517.9
. . , . . , . .


. , . . . .
: , , , .
Abstract. Electromagnetic diffraction problem on dielectric body located in rectangular waveguide is considered. The problem is reduced to volume singular integral equation on the body. Numerical collocation method for solving the equation is considered. The formulas of matrix coefficients for collocation method are presented. Keywords: boundary value problem, electromagnetic scattering, integral equations, numerical method.
- . , , ( ), . . , . , . (Ansis, Quikwave ..), , - [1].
[2-4]. . .
1
. = {: 0 < xi < a, 0 < 2 < b, < X3 < } - . Q (Q - ), ^- 3 3-- () (). () Q , e (Q), e 1 L(Q).
Q Q -. , ,
3
Q 0 ( R )
2 3
C - R , 0, 0nQ
( (1, 2, ) - , (, 0), r > 0, 0 S - -
3
R ). 1 > 0 (0 - ),
3
1 > 0, 2 > 0 (0 - ї ), R \ {1 > 0, 2 > 0} (0 - ї ), r > 0, 0 Q', Q' S - - Q' (0 - ї). , Q' - , 0 - ; Q' , 0 - . Q - Q . , Q - . , Q , Q = 0 . \ Q &0 (> 0), > 0).
E, H L2 loc (), e imt. - j0 L2 loc ().
3
R grad, div, rot .
ї () :
rot H = / + jE;
rot = /^. (1)
: || > C C > 0 (+ї + , -ї )
f E' H
=Z R
Z Q.
p±)eiY
X () pe iY 4^ / ( V2p )x
∞ ( V2p )x
^X(pVpe iY(p)V2Tp j
(2)

" ) = ^ $ X ) , ) > ( ) = , ' ) ^ X ^, ( , 2) X?) , * (, 2) ( & =2 - ^( ) - := {(1, 2): < < , < 2 < } , 2 = / + 2 /2 . -
(2)
Rp
Iq(+) = (pm )p
()
N.
(2) , , . (3) (2), ) .
, :
(4)
, (4) , . , (1), .
H 1()
1/2
H () . .
0 0 - Q , () = 0I, (I - ):
rot 0 = 70∞ + jE, rot 0 = /|100 (5)

0 0 (6)
E∞ P = , HV P = 0.
. . , - - .
7
(1)-(4) (5)-(6) [3]. .
1. ^ (). 0, 0 (). ,
2 1 1
, , (). |, | (0 \\,
|\ 11( \ ). , :
[]| = 0, []| = 0,
[ ] <2.
1 <2 1/2 1/2
1() (<2). , .
2
^ , 0 =2 , . [3]
_ ^ ^
^ 2 ^ _ '1 3 -31 . .
1 = X X- , 8~ 1 2 ^ 1 2 ; (7)
=0 =1 (1 + 80)
_ ^ ^
2 ^ ^ 3 3 . .
2 = X X------ 81 1^ 2 81 1^ (8)
=1 =0 (1 +80)
~
2 ^ 3 3 . . . .
3 = X X-----------------8^212 . (9)
^
=1 =1
=
^2 ( ^2 2
I +1---I 0 , -
) I )
, 1 ^ 0 .
= :
1 \|
=~- + gm (, ), , , (10)
4 | |
gm ( X ).
(, ) = (, ), ( = 1, 2, 3) .
2. Ge
1 eik0 I-1
Ge =-;------ I + g (, ), , , (11)
4 | |
- () g C(Q ) g C( Q).
, , .. g . ,
1 eik0| 1
---------------
4 I I
, .
3
- .
(1)-(4) (5), (6) . (1) :
rot = /^ + jE, ! = /0, (12)

jE = jE + jE . (13)
jE = -/(() 01) - .
, (4), (12)
= /0 A e grad div A e , = rot Ae , (14)
/0

Ae = {Ge (r)jE (y)dy - (15)

.
Ae
AAe + k0 AE = jE . (16)
, Ae , .
(14) (4), (12), .. jE . (13)-(15) -
( ) = 0( ) + 21 () <2
( )
-1
0
() +
- grad div | ()
2
( )
0
-1
()<, 2.
(17)
,
() = 0() + 21 ()
2
( )
0
-1
() +
- grad div | ()
2
( )
0
-1
(), \ 2.
(18)
(18) () \ Q , (), Q - (17). (17)
( ) = 0( ) - /0 1 ()
2
( )
0
-1
(), .
(19)
.

