NASA Tchnical Papr 3575 Elctromagntic Scattring Analysis of Arbitrarily Shapd Matrial Cylindr by FEM-BEM Mthod M. D. Dshpand VIGYAN, Inc. Hampton, Virginia C. R. Cockrll and C. J. Rddy Langly Rsarch Cntr Hampton, Virginia National Aronautics and Spac Administration Langly Rsarch Cntr Hampton, Virginia 23681-0001 July 1996
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1. Introduction Abstract A hybrid mthod that combins th finit lmnt mthod (FEM) and th boundary lmnt mthod (BEM) is dvlopd to analyz lctromagntic scattring from arbitrarily shapd matrial cylindrs. By this mthod, th matrial cylindr is first nclosd by a fictitious boundary. Maxwll s quations ar thn solvd by FEM insid and by BEM outsid th boundary. Elctromagntic scattring from svral arbitrarily shapd matrial cylindrs is computd and compard with rsults obtaind by othr numrical tchniqus. Th problm of lctromagntic scattring dtrmination from an arbitrarily shapd matrial cylindr is considrd in this papr. Elctromagntic scattring from an inhomognous matrial cylindr can b dtrmind by two basic approachs an intgral quation and a diffrntial quation. In th intgral quation approach, a surfac intgral quation or a volum intgral quation may b usd. In th surfac intgral quation formulation, conducting surfacs of a scattrr ar rplacd by quivalnt surfac lctric currnts, whras matrial surfacs ar rplacd by both quivalnt surfac lctric and magntic currnts. Coupld intgral quations ar thn formd by application of appropriat boundary conditions to th diffrnt fild componnts that ar producd by th quivalnt currnts. Ths coupld intgral quations ar solvd by th mthod of momnts for th unknown quivalnt currnts. Arvas and Sarkar (rf. 1) hav analyzd radar cross sctions of various two-dimnsional structurs with th surfac intgral quation approach. For a cylindr consisting of inhomognous matrial, th surfac intgral quation approach dos not corrctly modl th inhomognity of th matrial cylindr. In such cass th volum intgral quation formulation is thn usd to accuratly modl th inhomognous matrial cylindr. In th volum intgral quation formulation, th matrial cylindr is dividd into unit clls. Ths unit clls may b rctangular bricks or ttrahdral shaps and ar small nough so that th fild intnsity is narly uniform within ach cll. Th matrial cylindr is thn assumd to b rplacd by an quivalnt polarization currnt flowing in ths unit clls. Th coupld intgral quations, which ar formd by application of appropriat boundary conditions to th fild producd from th quivalnt currnts, ar thn solvd by th mthod of momnts. Using th polarization currnt concpt, Richmond (rf. 2) dtrmind scattrd fild pattrns of dilctric cylindrs of arbitrary cross-sctional shap. Th quivalnt currnt concpt has also bn usd by Sarkar and Arvas (rf. 3) and by Schaubrt, Wilton, and Glisson (rf. 4) to analyz lctromagntic scattring by arbitrarily shapd inhomognous dilctric bodis. Howvr, th mthod involvs th solution of a fully dns matrix quation, which may rquir prohibitivly larg computr mmory and long computation tim. In th diffrntial quation approach, th volum of th inhomognous scattrr is discrtizd, and Maxwll s quations in diffrntial quation form ar solvd. Som advantags of th diffrntial quation approach ovr th intgral quation approach ar: (1) a mthod mor convnint for handling complx inhomognous scattrrs, (2) a mthod that rsults in spars matrix quations, and (3) a mthod bttr suitd for closd-rgion problms. Howvr, for opn-rgion problms in lctromagntics, such as radiation and scattring, th volum must b proprly truncatd with an artificial or fictitious boundary with propr absorbing boundary conditions. Th accuracy of rsults obtaind with an artificial boundary with absorbing boundary conditions dpnds upon boundary locations as wll as th ordr of th boundary conditions. An altrnativ approach (known as th hybrid approach) prsntd in this papr rtains th advantags of both diffrntial quation and intgral quation approachs. Th gnral procdur for a hybrid tchniqu rquirs that th scattrr b nclosd by an artificial boundary. Maxwll s quations ar thn
