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  IEEE Antennas and Propagation Society Newsletter, April 1988 Finite-Difference Time-Domain (FD-TD) Modeling of Electromagnetic Wave Scattering and Interaction Problems bY Allen Taflove EECS Department Northwestern University Evanston, IL 60201 Abstract- This rticle, based pon invited apers at he XXII URSI General Assembly (Tel viv, ugust 1987) nd the URSI National adio cience eeting (Boulder, anuary 19881, reviews ecent pplications of he inite-difference time-domain (FD-TO) method for umerical odeling f lectromagnetic wave scattering and interaction problems. One of he goals of this article is to demonstrate that recent advances in FD-TD modeling oncepts and software mplementa- tion, combined with advances in computer echnology, have xpanded the scope, ccuracy, nd peed of FD-TD modeling o he oint where it may be the referred choice or structures hat cannot be easily reated by conventional ntegral quation and asymptotic approaches. As a class, such structures are electric- ally arge and ave omplex hapes, material composi- ti ons, apertures, and i teri or cavi ties 1. INTRODUCTION Contemporary igh-frequency lectromagnetic engineering roblems can involve wave interactions with complex, electrically-large hree-dimensional structures. These structures an ave shapes, material ompositions, pertures, r avities hich produce ear fields hat annot be resolved nto finite sets of modes or ays. Proper numerical model- ing f such ear fields equires ampling t sub- wavelength esolution o void liasing f magnitude ani phase information. The goal is o provide a self- consistent model of he utual oupling f he electrical ly smal 1 cell s corfipri sing he structure. A candidate numerical modeling approach or his purpose is he inite-difference time-domain (FD-TD) solution of Maxwell's url quations. his pproach is analogous to xisting inite-difference olutions of luid low problems ncountered in computational aerodynamics, in hat he umerical model is based upon a direct olution f he overning artial differential quation. ursuing his nalogy, FD-TD shares he omputational equirements of he luids codes and other imilar arge-scale artial iffer- ential equation solvers) n erms of computer floating point rithmetic ate, rimary random ccess memory size, and data andwidth o econdary memory. Yet, FD-TD is a non-traditional pproach o umerical electromagnetic odeling, where requency-domain approaches have dominated. One of he goals of his article s o demonstrate that ecent advances in FD-TD modeling oncepts and software mplementation, ombined ith dvances in computer echnology, have expanded he scope, accuracy and peed of FD-TD modeling to he point where it may be the preferred choice or certain ypes of scatter- ing and coupling roblems. With his in mind, this article wi succinctly eview he ollowing ecent FD-TD modeling alidations nd esearch rontiers: Continued o poge 6 Korada R. Umashankar EEC4 Department University of Illinois at Chicago Chicago, IL 60680 Introducing Feature Article Author Allen Taflove Allen Taflove was born in Chicago, IL on une 14, 1949. He received he B.S. (with ighest istinc- tion), M.S., and Ph.D. degrees in electrical engineer- ing rom orthwestern niversity, vanston, L, in 1971, 1972, and 1975, respectively. From 975 to 1984, e was a staff member at 111 7esearch Institute n Chicago, L, olding he osi- tions f ssociate ngineer, esearch ngineer, anc Senior ngineer. here, in ddition o esearch i 2lectromagnetic wave scattering and penetration, hc x-ovided echnical eadership or major rograms ir joint ight-of-way design or 60-Hz transmission ines Ind pipelines or ailroads; and extraction of oil ron shale, tar sand, nd slowly roducing ells sins novel in itu lectromagnetic eating echnology. H is one of hree principal co-inventors of he atter, and has been granted 10 U.S. patents in his area. In 1984, Dr. Taflove eturned o Northwestern a: an Associate Professor in he EECS Department. incc then, e as eveloped everal esearch rograms ir analysis and umeri cal methods for el ectromagneti ( dave interactions with arge, complex tructures. Hi2 related nterests nclude nverse cattering targel synthesi s; appl i ati ons of ecent ector uper. computers nd oncurrent rocessors in computationa' electromagnetics; and the new on-surface adiatior condition OSRC) theory or high-frequency scattering which e srcinated long with G. A. Kriegsmann n( K. R. Umashankar. Dr. Taflove s a member of Tau Beta