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QR
! !" #$%&'()*+$ &),-&.,/0)
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7 *-6&!,%.+,+ 8+!9+ % +,'!$%:&% - ,-';6!- *-'- -
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εr % εo &/-> .%&"%*$!A+'%6$%> +& "%.'!&&!A!,+,%& .%(+$!A+ % %&"%*;)*+0
R *+'"- %(7$.!*- E "-,% &%. %&*.!$- %' $%.'-& ,- "-$%6*!+( %(7$.!*- JφO
E = −∇φ. JP0PO
S
∇ · J = 0. JP0NO
S
-8$7':&% + A%.&/- &!'"(!)*+,+ ,+ %3 % Ω ,%6-$+ - ,-';6!- ,- ".-8(%'+
!"#$%&'()" *" +#",&-$' *- ."$"/#'0' 1"# 2$1-*3456' 7&89#65' !
!" #$%&'()*+)" , -!() .) $)*$ /%'()*+ 0 12%$+) !( *3()/ %*#*%+ .) .%$4
+/%5!%67)$ φ 8!) /)$ -,)( ' )8!'69 :;0$$%(" & *.%67)$ .) & *+ /* " 8!)
.)+)/(%*'( '-?!*$ @'/A()+/ $ .) & *+/ -) *' $!@)/BC&%) . . (C*% " $9 '@-%&'4
.'$ @'/' -%(%+'/ *3()/ .) $ -!67)$ : !"#" DEFG=0 H'% /)$ .)+'-I)$ $ 5/) '$
& *.%67)$ .) & *+ /* $9 '@/)$)*+'. $ ' $)?!%/0
> %*+)/'69 )*+/) . (C*% Ω ) '(5%)*+) )2+)/* & //) @ / ()% .) $)!
& *+ /* ∂Ω ) J .)#*%.' @)-'$ /)-'67)$ )*+/) )2&%+'69 ) /)$@ $+' )-J+/%&'$0 1(
+)/( $ @/K+%& $" %$$ @ .) $)/ B)%+ '@-%&'*. !(' .%$+/%5!%69 .) +)*$7)$ )-J+/%&'$
$ 5/) & *+ /* ) ().%*. ' .%$+/%5!%69 .) & //)*+)$ )-J+/%&'$ /)$!-+'*+)$ !" '
& *+/K/% " %*L)+'*. 4$) & //)*+)$ ) ().%*. '$ +)*$7)$ /)$!-+'*+)$0 >@)$'/ .) ()4
* $ !$!'-" IK !(' +)/&)%/' @ $$%5%-%.'.) 8!) & *$%$+) )( /)'-%M'/ '(5'$ '$ & %$'$
& *+!. )( @'/+)$ & (@-)()*+'/)$ . & *+ /* 0 N)$+) +/'5'-I " & *$%.)/ !4$)
'@)*'$ @/%()%/ &'$ " &!L' +/'*$&/%69 *' +)/(%* - ?%' ('+)(K+%&' '. +'.' J
.'.' @ /
φ(ξ, η) = φexc(ξ, η) @'/' (ξ, η) ∈ ∂Ω, :;0F=
q(ξ, η) = −σ∇φ · n|ξ,η @'/' (ξ, η) ∈ ∂Ω. :;0E=
N'$ )8!'67)$ :;0F= ) :;0E=" φexc .)* +' @)/#- .) +)*$7)$ .) )2&%+'69 :+'(4
5J( &I'('.' .) & *.%69 .) O%/%&I-)+ ! )$$)*&%'- * L'/?9 ('+)(K+%& =" q
/)@/)$)*+' P!2 /)$!-+'*+) :& *.%69 .) N)!('** ! *'+!/'-= ) n J ,)+ /
!*%+K/% " * /('- ' . (C*% Ω ) /%)*+'. @'/' B /'0 Q ,K-%. $'-%)*+'/ 8!)" #4
$%&'()*+)" & *$%.)/'*. 4$) !( @/ 5-)(' &'@'&%+%, " φexc /)@/)$)*+' +)*$7)$ )(
R -+$" q .)* +' &'/?'$ )-J+/%&'$ )( S !- (5 ) σ & //)$@ *.) ' @)/(%$$%,%.'.)" 8!)
