² Å ² Å Úò ² Å kÑ Œ m(A lossy half space) Berenger’s PML Ÿ TE œ TMdsec.pku.edu.cn/~tlu/lecture notes/apde6_12.pdf · 2017-04-03 · E= Em expj(ωt −kz), (1) H= Hm

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Åc,� ��¡�Åc.²¡Å, Åc´²¡�Å.Å�¡�´��,XJÅc´äkÓ�� ÚÓ��Ì�²¡.

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Cz,XJ|E1| = |E2|,KE;,´�,ÄK´ý�. ÷z¶��DÂ��54z�²¡Å�/ªXe

E = Em exp j(ωt − kz), (1)

H = Hm exp j(ωt − kz), (2)

Ù¥j = −√

(−1), k´Åê(wave number),k´¢�,XJvkP~(attenuation).

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r�54z�²¡Å�\Maxwell Equation,¿�b�gd>Öρf = 0,>6�ÝJf = σE��

−jkz ·E = 0,−jkz × E = −jωµH, (3)

−jkz ·H = 0, −jkz × H = σE + jωǫE, (4)

u´

z ·E = 0, E = − k

ωǫ − jσz × H, (5)

z ·H = 0, H =k

ωµz × E. (6)

ùL²EÚH´îÅ(transverse wave)�´��(orthogonal).E ×H��DÂ��.

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0��A�{|(characteristic impedance)

Z =|E||H| =

k

ωǫ − jσ=

ωµ

k. (7)

þª�±�ÑÅêk÷v��§

k2 = ω2ǫµ − jωσµ = ω2ǫµ

= ω2ǫ0µ0ǫrµr(1 − jσ

ωǫ),

(8)

Ù¥,ǫ0,µ0´ý��0>~ê(>NÇelectric permittivity)Ú^�Xê(^�Çmagnetic permeability).

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y

z

x

kI

θI

θR

θT

kT

kR

Medium 1 Medium 2

Plane of incidence

Interface

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Figure: \�>^Å30�1(Medium 1)Ú0�2(Medium 2)�m�.¡þ�)

��ÅÚ��DÑÅ.�þk´��A�ÅcR��DÂ�

�.�θI ,θR,θT©O´\��,��Úò��.

0�1¥´\�ÅÚ��Å,0�2¥´ò�Å. {üå�,·��Ä\�Å´�54z�.u´,\�Å

EI = EIm exp j(ωI − kI · r) (9)

ùp,kI´¢�,��\�Å�DÂ��.ù��§3¤k�mtÚ�mrþ½Â��²¡Å,�´�·û0�1. ��ÅÚò�Å�´�54z�²¡Å,/ªXe

ER = ERm exp j(ωRt − kR · r), (10)

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ET = ETm exp j(ωT t − kT · r). (11)

rþ¡�/ª�\Maxwell Equations�ÅÄ/ª

△E − ǫµ∂2

E

∂2t− σµ

∂E

∂t=

∆ρf

ǫ, (12)

b�σ = 0,ρf = 0,��

△ER + ǫ1µ1ω2ER = △ER + k1ER = 0, (13)

Ù¥

k1 = ω√

ǫ1µ1. (14)

Ïd

k2Rx + k

2Ry + k

2Rz = k2

1. (15)

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Ó��

k2Ix + k

2Iy + k

2Iz = k2

1, (16)

k2Tx + k

2Ty + k

2Tz = k2

2, (17)

(18)

\�Å,��ÅÚò�Å�m��,«3·û�.¡þz�:Úz��'X.Ïd

ωI = ωR = ωT , (19)

kI · rint = kR · rint = kT · rint, (20)

Ù¥,rint´�.¡þ:.u´k�²1�.¡�©þÑ��.