Pi, Eta Kapp; Nu, and Sigma Xi He is a Senior Member of IEEE anc a member of URSI Commission B. 5  IEEE Antennas and Propagation Society Newsletter, Aprll 1988 Featwe Article-Continued from page 5 1. Scattering models or hree-dimensional eentrant 2. Conformal models of urved urfaces; 3. Scattering models or wo-dimensional nisotropic structures; 4. Penetration odels or arrow lots and lapped joints n hick screens; 5. Coupling odels or ires and wire undles in free space and in arbitrary metal cavities; 6. Penetration models or he lectromagnetic ields within detailed, inhomogeneous ti ssue pproxima- tions f he omplete human body (at UHF frequencies 1 ; structures spanning up to 9 wavelengths; 7. Microstrip and microwave circuit models; 8. Scattering models or elativistically ibrating mirrors; Introducing Feature Article Author Korada R. Umashankar Korada R. Umashankar received he B.E. degree rom Mysore University, ndia, n 1962; the M.E. degree from he ndian nstitute f cience, angalore, India, n 1964; and the Ph.D. degree rom he University of Mississippi, niversity, MS, in 1974; all n electrical engineering. From 1964 to 1969, he was Assistant Professor and Head of he Department of lectrical ngineering, College f ngineering, arnatak niversity, ubli, India. uring 1974 and 1975, he was a Postdoctoral Research ssociate, and from 1975 to 1977, Assistant Professor of Electrical Engineering at he University of ississippi. From 1977 to 1979, he was the National Research Council Visiting Fellow at he U. S. Air Force Weapons Laboratory, irtland AFB, NM. During 1979 to 1984, he was Senior ngineer at IIT Research Institute, Chicago, L. Currently, Dr. Umashankar is Associate rofessor of Electrical Engineering and Computer Science at he University f llinois t Chicago. is rimary research is in he development of nalytical and numerical echniques in electromagnetic heory, EMP/EMC interactions, and EM simulation tudies. Research opics nclude he tudy f cattering rom bodies omprised f, r oated ith, nisotrGpic media; extension f he method of momenti to electrically ery arge argets; and the new on- surface adiation ondition (OSRC) theory or igh- frequency scattering. Dr. Umashankar is a member of Eta Kappa Nu and Sigma Xi He is a Senior Member of IEEE and a member of URSI Commission B. 9. Inverse cattering econstruction f one- dimensional, patially oincident rofiles f electrical permittivity and conductivity; 10. Inverse cattering econstruction of wo- dimensional onducti ng, homogeneous, and i homo- geneous dielectric argets rom inimal TM scattered ield pulse esponse data; and 11. Large-scale computer software. 2. GENERAL CHARACTERISTICS OF FD-TD As stated, FD-TD is a direct solution of Maxwell’s time-dependent curl quations. It employs o potentials. nstead, it applies imple, econd-order accurate central-difference approximations El] for he space nd time erivatives f he lectric and magnetic fields directly o he espective differen- tial operators of he drl equations. This achieves a sampled-data eduction f he ontinuous lectro- magnetic field n a volume of space, over a period of time. Space and time iscretizations re elected o bound errors n he sampling rocess, and to nsure numerical tability f he lgorithm [2]. Electric and magnetic field components are nterleaved n space to ermit a natural atisfaction f angential ield continuity onditions t media nterfaces. verall, FD-TD .is a marching-in-time procedure which simulates the ontinuous actual waves by ampled-data umerical analogs ropagating in a data space stored n a computer. At each time tep, he ystem f quations to update he ield components is ully explicit, so that here s no need to et up or olve a set f linear quations, and the equired omputer torage and running ime s roportional o he lectrical size of he volume modeled. Lottice Truncotion Plone (Invisiblb To All Waves) / x = 1/28 Figure 1 Time-Domain Wave-Tracking Concept of the FD-TD Method Fig. 1 illustrates he time-domain wave tracking concept of he FD-TD method. A region of space within the dashed lines s elected or ield sampling in space nd ime. At time = 0 , it is assumed that all fields ithin he umerical ampling egion re identically zero. An incident lane wave is assumed to nter he ampling egion at this point. Propaga- tion of he ncident wave is modeled by the commence- ment of ime stepping, which is simply he mplementa- tion f he inite-difference nalog f he url equations. Time stepping ontinues as the umerical analog of he ncident