COMPARISON OF TWO-DIMENSIONAL CONFORMAL LOCAL ...

COMPARISON OF TWO-DIMENSIONAL CONFORMAL LOCAL RADIATION BOUNDARY CONDITIONS1 Bernd Lichtenberg and Kevin J. Webb

School of Electrical Engineering Purdue University West Lafayette, IN 47907-1285 USA Douglas B. Meade

Department of Mathematics University of South Carolina Columbia, SC 29208 USA Andrew F. Peterson

School of Electrical Engineering Georgia Tech Atlanta, GA 30332 USA ABSTRACT Numerical solutions for open-region electromagnetic problems based on differential equations require some means of truncating the computational domain. A number of local Radiation Boundary Conditions (RBCs) for general boundary shapes have been proposed during the past decade. Many are generalizations of the Bayliss-Turkel RBC for circular truncation boundaries. This paper reviews several two-dimensional RBCs for general truncation boundaries. The RBCs are evaluated on the basis of their performance on two separate numerical tests: the annihilation of terms in the Hankel series and the comparison of near- eld and radar cross sections for nite element solutions to scattering problems. These tests suggest that the simpler RBCs can be very competitive with RBCs based on more sophisticated derivations. 1. INTRODUCTION Despite the rapid growth of computer resources over the last few years, the main concern in solving realistic open region electromagnetic problems is to reduce the size of the system of linear equations that must ultimately be solved without compromising accuracy. For nite element applications, this 1 This work was supported in part by the National Science Foundation under grant DMS-9404488 and the Semiconductor Research Corporation, under grant 94-DJ-211.

generally translates to a limit on the size of the computational domain. Thus, it is desired to truncate the open region as close and as conformal as possible to the scattering object. An accurate radiation boundary condition (RBC) must be applied on the truncation boundary. In recent years, a number of local RBCs for use on general boundaries have been proposed ([EM77], [EM79], [BT80], [BGT82], [Hig86], [KTU87], [MBTK88], [KM89], [KRM89], [MRK+89], [Jan92], [LC93], [MSPW92], [MSPW95], [Jin93]). An excellent summary of the RBCs for circular and planar truncation boundaries can be found in [Giv91]. The basic mathematical problem is to nd a solution to the scalar Helmholtz equation

r2u + k2u = f

(1)

in the two-dimensional exterior, , of a scatterer. The boundary conditions on the scatterer are either Dirichlet (TM polarization) or Neumann (TE polarization); the boundary condition \at 1" is the Sommerfeld radiation condition: lim 1=2(u , jku) = 0

!1

(2)

in two dimensions. Truncation of the open region must be accompanied by the selection of an appropriate boundary condition on the truncation boundary. This boundary condition, which replaces (2), should transmit waves arriving from inside the computational domain without generating spurious re ections. A rigorous analysis of the RBCs on domains created with general truncation boundaries is extremely dicult. The purpose of this paper is to present the results of several comparisons between the common local RBCs that have been proposed for general truncation boundaries. This begins, in Section 2, with a review of the derivation of these RBCs. Each of the boundary conditions will be written in a form compatible with implementation in a nite element method (FEM), e.g., un = Bu where un is the normal derivative of the (scalar) eld u and B is a tangential dierential operator. One comparison of the RBC is to examine jun ,Buj for u a term from the Hankel expansion of the solution to the exterior Helmholtz equation. This test and results are presented in Section 3. A full nite element implementation of these RBCs has been reported previously ([LRW+95], [Lic94]). Solutions obtained from the FEM are compared in Section 4. It is more than a little surprising to note that some of the simpler RBCs perform surprisingly well in both sets of tests.

2. RBCS FOR NONCIRCULAR BOUNDARIES We consider boundary conditions which are generalizations of RBCs on a circular boundary. These conditions all have the form

u = i()u + i ()u

(3)

where (; ) are polar coordinates, subscripts with  and  denote partial derivatives (e.g., u = @u @ ), and the subscript i denotes the order of the RBC (either 1 or 2). The coecients for three common families of RBCs are given in Table 1. The Bayliss-Turkel (BT) ([BT80], [BGT82]) and Khebir-RamahiMittra (KRM) RBC [KRM89] are both based on the two-dimensional far eld expansion by Karp [Kar61]. The KRM RBC has one more term in the numerators of 2 and 2 ; these coecients can also be considered as the leading terms of a Taylor series expansion in ,1 of the BT coecients. The Li-Cendes (LC) RBC, in contrast, is derived from the rst two terms of a cylindrical eigensolution expansion of the eld [LC93]. TABLE 1: Coecients of rst- and second-order RBCs for a circular boundary. The functions g1 and g2 are de ned in (20). Type BT KRM LC

1 ,jk ,

1

2 1

,jk , 2

1 (k) , kH H (k) 0

1 0 0 0

2

,jk , 2 + 8(1 + jk) 1

1

1 + (k),2

2 1

2(1 + jk) 1 + (k),2

,jk , 2 + 8(1 + jk) 2(1 + jk) 2 (g (k) , 1) 1 2 , k(kg (kg1(k) , 1) 1 (k) , 1) 1

Four generalizations of an RBC from a circular truncation boundary to a noncircular boundary are considered. These procedures, rst introduced by Kriegsmann et al.([KTU87], [MBTK88]), Li and Cendes (LC) [LC93], Khebir et al.(KRM) ([KM89], [KRM89], [MRK+89]), and Meade et al.(MPW) ([MSPW92], [MSPW95]), involve dierent sets of assumptions and simpli cations. While the second-order RBCs will receive the bulk of the attention in this analysis, the simpler rst-order RBCs clearly illustrate some of the key dierences between the dierent methods. The remainder of this section is devoted to a survey of all four methods, and some preliminary evaluation. All four methods are similar in that they use a change of coordinates to express radial derivatives in terms of the natural coordinates on the arti cial boundary. Three of the four procedures involve additional simplifying assumptions; only the MPW RBC succeeds in generalizing the rst-order