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XJkIy = 0,KkRy = 0, kTy = 0.u´��kI , kR,kT´�¡

�.ùn��þ¤3�²¡�¡�\�¡(the plane of incidence).k�x©þ��

kRx = kTx = kIx = k1 sin θI (21)

Xã1 uŃ´�

k2Rx + k

2Rz = k

2Ix + k

2Iz = k2

1 (22)

Úk2Rz = k

2Iz,kRz = −kIz. ·�ÀJKÒ,Ï��Å´lm�

.¡�. ¤±

θR = θI . (23)

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=��u\��(��½Æ).dã1,

kIx = k1 sin θI , (24)

kTx = k2 sin θT . (25)

dª(21),��

k1 sin θI = k2 sin θT (26)

ùÒ´ò�½Æ(Snell’s law ).ÏdXã1,

EI = EIm exp j(ωt − k1(x sin θI − z cos θI)), (27)

ER = EIm exp j(ωt − k1(x sin θR + z cos θR)), (28)

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ET = EIm exp j(ωt − k2(x sin θT − z cos θT )). (29)

dª(1.6),·��HÚE�'XXe

H =k× E

ωµ(30)

¦)��ÇΓ = EmHm

, ©ü«�¹, TE wave ÚTM wave. ?Û

²¡ÅÑ�±©)¡ùü«Å�Ú.TE wave, E²11u\�¡;TM wave, ER�u\�¡.dAmpere’s law(S�>´½Æ)�Ñ�EÚH 3�.¡üý��©þ��,? �±�Ñ��Ç.

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EIm

ERm

ETm

θI

θR

θT

Figure: TE wave, HR��¡�

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Xã2,TE wave��.¡��©þkHyÚEx.¤±

HIm − HRm = HTm (31)

½öEIm − ERm

Z1=

ETM

Z2(32)

¿�

(EIm + ERm) cos θI = ETm cos θI (33)

¦),�

Γ =ERm

ETm=

Z2 cos θT − Z1 cos θI

Z2 cos θT + Z1 cos θI(34)

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HIm

HRm

HTm

θI

θR

θT

Figure: TE wave, ER��¡�p

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Xã3,TM wave��.¡��©þkHxÚEy.¤±

EIm + ERm = ETm (35)

¿�

(HIm − HRm) cos θI = HTm cos θI (36)

½ö(EIm − ERm) cos θI

Z1=

ETM cos θT

Z2(37)

¦),�

Γ =ERm

ETm=

Z2 cos θI − Z1 cos θT

Z2 cos θI + Z1 cos θT(38)

o(:��½ÆÚ\�½Æé�«�¹Ñé§EÚH�.¡²

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(ε1,µ

1,0,0) (ε

2,µ

2,σ,σ*)

Region 1 Region 2

x

y

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�§·��ÄTEzÅ,E�Ü²1z¶§0�Xã4"Region 1(x < 0)´ÃÑ�0�§Region 2(x > 0)´�>�´σ§^�´σ∗.\�Å�^|H

inc = zH0 exp(−jβ1xx − jβ1yy)u´3Region 1�o�>^|£\�+��¤

H1 = zH0(1 + Γe2jβ1xx)e−jβ1xx−jβ1yy (39a)

E1 = [−xβ1y

ωǫ1(1 + Γe2jβ1xx) + y

β1x

ωǫ1(1− Γe2jβ1xx)]H0e

−jβ1xx−jβ1yy

(39b)>^|3Region 2¥

H2 = zH0τe−jβ2xx−jβ2yy (40a)

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E2 = [−xβ2y

ωǫ2(1 + σjωǫ2

)+ y

β1x

ωǫ2(1 + σjωǫ2

)]H0τe−jβ2xx−jβ2yy

(40b)Ù¥ΓÚτ´��'ÚDÑ'.