solvd by a diffrntial quation approach such as th finit lmnt mthod (FEM) insid th artificial boundary and by an intgral quation approach in discrtizd form such as th boundary lmnt mthod (BEM) outsid th artificial boundary. Although us of BEM on and outsid th artificial boundary rsults in a full dns matrix, convrgnc of an approximat solution to th xact solution is guarantd without a chang in th location of th artificial boundary. Th rmaindr of th papr is organizd as follows. Sction 3 contains basic formulation of th problm for FEM and BEM. Scattring amplituds of a fw sampl problms ar computd by th prsnt tchniqu in sction 4. Also in sction 4, rsults obtaind by th prsnt mthod ar compard with thos obtaind by othr analytical tchniqus. Th papr is concludd in sction 5 with commnts on th futur scop and xtnsion of th prsnt work. 2. Symbols A IJ a [B] b C [D] D Jp E 0 E z E in jwt f (I, J)th lmnt of FEM matrix for th subdomain cylindr radius; triangl sid lngth finit lmnt matrix boundary circl radius; triangl bas lngth fictitious boundary curv spars matrix (J, p)th lmnt of matrix [D] amplitud of incidnt lctric fild Z-axis componnt of total lctric fild incidnt lctric fild vctor tim convntion frquncy, Hz f s E or H scattrd far fild G (.) (.) [G] G pq [G] 1 [H] H pq ( 2) H 1 (.) ( 2) H 0 (.) H z H 0 Hin I, J intgrs J z i j k 0 m, n intgrs fr-spac Grn s function, whr rprsnts appropriat argumnt squar matrix of ordr P P (p, q)th lmnt of matrix [G] invrs of matrix [G] squar matrix of ordr P P (p, q)th lmnt of matrix [H] Hankl function of first ordr and scond kind Hankl function of zro ordr and scond kind Z-axis componnt of total magntic fild amplitud of incidnt magntic fild incidnt magntic fild vctor quivalnt lctric currnt rprsnting incidnt fild imaginary numbr qual to 1 propagation constant in fr spac 2
nˆ nˆ 1 P p Q p ( η) q unit normal vctor of curv C drawn outward unit normal vctor of curv C drawn inward numbr of nodal points on curv C intgr for pth nodal point picwis distribution along η dirction intgr for qth nodal point rˆ unit vctor T ( x, tsting function quivalnt to W I [ U] C U p W I ( x 1, y 1 ), ( x 2, y 2 ), ( x 3, y 3 ) ( x, ẑ η column matrix with nodal amplituds of normal drivativ on curv C amplitud of normal drivativ of on curv C xpansion function for Ith nod ovr th triangular subdomain coordinats of thr vrtics of triangl variabls of rctangular coordinats unit vctor along Z-axis α r ( x,, qual to rspctivly, for TM cas; qual to β r ( x, µ r ( x,, ε r ( x,, ε r ( x,, µ r ( x,, rspctivly, for TE cas [ Γ] column matrix for nodal amplitud [ Γ] C column matrix for nodal amplitud on curv C Γ I amplitud of ψ t at Ith nod p lngth of linar sgmnt btwn p and p + 1 nods on curv C p 1 lngth of linar sgmnt btwn p and p 1 nods on curv C δ IJ Kronckr dlta function ε( x, ε 0 ε r ( x, ε r ( x, ε 0 Θ p µ ( x, µ 0 µ r ( x, µ r ( x, µ 0 ( ρ, φ) ( ρ, φ ) σ 2D [ ϒ] C ϒ q φ in rlativ prmittivity at (x, prmittivity of fr spac variabl along linar sgmnts of curv C angl btwn two conscutiv linar sgmnts of curv C, dg rlativ prmability at (x, prmability of fr spac variabls of cylindrical coordinat systm sourc coordinat radar scattring width column matrix with nodal amplituds of incidnt fild on curv C amplitud of incidnt fild at qth nod on curv C ψ t incidnt angl of lctromagntic wav 3
