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  • 8/10/2019 Generelized Formulations - BLB

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    1722

    IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 36. NO 12, DECEMBER 1988

    Generalized Formulations for Electromagnetic

    Scattering from Perfectly Conducting and

    Homogeneous Material Bodies-Theory

    and Numerical Solution

    Abstract-Generalized E-field form ulatio n for three-dimen sional scat-

    tering from perfectly conducting bodies and generalized coupled operator

    equat ions for three-dimens ional scat ter ing f rom mater ial bodies are

    introduced. The suggested approach is to use a fict i t ious electric current

    flowing

    on

    a mathematical surface enclosed inside the body to simulate

    the scattered field and , in the material case, to use in addit ion a fict i t ious

    electric current flowing on a mathemat ical surface enclosing the body to

    simulate the field inside the body. Applica tion of th e respective bou ndar y

    condi t ions l eads to operator equat ions to be solved for the unknown

    ficti t ious currents which fac il i tate the fields in the various regions thro ugh

    the magnetic vector potential integral . Th e existence and uniqueness of

    the so lu t ion are d i scussed . These al t ernat ive operator equat ions are

    solvable via the method of moments. In particular, impulsive expansion

    funct ions for the currents in conjun ct ion wi th a poin t -matching tes t ing

    procedure can be used wi thout degrading th e capabi l ity of th e numerical

    solution to yield accurately near-zone and surface fields. The numerical

    solution is simple to execute, in most cases rapidly converging, and is

    general in that bodies of smooth but o therwise arb i t rary surface, both

    lossless and lossy, can be handled effectively. Boundary condit ion checks

    to see the degree to which the requi red b oundary condi t ions are sat i sf ied

    at any set of poin t s

    on

    the body surface are eas ily made for val idat ing the

    solution. Finally, results are given and compared with available analytic

    solu t ions , which demonst rate the very good accuracy

    of

    t h e m o m en t

    procedure.

    I . INTRODUCTION

    HREE-DIMENSIONAL problems of electromagnetic

    T

    cattering by perfectly conducting and material bodies

    have been a sub ject of intense investigation an d research to the

    electromagnetic community for many years. The study of

    electromagnetic scattering is not solely of academic interest,

    but of practical importance as well in many application areas.

    These efforts have led to a development of a large number of

    analysis tools and modeling technique s for quantitative evalua-

    tion of electromagnetic scattering by various objects. Among

    these methods, surface integral equation formulations are

    probably the most suitable ones for numerical solutions. The

    general procedure is to reduce the three-dimensional problem

    to two dimensions by cast ing the problem in terms of unknow n

    functions defined on the surface of the body rather than in

    terms of unknown volume functions. In considering scattering

    from a conducting body

    (Fig.

    l ) ,

    he problem is formulated in

    Manuscript received September 17, 1986; revised September 25, 1987.

    The autho rs are with the Departm ent of Electrical Engineering, Technion-

    IEEE Log

    Number 8823639.

    Israel Institute of Technology, Technion City, Haifa, 32000 Israel.

    J

    Mm

    1-

    I

    perfect ly conducting

    closed surface S

    Fig. 1. General problem

    of

    scattering by perfectly conducting body

    terms of the yet to be determined surface curre nt J, induced on

    the conducting body surface S. This can be done in two

    alternative ways discussed both by Poggio and Miller in [11.

    One formulation, known as the E-field integral equation, is

    derived by setting the component tangential to

    S

    of the sum of

    the incident electric field an d the electric field due to J,, both

    calculated with the condu cting body absent, equal to zero on S.

    The other formulation, known as the H-field integral equation,

    is derived by setting the comp onent tangential to S of the sum

    of the incident magnetic field and the m agnetic field due t o J,,

    both calculated with the conducting body absent, equal to zero

    just inside

    S.

    In considering scattering from a homogeneous

    material body (Fig. 2 ) , he problem can be formulated in terms

    of yet to be determined equivalent electric and magnetic

    currents

    J,,

    M, over the body surface S Application of

    bound ary conditions leads to a set of four integral equations to

    be satisfied. Linear combinations of these four equations leads

    to a coupled pair of integral equations to be solved. One choice

    of combination constants gives the formulation described by

    Poggio and Miller [ l j . Another choice of combination

    constants gives the formulation obtained by Muller

    [ 2 ] .

    If

    either the E-field or the H-field integral equation for the

    conducting body case were solved exactly, we w ould have the

    true solution. Similarly, if the coupled pair of integral

    equations for the material body case were solved exactly, we

    would have the true solution. To obtain approximate solutions,

    these equations are reduced to m atrix equations via the method

    of moments [3j. The solution of the matrix equations is then

    0018-926X/88/1200-1722 01.00

    988 I E E E

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    LEVIATAN

    et al. : GENERALIZED FORMULATIONS

    FOR

    ELECTROMAGNETIC SCATTERING

    1723

    Fig.

    2 .

    General problem of scattering by homogeneous material body

    carried out in the computer by inversion

    or

    elimination, and

    sometimes by i terative techniques. O nce the unknow n surface

    current

    J,

    in the conducting body case and the unknown

    equivalent surface currents J,, M, in the material body c ase are

    determined, the analysis of these scattering problems is

    completed as the fields and the field-related parameters may

    then be calculated in a straightforward manner.

    In this paper, we introduce alternative formulations for

    scattering from perfectly conducting and homogeneous mate-

    rial bodies with smooth surfaces. The novel idea is to

    formulate the problem in term s of unknown functions that for a

    conducting body are all defined on a mathematical surface

    enclosed in the body, and for material body are partially

    defined on a mathematical surface enclosed in the body and

    partially on a mathematical surface enclosing the body.

