Research Article The CFS-PML for 2D Auxiliary Differential Equation FDTD Method Using Associated Hermite Orthogonal Functions

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1 Hinawi International Antennas an Propagation Volume 7, Article ID 58738, 6 pages Research Article The CFS-PML for D Auxiliary Differential Equation FDTD Metho Using Associate Hermite Orthogonal Functions Feng Jiang, Xiao-Ping Miao, Feng Lu, Li-Yuan Su, 3 an Yao Ma 3 College of Defense Engineering, PLA University of Science an Technology, Nanjing 7, China Jiangsu Regulatory Bureau of Nuclear an Raiation Safety, Nanjing 9, China 3 National Key Laboratory on Electromagnetic Environmental Effects an Electro-Optical Engineering, PLA University of Science an Technology, Nanjing 7, China Corresponence shoul be aresse to Feng Lu; fenglu76@63.com Receive 5 October 6; Revise 8 February 7; Accepte 6 March 7; Publishe 8 May 7 Acaemic Eitor: Sotirios K. Gouos Copyright 7 Feng Jiang et al. This is an open access article istribute uner the Creative Commons Attribution License, which permits unrestricte use, istribution, an reprouction in any meium, provie the original work is properly cite. The complex frequency shifte (CFS) perfectly matche layer (PML) is propose for the two-imensional auxiliary ifferential equation (ADE) finite-ifference time-omain (FDTD) metho combine with Associate Hermite (AH) orthogonal functions. Accoring to the property of constitutive parameters of CFS-PML (CPML) absorbing bounary conitions (ABCs), the auxiliary ifferential variables are introuce. An one relationship between fiel components an auxiliary ifferential variables is erive. Substituting auxiliary ifferential variables into CPML ABCs, the other relationship between fiel components an auxiliary ifferential variables is erive. Then the matrix equations are obtaine, which can be unifie with Berenger s PML (BPML) an free space. The electric fiel expansion coefficients can thus be obtaine, respectively. In orer to valiate the efficiency of the propose metho, one example of wave propagation in two-imensional free space is calculate using BPML, UPML, an CPML. Moreover, the absorbing effectiveness of the BPML, UPML, an CPML is iscusse in a two-imensional (D) case, an the numerical simulations verify the accuracy an efficiency of the propose metho.. Introuction In the conventional finite-ifference time-omain (FDTD) metho,, the time step is constraine by the Courant- Frierichs-Lewy (CFL) stability conition. When fine structuressuchasthinmaterialanslotaresimulate,more computer memory an computation time are require. In orer to eliminate the CFL stability conition, some unconitionally stable FDTD methos have been evelope such as alternating-irection implicit (ADI) metho 3, Crank- Nicolson metho 4, locally one-imensional metho 5, anweightelaguerrepolynomials(wlp)fdtdmetho 6. Recently, the Associate Hermite (AH) FDTD methos to moel wave propagation have been introuce in 7, 8. The perfectly matche layer (PML) absorbing bounary conitions (ABCs) introuce by Berenger 9 have been wielyusefortruncatingfdtdomains.somepmlvariants have been propose to improve the absorbing effectiveness for various electromagnetic waves, like moifie PML (MPML), uniaxial anisotropic PML (UPML), generalize PML (GPML), an so forth. Recently, UPML-ABC for AH- FDTD metho is use in conuctive meium. Accoring tokuzuogluanmittrain,thecomplexfrequencyshifte (CFS) PML, 3 is the most accurate in all the PML. The frequency-omain coorinate-stretching variable of CFS- PML (CPML) has three ajustable variables, while there are two ajustable variables for UPML-ABC. The reflection error of CPML can be greatly reuce from ajusting variables, an it has been implemente in the Cartesian coorinate, perioic structures, cylinrical coorinates, ispersive materials, WLP-FDTD, an so on. In this paper, a D CPML for auxiliary ifferential equation (ADE) FDTD metho using AH orthogonal functions is propose. It is shown that the constitutive parameters of CPMLarecomplicate,anifthemethoin7isuse irectly here, the secon erivative fiel components woul be involve an the final matrix equations will also be complex. In orer to get a simple an easy formula, the

