An RWG Basis Function Based Near- to Far-Field Transformation for 2D Triangular-Grid Finite Difference Time Domain Method

Transcription

1 An RWG Basis Functin Based Near- t Far-Field Transfrmatin fr D Triangular-Grid Finite Difference Time Dmain Methd Xuan Hui Wu, Student Member, IEEE, Ahmed A. Kishk, Fellw, IEEE, and Allen W. Glissn, Fellw, IEEE Abstract A near- t far-field transfrmatin NTFT) based n the RWG basis functin is prpsed fr a D triangular-grid methd. The surface equivalence therem is applied and the equivalent current density, J z in the TM z case r M z in the TE z case, n the radiatin cntur is interplated by using RWG basis functins. The interplatin is valid fr the NTFT in either the time r frequency dmain. Frmulatins fr the interplatin scheme and the NTFT are presented fr the TM z case. The scattering by an infinite dielectric circular cylinder due t an electric line surce is simulated by using the triangular-grid as well as the prpsed NTFT. Numerical results in bth the time dmain and frequency dmain are demnstrated and verified by cmparisn with analytical slutins. Index Terms, RWG basis functin, near- t far-field transfrmatin, triangular grid. I. INTRODUCTION The finite-difference time-dmain ) methd prpsed by Yee [] has been widely and successfully used in cmputatinal electrmagnetics. An attractive feature f this methd is its capability f predicting the far-zne radiatin f a structure in bth time and frequency dmains. In a three dimensinal 3D) prblem, this is dne thrugh a near- t farfield transfrmatin NTFT), where the far-field is btained by integrating the equivalent current densities n a clsed surface enclsing the entire structure under simulatin, weighted by an apprpriate Green s functin. The equivalent current densities are btained by evaluating the tangential fields n the surface. In a tw dimensinal D) prblem, a clsed cntur, referred t as the radiatin cntur, is used instead f a clsed surface. This transfrmatin was first perfrmed in the frequency dmain FD) [], and was later realized in the time dmain TD) fr btd [3], [4] and D [5], [6], [7] prblems. Recently, a new transfrmatin based n a TD sphericalmultiple analysis was prpsed [8]. The cnventinal methd based n the Yee cell adpts a staggered rectangular grid scheme, and thus intrduces staircase errrs in simulating an arbitrarily shaped bject [9]. In 994, a D triangular-grid methd was prpsed t vercme this drawback [] and was successfully used t analyze micrwave circuits []. Thugh the triangular grid prvides mre flexibility in mdeling an bject with arbitrary shape, its NTFT is nt s straightfrward as that in the Yee cell case. In a D Yee cell based methd, the radiatin cntur is nrmally selected t be a rectangle, and either the tangential electric r magnetic field n the cntur is unknwn due t the half cell ffset f the tw fields. Frtunately, the unknwn field can be easily interplated frm the fields lcated n the tw sides f that cntur. This interplatin The authrs are with the Department f Electrical Engineering, University f Mississippi, University, MS 38677, USA is based n the fact that the field ne-half cell shifted frm the radiatin cntur is parallel t that cntur. In the D triangular-grid methd, this interplatin prcedure is nt applicable because the lcatin and directin f the fields arund the radiatin cntur are nt unifrm. In this paper, the RWG basis functin is brrwed frm the Methd f Mments MOM) in rder t slve the abvementined prblem in the NTFT f the triangular-grid methd. The RWG functin is a widely used basis functin in MOM fr expanding the current density n a pair f jined triangular patches, and evaluating the field due t the patches []. Herein, the RWG functin is nt used t get the cntributin f a triangle element, but instead t interplate the field n the triangle edges t btain the