    Specifically, in considering scattering from a perfectly con-

    ducting body, we simulate an equivalence for the region

    exterior to the body by m eans of a fictitious elec tric curren t

    J,,

    flowing on a smooth mathematical surface

    SI

    enclosed in

    S.

    This current is assumed to radiate in free space. The operator

    equation for

    J,,

    is then formally derived by setting the

    component tangential to

    S

    of the sum of the incident electric

    field and the electric field due to J,,, both calculated with the

    body absent, equal to zero on S . Similarly, in considering

    scattering from a material body, we simulate an equivalence

    for the region exterior to S by means of current J,, as we d o in

    the conducting-body case and, in addition, simulate an

    equivalence for the region interior to the body by means of a

    fictitious equivalent current J,, flowing on a smooth mathe-

    matical surface

    S ,

    enclosing

    S .

    This current is assumed to

    radiate in an unbounded space fi l led with the medium

    composing the object . The operator equations for

    J,,

    and J,,

    are then formally derived by an enforcement of the boundary

    condition, namely, the continuity of the tangential components

    of the electric and magnetic fields across S , which leads to a

    set of two operator e quations to be satisfied by the fields in the

    two simulated equivalent situations. It should be pointed out

    that it is certainly not claimed that exact solutions to the

    suggested operator equations are guaranteed for any selection

    of S, and

    S,.

    The existence of an exact solution is intimately

    related t o the analytic continuability of the scattered fields

    toward the interior region and of the internal field toward the

    exterior region. Exact solutions, if they exist, are actually

    equivalent currents, which produce the true fields in the

    respective regions. The existence question will be addressed

    further in Section

    111.

    To solve the proposed operator equations, we can apply a

    method of moments numerical solution. Being numerical, our

    solution will never be exact whether a mathematically admis-

    sable solution exists

    or

    not. Our objective is thus to match the

    boundary condit ion to some desired com putational accuracy

    and this can be effected for certain cho ices of

    Si

    and S, even if

    the existence of the solution cannot be guaranteed from a

    strictly mathem atical point of view . In pa rticular, the choice of

    impulsive sources as expansion functions for the unknown

    currents is well-suited. Good resu lts can be obtained using an

    expansion of impulsive currents that l ie a distance away from

    the surface because the fields these currents generate on the

    surface constitute a basis of smooth field functions. Being

    smooth field functions, they are suitable for representing

    smooth quanti ties on the boundary and a re l ikely to render the

    final solution ac curate, not only in the far zone but in the near

    zone and on the surface as well. The notable advantage of the

    displaced implusive curren ts is that they not only yield sm ooth

    field functions on the surface but also enable us to determine

    the fields anywhere analytically. T he quality that the fields are

    known anywhere analytically is appealing as one can save

    laborious surface current integrations when calculating the

    fields at the various stages of the solution. Note that there are

    quite a few field calculations involved. First, when construct-

    ing the generalized impedance matrix. Second, when testing

    the solution by checking the degree to which the required

    boundary conditions are satisfied over a denser set of points on

    the boundary. Third, when computing field-related quantities

    of interest after the solution has been established. Further-

    more, since we are actually using a basis of smooth field

    functions for representing fields on the boundary surface, a

    simple point-matching procedu re can be conveniently adopted

    for testing. Notice that the attractive combination of an

    impulsive current expansion and a point-matching testing

    procedu re which rende rs the solution trivial canno t be success-

    fully applied to the standard surface formulations. First,

    impulsive currents on the surface would inherently yield poor

    surface and near-zone field approximations. Second, even if

    one is interested merely in far-field quantities, one should

    refrain from testing procedu res, such as point matching, which

    give empha sis to surface quantities but rather resort to a testing

    procedure that will average out the inaccuracies.

    As already stated, our objective is to match the boundary

    condit ions over the surface to some desired accuracy. In our

    solution, however, we force the boundary condit ions to be

    obeyed only at a finite number of points on the boundary

    surface. Surely, the field between the match points might

    happen to be quite different from what is required by the

    boundary conditions. Therefore, the convergence of the solu-

    tion must be validated from a chec k on how well the boundary

    condit ions are matched between the points. A question that

    naturally arises is the relationship between the error in the

    boundary condition match and the errors in the exterior

    scattered field and in the interior field. We are not aware of a

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    LEVIATAN et al. GENERALIZED FORMULATIONS FOR ELECTROMAGNETIC SCATTERING

    1725

    free space

    (po, o)

    and excited by impressed sources

    (Ji, M i ) .

    Harmonic eJwtime dependence is assumed and suppressed.

    The geometry of the problem is shown in Fig. 2. The

    permeability of the material composing the body is

    p

    and its

    permittivity is E . Both

    p

    and E are considered complex to

    account for dissipation. T he total field on and exterior to S is

    the sum of the incident field

    Elnc,

    i ')and the field

    (ES,

    H s )

    scattered from the material body. The field inside the material

    body is denoted by

    E,

    H) .

    We primarily seek the scattered

    field in the region exterior to the obstacle as well as the field

    inside the obstacle.

    The proposed m ethod for solving the considered problem is

    to set up two simulated equivalent situations, one for the

    region exterior to S and the other for the region interior to S ,

    by means of two electric surface current distributions. In the

    simulated equivalence for the exterior region, we employ the

    model used in the metallic case and shown in Fig.

    3 .