J '.%()*$% *'-" )*8!'*+ 8!) @'/' !( @/ 5-)(' /)$%$+%, " φexc +'(5J( .)* +'
+)*$7)$ )( R -+$" q /)@/)$)*+' & //)*+)$ )-J+/%&'$ )( (%-%>(@T/) ) σ /)@/)$)*+'
& *.!+%,%.'.) )( (Ωm)−10
S *$%.)/'*. 4$) ' /)& *$+/!69 + ( ?/K#&'" φexc ) q $9 )$$)*&%'%$" !(' ,)M
8!) ?)/'( !( @'/ & *L!?'. @)- & *+/'$+) %*+)/* " '$$ &%'. .%/)+'()*+) U
.%$+/%5!%69 .'$ .%$+%*+'$ B'$)$ . )$& '()*+ * %*+)/% / . , -!() .) ().%.' Ω0
1( +)/( $ ('%$ )2'+ $" ' /)-'69 )*+/) )2&%+'69 " & *+/'$+) ) /)$@ $+' @ .) $)/
.)#*%.' @ / ()% . @)/'. / B /('- :@/ 5-)(' .%/)+ =
q = ℑ[φexc, σ], :;0DV=
! !" #$%&'()*+$ &),-&.,/0)
!"# #$%& ' # () *+#,-.&+/#+ # &,(0$%& () 10+0 23)* )/ &,(0$%& () 4)!/#,,
+).)+),*)- #& /)-/& 5+&63)/# () 7#3&+ () &,*&+,&8 9,:!#,*& :!) # +) &,-*+!$%&
() σ *),(& &/& (#(&- φexc ) q ' (#(# 5)3# 0,7)+-%& () ℑ ;5+&63)/# 0,7)+-&<
σ = ℑ−1[φexc,q]. ;=8>><
9/ *)+/&- /#*)/?*0 &- 0--& :!)+ (0@)+ :!)A ,& 5+&63)/# (0+)*& & &,*+#-*) σ
' &,2) 0(& ) φA & 5&*), 0#3 )3'*+0 &A ' &6*0(& !,0 #/),*) 5)3# ):!#$%& (0.)+), 0#3
;=8B< ) 5&+ !/# (#- &,(0$C)- () &,*&+,&A ):!#$%& ;=8D< &! ;=8E< ;,& #-& (#
&,(0$%& () 4)!/#,,A ' &6*0(& # /),&- () !/# &,-*#,*)
!"#$%&'()" *" +#",&-$' *- ."$"/#'0' 1"# 2$1-*3456' 7&89#65' !
! "#$%&$' (!) σreal *+,*-# $.+*$ /0 %#0 ,) e1 2)33)3 ")4$ 3' # 4)- +3"4!5
67 , - +"4#3") σreal # 8#4"*4 ,) )3".$!0 3 ) 4)38 3"#3 3 %4) - +" 4+ ∂Ω' !
$#*3 )9#"#$)+")' # 3 0!67 , 84 %0)$# *+:)43 & ;)*"# %!3-#+, 53) $*+*$*1?JA
2# )(!#67 =>1?JA' ψk 37 ;!+6C)3 ,) %#3) - +:)+*)+")$)+") )3- 0K*,#3 ) σk
3 4)38)-"*: 3 - )H-*)+")3 ,) ,)- $8 3*67 1 E / ' # 3 0!67 ,# )(!#67 =>1?@A
8 ,) 3)4 %"*,# "*$*
! !" #$%&'( )* +,-).%'/')$)* 0*1.-), , #2%,), ),0 34*5*-%,0 6'-'%,0
• !"#$%& '( )%)#*!" "+,-.$%/( ,*$"'" 0*+( !-#('( - )%!-#$%/"1 0()%#%2"3
'*45%'"1 *)0"$)" * 0('* #*$ ."5'"6 )%!*#$%" '" !"#$%& 7aij = aji 1 ∀ i, j81
" 0()%#%2%'"'* 7xT Ax = 01 ∀x 6= 08 * " *)0"$)%'"'* 0('*! )*$ /(54$!"'")