tangential field cmpnents required fr NTFT. This paper is rganized as fllws. Sectin II intrduces the RWG basis functin based interplatin scheme, and uses it t derive the NTFT frmulatins in bth the FD and TD. Sectin III prvides simulatins f the scattering by an infinite dielectric circular cylinder due t an electric line surce, and cmpares the results t exact slutins that are btained by the eigenfunctin expansin methd. Sectin IV presents the cnclusins f this wrk. II. THEORY Withut lsing generality, nly the TM z case is discussed in this paper. The TE z case can be btained in a similar manner. In the triangular-grid methd fr the TM z case, the ttal electric field E z is evaluated at triangle vertices, and the magnetic field H evaluated at the center f triangle edges, but nly fr the cmpnent nrmal t that edge, as shwn in Figure. The series f cnnected triangle edges shwn as a dashed line in the figure are assumed t be a part f the radiatin cntur, where bth the tangential electric and magnetic fields are needed t determine the equivalent current densities n the cntur. The tangential electric field E z n the radiatin cntur is btained during the iteratin, while the tangential magnetic field is unknwn, and shuld be btained thrugh a suitable interplatin scheme. In the next part, the interplatin f the magnetic field in a triangle is derived by using the RWG basis functin, thus btaining the expressin f the tangential magnetic field n the triangle edges. After that, NTFT frmulatins in bth the FD and TD are presented. A. Field expansin with the RWG basis functin Figure shws an arbitrary triangular mesh cell and the crrespnding magnetic field psitins in the triangular-grid methd. P, P, and P are triangle vertices. h, h, and h are triangle heights. H, H, and H are the magnetic

2 H E z φ ˆρ x P in Fig.. Triangular mesh and field lcatins fr TM z case S E z H H L in H L ut h h L ˆl l T E z ˆn ẑ P H 4 H 3 h 4 h ut φ l l x H H h P ut h h p Fig. 3. Apprximatin f tangential field n the radiatin cntur Fig.. p p M P P H Field expansin with RWG basis functins field cmpnents evaluated in the iteratin, each lcated at the center f a triangle edge and riented nrmal t that edge. M, an arbitrary psitin inside the triangle, defines three vectrs p, p, and p, each pinting frm a vertex with the same index t M. The well knwn RWG basis functin, nrmally used t expand the current density n a triangular patch, is used here t interplate the magnetic field. Frm [], each triangle edge is assciated with an RWG basis functin that defines a part f the field n the triangle, with the nrmal field cmpnent n that edge as the cefficient. With this cncept, the ttal field at the psitin M can be btained by the superpsitin f three RWG basis functins as HM) = i= H i p i M) h i. ) Cnsequently, the field inside the triangle can be interplated frm the nrmal field cmpnents n the triangle edges, which is knwn at each iteratin step. The tangential field cmpnent n a triangle edge can be easily btained by mving M nt that edge, and prjecting the result btained frm ) nt the directin alng the edge. B. Near- t far-field transfrmatin in frequency dmain Figure 3 shws tw adjacent triangles with a cmmn edge ST, which is a part f the radiatin cntur and has an angle f φ l with the x axis. P in is the triangle vertex inside the radiatin cntur, and P ut is the vertex utside that cntur. h, h,, h 4,, and h ut are the heights f the tw triangles with respect t the different edges. L in is the distance between the vertex S and the prjectin f P in nt the edge ST, while L ut is the distance between S and the prjectin f P ut nt the same edge. L is the length f the edge ST. The nrmal field cmpnents at the center f the triangle edges are dented by H, H, H, H 3, and H 4. E z and E z are the electric fields lcated at the vertices S and T. ˆl