    The

    scatterer is thus replaced by free space with yet unknown

    fictitious surface current distribution

    J,;

    flowing on a smooth

    mathematical surface Si enclosed in S . We denote the total

    field on and exterior to S in the simulated equivalence for the

    region exterior to S shown in Fig. 3by Einc+ E(J,,), Hinc+

    H(J.$J),

    where

    (E(J,;), H(J,;))

    is the field due to

    J,;,

    calculated

    in free space. This total field is simulating the total field

    (Einc

    + E , HI '

    +

    H S )

    present in the region exterior to

    S

    in the

    original situation shown in Fig. 2 . Similarly, in the simulated

    equivalence for the interior region, shown in Fig.

    4 ,

    the

    impressed sources are removed, the exterior region is filled

    with homogeneous material indentical

    to

    that composing the

    object, and yet unknown fictitious surface current J,, is

    distributed on a smooth mathematical surface S , enclosing S .

    This current, radiating in an unbounded homogeneous medium

    of constitutive parameters p and e , is assumed to simulate the

    electromagnetic field inside the material body. W e denote the

    interior to S in the simulated equivalence for the region

    interior to S shown in Fig. 4by (E(J,,), H(J,,)). This field is

    simulating the field (E, H ) present inside the body in the

    original situation shown in Fig.

    2.

    The question as to the existence of current J,; and J,, for

    arbitrarily selected surfaces S; and S, which exactly produce

    the scattered field (E,, H ) on and exterior to S and the

    field

    (E,

    H ) inside

    S ,

    respectively, will be discussed later.

    Like in the metallic case, it is impossible to guarantee the

    existence of such currents, in general, from a strictly

    mathematical point of view. Again, we will assume f or present

    purposes that for the considered choices of inner and outer

    surfaces S; and So there exist current distributions J,; and J,,

    which produce the true fields in the respective regions. Note

    that in this event the currents JSiand J,, are in fact equivalent

    currents.

    Hence, for such inner and outer surfaces S I and S o , he two

    simulated equivalent situations can be pieced together by

    enforcing the simulated fields in the two regions to obey the

    continuity conditions for the tangential components across the

    material boundary S . This leads to the o perator equations:

    iix [E(JSi)-E(J,,)] = x E' ' on S

    (2)

    ~ X [ H ( J , ~ ) - H ( J , , ) ] = i i x H I n C n S

    3 )

    unbounded homog eneous space

    P

    , * )

    mathemat ical closed

    surface So

    /

    mathemat ical closed

    sur f ace

    S

    Fig. 4.

    Simulated equivalence for region interior

    to

    S .

    where

    ii

    is a unit vector outward normal to

    S .

    Clearly, if

    2 )

    and 3 ) are satisfied, then by uniqueness [9, sec.

    3-31,

    the

    electromagnetic fields

    (E(J,,), H(J,,))

    in the region exterior to

    S

    and (E(J,,), H(J,,)) in the region interior to

    S

    will be

    exactly equal to the true scattered and total fields in these

    respective regions. Equations

    (2)

    and (3) thus constitute a

    generalized set of coupled op erator equation for the problem

    of Fig. 2in which

    Elnc,

    Inc) s known and J,, and J,,, for

    given

    SI

    and

    S o ,

    respectively, are the unknowns to be

    determined. On ce these currents are found, the analysis of the

    problem is completed as field and field related quantities can

    be readily calculated.

    111. EXISTENCE

    In this section, we first examine the strictly mathematical

    requirements that guarantee the existence of current distribu-

    tions J,, and J,, on arbitrarily selected S , and So , which

    produce the true fields in their respective regions.

    As discussed by Millar

    [lo],

    the exterior scattered field in

    both the metallic and material cases can be continued

    analytically into the region interior to S provided that in this

    process no singularity of the exterior scattered field is crossed.

    Note that the scattered field must have singularities within (or

    on) S for otherwise it would vanish identically

    [

    1 1 1 . The

    problem of locating singularities of the exterior scattered field

    has been studied by Millar

    [121.

    The location of the singulari-

    ties depends on the form of the scatterer surface and the

    smoothness of the incident field on this surface. Therefore,

    each problem would dem and detailed consideration on its own

    merits. Similarly, the interior field in the m aterial case can be

    continued analytically into the region exterior to

    S

    provided

    that in this process no singularity of this field is crossed. The

    internal field must also have singularities outside (or on) S for

    otherwise it would vanish indentically. Again, the location of

    these singularities depends on the form of the scatterer surface

    and the smoothness of the incident field on this surface and

    each problem would d emand detailed consideration on its own

    merits. Notice that the continuation of the scattered field

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    1726

    IEEE TRANSACTIONS ON ANlENN AS AND PROPAGATION, VOL.

    36,

    NO.

    12.

    DECEMBER 1988

    toward the region interior to

    S

    is effected in free space with the

    scatterer absent. Similarly, the continuation of the internal

    field toward the region exterior to S is effected in homogene-

    ous unbounded space filled with the same material composing

    the scatterer. Also, each analytic continuation, if exists, is, of

    course, unique.

    As an example, consider the simple two-dimensional

    scattering problem of a curr ent filament situated at distance p

    from the axis of a perfectly conducting cylinder of circular

    cross section of radius a (a

    < p ) . The solution to this

    problem is analytically derivable and can be found in [9, sec .

    5-91. Applying large order asymptotic expansions for Bessel

    and Hankel functions, it can be readily shown that the series

    [9, eq.