BT RBC. Each technique starts with one of the following total derivative expressions: u = unn + ut t un = un + un ; (4) where (t; n) are the local coordinates representing either the local Cartesian coordinate system aligned with the tangential (t) and normal (n) directions or the arclength (t) and distance (n) from the boundary (see Appendix A for details). 2.1. First-Order RBC The rst-order RBC on a circular boundary is u = 1u, with 1 from Table 1. We will now see four methods that have been proposed for the extension of the circular RBC to non-circular boundaries. Each of these boundary conditions will be of the form un = D1 u + D2ut (5) where D1 and D2 depend only on 1 and the geometry of the truncation boundary. The precise de nition of the coecients is summarized in Table 2. Note that while the linear system for the Helmholtz equation with a circular arti cial boundary is symmetric, (5) produces a symmetric linear system only when D2 = 0 on the arti cial boundary. This condition will be satis ed by two of the generalizations. TABLE 2: Coecients D1 and D2 used in rst-order generalized RBCs, (5). Procedure D1 D2 Kriegsmann 1(1=) 0 Li-Cendes 1()n = 1() sin( , ) 0 KRM 1()n = 1() sin( , ) tn = , 21 sin(2( , ) 1 =  () csc( , ) MPW , nt = , cot( , ) 1 n Kriegsmann. The simplest generalization has been proposed by Kriegsmann et al. ([KTU87], [MBTK88]). At each point on the arti cial boundary de ne a local polar coordinate system whose origin is the center of the osculating circle at this point. In other words, the origin of a local polar coordinate system is de ned by the curvature of the boundary at the considered point. Thus, the corresponding coordinate transformation can be described by @ ! @; 1 ! (6)

@

@n



where  denotes the curvature of the boundary. The corresponding rstorder radiation boundary condition is

un = 1(1=)u:

(7)

Kriegsmann originally proposed to use this RBC directly on the surface of the scatterer (this is the on-surface radiation condition, OSRC). While this RBC is not eective when applied directly on the scatterer [JVL89], the performance is signi cantly better when the truncation boundary is very close to the scatterer ([Lic94], [LLRW94]). We emphasize that this approach uses a dierent local coordinate system for each point on the arti cial boundary. This is a feature unique to the Kriegsmann procedure; the other three generalizations require the explicit choice of a single scattering center for the entire domain. This distinction will be explored further during the discussion of the computational results. Li-Cendes (LC). The algorithm proposed by Li and Cendes [LC93] is based on (4)2 with the simpli cation that all dependence on  is neglected. Thus,

un  n u = n 1u = 1 sin( , )u

(8)

where  is the angle between the x-axis and the tangent vector on the boundary (see Appendix A and Figure 10(a)). Note that (8) does reduce to the circular RBC when the truncation boundary is a circle ( ,  = =2). However, for non-circular boundaries, the omission of u n would appear to be a likely source of signi cant errors. Khebir-Ramahi-Mittra (KRM). Khebir et al. [KRM89] employ similar ideas in their derivation. The angular derivative is approximated by assuming the normal direction is independent of . That is,

u = unn + utt  utt :

(9)

Substituting (9) into (4)2 results in

un = 1n u + tn ut = 1 sin( , )u , 12 sin(2( , ))ut:

(10)

The assumption that n  0 is similar to the Li-Cendes approximation. In particular, both are reasonable approximations for quasi-circular boundaries, i.e.,  ,   =2. The presence of the rst-order tangential derivative in the KRM RBC should translate into a better agreement between the solutions on the truncated and in nite domains.

Meade-Peterson-Webb (MPW). The change of coordinates used by Meade et al. ([MSPW95], [MPW93]) is exact. From (4)1:  , ) u : un = n1 u , nt ut = sin(1, ) u , cos( (11) sin(  , ) t   This assumes only that n = sin( , ) 6= 0, i.e., no tangent line to the truncation boundary lies on a line through the origin. 2.2. Second-Order RBC The same principles can be applied to extend the second-order RBCs to a non-circular boundary. While the ideas are the same, the details are more involved and the analysis of the resulting RBCs is not as simple as for the rst-order RBCs. The general form of the RBC on a circular boundary is u = 2u + 2u ; (12) with 2 and 2 from Table 1. The search for generalized RBCs is limited to boundary conditions that express the normal derivative in terms of the function and its rst two tangential derivatives, that is: un = D1 u + D2ut + D3utt: (13) The only term that is missing from this form is the mixed derivative, utn; this term is omitted due complications created in a FEM implementation of this term. The critical step in generalizing the RBC from (12) to (13) is the approximation of u in terms of the normal derivative (un) and tangential derivatives. The coecients that result from the four dierent processes are summarized in Table 3. For the linear systems encountered when solving the Helmholtz equations with a (local) boundary condition, (13) will be sparse. Symmetry will be 3 observed only when D2 , @D @t = 0 on the arti cial boundary. This condition is satis ed on circular boundaries, since D2  0 and D3 = 2(). In general, this symmetry condition is not met. Kriegsmann. Kriegsmann's proposed method for expressing the second angular derivative in terms of tangential derivatives is based upon the local polar coordinate systems. Thus, the substitutions in (6) must be supplemented with 1 @ ! @: (14)

 @

This yields

@t

@  1 @u   1 u : u = 1 @t  @t 2 tt

(15)

