leitu
ng d
er d
iskr
eten
Gle
ichu
ngen
m m m e
(,
)(
,)
(,
)(
,)
(,
)(
,)
(,
)(
,)
(,
)(
,)
(,
)(
,)
(,
)(
,)
(,
)
(,
)
yz
xx
xz
yy
yx
zz
yz
xx
Et
Et
Ht
Jt
ty
z
Et
Et
Ht
Jt
tz
xE
tE
tH
tJ
tt
xy
Ht
Ht
Et
Jt
ty
z
tµ µ µ ε
∂
∂
∂=−
−−
∂∂
∂
∂
∂∂
=−
−−
∂∂
∂
∂
∂
∂=−
−−
∂∂
∂
∂
∂
∂=
−−
∂∂
∂
∂ ∂
RR
RR
RR
RR
RR
RR
RR
RR
e e
(,
)(
,)
(,
)
(,
)
(,
)(
,)
(,
)
(,
)
xz
yy
yx
zz
Ht
Ht
Et
Jt
zx
Ht
Ht
Et
Jt
tx
y
ε ε
∂∂
=−
−
∂
∂
∂
∂
∂=
−−
∂∂
∂
RR
RR
RR
RR
xEyE
zE
xH
zH
yH
m e
,1,
2,3
,1,
2,3
ii
ii
xx
xx
HJ
i
EJ
i
==
==
(,
)(
,)
(,
)(
,)
(,
)(
,)
xx
yy
zz
Bt
Ht
Bt
Ht
Bt
Ht
µ µ µ
= = =
RR
RR
RR
(,
)(
,)
(,
)(
,)
(,
)(
,)
xx
yy
zz
Dt
Et
Dt
Et
Dt
Et
ε ε ε
= = =
RR
RR
RR
Cons
titut
ive
equa
tion
for
hom
ogen
eous
isot
ropi
c m
ater
ials
/
Kons
titui
eren
de G
leic
hung
en fü
r ho
mog
ene
isot
rope
Mat
eria
lien:
6
3-D
FD
TD –
Der
ivat
ion
of th
e D
iscr
ete
Equa
tions
/
3D-F
DTD
–Ab
leitu
ng d
er d
iskr
eten
Gle
ichu
ngen
m(
,)
(,
)(
,)
(,
)y
zx
xE
tE
tH
tJ
ty
zµ
∂
∂
=−
−−
∂∂
RR
RR
(,
)(
,)
xx
Ht
Ht
t∂=
∂R
R
()d yE
()b zE
()m x
H
()f zE
()u yE
()
()
()
()
mm (
)(
)2
()
()
2
(,
)()
(,
)()
(,
)()
()
(,
)()
()
mx
x mx
x fb
zz
z
du
yy
y
Ht
Ht
Jt
Jt
Et
Et
Et
yy
y
Et
Et
Et
zz
z
µ= =
∂−
=+
∆
∂
∆
∂−
=+
∆
∂
∆
R R R R
○ ○
()d yE
()b zE
()m x
H(
)f zE
()u yE
()
()
()
()
()
()
m()
()()
()()
()d
uf
by
ym
mz
zx
xE
tE
tE
tE
tH
tJ
ty
zµ
−−
=−
+−
∆∆
A pa
rt o
f the
dis
cret
e cu
rl op
erat
or /
Ei
n Te
il de
s di
skre
ten
Rota
tions
oper
ator
s
7
2-D
EM
Wav
e Pr
opag
atio
n–
2-D
FD
TD –
TM a
nd T
E Ca
se /
2D E
M W
elle
naus
brei
tung
–2D
-FD
TD –
TM-
und
TE-F
all (
)ng
G∈
()n
gG
∈
1xx
= 3zx
=
2-D
TE
Case
/ 2
D-T
E-Fa
ll
Gnn
=
Gn(
)n xE
()n zE
()n y
H
2yx
=(
)ng
G∈
()n
gG
∈
1xx
= 3zx
=
2-D
TM
Cas
e /
2D-T
M-F
all
Gnn
=
Gn
()n x
H
()n z
H(
)n yE
2yx
=
m m
e
(,
)(
,)
(
,)
(,
)(
,)
(,
)
(,
)(
,)
(,
)
(,
)
yx
x
yz
z
xz
yy
xz
Et
Ht
Jt
tz
Et
Ht
Jt
tx
Ht
Ht
Et
Jt
tz
xx
z
µ µ ε
∂∂
=−
∂∂
∂∂
=−
−∂
∂
∂∂
∂
=
−−
∂∂
∂
=
+
RR
R
RR
R
RR
RR
Re
e
m
e e
(,
)(
,)
(,
)(
,)
(,
)(
,)
(
,)
(,
)(
,)