ψ in incidnt lctric or magntic fild scalar ψ t total lctric fild E z for TM cas; total magntic fild H z for TE cas ω angular frquncy (2πf ), rad/sc Abbrviations: in TE TM 3. Thory incidnt transvrs lctric transvrs magntic Considr a matrial cylindr infinit along th Z-axis and of arbitrary cross sction as shown in figur 1. For th purpos of analysis, th scattring structur is dividd into rgions I and II by contour C. Rgion II may consist in gnral of inhomognous matrial with prmability µ(x, and prmittivity ε(x,. It may also hav mbddd mtallic strips within th matrial. Rgion I is fr spac with prmability µ 0 and prmittivity ε 0 surrounding th cylindr. An incidnt lctromagntic wav is assumd in th dirction normal to th axis of th cylindr. Both transvrs lctric (TE) and transvrs magntic (TM) polarizations ar considrd, and jωt tim convntion is assumd. For TM and TE incidnc, rspctivly, th incidnt fild is givn by Ein ẑψ in ẑe 0 jk 0 ρ cos ( φ φ in) = = Hin ẑψ in ẑh 0 jk 0 ρ cos ( φ φ in) = = (1) whr E 0 = H 0 = 1 is th amplitud, φ in is th incidnt angl, and ( ρ, φ) ar th variabls of th cylindrical coordinat systm. Th total fild ψ t insid th rgion boundd by curv C can b dtrmind from th solution of th sourc-fr scalar wav quation givn by ------------------- ψ 1 α r ( x, t 2 + k 0βr ( x, ψ t = 0 (2) In quation (2) α r ( x,, β r ( x,, and ψ t ar qual to µ r ( x,, ε r ( x,, and E z, rspctivly, for TM Z-axis xcitation and ar qual to ε r ( x,, µ r ( x,, and H z, rspctivly, for TE to Z-axis xcitation. Multiplication of both sids of quation (2) by a tsting function T ( x, and intgration ovr th cross sction of rgion II yilds a wak form of th wav quation (Silvstr and Frrari, rf. 5) as 1 T ------------------- α r ( x, ψ 2 + β t r k 0( x, ψt dx d y = 0 Rgion II (3) With th us of vctor idntity T 1 α ------------------- r ( x, ψ t = T -------------------ψ 1 α r ( x, t + T -------------------ψ 1 α r ( x, t (4) and th divrgnc thorm, quation (3) can b writtn as 1 T -------------------ψ 2 k α r ( x, t 0βr ( x, T ψ t dx d y = T ψ t nˆ dc Rgion II C (5) whr nˆ is a unit vctor drawn outwardly to th curv C as shown in figur 1. 4
To construct an approximat solution of quation (5) by FEM, rgion II is approximatd by a union of triangls as shown in figur 2(a). On th th triangl ψ is rprsntd by a linar combination of t functions W I ( x, as ψ t = ψ t = 3 Γ I W I ( x, (on th triangl) I =1 0 (othrwis) (6) whr 1 x 1 y 1 W I ( x, = [ 1 x y] 1 x 2 y 2 1 x 3 y 3 1 δ I 1 δ I 2 δ I 3 and th Kronckr dlta function is dfind as δ IJ = 0 for I J and δ IJ = 1 for I = J. Th amplituds Γ 1, Γ 2, and Γ 3 ar th unknown valus of ψ t at th thr vrtics of th th triangl having coordinats ( x 1, y 1 ), ( x 2, y 2 ), and ( x 3, y 3 ), rspctivly. By Galrkin s mthod with T ( x, = W I ( x, and J = 1, 2, and 3, th lft sid of quation (5) ovr th th triangl can b writtn as 3 1 Γ I W -------------------W 2 J k α r ( x, I 0 β r ( x, W JW I dx d y ( J = 1, 2, 3) I =1 (7) Th prvious xprssion can b writtn in a matrix form as A 11 A 12 A 13 Γ 1 A 21 A 22 A 23 Γ 2 (8) A 31 A 32 A 33 Γ 3 whr A IJ = 1 W -------------------W 2 J k α r ( x, I 0βr ( x, W JW I dx Now considr a union of two triangls as shown in figur 2(b); an xprssion corrsponding to quation (8) for th union of two triangls can b writtn as dy (9) A 11 A 21 A 31 A 12 A 13 f f f ( A 22 + A 55 ) ( A 23 + A 56 ) A 54 f f f ( A 32 + A 65 ) ( A 33 + A 66 ) A 64 f 0 A 45 f A 46 0 0 Γ 1 Γ 2 Γ 3 Γ 4 (10) f whr A IJ ar dtrmind from quation (9) with suprscript rplacd by suprscript f and th intgration prformd ovr th f triangl. An appropriat alignmnt of common dgs, as indicatd in 5
