在三維色散介質中模擬電磁波的傳遞行為
117
0
0
全文
(2) Department of Engineering Science and Ocean Engineering College of Engineering. National Taiwan University Master Thesis. !"# Prediction of electromagnetic wave propagation in three-dimensional dispersive media. $%& Yu-Chieh Wang. Advisor : Tony Wen-Hann Sheu, Ph.D..
(3) 103 July, 2014.
(4)
(5) !"#$%&'()*+,-./0123 456789:;<= >?@ ABCDEFG==HI JK,-LM+ NOGPQRSTUV&'CWXMGYZ[ \@] ^G<=_`abJK$%&'Gcdef67g *+ \hCiG@ #Ojkl mn]copqrstuvb<=bwxy ==z{|b}~b}b} b} b*+b@<= Yannick Maxim GJK *+67@!!<= ¡@BC¢£G<=¤¥ef ¦(§+cd@<=¨©ª }«¬}®§v¯}°±²¯*+³´µ¶·@ ^G¸!¹ºb»¼½¾¿ÀÁ\ÂJK (§ ; ÃÄÅÆÇÈ U ÉÊ :;<=ËÌÍ@ Î Ï N;и!ÑÒÓ¼ÔÕ bugCÖ× Ø»¼ÙÚ UÛÜÝ}¶·UÞß==ËàáUâ}ãáUä FåUæçè æéê@ëì:;<=ííîîTUïðñòrstbó ô G@õÙö÷ ø* ÉùùúûÛüýþþµµy@ÿ UCC !ÿ A Û@==Ë y
(6) èU5 Ëy@ == ^G
(7) èUy@.
(8)
(9) ¼GÛ Ù ) (FDTD)* \ !"(#@
(10) ()¼ $C%W%&'() ( *+ 3, ) -.t \)/03,123,@3e $%()ô 4' (56/789. (Symplectic) :;<;=: Runge-Kutta ()N>b 01ì\?@*Aè 6!"(# BCDEF×G>)/0+12 H(#g I JK*L @M'NOGPQRS\@ T[U!ëI$%M'DE
(11) 3e $%() Gg G AÙPQ5VWXÙYÀZ[#\P\]$%P\^_@ *`áÀab[c Ù 3dve`À$%M'^*+f§gD^ @ h#[3e $%()789.]M'NOG7Ù45Aè Di¾jVNb !"(#$%01ì¾YkiT`á@
(12) *lmn7opD:M'qE Debye}Lornetz}Drude 0nCWr0 1G>&'he[
(13) s3Û:$%()ô !"(#D]tuv wtuPxGÙYD+b 01tRS@. Nyz{ !"(# ; ; M'NO ; #\P\$%P\ ; Debye qE ; Lorentz qE ; Drude qE.
(14) Abstract An explicit finite-difference scheme for solving the three-dimensional Maxwell’s equations in staggered grids is presented in time domain. The aim of this thesis is to solve the Faraday’s and Amp` ere’s equations in time domain within the discrete zero-divergence context for the electric and magnetic fields (or Gauss’s law). The local conservation laws in Maxwell’s equations are also numerically preserved all the time using proposed the explicit second-order accurate symplectic partitioned Runge-Kutta temporal scheme. Following the method of lines, the spatial derivative terms in the semi-discretized Faraday’s and Amp` ere’s equations are then properly discretized to get a dispersively very accurate solution. To achieve the goal of getting the best dispersive characteristics, this centered scheme minimizes the difference between the exact and numerical phase velocities with good rates of convergence are demonstrated for the problem. The significant dispersion and anisotropy errors manifested normally in finite difference time domain methods are therefore much reduced. The dual-preserving (symplecticity and dispersion relation equation) wave solver is numerically demonstrated to be efficient for use to get in particular long-term accurate Maxwell’s solutions. The emphasis of this study is also placed on the accurate modelling of EM waves in the dispersive media of the Debye, Lorentz and Drude types. Through the computational exercises, the proposed dual-preserving solver is computationally demonstrated to be efficient for use to predict the long-term accurate Maxwell’s solutions for the media of frequency independent and dependent types..
(15) KEY WORDS:. Maxwell’s equations; staggered grids; dispersion relation equa-. tion; exact and numerical phase velocities; Debye; Lorentz; Drude.
(16) E D H B J σ ²0 ²r µ0 µr λ c. (Electric field intensity) (Electric flux density) (Magnetic field intensity) (Magnetic flux density)
(17) (Electric current density) (Conductivity) (Permittivity of free space) (Relative permittivity) (Permeability of free space) (Relative permeability) (Wavelength) (Speed of light in free space). V/m C/m2 A/m W/m2 A/m2 S/m F/m. . H/m. m. m/s.
(18). |= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . }æ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ~e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2
(19) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5
(20) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0; CWr(# - !"(# 1.1. )/0 / 12 / (#D+¾K . . . . . . . . . . . . . . . . . . 2.2 )/0 / 12(#D:$D . . . . . . . . . . . . . . . . . . . . . 2.3 M'qE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 V . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 $%() 2.1. i ii iii v 1 1 2 3 4 4 6 6 7 9 10 20.
(21) . 2. : . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 789.: PRK 4' . . . . . . . . . . . . . . . . . . . . . . . . 3.3 g 4'(#:K . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 7M'NOAèD4'(): 3.1. 4.1. 4.2 4.3. FDTD. g 4' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Cr $: . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 t:O$ . . . . . . . . . . . . . . . . . . . . . . $%3D: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . f§gD (Anisotropy) ]$%M' (Numerical dispersion) : . 4.3.1 $%M'NO]#\M'NO: D (consistency) . 4.3.2 $%M'NO (Numerical dispersion analysis) . . . . . . 4.3.3 $%P\] \: (Numerical phase velocity and group velocity analysis) . . . . . . . . . . . . . . . . . . . . . .. $% :9k]Ê 0¡ $%01:9k 4.4. 5.1 5.2 5.3. . . . . . . . . . . . . . . . . . . . . . . . . . .. #:Oh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M'qEs CPML VDOh . . . . . . . . . . . . . . . . . . . . #¢: \ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 £% / '¤% . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 ¥¦§$) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 20 22 24 29 29 31 32 32 33 35 35 36. 36 38 53 54 56 57 58 58.
(22) . 3. 5.3.3. #¢01:9k. . . . . . . . . . . . . . . . . . . . . . . .. 0¨ 9 vk]Ê . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 ©ª«]¬ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
(23) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. 59 93 93 94. 79.
(24)
(25) . 2.1 2.2. 2.3 2.4 2.5. 2.6. 2.7. 2.8. 2.9. 3.1. V®¯():°±²@ . . . . . . . . . . . . . . . . . . . . . \CWrM'qEs³´(#+&'g ( *p° : ) :°±²@ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lM'qE:PVqCO$#µ@ . . . . . . . . . . . . . . . . lM'qE:PVqCO$¶µ@ . . . . . . . . . . . . . . . . * 17.5GHz Tstu+tà·¸ô 5GHz∼30GHz : ro@ (a) p°² (x=400∆x, t=0∼450∆t); (b) tp°² . . . . . . ¹º®¯» P}¼» Q *+½» R G CPML V®¯ :°±²@ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ;¾5*¹º®¯» P +¼» Q «TL¿»./ôM' +:M'qE°±²@ . . . . . . . . . . . . . . . . . . . . . . . . . CPML À ÁV®¯Â¤^@ (a) ÅÃg s ; (b) Debye qEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CPML À ÁV®¯Â¤^@ (a) Lorentz qE s ; (b) Drude qEs . . . . . . . . . . . . . . . . . . . . . . . . . . .. CPML. :F (b) s: µÄ»°±² +¾Å@ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 14. 14 15 15. 16. 17. 17. 18. 19. (a). 28.
(26) 3.2 4.1. 4.2. 4.3. 4.4. 5. (a). ]g Æ:°±² ; (b) tu]r$Æ:°±². Ç5rÈÉt³´ 3Ê ú½ (Zenith angle) θ( ¦ (Azimuth angle) φ@ . . . . . . . . . . . . . . . . . . . . . . . . . Ë3 Cr T 0.2 ·¸t:M'NO:#\]$%\ ÌÍ@ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ë3 ·¸ ∼ Cr %ÈÉt:M'NO:#\ ]$%\ÌÍ@ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . O$ a ∼ a ú½ (Zenith angle) θ ÈÉt¾%À(¦ (Azimuth angle) φ :@Qú½ (Zenith angle) θ T 0 ÈÉtO$ ÂÀ(¦ (Azimuth angle) φ _@ . tu (Angular frequencies) Vr$ (wavenumbers) k k :« ²øL¿!cf§gD3dv^@ (a) #\M'NO ; (b)
(27) 3ÎÛ7M'NOAè$%() ; (c) box scheme [26]; (d) −π 2. 1. 28. 42. 42. π 2. 43. 3. ◦. 4.5. x. y. symplectic scheme [26]; (e) Yee’s scheme [26]. . . . . . . . . . . . . . . 4.6. 44. 45. #\]$%tu^ ×100% Vr$ (wavenumbers) k k :«²øL¿
(28) 3e $%()s}br$ ( Ï ¼^Ðô 5% Ñ ) ;Ò$%()YÙYÀabcf§ gD3dv^@(a)
(29) 3ÎÛ7M'NOAè$%() ; (ωnum −ωexcat ) ωexcat. x. y. (b) box scheme [26]; (c) symplectic scheme [26]; (d) Yee’s scheme [26]. 46.
(30) 4.7. 4.8. 6. tu (angular frequency) ÓV k Ô}k Ô*+ κ = qk + k :¾5²@s}br$ (wavenumber) f$%()ÕÖ ×#\M'NOÀár$Ø$%M'dv^ÏÀ:Ø Ì;Ò$%()
(31) 3ÎÛ:7M'NOAèDE 4'VôZ[c$%M'dv^Yk5*+e [Vr$Aèý@ . . . . . . . . . . . . . . . . . . . . . . . . . . P\]B\Ì% ( ) V(¦ (Azimuth angle) :«²@Ã ²søÙ f$%()ÕpÚ c 4'3dvf§ gD (Anisotropy)
(32) 3ÎÛ:7M'NOAèDE4' P N ÈÉt;Ò$%()YZ[cf§gD (anisotropy) 3à^@ (a) N = 3.1; (b) N = 5; (c) N = 10;. 47. (d) Nλ = 20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 48. x. 2 x. y. 2 y. υp c. λ. λ. λ. λ. P\]B\Ì% ( ) Vrb »$ N = :«²Ë3( ¦ φ = @Àá N Ûf$%()c4'3dv$%M '^Õ@!`áÜÝ@
(33) 3ÎÛ:7M'NOAèDE 4' N %ÞÐ +ÙYÀZÐc$%M'3à ^_U!Í5M'NOAèYk@ . . . . . . . . . . . . . . . 4.10 ÈÉt7ß3Ùú½ φ *+(¦ θ #\]$% P\:^Vrb »$ N = :«²@Àá N Ûf $%()c4'3dv$%M'^Õà@`áÜÝ@
(34) 3ÎÛ7M'NOAèDE4'+á N %ÞÐ *ÙYÀZ[c$%M'3à^_U!Í5M'N OAèYk@ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . υp c. 4.9. π 4. λ. λ h. λ. λ. λ. λ h. 49. λ. λ. 50.