is a unit vectr alng the edge ST, riented in a cunterclckwise directin alng the radiatin cntur. ẑ is a unit vectr perpendicular t the triangle plane and pinting ut f the paper. ˆn = ˆl ẑ is a unit vectr nrmal t the edge ST, pinting utside the radiatin cntur. l is a lcal crdinate axis, taking the vertex S as its rigin. is the rigin f the glbal crdinate system, ˆρ is a unit vectr pinting t the radiatin directin f interest, and φ is the angle between ˆρ and the x axis. is a vectr pinting frm t S. The tangential field cmpnent H l n the edge ST is required fr the NTFT. With ), it can be interplated either frm H, H, and H in ), r frm H, H 3, and H 4 in 3). H in l l ) = H h l H h L l ) H l L in ) ) H ut l l ) = H 4 h 4 l + H 3 L l ) + H h ut l L ut ) 3) Because the RWG basis functin des nt guarantee the cntinuity f the tangential field acrss the edge ST, an averaging f Hl in and Hl ut is necessary, given as H l l ) = J z l ) = h uthl in l ) + Hl ut l ) = al + b + h ut + h ut H, where a = H H ) h ut h h + h ut H4 + H 3 + H ) h 4 h ut + h ut Lin H b = + H L h ) H3 L L uth h ut h ut + h ) ut + h ut. 4)

3 3 J z l ) in 4) is the equivalent electric current density in the ẑ directin. The weights fr the field averaging are selected such that the triangle clser t the radiatin cntur has a greater cntributin. It can be seen frm 4) that the tangential field H l n the edge has a linear relatinship with l. The far-zne electric field due t the J z in 4) can be derived as Ez J jk = η jk = η J z l ) expjk ρ ˆρ)dl al + b) exp[jk + l ) ˆρ]dl = ζ J al + b) exp[jk l csφ φ l )]dl [ al + b)ξ b ζ J + a aξ ] γ γ = if φ φ l π ) al ζ J + bl if φ φ l = π where ζ J = η jk expjk ˆρ), γ = jk csφ φ l ) and ξ = expγl). Althugh the result fr φ φ l π case cnverges t that fr the φ φ l = π case when φ φ l appraches t π, the tw cases shuld be treated separately fr the sake f numerical evaluatin. E z n the edge ST is equal t the equivalent magnetic current density M l in the directin f ˆl, and is assumed t be linearly varying with l as, 5) E z l ) = M l l ) = c l + d, 6) where c = E z E z and d = E z. Similar t 4), the farzne magnetic field due t the M l in 6) can be derived L as H M φ = sinφ l φ) η jk [ cl + d)ξ d ζ M = γ ) cl ζ M + dl + c cξ γ M l l ) expjk ρ ˆρ)dl ] if φ φ l π if φ φ l = π where ζ M = sinφ l φ) jk η expjk ˆρ), and γ as well as ξ are the same as thse in 5). With Ez J in 5) and Hφ M in 7), the ttal far-zne electric field cntributed by the equivalent current densities n the edge ST can be btained as, 7) E ST = E J z η H M φ. 8) The summatin f the far-zne electric field cntributed by all the triangle edges n the radiatin cntur gives the final far-zne radiatin. C. Near- t far-field transfrmatin in time dmain In the TD NTFT, the interplatin f ) is carried ut at each iteratin step n the radiatin cntur. Given the equivalent current densities in 4) and 6), the update f the far-zne field is perfrmed by using a duble Fast Furier Transfrm FFT) technique as prpsed in [5]. Fr each radiatin directin f interest, a pseud-wavefrm ft) is btained n-the-fly. ft) is the TD versin f 8) excluding the jk term. After finishing the iteratin, ft) is first cnverted int the FD by means f an FFT, then multiplied by the jk term, and finally cnverted back int the TD with an Inverse Fast Furier Transfrm IFFT) t get the physical far-zne wavefrm. By ignring the difference f phase delay n the edge ST and remving the jk term in 5) and 7), the cntributin f J z and M l n edge ST at a time instant t t the pseudwavefrm ft) can be btained as f ST t + τ) = L [ ) L sinφ φ l )M l, t η J z L, t )], where τ = + L ˆl ) ˆρ 3 8. M l and J z in 9) are evaluated at the center f the edge ST. That psitin is als used t calculate the time delay τ. In implementatin, f ST data is stred in an array fr values at discrete time