figur 2(b), is rquird to assur th corrct combination of trms in th matrix quation (10). To nsur continuity of filds across th common dg, Γ 6 = Γ 3 and Γ 5 = Γ 2 wr nforcd in th drivation of quation (10). With an assmbld msh of all triangls ovr th surfac boundd by curv C, th lft sid of quation (5) can b writtn as [ B] [ Γ] (11) whr th lmnts of th matrix [B] ar obtaind from A mn, [Γ] is th column matrix with lmnts givn by th valus of ψ t at th vrtics of triangular lmnts, m = 1, 2, 3,..., N, n = 1, 2, 3,..., N, and N is th total numbr of nods. To valuat th contour intgration on th right sid of quation (5), th normal drivativ of ψ t ovr curv C is rquird. To dtrmin th normal drivativ of ψ t on curv C, th following procdur is usd. Th boundary curv C is discrtizd into linar sgmnts as shown in figur 3 and th normal drivativ of th function ψ t on th curv is writtn as ψ t nˆ = P p=1 U p Q p ( η) (12) whr U p is th unknown amplitud at pth nod on curv C. Th function Q p ( η) varis linarly with η ovr th sgmnt as Q p ( η) Q p ( η) = = η ------------- ( 0 η p 1 ) p 1 p ( η p 1 ) ---------------------------------------- ( p 1 η p ) p (13) Th right sid of quation (5) can thrfor b writtn as C T ψ t nˆ dc P = U p T Q p ( η)dη = p=1 C [ D] [ U] (14) whr th lmnts of matrix [D] ar givn by D Jp = C W JQp ( η)dη (15) Th unknown amplitud U p is dtrmind as follows. Th function ψ t in rgion I is obtaind by th solution of th inhomognous wav quation 2 ψ t 2 i + k 0ψt = jωµ 0 J z (16) subjct to ssntial and natural boundary conditions on curv C. Ths ssntial and natural boundary i conditions ar th valus of th function and its normal drivativ on curv C. In quation (16), J z is th quivalnt currnt sourc producing th incidnt fild. 6
To obtain a solution, multiply quation (16) by G( ρ, φ; ρ, φ ), whr G(.) is th fr-spac Grn s function for a lin sourc locatd at ( ρ, φ ), and intgrat ovr rgion I, which givs ( 2 2 ψ t + k 0ψt )G(.) dx dy = jωµ 0 i J zg (.) dx dy Rgion I Rgion I (17) From th scalar Grn s thorm Rgion I [ G (.) 2 ψ t ψ t 2 G (.)] dx dy = G (.) ψ t nˆ 1 dc C C ψ t G (.) nˆ 1 dc (18) whr th unit normal vctor nˆ 1 is shown in figur 1. Equation (17) can thn b writtn as 2 ψ t 2 [ G (.) + k 0G (.) y = jωµ 0 i J zg (.) dx dy Rgion I Rgion I + ψ t G nˆ 1 dc ψ t nˆ 1G dc C C (19) 2 Bcaus 2 i [ G (.) + k 0G (.) ] = δ( ρ ρ )δ( φ φ )/ρ, jωµ J zg (.) d x d y = ψ i, and nˆ 1 = nˆ, quation (19) can b writtn as Rgion I ψ t ( ρ, φ) = ψ i ( ρ, φ) ψ t nˆ G (.)dη + ψ t G (.) C C nˆ dη (20) Th fr-spac Grn s function usd in th prvious quations is givn by G (.) = 1 ( 2) ----H 4 j 0 k0 ρ ρ (21) G (.) rˆ jk 0 ( 2) = -------H 4 1 k0 ρ ρ ( 2) ( 2) whr H 0 (.) and H1 (.) ar th Hankl functions of th scond kind of zro and first ordr, rspctivly, and rˆ is a unit vctor along ρ ρ. Th function ψ t on th curv in figur 3 can now b rprsntd by (22) ψ t = P p=1 Γ p Q p ( η) (23) whr Γ p is th unknown amplitud of ψ t at th pth nod on curv C. Substitution of quations (12) and (21) (23) into quation (20) yilds P jk 0 Γ q δ qq ϒ q ------- Γ 4 H ( 2) = + p 1 k0 ρ q ρ Qp ( η)rˆ nˆ p=1 C P j -- U 4 H ( 2) + p 0 k0 ρ q ρ Qp ( η) dη p=1 C dη (24) 7
whr ϒ q is th valu of incidnt fild at th qth nod on curv C. Equation (24) can b writtn in matrix form as H 11 H 12 H 1P Γ 1 ϒ 1 G 11 G 12 G 1P U 1 H 21 H 22 H 2P Γ 2 ϒ 2 G 21 G 22 G 2P U 2 =..................... H P1 H P2 H PP Γ P ϒ P G P1 G P2 G PP U P (25) whr jk 0 ( 2) H pq = ------- H 4 1 k0 ρ q ρ Qp ( η)nˆ rˆ C dη + δ qq (26) and G pq = j ( 2) -- H 2 4 0 k0 ρq ρ Qp ( η) dη C (27) Th normal drivativ of ψ t is thn obtaind by solution of quation (25) for [ U] C in trms of [ Γ] C to gt [ U] C = [ G] 1 [ H] [ Γ] C [ G] 1 [ ϒ] C (28) whr [ U] C and [ Γ] C rfr