TABLE 3: Second-order generalized RBCs. The de nition of g~1 and g~2 can be found in (21); the coecients A~, B~ , C~ , D~ are de ned in (35). Procedure D1 D2 D3 1 Kriegsmann 2(1=) 0 2 2 (1=) LC (orig.) k(1 , g~2)=g~1 0 1=(kg~1) LC (impr.) k2(1 , g~2 )=(kg~1 , ) 0 1=(kg~1 , ) KRM 2 sin( , ) , 21 sin 2( , ) 2 2 sin3( , ) ~ C~ ~ C~ ~ C~ MPW A= B= D= Thus, D1 and D2 are unchanged from the rst-order condition and D3 = 1 2 2 (1=). Note that a slightly more accurate condition should result from the elimination of the unnecessary approximation of (15); in either case, D2 , @D@t3 6= 0 on the arti cial boundary and the linear system will not be symmetric. Li-Cendes (LC). The derivation proposed by Li and Cendes is based on calculations in which all angular dependencies are ignored. Begin by considering a solution formed from the rst two terms in the Hankel expansion of the exact solution:

u = a0H0 (k) + a1H1 (k):

(16)

To determine appropriate values for the functions a0 and a1, compute the rst two normal derivatives of (16) ignoring all angular () dependencies: (8)

un  nu = kn a0H00 + a1H10 ; (8) @ unn  @n (nu) (17)  nnu + (n )2u = knn(a0H00 + a1H10 ) + k2(n)2(a0H000 + a1H100 ) ,




(17)

(18)

where each Hankel function, and derivative of a Hankel function, is evaluated at k. To express (18) in the desired form, note that (16) and (17) can be used to express a0 and a1 in terms of u, un, and known quantities. The nal generalized RBC is

unn + kg~1un + k2g~2u = 0

(19)

where

, H0 H1 g1 = HH1 HH0 , 0 1 H0 H1 nn g~1 = n g1 + k 00 0

00

0

n

and and

g2 = ,H1 H0 + H0 H1 H0 H1 , H0 H1 g~2 = (n)2g2 : 0

00

0

0

0

00

(20) (21)

Unfortunately, this is slightly dierent from the RBC published in [LC93]. In that original derivation, the authors assume n  1 and set g~1 = g1 and g~2 = g2 . For the computational tests in this paper, the exact coecients g~1 and g~2 have been implemented. It still remains to express the second normal derivative using terms that appear in (13). Li and Cendes suggest using the scalar Helmholtz equation in local Cartesian coordinates: (r2 + k2)u = unn + utt + k2u = 0: (22) The resulting RBC, which is referred to as the original LC RBC, is un = f~1u + f~2utt; (23) with (24) f~1 = k(1 g~, g~2 ) and f~2 = k1g~ : 1 1 Note that the LC original RBC does not reduce to the LC RBC for a circular boundary. The problem is that (22) represents the Helmholtz equation only on a planar segment of the boundary ( = 0). The general form of the Helmholtz equation in curvilinear coordinates is utt + unn + un + k2u = 0: (25) Using (25) leads to an \improved" LC RBC 2 , g~2) 1 un = kk(1 (26) g~1 ,  u + kg~1 ,  utt: The original LC RBC from (23) is recovered by setting  = 0, while the circular LC RBC is retrieved with  = ,1. Khebir-Ramahi-Mittra (KRM). The quasi-circular assumption of Khebir et al. that led to the approximation u  tut can be applied to obtain the approximation: (9) @ u  @ (tut ) = t ut + (t)2utt + n t utn: (27)

Note that for quasi-circular boundaries: n  0, t  1 and t  0. Thus u  (t)2utt: (28) This provides all the necessary ingredients to express un in terms of tangential derivatives: un = u n + u n (9)  u n + n tut (12) = (2u + 2 u)n + nt ut   (28) = (2n)u + (tn)ut + 2 n (t)2 utt: (29) Meade-Peterson-Webb (MPW). The MPW RBC uses the exact total derivative to express u in terms of the tangential and normal derivatives: u = (t )2 utt + 2t n utn + (n )2 unn + tut + n un: (30) The unn and utn terms in (30) are dicult to implement numerically. The second normal derivative can be expressed in terms of tangential terms by way of the curvilinear Helmholtz equation (25). This leads to the following exact generalization of a second-order RBC to a general boundary: 0 = Au + But + Cun + Dutt + Eutn ; (31) with coecients A = 2 , k2 2 (n )2 ; B = ,t + 2 t ;   C = ,n + 2 n ,  (n )2 ; (32)   D = 2 (t)2 , (n )2 ; E = 2 2 t n : Numerous tangential approximations to utn are proposed and studied in [MSPW95]. These results suggest that an eective approximation is obtained by taking the tangential derivative of the rst-order generalized MPW RBC (11): @ @ (11) @1 utn = @ (un)  @t u + 1 , @t (t) ut , @t (n) un , t utt : (33)

@t

n

The result is the approximate RBC ~ + Bu ~ t + Cu ~ n + Du ~ tt; 0 = Au

n

n

n

(34)

with coecients

@1

A~ = A + n@t E;  @ B~ = B + 1 ,n@t (t) E;  @ (n ) C~ = C , @tn  E;  t  D~ = D , n E:

(35)



To conclude this discussion of the generalized second-order RBCs, note that, with the exception of the generalized original LC RBC, each of the extended second-order RBCs reduce to the corresponding second-order RBC when the truncation boundary is circular. In lieu of rigorous analytical results for this class of boundary value problems, numerical tests are the best way to compare these conditions. 3. TEST 1: MODE ANNIHILATION The total number of boundary conditions that can be formed by combinations of the coecients for RBCs on a circular boundary (Table 1) with the methods for generalizing to a non-circular boundary (Tables 2 and 3) is quite large. The numerical experiments in this section and the next will be conducted using only the coecients as presented in the original papers. The rst test explicitly examines the error between the normal derivative and the combination of terms that appear on the right-hand side of (13) (i.e., jun , Bu)j), when u is a single term from the Hankel, or modal, expansion [Har61] of the solution to the exterior problem:

us =

1 X l=0

cl l ;