(,
)
xz
yy
yx
x
yz
z
xz
Et
Et
Ht
Jt
tz
x
Ht
Et
Jt
tz
Ht
Et
Jt
tx
xz
µ ε ε
∂∂
∂
=−
−−
∂∂
∂
∂∂
=−
−∂
∂∂
∂=
−∂
∂=
+
RR
RR
RR
R
RR
R
Re
e
GG
⊥
Dua
l ort
hogo
nal
grid
sys
tem
in s
pace
/
Dua
l-or
thog
onal
esG
itter
syst
em im
Rau
m
8
2-D
EM
Wav
e Pr
opag
atio
n–
2-D
FD
TD –
TM C
ase/
2D E
M W
elle
naus
brei
tung
–2D
-FD
TD –
TM-F
all
()n
gG
∈
()n
gG
∈
1xx
= 3zx
=
2-D
TM
Cas
e /
2D-T
M-F
all
Gnn
=
Gn
()n x
H
()n z
H(
)n yE
2yx
=
m
m
e
(,
)(
,)
(,
)
(,
)(
,)
(,
)
(,
)(
,)
(,
)
(,
)
yx
x
yz
z
xz
yy
xz
Et
Ht
Jt
tzE
tH
tJ
tt
x
Ht
Ht
Et
Jt
tz
xx
z
µ µ ε
∂∂
=−
∂∂
∂∂
=−
−∂
∂
∂∂
∂
=
−−
∂∂
∂
=
+RR
R
RR
R
RR
RR
Re
e
Two-
dim
ensi
onal
sta
gger
ed g
rid s
yste
m in
the
2-D
TM
cas
e /
Zwei
dim
ensi
onal
esve
rset
ztes
Gitt
ersy
stem
im2D
-TM
-Fal
l
9
Impl
emen
tatio
n of
Bou
ndar
y Co
nditi
ons
/ Im
plem
entie
rung
von
Ran
dbed
ingu
ngen
Boun
dary
con
ditio
n fo
r a
perf
ectly
ele
ctric
ally
con
duct
ing
(PEC
)mat
eria
l /
Rand
bedi
ngun
g fü
r ei
n id
eal e
lekt
risch
leite
ndes
Mat
eria
l
(,,
)
(,,
)
01
0
t tny
tt
nyE
nN
E
=
≤
≤
=
ii ii
Plan
e w
ave
boun
dary
con
ditio
n fo
r a
vert
ical
inci
dent
pla
ne w
ave
/ Eb
ene-
Wel
len-
Rand
bedi
ngun
g fü
r ei
ne v
ertik
al e
infa
llend
e eb
ene
Wel
le
(2,
,)
(3,
,)
(1,
,)
(2,
,2)
1 1
zt
zt
xz
tx
zt
nn
nn
yy
zz
Nnn
Nnn
tt
yy
EE
nN
nN
EE
−−
−
=
≤≤
≤
≤=
PW B
C /
EW-R
BPW
BC
/ EW
-RB
PEC
BC /
IE
L-RB
PEC
BC /
IE
L-RB
Slit
/ Sc
hlitz
PEC
BC /
IE
L-RB
Plan
e w
ave
exci
tatio
n /
Eben
e-W
elle
n-An
regu
ng
10
2-D
EM
Wav
e Pr
opag
atio
n–
2-D
FD
TD –
TM C
ase/
2D E
M W
elle
naus
brei
tung
–2D
-FD
TD –
TM-F
all
()
0n x
H=
()
0n z
H=
()
0n yE=
Gho
st c
ompo
nent
s w
hich
are
al
loca
ted
outs
ide
the
sim
ulat
ion
area
/
Gei
ster
kom
pone
nten
, wel
che
auße
rhal
b de
s Si
mul
atio
nsge
biet
es
liege
n
Gho
st g
rid c
ells
/
Gei
ster
gitt
erze
llen
Sim
ulat
ion
area
/
Sim
ulat
ions
gebi
et
11
2-D
EM
Wav
e Pr
opag
atio
n–
2-D
FD
TD –
TM C
ase/
2D E
M W
elle
naus
brei
tung
–2D
-FD
TD –
TM-F
all
Gho
st g
rid c
ells
/
Gei