(35) 4.11. 7. $% \â γ V(¦ (Azimuth angle) φ :«²@ò søÙ f$%()ÕpÚ c 4'3dvf§gD (Anisotropy)
(36) 3ÎÛ:7M'NOAèDE4' s}b$%r$t kh ;Ò$%()YZ[cf§g D (anisotropy) 3à^@ (a)
(37) 3ÎÛ:$%() ; (b) kh = π4 ; (c) kh = π2 ; (d) kh =. 4.12. 5.3. 5.4 5.5 5.6. 3π 4 .. . . . . . . . . . . . . . . . . . . . . . .. » 81×81 T 50 äÈÉÌÍ TM r3' :C% E %$%\#\@ . . . . . . . . . . . . . . . . . . . . . . . . . . T=50(s) ÌÍ; !"(# (TM wave) #\ $%\: å (Energy I) À :æç@ . . . . . . . . . . T=50(s) ÌÍ; !"(# (TM wave) #\ $%\: å (Energy II) À :æç@ . . . . . . . . . ;OhsHamiltonian À :æç@ . . . . . . . . . . . . ;Ohs 3,À :æç@ . . . . . . . . . . . . . . » 81×81×81 T 50 äÈÉtÌÍ ! "(#3' :C% E %$%\#\èT z = 0 :¾5 ²@ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . z. 5.2. . . . . . . . . . . . . . . . . . . . . . .. 51. \Ð%V(¦ (Azimuth angle) φ :«²@òs øÙ f$%()ÕpÚ c 4'3dvf§gD (Anisotropy)
(38) 3ÎÛ:7M'NOAèDE4' s}b$%r$ kh t;Ò$%()YãZ[cf§g D (anisotropy) 3à^@ (a)
(39) 3ÎÛ:$%() ; (b) kh = π4 ; (c) kh = π2 ; (d) kh =. 5.1. 3π 4 .. 52. 64. 65. 65 66 66. z. 67.
(40) . 8. T=50(s) ÌÍ !"(##\$%\: å (Energy I) À :æç@ . . . . . . . . . . . . . . . . . 5.8 T=50(s) ÌÍ !"(##\$%\: å (Energy II) À :æç@ . . . . . . . . . . . . . . . . 5.9 OhsHamiltonian À :æç@ . . . . . . . . . . . . 5.10 &'éêëg (1∼250 ) ]M'qE (251∼500 ) s ]7
(41) [29] :C% E $%\:ÌÍ@ (a) M'qE:PVq C;$Àtu² ; (b) &'ôëg] Debye qEs ; (c) &' ôëg] Lorentz qEs ; (d) &'ôëg] Drude qEs@ . 5.11 3D &'T 44 × 44 × 44 200 × 200 × 200 @M'qEs 01C% E P}Q R »:%« ÌÍ@ . . . . . . . . . . . . 5.12 44 × 44 × 44 200 × 200 × 200 »&'ì z = 0 : 2D ¾ 5°±²@ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.13 3D &'T 44 × 44 × 44(present) 200 × 200 × 200(reference) » Debye qEs01C% E P}Q R »:%*+PV ^ @ (a) P » E %:ÌÍ ; (b) Q » E %:ÌÍ ; (c) R » E %:ÌÍ ; (d) P}Q R »:PV^ @ . . . . . . . . . . 5.14 3D &'T 44 × 44 × 44(present) 200 × 200 × 200(reference) » Lorentz qEs01C% E P}Q R »:%*+P V^ @ (a) P » E %:ÌÍ ; (b) Q » E %:ÌÍ ; (c) R » E %:ÌÍ ; (d) P}Q R »:PV^ @ . . . . . . . . 5.7. 68. 68 69. z. z. 70. 71. 71. z. z. z. z. 72. z. z. z. z. 73.
(42) . 9. &'T 44 × 44 × 44(present) 200 × 200 × 200(reference) » Drude qEs01C% E P}Q R »:%*+PV ^ @ (a) P » E %:ÌÍ ; (b) Q » E %:ÌÍ ; (c) R » E %:ÌÍ ; (d) P}Q R »:PV^ @ . . . . . . . . . . 74 5.16 £%}'¤%+Ô§:íî² . . . . . . . . . . . . . . . 75 5.17 ï¤rôñ'¤ð :£% / '¤%:C% E % (∆t = 2668.5 f s)@ (a) ñò ; (b) ÁT 600; (c) ÁT 850; (d) ÁT 1200; (e) ÁT 1350; (f) ÁT 1600 . . . . . . . . . . . . . . . . . . 76 5.18 ód;$: ² . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.19 ¥¦§$Çô$T ξ = 1, 2 ¯5Çô:ÌͲ . . . . . 77 5.20 õö'¤:°±² . . . . . . . . . . . . . . . . . . . . . . . . 78 5.21 ï¤rô'¤ð :£% / '¤%:C% E % (∆t = 2668.5 f s)@ (a) ÁT 600; (b) ÁT 850; (c) ÁT 1200; (d) Á T 1350; (e) ÁT 1600; (f) ÁT 1900 . . . . . . . . . . . . . 79 5.22 ;õö (Mie scattering) '¤ ² T 12 :÷\\ . . . . . . 80 5.23 ;õö (Mie scattering) '¤ ² T 12 :$%\ . . . . . . . . 80 5.24 ;õö (Mie scattering) '¤ ² T 12 ÷\\]$%\¥ ø²:ÌͲ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.25 ;õö (Mie scattering) '¤ ² T 12 ÷\\]$%\¾5 ²:ÌÍ@ (a) E (x, 0); (b) E (0, y) . . . . . . . . . . . . . . . . . . . 82 5.15 3D. z. z. z. z. z. z. r. r. r. r. z. z.
(43) . 10. ;õö (Mie scattering) '¤ódDEO$ÈÉt: ÷\\]$%:ÌͲ@ (a) ² T 12 ùTú (Silicon) ódqC; $ ; (b) ² T 20 ùTûü (Acetone) ódqC;$ ; (c) ² T 50 ùT ýþ (Glycerin) ódqC;$ ; (d) ² T 100 ùT;ÿ (Barium binoxide) ódqC;$cdÅ www.engineeringtoolbox.com . 5.27 íîL¿5 (monitor plane) *C%À : . . . . . 5.28 õö'¤01 ï¤rô'¤ð :£% / '¤%:C % E %ôL5 1 (monitor plane 1) À :@ (a) ÁT 0(0 fs); (b) ÁT 560(2.8 fs); (c) ÁT 760(3.8 fs); (d) ÁT 850(4.25 fs); (e) ÁT 1160(5.8 fs); (f) ÁT 1800(9 fs) 5.29 õö'¤01 ï¤rô'¤ð :£% / '¤%:C % E %ôL5 2 (monitor plane 2) À :@ (a) ÁT 0(0 fs); (b) ÁT 560(2.8 fs); (c) ÁT 760(3.8 fs); (d) ÁT 850(4.25 fs); (e) ÁT 1160(5.8 fs); (f) ÁT 1800(9 fs) 5.30 õö'¤01 ï¤rô'¤ð :£% / '¤%:C % E %ôL5 3 (monitor plane 3) À :@ (a) ÁT 0(0 fs); (b) ÁT 560(2.8 fs); (c) ÁT 760(3.8 fs); (d) ÁT 850(4.25 fs); (e) ÁT 1160(5.8 fs); (f) ÁT 1800(9 fs) 5.31 õö'¤01 ï¤rô'¤ð :£% / '¤% :C% E %ôL5 1 (monitor plane 1) À :@ (a) ÁT 0(0 fs); (b) ÁT 560(2.8 fs); (c) ÁT 760(3.8 fs); (d) ÁT 850(4.25 fs); (e) ÁT 1160(5.8 fs); (f) ÁT 5.26. r. r. r. r. 83 84. z. 85. z. 86. z. 87. z. 5.32. 1800(9 fs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 88. Bð:°±². 89. . . . . . . . . . . . . . . . . . . . . . . . . . ..
(44) . 11. Q tu (normalized frequency) T 0.353 (c/a) C% E : (∆t = 0.05337 f s)@ (a) ÁT 575; (b) ÁT 1075; (c) ÁT 1750; (d) ÁT 2325 . . . . . . . . . . . . . . . . . . . . . . 5.34 Q ÁT 2325 ÌÍ tu (normalized frequency) T 0.353 (c/a) ] 0.206 (c/a) :9kÌͲ (∆t = 0.05337 f s)@ (a) 0.353 (c/a);. 90. (b) 0.206 (c/a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 91. 5.33. 5.35. z. Q ÁT 2325} tu (normalized frequency) T 0.353 (c/a) (∆t = 0.05337 f s)]
(45) ÌÍ:9k (a) Present method; (b) Mekis et al. [38] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 92.
(46) . 4.1. 4.2. &'Nb T T 30(s) ÈÉCr = 0.2 + Cr = 0.05 :^*+ 3
(47) CPU TIME(s) ÌÍ@ . . . . . . . . . . . . . . . . . . . . 39 ú½ (Zenith angle) θ T 0 &90 }30 &60 45 (¦ (Azimuth angle) φ T 0 &90 }6 &84 }9 &81 }12 &78 }22.5 &67.5 } 30 &60 }36 &54 }6 &84 }45 ÈÉt3V.O$ a ∼ a ÈÉ@Qú½ (Zenith angle) θ T 90 ø@!;ÈÉt O$FQú½ (Zenith angle) θ T 90 (¦ (Azimuth angle) φ T 0 Ó 90 ø@! ÈÉtO$@ . . . . . . . . . . . . . 40 ÈÉt
(48) 3e FDTD $%()] Yee ()3 D·¸ÌÍ@ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 pTl$%()$%M'NO@Å 1, 2, 3, 4 Óp °
(49) 3e $%()} box scheme [26]} symplectic method [26]} Yee schemes [26] . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ◦. ◦. ◦. ◦. ◦. ◦. ◦. ◦. ◦. ◦. ◦. ◦. ◦. ◦. ◦. ◦. ◦. ◦. ◦. ◦. ◦. 1. ◦. 3. ◦. ◦. ◦. 4.3. 4.4. 5.1. ◦. Q01 T T =5(s) »$T 51 }61 }71 81 / 4'()] Yee 4'()ÌÍ&'3@: L − error norms CP U @ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. 2. 2. 2. 2. 63.
