Time-Domain Developments in the Singularity Expansion Method. Douglas J. Riley. Thesis submitted to the Faculty of the

Transcription

1 Time-Domain Developments in the Singularity Expansion Method by Douglas J. Riley Thesis submitted to the Faulty of the Virginia Polytehni Institute and State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE in Eletrial Engineering APPROVED: W. A. Davis, Chairman I. M. Besieris W. L. Stutzman Deember, 1982 Blaksburg, Virginia

2 ACKNOWLEDGMENTS The author expresses his appreiation to W. A. Davis for his guidane in the development of this thesis. Without our numerous onversations many of the results obtained would not have been possible at this time. Appreiation is also given to I. M. Besieris and W. L.. Stutzman for their review and ritique of the text, and to Cherie Merix for aomplishing the painstaking task of typing the numerous equations. ii

3 TABLE OF CONTENTS ACKNOWLEDGMENTS ii Chapter I. II. INTRODUCTION FUNDAMENTAL INTEGRAL EQUATIONS OF ELECTROMAGNETICS Introdution..... Mathematial Formalism III. FUNDAMENTAL CONCEPTS OF THE SINGULARITY EXPANSION METHOD 14 IV. v. Introdution Spae-Frequeny Tehniques Natural Frequenies and Modes Coupling Coeffiients and Entire Funtions Spae-Time Tehniques Natural Frequenies, Natural Modes, and Stability Considerations Coupling Coeffiients EIGENSOLUTION METHODS FOR THE TRANSITION MATRIX Introdution... Root Searhing Methods Muller's Method.. Contour Integration Polynomial Matrix Redution Laguerre's Method... Matrix Eigenvalue Methods Similarity Transform Methods The LR Transforma~ion The QR Transformation Iterative Eigenvetor Methods Single Vetor Power Methods Multiple Vetor Power Methods WIRE-STRUCTURE ANALYSIS Introdution.. Spae-Frequeny Tehniques Exitation Models. Thin-Wire Kernel Approximation iii LL~ ~.L

4 Eletri Field Distrtbution Spae-Time Tehniques Transient Current Response 85 TD-SEM Pole Distribution Effet of Varying the Time Sampling Distane Pole Shift by Kernel Deoupling VI. TRANSIENT ANALYSIS OF THIN, PERFECT CONDUCTING RECTANGULAR PLATES.... In~rodution Mathematial Formalism..... Standard Gridding Sheme. Shifted Griddinq Sheme TD-SEM Pole Distribution VII. CONCLUSIONS Appendix A. SIMULTANEOUS ITERATION LISTING 131 REFERENCES 164 VITA 168 iv

5 TABLE OF FIGURES page Figure 5.1: Wire geometry Figure 5.2: Current distribution on a 2 meter dipole antenna with a radius of 0.01 meters. The number of unknowns was 31, and the frequeny of operation was 75 MHz Figure 5.3: Effet of the exat and approximate kernels on the urrent distribution of a 1 meter antenna with a radius of 0.02 meters. The number of unknowns was 49, and the frequeny of operation was MHz. 80 Figure 5.4: Figure 5.5: Sattered tangential eletri field distribution of a 1 meter antenna with a radius of 0.02 meters. The number of unknowns was 19, and the frequeny of operation was MHz Transient urrent distribution on one of two feed segments of a 1 meter antenna with a radius of meters. The number of unknowns was Figure 5.6: Pole distribution for 18 unknowns. The length of the satterer was 1 meter~ and the radius was meters Figure 5.7: Figure 5.8: Eigenvalue struture for 18 unknowns. The length of the satterer was 1 meter, and the radius was meters First layer pole distribution found by S. I. for 32 unknowns. The satterer was 1 meter in length, and the radius was meters Figure 5.9: Exeution ti.me requirements on an IBM 3032 omputer using FORTRAN H EXTENDED (OPT=2) for various eigensolution methods Figure 5.10: Storage requirements for various eigensolution methods v

6 Figure 5.11: Figure 6.1~ Figure 6. 2: Figure 6.3: Figure 6.4: Figure 6.5: Figure 6.6: Figure 6.7: Pole movement due to kernel deoupling. The satterer was 1 meter in length with a radius of meters. The number of unknowns was Plate geometry. 102 Annular propagation of ative regions due to the time differene (p-p'). 104 Standard gridding sheme 108 Current density omponent orresponding to the diretion of inident polarization alulated on the enter path of a 1 meter square plate. No boundary onditions have been enfored. A standard gridding sheme has been used with 16 pathes for eah urrent omponent. The pulse width was 4.9 light meters, and the hoie ~t=0.76 was made Current density omponent orresponding to the diretion of inident polarization alulated on the enter path of a 1 meter square plate. The normal omponent of the urrent density has been set to zero at the edges. A standard gridding sheme has been used with 16 pathes for eah urrent omponent. The width of the inident pulse was 4.9 light meters, and the hoie 6t=0.76 was made.... e 112 Current density omponent orresponding to the diretion of inident polarization alulated on the enter path of a 1 meter square plate. Normal omponents of the urrent density have been set to zero at the edges; parallel omponents have been extrapolated at the edges. A standard gridding sheme has been used with 16 pathes for eah urrent omponent. A omparison has been made with Bennett's model. The inident pulse was 4.9 light meters wide, and the hoie ~t=0.76 was made Shifted gridded sheme vi

7 Figure 6.8: Figure 6.9: Unstable urrent density distribution obtained from the shifted gridding sheme for ~t=~. The omponent orresponding to the diretion of inident polarization on the enter path is shown. A total of 12 pathes have been used for eah urrent density omponent Current density omponent orresponding to the diretion of inident polarization alulated on the enter path of a 1 meter square plate. A shifted gridding sheme has been used with 12 pathes for eah urrent omponent. The inident pulse was 4.9 light meters wide, and a omparison with the standard grid model has been made Figure 6.10: Figure 6.11: Poles for a 1 meter square plate disretized with a shifted gridding sheme. The total number of pathes for eah omponent of the urrent density was 2, and the hoie ~t=0.7~ was made... Poles for a 1 meter square plate disretized with a shifted gridding sheme. The total number of pathes for eah omponent of the urrent density was 6, and the hoie ~t=0.7l was made Figure 6.12: Figure 6.13: Poles for a 1 meter square plate disretized with a shifted gridding sheme. The total number of pathes for eah omponent of the urrent density was 12, and the hoie ~t=0.7~ was made Comparison of TD-SEM poles (shifted grid, 12 pathes for eah urrent density omponent, and ~t=0.7~) with available frequeny-domain poles. The results are for a 1 meter square plate.127 vii