ster
gitt
erze
llen
Sim
ulat
ion
area
/
Sim
ulat
ions
gebi
et
(2,
,,)
(3,
,,)
zt
zt
nn
nn
yy
EE
=
0yE=
0yE=
0yE=
0yE=
Plan
e w
ave
exci
tatio
n /
Eben
e-W
elle
n-An
regu
ngSl
it /
Schl
itz
12
2-D
TM
FD
TD –
Diff
ract
ion
on a
Sin
gle
Slit
/2D
-TM
-FD
TD –
Beug
ung
an e
inem
Spa
lt
13
2-D
TM
FD
TD –
Diff
ract
ion
on a
Sin
gle
Slit
/2D
-TM
-FD
TD –
Beug
ung
am S
palt
Wav
e fie
ld m
ovie
of t
he H
xfie
ld c
ompo
nent
/
Wel
lenf
eldf
ilmde
rH x
-Fel
dkom
pone
nte
Wav
e fie
ld m
ovie
of t
heH z
field
com
pone
nt /
W
elle
nfel
dfilm
der
H z-F
eldk
ompo
nent
e
Wav
e fie
ld m
ovie
of t
heE y
field
com
pone
nt /
W
elle
nfel
dfilm
der
E y-F
eldk
ompo
nent
e
14
2-D
TM
FD
TD –
Diff
ract
ion
on a
Dou
ble
Slit
/2D
-TM
-FD
TD –
Beug
ung
am D
oppe
lspa
lt
15
2-D
TM
FD
TD –
Diff
ract
ion
on a
Dou
ble
Slit
/2D
-TM
-FD
TD –
Beug
ung
am D
oppe
lspa
lt
Wav
e fie
ld m
ovie
of t
he H
xfie
ld c
ompo
nent
/
Wel
lenf
eldf
ilmde
rHx-
Feld
kom
pone
nte
Wav
e fie
ld m
ovie
of t
heH z
field
com
pone
nt /
W
elle
nfel
dfilm
der
H z-F
eldk
ompo
nent
e
Wav
e fie
ld m
ovie
of t
heE y
field
com
pone
nt /
W
elle
nfel
dfilm
der
E y-F
eldk
ompo
nent
e
16
Phot
onic
Cry
stal
s /
Phot
onis
che
Kris
talle
Joan
nopo
ulos
, J. D
., R.
D. M
eade
, J.
N. W
inn:
Phot
onic
Cry
stal
s –
Mol
ding
the
Flow
of
Ligh
t. Pr
ince
ton
Univ
ersi
ty
Pres
s, P
rince
ton,
199
5.
John
son,
S. G
.: Ph
oton
ic C
ryst
als:
The
Ro
ad fr
om T
heor
y to
Pr
actic
e .
Kluw
er A
cade
mic
Pr
ess,
200
1.
Link
s:
Phot
onic
Cry
stal
s Re
sear
ch a
t MIT
Hom
epag
e of
Pro
f. Sa
jeev
John
, Uni
vers
ity o
f Tor
onto
, Can
ada
17
2-D
TM
FD
TD –
Phot
onic
Cry
stal
s /
2D-T
M-F
DTD
–Ph
oton
isch
e Kr
ista
lle
()
r ()
r
Rela
tive
perm
ittiv
ity o
f the
bac
kgro
und
1
Rela
tive
Perm
ittiv
ität d
es H
inte
rgru
ndes
Rela
tive
perm
ittiv
ity o
f the
rods
11
.4Re
lativ
e Pe
rmitt
ivitä
t der
Stä
be
b r
ε ε
= =
18
2-D
TM
FD
TD –
Phot
onic
Cry
stal
s /
2D-T
M-F
DTD
–Ph
oton
isch
e Kr
ista
lle
Wav
e fie
ld m
ovie
of t
he H