(36)

where l = ejlHl(2) (k). The purpose of this experiment is to obtain an improved understanding of the impact of some of the approximations involved in the generalization of the RBC to a non-circular boundary. Note that while the ultimate goal is to nd the RBC with smallest error, the results of these tests are expected to be no better than those for a circular boundary, i.e., the accuracy of the circular boundary RBC is limited by its order and the applicability of the Karp expansion. Recall that the higher order Hankel functions exhibit rapid growth for small arguments. Thus, it will be necessary to examine both the absolute and

normalized (relative) errors. Let f(xi; yi )gNi=1 be N points on the boundary. Denote the absolute and normalized errors at the point (xi; yi) by:

ei = j( l )n(xi; yi ) , B l (xi; yi)j

eni = j( ) e(ix ; y )j ; (37) l n i i n

and

where Bu = D1 u + D2 ut + D3utt . Estimates of the absolute and normalized L1 (@ ) error on the truncation boundary are given by

E1 = N1

N X i=1

ei

E1n = N1

and

N X i=1

eni ;

(38)

respectively. x

Incident Plane Wave (TM) E

Scattered Field y

inc

s

k

H

∂Ω Circle

E

s

(i )

l

s

AAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAA x

inc

H

θ

y

w

inc

inc

(ii)

d

(iii)

d

k

d

Domain Ω Scatterer

a

y

x

b x

∂Ω Capped Strip

d (iv)

y

x y

w

AAA AAA s

(a)

pec

(b)

dielectric

FIGURE 1: Physical problems studied: (a) Capped strip and circular geometries; (b) Sample of scatterer geometries considered in numerical experiments: i. (6  0.1) pec scatterer (l = 6, w = 0:1), ii. (6  0.1) pec scatterer covered with a dielectric (d = 0:1, "r = 4:0), iii. (3  0.1) pec scatterer inside dielectric, iv. (6  0.1) pec scatterer with aperture (s = 2:06). Figure 1(a) shows the basic geometry of the capped strip and circular boundaries used in the tests. Figures 2{3 are representative of the results of this experiment. The rst test was conducted on a circular boundary with radius  = 5, and center at the origin. The same comparisons are repeated on a capped strip boundary with half-length a = 3 and radius b = 2 (with center at the origin). The normalized error for the rst thirty-one (l = 0; : : : ; 30) eigenfunctions is computed for the second-order Kriegsmann, Li-Cendes, KRM, and

2

10

1

10

Average Normalized Error in %

0

10

-1

10

-2

10

MPW

-3

10

Kriegsmann Li-Cendes KRM

-4

10

-5

10

-6

10

0

5

10

15 20 Order of Wave Function

25

30

FIGURE 2: Average normalized error on a circular boundary ( = 5) for the eigenfunctions of order 0 to 30. Note that the MPW and Kriegsmann curves are coincident.

3

2

10

10

0

10

2

Average Normalized Error in %

10 -2

Normalized Error ( in % )

10

-4

10

-6

10

-8

10

MPW

0

MPW

10

Kriegsmann Li-Cendes KRM -1

10

Kriegsmann

-10

10

1

10

Li-Cendes KRM -2

-12

10

0

50

100

150

200 250 Angle in degrees

(a)

300

350

400

10

0

5

10

15 20 Order of Wave Function

25

30

(b)

FIGURE 3: Results of the mode annihilation test for a capped strip boundary with (a = 3:0, b = 2:0): (a) normalized error of the rst eigenfunction and (b) normalized error for the eigenfunctions of order 0 to 30.

MPW boundary conditions; the results are displayed in Figure 2. Since the boundary is a circle, the MPW and Kriegsmann RBCs are expected to reduce to the classical second-order BT RBC. The slight dierences evident in the KRM RBC are due to the additional approximations used in the derivation. The LC RBC is completely dierent, and illustrates signi cantly degraded approximation properties for the lower order terms in the expansion. This is completely due to the simplifying approximations made in [LC93]. The improved LC RBC perfectly annihilates the two lowest order modes (l = 0 and l = 1), and is generally comparable to the other RBCs for higher modes. The capped strip boundary is entirely contained within the circle  = 5. Thus, as the boundary conditions will be applied at points closer to the origin, the computational results are expected to be somewhat less accurate than for the circular boundary. The capped strip is formed by two parallel plates with half-length a = 3 joined at each end by a semi-circle with radius b = 2 (see Figure 1(a)). The normalized error on the boundary is shown, for the rst eigenfunction (l = 0), in Figure 3(a). The boundary is parameterized in terms of the angle . Over most parts of the boundary, the MPW RBC is an order of magnitude better than the other generalized second-order RBCs. The LC RBC is of similar accuracy along the straight line segments of the capped strip. This is due to the vanishing of the curvature on these segments and hence the local Cartesian Helmholtz equation used in deriving the original LC RBC is valid. The dierences between the MPW, Kriegsmann, and KRM RBCs are due to the dierent approximations used in the derivations. Figure 3(b), containing the normalized error for the rst thirty-one modes, illustrates the potential signi cance of the dierent approximations used in the derivations of the extended RBCs. The bene ts of using the MPW RBC are most pronounced for the lowest modes. While it would be nice to have improved annihilation for higher modes, it should be noted that these modes typically do not contribute signi cantly to the scattered eld. The results presented here are representative of many other experiments ([Lic94], [PC95]). The general conclusion is that the second-order MPW extended RBCs are quite eective at annihilating single terms in the Hankel function expansion of the solution, particularly for the lower modes. For higher modes, and when the boundary is too close to the scattering center, the errors can easily exceed 100%. The other three generalized RBCs appear to suer from the additional approximations that are introduced in their derivations. The (original) LiCendes RBC suers additional performance losses from the use of the incorrect generalization of the Helmholtz equation in curvilinear coordinates. Comparable tests for the improved Li-Cendes RBC (not shown here), do not exhibit this behavior. In fact, the rst two modes are almost exactly