() = 0 () + 000 () + ^2 (), =| - |;
(20)
0( ) =
0
0 -1
. 1, ∞00() = 1--1, 0() = ^2, &0). (21)
4 4
, , :
~~ ----- () = v.p.
J 4, J
1 * 4
2
4
1 (^ -15^ (). (22)
, - (16) :
( ) +
1 - ( 1 ( ) - -^. 1(, ) " ( ) / ( ^
1 0 ] ! 2 _ 0 _
( , )
()
0
-1
(^ - ?2(, )
2
( )
0
-1
( ^ = 0( ). (23)
, 1, 2
f( x, y) = klGE (r) + ( - , grad) grad Go(r); fi( x, y) = ( - , grad) grad Goo(r);
(24)
(25)
(2)
, ( = 01) 2 ( 2 = 01). ( ) :
(, ) - , . (, ), .
4

J (GzU + graddiv(GEU))y,
Q
:
e~1nm\x3 _\
g =
ab n=im=0 Ynm (i + 50n )
sin (nXi )cos (mX2 )sin (nYi )cos (їY);
g3 =
3 ab
-sin (nXi) sin (mX2) sin (nYi) sin (mY2).
' n=i m=i Y nm

Pji2i3 = j(xi, x2, x3): ii < h < ii + i, i2 < -h2 < i2 + , < h. < + j,
( )
0 ^ ^ /*0 / \
Gi = ( 3) cos (nXi )sin (mX2 )cos (nH (( + 0,5))
n =im=i Ynmnm
Xsin
THHljsin(mH2 (2 + 0,5))sin)
mH2
2Hi ^ /o0m (x3) , ( X , , ( ( | 05) , (m2
2^^sin(mX2)sin[mH2(.2 + 0,5))smI ^ I;
m=i YQmm
S ^ ^ j0 (x )
G2=4, ^
sin (nXi) cos (mX2) sin (nHi (ii + 0,5))
n =im=i Ynmnm
sin
nHi
cos
(mH2 (2 + 0,5)) sin
mH2
^ /n02 (x3 ) sin (nXi )sin ^nHi (. + 0,5))^ ;
n=i Yn0m V 2 J
~ ~ J0 (x \
Jnm\x3)
G3=T
n=i m=i Ynmnm
sin
(nXi )sin(mX2 )sin(nHi (ii + 0,5))
sin
tHl j sin (mH2 (2 + 0,5)) sin ^
mH2

-xi
-x2

Xl = X2 =- Yl = -^, Y2 = b
-2
h
Hi = ^,H2 =
, > i------.2
b a
=h2 ,
aba
xi = Jihi, x2 = J2h2, = kh, 2 = i2h2 ;
b
Y nm
Jnm(x3) = <
(-exP (-(x3 -(z3 + *) ( Ynm )- exP (-(x3 - i3h3 )Y nm )) x3 > (( +
(exp(-(/'3h3 - (Y nm ) exP(-((( + !)h3 - x3 )Ynm )) x3 < i3h3,
(2 - exp (-(x3 - /33 ) Y nm) - exP (-((z3 + !)h3 - x3 )y nm ))
i3h3 < x3 <(( + !)h3- :
sin nx cos ny
r (x, y; A) = y- '
=L n(n2 2)
= p(x, y;)y q(x, y;) (0 < x, y < );
p(x, y; ) =

42 (i _ "
-2A
XI e-M2*-x-y) -e-^(x+y)+ sign(x-y)g-AMx-yl) -e-Ax-yl Yl-q (x, y; A) =-^y (71- x - y + sign (x - y )(-| x - y|));
4
p2( x, y; ) = p( X, y; ) =
_

4 sin A
( (A(x y _))y sign (x _ y )sin (|x _ y| _))); pi(x, y; ) = p(x, y; );
_ sin nx sin ny _
y( X, y; ) = -
n=1
e-Al x_y| y -(2_ x_y) _ e~A(xyy ) _ e-A(2_x_y)
-2A
,(x> 0,y >0, x y <2);
4 i _
S2 (x, y; ) = s( x, y; ) = 4 (cos ( _ x _ ))_ cos (A(_ |x _ y
si( x, y; A) = s( x, y; A);
d (x, y; ) = ds = _

dx 4 _ 2A
_e-A(xyy) y e-A(2_x_y)
,_AIx_yl _eA(2_lx_^sign(x_y)_
(x > 0, y > 0, x y < 2);
d2( x, y; A) = d (x, y; iA) =
a (sin (A(_x _ y ))_sign (x _ y) sin (A(_| x _ y|
n
i
/(3)
, ; ) = (, ; );
(-(-(3 _(73 + 1)3 ) )-(-(3 -/3%3 ) )) 3 >(( + 1)h3, ((-(/33 - 3 ( )-(-((73 + 1)3 - 3 ) )), 3 < /3%3-
(-(-(3 - /3%3 ) )- (-((73 + 1)3 - 3 ) )),
/3%3 < 3 <(/3 + 1)3.
1
Qj1 }2} - \ 1 -| + | h1, 2 -| 72 + | h2, 3 -| 73 + I %3
1
1
( )
1 | %3 | 73 +
∞1 -4II-
-1 -1
1 I ^ 1 I ^ 1
008 I ?1 + I1008 I 71 + I1 ^
| /'2 + |21| 72 + |21 2
2 / ( I 73 + 2 | %3 I 1 1
+ 211--------^1 | /2 + - || 72 + - |