8 Chapter I INTRODUCTION The alulation of natural frequenies (poles) and natural modes (free osillations) of strutures is a fundamental problem of many disiplines. Until reently, the mathematial study of these parameters has unfortunately been limited to anoni geometries whih lend themselves to eigensolution by separation of variable tehniques. The singularity expansion method (SEM) removes this geometrial restrition by enabling one to obtain the natural frequenies and natural modes of an arbitrary objet. The SEM also enables one to determine the response of the objet to an arbitrary foring funtion diretly from an appropriate expansion of the modal and pole struture. The basi theoretial foundations of the SEM were initially presented using frequeny-domain tehniques applied to eletromagneti equations by Baum [1]. Baum's development was subsequently extended by Marian and Latham [2]; and rigorous mathematial justifiation of some of the basi foundations has reently been presented by Ramm [3]. Analyti frequeny-domain SEM results for the perfet onduting spherial satterer were originally obtained by Baum [1], and numerial frequeny-domain results for thin, per- 1

9 2 feet onduting ylindrial surfaes were initially presented by Teshe [4]. Interest in time-domain tehniques in the SEM has not been as widespread as frequeny-domain methods; however, several varied ontributions have reently been made toward establishing the versatility of time-domain methods. A time-domain method analogous to the original frequeny-domain method may be found in Baum [5]. The appliability of this method, however, has been somewhat limited due to the level of diffiulty of the desribing equations. Van Blarium and Mittra [6] developed a rather unique method whereby the natural responses may be obtained using Prony's method [7] one the transient response of the objet is known. An obvious ompliation with this method is that the determination of the transient response an be a non-trivial problem. An alternate time-domain method whih sidesteps the ompliations of the above methods has been introdued by Cordaro and Davis [8]. This method, known as time-domain SEM (TD-SEM), enables one to find the natural responses diretly from the finite differene representation of the governing integral equations ast in a matrix eigenvalue form. Unfor- ~unately, the matries generated tend to be quite large, and hene the previous work has been limited to one-dimensional geometries disretized with relatively few unknowns. The

10 3 results whih have been obtained, however, indiate that the Cordaro-Davis method is apable of produing a great deal of information quite effiiently. The intent of this study is to extend the appliability of TD-SEM, and extend numerial time-domain tehniques in general. Toward establishing this intent, the following (prinipal) set of tasks are defined: (1) determine stability riteria for various finite differene representations of eletromagneti equations, (2) develop simple time-domain expressions for determining the SEM oupling oeffiients (these are parameters whih ouple the natural frequenies and modes to the inident foring funtion), (3) develop an eigensolution algorithm whih will solve the large sale matries generated by TD-SEM, (4) obtain a pole distribution for the linear satterer disretized with a large number of unknowns, (5) apply TD-SEM to the two-dimensional retangular plate problem. The outline for establishing these tasks is as follows. The fundamental governing equations of eletromagnetis are developed in integral form in Chapter 2 using dyadi Green's funtion theory. The equations are initially developed in general, and are then speialized to desribe thin, perfet onduting surfaes. An effort has been made to keep the development brief by leaving several intermediate

11 4 steps to the referenes. This is done sine the appliation and solution of the final results is the prinipal intent of this work and not the mathematial subtleties of the development. The basi onepts of the frequeny-domain SEM and the time-domain method of Cordaro and Davis are presented in Chapter 3. Numerial solution tehniques and stability tehniques for the time-domain equations are also presented. The stability disussion, as applied to these equations, is presented for the first time. A variety of eigensolution methods appliable to the matries generated by the Cordaro-Davis method are presented in Chapter 4. The disussion ulminates with the development of an eigensolution algorithm for large matries in blok ompanion form. The tehniques developed in the previous hapters are applied in Chapters 5 and 6 to two anoni examples. In Chapter 5, the one-dimensional, thin, perfet onduting wire is onsidered in both the frequeny- and time-domains; urrent, eletri field, and pole distributions for a large number of unknowns are presented. In Chapter 6, the two-dimensional retangular plate is analyzed in the time-domain; time-domain pole distributions are introdued for the square plate.

12 Chapter II FUNDAMENTAL INTEGRAL EQUATIONS OF ELECTROMAGNET I CS 2.1 INTRODUCTION Singular integral equations (or singular integro-differential equations) represent a powerful and widely used approah to the solution of both antenna and eletromagneti sattering problems. A variety of methods may be used to obtain these equations. Poggio and Miller [9] rigorously develop the neessary results using the vetor Green's theorem [10]. In this formalism, the onept of inident and sattered fields in onjuntion with equivalent soures develops in a natural way. In this hapter, the frequeny-domain equations are developed frm linear system foundations. Although, perhaps, this approah is less rigorous than the method of Poggio and Miller, it yields fundamental results readily, without extensive vetor manipulations. The timedomain representations of these equations are then obtained by inverse Fourier transform tehniques. These general frequeny- and time- domain results are finally speialized to desribe thin, perfetly onduting surfaes. 5

13 6 2.2 MATHEMATICAL FORMALISM The mathematial formulation of eletromagneti phenomena is fundamentally dependent on a onise set of equations known as Maxwell's equations. The omplexity of these equations is highly dependent on the host medium. We will restrit our disussion throughout to a homogeneous, linear, and isotropi medium. For suh a medium, Maxwell's equations may be written in differential form in the frequenydomain as (a vetor will be denoted by a single bar; a frequeny-domain quantity will be denoted by a tilde) v x Et (r;w) v x lit (r;w) '\/. E = p =t = -jwµ H (r;w) - M (r;w) 0 =t = jwe: E (r;w) + J (r;w) 0 (2-1) =t (r;w) -t (r;w)/e:0 v. =t (r;w) -t H = m (r;w)/µ 0 Note that the time dependene, exp{jwt}, has been suppressed. The total eletri and magneti field intensities =t - =t - are denoted by E (r;w) and H (r;w), the total eletri and =t - magneti urrent densities are denoted by J (r;w) and =t - _t - -t - M (r;w), and the parameters p (r;w), m (r;w), e 0, µ 0, w, and ~ denote, respetively, total eletri and magneti harge densities, eletri permittivity, magneti permeability, frequeny, and observation position.