    (5-120)]

    representing the scattered field on and exterior

    to the cylinder p

    a

    is also uniformly convergent in the

    interior annular region

    a 2 / p

    < p

    6) [lo]. For this case, the expansion in

    terms of Mathieu functions conve rges uniformly on and within

    the cylinder as far as the interfocal segment. Thus excluding

    the interfocal segmen t, the analytic continuation of the exterior

    scattered field is valid everywhere throughout the interior

    region.

    We now assume, without further comment, that the scat-

    tered field (ES, Hs) can be analytically continued to some

    extent into the region inside S and, in the material case, that

    the internal field

    (E, H)

    can be analytically continued to some

    extent into the region exterior to S. Notice again that the

    continuation of the scattered field towa rd the region interior to

    S is effected in free space (with the sca tterer absent) and that of

    the internal field is effected in homog eneou s unbounded space

    filled with the same material composing the scatterer. We

    denote the former by (E Sp, Hp)and the latter by (EP,

    HP)

    as

    shown, respectively in Figs. 5and

    6.

    Now given a smooth

    mathematical surface SI inside S lying within the region

    through which the continuation of the scattered field (ESPP,

    Hsp)

    is valid an d enclosing the re gion containing the singulari-

    ties, we can use the equivalence principle [9, sec. 3-51 to set up

    an equivalent situation. A pertaining illustration is shown in

    Fig. 5 . Let the original scattered field (ES, HS) exist on and

    external to S, the analytic continuation of the scattered field

    (ESPp, p)exist between S and SI, and let the sou rce-free field,

    denoted by (ES,H),having a tangential electric field ove r

    SI

    equal to that of ESP, exist internal to SI . To suppor t these

    fields, there must be a surface equivalent current J,, over S, to

    account for the discontinuity in the magnetic field across

    SI .

    This current is

    (4)

    where

    fi,

    is a unit vector outward normal to

    SI.

    From the

    uniqueness theorem [9, sec.

    3-31, we know that the field

    interior to SI produced by J,, radiating in free space will be

    (E,H),

    between

    SI

    and

    S

    the field will be ESP,H), and

    farther out , exterior to S, the field will be (ES, H). In a

    J,,

    =

    fi,

    X (HSP

    -

    HS

    on

    SI

    u n b o u n d e d h o m o g e n e o u s s p a c e

    P o C O )

    ECE ,C.

    ??I

    m a th e m a t i ca l c l o se d

    surface S

    m a th e m a t i ca l c l o se d

    surface

    S ,

    Fig. 5 .

    Analytic continuation

    of

    scattered field towards region interior to

    S.

    u n b o u n d e d h o m o g e n e o us sp a ce

    ( P *

    ( N '

    +

    N )/2 field points

    r ; ,m = 1 , 2 , - . . , N S o n S . T h e r e s u l t i s

    [ z ~ I & =

    B

    (24)

    where

    L L J

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    LEVIATAN er al . . G EN ER A LI ZED FO R M U LA T I O N S FO R ELEC TR O M A G N ETI C SC A TTERI N G

    1729

    In

    (24), [Z,] i s a 4N by 2(N' + N ) generalized impedance

    matrix, is a 2( N' + N )-element generalized unknow n

    current column vector, and

    is a 4Ns-element generalized

    voltage source column vector. In (25), the matrices [Z& ],

    p ,

    q =

    1 ,

    2, are precisely the matrices introduced in (19) and

    defined in detail thereafter. Specifically, each

    [Z ,q]

    enotes

    an Ns by NI matrix whose (m, n ) element is the

    iirn

    component of the electric field at observation point rk on S

    due to a current element I l b of unit moment

    Zl;, =

    1).

    Similarly, each [ 2 ] , p , q = 1, 2, denoted an Ns by N

    matrix whose (m, n) element is the negative of the i:,,

    component of the electric field at observation point r;, on S

    due to a current element II;, of unit moment (Zlf, = 1).

    Further, each [ZLp,], , q

    =

    1, 2, denotes an S y

    N'

    matrix

    whose (m, n) element is the ii,, component of the magnetic

    field at observation point

    r;,

    on

    S

    due to a current element

    ll;,

    of unit moment

    Z/q,l

    =

    I). Similary,

    [Z','pq],

    ,

    q

    =

    1, 2 ,

    denotes an N by

    NI'

    matrix whose (m,

    n)

    element is the

    negative of the i;, component of the magnetic field at

    observation point r; on S due to a current element llf,, of unit

    moment (I/;,

    =

    1). In (26), the vectors ,q

    =

    1 ,

    2 ,

    are

    precisely the vectors introduced in (20) and defined in detail

    thereafter. Specifically, each T; enotes an -element colum n

    vector whose nth element is

    I/:,.

    Similarly, each F: q

    =

    1,

    2, denotes an NI'-element colum n vector whose nth elemen t is

    I/;, , . Finally, in (27), the vectors Vep , = 1, 2, are precisely

    the vectors introduced in (21) and defined in detail thereafter.

    Specifically, each

    vep

    enotes the Ns-element colum n vector

    whose mth element is the negative of the i;,,, component of

    E' ' at observation point rk on S . Similarly, each

    Fhp,

    p =

    1 ,

    2, denotes an Ns-element column vector whose mth element is

    the negative of the

    iirn

    omponent

    HI '

    at observation point

    rk

    on S .