β1x = k1 cos θ; β1y = k1 sin θ x < 0 (41a)

β2x =

√

(k2)2(1 +σ

jωǫ2)(1 +

σ∗

jωµ2) − (β2y)2 x > 0 (41b)

Kki = ω√

µiǫi i = 1, 2.��þHz3�.¡üý��,

1 + Γ = τ (42)

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��þEy3�.¡üý��,

β1x

ωǫ1(1 − Γ) =

β2x

ωǫ2(1 + σjωǫ2

)τ (43)

d(42),(43)�,

Γ =

β1xωǫ1

− β2x

ωǫ2(1+σ

jωǫ2)

β1x

ωǫ1+ β2x

ωǫ2(1+σ

jωǫ2)

(44)

�,β2y = β1y = k1 sin θ £ò�½Æsnell’slaw¤. �±w�Γ´θ�¼ê,��5`,é?¿\��θ, Γ 6= 0, 3AÏ�¹e(\��θ = 0),·�k

Γ =η1 − η2

η1 + η2(45)

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Ù¥

η1 =√

µ1

ǫ1

η2 =

√

µ2(1+σ∗

jωµ2)

ǫ2(1+σ

jωǫ2)

(46)

u´,XJ�µ2 = µ1, ǫ2 = ǫ1 �

σ∗

µ2=

σ

ǫ2(47)

Kk1 = k2, η1 = η2. ù�Γ = 0,��Ã��Å.ù�,�±¦Ñ

β2x = (1 +σ

jωǫ1)k1 = k1 − jση1 (48)

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u´3Region 2¥�Å

E2 = yη1H0e−jk1xe−ση1x (49a)

H2 = zH0e−jk1xe−ση1x (49b)

wþª§3R�\��¹e§3Region 2¥�Å3R�u.¡��þ�êP~"?�Ú�§¦+3Ñ�0�¥DÂ§Å%´

ÃÚÑ�£dispersionless¤.�Ò´`§Å�´Ø�6uªÇ�§Ïd§éR�\�Å ó,Region 2 (Ô�ëêX(47)½Â)´��Region 1�"

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�¤õ§(J3uT0��U��áÂR�\��Å§Ïd§�

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�§éu��O�«�§ù´Øy¢�"BerergerJø#��Ôn�áÂ�§�±áÂ(Ã��)?Û�Ý\��«ªÇ�Å§¦¦^�©�|��{§æ^�õ�gdÝ§Berengerreported local refection coefficients for his PML 1/3000th that ofthe Mur ABC .

�ÄTEz²¡Å(Xã3)§0��.¡´x = 0.3Region2,Maxwell’s curl equations��¤(time-dependent form)

ǫ2∂Ex

∂t+ σyEx =

∂Hz

∂y(50a)

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ǫ2∂Ey

∂t+ σxEy = −∂Hz

∂x(50b)

µ2∂Hzx

∂t+ σ∗

xHzx = −∂Ey

∂x(50c)

µ2∂Hzy

∂t+ σ∗

yHzy =∂Ex

∂y(50d)

Ù¥

Hz = Hzx + Hzy (51)

σx, σy L«electric conductivities, and the parameters σ∗x and σ∗

y

L«magnetic losses.σx = σy = 0, σ∗

x = σ∗y = 0 lossless medium.

σx = σy = σ, σ∗x = σ∗

y = 0 electrically conductive medium.

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if ǫ1 = ǫ2, µ1 = µ2, σx = σy = σ, σ∗x = σ∗

y = 0 �÷v σǫ2

= σ∗

µ2Ò

ć�Ü©ù�UÃ��áÂR�\�Å�áÂ�"

(50d)L¤Time-harmonic form

jωǫ2(1 +σy

jωǫ2)Ex =

∂

∂y(Hzx + Hzy) (52a)

jωǫ2(1 +σx

jωǫ2)Ey = − ∂

∂x(Hzx + Hzy) (52b)

jωµ2(1 +σ∗

x

jωµ2)Hzx = −∂Ey

∂x(52c)

jωµ2(1 +σ∗

y

jωµ2)Hzy =

∂Ey

∂y(52d)

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Ù¥L«phasor quantity( letting the hat symbol denote a phasorquantity){üå�§Ú\e¡�ÎÒ§

sω = (1 +σω

jωǫ2); s∗ω = (1 +

σ∗ω

jωµ2) : ω = x, y (53)

u´(52)Ú(52b)�±#�¤

jωǫ2syEx =∂


jωǫ2sxEy = − ∂

∂x(Hzx + Hzy) (54b)