14 In the ase of sattering by an obstale, we may deompose the total fields and soures as "'t (r;w) :::in (r;w) "'S E = E + E (r;w) (2-2a) lit (r;w) :::in (r;w) + ;:;s = H n (r;w) (2-2b) "'t (r;w) 3s (r;w) + J (r;w) (2-2) J = Mt (r;w) ~is (r;w) +M (r;w) (2-2d) _t p (r;w) _s = p (r;w) + p (r;w) (2-2e) _t m (r;w) _s m (r;w) +m. (r;w) (2-2f) "'in "'in "'S "'s where E I H denote the inident fields whih J I M I -S p and -s give rise and "'s "'s I m to, E I H are the sattered fields due to the soures J, M, - - p I and m indued on the satterer {for dieletri sattering these soures are interpreted as effetive soures that replae the obstale}. :::s "'s fields E, H The sattered obey, then, the vetor Helmholtz equations -jwµ 0 J (r;w) - v x M (r;w) (2-Ja) and -jw M (r;w) + v x J (r;w) 0 (2-Jb) where k is the wavenumber, w~(e 0 µ 0 ). The fields whih satisfy (2-3) may be found by onvolving the impulse response of (2-3) with the foring funtions

15 8 present. The impulse response is,obtained by determining the dyadi Green's funtion, f(r,r' ;w) (a double bar will denote a dyadi), whih satisfies 'V X " v -X' f (-r,::t, w) - k 2 l. '"'1 (r-,i., ::, w) - v " (-r,,_; :J\ I (2-4) Here, I denotes a unit dyadi, r' denotes the soure position, and o(r,r') denotes a three-dimensional Dira delta distribution. The solution of equation (2-4) is given by [11] f (r,?;w) = (Y + ~ ilil) G (r,?;w) k (2-5) where G(r,r' ;w) is the free-spae Green's funtion e -jklr - r'i 41T Ir r I I (2-6) The sattered fields E 5 (r;w) and H3(r;w) may now be expliitly represented by = f f (r~r';w) [-jwµ 0 J (?;w) - vtx M (?;w)] dr' ) V' (2-7a) and 8 5 (r;w) =,l, f (r,r';w) [-jws 0 ~r (?;w) + 'i/x J C?;w)] dr' (2-7b) where V' denotes the volume oupied by the satterer.

16 9 We substitute, next, equations (2-7a,b) into equations (2-2a,b).,.., - -~ Using the vetor identities GV'xM=V'x(GM)-V'GxM, (V'G) V'xM=-V' (V'GxM), and the relation VG(r,r' ;w)= ' - - -, -V G(r,r ;w), we obtain the following spae-frequeny representations for the total elet~i and magneti fields: Et(r;w) = Ein(r;w) ~ J [k2y+vv] J(r')G(r,r';w)dr' JWE o V' _...±_ [k 2 i+vv1. { f n xm(r')g(r r' w)ds' k2 out ' ' s and + J ~(r') x v'a(r,r';w)dr'} rt V' (2-8a) V' ::ot - H (r;w) = liin(r;w) ~ J [k2y+vv] M(r')G(r,r';w)dr' ( JWµ o V' t f n out xjcr')g(r,r';w)ds' s + f J(;:'> x o G;:,r ;wld;:' 1 V' J r t 't"..,, (2-8b) The integration over the surfae, S, denotes integration over the surfae bounding the volume V'. The spae-time representations of the eletri and magneti fields may be obtained by inverse Fourier transforming the frequeny dependene found in expressions (2-8a) and (2-8b) [9]. They are given by

17 10 a -t - : - E (r t) o at ' i ~ 2 J M(r' ;-r) [ V'v] [.;_ x R- ds' 2 at2 s out + J (M(r' ;T) x <r-r') + aa-r fi<r' ;T) x <r-r))dr'] V' R 3 R i: t V' (2-9a) i:=t-r/ and + j' <Jr' ;-r) x V' (r-r') a-<-,, <r-r;))d-r'l R 3 + at J r ;TJ X R -r=t-r/ - r V' (2-9b) where R=lr-r' I, denotes the speed of light in vauum, and the parameter T denotes the time delay assoiated with a wave propagating over a satterer.the notation (a/a1)~(~' ;T) should be interpreted as (a/at)m(r' ;t) evaluated at t=1. Expressions (2-9a,b) are known as the spae-time eletri and magneti field integral expressions (EFIE, MFIE); whereas expressions (2-8a,b) are the spae-frequeny representations of the EFIE and MFIE.

18 11 The representations presented are general expressions whih are valid for an arbitrary satterer positioned in the previously assumed medium for all r suh that f~r'. At the offending point r=r' I the expressions beome singular and hene must be evaluated by onsidering the limit as r appreahes r' [9,12]. The Cauhy or Hadamard prinipal value [13] is typially used for the desription of these integrals. The frequeny- domain representations of the eletri and magneti fields beome in the Cauhy prinipal value sense (a si~gle bar through the integral will denote a Cauhy integral) "'t - E (r;w) = 2E =<in (r;w) Jr 2"" - - FP [k r+vvj J(r')G(r,r';w)dr' jw:o V' _.1._2 [k 2Y+vvJ { f D. k S out x M{r')G(r,?;w)as' r + + M(r') x v'(r,i'.i';w)ar' JV' } r EV' (2-lOa) and Ht(r;w) = 2liin(r;w) ~ FP J [k2y+vv] MCr')G(r,r';w)dr' JW).lo V' + ~ [k 2 Y+vvJ {Jn x JCr')G(r,r';w)as' k S out + 1 J(r') x v'g(r,r';w)dr' Tv' i i' < V' I J (2-lOb)

