Recent Advances in Time-Domain Maxwell s Equations in Metamaterials

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1 Recent Advances in Time-Domain Maxwell s Equations in Metamaterials Yunqing Huang 1, and Jicun Li, 1 Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Cina Department of Matematical Sciences, University of Nevada, Las Vegas, Nevada , USA Abstract. In tis paper, we present tree second-order finite element metods solving Maxwell s equations wen metamaterials are involved. Te first metod is based on integral-differential equations transformed from te governing equations; te oter two metods solve te original governing differential equations. Numerical results are presented for te last two metods. 1 Introduction Te double negative (DNG metamaterials are artificially structured electromagnetic materials wit bot negative permittivity and permeability. Te successful construction of DNG metamaterials in triggered a new wave in te study and design of DNG metamaterials and exploration of applications in diverse areas suc as sub-wavelengt imaging and cloaking. Since, engineers and pysicists ave proposed some matematical models and initiated some numerical simulations for DNG metamaterials. Most metods are eiter based on te finite-difference timedomain (FDTD metods or finite element metods on frequency domain (mainly using commercial packages suc as HFSS and FEMLAB. Toug tere exist many work (e.g., [5] and references cited terein devoted to Maxwell s equations, to our best knowledge, most are for te simple media case. Tere are very few numerical analysis work wen metamaterials are involved except our recent effort in time-domain finite element metods [4,3,1]. In tis paper, we systematically present tree second-order metods: one solved as integral-differential equations; te oter two solved directly from te original differential equations eiter implicitly or explicitly. Te Governing Equations Te DNG metamaterials can be simulated using lossy Drude polarization and magnetization models. Te governing equations for modeling wave propagation in metamaterials are [4]: Supported by te NSFC for Distinguised Young Scolars (16516 and National Basic Researc Program of Cina under te grant 5CB3171. Supported by National Science Foundation grant DMS W. Zang et al. (Eds.: HPCA 9, LNCS 5938, pp , 1. c Springer-Verlag Berlin Heidelberg 1

2 Recent Advances in Time-Domain Maxwell s Equations in Metamaterials 49 E H ɛ = H J, t μ = E K, (1 t 1 J ɛ ωpe t + Γe 1 K J = E, ɛ ωpe μ ωpm t + Γm K = H, ( μ ωpm were ɛ and μ are te vacuum permittivity and permeability respectively, ω pe and ω pm are te electric and magnetic plasma frequencies respectively, Γ e and Γ m are te electric and magnetic damping frequencies respectively, E(x,tand H(x,t are te electric and magnetic fields respectively, and J(x,tandK(x,t are te induced electric and magnetic currents respectively. For simplicity, we assume tat te modeling domain be Ω (,T, were Ω is a bounded Lipscitz polyedral domain in R 3 wit connected boundary Ω. Furtermore, we assume tat te boundary of Ω is perfect conducting so tat n E = on Ω, (3 were n is te unit outward normal. Also we assume te initial conditions: E(x, = E (x, H(x, = H (x, J(x, = J (x, K(x, = K (x. Lemma 1. [1] Tere exists a unique solution for te system (1-(. Furtermore, te solution of (1-( satisfies te following stability estimate ɛ E(t + μ H(t + 1 K(t μ ωpm + 1 J(t ɛ ωpe ɛ E( + μ H( + 1 K( μ ωpm + 1 J( ɛ. (4 ωpe 3 Tree Different Numerical Metods To construct a finite element metod, we partition Ω by a family of regular tetraedral meses T wit maximum mes size. For simplicity, consider te lowest order Raviart-Tomas-Nédélec s spaces: for any K T, U = {u H(div; Ω u K = c K + d Kx}, V = {v H(curl; Ω v K = a K + b K x}, V = V {n v = on Ω}, were a K, b K, c K are constant vectors in R 3, and d K is a real constant. For any v H α (curl; Ω, 1 <α 1, it is known tat we can define te Nédélec interpolant Π v V on eac tetraedron K and ave ([5]: v Π v + (v Π v C α v α,curl. (5 Denoting by P u U te standard (L (Ω 3 -projection onto U,weave u P u C α u α u H α (Ω. (6 To define a fully discrete sceme, we divide te time interval (,TintoM uniform subintervals by points t k = k, were = T M. Moreover, we denote te k-t subinterval =[t k 1,t k ], and define u k = u(,k.