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Berenger’s0�¥�²¡Å) e¡ò�ÑBerenger’s0�¥�²¡Å). (54)ü>éy¦�£kü>Ø±sy¤

jωǫ2∂Ex

∂y=

∂

∂y

1

sy

∂

∂y(Hzx + Hzy) (55)

r(52d)�∂Ex∂y �L�ª�\þª

− ωǫ2µ2Hzy =1

s∗y

∂

∂y

1

sy(Hzx + Hzy) (56)

aq�§

− ωǫ2µ2Hzx =1

s∗x

∂

∂x

1

sx(Hzx + Hzy) (57)

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üª�\§¿|^(51)

1

s∗x

∂

∂x

1

sx

∂

∂xHz +

1

s∗y

∂

∂y

1

sy

∂

∂yHz + ω2µ2ǫ2Hz = 0 (58)

ù��§ke¡/ª�)£this wave solution supports thesolutions ¤

Hz = H0τe−j√

sxs∗xβ2xx−j√

sys∗yβ2yy (59)

with the dispersion relationship

(β2x)2 + (β2y)2 = (k2)

2 = ωǫ2µ2 (60)

=⇒ β2x =√

(k2)2 − (β2y)2 (61)

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u´§l(56),(57)Ú(51),·��

Ex = −H0τβ2y

ωǫ2

√

s∗ysy

e−j√

sxs∗xβ2xx−j√

sys∗yβ2yy (62a)

Ey = H0τβ2x

ωǫ2

√

s∗xsx

e−j√

sxs∗xβ2xx−j√

sys∗yβ2yy (62b)

¦+æ^|©�§£field splitting¤§EÚH²1u.¡�©

þ��E,´é�§dd§·��±¦Ñ��'"Äk§du

3x = 0�,é?�y§��þ��Ñ¤á§�Snell’s lawaq§ATk

√

sys∗yβ2y = β1y = k1 sin θ (Snell′slaw) (63)

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·��±�sy = s∗y = 1,=⇒ σ = σ∗ = 0 d3x = 0�,.¡üý��þëY§�

Γ = (β1x

ωǫ1− β2x

ωǫ2

√

sx

s∗x) · (β1x

ωǫ1+

β2x

ωǫ2

√

sx

s∗x)−1 (64)

τ = 1 + Γ (65)

Ã��^�

y3b�ǫ1 = ǫ2, µ1 = µ2, sx = s∗x.ù�duk1 = k2, η1 =

√

µ1/ǫ1 =√

µ2/ǫ2,�σx/ǫ1 = σ∗

x/µ1�±��β2y = β1yd(61)ª�β2x = β1x�

\(65)ª�, é¤k�\��θ, Γ = 0 . 3ù«�¹e§3x > 0�«�§>^|

Hz = H0e−jsxβ1xx−jβ1yy = H0e

−jβ1xx−jβ1yye−σxxη1 cos θ (66a)

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Ex = −H0η1 sin θe−jβ1xx−jβ1yye−σxxη1 cos θ (66b)

Ey = H0η1 cos θe−jβ1xx−jβ1yye−σxxη1 cos θ (66c)

3matched Berenger’s medium,DÂ�Å(the transmitted wave)�DÂ��Ú�Ý�\�Å�Ó§Ó�÷Xx��êP~§P~Ïf(the attenuation factor)´Ø�6uªÇ�§Ø�DÚ�Ñ�0�§ù«5�·û�«�Ý\��Å"

Ez = Ezx + Ezy (67)

µ2∂Hx

∂t+ σ∗

yHx = −∂Ez

∂y(68a)

µ2∂Hy

∂t+ σ∗

xHy =∂Ez

∂x(68b)

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ǫ2∂Ezx

∂t+ σxEzx =

∂Hy

∂x(68c)

ǫ2∂Ezy

∂t+ σyEzy = −∂Hx

∂y(68d)

�TE caseaq§�±�ÑPML5�§(J�ké��Cz§�Ü©�§�´ǫ2Úµ2p�§σÚσ∗p�,, PML ��^�´��"