19 12 Note that an interhange of primed to unprimed oordinates has been made. Similar fators of two appear in the timedomain representations, and FP denotes 'the finite part of'. These results may be speialized to desribe thin, perfeet onduting surfaes [12]. On suh a surfae, the appropriate boundary onditions [10] are that the tangential A "'t A total eletri field is zero, i.e., nxe =O (n defined to be the outward normal unit vetor on S), and that the tangential total magneti field is equal to an equivalent surfae urrent soure, i e / A -"'t "' nxh =J 5 With these boundary onditions, we may immediately write the spae-frequeny representations for the eletri and magneti fields on the surfae S as -fix Ein(r;w) = ~FP J [k 2 Y + vv];j (r';w) (r,r';w) dr' JW 0 S S (2-lla) and 1 (r;w) s = 2il x ~in(r;w) - 2nxf 1 (t;w) x VG(r,r';w)ds'. s s (2-llb) Similarly, the spae-time representations are given by n = -x 41T = 2 J [ -I a ] J ( r I ; T ) d-r ' FP 2? + VV R s at- T = t-r/ (2-12a)

20 13 and - - A -in - n ( - - (r-r I) J 6(r;t) = 2 n x H (r;t) + 2 ~ x Js [J6(r';T) x R3 + ;L Js(r';T) x (r-r;>1 ds' I R T=t-R/ Note that the equivalent magneti surfae urrent, M 6, (2-12b) does not appear in these expressions. eletri field by M is related to the total s M = s =<t -ft x E (2-13) whih vanishes for perfet onduting surfaes. Note, also, that for good ondutors the effetive urrent soure J may be replaed by a'et (a denotes the ondutivity of the obstale), and therefore terms involving nxj also tend to zero. A As a final remark, we note that the term 2nxH =<in appearing in expression (2-12b) is ommonly known as the physial optis approximation for the urrent density J~. This approximation is useful for testing the validity of results obtained from expressions (2-12a,b) when no results for omparison exist.

21 Chapter III FUNDAMENTAL CONCEPTS OF THE SINGULARITY EXPANSION METHOD 3.1 INTRODUCTION The motivation for the singularity expansion method (SEM) is essentially based on experimental observations whih have established that the transient surfae urrents generated on strutures (satterers) by arbitrary exitation are primarily in the form of damped sinusoids; the partiular shape being dependent on the form of exitation and the speifi geometry of the struture under onsideration. By assuming that a satterer an be uniquely speified mathematially by an assoiated modal and pole struture, and that the form of the exitation is known, the SEM enables one to determine the surfae urrents diretly from an appropriate expansion of these parameters. Speifially, the expansion was found to require knowledge of four parameters [l]: the natural frequenies and orresponding natural modes, the struture of the inident wave, and salar oeffiients that ouple the natural resonanes to the inident wave (oupling oeffiients). Sine the form of the exitation is assumed to be known, the natural frequenies, natural modes, and oupling oeffiients need to be determined in order to establish an SEM representation of the problem. 14

22 15 Mathematially, the expansion for the spae-frequeny surfae urrents indued by delta funtion exitation on finite, perfet onduting objets in free spae is given by [ 1 ] -m u<r;s) =I na ~a<r)cs-sa) a+ wr;s) (3-la) a In the time-domain, this representation beomes s t u(r;t) = l na va(r) a + w(r;t) a (3-lb) In these equations, s is a omplex variable whih is related to the frequeny, w, by Im{s}=w, U(r;s), U(r;t) denote the spae-frequeny and spae-time surfae urrents, na denotes the oupling oeffiient assoiated with the pole sa' ~a(r), - - va(r) denote the natural mode vetors assoiated with sa, ~ - W(r;s) denotes an entire funtion and W(r;t) denotes the orresponding time-transf orrned funtion, ma denotes the multipliity of the pole sa, and the summations are over all poles. In Setion 3.2.1, we onsider spae-frequeny tehniques for obtaining the natural frequenies, and natural modes. In Setion 3.2.2, we present spae-frequeny tehniques for obtaining the oupling oeffiients, and briefly disuss entire funtions. In Setion 3.3.1, we develop the Cordaro-Davis method for obtaining the natural responses.

23 16 Available tehniques for analyzing the stability of various finite differene approximation shemes are also disussed in that setion. And in Setion 3.3.2, we present transient matrix methods for determining the oupling oeffiients. 3.2 SPACE-FREQUENCY TECHNIQUES Natural Frequenies and Modes An arbitrary Fred..~olm integral equation of the first kind (e.g., expression (2-lla)) may be ast in the general form J ~(r,r' ;s) u<r' ;s)dr' = r(r;s) R3 (3-2) where f(r,r' ;s) denotes a dyadi kernel, U(r' ;s) denotes the desired unknown, and f (r;s) denotes an arbitrary foring funtion. For simpliity, we will write these integral equations using the inner produt notation [1] <r<r,r';s); u<r'.;s)> - r(r;s> (3-2a) where the appropriate operation between the kernel and unknown will be given above the omma separating these parameters, and the integration is with respet to the ommon spatial variable.

24 17 A natural mode, ~0 (r), satisfies equation (3-2a) in the absene of a foring funtion. We may write <r<r r' s ) ~ ' ' CL ' CL <r')> = o (3-3) where sa denotes the orresponding omplex natural frequeny. = The parameters sa and v 0 (r) may be found by disretizing equation (3-3) using a method of moments [14] formalism. We obtain (f (s )) (\i ) = o ) n,m a n a n (3-4) Where n, mare positive integers, (rn,m (SCL)) denotes an n by m matrix, (vn)cl denotes the unknown mode vetor of length n, and (On) is a zero vetor of length n. The magnitude of both n and m is dependent on how refined the disretization is. Equation (3-4) represents a homogeneous system of equations. Suh a system has a solution if and only if the matrix (rn,m (s 0 )) is singular. Hene, the natural frequenies, sa, may be found by solving det [f (s )] = 0. n,m CL (3-5)

25 18 The natural modes may now be found from equation (3-4) using the results of equation (3-5). Equation (3-5) is, in general, extremely ompliated to solve. Numerial solution tehniques typially use either a funtion iteration root searhing tehnique (Setion 4.2.1) or a ontour integration (15] (Setion 4.2.2) method. The use of ontour integration allows one to loate desired roots by partitioning the omplex plane Coupling Coeffiients and Entire Funtions The following derivation for obtaining the SEM oupling oeffiients patterns a development due to Baum [5]. Assoiated with the oupling oeffiient, na' is a oupling vetor, µ (r). The oupling vetor is defined to be a "" - the onjugate adjoint of the natural mode, "a (r), and hene satisfies <µ a <r'), r<r, r' s, a. )> = o. (3-6) By applying the method of moments, we have Ciln)a. (f (s )) = (0 ). ' n,m a. n (3-7) The kernel is now expanded in a Taylor series about s=s a. as