2 International Antennas an Propagation auxiliary ifferential variables are introuce. Base on the ADE technique 4, 5 an Galerkin temporal testing proceure, one relationship between fiel components an auxiliary ifferential variables is erive. Accoring to auxiliary ifferential variables an Maxwell s equations of CPML, the other relationship between fiel components an auxiliary ifferential variables is erive. Then the formulations of all orers of AH functions are obtaine to calculate the magnetic fiel expansion coefficients. Accoring to 7, the matrix equations of CPML, Berenger s PML (BPML), an free space can be unifie. At last, the electric fiel expansion coefficients canalsobeobtaine,respectively.tovaliatetheefficiency of the propose metho, a D case is calculate. An the efficiency of the propose metho is verifie through the comparison with BPML an UPML ABCs.. Formulation With free space, the frequency-omain Maxwell s equations of CPML for D TE z moel case are jωε E x = R y H z Accoring to (4), the first erivative of fiel components u(r, t) with respect to t is 7 u (r, t) = (u n+ (r) n+ u n (r) n ) φ n ( t), where u (r) =. Using ADE scheme 3, 4, four auxiliary ifferential variables are introuce: ψ Ex = ψ Ey = ψ Hzx = (5) jωe x α y + jωε, (6) jωe y α x + jωε, (7) jωh zx α x + jωε, (8) jωε E y = R x H z jωμ H zx = R x E y () ψ Hzy = jωh zy α y + jωε. (9) With the transition relationship from frequency omain to timeomain,(6)canbewrittenas jωμ H zy = R y E x where ε is the electric permittivity an μ is the magnetic permeability of free space; R Φ =κ Φ +σ Φ /(α Φ + jωε ), Φ= x, y. ForCPML,α Φ an σ Φ areassumetobepositivereal an κ Φ is real an. An orthonormal set of basis functions {φ,φ,φ,φ 3,...} canbeefineas φ n ( t) = ( n n!π / ) / e t / H n ( t), () where H n ( t) = ( ) n e t ( n /t n )(e t ) are Hermite polynomials, t =(t T f )/, T f is a time-translating parameter, an is a time-scaling parameter. Choosing a finite orer of basis functions 7 an proper parameters for T f an, an then using these transforme basis functions, the causal fiel components, taking u for example, can be expane as u (r, t) = u n (r) φ n ( t). (3) The time erivative of the nth orer AH function is t φ n ( t) { φ ( t), (n =), = { { n φ n ( t) n+ φ n+ ( t), (n ). (4) α y ψ Ex +ε ψ Ex = E x. () Applying (3) an (5) to (), the fiel functions in () can be expane by (α y ψ p Ex (r) +ε φ n ( t) = φ n ( t). (ψ n+ Ex (r) n+ ψ n Ex (r) n )) (E n+ x (r) n+ E n x (r) n ) () Multiplying both sies by φ n ( t) an integrating over (, ),weget n+ ε ψq+ Ex (r) +α yψ q Ex (r) n = n+ Eq+ x (r) n Eq x (r). ε ψq Ex (r) ()

3 International Antennas an Propagation 3 Applying the same above proceure with (7) (9), we have n+ ε ψq+ Ey (r) +α xψ q Ey (r) n = n+ Eq+ y (r) n Eq y (r), ε ψq Ey (r) n+ ε ψq+ Hzy (r) +α yψ q Hzy (r) n ε ψq Hzy (r) = n+ zy (r) n Hq+ Hq zy (r). (3) Similarto7,8,wecanrewrite()-(3)inamatrixform: ξ y ψ Ex = ς E x, n+ ε ψq+ Hzx (r) +α xψ q Hzx (r) n ε ψq Hzx (r) ξ x ψ Ey = ς E y, ξ x ψ Hzx = ς H zx, (4) = n+ zx (r) n Hq+ Hq zx (r), where ξ y ψ Hzy = ς H zy, α Φ ε ε α Φ + ε ξ Φ = ε ς = + α Φ ε + ε α Φ. (Φ = x, y) (5) Equation(4)canberewritteninanewmatrixform: ψ Ex =ξ y ς E x, ψ Ey =ξ x ς E y, ψ Hzx =ξ x ς H (6) zx, ψ Hzy =ξ y ς H zy. Applying (6) (9) to () an transferring from frequency omain to time omain, we have ε (κ y E x +σ yψ Ex )= H z ε (κ x E y +σ x ψ Ey )= H z μ (κ x H zx μ (κ y H zy +σ x H zx )= E y +σ y H zy )= E x y. (7) Applying the same above proceure with (7), we have κ j α E q x (r) +σ j β ψ q Ex (r) = ε β Hq z (r) κ i α E q y (r) +σ i β ψ q Ey (r) = ε β Hq z (r) κ i+.5 α H q zx (r) +σ i+.5 β ψ q Hzx (r)