-1

. 2
2 + 8, , +
/3 73
I 1 I/1 + 2 I1008 I 71 + 1 I1
1
-1
. 1 1
72 + 2 I2,/22,
72 + ^ | 2, (/2 + 1)2,
+8,-
26 2 1
/3 \ 4

1 1
2(2(72 + 2), 2/2; ) - 2(2(72 + 2), 22 + 1); )
2 2
2 + 72 - /2 + 2
72 - /2 +1
72 - /2
./2 - /2

X <1

21 1 (/1 +l),11 71 + ^; |-2 | 1/1171 + ;
4 2 2
1 + | - /! - ^
(
-
- /- 2
-/ +-
1
(
-
1 - /1 +
1

222.
./ 3 I 73 ^
=1 =1
(. 1 ^ 1 ^ 1
I /1 + I I 1 + I1 ^ X
1 ( 1 2
X | /2 + ^ I20 I 72 + ^ I28+
+ 2251 /1 + 2 |15 +2]151 +
2 =1
. 2 X ----2
2
2 <3 ^ 00
=1
((
71 2
2 2 2
1\ 4

2\ \ +21,1/1;

+ | ] , (/1 + ), 1 ( 71 + 2 ^, 1 (/1 +1);
+ | - / + 2
71 - / + 2
-8| -/1 -2
(
-

2
2 (/2 + l),2 72 + 2^;
2
2/2, 2 72 +

4 2 2
2 + ^| 72 - /2 - 2
72 - /2 - 2
72 - /2 +
1
(
72 - /2 +
2
22 2
= " !!
/ 3 73 ^
=1 =1
( 1 ^ ( 1 ^ 1
I / + ^ I I 7 + -2! 2 X
I . 1 . mH2
J2 + ^ I2"'
X sin ml /'2 + | H2sin ml j + | H 2sin+
+ 8;
8b
f
4 -1 n
^ n=1
1
2 sinn I ;i + 11 Hisin n 1 ji + 11 Hisin nHHl x
H2 I J2 + ^J,H2/2;An -p H2 |J2 + 2 IH2 (/2 + i);
1
a2
+ / J3 _4
H2 + sign| J2 - ;2 + 2
J2 - /2+2
\
sign| J2 - ;2
J2 - ;2
X
P2
ka
\ A
Hi| J1 + | , H1/1; -p2 Hi| J1 + | , H1 (/1 + 1);
ka

2
4k 2 a 2
H1 + sign| Ji - ;1 +
1
- H
J1 - ;1 +
1
-sign| Ji -;1 -2 Il -H1
J1 - * - 2
/J
: graddiv(Gt7) = graddiv(GiUi, 2, G3U 3) =
= grad
Ui +2 U2 +
1 d%2 6x3
U
+
32G. Ui +^^^ U2 +^ u3
\
3xi
12
13
xi +
21
322
23
2 +
32<G Ui +^2. u2 +33'
31
32
3.
Gi, G2, G3
32Gl 32Gl 32Gl 32G2 32G2 32G2 32G3 32G3
32 2^ 30^ 10^ 2 30^ 3x^3 23
32G
3
3
2

3 2G
3x2
1 8a ^ 1 . .. 1 . . . 1
1 = --3 2 sin m(/2 + -)H sin m(J2 + -H
. HH
3 m m=1
sin
X
X
( (
5/ .-
/3 .3
I V
( -
.1 + 2 |^,/1H1,
//
1
3 ^ / ^3173 + 2II ( _ 1 ( . 1 . 1
+ ------- 2-------- | / + | ^ | 7 + -1
=1
V
22
8 1 . 1 . 1 . 1
= 3 - (/ +2> (71 + -)
X
X
( (
5/ 7
/3 .3
I V
=1
1
2 + 2 2,(/2 + 1)2,
-

.2 + 2 2,/22,
/
3 - / [ 3 [ 73 + 2
+
3 , 2
^ =1 *
1 ( 1 2
| /2 + 2 20 I 72 + 2 2 sin2^
(, ; ) = (, ; ) - , (, ; ) = ?2 (, ; ) - ;
20