instants n t, where n is a nn-zer integer and t is the iteratin time increment. Because t + τ in 9) may nt be lcated at thse specific time instants, the value f f ST t+τ) shuld be mapped nt tw adjacent time steps as f ST m t) = m + t + τ ) f ST t + τ) t ), ) t + τ f ST m t + t) = m f ST t + τ) t assuming m is the largest integer n larger than t + τ t. This mapping was first prpsed in [4]. The final pseud-wavefrm ft) can be btained after evaluating the cntributin f all the triangle edges n the radiatin cntur. III. RESULTS AND DISCUSSION The scattering by an infinite circular cylinder due t an electric line surce is studied by using the triangular-grid methd, and the far-zne radiatin is btained by the methd presented befre. The gemetry under simulatin and the crdinate system are illustrated in Figure 4. The shadwed disc with a radius f 4 mm represents an infinite dielectric cylinder whse center is lcated at the rigin f the crdinate system. The cylinder is made f a material with ɛ r = 5. and µ r =.. An electric line surce is lcated at the left f the cylinder, with a distance f 6 mm frm the disc center. The variatin f the electric current I z fllws a Gaussian functin defined as [ ) ] t 4σ I z t) = exp, ) σ 9)

4 4 Fig. 4. Gemetry I z Absrbing bundary Radiatin cntur 4 mm 6 mm ɛ r = 5. µ r =. φ ˆρ x 8 mm mm where t s and σ = ps. Mur s secnd rder absrbing bundary as described in [] is implemented n a circle with a radius f mm, t truncate the cmputatinal dmain. The circle shwn as a dashed line has a radius f 8 mm and is the radiatin cntur where the tangential electric and magnetic fields are used t calculate the far-zne radiatin. φ defines the directin where the far-field radiatin is t be evaluated. In rder t use the triangular-grid methd, the entire dmain is meshed int 5766 triangles. The gemetry in Figure 4 is simulated twice by using the methd, ne with the dielectric cylinder, and ne withut it. The scattered fields are then btained by subtracting the results btained in the abve tw cases. Fr cmparisn, the scattered field is als analytically derived in the FD by the eigenfunctin expansin methd EFEM), given as E s zω) = k I z ω) 4ωɛ + n= C n H ) n k ρ) exp[jnφ φ )], where C n = M N P Q M = ɛ r µ r J n k R)H n ) k ρ )J n ɛ r µ r k R) N =µ r J nk R)H n ) k ρ )J n ɛ r µ r k R) k R)J n ɛ r µ r k R) P =µ r H n ) Q = ɛ r µ r H n ) k R)J n ɛ r µ r k R). ) In ), R = 4 mm is the cylinder radius, and ρ = 6 mm as well as φ = π determine the psitin f the electric line surce. The results btained by this way are referred t as exact results in later discussin. Furthermre, fr a specific directin, the exact results btained in ) ver a brad band are transfrmed int the TD by applying the IFFT. The cmparisn between the -NTFT and EFEM methds are carried ut in bth the FD and TD. At a frequency f 8.75 GHz, the scattered H φ n the radiatin cntur are interplated frm ), and pltted in Figure 5 fr magnitude and phase. The scattered E z n the radiatin cntur is btained frm the iteratin withut any interplatin, and is pltted in Figure 6 fr magnitude and phase. The farfield E z calculated frm 8) is pltted in Figure 7, als fr magnitude and phase. Mrever, the TD wavefrms f the scattered E z at a distance f 8 m frm the rigin are pltted in Figure 8 fr different φ angles. Excellent agreement between the -NTFT and exact slutin is bserved in all the results, especially thse f the far-field cases. IV. CONCLUSIONS An NTFT based n RWG basis functins was prpsed fr a D triangular-grid methd. The surface equivalence therem was applied, and the equivalent current density, fr example, J z in the TM z case, n the radiatin cntur was interplated by using RWG basis functins. Frmulatins fr the interplatin and the NTFT were presented fr the TM z case. In the FD, the radiatin due t the equivalent current densities