4 4 International Antennas an Propagation = μ β Eq y (r) κ j+.5 α H q zy (r) +σ j+.5 β ψ q Hzy (r) = β Eq x (r) μ (8) a a b b b 3 b 4 b 5 PEC PML where α = β =., (9) Using central ifference scheme, substituting (6) into (8), grafting (8), an using H z =H zx +H zy,wehave E x i+.5,j = C E y i,j η j β (H z i+.5,j+.5 H z i+.5,j.5 ), E y i,j+.5 = C E x i,j η i β (H z i+.5,j+.5 H z i.5,j+.5 ), H z i+.5,j+.5 = C H y i,j η j+.5 β (E x i+.5,j+ E x i+.5,j ) C H x i,j η i+.5 β (E y i+,j+.5 E y i,j+.5 ), () () () where η Φ =κ Φ α + σ Φ βξ Φ ς (Φ = x, y), CE x i,j = /ε Δx i, C E y i,j = /ε Δy j, C H x i,j = /μ Δx i, C H y i,j = /μ Δy j,anδx i an Δy j are the lengths of the cell ege where the electric fiels are locate. Δx i an Δy j are the istances between the center noes where the magnetic fiels a 3 a 4 a 5 y a 6 x p c c c 3 J y Figure : Computational omain. arelocate.fortheabove(i, j) subscripts for C x an C y in () (), (i, j) is not a real position but an array inex of each fiel variable, as shown in Figure of 6. In orer to form the matrix equations only containing magnetic fiel, we apply ()-() to (): where b l H z i.5,j+.5 +b r H z i+.5,j+.5 +ah z i+.5,j+.5 +b u H z i+.5,j+.5 +b H z i+.5,j.5 =, b l = C H x i,j CE x i,j η i+.5 β η i β, b r = C H x i,j CE x i+,j η i+.5 β η i+ β, b u = C H y i,j CE y i,j+ η j+.5 β η j+ β, b = C H y i,j CE y i,j η j+.5 β η j β, a= (b u +b +b l +b r +I). (3) (4) I is unit matrix. From (3), only magnetic fiel vector variables remaine. An it can also be foun that each magnetic fiel variable is relate to the four ajacent magnetic fiels. If R=in (), the final matrix equations of CPML will egrae to the matrix equationsoffreespaceinah-fdtdmetho.ifκ=an α Φ =in (), the final matrix equations of CPML will egrae to the matrix equations of BPML in AH-FDTD metho. So the matrix equations of CPML an BPML an free space can be unifie. An this greatly facilitates the programming. Finally, a bane sparse coefficient matrix is obtaine like 7, whichcontainsallthepointsinfreespaceancpmlabc. An the paralleling-in-orer solution scheme in 8 is applie to inirectly but efficiently calculate all of the expansion

5 International Antennas an Propagation 5 coefficients. If all of the expansion coefficients of the magnetic fiel are calculate, the expansion coefficients of the electric fiel can be obtaine from ()-(). Finally, E x (r, t), E y (r, t), an H z (r, t) canbereconstructebyusing(3). 3. Numerical Demonstration In orer to valiate the efficiency of the presente metho, wave scatter from a PEC meium is simulate. The BPML, UPML,anCPMLABCsareusetotruncatetheFDTD omains. The PML constitutive parameters are scale using mth orer polynomial scaling 7: σ(ρ Φ )=σ Φ,max ( ρ m Φ ) D Φ σ opt = (m+) 5πΔρ, (Φ = x, y), κ(ρ Φ )=+(κ Φ,max )( ρ m Φ ) D Φ (Φ = x, y), (5) where ρ inicates the istance from the ispersive meium- PML interface into the PML, D is the epth of the PML, an m=4is the orer of polynomial. As illustrate in Figure, the computational omain is truncate by aitional PML in x-irection an yirection, respectively. An we choose b = b5 = a = a6 = mm, a = mm, a3 = 6 mm, a4 = mm, a5 = mm, b = mm, b3 = 3 mm, an b4 = 5 mm (in Figure ), which results in a 5 5 cells lattice with Δx = Δy = mm. For the PEC rectangle s scatter, we choose c = 6 mm, c = 8 mm, an c3 = mm. A sinusoial-moulate