    Having formulated the matrix equation

    (lo),

    the unknown

    current vector can be found in a simple manner. If the

    boundary condition is imposed at

    Ns = 1

    2(N'

    + NI')

    points

    on S , then the solution will be, in analogy to (22),

    If, on the other hand, the boundary condition is forced at N

    >

    1/2(N1 +

    N )

    points on S , then the solution will be, in

    analogy to

    (23),

    This completes the solution of matrix equation (24). Once the

    unknown current is derived, either from (28) or (29), one can

    readily proceed to evaluate an approximate scattered field

    (E ', H ')

    in the exterior region, an approximate field

    (E ,

    HI') in the interior region, and, of course, any other field-

    related quantity of interest.

    VI.

    NUMERICALESULTS

    A

    versatile computer program has been developed using the

    formulation of the preceding se ction. To check the accuracy of

    the suggested method, we consider a conducting sphere, a

    conducting cylindrical rod with rounded ends whose axis of

    symmetry coincides with the z axis, and a dielectric sphere

    illuminated by an incident plane wave of unit magnitude.

    E' '

    = U, exp ( - j k , z )

    (30)

    1

    70

    HI '=

    uy xp (

    - j k ,z )

    (31)

    propagating in the z direction. For the spherical cases, the

    exact solution can be foun d in

    [9,

    sec. 6-91. For the conducting

    cylindrical rod with rounded ends case, a numerical solution is

    available in [141. Some computational results obtained with

    the program are given in this section and compared with the

    available solutions. To limit the data displayed, respresenta-

    tive results will be shown, without loss of generality, in the

    principal xz plane.

    The location of the sources may affect the rate of conver-

    gence. Based

    on

    previous studies

    [4]-[7],

    one would expect

    the numerical results to converge faster to a sufficiently

    accurate value when the sources are situated on surfaces

    concentric with S and of figure similar to

    S .

    For the spheres,

    the mathematical surfaces S , and So are thus taken to be

    spherical surfaces of radii r' and rI1, respectively, concentric

    with S . The current elements are evenly spaced along the

    latitudinal and longitudinal lines of the respective spherical

    surfaces. The match points are also evenly spaced on S . It was

    found that for a sphere of radius rs selections of

    r

    between

    0.2rs and 0.W and of rl 'greater than 1 5rs have a comparable

    rate of convergence. In contrast, the rate of convergence

    deteriorates when the inner sources approach either the sphere

    center or the sphere surface, and when the outer sources

    approach the sphere surface. For the numerical examples, the

    inner source points are located on a spherical surface of radius

    rl =

    0.2rS and the outer source points are located on a

    spherical surface of radius

    rl' =

    2.0rs.

    In

    line with the above

    criteria, for a conducting cylindrical rod with rounded ends of

    diameter

    d'

    and length

    I

    the mathematical surface

    S,

    is taken

    to be a finite hemisphere-capped cylinder of diameter 0.2d'

    and length 1 - 0.8d (i.e., the distances between the centers of

    hemispherical caps of Si nd of S are equal). Furthermore,

    with regards to the option of imposing the boundary condition

    in the least square error sense, it was empirically found that,

    although in some cases one can achieve the same accuracy

    using fewer sources, thereby gaining the advantage of invert-

    ing smaller matrices, in general this option is redundant. In the

    metallic case, we thus take an equal number of inner source

    points

    N'

    and match points

    Ns.

    In the material case, w e take

    an equal number of inner source points NI , outer source points

    N , and match points

    Ns.

    This common num ber is denoted for

    convenience by N. ote that in the material case the value of

    N

    represents twice as many sources as in the metallic case. It

    should be remarked that the sources do not have to be split

    equally between the inner and outer regions. Other combina-

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    .......

    N = 16

    .25

    N: 6

    _ _ _

    \

    0

    45 90

    135 I80

    e ( d e v p s )

    Fig.

    7.

    Plots

    of

    boundary condition error AE versus 8 in x z plane

    for

    metallic sphere

    of

    radius

    r s =

    0.2X, for various numbers of sources and

    match points

    N .

    t ions can be used and may even yield a more rapidly

    converging solution. Clearly, in any event, o ne should test the

    solution by increasing the number of sources and match points

    and verify the fulfillment of the boundary conditions between

    the match points. Furthermore, for any desired quantity of

    interest , one should examine the numerical convergence by

    comparing the results for an increasing number of sources and

    match points. If the compu ted results are sufficiently close, the

    solution can be taken as satisfactory.

    A . Perfectly

    Conducting

    Sphere

    Results for the problem of plane wave scattering by a

    perfectly conducting sphere are shown in Figs.

    7-10.

    The

    conducting sph ere in Figs. 7-9 is of radius rs = 0.2X, where X

    is

    the wavelength in free space. In F ig. 10,a larger conducting

    sphere of radius rs = 1 .OX is examined.

    First, we study the convergence of the boundary condition

    error AE,, defined by

    (f ix

    E '+ ElnC / n S

    IEincI

    E , = (32)

    This quantity reveals how well the boundary condition is

    satisfied between the match points. Plots of AI? as a function

    of the polar an gle

    8

    in the xz plane for various values of the

    parameter

    N

    re presented in Fig.

    7 .

    Cases considered are

    N

    = 16, 25, and 36. T he boundary condit ion error, which by (1)

    is zero at the match points, increases smoothly and reaches a

    maximum between the points. A s the number of sources and

    match points increases, the m aximum of

    AEbc

    on the surface

    falls sharply. No te that even forN s small as 36 the maximum

    is smaller than 0.3 5 percent. Th is nature of convergenc e has

    been observed in other cases involving spheres

    of

    other radii

    .......