"'%"5#*1 0($ !*%( '" *9:";?@ "0+%/"'" "( 0$(.+*!" '* AB? - '*)/$%#" :)"5'(3)* ")
*9:";C*) ,(2*$5"5#*) '" AB?1 '"'" 0*+") *9:";C*) 7D6E837D6F86 G 0(#*5/%"+ *+-3
#$%/( 0('* )*$ '*#*$!%5"'( *! /"'" 0(5#( '( '(!H5%( '%)/$*#( )*,:5'( ( !('*+(
'( >?@ 7 !"#$% '#(#1 IFJK86 L*)#* #$"."+M(1 :! '(!H5%( 0+"5( 72D8 -
/(5)%'*$"'( 0"$" " )*;
!"#$%&'()" *" +#",&-$' *- ."$"/#'0' 1"# 2$1-*3456' 7&89#65' !
! "!#$%&'() $)*#+'&!, -$./% ! ( #$!+'( ! 0123 !4 .+(/4 $ )'5$+ ( $ $ /6
$)$6$%#! 7%'#! $4#8! +$)(&'!%( !4 &!6 !4 "!#$%&'('4 $)*#+'&!4 %! ('4 4$./% ! (
$9"+$448!
φ = Nψ, :;,
$6 ?/$ φ * ! @$#!+ ?/$ &!%#*6 !4 "!#$%&'('4 $)*#+'&!4 $6 &( ( @*+#'&$ ! $)$6$%#!3
N * ( 6(#+'A $ B!+6( ! $)$6$%#! $ ψ * ! @$#!+ ?/$ &!%#*6 !4 "!#$%&'('4 $)*#+'&!4
%! ('4,
C! $D4$ $#$+6'%(+ ! "!#$%&'() $)*#+'&! %! $)$6$%#! #+'(%./)(+ "!+ /6 "!)'%ED
6'! $ "+'6$'+( !+ $6 : !"#$3
φ(e) = a1 + a2x + a3y, :;,
#() ?/$ φ(e) * ! "!#$%&'() $)*#+'&! ! $)$6$%#!3 a13 a2 $ a3 48! !4 &!$7&'$%#$4 !
"!)'%E6'! )'%$(+3 x $ y 48! (4 &!!+ $%( (4 .)!5('4 ! $)$6$%#!, H$44$ 6! !3
&!%4' $+(% ! ! @()!+ ! "!#$%&'() $)*#+'&! $6 &( ( %I ! $)$6$%#! #+'(%./)(+
: $%!6'%( !4 %I4 '%#+J%4$&!4 !/ )!&('43 %/6$+( !4 )!&()6$%#$ %! $)$6$%#! .$+()D
6$%#$ $ 6! ! (%#'DK!+L+'! &!6! %( 2'./+( ;,>3 #$6D4$
ϕ(e)1 = a1 + a2x1 + a3y1,
ϕ(e)2 = a1 + a2x2 + a3y2, :;,MN>
ϕ(e)3 = a1 + a2x3 + a3y3.