to unknown nodal amplituds on curv C. With substitution of quation (28) into quation (14), th rightsid of quation (5) rducs to C T ψ t nˆ dc = [ D] [ G] 1 [ H] [ Γ] C [ D] [ G] 1 [ ϒ] C (29) From quations (11) and (29), quation (5) can b writtn in matrix form as [ B] [ Γ] [ D] [ G] 1 [ H] [ Γ] C = [ D] [ G] 1 [ ϒ] C (30) and solvd for [ Γ], which also includs additional [ Γ] C trms. Thn [ U] C is dtrmind by substitution of [ Γ] C into quation (28). Th scattrd far fild in th ϕ dirction is thn dtrmind by th last two trms of quation (24) with ρ q = ρ and th asymptotic valuations of th Hankl functions as ρ k 0 f s ---- 2 j ----------- 1/2 jk 0 ρ P Γ 4 πk 0 ρ p Q p ( η) jk 0ρ cos( ϕ ϕ = ) dη p=1 C j -- 2 j ----------- 1/2 jk 0 ρ P U 4 πk 0 ρ Q p p ( η) jk 0ρ' cos ϕ ϕ + ( ) dη p=1 C (31) 8
whr ( ρ, ϕ ) is a function of η. For TM and TE incidnc, rspctivly, th radar cross sction is thn obtaind from σ 2D limit ρ 2π ρ f 2 s = ------------ 2 E 0 σ 2D limit ρ 2π ρ f 2 s = ------------ 2 H 0 (32) 4. Numrical Rsults In this sction, numrical rsults for th scattring width of various 2-D matrial cylindrs ar prsntd. To solv quation (30), th matrics [ B], [ D], [ G], and [ H] must b dtrmind. Matrics [ B] and [ D] ar spars matrics, and thir lmnts ar dtrmind by quations (9) and (15), rspctivly. Matrics [ H] and [ G] ar dns matrics, and thir lmnts ar dtrmind by quations (26) and (27), rspctivly. Equations (26) and (27) ar valuatd with th us of Gauss quadratur numrical intgration. Whn p = q, th intgration in quation (26) rsults in H qq = ( Θ q ( 2π) ), whr Θ q is th intrnal angl at th qth nod shown in figur 3. Th column matrix [Γ], which is obtaind aftr th solution of quation (30), is usd in quation (28) to dtrmin [ U] C. From known [Γ] and [ U] C on curv C, th scattrd far fild, and hnc, th scattring width ar dtrmind by quations (31) and (32). To validat th computr cod, th bistatic scattring width of a prfctly conducting cylindr of radius 1.0λ, whn xcitd by a TM-polarizd plan wav, is calculatd by th prsnt formulation and compard with th calculatd rsults givn in rfrnc 1. (S fig. 4.) Rsults of th two calculation mthods ar in good agrmnt. For calculations by th prsnt mthod, th circular cylindr is assumd to b nclosd by an artificial circl with radius qual to 1.2λ. Th rgion nclosd by th artificial circl is analyzd by th finit lmnt mthod, and th rgion outsid th circl is analyzd by th boundary lmnt mthod. For th BEM th circl is dividd into 76 points. Slction of th location of th artificial boundary around th cylindr is arbitrary. For th conducting cylindr, whn th location of th artificial boundary coincids with th objct boundary, th problm can b solvd by th us of only BEM. For validation of th FEM formulation, th location of th artificial boundary was slctd at 1.2λ radius to provid a minimum on-cll-thick FEM rgion. If th artificial boundary is movd farthr away from th objct boundary, th computational ara is unncssarily incrasd. Th circumfrnc of th artificial boundary was dividd into 76 linar sgmnts so that th lngth of ach small sgmnt was lss than 0.1λ. In figur 5 th bistatic scattring width of th circular cylindr, whn xcitd by a TE-polarizd plan wav, is shown and compard with arlir publishd rsults. (S rf. 1.) To chck if dilctric and magntic matrials ar proprly handld by th prsnt mthod, th bistatic scattring width of a matrial-coatd circular cylindr is calculatd for both TE and TM xcitations and prsntd in figurs 6 8 along with arlir publishd rsults. (S rf. 1.) Ovrall arlir rsults and th prsnt mthod rsults ar in good agrmnt. For th rsults prsntd in figurs 6 8, th artificial circular boundary was takn at radius b. Th artificial boundary