26 19 C'J r(r,r' ;s) = I (s-s )t ~~ (r,r') =O a.,a i a r: (r,r') =If~- rr,r';s)i.,a as s=s a (3-8a) (3-8b) The foring funtion is similarly expanded as 00 I(r;s) = I =0 (s-s / Ci. I,a. Cr) (3-9) ri (r) =...!..._a_ rr s) I Ci. n.r ' ' ~ as. s=s a Assuming only a first order pole, we may write the response from equation (3-la) as ij(r;s) ~ - -1 ~ - n v (r)(s-s ) + U'(r;s) a. a a (3-10) where U' (r;s) denotes some analyti funtion about s=sa. By substituting (3-9), (3-10) and (3-11) into the basi equation (3-2a) and mathing powers of (s-sa), we obtain <f 0 <r,r'); n v Cr)> o,a a. a. (3-lla) and (3-llb) Operating on (3-12b) from the left by ~ (r) yields a

27 20 <µ <r); r 1 <r,r'); n ~ <r)> = <~ <r) 1 (r)> a,a a a a ' o,a (3-12) sine <~ (r); a ro (r,r');,a u'(r';s)> = r I ar' lf ar: ~ <r).. u'(r';s) = 0 R3 R3 Cl. f o r,;: ~ (3-13) by equation (3-6). Therefore, <µ (r); 1 <r)> a o,a <µ (r); r, <r,r'); ~ <r)> a -,a a (3-14) is the expression for the oupling oeffiient at s=sa. The oupling oeffiients relate the inident waveform to the modal struture of an objet. They indiate whih modes are exited and the extent to whih they are exited. Baum [S] has disussed two different, but ultimately equivalent, types of these oupling oeffiients in order to treat two different philosophial interpretations as to how modes are ativated. In one interpretation, all modes are exited simultaneously aross an objet no matter where on the objet the exitation originated. In the other interpreta-

28 21 tion, modes in various regions annot be exited until the inident wave has reahed those regions. We will not pursue these types further here. The entire funtion, W, assoiated with the pole, sa' is neessary for equation (3-la) to be mathematially valid. Its form and use are not well understood, however. Typially, the entire funtion is omitted by the empiri justifiation of obtaining urrent distributions diretly from a set of poles whih are in good agreement with the distributions obtained by standard methods [S,16]. The physial signifiane of the inlusion or omission of the entire funtion requires further onsideration. 3.3 SPACE-TIME TECHNIQUES 3.3;1 Natural Frequenies, Natural Modes, and Stability Considerations In the time-domain, eletromagneti integral equations of the first kind may be written in general form as = - J rr,r';t-t'). ur';t')dt'dr' = r<r;t) Rl (3-15) where r(r,r' ;t-t') denotes a retarded dyadi Green's f~ntion [11], U(r' ;t) denotes the desired unknown, and I(r;t) denotes an arbitrary foring funtion.

29 22 For illustrative purposes, we will restrit the disussion in this setion to thin, perfet onduting surfaes for whih integral expression (2-12a) is appropriate. The disussion will also be limited to retangular (x,y,z) oordinate systems. A similar development applies to other expressions whih may be ast in the form of equation (3-15), and other oordinate systems. Sine the spatial differential operators appearing in expression (2-12a) are with respet to the unprimed oordinates, the following variation of this expression is valid: a -in - [ i EQft x at E (r;t) = ft x z I J 8 s (r'; t-r/) 41TR dr' (3-16) A Here, n is the outward normal on some arbitrary surfae S. The integral over this surfae is ommonly known as the magneti vetor potential. By letting A(r;t) denote this potential, we may write (3-16) as (3-16a) The urrent density, J (r;t-r/), appearing in (3-16) is s typially the unknown whih is desired. However, for notational purposes, and stability analysis, expression (3-16a) is also of interest. This will beome apparent as we progress.

30 23 In general, when a desired unknown appears buried within the integrand of an integral equation it is not possible to determine it analytially. To obtain a numerial solution, one generally begins by expanding this unknown in some suitable set of basis funtions. If the funtion to be expanded is at least pieewise ontinuous over the region of interest, a suitable basis set would be a pulse expansion for the spatial variables. If the funtion is also reasonably well behaved through time, the temporal dependene may also be expanded as pulses. The funtion J (r;t-r/) generally s satisfies these requirements, and hene a pulse expansion in both spae and time is appropriate. It should be noted, however, that this approximation an beome quite poor at surf ae edges due to the singular behavior of the urrent omponent parallel to the edge. Speial are is required for suh strutures (Chapter 6). The expansion of the urrent density may be written as J (r;t-r/) = s N "" Z I J. PA (t-pf:.t-r/)s.(r) ip ut 1 i=l p=-00 (3-17) where Pt.t(t-p/::.t-R/) = s. Cr) = 1 1, for t in the time interval entered at pf:.t+r/ { 0, elsewhere 1, for r in the spae segment entered at if:.r { 0, elsewhere.