wave strikes he modeled target embedded within he ampling egion. A outgoing Continued on page 7 5  IEEE Antennas and Propagation Society Newsletter, April 1988 Feature Article-Continued from page 6 scattered wave analogs deally ropagate hrough he 1 atti e truncation pl anes with negl i i bl ref 1 ecti on to xit he sampling egion. Phenomena such s induction of surface currents, scattering and multiple scattering, enetration hrough pertures, and cavity excitation are modeled ime-step y ime-step by the action of he url equations analog. Self-consistency of hese modeled phenomena is enerally assured if their patial and emporal ariations re ell resolved by the space and time sampling process. Time stepping is continued until he desired ate- time ulse esponse r teady-state ehavior is achieved. An important example of he atter s he sinusoidal teady tate, wherein he ncident wave is assumed to have a sinusoidal dependence, and time stepping is continued until all ields n he sampling region xhibit inusoidal epetition. his s a consequence of he imiting amplitude principle C31. Extensive umerical xperimentation ith FD-TD has shown that he number of complete ycles f he incident wave required o be time-stepped to achieve the inusoidal teady tate is approximately qual o the Q factor f he tructure r phenomenon being model ed. EV 'Y X Figure 2. Positions of the Field Components about a Unit Cell of the YEE Lattice 111 Fig. 2 illustrates he positions of he electric and magnetic ield components about a unit cell of he FD-TD lattice n Cartesian coordinates C11. Note that each magnetic field vector component is surrounded y four circulating electric ield vector components, and vice ersa. his rrangement ermits ot nly a centered-difference analog o he space derivatives of the url quations, ut lso a natural eometry or implementing he ntegral orm f araday's Law and Ampere's Law at he pace-cell evel. his ntegral interpretation permits a simple but effective modeling of he hysics of moothly urved arget urfaces, penetration hrough narrow slots having sub-cell gaps, and coupling to thin wires having ub-cell diameters, as w be een later. Fig. 3 illustrates how an arbitrary hree- dimensional catterer is embedded in an FD-TD space lattice comprised of he nit ells f ig. 2. Simply, desired alues of electrical permittivity and conductivity re ssigned o ach lectric ield component of he attice. orrespondingly, esired values f agnetic erineabi 1 i y cna quivalent conductivity re ssigned o ach agnetic ield component of he attice. The media arameters re interpreted by the FD-TD program as local coefficients for he ime-stepping lgorithm. pecification f media properties in this component-by-component manner FD TD UNIT CELL - Y Figure 3. Arbitrary Three-Dimensional Scatterer Embedded n a FD-TD Lattice results n a stepped-edge, or taircase, pproxima- tion f urved urfaces. ontinuity f angential fields s assured at he nterface of dissimilar media with his rocedure. here s no eed for pecial field matching t media interface oints. Stepped- edge approximation f urved urfaces has een found to be adequate in he FD-TD modeling problems tudied in he 1970's and early 1980's, including wave inter- actions with biological issues C41, penetration nto cavities [SI, [SI, and electromagnetic ulse EMPI interactions ith complex structures [7] - C91. However, recent nterest n wide dynamic ange models of cattering by curved argets has rompted the development of urface-conforming FD-TD approaches which eliminate taircasing. These wi be ummarized later n his article. 1 1 Reqion I : Toial Fields Region 2 : Scattered Fields Source Lattice Truncation (a 1 Region I : Totai ields E Region 2 : Scottered Flelds x jo +... .. ~~...t..,~...t...~...t.,.~ ,,. ,.._._ ---- -HY e & - H iI (b) Figure 4. Division of FD-TD Lattice into Total-Field and Scattered- Field Regions. a) Lattice division; (b) Field component geometry at connecting plane y = joS [lo] 1111 Fig. 4 illustrates he ivision f he FD-TD lattice nto otal-field and scattered-field egions. This division has een ound to be ery seful ince it permits he fficient imulation f an incident plane wave in he otal-field egion with rbitrary Continued on poge 8 7  -- IEEE Antennas and Propagation Society Newsletter. April 1988 Feature Article-Continued from page 7 angle f ncidence, olarization, ime-domain waveform, nd duration [lo], C111. Three dditional important enefits rise rom his attice ivision. a. b. C. A large near-field computational