3 5 Y. Huang and J. Li 3.1 Treated as Integral-Differential Equations Solving ( wit initial currents J (x andk (x, we obtain J(x,t; E =e Γet J (x+ɛ ω pe t K(x,t; H =e Γmt K (x+μ ω pm e Γe(t s E(x,sds J N (x+j N(E, (7 t e Γm(t s H(x,sds K N (x+k N (H. Hence, te governing equations (1-( become: find (E, H suc tat E ɛ t H + J N(E = J N (x (x,t Ω (,T, (8 H μ t + E + K N (H = K N (x (x,t Ω (,T. (9 Now we can formulate a Crank-Nicolson mixed finite element sceme for (8-(9 as follows: for k =1,,,M, find E k V, Hk U suc tat ɛ (δ E k, φ (H k, φ +(J k, φ = ( J N, φ, (1 μ (δ H k, ψ +( E k, ψ +(K k, ψ = ( K N, ψ (11 for any φ V, ψ U and <t T, subject to te initial conditions E (x =Π E (x and H (x =P H (x. (1 Here we denote ũ = 1 I u(tdt for any u. Te J k k is defined as J k = 1 (1 + e Γe J k 1 J =, J k = e Γe J k 1 Similar formula will be used for K k. + ɛ ω pe 1 (e Γe E k 1 + E k, 1 k M, (13 + ɛ ω pe (e Γe E k 1 + E k, 1 k M. (14 Lemma. [1] For S = H 1 (curl; Ω or S =(H α (Ω 3 wit α, we ave (i δ u k S 1 (ii u k 1 t k 1 u t(t Sdt u(tdt S 3 t k 1 4 u H 1 (,T; S, t k 1 u tt(t Sdt u H (,T; S. be defined by (7 and (13, respec- Lemma 3. Let J k N J N (E(,t k and J k tively. Ten for any 1 k M, we ave J k J k N C[ E l E l + 4 ( E(t + E t (t + E t (t dt]. l=

4 Recent Advances in Time-Domain Maxwell s Equations in Metamaterials 51 Proof. Recall tat J N (E =ɛ ω pe t e Γe(t s E(sds. For clarity, we denote J k = J N (E(t k. By definitions of J and J k,weave J k J k = 1 (1 + e Γe (J k 1 J k 1 +ɛ ωpe [ 1 e Γ e(t k s E(sds 1 e t k 1 (e Γ E k 1 + E k ] +ɛ ωpe 1 [(Ek E k +e Γe (E k 1 E k 1 ] J k 1 J k 1 + ɛ ωpe t k 1 s (e Γe(t k s E(s ds +ɛ ωpe 4 ( Ek E k + E k 1 E k 1, (15 were in te last step we used te inequality 1 (uk + u k 1 1 u(tdt t k 1 for u(s =e Γe(tk s E(s. From [, (Eq. (7], for k, we ave l= t k 1 u tt (t dt k 1 1 J k 1 J k 1 ɛ ωpe[ E l E l + C ( E + E s + E ss ds]. (16 Wen k =1,weaveJ = J =. Hence (16 olds true for k =1. Combining (15-(16, we obtain (for any k 1 J k J k C[ E l E l + ( E + E t + E tt dt], (17 l= were we absorbed te dependence of ɛ,ω pe,γ e into te generic constant C. Squaring bot sides of (17 and using Caucy-Scwarz inequality leads to J k J k C[ ( C[ E l E l + 4 ( l= ( E + E t + E tt dt ] E l E l + 4 ( E + E t + E tt dt] l= wic concludes te proof. Teorem 1. Let (E n, H n and (E n, H n be te solutions of (1-( and (1- (11 at time t = t n, respectively. Assume tat for 1 <α 1, E, E t L (,T; H α (curl; Ω, H, H t L (,T; H α (Ω, E tt, E tt, H tt, H tt L (,T;(L (Ω 3. Ten tere is a constant C = C(T,ɛ,μ,ω pe,ω pm,γ e,γ m, E, H, independent of bot te time step and te mes size, suc tat max 1 n M ( En E n + H n H n C( + α.

5 5 Y. Huang and J. Li Proof. Multiplying (8-(9 by test functions φ and ψ, integrating te resultants in time over, ten coosing φ = 1 φ, ψ = 1 ψ and subtracting (1 and (11, respectively, we obtain te error equations ɛ (δ ξ, k φ (η k, φ =ɛ (δ (Π E k E k, φ (P H k 1 H(sds, φ +(J k 1 J N (sds, φ, (18 μ (δ η, k ψ +( ξ k, ψ =μ (δ (P H k H k, ψ +( (Π E k 1 E(sds, ψ +(K k 1 K N (sds, ψ, (19 were we introduced te notation ξ k = Π E k E k and ηk = P H k H k. Coosing φ = (ξ k + ξk 1 =ξ k, ψ = (η k + ηk 1 =η k in (18-(19, and adding te resultants togeter, we ave ɛ ( ξ k ξ k 1 +μ ( η k η k 1 =ɛ (δ (Π E k E k, ξ k (P H k 1 H(sds, ξ k +(J k 1 J N (sds, ξ k +μ (δ (P H k H k, η k +( (Π E k 1 E(sds, η k +(K k 1 K N (sds, η k 6 = (Err i. i=1 Using te inequality ab δa + 1 4δ b and Lemma, we ave ( (Err 1 δ 1ɛ ξ k + ɛ δ (Π E k E k δ 1 δ 1ɛ ( ξ k + ξ k 1 + ɛ t(π E k E k dt. (1 δ 1 Similarly, by te projection property of P and integration by parts, we ave (Err = ( (H k 1 H(sds, ξ k δ ξ k + (H k 1 H(sds, δ δ ( ξ k + ξ k H tt(t dt. ( 8δ Similarly, by Lemmas and 3, we can obtain (Err 3 =(J k J k N + J k N 1 J N (sds, ξ k δ 3( ξ k + ξ k 1 + ( J k J k δ N + J k N 1 J N (sds 3 δ 3( ξ k + ξ k 1 + C E l E l δ 3 l=