31 24 Here, J. 1,p denotes the urrent amplitude oeffiients; i denotes a general spatial index, i.e, i may represent one, two or three integer variables depending on the geometry of the problem; N denotes a general upper bound for the summations orresponding to eah of integer variables whih i represents; and Ar denotes a general spatial sampling distane, i.e., Ar= (Ax,Ay,Az). Expansion (3-17) enables one to write the vetor potential appearing in expression (3-16a) expliitly as Nm Nn Nk m A(m6,nt..,kt.., p6t) - 'i' 'i' 'i' 'i' J - mf=l nf=l kf=l p'=-= m',n',k',p' Glm-m'I,jn-n'l,lk-k'!,(p-p') (3-18) where G = I a.,b,o,n (a.+l/2)6 (f3+1/2)6 J {a.-1/2)6 (f3-1/2)6 m, n, and k beinq positive integers whih are bounded by Nm, Nn' and Nk respetively, and p is an unbounded integer (by ausality, p may be restrited to positive integers). Note that in this expansion we have taitly assumed that the spatial sampling distane is uniformly equal to some onstant, A, so that the ontinuous variables, (x,y,z), orrespond to

32 25 the disrete variables, (ma,na,ka). This is typially, but not neessarily, done. The urrent oeffiients appearing in (3-18) are the desired parameters. They may be extrated by approximating the ontinuous differential operators appearing in (3-16a) by entral finite differene operators. A thorough disussion of finite differene approximation (FDA) tehniques may be found in Ames (17]. In passing, it is worthwhile to note that it is possible to establish an analyti equivalene between the finite differene formulation of a time-domain problem and the basis set formulation of the equivalent frequeny-domain problem by using inverse transform tehniques. This equivalene is satisfying sine it establishes that finite differene tehniques are not simply onvenient mathematial tools for the solution of time-domain problems, but are appropriate, physially meaningful, methods of solution. By using finite differenes, the time derivative of the vetor potential may be written as 1 a z- 2 A(r;t) = at 1 [A(r; (p+1)tit) + X<r; (p-1)tit) (tit) ( 2) - 2 A(r;ptit)J + 0 (Lt) (3-19) where O((At) 2 ) denotes the order of the trunation error intradued in the FDA. The spatial operators may be similarly differened (Setion 5.3, and Setions 6.3, 6.4).

33 26 Expression (3-16a) may now be written as (p=l,2,... ) n x A(r;(p+l)bt) = (bt) 2 [n x v(v A(r;pbt))J -n x X<r;Cp-l)bt) + 2n x Xr;pbt) + ' ) 2 A a -in - ) ( ) 2) 0 tbt n x at E (r;pbt + O(bt (3-20) This formulation establishes an expliit or time-marhing finite time-differened sheme for the vetor potential. An expliit sheme allows one to find future values in terms of previous results without the need for a matrix inversion. Note that the values of the vetor potential at two previous times are required. By substituting expression (3-18) into the differene equation for the vetor potential and manipulating the summations, we may obtain an expliit expression for the ur- rent density oeffiients, J. 1,p - (general spatial index i}. An expliit expression for these oeffiients for the linear thin-wire problem may be found in Setion 5.3. In this setion, we onsider a general expression for these oeffiients whih is suitable for an arbitrary geometry. The formulation will naturally lead into a disussion of stability methods for finite differene shemes. By translating the ontinuous temporal and spatial operators appearing in expression (3-16a) to entral finite dif-

34 27 ferene operators, and by using expansion (3-18) to represent the vetor potential, the following general representation for the urrent density oeffiients is obtained (note - that the notation, Tp+l, used to represent all disrete funtions is interpreted as T(mA,nA,kA,(p+l)At)): NT 0 = I BP, Jp'-p, + Fp+l p'=-1 or -1 NT - - = B -1 [ I B I JI - I + F +l p'=o P P P p J (3-21) where Bp' denotes oeffiient matries orresponding to different times (B_1 is a diagonal matrix orresponding to - - J 'p+l); Fp+l denotes the foring funtion at the (p+l )-th time step; and NT denotes an integer whih is one fewer than the number of time steps required for a wave to propagate aross the maximum distane of the struture; in other words, if, for example, six time steps are required for a wave to travel this maximum distane, NT would be five sine the summation begins at zero. The prime, J'p+l' indiates a vetor of the urrent density oeffiients of every spatial point of interest on the struture. And as a final remark, we note that the rank of the B matries is dependent on the partiular geometry of the problem being studied. For on-

35 28 veniene, we define the rank of these matries to be some integer, N. To obtain the natural frequenies and natural modes, we are interested in equation (3-21) in the absene of a foreing funtion, i.e., C Jr p' p-p' (3-22) -1 where C,=(B 1 ) B,. The solution of this differene equa- P - p tion may be obtained by z transform tehniques. For simple poles, the solution is given by J~+l P+l - = z v a a (3-23) where za=exp{sa6t} denotes the transient representation of the pole, sa, and va denotes a vetor spatially desribing the natural mode. For poles of multipliity ma,ma=l, the solution is given by (J' m m ) a = (p+l) a p+l (3-24) where va m denotes the natural mode vetor orresponding to ' a a pole of multipliity ma. Note that entire funtions do not appear in this development; a pole struture only is the basis for this method. Pole lusters may attempt to model an entire funtion however, and therefore entire funtions

36 29 may still be signifiant although they are net expliitly represented in the formalism. By substituting equation (3-23) (assuming first order poles) into (3-22), we obtain ~T -(p '+10- L C, z 'Iv = p '=O p a J a 0. (3-25) This is a homogeneous system of equations. The poles may be found from det ~ - J NT, I I z: (p +l) = 0 p'=o p (3-26) The modes may now be found from (3-25). There is an alternative to this z transform solution tehnique. Any finite differene sheme in the form of equation (3-22) may be ondensed into an equivalent two-level matrix form [17] by introduing a state vetor, K for p the p-th time step, suh that (T denotes transpose) J -T K..., J' - T N ' p p-~t (3-27) a state transition matrix, ~' suh that

37 lo 30 l."... CN T I p = 0 I (3-28) :10 and forming (3-29) A disussion of error propagation, or a stability analysis of finite differene shemes is appropriate at this time. A disrete finite differene representation of a ontinuous problem may yield an unstable (unbounded) solution when ertain relationships between the sampling distanes used for different variables are not satisfied. For hyperboli equations (wave equations, e.g., equation (3-16)) the relation between the time sampling (6t) and spatial sampling (6, assuming a uniform sampling distane in all diretions) distanes are of interest. It has been shown by Courant, Friedrihs and Lewy (CFL) [17] that the time sampling distane for these equations an be at most equal to the spatial distane, i.e., ~t=~. This is the most lax restrition possible; it an tighten onsiderably depending on how the