(ǫ∂

∂t+ σy)Exy =

∂


(ǫ∂

∂t+ σz)Exz = − ∂

∂z(Hyx + Hyz) (69b)

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(ǫ∂

∂t+ σz)Eyz =

∂

∂z(Hxy + Hxz) (69c)

(ǫ∂

∂t+ σx)Eyx = − ∂

∂x(Hzx + Hzy) (69d)

(ǫ∂

∂t+ σx)Ezx =

∂

∂x(Hyx + Hzy) (69e)

(ǫ∂

∂t+ σy)Ezy = − ∂

∂y(Hxy + Hxz) (69f)

Ó�§Faraday’s Law is given by

(µ∂

∂t+ σ∗

y)Hxy = − ∂

∂y(Ezx + Ezy) (70a)

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(µ∂

∂t+ σ∗

z)Hxz =∂

∂z(Eyx + Eyz) (70b)

(µ∂

∂t+ σ∗

z)Hyz = − ∂

∂z(Exy + Exz) (70c)

(µ∂

∂t+ σ∗

x)Hyx =∂

∂x(Ezx + Ezy) (70d)

(µ∂

∂t+ σ∗

x)Hzx = − ∂

∂x(Eyx + Ezy) (70e)

(µ∂

∂t+ σ∗

y)Hzy =∂

∂y(Exy + Exz) (70f)

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|©��§�±�¤E�I�mþ�Maxwell’s �§e¡��IC�§

x →∫ x

0sx(x

′

)dx′

; y →∫ y

0sy(y

′

)dy′

; z →∫ z

0sz(z

′

)dz′

(71)3(71)¥§b�sω3÷�I¶��´ëY¼ê§3� �IX¥

� �

∂

∂x=

1

sx

∂

∂x;

∂

∂y=

1

sy

∂

∂y;

∂

∂z=

1

sz

∂

∂z(72)

u´§½ÂFÝ�f

∇ = x∂

∂x+ y

∂

∂y+ z

∂

∂z= x

1

sx

∂

∂x+ y

1

sy

∂

∂y+ z

1

sz

∂

∂z(73)

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5µ∇f(x, y, z) 6= ∇f(x, y, z),∇ Ø´�5�FÝ�f3#�IXe�/ª§ ½Â�#��f.�mNÚ�Maxwell’s equations 3E�� I�mÒ�L�¤

jωǫE = ∇ × H

= x(1

sy

∂

∂yHz −

1

sz

∂

∂zHy) + y(

1

sz

∂

∂zHx − 1

sx

∂

∂xHz)

+ z(1

sx

∂

∂xHy −

1

sy

∂

∂yHx)

(74)

− jωµH = ∇ × E

= x(1

sy

∂

∂yEz −

1

sz

∂

∂zEy) + y(

1

sz

∂

∂zEx − 1

sx

∂

∂xEz)

+ z(1

sx

∂

∂xEy −

1

sy

∂

∂yEx)

(75)

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5¿µ

∇ · (D) = ∇ · (ǫE) 6= 0 (76)

∇ · (D) = ∇ · (ǫE) = 0 (77)

½Â#�f§¢Sþ´½Â#��É5e�MaxwellEquation.�ksx = sy = sz�,â�DÚÔ��Maxwell Equation�Ó§þ¡ª¤á�^�´>Ö�"§Ó�sx independent on y, z§sy

independent on x, y, sz independent on x, y, (74)Ú(75)¢Sþ�d(69f)Ú(70f)�Ñ"~X:(69f) �Time-harmonic �/ª

jωǫsyExy =∂


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jωǫszExz = − ∂

∂z(Hyx + Hyz) (78b)

jωǫszEyz =∂

∂z(Hxy + Hxz) (78c)

jωǫsxEyx = − ∂

∂x(Hzx + Hzy) (78d)

jωǫsxEzx =∂

∂x(Hyx + Hyz) (78e)

jωǫsyEzy = − ∂

∂y(Hxy + Hxz) (78f)