- = - -
^ ^
/ 3 I 73 + ~
2 I
2 (7 +-'|Hl0s(/' +-2 |X
21
z 1 =1 =1 I,
1 . ( 1 (. 1 . 2
Xsin----- /2 + |2 72 + |2Sin
2 21 2I 2
2
/373 2 ^
=1
-5/373 ^ Sn I /2 + -2 I2 72 + -2 I2 sin
X
X
171 + 2 , (/1 +1); ,

(
71 + I,/;
V
,
(. . . . ) 20 ( .... )
(//2, 7 72) = ^^~ (7l, 72, /1, /2)
12 21
5(, ; ) = 5 (, ; ) - , 5(, ; ) = ^ (, ; ) - ;
20

=
V. 1
/ ( )
31 2
3 1 =1 =1 '
- () (2) ( (/' +0,5)) X
X 81
1 sin ( 2 (/2 + 0,5)) (

2 1.
202 8
32
=-
/ ( 3)
3
=1 =1

sin () (2) ( (/ + 0,5)) X
X 81

2G3 = _8_
13

/ ( 3)
3
008
=1=1

( )sin(2 )(( (/' + 0,5))X
X 81

2 .
2G3 = _8_ 23
/ (3)
_
=1 =1

3
sin () (2) ( + 0,5)) X
X 81
^ sin ( 2 (/2 + 0,5)) (

2 .
2
/ ( 3)
______3 = ^_______________-1 3 3
- 2 2 2- 2- 2
3 =1 =1

( )sin(2 )( (/' + 0,5))X
X 81
2
, .. ().
5
. , .
. ,
-
Q,
( )
0
N-1
- I
(0, I - .

=
1
0
:
0

= /() - 010(, )3(- < < | (, )3() = 0 () . (27)

:
3 ^
^ () - 01(, )31 ()- | (,)3() = 01 (), / = 1,2,3 (28)
1=1
3 :
_ N _ N _ N
31 = ^ / ), 3 2 = ^ /2( ), 3 3 = ^ (, ),
=1 =1 =1
/ - -ї, I.
/ . , - : = {: < 1 < 2, >1 < 2 < 2, 1 < < 2} . :
/ = { : 1, < 1 < 1,+1, 2,/ < 1 < 2,1+1, 3, < 3 < 3,+1};
2 - 1 1 2 - , 2 -
1 = 1 +-----1 k, 2 / = 1 +-1 /, 3 = 1 +-1 m,

= 1,..., -1; /, = 1,..., -1.
1 :=| 1 - 1 -1 |, / :
1, ,
0, * /.
23
/[, / , 2 , . 2 . :
/1, /, /, = 1,..., N,
N = !(3 - 2).
^, 0^ :
11 12 13
21 22 23 B2
31 32 33 3 J
Aki :
4 = 4 (x); 4=! (xj) - k/ko2 j Gk (xj, j) fi (-
Q
-^- j 4^Gl (Xj, y)f (y)dy, (29)
Oxk J OXi
Q 1
:
xi = (xi1,xi2,xi3 ) Xi1 = (( + 1/2) xi2 = (( + 1/2)h2 xi3 = (( + 1/2)h3 , , l = 1,2,3; i, j = 1,..., .
, . .

1. Shestopalov, Yu. V. Development of Mathematical Methods for Reconstructing Complex Permittivity of a Scatterer in a Waveguide / Yu. V. Shestopalov, Yu. G. Smirnov, V. V. Yakovlev // Proceedings of 5th International Workshop on Electromagnetic Wave Scattering, October 22-25. - Antalya, Turkey, 2008.
2. , . . - / . . , . . // . . - . - 2008. - 2. - . 2-14.
3. , . . / . . // . . - . - 2009. - 1. - . 11-24.
4. , . . - / . . // . . - . - 2008. - 3- . 2-10.

- , , ,
E-mail: mmm@pnzgu.ru

- , , ,
E-mail|: _medv@mail.ru
,
E-mail: mmm@pnzgu.ru
Smirnov Yury Gennadyevich Doctor of physico-mathematical sciences, professor, head of sub-department of mathematics and supercomputer modeling, Penza State University
Medvedik Mikhail Yuryevich Candidate of physico-mathematical sciences, associate professor, sub-department of mathematics and supercomputer modeling, Penza State University
Vasyunin Denis Igorevich
Post graduate student, Penza State University
517.9 , . .
I . . , . . , . . II . . - . -2009. - 3 (11). - . 71-87.