38 31 disretization is implemented. Two methods are available to analyze the stability of finite differene shemes for linear equations. We onsider these now. The state transition matrix appearing in equation (3-29) ontains all the information of the finite differene approximation (inluding boundary onditions). A stability analysis of the differene sheme may be done by examining the magnitude of the eigenvalues of this matrix. If all the eigenvalues are less or equal to one in magnitude, errors will not grow through time and hene the solution will be bounded. This tehnique is known as matrix stability analysis. The matrix stability method is useful for testing if a known CFL ondition yields a stable solution. It does not predit, in general, the speifi numerial value required for stability. An alternate method may be used to determine, or at least approximate, this value. A simple method known as Fourier stability analysis may be used to determine the stability riterion for an unompressed differene sheme (e.g., equation (3-20) or (3-22) instead of equation (3-29)). The method analyzes only.the speifi differene equation and hene ignores the influene f boundary onditions. Sine boundary onditions an influene the stability of a sheme, the Fourier method is not onsidered as thorough as the matrix method. However, sine

39 32 a speifi number, whether exat or approximate, for the CFL ondition is readily produed, this method provides useful a priori information about a partiular differene formulation. The matrix method may always be used to onfirm the stability riterion given. In brief, the Fourier method examines the propagating effeet f a single row of errors along some arbitrary line of the FDA. This is aomplished by determining an exponential solution for the differene sh~me from disrete separation of variable tehniques. For a stable solution, restritions on the exponential solution must be enfored. A one-dimensional example may be found in Setion 5.3. Two-dimensional examples may be found in Setions 6.3, 6.4. Stability alone does not imply onvergene of the FDA to the true solution. For a thorough disussion on matrix and Fourier stability methods and onvergene requirements one should refer to Ames [17]. The stability of physial problems is mathematially desribed by the loation of poles in the omplex plane. The I stability of the finite differene representation of eletromagneti expressions is dependent on the magnitude of the eigenvalues of the state transition matrix. Hene, we antiipate some relation to exist between these eigenvalues and the true poles.

40 33 The relation follows simply by onsidering the solution of the differene equation, equation (3-24), applied to the state transition formulation. For first order poles, we may write (3-30) or z K = q; K a a a (3-31) This represents an algebrai eigenvalue problem for the eigenvalue, z, and the eigenvetor, K. It an be shown that a a the natural mode, va' and K are related to one another by a N -1 T z -\) T,...,\} -TJ a CL a. (3-32) The poles, sa.' may be found by solving s = ln(z )/1::.t a. a (3-33) Coupling Coeffiients A method has been presented whih determines the natural frequenies and natural modes. To omplete the SEM form of solution we need to determine the oeffiients that ouple

41 34 the natural frequenies and modes to the inident foring funtion. Two different formulations of these oupling oeffiients are possible in the time-domain. Cordaro [18] has suggested a method to obtain an exat representation of the oeffiients when a omplete set of distint eigenvalues (first order poles) is known. This is aomplished by using eigenvetor deomposition tehniques. The basi method may be extended to obtain an approximation to the oeffiients when only a partial set of distint eigenparameters is known. An alternate formulation for a partial set of poles is a time-domain analog of the frequeny-domain tehnique previously presented (Setion 3.2.2). We will initially onsider Cordaro's method. Define a state vetor, U 0, to represent a normal inident foring funtion at the p-th time step as UT p = (F p' 0 '.., O]T. Here,!. and 0 are Nxl row vetors~ p - Note that for a delta or impulse exitation only U 0 is (3-34) nonzero. The state urrent distribution may now be written expliitly as (a state representation of the foring funtion has been added to equation (3-29))

42 35 (3-35) where m is some arbitrary time step. By assuming a full set of distint eigenvalues, A, and orresponding eigenvetors, r, we may deompose! as (3-36) Equation (3-35) may now be written as m-l m-i K = r l A r u. m i=l 1 (3-37) Sine only the first N omponents of the eigenvetors orrespond to the natural modes, we introdue a vetor, TP, to spatially desribe only the first N omponents of the state urrent vetor KP (the first N omponents define J;). We may write {assuming a.n impulse exitation) - p Tp =[I, 0,., O]rA r uo (3-38) Here, I and 0 denote NxN identity and zero matries. Next, we let M be defined to be the unnormalized natural mode matrix and C be defined to be the unnormalized vetor of oupling oeffiients. They are given respetively by (3-39)

43 36 (3-40) Combining these results yields (3-41) or equivalently (3-42) where the pole sa is related to Za by equation (3-33}, na - denotes the oupling oeffiients, and va denotes_ the orresponding natural modes. Equation (3-42) is the desired SEM representation. When only a partial set of eigenvalues is known, the deomposition of t as given by equation (3-36) is not possible exatly sine the inverse whih appears only exists in a generalized or pseudo inverse sense. Therefore, only a least squares approximation to the oupling oeffiients is possible in this ase. This ompliation may be avoided by developing a time-domain formulation for the oupling oeffiients analogous to the frequeny- domain method (Setion 3.2.2). We begin by = I replaing r(r,r ;s) by a matrix funtion T(z) defined by ~ NT+l ~T (NT-i~ T(z) = z I - l C. z i=o i (3-43)

44 37 By following similar power series expansions, we then define T 1 (z ) in analogy with f.i_ (r,r') to be,a. a.,a. NT (NT-;:l) (NT-i-lj T 1 (z ) = [ (NT+l)z I - l (NT-i)C.z,a. a. a. i=o 1 a. (3-44) We define, next, the oupling vetor, µa.(r), to be the first blok of N elements of the left eigenvetor1 orresponding to the a-th eigenvalue of the state transition matrix. By replaing the frequeny-domain inner produt operations by matrix multipliations~ we may write the oupling oeffiient at z=za.=expf sa.at} as i? (T 1 Cz )v ) a.,a. a a (3-45) where I(za) is the foring funtion vetor evaluated at Za. It should be noted that when eah of the sub-matries of the state transition matrix are symmetri, the first N rows of the left and right eigenvetors are idential to a normalization fator. This is not true for the remaining portion of these vetors, however, sine it an be shown that 1 Let p be the right eigenvetor of the transpose of some matrix A orresponding to the eigeqyalu~ A. Then p satisfies.a.tp=ap. Now onsider (ATpl=p 1 A=pJ.A. In this ase, p is known as the left eigenvetor of the matrix A. Hene, p is either the right eigenvetor assoiated with the matrix AT or the left eigenvetor assoiated with the matrix A orresponding to the eigenvalue \.