n each triangle edge was derived analytically, while in the TD, a pseud-wavefrm was updated n-the-fly, and the physical far-zne field was btained thrugh a duble FFT technique. As an example, the scattering f an infinite dielectric circular cylinder due t an electric line surce was simulated by using the triangular-grid as well as the prpsed NTFT. In the FD, the tangential scattered electric and magnetic fields n the radiatin cntur were examined at the frequency f 8.75 GHz, and were used t btained the far-zne electric field. In the TD, the wavefrms at six different φ angles are demnstrated. All the results were verified by analytical slutins. This idea f apprximating the field inside the triangular grids by using the RWG basis functins is nt nly useful fr the NTFT, but may als be used in the iteratin f the triangular-grid where the updating f the transverse field assciates with a cntur integratin. In [], such integratin is evaluated alng a series f clsed line segments, each perpendicular t a triangle edge, and the field cmpnent alng the line segment is assumed t be cnstant n that line segment. By using the RWG basis functins, suctegratin can be evaluated analytically, and different integratin cnturs can be chsen. Mrever, this idea can be further extended t a three dimensinal 3D) prblem where the basis functin prpsed in [3] may be adpted t apprximate the field in the tetrahedral elements. REFERENCES [] K. S. Yee, Numerical slutin f initial bundary value prblems invlving Maxwell s equatins in istrpic media, IEEE Trans. Antennas Prpagat., vl. AP-4, pp. 3-37, May 966. [] A. Taflve, Applicatin f the finite-difference time-dmain methd t sinusidal steady state electrmagnetic-penetratin prblems, IEEE Trans. Electrmagn. Cmpat., vl. EMC-, pp. 9-, Aug. 98. [3] K. S. Yee, D. Ingham, and K. Shlager, Time-dmain extraplatin t the far field based n calculatins, IEEE Trans. Antennas Prpagat., vl. 39, pp. 4-43, Apr. 99. [4] R. J. Luebbers, K. S. Kunz, M. Schneider, and F. Hunsberger, A finitedifference time-dmain near zne t far zne transfrmatin, IEEE Trans. Antennas Prpagat., vl. 39, pp , Apr. 99.

5 5 Magnitude, A/m Magnitude, V/m Magnitude, V/m Fig. 5. The magnetic field H φ at ρ = 8 mm Fig. 6. The electric field E z at ρ = 8 mm Fig. 7. The electric field E z at ρ = 8 m φ= φ= φ= φ= φ= φ=8 Fig. 8. Scattered E z wavefrm at ρ = 8 m [5] R. J. Luebbers, D. Ryan, and J. Beggs, A tw-dimensinal time-dmain near-zne t far-zne transfrmatin, IEEE Trans. Antennas Prpagat., vl. 4, pp , Jul. 99. [6] M. Kragltt, M. S. Kluskens, and W. P. Pala, Time-dmain fields exterir t a tw-dimensinal space, IEEE Trans. Antennas Prpagat., vl. 45, pp , Nv [7] S. G. Garcia, B. G. Olmed, and R. G. Martin, A time-dmain near- t far-field transfrmatin fr in tw dimensins, Micrwave Opt. Technl. Lett., vl. 7, n. 6, pp , December. [8] C.-C. Oetting, L. Klinkenbusch, Near-t-far-field transfrmatin by a time-dmain spherical-multiple analysis, IEEE Trans. Antennas Prpagat., vl. 53, pp , Jun. 5. [9] R. Hlland, Pitfalls f staircase meshing, IEEE Trans. Electrmagn. Cmpat., vl. 35, pp , 993. [] C. F. Lee, B. J. McMartin, R. T. Shin, and J. A. Kng, A triangulargrid finite-difference time-dmain methd fr electrmagnetic scattering prblems, J. Electrmagn. Waves Applicat., vl. 8, n. 4, pp , 994. [] Tatsu Ith, Full wave time-dmain CAD tls fr distributed nnlinear micrwave circuits, Final Reprt fr MICRO Prject 98-6, Industrial Spnsr: Raythen Systems Cmpany. [] S. S. M. Ra, D. R. Wiltn, and A. W. Glissn, Electrmagnetic scattering by surfaces f arbitrary shape, IEEE Trans. Antennas Prpagat., vl. AP-3, pp , May 98. [3] D. H. Schaubert, D. R. Wiltn, and A. W. Glissn, A tetrahedral mdeling methd fr electrmagnetic scattering by arbitrarily shaped inhmgeneus dielectric bdies, IEEE Trans. Antennas Prpagat., vl. AP-3, pp , Jan. 984.