(50) 5.2. 5.3. 0. Q01 T T =5(s) &'P×: L − error norms Ì Í/4'()] Yee 4'()3 »$]
(51) : CP U @ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Q01 T T =5(s) »$T 51 }61 }71 81 ÌÍ /4'()] Yee 4'()&'3@: L − error norms CP U @ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Q01 T T =5(s) &'P×: L − error norms Ì Í/4'()] Yee 4'()3 »$]
(52) : CP U @ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. 3. 3. 3. 63. 3. 2. 5.4. 63. 2. 63.
(53) . 1.1. !" (Maxwell) RwCW%m(#*CWr- ]. /ÙPQb@wCW- +CWr],: P«/ kï!f¾./5*:;ñøCr³}B ] } U Ï}Ir}úø}!"}ÀtCW#}CW$%¥¥@ CWr#¢rss³>#& À'(fl'(,'¤'( 9.úø)¤rIà9.s³#¢ s*+rs}'(À ,+-5VCWr³ÀtCW#À./0VCWr.¥ 1kïÀ CWrDÙá& 2æ±Ê@Ãôrs'(D33. æG>$%01 (@!#¢rstCWrD@cè&'CW (Computational Electromagnetics) 9 4&'}&'5ð Ï}CW ¥PN4¢£67vT 82æÂNá9:gÛ @ 4&' (Scientific Computing) wNvT 82æ;< ./ô4}ª#}N=}>-¥f?@æä)¼A*BC$0 nDE#¢,-ÚFG>GQ:$%()CiG&'@9kÂHI Fcè$%()ø¼ë#±¯]4&': 2æJKL@VôC W- +¾./ O*CW- TmnG>D&' ÏTMN O/$% 3effl()NÛv 89P./4@Q&' ;/ô&'CW$%()ø Tt)] )R?@t 1873.
(54) 1.2. . 2. )@æÙtÙ (Frequency Dependent Finite Difference, FDFD) ()}S ) (Moment Method, MoM) ÙTU) (Finite Element Method, FEM) ¥F _ )@æÙ Ù (Finite Difference Time Domain, FDTD) ()} Ùð (Finite Volume Time Domain, FVTD) )} (# (Time Domain Integral Equation, TDIE) )@]V()s Ù (FDTD) ( )¼ lV !"(#DHIºK \$%();Wø* X %Y&'Ztà[\% Ï]-'(^nCW'¤Ó: /0}f§gDM'qECWVCW'¤ Ù¾v!_ç vT]-f?qECW'¤@æ()`L©YHÁCi5ðrs tÙYaÛ]./w@ 1.2. . Yee [1] bAe [ Ù (Finite Difference Time Domain, FDTD) ()Â./ôCWr &'è ()@!:;c\Û] ./@ Ù (Finite Difference Time Domain, FDTD) ()OºKà !"H(# Û>VC% E W% H f g Gd e4'(+ E ( Ó H) % f¸Ù H ( Ó E) % r gè C%]W%(<hT (Yee’s cell)@./]l4'(iê !"H(#jT D :4' (#7ìk Ô lÁKHÀ \g CW%è ()mnÀ`°[CWr³´>#*á ô ]í&@ 1975 Taflove ¥/ FDTD &':/0qEorï¤ \ X%CW'¤Âe [$%3 ()@ Taflove ì 1988 p À [ Yee 'q$%M'^@ Q à Yee 3e Yee’s cell r/01CWrg G³´FT[ Ù&'Ñ01ñg sCWst&'Ñuv®¯Gíî V®¯()Å FDTD ()w*VôV®¯ º¼x8 @@V®¯()ëñì 1969 à Taylor 3e :y%()V®¯ ()ìÂÀ<d/ Mur V®¯ (Absorbing Boundary Condition ABC) [2]+ôG±z 90 o3Û£ (Perfectly Matched Layer PML) [3, 4]¾VYkÛÛ5@ ×Õy* FDTD ()./ôM'qEÂHI[ @ w FDTD G/ôuvM'qEë;/ÙYV®¯()hT£ 1966.
(55) 1.3.
(56) . 3. (Convolutional Perfect Match Layer, CPML) [5]@ 3{M'qE¼qED ÐÀtu_Ù3qE;¼ |¦CWqEs l@ÃôÅ7¯}µ ,E~}}}} <}Í},ðD}¥4ð}ÕªqE¥¾CW D]tuÙ N1 ôM'qE@×Ù V,E 3tuCWr«/tÏp Ú M'DBrN ¥@×Õy M'qE[ \CWD±ÛÛ ÛÛ@] (5¼cTÀáC Ïc\Û ¦sCWÛÛæ§ôtZàqE M'DVCW³}'¤})¤¥ÏÛÛ@× (5Àá Õy#Å7vÀkïvÀc_¸![ ¦9 1e [ ¦9 g #s3¸!ÍC4}¥4ð !"}9ód4 Ï¥+M'qE@cèRwM'qE[ CWD ()V}g #},C4}}<} Ï}À E#}ód4¥ ¦Û+./Ùk2æ±Ê@ è^ 1982 Umashankar Taflove d/ FDTD &' U'¤u5 (RCS)i&'Ñ T»% / '¤%Ñ (Total field / Scatter fieldTF/SF) Èɼ'¤&'sï¤ríî láÙY()@ 1.3. . ,-±Ê_CWr³´IT: ]g Ù 3PND+t u (Angular frequency) ]r$ (Wave number) NOF³ FDTD () d/ ( +g ] 4'=T;=RS:s )BÙ ¾36/$%()],-±ÊG`7Âñ#EJ9T[ ]V() ,-±ÊG$
(57) ¡¢*CWr:M'NOT()í&£i ]g 4'G@ºKJ9Â/7BCDE 4'g G¤ 7ëIM'NOAè:g 4'*LCWr³´IT ] g ÙáYS¥å9 ¦SA 01 Á>#sÕ@! 7,-±ÊëIM'NO\@ Å7¯s}µ ,E¾PVqC;$ (Relative permittivity) ÕÐÀ tu_cñ)ÃPVqC;$]tuñN:ÅÃg !"(# D \1× Ú[ ]-M'qE FDTD ()é§ Lubbers ¥e RC-FDTD () [6]} Kelley ¥d/C% NøDרÜÝ[ RC-FDTD ().
(58) 1.4. . 4. &'© PLRC-FDTD () [7]} Siushansian ¥d/4'ª,´ ר *ÜÝ RC-FDTD ()&'© TRC-FDTD () [8]}+
(59) 36/«¬( # ADE-FDTD () [9]. «¬(# ADE-FDTD -¼ittPVqC;$:§$ä ®w¯ °N±-ì j *´²§$Ó³ I ,Y9C% i !"(#sI G,H(#«4' *C}W%eÀ&' *á@!À ±g CW%%@ 1.4. . w Ù )¦k/ Yee 3e : (Yee’s cell) \ !"(#DGQÀ01CWrIT@´µ Ù () ¾ 4']g 4'µ¼¶èvw_CWr³´,-ITG ] g st¼PxNOF°:tu]r$: NO ( *hTM'NO (Dispersion relation)) æ<7ßר01>#s@ Uy*-· !"(#T£XM6/789. (symplecticity) 4'()*SA$(#s3Q7ÙBCDEÂ* BC L»¸O ¼¹$BCDE@3^º»ìM'qEs³CW r(#H_]-M'qE3dvC¼J (Polarization current density)@ Vôr³´: ]g PxNOUy*M'NOAè (Dispersion relation preserving) «T£½ïÇ5rô(#s6@ ]g &'G@ ¾9¿è*Û 7ëIM'NO: !"(#4'()ÂjV f§gD:P\}]M'NOAèGä ± ÂH ÁÀiè ( )./ôM'qEPN \ëìÂ./ô#¢ \@ 1.5.
(60) .
(61) À-· !"(#]M'NO(#£XMÛ 7ëI$%M'NO:g 4'()@bAô0 sqÁ &'CWrPNÂ]Û:
(62) *+ ]@
(63) 0;iDE !"(#D½/LMôñ 01@0iQ !"(#D:,-DE*,- / $Õ Ã: 4'K 7M'NOAè:g 4'@0jV7M'NO.
(64) 1.5. . 5. Aè:4'HI$%M'NO*+f§gD: F0¡i3ÎÛ : g 4'HI#Oh*he¾øÄD]RSD@è^i./NOhñ^ # \ŧï¤Ç5r:£% / 'í%¥#¢@lmnM'qE Debye}Lorentz}Drude qE0nCWr01*
(65) ·Æ ëì 0¨siä 9 Âe©øÛ@.
(66) - 2.1. / / . !"(#DOÃ)/03,}123,+ 3,3Dv@ t ;yø* Óp°vt123,})/03,p° ~ ~ = − ∂ B − J~m , ∇×E ∂t ~ ~ = ∂ D + J, ~ ∇×H ∂t. (2.1.1) (2.1.2). *+ 3,3DvÇ(#. ~ = 0, ∇·B. (2.1.3). ~ = ρ. ∇·D. (2.1.4). GE(#Ds H~ }E~ }B~ D~ ÓTW% (Magnetic field intensity)}C% (Electrical field intensity)}W¦ (Magnetic displacement) C¦ (Electrical displacement)@ (2.1.2) ] (2.1.1) s: J~ J~ ÓopC¼å (Electric current density) W¼å (Equivalent magnetic current density)@ì (2.1.4) : ρ opCÈå (Electrical charge density)@f§D (Isotropic) øDqEs. $t.NO m. ~ = ² E, ~ B ~ = µ H, ~ J~ = σ E, ~ J~m = σm H. ~ D. (2.1.5). Gs² µ Óp°qCO$ (Electric permittivity) WO$ (Magnetic permeability) σ σ ÓTCu (Electric conductivity) Wu (Equivalent magnetic loss)( ÓopqEÉÊWÉÊ )@
(67) 3K ! m.
(68) 2.2. / . 7. "(#:4'()Ëì`]¼/01CWrôÅqE (Simple medium) s³ÈÉ3*opódDEO$WO$ (Magnetic permeabilityµ) *+qCO$ (Electric conductivity²) iT;$@RódDEO$ @æDEód:qCDE*+WDXÕT@¾s ² = ² ·² _ µ = µ ·µ F RódDEO$Ì3[r:³\ c(c ≡ 1/√²µ(≈ 3.0 × 10 m/s))
(69) iCð (Perfectly electric conducting) g t \GE (#Í[WDód^PVWu (Relative magnetic permeability) *Î íT µ = 1; VôÅÃg PVqCO$ ² = 1Vôód:qE¾ ² %$ ² ≥ 1@ GE:ÏÐøn(# (Hyperbolic equations) iôÙ \:1. uv,-g @ôuv®¯ (truncated boundary) GÑÕÎíTð 0. r. 0. r. 8. r. r. r. r. (perfectly conducting) ~ = 0, ~n × E. (2.1.6). ~ = 0, ~n · H. (2.1.7). ¾s ~n op§^Ŧ)§ (unit outward normal vector)@ 2.2. / !. xGÄ3E:ÎíñoX/0f§DÅqEt: !"(#Dø *ÜÒT ~ 1 ~ = − ∂H , (∇ × E) µ ∂t ~ 1 ~ = ∂E , (∇ × H) ² ∂t. (2.2.1) (2.2.2). ~ = 0, ∇·B. (2.2.3). ~ = 0. ∇·D. (2.2.4). (2.2.3-2.2.4) *+ (2.1.5) Â:BÀvwô !"(#D [10] :^F+ (2.2.3-2.2.4) øT !"(#D:Ç()@i (2.2.1) + (2.2.2) RÓ Ó'' (divergence operator)ø@¢ 3,O!êô)/03, ]123,:s@.