annihilated; the errors for higher order modes are generally somewhat larger than for the corresponding MPW RBC. To understand the relatively poor performance of the Kriegsmann RBC on the capped strip, note that the scattering center of the eigenmode is not close to the local scattering centers of the semi-circles or the segments. (The scattering centers coincide for the circular boundary.) It will be most interesting to see how this boundary condition performs in the scattering tests. Experience with numerical RBCs ([RKM91], [MPC+94], [JL94]) suggests that the distributed scattering origins used in the Kriegsmann RBC will perform well in these cases. 4. TEST 2: FEM COMPUTATIONS The mode annihilation test was very simple. A more realistic test is the problem of nding the scattered wave produced when a plane wave interacts with a long, thin scatterer. Of particular interest are the creeping and evanescent waves that develop as the truncation boundary is brought closer to the scatterer. Solutions will be compared using three separate criteria. The radar cross section (RCS) is a far- eld measure of the quality of a solution that is computed as an integral of near- eld values. This averaging may result in some insensitivity to errors in the near- eld. Thus, the RCS is a good measure of the quality of the solution in the far- eld, but not close to the scatterer. The second measure is a comparison of the magnitude and phase of the complex-valued solution. Many dierences are dicult to detect visually. It is typically better to look at the magnitude and phase of the error between the computed solution and a reference solution. Another measure of the quality of a solution is the relative Lp( ), 1  p  1, norm of the error between the computed solution and a reference solution:

Ep =

calc

u

, uref

Lp ( ) kuref kLp ( )

(39)

where is the intersection of the domains on which ucalc and uref are computed. For our rst example, consider a rectangular scatterer made from a perfect electric conductor (pec). Assume the dimensions of the scatterer are (6  0.1) and that the incident wave is a unit magnitude monochromatic plane wave arriving with an incident angle of 315 (measured from the positive x-axis; see Figure 1(a)). The reference solution for these tests is the solution computed using the second-order BT RBC on a circular truncation boundary located a distance

 = 5 from the center of the scatterer. This solution, shown in Figure 4,

was determined to be a reference solution after conducting convergence tests on the FEM solution and comparing the solution with those obtained using a method of moments (MoM) integral equation technique ([Jin93], [LRW+ 95], [Lic94], [Har68], [PRMar]); about 20 nodes per wavelength with linear nodebased elements were necessary. Note that the incident wave is partially re ected by the plate and partially diracted around the front end of the plate. The standing wave pattern above the plate can be seen in both the magnitude and phase plots. Also, the eld lines satisfy the boundary condition on the pec.

4

4

3

3

2

2

1

1

0

0

-1

-1

-2

-2

-3

-3

-4

-4

-5 -5

-4

-3

-2

-1

0

1

min = 0

2

3

max = 2.2 a) Magnitude of Ez

(a) magnitude

4

-5 -5

-4

-3

-2

-1

0

1

min = -3.142

2

3

4

max = 3.142 b) Phase of Ez

(b) phase

FIGURE 4: Magnitude and phase of the Ez eld for TM polarization, wave incident from inc = 315 , circular boundary with  = 5:0. The four generalized RBCs (Kriegsmann, LC, KRM, and MPW) are used to compute the solution for the TM polarization on the truncated domain with a capped strip outer boundary with half-length a = 3 and radius b = 2. The bistatic RCS curves (Figure 5) for the Kriegsmann, MPW, and LC solutions are all in close agreement with the bistatic RCS curve of the reference solution (which is omitted to avoid further cluttering the gure). Only the KRM curve diers signi cantly at several observation angles. Figure 6 provides a comparison of the magnitudes of the Ez eld obtained using the Kriegsmann (a) and KRM (b) RBCs. The magnitudes of the LC and MPW solutions are virtually indistinguishable from the Kriegsmann so-

25 MPW

20

Kriegsmann Li-Cendes KRM

RCS in dB

15

10

5

0

-5

-10 0

50

100

150 200 250 Observation Angle

300

350

400

FIGURE 5: Bistatic RCS values (in dB) for the (6  0.1) pec plate with incident angle inc = 315 , TM polarization; all computations performed using a capped-strip boundary with a = 3 and b = 2. lution; the phase plots are omitted because they do not add much to the understanding of this problem. Notice the eld distortions in the KRM solution on the right side of the domain where strong elds occur close to the circular section of the boundary. An eective way to use the normalized L1 errors to compare computed solutions is to consider a sequence of capped strip boundaries with successively smaller half-widths (b = 2, 1:5, 1, 0:5, 0:25) for the TM polarization. The normalized L1 errors for each of the four extended RBCs are shown in Figure 7(a). Similar results are obtained the Lp errors for p > 1; the error in the KRM solution is typically 20% larger than the corresponding error in the LC and MPW solutions. These errors typically grow as the boundary moves closer to the scatterer. This behavior is expected as neither the far- eld expansion nor the low-order eigenmodes are sucient to accurately describe this portion of the eld. The Kriegsmann solution is dierent. The Lp errors remain small until the boundary is much closer to the scatterer (e.g., the error for b = 0:25 is not appreciably larger than the error for b = 2). The same trends are observed for the TE polarization (Figure 7(b)). The ndings for the numerical example and the earlier analytical conclusions appear to be somewhat contradictory. The (original) LC RBC is the worst RBC in the eigenfunction test, but is comparable to the MPW condition in the scattering tests. The KRM RBC gives mediocre eigenmode results but has large scattering errors. The Kriegsmann RBC, which performed moderately well in the annihilation tests, surprisingly outperforms the rest in these scattering examples. One explanation of the superior performance of the Kriegsmann RBC in

5

5

4

4

3

3

2

2

1

1

0

0

-1

-1

-2

-2

-3

-3

-4

-4

-5 -5

-4

-3

-2

min = 0

-1

0

1

2

3

4

5

-5 -5

-4

max = 2.103

-3

-2

min = 0

(a) Kriegsmann RBC

-1

0

1

2

3

4

5

max = 2.42

(b) Khebir-Ramahi-Mittra RBC

FIGURE 6: Magnitude of the Ez eld for TM polarization, wave incident from inc = 315 , capped strip boundary with a = 3:0 and b = 2:0 for the (a) Kriegsmann and (b) Khebir-Ramahi-Mittra RBC.