xfie
ld c
ompo
nent
/
Wel
lenf
eldf
ilmde
rH x
-Fel
dkom
pone
nte
Wav
e fie
ld m
ovie
of t
heH z
field
com
pone
nt /
W
elle
nfel
dfilm
der
H z-F
eldk
ompo
nent
e
Wav
e fie
ld m
ovie
of t
heE y
field
com
pone
nt /
W
elle
nfel
dfilm
der
E y-F
eldk
ompo
nent
e
19
2-D
TM
FD
TD –
Phot
onic
Cry
stal
s /
2D-T
M-F
DTD
–Ph
oton
isch
e Kr
ista
lle
Wav
e fie
ld m
ovie
of t
he H
xfie
ld c
ompo
nent
/
Wel
lenf
eldf
ilmde
rH x
-Fel
dkom
pone
nte
Wav
e fie
ld m
ovie
of t
heH z
field
com
pone
nt /
W
elle
nfel
dfilm
der
H z-F
eldk
ompo
nent
e
Wav
e fie
ld m
ovie
of t
heE y
field
com
pone
nt /
W
elle
nfel
dfilm
der
E y-F
eldk
ompo
nent
e
20
2-D
TM
FD
TD –
Phot
onic
Cry
stal
s /
2D-T
M-F
DTD
–Ph
oton
isch
e Kr
ista
lle
21
2-D
TM
FD
TD –
Phot
onic
Cry
stal
s /
2D-T
M-F
DTD
–Ph
oton
isch
e Kr
ista
lle
22
FDTD
and
FIT
/ F
DTD
und
FIT
FDTD
:
Fi
nite
Diff
eren
ce T
ime
Dom
ain
/ Fi
nite
Diff
eren
zen
im
Zeitb
erei
chFI
T
:
Fi
nite
Inte
grat
ion
Tech
niqu
e/
Fini
te In
tegr
atio
nste
chni
k
m e
e m
(,
)(
,)
(,
)
(,
)
(,
)(
,)
(,
)(
,)
(,
)(
,)
tt
tt
tt
tt
tt
tt
ρ ρ
∂=−∇
−∂ ∂
=∇
−∂ ∇
=
∇=
BR
×ER
JR
DR
×HR
JR
DR
R
BR
R
i i
m e
e m
d(
,)
(,
)(
,)
d d(
,)
(,
)(
,)
d
(,
)
(,
)
(,
)
(,
)
SC
SS
SC
SS
SV
V
SV
V
tt
tt
tt
tt
ttdV
ttdV
ρ ρ=∂ =∂
=∂ =∂
=−
−
=−
= =
∫∫∫
∫∫
∫∫∫
∫∫
∫∫∫∫∫
∫∫∫∫∫
BR
dSER
dRJ
RdS
DR
dSHR
dRJR
dS
DR
dSR
BR
dSR
ii
i
ii
i
i i
FDTD
Max
wel
l’s e
quat
ions
in d
iffer
entia
l for
m /
M
axw
ells
che
Gle
ichu
ngen
in D
iffer
entia
lform
FIT
Max
wel
l’s e
quat
ions
in in
tegr
al fo
rm /
M
axw
ells
che
Gle
ichu
ngen
in In
tegr
alfo
rm
0
00
,,
22
(,
) zz
zz
fz
tfz
tfzt
zz
=
∆∆
+−
−
∂
≈∂
∆
0
00
(,
)d,
2z
z
zz
zfztz
fz
tz
+∆
=
∆
≈
+∆
∫
FD a
ppro
xim
atio
n of
spa
tial a
nd
tem
pora
l der
ivat
ives
/ F
D-
Appr
oxim
atio
n vo
n rä
umlic
hen
und
zeitl
iche
n Ab
leitu
ngen
FIT
appr
oxim
atio
n of
spa
tial a
nd
tem
pora
l int
egra
ls /
FIT
-App
roxi
mat
ion
von
räum
liche
n un
d ze
itlic
hen
Inte
gral
enCe
ntra
l diff
eren
ce a
ppro
xim
atio
n /
Zent
rale
Diff
eren
zen