    N = 4

    ~ N.16

    N.36

    _ _ -

    **** exac t

    0

    45

    90 135

    180

    e d e g m s )

    Fig. 8.

    Plots of surface current Je on a metallic sphere of radius r s = 0.2X

    versus

    8

    in

    xz

    plane, for various numbers

    of

    sources and match points

    N.

    ......

    N.8

    N.16

    _ .

    .36

    ****

    e x o c i

    o m i

    I

    I

    154

    00

    0 4 5 90 135 180

    9 d e g r e e s )

    Fig. 9.

    Plots of scattering cross section

    U

    versus

    8

    in

    x z

    plane for various

    numbers

    of

    source s and match points

    N or

    case

    of

    metallic spher e of radius

    r s =

    0 . 2 h

    and when the sources were distributed differently. Thus, by

    forcing (1) to be obeyed at a sufficiently dense set of points on

    S ,

    (1)

    can be satisfied within a low error between the match

    points as well, and consequently fields and field-related

    quantities of interest can be approximated to a satisfactory

    degree of accuracy.

    Fig. 8shows plots

    of

    the

    8

    componen t of the surface induced

    current J given by

    J = i x (H '+ HInc)

    (33)

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    GENERALIZED FORMULATIONS FOR ELECTROMAGNETIC SCATTERING

    1731

    160

    120

    x

    \ 0

    40

    0

    N.25

    N

    =49

    __

    N.100

    **** e x o c t

    .

    .

    .

    _ _ - _

    0

    45

    D O 1:15 1 8 0

    0 (dryrce.)

    Fig.

    10.

    Plots of scattering cross section U versus 6 in xz plane for various

    numbers

    of

    sourc es and match points Nfor case

    of

    metallic sphere of radius

    rs = 1 O X

    versus 8 in the xz plane for various values of N . Here, the

    interval from

    0

    to 90 on 8 is in the shadow region of S

    while 90 to 180 interval is in the lit portion of

    S .

    Note

    that for N = 36 the results are in excellent agree men t with the

    exact eigenvalue solution [9, sec. 6-91. It should be remarked

    that the accuracy obtained here is equivalent to that obtained

    by Rao et al. [15] using 96 t r iangular patches to model the

    sphere.

    Next, we compute the scattering cross section defined by

    IE

    ( E i n c

    U =

    lim 4ar2

    7- m

    (34)

    where r is the radial distance from th e origin. Plots of U versus

    8

    in the

    xz

    plane for various values of N are shown in Fig. 9

    and compared with the exact solution. Again, very good

    agreement with the exact solution is seen for N = 36.

    Finally, we co mpu te the scattering cross section (34) for the

    larger sphere. Plots of

    U

    versus 8 in the xz plane for various

    values of N are depicted in Fig. 10 and compared with the

    exact solution. Of course, we expect that more sources wil l

    now be required to render the solution accurate. Here the

    results converge to the exact solution for

    N

    not larger than

    100.

    B. Perfectly Conducting Cylindrical Rod with

    Hemispherical Caps at the Ends

    Results for the problem of plane wave scattering by a

    perfectly conducting cylindrical rod with hemispherical caps at

    the ends are shown in Fig. 11. The rod is of diameter

    d' =

    0.4X and length I =

    1.5h.

    Plots of

    U

    as a function of 8 in the

    xz

    plane for various values of

    N

    are dipicted in F ig. 11and

    compared with Andreasen's numerical result [141. Observe

    0 D O

    0 7 5

    0 60

    4

    \ o 45

    0 JC

    0 1:

    0 O

    N.65

    N.96

    =192

    * *

    *

    Andreasen

    . ___.

    ..

    _ _ _ - -

    0

    45 00 135 180

    9

    (dcyrerr)

    Fig.

    1 1

    Plots

    of

    scattering cross section

    U

    versus 0 in xz plane for various

    numbers of sources and match points for case of perfectly conducting

    cylindrical rod with rounded ends

    of

    diameter

    d

    =

    0.41

    and length I =

    1.51.

    that for N = 192 our result is in excellent agreement with

    Andreasen's result. Note also that even for N as small as 96,

    our result is quite close to that of Andreasen.

    C. Dielectric Sphere

    Results for the problem of plane wave scattering by a

    dielectric sphere are shown in Figs.

    12-14. T he sphericai

    scatterer considered is of radius rs

    =

    0.2X. The sphere is of

    permeability

    p

    =

    po

    and permittivity E = 3eO.

    In a manner analogous to that presented in the metallic case,

    we first carry out a study of the con verge nce of the boundary

    condition erro rs

    AE

    and AHk defined by

    Ifi

    x (E5'+E' '-E )I on S

    IE' 'I

    l fi x (H + HI '- HI1) on S

    (H' ')

    AEb, = (35)

    AHbc= (36)

    Plots of U and AHk as a function of 8 in the xz plane for

    various values of parameter

    N

    are presented in Fig. 12.Cases

    considered are N

    =

    25, 36 , and 49. The boundary condition

    er rors

    AE,

    and

    AHbc,

    which by (2) and (3) are zero at the

    match points, increase smoothly and reach a maximum

    between the points. As the numb er of sources and match points

    increases, the maxima of AE, and AH, on the surface fall

    sharply. Note that even

    for N

    as small as 49, the maxima of

    AEk

    and

    AH,

    are sm aller than 0.7 percent.

    To give some addit ional information on the conv ergence as

    the number of expansion functions and match points is

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    1732

    4

    3

    L

    >

    I

    0

    ........