1.2

1.2 Kriegsmann

1

Kriegsmann

Li-Cendes

1

KRM

KRM

MPW

MPW 0.8

L1 error

L1 error

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0 0

Li-Cendes

0.5 1 1.5 Radius b of Half Circle in Multiples of the Wavelength

(a) TM polarization

2

0 0

0.5 1 1.5 Radius b of Half Circle in Multiples of the Wavelength

2

(b) TE polarization

FIGURE 7: Relative L1 error in the (a) TM polarized and (b) TE polarized scattered wave from a (6  0.1) pec scatterer with a incident plane wave with inc = 315 .

this scattering test is based on a conjecture that the best way to represent an outward going wave along a straight line is by plane waves (i.e., the scattering center is at 1 or  = 0). On the other hand, for a circular outer boundary a circular wave centered at 1= =  would be preferred. While this reasoning is simplistic, it ts precisely with the application of the Kriegsmann RBC on a capped strip boundary. The ndings of the preceding example are fairly general. Similar results are obtained, for both TE and TM polarizations, on the structures displayed in Figure 1(b)(ii{iv). The reference solution is obtained as before { FEM solution using the second-order Bayliss-Turkel RBC on a circular domain with  = 5. (Since these are mixed dielectric problems, veri cation with integral equation methods is not convenient.) Consider now the (6  0.1)  pec strip covered with a 0.1  thick dielectric of relative permittivity "r = 4:0. This problem is designed so that the incident wave excites a creeping wave in the dielectric layer. The creeping wave is concentrated at one end of the dielectric layer and dissipates to the other end of the dielectric. The monostatic RCS values from 0 to 180 for capped strip boundaries of b = 2:0 and b = 1:0 are examined for TE polarization in Figure 8. The Kriegsmann RBC handles the creeping wave, which creates evanescent waves close to the scatterer, better than the MPW RBC. It could be argued that an obliquely incident wave couples most of its energy into the creeping wave in the dielectric, which guides the wave until it hits one of the ends of the guiding layer. There the wave is partly re ected, bent or radiated. Hence, most of the scattering eects in the domain are due to the radiating ends of the scattering object. Clearly, these locations coincide with the local scattering origins of the Kriegsmann RBC for the circular outer boundary segments of the capped strip. Thus, it seems that the Kriegsmann RBC is, once again, better equipped to absorb the resulting radiation modes. The results for the TM polarization are comparable. Similar conclusions can be drawn from the eld plots and the Lp errors, which are omitted (see [Lic94] for a full set of gures). The LC RBC performs very much like the MPW condition, the KRM RBC is noticeably inferior. In the next example, Figure 1(b)(iii), the pec plate is shortened, and the resulting scattering geometry is unsymmetric. The elds at the dielectric end of the structure are passing through the dielectric and gain a phase shift which results in a dierent direction of the exiting elds. Therefore, the origins of the scattered eld should not be at the origin of the coordinate system any longer. Thus, the distribution of scattering origins should be advantageous. This conjecture is further supported by the monostatic RCS plots for capped strip boundaries with b = 2 and b = 1 (see Figure 9). Consistent with previous observations, the reformulation using the Kriegsmann RBC is more accurate than the MPW RBC, particularly as the

25

25 MPW (b=2.0) Kriegsmann (b=2.0) circular BT

20

MPW (b=1.0) Kriegsmann (b=1.0) circular BT

20 15

15

10

RCS in dB

RCS in dB

10

5

5 0

0 -5 -5

-10

-10

-15 0

-15

20

40

60

80 100 Observation Angle

120

140

(a) capped strip b = 2:0

160

180

-20 0

20

40

60

80 100 Observation Angle

120

140

160

180

(b) capped strip b = 1:0

FIGURE 8: Monostatic Radar Cross Section for the (6 x 0.1)  pec scatterer covered with dielectric ("r = 4:0): capped strip domains of (a) b = 2:0 and (b) b = 1:0, circular BT RBC, MPW and Kriegsmann RBC, TE polarization. boundary moves closer to the scatterer. Again, the comparison with the other RBCs is omitted for simplicity of the gures. Similar conclusions also apply to the TM polarization. The last example, Figure 1(b)(iv), has an aperture of 2.06  in the middle of the original (6  0.1)  pec plate and operates as a rst-order Fresnel zone plate, focusing the perpendicularly incident light at a point approximately 2 behind the plate. This example is designed to test two features. First, what happens if large elds are encountered directly on the boundary (b = 2:0) or the point with highest light concentration is outside the boundary (b = 1:0). Second, the assumption of diverging waves absorbed by the boundary is violated. Actually, the wavefronts impinging on the boundary are converging towards the focal point. The plots for this test are omitted altogether. The results are, however, consistent with all of the previous examples: the Kriegsmann RBC produces the most reliable results for arti cial boundaries close to the scatterer. These nal examples reinforce the original observations in the simplest tests. And, in hindsight, it is not so surprising that the Kriegsmann RBC outperforms the other RBCs. The distributed scattering origins should actually improve the numerical solution. The fact that we have been able to obtain comparable solutions using a non-circular boundary is only half of the story. The other half is that this solution requires signi cantly less computer memory and time to compute than the solution on a circular domain. The savings reported in Table 4 are typical.

20

20

15 10 10 5 0

RCS in dB

RCS in dB

0 -5

-10

-10 -20 -15 -20 -25 -30 0

MPW (b=2.0) Kriegsmann(b=2.0) circular BT 20

40

60

-30

80 100 Observation Angle

120

140

(a) capped strip b = 2:0

160

180

-40 0

MPW (b=1.0) Kriegsmann(b=1.0) circular BT

20

40

60

80 100 Observation Angle

120

140

160

180

(b) capped strip b = 1:0

FIGURE 9: Monostatic Radar Cross Section for a (3  0.1)  pec scatterer enclosed unsymmetrically inside a block of dielectric with "r = 4:0: capped strip domains of (a) b = 2:0 and (b) b = 1:0, circular BT RBC, MPW and Kriegsmann RBC (TE polarization).