Appr
oxim
atio
n
Mid
poi
nt r
ule
appr
oxim
atio
n of
a 1
-D in
tegr
al /
M
ittel
punk
tsre
gel-
Appr
oxim
atio
n ei
nes
1D-
Inte
gral
s
23
Def
initi
on o
f Mat
eria
l Cel
ls /
D
efin
ition
der
Mat
eria
lzel
len
1xx
= 3zx
=2
yx
=1
xx
=
3zx
=
2yx
=
1xn=
xx
nN
=1
yn=
yy
nN
=
zz
nN
=1zn=
()n
mM
ater
ial c
ell /
M
ater
ialz
elle (
)
1
Nn
nM
m=
=∑
()
()
()
()
()
()
nn
N
nn
N
mn
mn
→∈
∈
→∈
∈
εR
ε
νR
ν
()
()
()
11
11
1,2,
,
1
xx
yy
zz
xyz
x yx
zx
y
nM
nM
nM
n
nN
NNN
M MN
MNN
=+
−+
−+
−
==
= = =
…
24
3-D
FIT
–D
eriv
atio
n of
the
Dis
cret
e G
rid E
quat
ions
/
3D-F
IT –
Able
itung
der
dis
kret
en G
itter
glei
chun
gen
dd
dx
SS
R=
=
=
dSn
edR
s
()d yE
()b zE
()m xB
()f zE
()u yE
()d yE
()b zE
()m xB
()f zE
()u yE
md
(,
)(
,)
(,
)d
SC
SS
tt
tt
=∂=−
−∫∫
∫∫∫
BR
dSER
dRJ
RdS
ii
i
()
()
()
()
33
()
33
()
(,
)(
,)d
(,
)d
()d
()xS
S
xS m x
S yz
m x
tdS
tS
BtS
Bt
Sy
zyz
Btyz
yz
yz
=∆∆
= =
=+Ο
∆∆
+∆
∆
=∆∆
+Ο
∆∆
+∆
∆
∫∫∫∫ ∫∫
∫∫
nBR
eBR
R
ii
1xx
= 3zx
=2
yx
=
1xx
=3
zx
=
2yx
=
inte
grat
ion
cell
/
-In
tegr
atio
nsze
lle(
)m xBI
()m xBI
()
()
()
()
33
()
33
()
(,
)d()
dd
()
mS
Syz
m
ftS
ft
yz
yz
yz
ftyz
yz
yz
=∆∆
=+Ο
∆∆
+∆
∆
=∆∆
+Ο
∆∆
+∆
∆
∫∫∫∫
R
x∆
z∆
Fiel
d co
mpo
nent
in th
e m
iddl
e /
Feld
kom
pone
nte
in d
er M
itte
Appr
oxim
atio
n er
ror
/ Ap
prox
imat
ions
fehl
er
z∆
y∆
y∆
25
3-D
FIT
–D
eriv
atio
n of
the
Dis
cret
e G
rid E
quat
ions
/
3D-F
IT –
Able
itung
der
dis
kret
en G
itter
glei
chun
gen
dd
dd
d
dd
x
yy
zz
Syz
Ry
Rz
==
==
==
dSn
edR
se
dRs
e
()d yE
()b zE
()m xB
()f zE
()u yE
md
(,
)(
,)
(,
)d
SC
SS
tt
tt
=∂=−
−∫∫
∫∫∫
BR
dSER
dRJ
RdS
ii
i
1xx
=3
zx
=
2yx
=
inte
grat
ion
cell
/
-In
tegr
atio
nsze
lle(
)m xBI
()m xBI
y∆
z∆
()
()
()
()
()
()
()
()
()
()
()
(,)
(,)
(,)
(
,)(
,)
(,)
d(
,)d
(,)
d(
,)d
(,)
d(
,)d
(
,)d
(,)
d
uf
db
uf
db
uf
d
CS
CC
CC
yz
CC
yz
CC
yz
CC
yz
C
tt
t
tt
ty
tz
ty
tz
Ety
Etz
Ety
Et
=∂=
+
++
=+
−−
=+
−−
∫∫
∫∫
∫
∫∫
∫∫
∫∫
∫
ER
dRER
dRER
dR
ER
dRER
dR
ER
eER
e
ER
eER
e
RR
RR
ii
i
ii
ii
ii
()b
Cz
∫
()u
C
()f
C(
)dC
()b
C
(,
)?