was dividd into 90 sgmnts. Any furthr incras in th discrtization of th artificial boundary C rsultd in insignificant changs in th rsults prsntd in figurs 6 8. In gnral, th artificial boundary curv C nclosing th cylindrical scattring structur may b of any shap. To chck this aspct, th bistatic scattring widths of cylindrs with triangular cross sctions, as shown in figur 9(a), ar considrd. Th triangular cylindrs ar nclosd by th following boundaris: (1) circular boundary (fig. 9(a)), (2) lliptical boundary (fig. 9(b)), (3) conformal boundary with 9
blndd cornrs (fig. 9(c)), and (4) conformal boundary with sharp cornrs (fig. 9(d)). Th bistatic scattring widths, which wr obtaind for th four xampls, ar shown in figur 10 for comparison with arlir rsults publishd by Ptrson and Castillo in rfrnc 6. Rsults prsntd in figur 10 ar in good agrmnt with ach othr. Th rsults prsntd in figur 9 show advantags from th artificial boundary slctd to b as clos as possibl to th outr boundary of th cylindr. This rducs th computational surfac and, hnc, th numbr of unknowns. For th xampls considrd, th artificial boundary slctd in figur 9(d) is th optimum bcaus it has th last numbr of unknowns whn compard with th artificial boundaris considrd in figurs 9(a) 9(c). Othr gomtris and thir rsults ar shown in figurs 11 13. Th gomtris that wr considrd includd th conducting strip shown in figur 11(a), th microstrip transmission lin shown in figur 12(a), and th conducting cylindr of von Karman shap shown in figur 13(a). For th calculation of th scattring widths of a conducting strip, a microstrip transmission lin, and a conducting cylindr of von Karman shap, th artificial boundaris wr slctd to b lliptical nclosurs of th outr boundaris of th structurs as shown in figurs 11(a), 12(a), and 13(a), rspctivly. Th scattring widths of ths structurs calculatd by th prsnt mthod ar prsntd along with arlir publishd rsults (rf. 1) in figurs 11(b), 11(c), 12(b), and 13(b). Th rsults of all gomtris ar in good agrmnt. 5. Conclusion A hybrid tchniqu that combins th finit lmnt mthod and boundary lmnt mthod has bn usd to dtrmin backscattrd filds from arbitrarily shapd matrial cylindrs. Validity of th computr cod dvlopd with th prsnt hybrid tchniqu has bn dmonstratd with various arbitrarily shapd matrial cylindrs. Although us of th boundary lmnt mthod in th prsnt hybrid tchniqu lads to a partly spars and partly dns matrix, th prsnt mthod is guarantd to convrg irrspctiv of th shap of th trminating or artificial boundary nclosing th arbitrarily shapd matrial cylindr. Th prsnt hybrid tchniqu can b applid to study th transmission lin charactristic of cylindrical strip lins. NASA Langly Rsarch Cntr Hampton, VA 23681-0001 Fbruary 26, 1996 6. Rfrncs 1. Arvas, Ercumnt; and Sarkar, Tapan K.: RCS of Two-Dimnsional Structurs Consisting of Both Dilctrics and Conductors of Arbitrary Cross Sction. IEEE Trans. Antnnas & Propag., vol. 37, May 1989, pp. 546 554. 2. Richmond, J. H.: Scattring by a Dilctric Cylindr of Arbitrary Cross Sction Shap. IEEE Trans. Antnnas & Propag., vol. AP-13, May 1965, pp. 334 341. 3. Sarkar, Tapan K.; and Arvas, Ercumnt: Scattring Cross Sction of Composit Conducting and Lossy Dilctric Bodis. IEEE Proc., vol. 77, May 1989, pp. 788 795. 4. Schaubrt, D. H.; Wilton, D. R.; and Glisson, A. W.: A Ttrahdral Modling Mthod for Elctromagntic Scattring Arbitrarily Shapd Inhomognous Dilctric Bodis. IEEE Trans. Antnnas & Propag., vol. AP-32, Jan. 1984, pp. 77 85. 5. Silvstr, P. P.; and Frrari, R. L.: Finit Elmnts for Elctrical Enginrs. Cambridg Univ. Prss, 1990. 6. Ptrson, Andrw F.; and Castillo, Stvn P.: A Frquncy-Domain Diffrntial Equation Formulation for Elctromagntic Scattring From Inhomognous Cylindrs. IEEE Trans. Antnnas & Propag., vol. 37, May 1989, pp. 601 660. 10