(70) 2.2. / . 8. 4sÑÕøºK \ÏÐø:(# (2.2.1-2.2.2)+ ø@![ :C%]W%@Vô (2.2.3-2.2.4) :ñ'() (divergence-free constraint conditions) :ÇÑÕiô#OhsHI @ !"(#Dø*/Ô "Õý (Hamiltonian dynamical system) p°: ∂ ∂t. ~ H ~ E. =. 0 −I I. . 0. ~ δH / δ H ~ δH / δ E. ,. (2.2.5). ¾sÔ "Õ(# (Hamiltonian functionH) ø*tp°: [11] Z. ~ E) ~ = H(H, Ω. 1 1 ~ ~ +1 E ~ · ∇ × E) ~ dΩ. ( H ·∇×H 2 ² µ. (2.2.6). ~ ~ ¾s Q Kole ¥Õ [12]³: (2.2.1-2.2.2) øÒv ψ(t) = G ψ(t) ´ ~ ≡ (m(t), ~ ~ ψ(t) ~ ~n(t)) = µ H(t), ² E(t) @GE: G SÖø*t:ÂVhS Ö (skew-symmetric matrix) p: [12] T. 1/2. ∂ ∂t. T. 1/2. . G=. . 0. −µ−1/2 ∇ × (²−1/2 ). ²−1/2 ∇ × (µ−1/2 ). 0. .. (2.2.7). ~ p°:èÑÕø@×(#:\ ~ !"(#Døà ∂ ψ(t) / ∂t = G ψ(t) ³ ´ ~ = exp t G ψ ~ (t = 0) ¾s exp(t G) `°[ !"(#D:\O Ø ψ(t) ~ ;]CW%: HÙ$@i§ ψ~ ÚT R ² E~ · E~ + µ H~ · HdΩ å (energy densityw(t)) ÙºK:N¾+ Ω. Z. ~ ·E ~ +µ H ~ ·H ~ dΩ. ²E. w(t) = Ω. (2.2.8). (2.2.8) : åÐÀ Ü@Vô (2.2.6) + (2.2.8) À Õ: DEÑÕøi¾«T#Oh:/@.
(71) 2.3. !". 2.3. 9. "#$%. ;ÛM'qE0né§ÜÝ0n (Debye model)}Þßà0n (Lorentz model) Üá0n (Drude);yÅqCO$pUt : (1) Debye qE ²r (ω) = ²∞ +. (2) Lorentz. ²s − ²∞ σ ∆² σ + ≡ ²∞ + + 1 + Iωτ Iω²0 1 + Iωτ Iω²0. (2.3.1). ∆ωp2 σ + 2 2 ²p + 2Iωδp − ω Iω²0. (2.3.2). ωp2 σ + 2 Iωγp − ω Iω²0. (2.3.3). qE ²r (ω) = ²∞ +. (3) Drude. qE ²r (ω) = ²∞ +. ¾s∆² = ² − ² @² Tâ\Ó&t PVqCO$}² Tñãtu PVqCO$} δ TäåO$}ω TæEXçtu (Resonant frequency)}τ T»èé (Relaxation time)}X γ = 1/τ T»èé ê$@ 7ß (2.3.1-2.3.3) (2.1.2) 123,H(#/øp°T s. ∞ p. s. ∞. p. p. ~ ~ = ²0 ²∞ ∂ E + σ E ~ + J~d , ∇×H ∂t. (2.3.4). ¾s J~ JTqEM'3½ÉC¼J (Polarization current density) Vô?nM'qE;y Ó$t«¬ëI (# d. ~ ∂ J~d ∂E J~d + τ = ²0 ∆² , ∂t ∂t ~ ∂ J~d ∂ 2 J~d ∂E ωp2 J~d + 2δp + = ²0 ∆²ωp2 , 2 ∂t ∂t ∂t ~ ∂ J~d ∂ 2 J~d 2 ∂E γp + = ² ω . 0 p ∂t ∂t2 ∂t. (Debye). (2.3.5). (Lorentz). (2.3.6). (Drude). (2.3.7). M'qE FDTD &'(5w3EÙ´K (RC) )}«¬(# (ADE) )}Z °)} NøD´K (PLRC) ¥()@
(72) M'qE id/«¬(# (ADE) )ÂjVGEmnlM'qE« pK :@.
(73) 2.4. #$%&'()*+. 2.4. 10. &'()*+,-.. Å7¯sµ CWr æñt01(ìí!¾ë# ,-EîôCiïððÇ- GøHI/ñÇ&'+ñò $Q@Qñ Yee 3e FDTD () &'ÑÕTuv3 $%r+`Ú]Ï[ FDTD ëñ&¦Û& Àóô º! 1981 Èõ4ÿ G.Mur e [ Mur V®¯() [2], 1994 Berenger e [VY (Perfectly Matched Layer, PML) ®¯() [3] 6@ FDTD ()öÛ 9÷ý@wTøÛ $l PML () + Gedney 3e Ô§ (uniaxial PML, UPML) [13] Abarbanel 3 e f§gD (anisotropic PML, APML) [14] ¥()@
(74) 6/ V®¯T (convolution PML, CPML) [5]¾_»T¼;Vôù úr (evanescent waves)7ÙÞ5Výè^ûüZ[ïðð ý/@Í[VYk5:^X7þGDCPML V®¯£vwô FDTD &' qEø*äÿ ÜÀ./!f§D}f§gD}V}M 'Ó¼:øDqE&'s@cèCMPL PVì UPML V®¯YG/ ô:ÅÃg s'¤D @èUyJ [¾ O$ (tensor coefficient)s , [15] w. sw = κw +. σw ; aw + jω²0. w = x, y, z. (2.4.1). 2.4.13DE¼C$k(§ g §$½ïÆ (stretchedcoordinate) j°NO Z. w ˜→ 0. w. 0. 0. sw (w ) dw ;. w = x, y, z .. (2.4.2). G>®w¯j°NO = T®w¯°'3Ê −1. µ. ¶ 1 s¯w = = κw + σw ¶ ¸ · µ σw aw δt δ(t) σw − + t u(t) ≡ + ζw (t) = exp − 2 κw ²0 κw ²0 κw ²w κw −1. (. (2.4.3). ¾s u(t) δ(t) ӼŦÁ=§$ (Unit step function) §$ (Delta.
(75) 2.4. #$%&'()*+. function)). 11. ø@ !"(#D , :. µ ¶ ∂Hy ∂Dx 1 ∂Hz 1 ∂Hy ∂Hz = − + ζy ∗ − ζz ∗ , ∂t kx ∂y kz ∂z ∂y ∂z µ ¶ ∂Dy 1 ∂Hx 1 ∂Hz ∂Hx ∂Hz = − + ζz ∗ − ζx ∗ , ∂t kz ∂z kx ∂x ∂z ∂x µ ¶ ∂Hy ∂Dz 1 ∂Hy 1 ∂Hx ∂Hx = − + ζx ∗ − ζy ∗ , ∂t kx ∂x ky ∂y ∂x ∂y. µ ¶ ∂Ey ∂Bx 1 ∂Ez 1 ∂Ey ∂Ez =− − + ζy ∗ − ζz ∗ , ∂t kx ∂y kz ∂z ∂y ∂z ¶ µ ∂By 1 ∂Ex 1 ∂Ez ∂Ex ∂Ez =− − + ζz ∗ − ζx ∗ , ∂t kz ∂z kx ∂x ∂z ∂x ¶ µ ∂Ey ∂Bz 1 ∂Ey 1 ∂Ex ∂Ex . =− − + ζx ∗ − ζy ∗ ∂t kx ∂x ky ∂y ∂x ∂y. (2.4.4). (2.4.5). FDTD s CPML V®¯ CW%o&'øG> (2.4.4) (2.4.5) 4' #Ú@&'@æÑÒ¼4' ζ ∗ (∂ /∂w) #ÚcTºKHI&' æùÊ ïðð&' @G> Luebbers Hunsberger 3d/´K (RC) [6] Ïø*ÙY\Ìè @ 3Ê (2.4.3) s ζ(t) 4'. Z (m) T w. v. w. Z Zw (m) =. ¶ ¸ · µ Z (m+1)∆t σw aw σw ζw (τ ) dτ ≈ − + τ dτ exp − ²0 κ2w m∆t ²0 κw ²0 · µ ¶ ¸ σw aw = cw exp − + m∆t ²0 κw ²0. (m+1)∆t. m∆t. ¾s. · µ µ ¶ ¶ ¸ σw aw σw cw = exp − + ∆t − 1 σw κw + κ2w aw ²0 κw ²0. (2.4.6) (2.4.7). (2.4.8). (2.4.4) s ζ ∂H /∂w) 4',ø*רv w. v. ψw,v (n) = ζw (t) ∗. n−1 X ∂ ∂ Hv (t)|t=n∆t ≈ Zw (m) Hv (n − m) ∂w ∂w. (2.4.9). m=0. à (2.4.9) øÙ &' n∆t 4' ψ N n A)Z)] #¢&'siB2À&'\@ Luebbers Hunsberger i (2.4.9) ÒT *t´K, : ψw,v (n) = bw ψw,v (n − 1) + cw. ∂ Hv (n) ∂w. (2.4.10).