CS
t=∂
=∫
ER
dRi
26
3-D
FIT
–D
eriv
atio
n of
the
Dis
cret
e G
rid E
quat
ions
/
3D-F
IT –
Able
itung
der
dis
kret
en G
itter
glei
chun
gen
()
()
()
()
(,
)(
,)d
(,
)d(
,)d
(,
)du
fd
by
zy
zC
SC
CC
Ct
Ety
Etz
Ety
Etz
=∂=
+−
−∫
∫∫
∫∫
ER
dRR
RR
Ri
()
() (
)
() (
)
()
()
()
()
()
()
3(
)
3(
)
3(
)
3(
)
3(
)
()
(,
)d()
d
()
(,
)d()
d
()
(
,)d
()d
(
uu
ff
dd
uy
yC
Cy
u y fz
zC
Cz
f z dy
yC
Cy
d y
EtyE
ty
y
Ety
y
EtzE
tz
z
Etz
z
EtyE
ty
y
Et
=∆ =∆ =∆
=+
∆
=∆
+∆
=
+∆
=∆
+∆
=
+∆
=
∫∫
∫∫
∫∫
R R R
○
○
○
○
○
() (
)
()
()
()
3
3(
)
3(
)
)
(,
)d()
d
()
bb
bz
zC
Cz
b z
yy
EtzE
tz
z
Etz
z
=∆
∆+
∆
=+
∆
=∆
+∆
∫∫
R
○
○
○
()
()
()
()
3(
)
3(
)
(,
)d()
d
()
uu
mC
Cy
m
ftR
ft
yy
fty
y
=∆
=+
∆
=∆
+∆
∫∫
R○
○
Fiel
d co
mpo
nent
in th
e m
iddl
e /
Feld
kom
pone
nte
in d
er M
itte
Appr
oxim
atio
n er
ror
/ Ap
prox
imat
ions
fehl
er
()mf
1xx
=3
zx
=
2yx
=
y∆
()u
C
yα=
27
3-D
FIT
–D
eriv
atio
n of
the
Dis
cret
e G
rid E
quat
ions
/
3D-F
IT –
Able
itung
der
dis
kret
en G
itter
glei
chun
gen
()
()
()
()
()
()
()
()
()
()
()
()
()
()
33
(,
)(
,)d
(,
)d
(,
)d(
,)d
()d
()d
()d
()d
uf
db
uf
db
yz
yz
CS
CC
CC
uf
db
yz
yz
CC
CC
yz
yz
tE
ty
Etz
Ety
Etz
Et
yE
tzE
tyE
tz
yz
=∂
=∆=∆
=∆=∆
=+
−−
=+
−−
+∆
+∆
∫∫
∫∫
∫
∫∫
∫∫
ER
dRR
RR
Ri
○○
()
()
33
()
()
()
()
(,
)()
()
()()
u
fd
by
zy
zC
St
EtyE
tz
EtyE
tz
yz
=∂
=
∆+
∆−
∆−
∆+
∆+
∆
∫
ER
dRi○
○
()d yE
()b zE
()m xB
()f zE
()u yE
1xx
=3
zx
=
2yx
=in
tegr
atio
n ce
ll /
-
Inte
grat
ions
zelle
()m xBI
()m xBI
y∆
z∆
()u
C
()f
C(
)dC
()b
C
28
3-D
FIT
–D
eriv
atio
n of
the
Dis
cret
e G
rid E
quat
ions
/
3D-F
IT –
Able
itung
der
dis
kret
en G
itter
glei
chun
gen
md
(,
)(
,)
(,
)d
SC
SS
tt
tt
=∂=−
−∫∫
∫∫∫
BR
dSER
dRJ
RdS
ii
i
()
()
()
()
mm
m
33
()
m
33
()
m
(,
)(
,)d
(,
)d
()d
()xS
S
xS m x
S yz
m x
tdS
tS
JtS
Jt
Sy
zyz
Jtyz
yz
yz
=∆∆
= =
=+Ο
∆∆
+∆
∆
=∆∆
+Ο
∆∆
+∆
∆
∫∫∫∫ ∫∫
∫∫
nJ
ReJ
R
R
ii
()
()
()
()
()
()
md
()()
()()
()()
dm
uf
db
mx
yz
yz
xB
tyz
EtyE
tzE
tyE
tz
Jtyz
t
∆∆
=−
∆+
∆−
∆−
∆−
∆∆
inte
grat
ion
cell
/
-In
tegr
atio
nsze
lle(
)m xBI
()m xBI
()
()
()
()
()
()
33
(,
)()
()()
()
uf
db
yz
yz
CS
tE
tyE
tzE
tyE
tz
yz
=∂=
∆+
∆−
∆−
∆
+∆
+∆
∫ER
dRi
○○
inte
grat
ion
cell
/
-In
tegr
atio
nsze
lle(
)m xBI
()m xBI
inte
grat
ion
cell
/
-In
tegr
atio
nsze
lle(
)m xBI
()m xBI