Incidnt fild E in ( H in ) Y ρ Rgion I µ 0, ε 0 Rgion II µ r, ε r Z φ nˆ 1 Curv C intrfac btwn matrial cylindr and fr spac nˆ Figur 1. Matrial cylindrical scattrr and rlatd coordinat systm. 11
f th th (a) Typical th and fth triangls. 4 f th 3 6 5 2 th 1 (b) Gomtry of union of two triangls. Figur 2. Rgion II discrtizd into triangular subdomains. 12
(p+1)th p pth p 1 (p 1)th Θ p Y Z η P 2 Nod p = 1 Figur 3. Gomtry of discrtizd boundary curv C. 13
25 20 Rfrnc Rf. [1] 1 Prsnt mthod Mthod E z a φ Conducting cylindr 15 10 σ/λ, db 5 0-5 -10-15 0 30 60 90 120 150 180 φ, dg Figur 4. Bistatic scattring width of conducting circular cylindr xcitd by TM-polarizd plan wav. α = 1.0λ and φ in = 180. 14
20 10 Rf. Rfrnc [1] 1 Prsnt Mthod mthod H z a φ Conducting cylindr 0 σ/λ, db -10-20 -30 0 30 60 90 120 150 180 φ, dg Figur 5. Bistatic scattring width of conducting circular cylindr xcitd by TE-polarizd plan wav. α = 1.0λ and φ in = 180. 15
20 Rfrnc Rf. [1] 1 Prsnt mthod Mthod Conducting cylindr ε r = 2.0 µ r = 1.0 E z b φ 2a 10 σ/λ, db 0-10 0 30 60 90 120 150 180 φ, dg Figur 6. Bistatic scattring width of coatd conducting circular cylindr xcitd by TM-polarizd plan wav. a = 1.0λ; b = 1.5λ; ε r = 2.0; µ r = 1.0; and φ in = 180. 16
Conducting cylindr ε r = 2.0 µ r = 1.0 H z φ 20 10 Rfrnc Rf. [1] 1 Prsnt mthod Mthod 2a b 0 σ/λ, db -10-20 -30 0 30 60 90 120 150 180 φ, dg Figur 7. Bistatic scattring width of coatd conducting circular cylindr xcitd by TE-polarizd plan wav. a = 1.0λ; b = 1.5λ; ε r = 2.0; µ r = 1.0; and φ in = 180. 17
20 10 E z Rf. Rfrnc [1] 1 Prsnt mthod Mthod Conducting cylindr ε r = 2.0 µ r = 2.0 φ 2a b 0 σ/λ, db -10-20 -30 0 30 60 90 120 150 180 φ, dg Figur 8. Bistatic scattring width of coatd conducting circular cylindr xcitd by TE-polarizd plan wav. a = 1.0λ; b = 1.5λ; ε r = 2.0; µ r = 2.0; and φ in = 180. 18
Y Y Z Z Conducting triangl b a COSMOS/M Vrsion : 1.6 (a) Artificial circular boundary with radius 1.2λ 0. Figur 9. Isoscls triangular mtallic cylindr with a = 1.85λ 0 and b = 1.404λ 0. FEM mthod is usd insid circular boundary and BEM mthod is usd outsid circular boundary. 19
Y Y Z Z Conducting triangl (b) Artificial lliptical boundary with major axis 1.3λ 0 and minor axis 1.1λ 0. Figur 9. Continud. 20
Y Y Z Z Conducting triangl (c) Artificial conformal boundary with blndd cornrs 3 to 4 clls away from triangl. Figur 9. Continud. 21
YY Z Z Conducting triangl (d) Artificial conformal boundary with sharp cornrs 1 to 3 clls away from triangl. Figur 9. Concludd. 22
15 Rf. Rfrnc [ ] 6 Prsnt Circular Mthod outr boundary ( With Circular outr boundar Prsnt Elliptical Mthod outr boundary ( With Ellipitical outr boundar Prsnt Conformal Mthod outr boundary ( With Confarmal outr boundar 10 Prsnt Conformal Mthod outr boundary ( With Confarmal with sharp outr cornrs boundary sharp cornr) Y σ/λ, db 5 E z A Z C B φ 0-5 -10 0 30 60 90 120 150 180 φ, dg Figur 10. Bistatic scattring cross sction of isoscls triangular cylindr shown in figur 9 and xcitd by TM-polarizd plan wav with angl of incidnc φ in = 180. Computd boundaris as shown in figur 9. 23
Y Y Z Z Conducting strip (a) Gomtry and triangular msh. Figur 11. Conducting strip (infinit along Z-axis) with artificial lliptical boundary with major axis 1.3λ 0 and minor axis 0.3λ 0. 24
20 15 Rf. Rfrnc [1] 1 Prsnt mthod Prsnt Mthod Y Incidnt wav E z.04λ 0 φ 10 2.2λ 0 σ/λ, db 5 0-5 -10 0 10 20 30 40 50 60 70 80 90 φ, dg (b) Monostatic scattring width of conducting strip xcitd by TM-polarizd plan wav. Figur 11. Continud. 25