6 Recent Advances in Time-Domain Maxwell s Equations in Metamaterials 53 + C5 tk ( E + E t + E tt δ dt + 4 J N 3 4δ 3 t ds δ 3( ξ k + ξ k 1 + C ( ξ δ l + Π E l E l 3 l= + C5 tk ( E + E t + E tt δ dt + 4 J N 3 4δ 3 t ds. (3 Similarly, we can easily obtain (Err 4 δ 4μ ( η k + η k 1 + μ t(p H k H k dt, δ 4 (Err 6 δ 6( η k + η k 1 + C ( η l + P H l H l δ 6 l= + C5 tk ( H + H t + H tt dt + 4 K N ds, δ 6 4δ 6 t (Err 5 =( (Π E k E k + E k 1 E(sds, η k δ 5 η k + ( (E k 1 E(sds + Π E k E k δ 5 δ 5( η k + η k ( E 4δ 5 t dt + Π E k E k δ. 5 Finally, summing up bot sides of ( from k =1uptoanyn M, usingte estimates of (Err i, te fact tat ξ = η =, and coosing δ i small enoug so tat ξ n and ηn can be controlled by te left and side, we ave ɛ ξ n + μ η n n 1 C 1 ( ξ k + η k + ɛ δ δ + C4 δ 3 + μ δ δ 5 + C4 δ 6 k=1 tn tn n k=1 tn tn n k=1 H tt(t dt + C δ 3 t(π E k E k dt n Π E l E l l=1 ( E + E t + E tt dt + 4 4δ 3 tn t(p H k H k dt + δ 5 Π E k E k ( E t dt + C δ 6 n P H k H k k=1 ( H + H t + H tt dt + 4 4δ 6 tn J N t ds K N t ds, (4 wereweusedtefactk T and absorbed T into te generic constant C. Using interpolation estimates (5 and (6 to (4 and te inequality J T N t C( E(s ds + E(t + E t(t, (5

7 54 Y. Huang and J. Li we obtain n 1 ɛ ξ n + μ η n C 1 ( ξ k + η k +C ( 4 + α, (6 k=1 wic concludes te proof by te standard tecnique [1]. 3. Treated as Differential Equations: Crank-Nicolson Metod Recently, we found tat we can solve (1-( directly and efficiently by constructing a Crank-Nicolson sceme[3]: for k =1,,,M, find E k V, J k V, H k, K k U suc tat ɛ (δ E k, φ (H k, φ +(J k, φ =, (7 μ (δ H k, ψ +( E k, ψ +(K k, ψ =, (8 1 (δ ɛ ωpe J k, φ + Γe (J k, φ ɛ ωpe =(E k, φ, (9 1 (δ μ ωpm K k, ψ + Γm (K k μ, ψ ωpm =(H k, ψ, (3 are true for any φ, φ V, and ψ, ψ U. In practical implementation, at eac time step we just need to first solve a smaller system ( ɛ =( ɛ ( μ =( μ + ɛω pe +Γ e (E k, φ (H k, φ ɛω pe (E k 1 +Γ e, φ +(H k 1, φ + μω pm +Γ m (H k, ψ +( E k, ψ μω pm 4 +Γ e (J k 1, φ, (31 (H k 1, ψ +Γ ( E k 1 4, ψ (K k 1, ψ m +Γ. (3 m for E k and H k,tenupdatej k and K k as follows: J k = ɛω pe (E k + E k 1 + Γe J k 1, +Γ e +Γ e (33 K k = μω pm (H k + H k 1 + Γm K k 1. +Γ m +Γ m (34 In [3], we proved te following optimal error estimate: max 1 n M ( En E n + H n H n + J n J n + K n K n C( + α.