TABLE 4: Area and computational time savings for a circle and capped strip computational domain and a (6  0.1)  pec scatterer. Computational Area # Total Time 2 Domain Area/ Savings Unknowns Time (s) Savings Circle (r=5) 77.93 | 36,818 2,724.99 | CS (b=2.0) 35.97 53.38% 17,062 566.91 79.2% CS (b=1.5) 24.47 68.6% 11,650 254.88 90.6% CS (b=1.0) 14.54 81.3% 7,012 88.64 96.7% CS (b=0.5) 6.19 91.0% 3,064 17.33 99.4%

5. CONCLUSION In this paper, several common local radiation boundary conditions (RBCs) for general two-dimensional truncation boundaries were compared. The comparisons focused on the assumptions involved in their derivation and on their performance in two dierent computational settings. The rst test examined the errors produced when the boundary conditions were applied to eigenfunctions of the open space problem, i.e., Hankel functions. The second set of tests compared nite element solutions to realistic scattering problems with dierent RBCs. While the Meade-Peterson-Webb (MPW) RBC performed well in the mode annihilation experiments, it did not produce the most accurate results in the scattering tests. In the scattering tests, the Kriegsmann RBC was found to be the most accurate, even when applied close (as close as 0:25) to the scatterer. The dierences in the derivations of these dierent RBCs suggests possible explanations for these observations. The accuracy of the Kriegsmann RBC, which is based on very simple ideas, is largely due to the distributed scattering origins; all of the other RBCs use a common scattering origin for the entire truncation boundary. The superior performance of the Kriegsmann RBC is observed even in the presence of creeping and evanescent waves. This performance means that the truncation boundary can be typically be placed within one wavelength of the scatterer without creating unreasonable errors. The computational savings observed from the reduction of the size of the computational domain exceed 90%, compared with a solution computed on a circular boundary. A. Local Coordinate Systems The extension of an RBC on a circular truncation boundary to non-circular truncation boundaries is typically based on a coordinate transformation. The RBCs discussed in this paper can be realized by a change of variables into either a local Cartesian coordinate system (Section A.1) or into Fermi coordinates, that is, in terms of the arclength and normal distance from the boundary (Section A.2). A.1. Local Cartesian Coordinates The rst approach corresponds to a simple translation and rotation of the given coordinate system. The equations which de ne this transformation are easily seen to be (see also, Figure 10(a)):

n = (x , x0) cos ~ + (y , y0) sin ~

(x , x0) sin  , (y , y0) cos  (40)  sin( , ) , 0 sin( , 0)  sin( , ) , a; ,(x , x0) sin ~ + (y , y0) cos ~ (x , x0) cos  + (y , y0) sin  (41)  cos( , ) , 0 cos( , 0 ) qcos( , ) , b; (n + a)2 + (t + b)2; (42)  =  , arctan nt ++ ba (43) where 0 , 0 and  are considered constant within each local coordinate system (only  and  are allowed to vary). This technique assumes no knowledge of the entire boundary, but instead assumes that the boundary is locally linear. Then, the discontinuities between line segments are ignored; this assumption is reasonable when the curvature is not too large, i.e., the angle between segments is close to 180 . A.2. Fermi Coordinates Fermi coordinates are based on a coordinate system de ned in terms of t, the arclength (measured counterclockwise from the positive x-axis) along @

and n, the distance from @ , measured along (outward) normal vectors to @ . Assuming the domain, , is smooth and convex, this is a global change of coordinates in the complement of . Let @ = f(X (t); Y (t)) : 0  t  Lg, with L denoting the length of @ , be the arclength parameterization of @ (see Figure 10(b)). The direction vectors t^ and n^ can be expressed as: = = = t = = = =  =

t^ = n^ =

" "

#

"

#

X 0(t) = cos (t) Y 0 (t) sin (t) # " # Y 0 (t) = sin (t) : ,X 0(t) , cos (t)

(44) (45)

Recall that X 0 (t)2 + Y 0(t)2 = 1, since t is arclength. Any point (x; y) in the exterior region can now be expressed as

x = X (t) + nn^ x = X (t) + nY 0(t) y = Y (t) + nn^ y = Y (t) , nX 0 (t):

(46) (47)

All derivatives needed to complete the change of variables can be obtained from these equations.

These two coordinate systems yield very similar rst-order results (e.g., n ) on the boundary (n = 0). Higher order expressions, which are essential to the derivation of the generalized second-order BT RBC proposed by Meade et al. ([MSPW95], [MPW93]), are noticeably dierent. This is most apparent in the natural way in which curvature, , is incorporated into the Fermi coordinates (viz., the Jacobian of the change of coordinates de ned in (46) and (47) is Q(t; n) = 1 + n(t)). y

y (x,y) = (t,n) = (ρ,φ)

y

(x,y) = (t,n) = (ρ,φ)

y n

n ∂Ω

t

y0 ρ0 φ0

θ

^ n ∂Ω

~ θ

AAA AAA

(a)

θ(t)

Y(t) t

t

ρ φ

domain Ω

x x0

^t

x

AAA

t

domain Ω

X(t) x

x

(b)

FIGURE 10: Transformation to local coordinate system using (a) local tangential and normal directions (, 0 and 0 are constants) and (b) a parameterization of the outer boundary ( is a function of t). References [BGT82] A. Bayliss, M. Gunzburger, and E. Turkel. Boundary conditions for the numerical solution of elliptic equations in exterior regions. SIAM J. Appl. Math., 42(2):430{451, April 1982. [BT80] A. Bayliss and E. Turkel. Radiation boundary conditions for wave like equations. Comm. Pure Appl. Math., 33:707{725, 1980. [EM77] B. Engquist and A. Majda. Absorbing boundary conditions for the numerical simulation of waves. Mathematics of Computation, 31(139):629{651, July 1977. [EM79]

B. Engquist and A. Majda. Radiation boundary conditions for acoustic and elastic wave calculations. Comm. Pure Appl. Math., 32:313{357, 1979.