(76) 2.4. #$%&'()*+. 12. ¾sc (2.4.8)b T w. w. · µ ¶ ¸ σw aw bw = exp − + ∆t . ²0 κw ²0. (2.4.11). è (2.4.10) s ψ ø*G>Å ro#Ú./ CPML V®¯:/@ëìUyi ζ ∂H /∂w 4'à ψ (n) _o: º»ódM'qE:D./ø@!t7 CPML V®¯: !"(#D , : w,v. w. v. w,v. µ ¶ ∂Ex 1 1 ∂Hz 1 ∂Hy = − − J d,x + ψEx,y − ψEx,z , ∂t ²0 ²r ky ∂y kz ∂z µ ¶ ∂Ey 1 ∂Hx 1 1 ∂Hz = − − J d,y + ψEy,z − ψEy,x , ∂t ²0 ²r kz ∂z kx ∂x µ ¶ ∂Ez 1 1 ∂Hy 1 ∂Hx = − − J d,z + ψEz,x − ψEz,y , ∂t ²0 ²r kx ∂x ky ∂y. (2.4.12). µ ¶ 1 1 ∂Ez 1 ∂Ey ∂Hx =− − + ψHx,y − ψHx,z , ∂t µ0 µr ky ∂y kz ∂z µ ¶ ∂Hy 1 1 ∂Ex 1 ∂Ez =− − + ψHy,z − ψHy,x , ∂t µ0 µr kz ∂z kx ∂x µ ¶ ∂Hz 1 1 ∂Ey 1 ∂Ex =− − + ψHz,x − ψHz,y . ∂t µ0 µr kx ∂x ky ∂y. ¾s. ∂Hzn , ∂y ∂Hzn n−1 = bx · ψE + c · , x y,x ∂x ∂Hyn n−1 , = bx · ψE + c · x z,x ∂x. ∂Hyn , ∂z ∂Hxn n−1 = bz · ψE + c · , z y,z ∂z ∂Hxn n−1 = by · ψE + c · , y z,y ∂y. n−1 n ψE = by · ψE + cy · x,y x,y. n−1 n ψE = bz · ψE + cz · x,z x,z. n ψE y,x. n ψE y,z. n ψE z,x n+ 1 ψHx,y2 n+ 1 ψHy,x2 n+ 1 ψHz,x2. n+ 12. = by ·. n− 1 ψHx,y2. ∂Ez + cy · ∂y. = bx ·. n− 1 ψHy,x2. ∂Ez + cx · ∂x. = bx ·. n− 1 ψHz,x2. ∂Ey + cx · ∂x. n ψE z,y. ,. n+ 1 ψHx,z2. ,. n+ 1 ψHy,z2. ,. n+ 1 ψHz,y2. n+ 12. n+ 12. (2.4.13). n+ 12. = bz ·. n− 1 ψHx,z2. ∂Ey + cz · ∂z. = bz ·. n− 1 ψHy,z2. ∂Ex + cz · ∂z. = by ·. n− 1 ψHz,y2. ∂Ex + cy · ∂y. ,. n+ 12. ,. n+ 12. . (2.4.14). (2.4.14) s3E: b + c (w = x or y)øp°v : w. bw = e. −. σw ²0 kw. + a²w ∆t 0. w. ; cw =. σw 2a σw kw + kw w. µ e. −. σw ²0 kw. + a²w ∆t 0. ¶ −1 .. (2.4.15).
(77) 2.4. #$%&'()*+. ¾s. 13. µ. ¶ d−w m σw = σw,max , d µ ¶ d − w ma aw = aw,max ; 0 ≤ w ≤ d, d µ ¶ d−w m kw = 1 + (kw,max − 1) · . d. (2.4.16). 2.4.16s d op CPML l_ σ :Ðø*t&': max. σw,max =. 0.8(m + 1) , √ η 0 ∆ ²r µ r. (2.4.17). ¾s η = pµ /² Tëgsrä @ ² 2.1T CPML :°±²/² 2.2e[*-· !"(#T£ XMº»ódM'qED./ CPML XM²@
(78) Í[#ÊÅ Ãg (² = 1) CWr³´^ÓjVlmnM'qEód:CWr ³´ 1è ² Ãñãtu PVqCO$ ² JïTódD EHI3tà·¸tCWr³´ 7,-D@Q [16] UyJ [lM'qE:qCO$Àtu:$ ( ² 2.3-2.43° ) ÈÉt¸O¾VYk@ FDTD &'sg 4' T ∆x = ∆y = ∆z = 8.565E − 004+r b 7 20 » :4' ∆t = ∆x/5c@íît: ro 0. 0. 0. r. r. Gaussian pulse = cos(2πf t) e. ∞. −. 4π(t−t0 )2 d2. (2.4.18). ¾ d=200dt ] t = 6/f @×í3: ro:stuT 17.5GHzXtà· ¸ô 5GHz∼30GHz : +lM'qE«/:·¸( ² 2.53° )@&' g 90×90×90 TXroîôsV®¯lT 10∆xÂjVÇ 5®¯» P}¼®¯» Q *+½®¯» R «T¤C% E L¿» ( ² 2.6-2.73° ) ¸OVYk² 2.8-2.9éê[ÅÃg }Debye qE} Lorentz qE] Drude qEV®¯Â¤
(79) É@ 0. z.
(80) 2.4. #$%&'()*+. 14. 2.1: CPML !"#$%&'( ). 2.2: *+ ,-./0123456789 ( :'& ) &'( ).
(81) #$%&'()*+. 15. 8. 6. Eps_re Debye Eps_re Lorentz Eps_re Drude. Real part of εr. 4. 2. 0. -2. -4. -6. -8 5E+09. 1E+10. 1.5E+10. 2E+10. 2.5E+10. 3E+10. Frequency (Hz). 2.3: ;<-./&=>). 8 Eps_im Debye Eps_im Drude Eps_im Lorentz. 6 4. Imaginary part of εr. 2.4. 2 0 -2 -4 -6 -8 5E+09. 1E+10. 1.5E+10. 2E+10. 2.5E+10. 3E+10. Frequency (Hz). 2.4: ;<-./&?>).
(82) 2.4. #$%&'()*+. 16. 1 0.8 0.6 0.4. E z(t). 0.2 0. -0.2 -0.4 -0.6 -0.8 -1. 5E-10. 1E-09. 1.5E-09. 2E-09. 2.5E-09. Time(s). (a). 0.04. Center frequency = 17.5 GHz Band width = 5E09 ~ 30E09 GHz. 0.03. |E z(f)|. 0.02. 0.01. 0. -0.01. 5E+09. 1E+10. 1.5E+10. 2E+10. 2.5E+10. 3E+10. f/Hz. (b). 2.5: : 17.5GHz @AB7BCDEF 5GHz∼30GHz &GHI) (a) JK' (x=400∆x, t=0∼450∆t); (b) BK' .
(83) 2.4. #$%&'()*+. 17. 2.6: ,LM"#N POPQN Q :7PRN R S;T CPML !"#&'( ). 2.7: UTVW2:LM"#N P 7PQN Q X@YZN2[\F-.7]-./ '( ).
(84) 2.4. #$%&'()*+. 18. 10. 0. Reflective wave amplitude. Point P for Vacuum Point Q for Vacuum Point R for Vacuum. 10. -5. 10-10. 10-15. 10-20. 10. -25. 0. 1000. 2000. 3000. 4000. 5000. Time step. (a). 10. 0. Reflective wave amplitude. Point P for Debye medium Point Q for Debye medium Point R for Debye medium. 10. -5. 10-10. 10-15. 10-20. 10. -25. 0. 1000. 2000. 3000. 4000. 5000. Time step. (b). 2.8: ;T CPML ^J_`a !"#bcde) (a) ; (b) Debye /.
(85) 2.4. #$%&'()*+. 19. 10. 0. Reflective wave amplitude. Point P for Lorentz medium Point Q for Lorentz medium Point R for Lorentz medium. 10. -5. 10-10. 10-15. 10-20. 10. -25. 0. 1000. 2000. 3000. 4000. 5000. Time step. (a). 10. 0. Reflective wave amplitude. Point P for Drude medium Point Q for Drude medium Point R for Drude medium. 10. -5. 10-10. 10-15. 10-20. 10. -25. 0. 1000. 2000. 3000. 4000. 5000. Time step. (b). 2.9: ;T CPML ^J_`a !"#bcde) (a) Lorentz / ; (b) Drude /.
(86) ! "#$ CWr$%()@æø T tR?t)é§ÙS ) (Moment method, MoM)}ÙTU) (Finite element method, FEM) ¥F )é§Ù Ù ) (Finite difference time domain, FDTD)} ( #) (Time domain integral equation, TDIE) ¥@
(87) Q - (difference theory)/ <$Î (Taylor series) DEr³IT:I (#j°v (#@&'Ñi vÙ»7ìô»GV(#HI4 ']&'@ 3.1. FDTD. /01234. @æi !"(#I (# *Ù (finite difference) ,Î7ì (time domain) s p°@T[½Ésæ::,-D^ ( AJ3dv )ÑÕ ½/ä&'@1966 K. S. Yee bAÛp* FDTD )\g s CW% [17]e E}H % Ä» g Gde E( Ó H) % f¸Ù H( Ó E) % rg² 3.1² (a) p ° Yee’s cell ./]l4'(À&'C%W %@ öÆtì-·qE()t7ßño: !"(# (2.2.1FDTD(finite difference time domain).
(88) 3.1 FDTD. 2.2.2). ,-./0. 21. @ÑÕøi123,+)/03,:H(#Îv ∂Ex ∂t ∂Ey ∂t ∂Ez ∂t ∂Hx ∂t ∂Hy ∂t ∂Hz ∂t. µ ¶ ∂Hy 1 ∂Hz = − ² ∂y ∂z µ ¶ 1 ∂Hx ∂Hz = − ² ∂z ∂x µ ¶ 1 ∂Hy ∂Hx = − ² ∂x ∂y µ ¶ 1 ∂Ey ∂Ez = − µ ∂z ∂y µ ¶ 1 ∂Ez ∂Ex = − µ ∂x ∂z µ ¶ 1 ∂Ex ∂Ey = − µ ∂y ∂x. (3.1.1). Ã (3.1.1) ø¢ !"(#ø*Ñ T§C% (transverse electric polarizationTE mode) ?Ó¾(#Ã (E , E , H ) 3Dv x. ∂Ex 1 ∂Hz = ∂t ² ∂y ∂Ey 1 ∂Hz =− ∂t ² ∂x 1 ∂Ey ∂Ex ∂Hz = ( − ) ∂t µ ∂x ∂y. y. z. (3.1.2). ÀÃ (3.1.1) :(#D*ø@§W% (transverse magnetic polarizationTM mode) ¾(#Ã (H , H , E ) 3Dv x. y. z. ∂Hx 1 ∂Ez =− ∂t µ ∂y ∂Hy 1 ∂Ez = ∂t µ ∂x ∂Ez 1 ∂Hx ∂Hy = ( − ) ∂t ² ∂y ∂x. (3.1.3).