    N: 25

    ~ N = 3 6

    N=

    49

    ;

    _ _ _ _

    , '

    IEEE TRANSACTIONS ON ANTENN AS AND PROPAGATION. VOL

    36, NO. 12, DECEMBER 1988

    0 45 90 I35 I80

    O( d e g i r e \ )

    (b )

    Fig. 12.

    (a)

    Plots of

    boundary condition error

    AEh

    versus 8 in

    xz

    plane for

    dielectric sphere of radius rs = 0.2X and pe rmittivity t =

    3to,

    for various

    numbers

    of

    sources and match points

    N.

    b) Plots

    of

    boundary condition

    error AH h versus

    0

    in xz plane for dielectric sphere of radius r = 0.2X and

    permittivity E = 3t0, for various numb ers of sources and match points N.

    increased, we investigate the convergence of the approximate

    scattered field to the exact solution on the surface of the

    sphere. For this purpose, we define the scattered field erro rs

    A E and A H as follows:

    (ESr-EZxactIn S

    IEinCI

    E = (37)

    R

    I

    . 3 1

    w 4

    - I

    ........

    N =

    25

    __ N.36

    N =

    49

    8

    6

    -

    X 4

    a

    2

    0

    0 45 90 135 180

    degrr6i)

    ( b )

    Fig. 13. (a)

    Plots

    of scattered field error

    A E

    on boundary versus

    8

    in xz

    plane for dielectric sphere

    of

    radius

    r s

    = 0.21 and perm ittivity

    t

    = 3to ,

    for

    various numbers

    of

    sources and match points N. b) Plots

    of

    scattered field

    e r ro r A H on boundary versus 0 in xz plane for dielectric sphere

    of

    radius

    rs

    =

    0.2X

    and perm ittivity t = 3to,

    for

    various numbers

    of

    source s and r.atch

    points N.

    (HSr-H&actI

    n S

    IHinCI

    H = (38)

    where

    (E ,xact,

    ,x,c,) enote the exact values of the scattered

    field obtained using the result of

    [9,

    sec. 6-91.

    Plots

    of A

    E

    and

    A H as a function

    of 19

    in the xz plane are depicted in Fig. 13.

    I ,

    Note the convergence of the fields as the number

    N

    ncreases.

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    LEVIATAN et al. : GENERALIZED FORMULATIONS FOR ELECTROMAGNETIC

    SCATTERING

    1733

    0

    16

    0 0 8 1

    o o o ,

    , ,

    , ,

    4 5 9 0 135 180

    8

    deg iee3 )

    The operator equations are solvable by the method of

    moments. In particular , an impulsive expansion for represent-

    ing the unknown current can be used. The noticeable

    advantage of the impulsive expansion is that it enables

    us

    to

    determine the fields anywhere in space analytically, thereby

    rendering the quite a few field calculations involved in the

    various stages of the solution trivial. At the same time, this

    expansion yields accurate results not only in the far-zone but in

    the near-zone and

    on

    the surface a s well because the displaced

    impulsive sources generate smooth fields on the surface

    suitable for representing smooth quantities on the surface.

    Finally, s ince we are using, though indirectly, a smooth

    expansion for the fields

    on

    the surface, a point-matching

    procedure can be selected for testing.

    Th e proposed method has already been applied successfully

    to two-dimensional waveguide

    [4]-[6]

    and free-space

    [7]

    scatter ing problems,

    where the various fields have been

    approximated using filamentary currents situated on suitably

    selected surfaces. The numerical procedure was found to be

    simple to apply, of wide range of applicability, and rapidly

    converging. An application of this method to three-dimen-

    Fig. 14. Plots

    of

    scattering cross section U versus

    e

    in xz plane

    for

    various

    sional scattering problems where the various fields are

    approximated using impulsive

    current

    elements has been

    presented in this paper. The numerical solution for three-

    numbers

    of

    sources and match points

    N or

    case of dielectric sphere of

    radius rs = 0 2X and permittivity E

    =

    3to.

    A similar convergence to the exact solution is also found for

    the scattered field outsid e the sphere and for the field inside the

    sphere. These plots will not be shown here.

    Finally, the scattering cross section given by (34) is

    computed for the dielectric sphere case. Plots of

    U

    versus

    0

    in

    the xz plane for various values of N are shown in Fig. 14 and

    compared with the exact solution. Very good agreement with

    the exact solution is seen for N =

    36.

    VII . CONCLUSION

    Generali zed formulations for three-dimensional problem s of

    scattering by perfectly conducting and homogeneous material

    bodies have been proposed. The innovative approach is to use

    the field of a fictitious electric current flowing o n a mathe mati-

    cal surface enclosed within the body to simulate the exterior

    scattered field. In the material body case, the field of an

    additional fictitious electric current flowing on a mathematical

    surface enclosing the body is used to simulate the internal

    field. Application of the boundary conditions at the body

    surface leads to alternative operator equations to b e solved for

    the unknown currents which facilitate the fields in the various

    regions via the magnetic vector potential integral.

    Attention has been paid

    to the questions of existence and

    uniqueness of the solution. It is found that from a strictly

    mathematical point of view, one cannot, in general, guarantee

    the existence of current distributions, for arbitrary selected

    surfaces inside and outside the body, that will produce the true

    fields in the respective regions. The existence of an exact

    solution is intimately related to the analytic continuability of

    the scattered field towards the interior region and the

    internal field, in the material body case, towards the exterior

    region. Exact solutions, if they exist, are actually equivalent

    currents which produce the true fields in the respective

    regions.

    dimensional problems is also simple to execute, rapidly

    convergin g, and general in that bodies of smooth but otherwise

    arbitrary surface both lossless and lossy can be handled

    effectively. It should be clear that it is almost impossible to

    state a rule of thumb as to the choice of source location and

    number.