dynamic ange is achieved, ince he catterer f nterest is embedded in he otal-field egion. Thus, low ield evels n shadow regions or within shielding nclosures re computed directly without uffering ubtraction noise as would be he ase if scattered ields n such regions were ime-stepped ia FD-TD, and then added to a cancel ing ncident ield o obtail, the ow otal-field evels). Embedding the catterer n he otal field region ermits a natural atisfaction f tangential ield ontinuity cross media interfaces, as iscussed arlier, ithout having o compute the ncident ield t possibly numerous points along a complex locus that s unique o each scatterer. The zoning arrangement of Fig. 4 requires computatjon of the ncident ield only along he ectangblar connecting surface between the otal-field and scattered-field egions. his urface s .fixed, .e., ndependent f he shape or composition f he nclosed catterer eing modeled. The provision f a well-defined cattered- field egion n he FD-TD lattice permits he near-to-far ield ransformation llustrated in Fig. 5. The dashed virtual urface shown in ig. 5 can e ocated long onvenient lattice lanes n he cattered-field egion of Fig. 4. Tangential cattered E and H fields computed via FD-TD at his irtual surface can hen be weighted by he ree-space Green's unction and then ntegrated (summed) to rovide he ar-field esponse and radar cross ection full bistatic esponse or he assumed illumination ngle) [lll - C131. The near-field ntegration urface has a fixed rectangular shape, nd thus is independent of the shape or omDosition f he nclosed - I I I I I I I I scattered being modeled. ( P, ?is) d I NO SOURCES 8 ZERO FIELDS --I I I I I I I (a) (bJ Figure 5. Near-to-Far Field Transformation Geometry (a) Original problem; (b) Equivalent problem external o the virtual surface, Sa [ll] Fig. 4 uses the erm lattice runcation o designate he utermost 1 attice planes n he cat- tered-field egion. The fields at hese planes cannot be omputed sing he entered-differencing pproach discussed arlier ecause f he assumed absence of known field ata t oints utside f he attice truncation. These data are needed to orm he central differences. herefore, n uxiliary attice runca- tion ondition s necessary. his ondition must be consistent with Maxwell's auations n hat n ut- going cattered-wave umerical nalog triking he lattice runcation must exit he attice ithout appreciable on-physical eflection, ust as if the lattice runcation was invisible. It has een shown that he equired attice runcation ondition s really a radiation ondition n he near ield [lo], C141 - C171. Further, it has een shown that convenient ocal approximations of he exact adiation condition an e enerated and applied ith good results [IO] - [171. Based upon this esearch, he procedure or constructing more precise ocal approxi- mations of he exact adiation condition is reasonably well nderstood. hese pproximations re urrently under tudy for numerical mplementation in he FD-TD computer programs C181. 3. THREE-DIMENSIONAL FD-TD SCATERINC MODELS Analytical and experimental Val i ati ons have been obtai ne6 relative o FD-TD modeling of anonical three-dimensional conducting argets spanning 1/3 to 9 wavelengths C121, C131, C191, C201. For revity, nly one such validation wi1,l be eviewed here. Fin E2 f ky HX 0.37 m Corner Reflector Looks / o'< 0'. 1800 Fin Center I I I I I I I L- 0' = 90' r__-________________ / LATTICE EDGE 9 . 00 Fig. 6 Geometry of Crossed-Plate Scatterer and Illumination 13], 1191, 1201 Fig. 6 depicts he geometry of a crossed-plate scatterer omprised f wo flat plate:, Zlectrically bonded together o orm he shape of a T . The main plate has the dimensions 30 cm x 10 cm x 0.33 cm, and the bisecting in has the dimensions 10 cm x 10 cm x 0.33 cm. The illumination s a plane wave at 0 elevation ngle and TE polarization elative o he main late, and at he requency 9.0 GHz. Thus, the main plate spans 9.0 wavoelengths. ote that ook angle zimuths etween 90 and 180 provide ubstan- tial corner eflector physics, n addition o he edge diffraction, orner iffraction, and other ffects found or an isolated lat plate. For he 9-GHz D-TD model, the attice ell ize is 0.3125 cm, approximately 1/11 wavelength. The main plate s formed y 32 x 96 x 1 cells; he bisecting fin s formed by 32 x 32 x 1 cells; and the overall lattice s comprised of 48 x 112 x 48 cells (1,548,288 unknown field components) ontaining 212.6 cubic Continued on page 9
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