(89) 3.2. 1234 PRK 567. (3.1.1). 22. ~ : !"(#Dø*±-v§ (#p° ∂φ/∂t = f~¾s ~ = (Ex , Ey , Ez , Hx , Hy , Hz )T φ ³ ´ ∂Hy 1 ∂Hz − ∂z ² ∂y 1 ¡ ∂Hx ∂Hz ¢ ² ³ ∂z − ∂x ´ 1 ∂Hy x − ∂H ² ∂x ∂y ³ ´ f~ = 1 ∂Ey − ∂Ez µ ∂z ∂y ¢ 1 ¡ ∂Ez x µ ∂x − ∂E ∂z ³ ´ ∂Ey 1 ∂Ex µ ∂y − ∂x. (3.1.4). (3.1.5). T[e$%:RSDÑÕ½/7»T H | H | f »%4'¾s a ∼ a T3:O$/± a ∼ a %Ð ab^6¾ø*@!Í5 Ç [18, 19]@ ² 3.1² (b) | Ð (mesh size) dx}dy 4'tøרv{ | y i,j,k+ 1 2. 1. 3. x i,j,k+ 1 2. 1. 3. ∂Hy ∂x i,j,k+ 12. ∂Hx ∂y i,j,k+ 12. · ³ ´ ³ ´ ∂Hy 1 |i,j,k+ 1 = a1 Hy |i+ 5 ,j,k+ 1 − Hy |i− 5 ,j,k+ 1 + a2 Hy |i+ 3 ,j,k+ 1 − Hy |i− 3 ,j,k+ 1 2 2 2 2 2 2 2 2 2 ∂x dx ³ ´¸ + a3 Hy |i+ 1 ,j,k+ 1 − Hy |i− 1 ,j,k+ 1 (3.1.6) 2. 2. 2. 2. · ³ ´ ³ ´ 1 ∂Hx |i,j,k+ 1 = a1 Hx |i,j+ 5 ,k+ 1 − Hx |i,j− 5 ,k+ 1 + a2 Hx |i,j+ 3 ,k+ 1 − Hy |i,j− 3 ,k+ 1 2 2 2 2 2 2 2 2 2 ∂y dy ³ ´¸ + a3 Hx |i,j+ 1 ,k+ 1 − Hx |i,j− 1 ,k+ 1 (3.1.7) 2. 2. 2. 2. T[6 AI J: \Y~ ,-±ÊstiCWr³´>#s ]g G@¾91ÑÕi/ ÎÂG>0;?n: (# (Modified equation of second kind) *+M'NOAè (Dispersion relation preserving, DRP) LM*L @7ëIM'NOAè4'(#:O$@ 3.2. 5678 PRK 9:;#. !"(# ECWrôg :³;yst$Ô "Õ 9. (Hamiltonian structure) DE [20, 21]@ô [22, 23] ¥
(90) se[ æ@ !7Ô "ÕDE1 ½ï789. (symplectic) : (Ì©½.
(91) 3.2. 1234 PRK 567. 23. ï: @!ÍI:- BCDE@
(92) i6/789.DE: 4 '@ 4'@*Aè89.DE ÑÕ*7ß½ï Aèg 4 ':M'DE@
(93) 4'>#s3d/£LM¼ øÀÜÝÓZ[$%M 'NO(#]#\M'NO(#: ^@°:ôr$g :tiº K $%tu ω ] x}y + z (§:r$NOÂ6$%:M'NO@ * øÀÖ×#\:M'NO@ ôöÆGì-·qEs:ño !"(#D (2.2.1-2.2.2)ò tÔ "ÕNO num. ∂E ∂H = ∂t ∂H ∂H ∂H =− ∂t ∂E. (3.2.1). ¾sÔ "Õ H øÒvt 4 H(p, q) = T (p) + V (q) dp ∂H V (q) =− =− dt ∂q dq dq ∂H T (p) = = dt ∂p dp. T[Aè ÁØH>#sÕ7ëI:M'NOÑÕ6/` (explicit) : FVô7 49.:Ô "Õ: ØHø*6/ `789. (explicit symplectic time integrator) [24]@
(94) 4 '6/;< (two-stage) ;=RS (second-order accurate) :`789.: 4 Runge-Kutta () (explicit symplectic partitioned Runge-Kutta method)+ Q0 = q n P1 = pn 1 ∂T Q1 = Q0 + ∆t (P1 ) 2 ∂p ∂V P2 = P1 − ∆t (Q1 , tn+ 1 ) 2 ∂q 1 ∂T Q2 = Q1 + ∆t (P2 ) 2 ∂p q n+1 = Q2 pn+1 = P2.
(95) 3.3. 867 9:. 24. ¾s p}q op#¢:4'%G n + n + 1 op ÁF P }Q op!ò» :%t 1 + 2 op<$@ ño+ÅqEÎít !"(#Døp°vt:÷4 ' (semi-discretization) H n+1/2 = H n −. dt ∇ × En 2µ. (3.2.2a). dt ∇ × H n+1/2 ² dt = H n+1/2 − ∇ × E n+1 2µ. E n+1 = E n +. (3.2.2b). H n+1. (3.2.2c).
(96) i*èTmnôÄ 3.3 sK !":0;?(# (Modified equation of second kind)@ 3.3. <:;# . Äig 4' (3.1.6-3.1.7) *+789.: `°4' (explicit symplectic partitioned Runge-Kutta) mnt/0;?(#: (Modified equation of second kind analysis) *K g 4':(# bAi (3.2.2c)K Áø@ H = H − ∇ × E @ iGE(#oï (3.2.2a)ø@ H = H − ∇ × E @Ài =E + ∇ × H @cèÑÕøi1 (3.2.2b) K÷ Áø@ E 23,ÆtÒ t z}x y p° n. n+1/2. n+1/2. n+1/2 Ez |i,j,k+ 1 2 n+1/2. Ex |i,j,k+ 1 2. n+1/2. Ey |i,j,k+ 1 2. n−1/2. n−1/2. dt ². n−1/2. dt µ. n. n. n. µ ¶ dt ∂Hy n ∂Hx n = + | | , 1 − 1 ² ∂x i,j,k+ 2 ∂y i,j,k+ 2 µ ¶ ∂Hy n dt ∂Hz n n−1/2 | | , = Ex |i,j,k+ 1 + 1 − 1 ² ∂y i,j,k+ 2 ∂z i,j,k+ 2 2 µ ¶ dt ∂Hx n ∂Hz n n−1/2 = Ey |i,j,k+ 1 + |i,j,k+ 1 − | . 1 ² ∂z 2 ∂x i,j,k+ 2 2 n−1/2 Ez |i,j,k+ 1 2. dt 2µ. (3.3.1a) (3.3.1b) (3.3.1c). ÑÕ z (§:123, (3.3.1a)jVg JAx 3.1 Ä3K :4'.
(97) 3.3. 867 9:. 25. "«ô (3.1.6-3.1.7) sø@ n+1/2 Ez |i,j,k+ 1 2. =. " dt 1 + a1 (Hy |ni+ 5 ,j,k+ 1 − Hy |ni− 5 ,j,k+ 1 ) + a2 (Hy |ni+ 3 ,j,k+ 1 − Hy |ni− 3 ,j,k+ 1 ) ² dx 2 2 2 2 2 2 2 2 # " dt 1 − Hy |ni− 1 ,j,k+ 1 ) − a1 (Hx |ni,j+ 5 ,k+ 1 − Hx |ni,j− 5 ,k+ 1 ) 2 2 ² dy 2 2 2 2 #. n−1/2 Ez |i,j,k+ 1 2. + a3 (Hy |ni+ 1 ,j,k+ 1 2. 2. + a2 (Hx |ni,j+ 3 ,k+ 1 − Hx |ni,j− 3 ,k+ 1 ) + a3 (Hx |ni,j+ 1 ,k+ 1 − Hx |ni,j− 1 ,k+ 1 ) . 2. 2. 2. 2. 2. 2. 2. (3.3.2). 2. / <$ (Taylor series) i (3.3.2) sJA E | H | H | H | H | H | H | V n +g i, j, k »Î :ø@ n±1/2 z i,j,k+ 1 2. n y i± 1 ,j,k+ 1 2 2. n x i,j± 5 ,k+ 1 2 2. n x i,j± 3 ,k+ 1 2 2. n y i± 5 ,j,k+ 1 2 2. n y i± 3 ,j,k+ 1 2 2. n x i,j± 1 ,k+ 1 2 2. 3 1 ∂Ez n 1 ∂ 2 Ez n 1 n±1/2 3 ∂ Ez n Ez |i,j,k+ 1 = Ez |ni,j,k+ 1 ± ∆t |i,j,k+ 1 + ∆t2 | ∆t | 1 ± 1 + ... 2 2 ∂t 2 8 ∂t2 i,j,k+ 2 48 ∂t3 i,j,k+ 2 2 ∂Hy n ∂ 2 Hy n ∂ 3 Hy n 5 25 125 Hy |ni± 5 ,j,k+ 1 = Hy |ni,j,k+ 1 ± ∆x |i,j,k+ 1 + ∆x2 ∆x3 |i,j,k+ 1 ± | 1 + ... 2 2 2 2 2 2 2 ∂x 8 ∂x 48 ∂x3 i,j,k+ 2 ∂Hy n ∂ 2 Hy n ∂ 3 Hy n 3 9 9 Hy |ni± 3 ,j,k+ 1 = Hy |ni,j,k+ 1 ± ∆x |i,j,k+ 1 + ∆x2 |i,j,k+ 1 ± ∆x3 | 1 + ... 2 2 2 2 2 2 2 ∂x 8 ∂x 16 ∂x3 i,j,k+ 2 ∂Hy n ∂ 2 Hy n ∂ 3 Hy n 1 1 1 Hy |ni± 1 ,j,k+ 1 = Hy |ni,j,k+ 1 ± ∆x |i,j,k+ 1 + ∆x2 |i,j,k+ 1 ± ∆x3 | 1 + ... 2 2 2 2 2 ∂x 2 8 ∂x 2 48 ∂x3 i,j,k+ 2 5 ∂Hx n 25 ∂ 2 Hx n 125 3 ∂ 3 Hx n Hx |ni,j± 5 ,k+ 1 = Hx |ni,j,k+ 1 ± ∆y |i,j,k+ 1 + ∆y 2 ∆y |i,j,k+ 1 ± | 1 + ... 2 2 2 2 2 2 2 ∂y 8 ∂y 48 ∂y 3 i,j,k+ 2 3 3 ∂Hx n 9 ∂ 2 Hx n 9 3 ∂ Hx n |i,j,k+ 1 + ∆y 2 ∆y Hx |ni,j± 3 ,k+ 1 = Hx |ni,j,k+ 1 ± ∆y | | 1 ± 1 + ... 2 2 2 2 ∂y 2 8 ∂y 2 i,j,k+ 2 16 ∂y 3 i,j,k+ 2 1 ∂Hx n 1 ∂ 2 Hx n 1 ∂ 3 Hx n Hx |ni,j± 1 ,k+ 1 = Hx |ni,j,k+ 1 ± ∆y |i,j,k+ 1 + ∆y 2 |i,j,k+ 1 ± ∆y 3 | 1 + ... 2 2 2 2 2 2 2 ∂y 8 ∂y 48 ∂y 3 i,j,k+ 2. NÃ4'ìi*GÎ: <$oï (3.3.2)N>±-ìø@ ∂Ez n dt2 ∂ 3 Ez n dt4 ∂ 5 Ez n dt6 ∂ 7 Ez n |i,j,k+ 1 + + ... = | | | 1 + 1 + 5 i,j,k+ 2 7 i,j,k+ 12 2 ∂t 24 ∂t3 i,j,k+ 2 1920 322560 µ ∂t · ¶ ∂t 3 ∂Hy n ∂ Hy n 125 1 9 1 (5a1 + 3a2 + a3 ) | a1 + a2 + a3 dx2 | 1 + 1 ² ∂x i,j,k+ 2 24 8 24 ∂x3 i,j,k+ 2 µ ¶ µ ¶ 5 7 625 81 1 15625 243 1 4 ∂ Hy n 6 ∂ Hy n + a1 + a2 + a3 dx | + a + a + a dx | 1 1 1 2 3 384 640 1920 ∂x5 i,j,k+ 2 64512 35840 322560 ∂x7 i,j,k+ 2 ¸ · µ ¶ 1 ∂Hx n 125 9 1 ∂ 3 Hx n + ... − (5a1 + 3a2 + a3 ) |i,j,k+ 1 + a1 + a2 + a3 dy 2 | 1 2 ² ∂y 24 8 24 ∂y 3 i,j,k+ 2 µ ¶ µ ¶ 7 5 625 81 1 15625 243 1 4 ∂ Hx n 6 ∂ Hy n + a1 + a2 + a3 dy | a + a + a dx | 1 + 1 1 2 3 384 640 1920 ∂y 5 i,j,k+ 2 64512 35840 322560 ∂x7 i,j,k+ 2 ¸ + ... . (3.3.3). (3.3.3) s:= I J | | | Nà !"(#DøK C%ÓW%r(# (wave equation)/ ∂ 3 Ez n ∂t3 i,j,k+ 12. ∂ 5 Ez n ∂t5 i,j,k+ 12. ∂ 7 Ez n ∂t7 i,j,k+ 12.