    In

    any case, one shou ld test the solution by examining

    both the degree to which the boundary conditions are satisfied

    over a denser set of points on the boundary and the numerical

    convergence of the considered quantity as the number of

    sources and match points is increased. Som e choices of source

    location may speed up the convergence, but even for choices

    less than optimal the solution will usually converge to the

    appropriate limiting value without seriously taxing the com-

    puting system. In any case, it is immediately known if the

    results are inaccurate by using the boundary condition check.

    As presented here, the formulation deals exclusively with a

    single scatterer. The extension of this formulation to encom-

    pass the multiscatterer case, say Mo ne s, is s traightforward. In

    this case, the exterior scattered field is simulated by the field

    of M sets of sources, each situated inside its corresponding

    body, while the field inside each of the material bodies is

    simula ted, as before, by the field of an appropriate set situated

    outside the body. Boundary conditions are then sim ultaneously

    applied at selected points

    on

    the M surfaces.

    The suggested technique is mainly applicable to scatterers

    with smooth surface. A lack of smoothness can cause the

    method to fail since the fields generated on the surface by the

    impulsive current elements that lie a distance away from the

    surface are smooth, an d clearly a sum of such sm ooth fields is

    not best suited for representing fields near edges where the

    fields are singular . This deficiency can in pr inciple be

    overcome by incorporating, in addition to the impulsive

    current elements, surface currents capable of representing the

    correct edge singularity in subdomains near the edges.

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    1734

    IEEE TRANSAC TIONS ON ANTENNAS AND PROPAGAT ION, VOL. 36, NO. 12, DECEMBER 1988

    Howev er, it might even be possible

    to

    use impulsive Sources

    near

    the

    edges to the edge behavior to accuracy that

    may be sufficient for engineering need s. These appro aches are

    [9] R.

    F.

    Harrington,

    Time-Harmonic Electromagnetic Fields.

    New

    York: McGraw-Hill , 1961.

    R. F. Millar, Rayleigh hypothesis in scattering problem,

    Electron.

    Lett.,

    vol. 5 ,

    pp.

    416-418, Aug. 1969.

    [ IO ]

    currently under investigation by us.

    REFERENCES

    111

    I21

    131

    A . J. Poggio and E.

    K .

    Miller, Integral equation solutions of three-

    dimensional scattering problem s, in

    Computer Techniques

    for

    Electromagnetics,

    R. Mittra, Ed. Oxford, England: Pergamon,

    1973, ch. 4.

    C. Muller,

    Foundations

    of

    the Mathematical Theory

    of

    Electro-

    magnetic Waves.

    R. F. Harrington,

    Field Computation by Moment Methods.

    New

    York: Macmillan, 1968.

    New York: Springer Verlag, 1969.

    [ I ]

    [

    121

    R. C ourant and-D. Hilbert ,

    Methods

    of

    Mathematical Physics,

    vol.

    2 .

    R. F. Millar, The Rayleigh hypothesis and singularities of solutions to

    the Helmholtz equation,

    Bull. Radio Elec. Eng. Div. Nut. Res.

    COumnC. Ca n. .

    Vol. 20.

    DD.

    23-27. Aor. 1970.

    New York: Interscience, 196 2, pp. 317-318.

    ..

    J. R. Mautz and R.

    F.

    Harrington, H-field, and E-field, and

    combined-fields solutions for conducting bodies of revolution,

    Arch.

    Elek. Ubertragung.,

    vol. 32, pp. 157-164, Ap r. 1978 .

    M . G . Andreasen, Scattering from bodies of revolution, IEEE

    Trans. Antennas Propagat.,

    vol. AP-13, pp. 303-310, Mar. 1965.

    S . M. Rao, D. R. Wilton, and A. W. Glisson, Electromagnetic

    scattering by surfaces

    of

    arbitrary shape,

    IEEE Trans. Antennas

    Propagat.,

    vol. AP- 30, pp. 409-418, M ay 1982.

    [4]

    Y. Leviatan, P. G. Li, A. T. Adams, and J . Perini, Single-post

    inductive obstacle in rectangular waveguide, IEEE Trans. Micro-

    wave Theory Tech.,

    vol. MTT-31,

    pp.

    806-811, Oct. 1983.

    ..

    1984.

    Y. Leviatan and G. S . Sheaffer, Analysis of inductive dielectric posts

    in rectangular wavegu ide,

    IEEE Trans. Microwave Theory Tech.,

    vol. MTT-35, pp. 48-59, Jan. 1987.

    Y. Leviatan and A. Boag, An alysis of electromagne tic scattering from

    dielectric cylinders using a multifilament current model,

    IEEE

    Trans. Antennas Propagat.,

    vol. AP-35, pp. 1119-1 127, Oct. 1987.

    A. C. Ludwig, A comparison of spherical wave boundary value

    matching versus integral equation scattering solutions for a perfectly

    conducting bod y,

    IEEE Trans. Antennas Propagat.,

    vol. AP-34,

    pp. 857-865, July 1986.

    [6]

    Amir

    Boag,

    for a photograph and biography please see page 1127 of the

    October 1987 issue of this TRANSACTIONS,

    71

    [8]

    Alona b a g ,

    for

    a photograph and biography please see Page 1607 of the

    November 1988 issue

    of

    this TRANSACTIONS.