8 Recent Advances in Time-Domain Maxwell s Equations in Metamaterials Treated as Differential Equations: Leap-Frog Metod Inspiring from te classic FDTD sceme, we can formulate a leap-frog sceme: Given initial approximations E, K, H 1, J 1, for k = 1,,, find E k V, J k+ 1 V, H k+ 1, K k U suc tat ɛ ( Ek E k 1, φ (H k 1, φ +(J k 1, φ =, (35 μ H k+ 1 H k 1 (, ψ +( E k, ψ +(K k, ψ =, (36 1 ( J k+ 1 J k 1, φ ɛ ωpe + Γe ( 1 ɛ ωpe (J k+ 1 + J k 1, φ =(E k, φ, (37 1 μ, ψ ωpm + Γm ( 1 μ ωpm (K k + K k 1, ψ =(H k 1, ψ, (38 for any φ V, ψ U, φ V, ψ U. Conditional stability and te following optimal error estimate were obtained in [1]: max ( E n E n + H n+ 1 n+ 1 H + J n+ 1 n+ J 1 + K n K n 1 n C( + α +C ( E E + H 1 H 1 + J 1 J 1 + K K. 4 Numerical Examples Here we sow some numerical results obtained by te Crank-Nicolson (CN sceme (7-(3 and Leap-Frog (LP sceme (35-(38. We implemented our algoritms using te lowest order rectangular edge element basis functions [1]. Our tests were carried out using MATLAB 7. running on Dell Latitude D63 laptop wit 1GB memory and. GHz CPU. In order to test our algoritm wit an analytical solution, we add source terms to te original governing equations, i.e., te LP sceme (35-(38 becomes: (E k, φ =(E k 1, φ + ɛ [(H k 1, φ (J k 1, φ +(f k 1, φ ], (39 (H k+ 1, ψ =(H k 1, ψ μ [( E k, ψ +(K k, ψ (g k, ψ ], (4 J k+ 1 = ɛω pe E k + Γe J k 1, (41 +Γ e +Γ e K k = μω pm H k 1 + Γm K k 1, (4 +Γ m +Γ m were f k 1 and g k are added source terms. Wile for CN sceme, we add f k and g k to te rigt and side of (7 and (8, respectively. To compare te performance between CN and LP scemes, we used te exact solution developed for te -D transverse electrical model in [1]. Troug many

9 56 Y. Huang and J. Li Table 1. L errors obtained by te CN sceme wit =1 8 Time steps nt =1 nt = 1 DOF Mes sizes E x H z E x H z Time (sec e e e e e e e e e e e e e e e e e e e e Table. L errors obtained by te CN sceme wit =1 1 Time steps nt =1 nt = 1 DOF Mes sizes E x H z E x H z Time (sec e e e e e e e e e e e e e e e e e e e e Fig. 1. Numerical solution on mes: (Left electric field; (Rigt magnetic field tests, we found tat even for CN sceme good results can only be acieved until =1 8 or smaller. Our numerical results obtained wit CN sceme using =1 8 and =1 1 are presented in Table 1 and Table, respectively. Also presented are te total number of degree of freedom (DOF and computational time. Tables 1- sow clearly O( convergence for E, wic is superconvergent for rectangular meses [1]. Te accuracies for H are really excellent, but it sows O( as mes gets finer. We tink tis is due to te larger sampling errors coming wit finer meses since H are approximated by piecewise constants.

10 Recent Advances in Time-Domain Maxwell s Equations in Metamaterials 57 Comparing Tables 1- to Tables -3 of [1] (results obtained by LP sceme, we see tat wit te same mes and time step CN sceme acieves similar accuracy for E as LP sceme and as muc more accurate results for H tan LP sceme. However, CN sceme takes muc longer computational time tan LP sceme. 5 Concluding Remarks In tis paper, we presented tree second order metods for solving te metamaterial Maxwell s equations in time-domain. Considering te accuracy and computational cost, it seems tat te LP sceme is te best coice of te tree metods. More rigorous comparisons will be carried out in our future work. For practical applications, we ave to work on ig performance computers, since our laptop runs out of memory for 3 3 mes. References 1. Li, J.: Numerical convergence and pysical fidelity analysis for Maxwell s equations in metamaterials. Comput. Metods Appl. Mec. Engrg. 198, (9. Li, J., Cen, Y.: Analysis of a time-domain finite element metod for 3-D Maxwell s equations in dispersive media. Comput. Metods Appl. Mec. Engrg. 195, 4 49 (6 3. Li, J., Cen, Y., Elander, V.: Matematical and numerical study of wave propagation in negative-index materials. Comput. Metods Appl. Mec. Engrg. 197, (8 4. Li, J., Wood, A.: Finite element analysis for wave propagation in double negative metamaterials. J. Sci. Comput. 3, (7 5. Monk, P.: Finite Element Metods for Maxwell s Equations. Oxford University Press, Oxford (3