(98) 3.3. 867 9:. <$ÎøiGE:= I Jj°Tg I JT*t : ∂ 3 Ez n | 1 = ∂t3· i,j,k+ 2 µ ¶ 5 ∂ 3 Hy n 1 125 9 1 2 ∂ Hy n (5a + 3a + a ) | a + a + a dx | 1 + 1 1 2 3 1 2 3 ²2 µ ∂x3 i,j,k+ 2 24 8 24 ∂x5 i,j,k+ 2 µ ¶ ¸ 7 625 81 1 4 ∂ Hy n a1 + a2 + a3 dx | + 1 + ... 384 640 1920 ∂x7 i,j,k+ 2 · µ ¶ 5 1 ∂ 3 Hx n 125 9 1 2 ∂ Hx n − 2 (5a1 + 3a2 + a3 ) | a + a + a dy | 1 + 1 1 2 3 ² µ ∂y 3 i,j,k+ 2 24 8 24 ∂y 5 i,j,k+ 2 µ ¶ ¸ 7 625 81 1 4 ∂ Hx n + a1 + a2 + a3 dy | 1 + ... 384 640 1920 ∂y 7 i,j,k+ 2 ∂ 5 Ez n | 1 = ∂t5 · i,j,k+ 2 µ ¶ 7 ∂ 5 Hy n 1 125 9 1 2 ∂ Hy n + a + a + a (5a + 3a + a ) | | 1 1 1 2 3 dx 1 2 3 3 2 5 i,j,k+ 2 ² µ ∂x 24 8 24 ∂x7 i,j,k+ 2 ¶ ¸ µ ∂ 9 Hy n 81 1 625 a1 + a2 + a3 dx4 + | 1 + ... 384 640 1920 ∂x9 i,j,k+ 2 µ ¶ · 7 1 125 9 1 ∂ 5 Hx n 2 ∂ Hx n − 3 2 (5a1 + 3a2 + a3 ) a + a + a dy | | 1 + 1 1 2 3 ² µ ∂y 5 i,j,k+ 2 24 8 24 ∂y 7 i,j,k+ 2 µ ¶ ¸ 625 81 1 ∂ 9 Hx n + a1 + a2 + a3 dy 4 + ... | 1 384 640 1920 ∂y 9 i,j,k+ 2 ∂ 7 Ez n | 1 = ∂t7 · i,j,k+ 2 µ ¶ 9 ∂ 7 Hy n 1 125 9 1 2 ∂ Hy n a + a + a dx (5a + 3a + a ) | | 1 + 1 1 2 3 1 2 3 ²4 µ3 ∂x7 i,j,k+ 2 24 8 24 ∂x9 i,j,k+ 2 µ ¶ ¸ 11 625 81 1 4 ∂ Hy n + a1 + a2 + a3 dx | 1 + ... 384 640 1920 ∂x11 i,j,k+ 2 µ ¶ · 9 1 ∂ 7 Hx n 125 9 1 2 ∂ Hx n − 4 3 (5a1 + 3a2 + a3 ) + a + a + a | | 1 1 1 2 3 dy 7 i,j,k+ ² µ ∂y 2 24 8 24 ∂y 9 i,j,k+ 2 µ ¶ ¸ 625 81 1 ∂ 11 Hx n + a1 + a2 + a3 dy 4 | 1 + ... 384 640 1920 ∂y 11 i,j,k+ 2. 26.
(99) 3.3. 867 9:. 27. 3@o°9kt ∂Ez n |i,j,k+ 1 = 2 ∂t µ · ¶ ∂Hy n ∂ 3 Hy n 1 125 9 1 Crx2 (5a1 + 3a2 + a3 ) |i,j,k+ 1 + a1 + a2 + a3 − (5a1 + 3a2 + a3 ) dx2 | 1 2 ² ∂x 24 8 24 24 ∂x3 i,j,k+ 2 µ µ ¶ ¶ ∂ 5 Hy n 625 81 1 Crx2 125 9 1 Crx4 a1 + a2 + a3 − a1 + a2 + a3 − (5a1 + 3a2 + a3 ) dx4 | + 1 384 640 1920 24 24 8 24 1920 ∂x5 i,j,k+ 2 ¸ + ... à ! · Cry2 1 ∂Hx n 125 9 1 ∂ 3 Hx n − (5a1 + 3a2 + a3 ) |i,j,k+ 1 + a1 + a2 + a3 − (5a1 + 3a2 + a3 ) dx2 | 1 2 ² ∂y 24 8 24 24 ∂y 3 i,j,k+ 2 à ! µ ¶ Cry4 Cry2 125 625 81 1 9 1 ∂ 5 Hx n + a1 + a2 + a3 − a1 + a2 + a3 − (5a1 + 3a2 + a3 ) dy 4 | 1 384 640 1920 24 24 8 24 1920 ∂y 5 i,j,k+ 2 ¸ + ... . (3.3.4). Ká (3.3.4) $(# ÉtR(o$(#. ∂Ez ∂t. =. 1 ². ³. ∂Hy ∂x. −. ∂Hx ∂y. è+øK È. ´. 5a1 + 3a2 + a3 = 1. (3.3.5). 9 1 Crx2 125 a1 + a2 + a3 − (5a1 + 3a2 + a3 ) = 0 24 8 24 24 or Cry2 125 9 1 a1 + a2 + a3 − (5a1 + 3a2 + a3 ) = 0 24 8 24 24. (3.3.6). ¾s Cr ≡ Cr ≡ T@(#:ñcA$@G>¾Ò(§ :1 23, ((3.3.1b)-(3.3.1c)), ÀK GE:o$(#D@è ?ò [ (o$(#(@* \©¢$@ ëì ((#KO½ïÇ5rXM*K $%M'NO( # (Dispersion relation equation)@i&' }g ÁÀïts tu]r$*eVr$Aèý ² 3.2@Nôr(#4' *+M'NO iôt ÄHIÊ @ x. c∆t ∆x. y. c∆t ∆y.
(100) 3.3. 867 9:. 28. (a). (b). 3.1: (a) fgh&ijklm (b) ,fgijkl&n>oN'( 7pq). ω. t x (a). κ (b). 3.2: (a) Jrstk&'( ; (b) PBrstk&'( .
(101) % &'()*+,-.($/01
(102) ièÄs½ïì03K: (3.1.6-3.1.7)jVr(# (Wave equation) ±- 4'@Vô7M'NO4'(# (Dispersion relation equation, DRE)½ïÇ5rXMøi (time) V.g (space) :Æj°Ttu (Angular frequency) V.r$ (Wavenumber) Æ6@ ]g &'G@*@Y¥å¾9@T[#$$%M '^ (dispersion error) ÷
(103) iïø*©SDEP\XMH_ @7ëIM'NOAè4'(#*eg 4'Vr$AèD ÂXV (#3dv3D}M'^*+f§gDä @ 4.1. =><:;#?@. G>ñoX/0f§DÅqEt: !"(#D (2.2.1-2.2.2)ø K t;=I ,r(# 1 ∂2φ = ∇2 φ c2 ∂t2. (4.1.1). 1 ∂2φ ∂2φ ∂2φ ∂2φ = + 2 + 2 2 2 c ∂t ∂x2 ∂y ∂z. (4.1.2). ¾s c Tr:³\@φ øÿ±o°TC% E~ ÓW% H~ @7ßÈÉt:r (#;ÆtøÎT oïÇ5r\ φ = φ e 0. −I(ωt−kx x−ky y−kz z). ô (4.1.2)ø@#\M'NO. ¡ ¢ 2 ωexact = c2 kx2 + ky2 + kz2. (4.1.3).