-!)/&'!%(% ! ! 4'4#$6( $ $?/(OP$4 :;,MN> "(+( a13 a2 $ a3 !5#*6D4$
a1 =1
2A(e)[ϕ
(e)1 (x2y3 − x3y2) + ϕ
(e)2 (x3y1 − x1y3)
+ ϕ(e)3 (x1y2 − x2y1)], :;,M
a2 =1
2A(e)[ϕ
(e)1 (y2 − y3) + ϕ
(e)2 (y3 − y1) + ϕ
(e)3 (y1 − y2)], :;,MM>
a3 =1
2A(e)[ϕ
(e)1 (x3 − x2) + ϕ
(e)2 (x1 − x3) + ϕ
(e)3 (x2 − x1)], :;,M;>
$6 ?/$ A(e) * ( L+$( ! $)$6$%#! #+'(%./)(+ ( ( "!+
A(e) =1
2[(x2y3 − x3y2) + (x3y1 − x1y3) + (x1y2 − x2y1)] . :;,MQ>
-/54#'#/'% !D4$ (4 $?/(OP$4 :;,MD:;,M;> %( $?/(O8! :;, $ (.+/"(% !D4$
&!%@$%'$%#$6$%#$ ()./%4 #$+6!43 ( +$)(O8! ( $?/(O8! :;, * 4(#'4B$'#( $ ( 6(#+'A
! !" #$%&'( )* +,-).%'/')$)* 0*1.-), , #2%,), ),0 34*5*-%,0 6'-'%,0
!" #$%&'(" ( #)*+! ,"- . ! ! /)*
N1 = α1 + β1x + γ1y, ,0123-
N2 = α2 + β2x + γ2y, ,0124-
N3 = α3 + β3x + γ3y, ,0125-
(+ 6$(
α1 = x2y3 − x3y2, α2 = x3y1 − x1y3, α3 = x1y2 − x2y1,
β1 = y2 − y3, β2 = y3 − y1, β3 = y1 − y2, ,0127-
γ1 = x3 − x2, γ2 = x1 − x3, γ3 = x2 − x1.
8( !9)* ) 9)+ ! :()*;! ) ? /) (@"( )A:(* ! +!:*;B ( 9)%:*!":( (C.:*;9)
) (C(+(%:) ! /!*:;* ! (D/*(""E) ) F*;%9G/;) )" F):(%9;!;" H;*:$!;"
∫
V(e)
∇φTσ∇φdV(e) =
∫
Γq(e)
φqIndΓ(e), ,012I-
(+ 6$( V . ) J)C$+( ) (C(+(%:)? Γq . ! #*)%:(;*! ) )+G%;)? (+ 6$( ! 9)% ;@
&E) ( 9)%:)*%) ( :(%"'(" (C.:*;9!" . !/C;9! !? φ . ! ;":*;A$;&E) ( /):(%9;!;"
(C.:*;9)" J;*:$!;"? φq . ! ;":*;A$;&E) ( /):(%9;!;" (C.:*;9)" J;*:$!;" (+ Γq? In . !
9)+/)%(%:( ! 9)**(%:( (C.:*;9! ,)$ J)C:!K(+- %! ;*(&E) ) J(:)* %)*+!C n !)
9)%:)*%) ) )+G%;) ( σ . ) 9)%:*!":( (C.:*;9)1
8;"9*(:;B!% )@"( ) )+G%;) ,"(&E) ! :$A$C!&E) "$A+(:; ! L :)+)K*!M!- /(C)
? ! (6$!&E) ,012I- :)*%!@"(
M∑
e=1
∫
V(e)
∇φT
(e)σ(e)∇φ(e)dV(e) =M∑
e=1
∫
Γq(e)
φq(e)In(e)dΓ(e), ,010N-
(+ 6$( M . ) %O+(*) :):!C ( (C(+(%:)" ( 9! ! :(*+) ! (6$!&E) . :)+! ) (+
$+ (:(*+;%! ) (C(+(%:) ,(-1 P)%"; (*!% )@"( ! (6$!&E) ,01Q7- ( ! ;K$!C ! (
∇φ(e) =
∂φ(e)
∂x
∂φ(e)
∂y
, ,010Q-
:(+@"(
∂φ(e)
∂x
∂φ(e)
∂y
=
(∂N1∂x
∂N2∂x
∂N3∂x
∂N1∂y
∂N2∂y
∂N3∂y
)
∂ϕ(e)
∂x
∂ϕ(e)
∂y
= B(e)ϕ(e), ,0102-
!"!#$%& !' "(#'$%& )*+,-. )!"#$%/ 0--1./ ' 2'3453 !' "(#'$%& )*+6./ 7 '23"8"94'!' 9&
9":& +
!"#$%&'()" *" +#",&-$' *- ."$"/#'0' 1"# 2$1-*3456' 7&89#65' !