45 38 the left eigenvetors have a muh more ompliated struture than the right eigenvetors. The matrix deomposition formulation is reommended when a full set of distint eigenparameters is known; the leftright eigenvetor formulation is reommended when a partial set is known. In onlusion, we note that although the TD-SEM formulation for obtaining the natural frequenies and natural modes is relatively straightforward, a fundamental ompliation does underlie the method. Sine the size of the transition matrix is highly dependent on the geometry and the level of disretization of a partiular problem, it is possible, even for simple geometries, to generate a transition matrix whih surpasses the high speed storage apabilities of the largest omputers. A variety of tehniques whih attempt to handle this ompliation by taking advantage of the form of this matrix are presented in Chapter 4.

46 Chapter IV EIGENSOLUTION METHODS FOR THE TRANSITION MATRIX 4.1 INTRODUCTION The TD-SEM model is a straightforward and effiient method for determining the SEM parameters for simple geometries disretized with relatively few unknowns. This is aomplished by transforming the pole searhing problem into an algebrai eigenvalue problem (Setion 3.3.1). As the number of unknowns inrease, however, the matrix whih TD-SEM generates beomes unmanagably large, thereby making the searh for eigenvalues diffiult and ompliated. The matrix!, whose eigensolution is sought, is given by equation (3-28). Some omments are in order about the form and properties of this matrix.! is known as a sparse matrix sine it ontains a large number of zero elements. It is in blok upper Hessenburg, or more speifially, blok E'robenius form [19]. A matrix in Frobenius form possesses no symmetry properties, and therefore, t unfortunately falls into the lass f unsymmetri real matries, or general real matries. This is indeed a ompliation sine the field of eigensolution methods is both narrowed and ompliated for unsymmetri matries due to the possibility of obtaining omplex eigenvalues and generalized eigenvetors. 39

47 40 A matrix in E'robenius form possesses the property that it is its own ompanion matrix. In other words, the problem of determining the eigenvalues, X, of i may be done in either of two possible forms. First, we may onsider the full matrix and solve det [~ - AI] = 0 (4-1) where I is the identity matrix; alternatively, we may solve o. (4-2) where 0, 1,...,CNT denote the sub-matries of the top row of the transition matrix. The former sheme generally leads to eigensolution methods, whereas the latter generally leads to root searhing methods. An exeption is an appliation of Laguerre's root searhing method to a matrix in Hessenburg form [ 20]. Laguerre's method and various other root searhing methods are disussed in Setion 4.2. Eigensolution methods for unsymmetri matries are presented in Setion 4.3. A survey of eigensolution methods for symrnetri matries may be found in (21].

48 ROOT SEARCHING METHODS ~~ - that We are required to find~., i=l,2,... N (N +l), suh i T T det [B.] = 0 J. (4-3 where (4-4) is an N xn polynomial matrix. We onsider three tehniques for obtaining the roots of equation (4-3). In the first approah, (4-3) is solved diretly. This requires root searhing methods whih utilize funtion iteration sine the expliit oeffiients of the harateristi equation are not known. Muller's method [22] represents a logial method for solution and is disussed in Setion An alternate method for obtaining these roots is to use the omplex ontour integration method of Singaraju, Giri, and Baum [15]. This tehnique is presented in Setion The third approah is to exploit polynomial matrix redution methods [23] whereby the polynomial matrix (4-4) is iteratively redued into a triangular polynomial matrix. The expliit harateristi equation is then the produt of the diagonal polynomials. A wide seletion

49 42 of effiient polynomial zero searhing methods may then be used to find the roots. A polynomial matrix redution method is disussed in Setion An appliation of Laguerre's method is presented in Setion Sine this tehnique does not utilize either form (4-3) or (4-4) we onsider it to be independent of the methods previously mentioned. However, sine the method is a zero searhing method it logially belongs within Setion Muller's Method The following is a brief summary of the work due to Muller [ 22]. We are interested in determining the values of X whih satisfy f(x)=o, for some funtion f. One begins the proess with the values X 1., h., k., f(l), f(x. 1 ), and f(x. 2), l. l. l. l.- l.- where A.i, hi, and k i are some judiious initial guesses, and i is an iterative index; ki+l is then determined by the formula -2 (;\.)o. l. l. k I 1 = 2 1 ( 4-5) l.t g.+(g.-4f(a.)o.k.[f(a. 2)k.-f(A. l)o.+f(a.)] /2 i- l. l. l. l. 1- l. l.- l. l. where

50 43 o.=l+k., 1 1 (4-Sa) and g 1. = (\. 2 )k~-f(a. 1 )o~+f(a.)(k.+o.) 1-1 i- i 1 i i (4-5b) Then Ai+l = Ai + hi+l hi+l = ki+l hi ' (4-5) (4-Sd) and f(ai+l) (4-5e) are omputed. The sign of the square root in the formula for ki+l is hosen to make the denominator have the greater modulus. The formulas are derived by fitting a quadrati of the form, b 0 A2 +b 1 A+b 2, through the following three points (Ai,f(Ai)), <\_ 1,f().i_ 1 )), (Ai_ 2,f(Ai_ 2 )). The oeffiients b 0, b 1, b 2 satisfy b 2 > bl > b2 f (>... 1) 0 i- i- 1-? b ' - +bl > b2 = f(j...,.) 0 t\i-2 i- i-.:. (4-5f)

51 44 The proess iteratively ontinues until some speified riterion for aeptane of the root estimate is satisfied or an upper limit on the number of allowed iterations has been reahed. It is interesting to note that the onvergene properties of Muller's method have never been proven for polynomials with orders greater than two. Nevertheless it is ommonly used on relatively large polynomials with exellent results. For our purpose, equation (4-3) denotes the funtion f disussed above. In general, only one determinant evaluation is required for eah estimate at Ai. Exellent results (eight to ten digit agreement with known solutions) were obtained using this tehnique for systems whih possessed approximately 110 roots. For higher order systems, however, fewer and fewer of the predited roots had any relation to the atual roots. In partiular, for a system whih was known to have 225 roots, only 4 of the predited roots had any relation to the atual roots. This breakdown is attributed to dereased separation in the roots of large systems (sine by stability, all the eigenvalues must fall within the unit irle in the omplex plane), oupled with the numerial roundoff errors assoiated with evaluating equation (4-3).