[Giv91]

D. Givoli. Non-re ecting boundary conditions. J. Comput. Phys., 94:1{29, 1991.

[Har61]

R.F. Harrington. Time-Harmonic Electromagnetic Fields. McGraw-Hill, New York, 1961. [Har68] R.F. Harrington. Field Computation by Moment Methods. The Macmillan Company, New York, 1968. [Hig86] R.L. Higdon. Absorbing boundary conditions for dierence approximations to the multidimensional wave equation. Mathematics of Computation, 47(176):437{459, Oct. 1986. [Jan92] R. Janaswamy. 2-d radiation boundary conditions on an arbitrary outer boundary. Microwave and Optical Tech. Lett., 5(8):393{395, July 1992. [Jin93] J. Jin. The Finite Element Method in Electromagnetics. J. Wiley & Sons, New York, 1993. [JL94] J.O. Jevtic and R. Lee. A theoretical and numerical analysis of the measured equation of invariance. IEEE Trans. Antennas Propagat., 42(8):1097{1105, Aug. 1994. [JVL89] J.-M. Jin, J.L. Volakis, and V.V. Liepa. A comparative study of the osrc approach in electromagnetic scattering. IEEE Trans. Antennas Propagat., 37(1):118{124, Jan. 1989. [Kar61] S.N. Karp. A convergent far- eld expansion for two-dimensional radiation functions. Comm. Pure Appl. Math., 14:427{434, 1961. [KM89] A. Khebir and R. Mittra. Absorbing boundary conditions for arbitrary outer boundary. In IEEE Antennas and Propagation Society, editors, AP-S International Symposium (Digest), pages 46{49. IEEE, IEEE Service Center, Piscataway, NJ, USA, June 1989. [KRM89] A. Khebir, O.M. Ramahi, and R. Mittra. An ecient partial differential equation technique for solving the problem of scattering by objects of arbitrary shape. Microwave and Optical Technology Letters, 2(7):229{233, July 1989. [KTU87] G.A. Kriegsmann, A. Ta ove, and K.R. Umashankar. A new formulation of electromagnetic wave scattering using an on-surface radiation boundary condition approach. IEEE Trans. Antennas Propagat., 35(2):153{161, Feb. 1987. [LC93] Y. Li and Z.J. Cendes. Modal expansion absorbing boundary conditions for two-dimensional electromagnetic scattering. IEEE Trans. Magnetics, 29(2):1835{1838, March 1993.

[Lic94]

B. Lichtenberg. Finite Element Modeling of Wavelength-Scale Diractive Elements. PhD thesis, Purdue University, School of Electrical Engineering, West Lafayette, IN 47907, December 1994.

[LLRW94] B. Lichtenberg, Y.-S. Liu, J.S. Reynolds, and K.J. Webb. Applications and performance of a local conformal radiation boundary condition. In Digest of the IEEE Antennas Propagat. Society International Symposium, Seattle, WA. IEEE Antennas Propagat. Society, IEEE Service Center, Piscataway, NJ, USA, June 1994. [LRW+95] B. Lichtenberg, J.S. Reynolds, K.J. Webb, A.F. Peterson, and D.B. Meade. Numerical study of a conformal two-dimensional radiation boundary condition. submitted to IEEE Trans. Antennas Propagat, April 1995. [MBTK88] T.G. Moore, J.G. Blaschak, A. Ta ove, and G.A. Kriegsmann. Theory and application of radiation boundary operators. IEEE Trans. Antennas Propagat., 36(12):1797{1812, Dec. 1988. [MPC+94] K.K. Mei, R. Pous, Z. Chen, Y.-W. Liu, and M.D. Prouty. Measured equation of invariance: A new concept in eld computations. IEEE Trans. Antennas Propagat., 42(3):320{328, March 1994. [MPW93] D.B. Meade, A.F. Peterson, and K.J. Webb. Radiation boundary conditions for the vector helmholtz equation. In Digest of 1993 URSI Radio Science Meeting, Ann Arbor, MI, page 255. URSI, June 1993. [MRK+89] R. Mittra, O.M. Ramahi, A. Khebir, R. Gordon, and A. Kouki. A review of absorbing boundary conditions for two and threedimensional electromagnetic scattering problems. IEEE Transactions on Magnetics, 25(4):3034{3039, July 1989. [MSPW92] D.B. Meade, G.W. Slade, A.F. Peterson, and K.J. Webb. Analytic evaluation of the accuracy of several conformable local absorbing boundary conditions. Digest of the IEEE Antennas Propagat. Society International Symposium, Chicago, IL, 1:540{ 543, July 1992. [MSPW95] D.B. Meade, G.W. Slade, A.F. Peterson, and K.J. Webb. Comparison of local radiation boundary conditions for the scalar helmholtz equation with general boundary shapes. Trans. Antennas Propagat., 43(1):6{10, Jan. 1995.

[PC95]

C. Piellusch-Castle. The derivation, implementation, and comparison of arti cial boundary conditions for the helmholtz equation. Master's thesis, University of South Carolina, Department of Mathematics, Columbia, SC 29208, August 1995. [PRMar] A. F. Peterson, S. L. Ray, and R. Mittra. Computational Methods for Electromagnetic Scattering. IEEE Press, Piscataway, NJ, to appear. [RKM91] O.M. Ramahi, A. Khebir, and R. Mittra. Numerically derived absorbing boundary condition for the solution of open region scattering problems. IEEE Trans. Antennas Propagat., 39(3):350{353, March 1991.