29
3-D
FIT
–D
eriv
atio
n of
the
Dis
cret
e G
rid E
quat
ions
/
3D-F
IT –
Able
itung
der
dis
kret
en G
itter
glei
chun
gen
()r zE
()d xE
()r zE
()m yB
()l zE
()u xE
1xx
= 3zx
=2
yx
=
1xx
=
3zx
=2
yx
=
inte
grat
ion
cell
/
-In
tegr
atio
nsze
lle(
)m yBI
()m yBI
x∆
z∆
()m yB
()l zE
()d xE
()u xE
()
()
()
()
()
()
m
d()
d
()()
()()
()
m y
ul
dr
xz
xz
m y
Btyz
t
EtxE
tzE
txE
tz
Jtyz
∆∆
=−−
∆+
∆+
∆−
∆
−∆∆
md
(,
)(
,)
(,
)d
SC
SS
tt
tt
=∂=−
−∫∫
∫∫∫
BR
dSER
dRJ
RdS
ii
i
inte
grat
ion
cell
/
-In
tegr
atio
nsze
lle(
)m yBI
()m yBI
30
3-D
FIT
–D
eriv
atio
n of
the
Dis
cret
e G
rid E
quat
ions
/
3D-F
IT –
Able
itung
der
dis
kret
en G
itter
glei
chun
gen
1xx
= 3zx
=2
yx
=
1xx
=
3zx
=2
yx
=
inte
grat
ion
cell
/
-In
tegr
atio
nsze
lle(
)m zBI
()m zBI
y∆
()f xE(
)m zB
()b xE
()r yE
()l yE
()f xE
()m zB
()b xE
()r yE
()l yE
x∆
()
()
()
()
()
()
m
d()
d
()()
()()
()
m z
br
fl
xy
xy
m z
Btxy
t
EtxE
tyE
txE
ty
Jtxy
∆∆
=−
∆+
∆−
∆−
∆
−∆∆
md
(,
)(
,)
(,
)d
SC
SS
tt
tt
=∂=−
−∫∫
∫∫∫
BR
dSER
dRJ
RdS
ii
i
inte
grat
ion
cell
/
-In
tegr
atio
nsze
lle(
)m zBI
()m zBI
31
3-D
FIT
–D
eriv
atio
n of
the
Dis
cret
e G
rid E
quat
ions
/
3D-F
IT –
Able
itung
der
dis
kret
en G
itter
glei
chun
gen
md
(,
)(
,)
(,
)d
SC
SS
tt
tt
=∂=−
−∫∫
∫∫∫
BR
dSER
dRJ
RdS
ii
i
()
()
()
()
()
()
m
()
()
()
()
()
()
m
()
()
()
(
d()
()()
()()
()d d
()()
()()
()()
d d()
()()
d
mu
fd
bm
xy
zy
zx
mu
ld
rm
yx
zx
zy
mu
fd
zy
zy
Btyz
EtyE
tzE
tyE
tz
Jtyz
t Btyz
EtxE
tzE
txE
tz
Jtyz
t Btyz
EtyE
tzE
t
∆∆
=−
∆+
∆−
∆−
∆−
∆∆
∆∆
=−−
∆+
∆+
∆−
∆−
∆∆
∆∆
=−
∆+
∆−
)(
)(
)m
()()
()b
mz
ztyE
tz
Jtyz
∆+
∆−
∆∆
32
Dua
l-O
rtho
gona
l Grid
Sys
tem
in S
pace
/D
ual-
orth
ogon
ales
Gitt
ersy
stem
im R
aum
1xx
= 3zx
=2
yx
=
3-D
/
3D
()
()n n
gG
mM
∈ ∈ ()n
gG
∈
G
G⊥
Prim
ary
grid
/Se
cond
ary
(dua
l) gr
idPr
imär
es G
itter
Seku
ndär
es (d
uale
s) G
itter
Gnn
=
Gn
()n xE
()n yE
()n zB
()n yB
()y
nM
zE−
() z
nM
yE−
() z
nM
xE−
() x
nM
zE−
() x
nM
yE−
()y
nM
xE−
()
()
()
11
11
1,2,
,
1
xx
yy
zz
xyz
x yx
zx
y
nM
nM
nM
n
nN
NNN
M MN
MNN
=+
−+
−+
−
==
= = =
…
GM
=Pr
imar
y gr
id /
Mat
eria
l grid
Prim
äres
Gitt
erM
ater
ialg
itter
Glo
bal n
ode
num
berin
g /
Glo
bale
Gitt
ernu
mm
erie
rung
()n zE
()n xB
33
End
of L
ectu
re 8
/En
de d
er 8
. Vor
lesu
ng