!" #$% B(e) & ! '! ()* +% ,-%(!+,(%. +)/%(%01)!). +,. -,")02'),. ")0%!(%. +! '! ()*
+% /$034%. +% /,('! N 5%#$!34%. 56789: ; 5678%0+,;.% #$% !0 , -!(! ,. -, %01)!. φ #$!0 , -!(! ,. -, %01)!). φ? & @A")+!
! (%"!3B, +! %#$!3B, 567CD:? %0 B, -,+%;.% (%%.1(%@%( ! %#$!3B, 5676E:
M∑
e=1
ϕT(e)
∫
V(e)
BT(e)σ(e)B(e)dV(e)ϕ(e) =M∑
e=1
∫
Γq(e)
φq(e)In(e)dΓ(e), 56766:
,$
M∑
e=1
ϕT(e)k(e)ϕ(e) =M∑
e=1
∫
Γq(e)
φq(e)In(e)dΓ(e), 5676F:
%' #$% k(e) & ! '! ()* +% 1,0 (!. % %"& ()1, +, %"%'%0 ,? +%G0)+! -,( $'! )0 %H(!"
+% @,"$'%7
I,0.)+%(!0+, ! %.-%..$(! 5h: +, %"%'%0 , 1,0. !0 %? %';.%
dV(e) = h(e)dA(e). 56769:
J%%.1(%@%0+, ! %#$!3B, #$% +%G0% ! '! ()* k(e) ,> &';.%
k(e) =
∫
V(e)
BT(e)σ(e)B(e)h(e)dA(e). 5676K:
L..)'? +%G0%;.% ! '! ()* +% 1,0 (!. % %"& ()1, +, %"%'%0 , %' %(',. +!.
1,,(+%0!+!. H",>!). (x, y)7 =!>%0+,;.% #$% ,. -,")02'),. $.!+,. 0! +% %(')0!3B,
+, -, %01)!" %"& ()1,? 0, %"%'%0 , ()!0H$"!(? .B, +% -()'%)(! ,(+%'? %';.% #$% !
'! ()* +,. ,-%(!+,(%. +)/%(%01)!). B(e) +! '! ()* +% /,('! N & 1,0. !0 %7 M,H,?
-,+%;.% (%.,"@%( ! )0 %H(!" +! %#$!3B, 5676K: !0!") )1!'%0 %
k(e) =σ(e)
4A(e)
β21 β1β2 β1β3β1β2 β
22 β2β3
β1β3 β2β3 β23
+
γ21 γ1γ2 γ1γ3γ1γ2 γ
22 γ2γ3
γ1γ3 γ2γ3 γ23
. 5676 %03B, +,. -, %01)!).
%"& ()1,. 0,+!).? 0, +,'U0), +).1(% )*!+, %' %"%'%0 ,. G0) ,. ()!0H$"!(%.7 N%
! ! "#$#%&'()*+, -,. /,$#(0')'. 123$%'0,. 4,-)'.
!"#$" !"% &'"#( $" )*+, % (- $' !". /&' '01(!" $" '0'%'.&" 1 (./'#($
'% 2% % (- $' !". /&' 30"4 0 K 52' #'6#'/'.& &"$" " $"%7.("8 9 (./'#:;"
"!"##' 6"# %'(" $' 2% % (- $' !".'!&( Bi, jC,
6"# 'G'%60", $'/, 52' !"##'/6".$'% "/
!"#$%&'()" *" +#",&-$' *- ."$"/#'0' 1"# 2$1-*3456' 7&89#65'
!"#$%"&'"()* !* +*,$%-#$. / !-!- 0.1* 2.$*%
σ =
σ1σ23
3
3
σG3
3
3
σG+(2(Nx−1)−1)σG+2(Nx−1)
3
3
3
σM
453567
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Aφij−1 + Bφi−1j + Cφij + Dφi+1j + Eφij+1 = 0, 453FG7
+*:
A = −1
2. (σL1 + σL2) , 453FH7
B = −1
2. (σL1 + σL4) , 453FB7
C = 1. (σL1 + σL6) +1
2. (σL2 + σL3 + σL4 + σL5) , 453F57
D = −1
2. (σL3 + σL6) , 453FF7
E = −1
2. (σL5 + σL6) . 453FI7
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β =
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2, 12325
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