Gaussian pulse is chosen as the y-irection excitation source: J y (t) =e (t t c) /t sin (πf c (t t c )), (6) where t = /(f c ), t c = 3t,anf c =.8 GHz. The time uration of interest for the analyze fiels is chosen as T s =5nsanthebanwithislimiteuptothefrequency W s =3GHz. We choose the orer = 3 antherangeof =.3 9 from the conition above. The time-translating parameter T f =5ns. In orer to compare the performance of the propose CPML with that of BPML, a reference solution was also simulate with no reflection coming from the bounary. An the same mesh is extene by cells in x-irection an yirection,leaing to a 5 5 cells lattice. An the reference fiel is calculate using AH-FDTD metho. The reflection erroriscalculateasfollows: R B =log ( EB x (t) ET x (t) max EB x (t) ), (7) where E T x (t) isthefielcomputeinthetestomainaneb x (t) is the reference fiel. In Figure, the waveform of electric fiel at points P is graphe by reference fiel, BPML, UPML, an CPML. An Ex (V/m) Reference value BPML Time (ns) UPML CPML Figure : Transient electric fiel of x component at the observation point. Reflection error (B) BPML UPML CPML Time (ns) Figure 3: Reflection error in the electric fiel intensity relative to the fiel s maximum amplitue versus time for BPML (with D = cells, σ max /σ opt =,anm = 4), UPML (with D =cells,κ max =, σ max /σ opt =.,anm =4),anCPML(withD =cells,α =.3, κ max =7,σ max /σ opt =.,anm =4)atpointp. it shoul be pointe out that parameters with ifferent PML are ifferent. Using (7), the reflection error is compute at the measurement point P. In Figure 3,it is note that the maximum relative error is 43 B, 58 B, an 7 B for the BPML, UPML, an CPML, respectively. An it is obvious that, with the CFS factor, the CPML is superior to

6 6 International Antennas an Propagation Table:ComparisonofCPUresourceforvariousmethos. Methos Δt (ps) Memory (Mb) CPU time (s) Conventional FDTD UPML of WLP-FDTD CPML of propose metho BPML an UPML. In Figures an 3, the computation time is much higher than the time-support of source wave. An late-time reflections of CPML are much lower than those of UPML an BPML. The CFL stability conition of this moel is Δt.33 ps. The time step size for the propose metho is 3.3 ps. In Table, comparison of CPU resource for four methos is presente. In orer to guarantee computation accuracy, the CPML, 8 is also aopte in WLP-FDTD an conventional FDTD metho. The total memorystoragefortheproposemethoisincreaseto 5. Mb, about 57.8 times of the conventional FDTD metho an 3.8 times of WLP-FDTD metho, while the total CPU time for the propose metho can be reuce to about.% of the conventional FDTD metho an 5.9% of WLP-FDTD metho. 4. Conclusion Using ADE scheme, the CPML for D AH-FDTD metho has been presente in this paper. This metho is free from CFL stability conition for it has eliminate the time variable in calculation. The final matrix equations of CPML an BPML an free space can be unifie. By solving the bane sparse coefficient matrix, magnetic fiel expansion coefficients of all orers of AH functions can be obtaine. Then the electric fiel expansion coefficients can also be obtaine, respectively. Numerical results show that this implementation is very effective in absorbing the electromagnetic waves, which means that the propose metho can save more computation time an computer memory. Conflicts of Interest The authors eclare that there are no conflicts of interest regaring the publication of this paper. References K. S. Yee, Numerical solution of initial bounary value problems involving Maxwell equations in isotropic meiia, IEEE Trans. on Antennas an Propagat, vol.4,no.3,pp.3 37, 966. R. Xiong, B. Chen, Z. 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