(104) 4.1. ;<867 =>. 30. T[T ]g @Y¥å¾94'>#s½ïÇ5rX M@i4'»:j° φ| = φ e oï
(105) 3 K 4' | = g 4'øÒv | = · ³ ´ ³ ´ ³ ´¸ a φ| − φ| +a φ| − φ| +a φ| − φ| } | · ³ ´ ³ ´ ³ ´¸ a φ| − φ| +a φ| − φ| +a φ| − φ| } | · ³ ¸ ³ ³ ´ ´ ´ a φ| − φ| + a φ| − φ| + a φ| − φ| @ èø@ n i. ∂φ n ∂t i. 0. −I(ωnum n∆t−kx i∆x−ky i∆y−kz i∆z). n+1/2. φ|i. n−1/2. −φ|i ∆t. ∂φ n ∂x i,j,k. 1 ∆x. 1. n i+ 52 ,j,k. n i− 52 ,j,k. 2. n i+ 32 ,j,k. n i− 32 ,j,k. 3. n i+ 12 ,j,k. n i− 12 ,j,k. 1 ∆y. 1. n i,j+ 52 ,k. n i,j− 52 ,k. 2. n i,j+ 32 ,k. n i,j− 32 ,k. 3. n i,j+ 12 ,k. n i,j− 21 ,k. 1 ∆z. 1. n i,j,k+ 52. n i,j,k− 52. n i,j,k+ 32. n i,j,k− 32. ∂φ n | ∂t i,j,k ∂φ n | ∂x i,j,k ∂φ n | ∂y i,j,k ∂φ n | ∂z i,j,k. 2. 3. n i,j,k+ 12. 2 sin (ωnum ∆t/2) φ ¡ 5∆t ¢ ¡ ¢ ¡ ¢ a1 sin 2 kx ∆x + a2 sin 32 kx ∆x + a3 sin 21 kx ∆x = 2I φ ¡5 ¢ ¡∆x ¢ ¡ ¢ a1 sin 2 ky ∆y + a2 sin 23 ky ∆y + a3 sin 21 ky ∆y = 2I φ ∆y ¡ ¢ ¡ ¢ ¡ ¢ a1 sin 52 kz ∆z + a2 sin 32 kz ∆z + a3 sin 12 kz ∆z = 2I φ ∆z ∂2φ ∂t2. ∂2φ ∂x2. ∂2φ ∂y 2. ∂φ n ∂z i,j,k. = =. n i,j,k− 12. = −I. Káø @t;=I J } } }. ∂φ n ∂y i,j,k. (4.1.4a) (4.1.4b) (4.1.4c) (4.1.4d). ∂2φ ∂z 2. ¸ · ∂2φ n sin (ωnum ∆t/2) 2 | =4 φ ∂t2 i,j,k ∆t ¢ ¡ ¢ ¡ ¢¸ ¡ · a1 sin 25 kx ∆x + a2 sin 32 kx ∆x + a3 sin 21 kx ∆x 2 ∂2φ n | = −4 φ ∂x2 i,j,k ∆x ¡ ¢ ¡ ¢ ¡ ¢¸ · a1 sin 25 ky ∆y + a2 sin 23 ky ∆y + a3 sin 12 ky ∆y 2 ∂2φ n | = −4 φ ∂y 2 i,j,k ∆y ¡ ¢ ¡ ¢ ¡ ¢¸ · a1 sin 25 kz ∆z + a2 sin 23 kz ∆z + a3 sin 12 kz ∆z 2 ∂2φ n | = −4 φ ∂z 2 i,j,k ∆z. (4.1.5a) (4.1.5b) (4.1.5c) (4.1.5d). iGE;=I J (4.1.5a-4.1.5d) oï (4.1.2)ø@t$%M'NO (Dispersion relation equation). ¡ ¢ ¡ ¢ ¡ ¢¸ · a1 sin 52 kx ∆x + a2 sin 23 kx ∆x + a3 sin 12 kx ∆x 2 1 sin2(ωnum ∆t/2) = c2 ∆t2 ∆x ¡ ¢ ¡3 ¢ ¡ ¢¸ · 5 a1 sin 2 ky ∆y + a2 sin 2 ky ∆y + a3 sin 12 ky ∆y 2 + ∆y ¡5 ¢ ¡ ¢ ¡ ¢¸ · a1 sin 2 kz ∆z + a2 sin 32 kz ∆z + a3 sin 21 kz ∆z 2 + (4.1.6) ∆z. 01r³ ÑÕ 3Ê r$§ (wavenumber vector)~k = (k , k , k ) = |k|(sin θ cos φ, sin θ sin φ, cos θ)*DErIH(§¾s θ Tú½ (Zenith anx. y. z.
(106) 4.1. ;<867 =>. 31. φ T(¦ (Azimuth angle)7² 4.1@cè (4.1.6) N±-ìø@. gle). ωnum ∆t. ·µ. µ µ ¶ µ ¶ µ ¶¶2 5 3 1 Crx2 a1 sin ksinθ cos φ∆x + a2 sin ksinθ cos φ∆x + a3 sin ksinθ cos φ∆x 2 2 2 µ µ ¶ µ ¶ µ ¶¶2 5 3 1 + Cry2 a1 sin k sin θ sin φ∆y + a2 sin k sin θ sin φ∆y + a3 sin k sin θ sin φ∆y 2 2 2 µ µ ¶ µ ¶ µ ¶¶2 ¶1/2 ¸ 5 3 1 2 + Crz a1 sin k cos θ∆z + a2 sin k cos θ∆z + a3 sin k cos θ∆z 2 2 2 = sin−1. (4.1.7). ¾sCr ≡ Cr ≡ Cr ≡ F/0 (∆x = ∆y = ∆z = h) È ÉtCr = Cr = Cr = Cr@$%P\øà (4.1.6) oï3Ê (| |) _ @ T[h eÂAèRSM'NObA3Ê^(# (Error funci tion) T | |−| | @^(#i ô 0 ≤ hk ≤ π Ñ AèëÐ%@K ár$g t7ß3Ù^%+ · ¸ Z x. c∆t ∆x. x. ωnum ~k. c∆t ∆y. y. y. z. c∆t ∆z. ωnum ~k. z. ωexact ~k. 2. mπ. E=. | −mπ. ωnum ωexact |−| | ~k ~k. 2. W (kh) dkh. (4.1.8). ¾s W (kh) T%2§$@Kt& = 0ø @3 0 (o$(#@ (3.3.5) + (3.3.6)+ø 7 DRP DE: a , a a O $@3 7 DRP DE: a , a a O$ m}Cr}ú½ (zenith angle) θ ](¦ (azimuth angle) φ :'iëì9k@tijV; y« @ ∂E(a1 ,a2 ,a3 ,m,Cr,θ,φ) ∂a3. 1. 1. 2. 2. 3. 3. : Ì3[^(#ìVô '(T[ ÏÐø(#ôg 4' D ( \:4'»TVhÎ:n )1^(# (Error function) · ¸st ÏÐø(#:D+i í&TVh:n@ >#s ; )Àá*+^ (aliasing error)FT[DE ±r,± ·¸ ÀAèÓÖ× π@² 4.2TË3 Cr = 0.2 m T } } } * + ]#\:M'NO:ÌͲÔ+,Ô Óop[r$ (modified wavenumber)kh tu (angular frequency)@ ÑÕ¼-øÀ6$%M'NO]#\M'NO@*AèÖ ×*er$Aèý@Q Ûæ×±r, π ÛAè4'( #M'NO@ 4.1.1. 1 2. 1 3. 3 7. 2 5. 3 8.
(107) 4.1. ;<867 =>. 32. Cr $: Cr ≡ O$T ñcAJ;op Áb ∆t ]b h :NOVô` (explicit) 4'()'$3D() Cr % 7ß Cr %Vô$%M'NO:@
(108) 3e 7 DRP DE:g 4'T (#!ê]3DÙN: Cr O$F² 4.3TQ m T Cr T 0.6}0.5}0.4}0.3}0.2 0.05 :M' NO#\]$%\:ÌͲFQ Cr %ÛÐ $%M'NO]#\M'N OÛ,Ö×;op[(#:M'NOiÛ@*ÍIÀAèF Cr %T 0.2 ] Cr %T 0.05 ÌÍø¢æçK×FQ$%r$]tu#\:æçK × 3# &'vá .ï7ßp 4.1e&'b (T = 30(s)) : OhQ Cr % ÓT 0.2 ] 0.05 Cr %T 0.2 3
(109) &' (CPU TIME) rT Cr %T 0.05 : 25%@cè 7 $%M'NOAè#*+ &' :v
(110) RS3'(6/: Cr %T 0.2@ 4.1.2. c∆t ∆x. 1 2. t:O$ Vôg =I J*4'( [18, 19] ÎÐÙ a ∼ a 3O $@à Îoï4'Âi I JjTg I JNb= I JT&ìø@! (3.3.5-3.3.6) R((#@xr³´L»_ /BÇÀù <$ (Taylor series) sb=^JAÂi ]t « ,-¾9@cè `789. (explicit symplectic time integrator) [24]øK 7M'NO:4'(# (Dispersion relation equation DRE)*è(# /$%P\]#\P\: ^_LMø* @7M'NOAè0((#Ãè((#ø \ 7ëIM'NO Aè: a ∼ a O$@ @0(M'NOAè(#>#sG>GEÊ 7 [ m *+ Cr V±ð$%M'NOpÚ:+&'vÑÕ'3[ m = *+ Cr = 0.2@èø*±- jVú½ (zenith angle)θ ](¦ (azimuth angle)φ Õ7ÙëI$%M'NOO$ a ∼ a @ p 4.2T'ú½ (Zenith angle)θ T 0 &90 }30 &60 45 (¦ (Azimuth angle)φ T 0 &90 }6 &84 }9 &81 }12 &78 }22.5 &67.5 }30 &60 }36 &54 } 6 &84 }45 3 @O$ a ∼ a :ÈÉ@(¦ (Azimuth angle)φ T 90 ∼ 360 ʧ$3Êøi (4.1.7) C¥Àj° 0 ∼ 90 :Ñ 4.1.3. 1. 1. 3. 3. 1 2. 1. ◦. ◦. ◦. ◦. ◦. ◦. ◦. ◦. ◦. ◦. 1. ◦. ◦. 3. ◦. ◦. ◦. ◦. ◦. ◦. ◦. ◦. ◦. ◦. 3. ◦. ◦. ◦. ◦.
(111) 4.2. ?@A =>. 33. @_G>p 4.2øÙ Qú½ (Zenith angle)θ T 90 ø@!T; ÈÉtO$FQú½ (Zenith angle)θ T 0 ø@!T ÈÉt O$@² 4.4TO$ a ∼ a ú½ (Zenith angle)θ ÈÉt¾%À (¦ (Azimuth angle) φ :òsøÙ Qú½ (Zenith angle) θ T 0 ÈÉtO$: ÂÐÀ(¦ (Azimuth angle) φ _@ ◦. ◦. 1. 3. ◦. 4.2. ABC! ?@. Ãô
(112) 3d/ PRK-DRP $%()T `°: &'> #s Á ∆t g Á ∆x}∆y ∆z Â:£Àvw;y%st 0!Ç*#$ Ú$%G3D@T[@!
(113) 3d/$%() q ~ = ~ ÓW%j° 3D()bA7ßÅÃqEsC%j° E E q ~ = ~ i (2.2.1-2.2.2) * V ~ = H ~ + IE ~ p°:@1g H H (Normalized space)øÒvt, µ0 ²0. ²0 µ0. ∗. ∗. ∗. ∗. ~ 1 ∂V ~ = I∇ × V c ∂t. (4.2.1). ¾s I = √−1c = GOvwô ² = µ = 1 ÈÉt@Q Taflove Brodwin [25] K()i
相關文件
In the inverse boundary value problems of isotropic elasticity and complex conductivity, we derive estimates for the volume fraction of an inclusion whose physical parameters
Since we use the Fourier transform in time to reduce our inverse source problem to identification of the initial data in the time-dependent Maxwell equations by data on the
“Find sufficiently accurate starting approximate solution by using Steepest Descent method” + ”Compute convergent solution by using Newton-based methods”. The method of
Second graders’ concepts of place value represented by problems involving cuisenaire rods, coins, and cherries..
In this study, we compute the band structures for three types of photonic structures. The first one is a modified simple cubic lattice consisting of dielectric spheres on the
For problems 1 to 9 find the general solution and/or the particular solution that satisfy the given initial conditions:. For problems 11 to 14 find the order of the ODE and
For R-K methods, the relationship between the number of (function) evaluations per step and the order of LTE is shown in the following
To improve the convergence of difference methods, one way is selected difference-equations in such that their local truncation errors are O(h p ) for as large a value of p as