20 10.04λ 0 Rfrnc Rf. [1] 1 Prsnt mthod Mthod Y Incidnt wav H z φ σ/λ, db 0 2.2λ 0-10 -20-30 0 10 20 30 40 50 60 70 80 90 φ, dg (c) Monostatic scattring width of conducting strip xcitd by TE-polarizd plan wav. Figur 11. Concludd. 26
Y Dilctric slab Y Z Z (a) Gomtry and triangular msh. Figur 12. Microstrip transmission lin with artificial lliptical boundary with major axis 0.6λ 0 and minor axis 0.3λ 0. 27
10 5 Rf. Rfrnc [1] 1 Prsnt mthod Mthod 0-5 σ/λ, db -10 Y Incidnt wav E z -15-20 W 2 ε r = 6.25 W 2 t 3 t 2-25 W 1 t 1-30 -90-60 -30 0 30 60 90 φ, dg (b) Monostatic scattring width of microstrip transmission lin xcitd by TM-polarizd plan wav. W 1 = 0.9λ 0 ; W 2 = 0.15λ 0 ; t 1 = 0.02λ 0 ; t 2 = 0.1λ 0 ; and t 3 = 0.05λ 0. Figur 12. Concludd. 28
Y Y 2.0λ 0 2.2λ 0 Z Z 1.0λ 0 1.1λ 0 (a) Gomtry and triangular msh. Figur 13. Conducting cylindr of von Karman shap with conformal artificial boundary. 29
20 Rf. Rfrnc [1] 1 Prsnt mthod Mthod 10 0 σ/λ, db -10 Y Incidnt wav E z φ -20 Z D L -30 0 30 60 90 120 150 180 φ, dg (b) Monostatic scattring width of conducting cylindr of von Karman shap xcitd by TM-polarizd plan wav. L = 2.0λ 0 and D = L/2. Figur 13. Concludd. 30
REPORT DOCUMENTATION PAGE Form Approvd OMB No. 0704-0188 Public rporting burdn for this collction of information is stimatd to avrag 1 hour pr rspons, including th tim for rviwing instructions, sarching xisting data sourcs, gathring and maintaining th data ndd, and complting and rviwing th collction of information. Snd commnts rgarding this burdn stimat or any othr aspct of this collction of information, including suggstions for rducing this burdn, to Washington Hadquartrs Srvics, Dirctorat for Information Oprations and Rports, 1215 Jffrson Davis Highway, Suit 1204, Arlington, VA 22202-4302, and to th Offic of Managmnt and Budgt, Paprwork Rduction Projct (0704-0188), Washington, DC 20503. 1. AGENCY USE ONLY (Lav blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED July 1996 Tchnical Papr 4. TITLE AND SUBTITLE 5. FUNDING NUMBERS Elctromagntic Scattring Analysis of Arbitrarily Shapd Matrial Cylindr by FEM-BEM Mthod WU 505-64-52-04 6. AUTHOR(S) M. D. Dshpand, C. R. Cockrll, and C. J. Rddy 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) NASA Langly Rsarch Cntr Hampton, VA 23681-0001 8. PERFORMING ORGANIZATION REPORT NUMBER L-17469 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) National Aronautics and Spac Administration Washington, DC 20546-0001 10. SPONSORING/MONITORING AGENCY REPORT NUMBER NASA TP-3575 11. SUPPLEMENTARY NOTES Dshpand: ViGYAN, Inc., Hampton, VA; Cockrll: Langly Rsarch Cntr, Hampton, VA; Rddy: NRC-NASA Rsidnt Rsarch Associat, Langly Rsarch Cntr, Hampton, VA. 12a. DISTRIBUTION/AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE Unclassifid Unlimitd Subjct Catgory 17 Availability: NASA CASI (301) 621-0390 13. ABSTRACT (Maximum 200 words) A hybrid mthod that combins th finit lmnt mthod (FEM) and th boundary lmnt mthod (BEM) is dvlopd to analyz lctromagntic scattring from arbitrarily shapd matrial cylindrs. By this mthod, th matrial cylindr is first nclosd by a fictitious boundary. Maxwll s quations ar thn solvd by FEM insid and by BEM outsid th boundary. Elctromagntic scattring from svral arbitrarily shapd matrial cylindrs is computd and compard with rsults obtaind by othr numrical tchniqus. 14. SUBJECT TERMS 17. SECURITY CLASSIFICATION OF REPORT 18. SECURITY CLASSIFICATION OF THIS PAGE 19. SECURITY CLASSIFICATION OF ABSTRACT 15. NUMBER OF PAGES Elctromagntic scattring; Hybrid FEM-BEM tchniqu 31 Unclassifid Unclassifid Unclassifid 16. PRICE CODE 20. LIMITATION OF ABSTRACT NSN 7540-01-280-5500 Standard Form 298 (Rv. 2-89) Prscribd by ANSI Std. Z39-18 298-102 A03