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1

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c 0 00 d 00 0 d

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k22c − (β2y)

2b−1 β2xβ2yb−1 0

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2d − (β2x)2b−1 − (β2y)2a−1

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Hy

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2b−1c−1 = 0 : TEz(Hz = 0) (85)

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β1x

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(86)

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(87)

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2β1x

β1x + β2xb−1(88)

?�Ú,3�.¡x = 0?,

β2y = β1y (89)

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r(89)�\(84),¦β2x

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β2x =√

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2 − (β1y)2b2 = b√

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lRegion 1D�Region 2�,Ã��^�´Region 2¥�0�äkXe�ǫÚµÜþ.

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s−1x 0 00 sx 00 0 sx

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�sxÑ´k��. ~X,�sx = 1 + σx/jωǫ1 = 1 − jσx/ωǫ1. u´l(92)�

β2x = (1 − jσx/ωǫ1)β1x (93)

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(94)

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∂

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Tiao Lu PML





3�.¡x = 0?vk (source),KDx = ǫs−1x Ex�½´ëY

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∂

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ǫ3�.¡?ëY,s−1x �3�ê�fÜ,¤±DxÚExÑ´ëY

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∇× H = jωǫ¯sE (98a)

∇× E = −jωǫ¯sH (98b)

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¯s =

s−1x 0 00 sx 00 0 sx

s−1y 0 0

0 sy 00 0 sy

s−1z 0 00 sz 00 0 sz

=

syszs−1x 0 0

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0 0 sxsys−1z

(99)

#N�ü ¢Üκ,

sx = κx +σx

jωǫ(100a)

sy = κy +σy

jωǫ(100b)

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sz = κz +σz

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�sy = 1.3ymin,ymaxÚzmin,zmaxü¡��?�UPML�sx = 1.3n¡��UPML¦^(99)ª�Ñ�Üþ.

d(99) �E�ÜþØ´ü¶�, ´��É5�., ,ù«PMLE,��UPML,Q,§3vk��?�«�´ü¶��É5�.

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sx = κx +σ

′

x

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vGauss’s Law�¿©^��Ï)º�ØÐ. ·��Ä�x¶R��UPML>.,�PML��´3yz��´©¡~ê��Nǫ(y, z), UPML�0>Xê

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.

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∂

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−jsxβ1xx

XJsxØ�6y,

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|^β1x = k1 cos θ, β1y = k2 sin θ,

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6yÚPML¥�D ÷vGauss’s Law��¿©^�.XJ��

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ê�Kz5¢y§,�sω = κω + σ′

ω/jωǫ0ùp´÷î�´ØC�.

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3,3θ = 90o�,�¹¬C�é�,ù�,R = 1, PML´��Ã�

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�,·�F"R(θ)¦�U��,²w�,éu��PML�,7L4σω�,AO´θ �C90o�.

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R(θ) = e−2η cos θR d0 σx(x)dx (104)

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õ�ªCz

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κx(x) = (g1∆ )x (108b)

Tiao Lu PML





ùpσx,0´PMLL¡�>�,g´'~Ïf,∆´FDTD��Ý.ùpPML>�dσx,0O\�gd/∆σx,0, r(108)�\(104),�

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σx,0 = − ln[R(0)] ln(g)

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��5`,2 ≤ g ≤ 3é�õêFDTD�[5`´Ð�.lÑØ�

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σx,opt = −(m + 1) · (−16)

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0.8(m + 1)

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ù�®²�y¢ééõA^Ñ´érobust.

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ùÜ©?ØUPML3FDTDlÑ¥XÛA^.FDTDlÑ��§´(98).�§�Ô�ëêd(100)½Â.

l(98a)Ú(99)XÃ,Ampere’s Law 3UPML ¥�L«¤

∂Hz∂y −∂Hy

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∂x∂Hy

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= jωǫ

sysz

sx0 0

0 sxszsy

0

0 0sxsy

sz

Ex

Ey

Ez

(112)

ùp�L«�B,·�E(100),½Âsx, syÚsz.

sx = κx +σx

jωǫ(113a)

sy = κy +σy

jωǫ(113b)

Tiao Lu PML





sz = κz +σz

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��r(113)�\(112),,�C��m«�(time domain)¬��ÜþXêÚE |�òÈ,ù´ØÐ�,Ï�òÈ�¢yO�þé�,��k��{´½Â#�'X, ØªÇ��ÍÜ.AO�,-

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sxEx (114a)

Dy = ǫsx

syEy (114b)

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szEz (114c)

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Tiao Lu PML





∂Hz∂y −∂Hy


∂x∂Hy

∂x −∂Hx∂y

= jω

sy 0 00 sz 00 0 sx

Dx

Dy

Dz

(115)

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κy 0 00 κz 00 0 κx

Dx

Dy

Dz

+1

ǫ

σy 0 00 σz 00 0 σx

Dx

Dy

Dz

(116)

Tiao Lu PML





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2ǫκy + σy∆t)Dx|ni+1/2,j,k + (

2ǫ∆t

2ǫκy + σy∆t)·

(Hz|n+1/2

i+1/2,j+1/2,k − Hz|n+1/2i+1/2,j−1/2,k

△y

−Hy|n+1/2

i+1/2,j,k+1/2 − Hy|n+1/2i+1/2,j,k−1/2

△z)

(117)

Tiao Lu PML





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σz

jωǫ)Ex (118)

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∂

∂t(κxDx) +

σx

ǫDx = ǫ

∂

∂t[(κzEx) +

σz

ǫEx] (119a)

aq�,l(114b)Ú(114c),·��

∂

∂t(κyDy) +

σy

ǫDy = ǫ

∂

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σx

ǫEy] (119b)

Tiao Lu PML





∂

∂t(κzDz) +

σz

ǫDz = ǫ

∂

∂t[(κyEz) +

σy

ǫEz] (119c)

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Ex|n+1i+1/2,j,k =(

2ǫκz − σz∆t

2ǫκz + σz∆t)Ex|ni+1/2,j,k + [

1

(2ǫκz + σz∆t)ǫ]·

[(2ǫκx + σx∆t)Dx|n+1i+1/2,j,k − (2ǫκx − σx∆t)Dx|ni+1/2,j,k]

(120)

Tiao Lu PML





o�,�#UPML¥EI�üÚ(1)d(117)�D (2)d(120)dD¦E.aq�Ú½·��±�#UPML¥�H. l(109)Ú(99)·��±��B��#, ±Bx�~,

Bx|n+3/2i,j+1/2,k+1/2 =(

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2ǫκy + σy∆t)Bx|n+1/2

i,j+1/2,k+1/2 − (2ǫ△t

2ǫκy + σy∆t)

(Ez|n+1

i,j+1,k+1/2 − Ez|n+1i,j,k+1/2

△y

−Ey|n+1

i,j+1/2,k+1 − Ey|n+1i,j+1/2,k

△z)

(121)

Tiao Lu PML





1�Ú,dB¦H±Hx�~

Hx|n+3/2i,j+1/2,k+1/2 =(

2ǫκz − σz∆t

2ǫκz + σz∆t)Hx|n+1/2

i,j+1/2,k+1/2 + [1

(2ǫκz + σz∆t)µ]·

[(2ǫκx + σx∆t)Bx|n+2/3i,j+1/2,k+1/2

− (2ǫκx − σx∆t)Bx|n+1/2i,j+1/2,k+1/2]

(122)

3CFL^�e, ¦^(117),(120),(121)Ú(122) �O��ê�½5®�ê�¢��(shown). ?�Ú,�±L²(shown)lÑ|÷vGauss’s Law,UPML´·½�(well posed).

Tiao Lu PML

Documents

² Å ² Å Úò ² Å kÑ Œ m(A lossy half space) Berenger’s PML Ÿ TE œ TMdsec.pku.edu.cn/~tlu/lecture notes/apde6_12.pdf · 2017-04-03 · E= Em expj(ωt −kz), (1) H= Hm