ELABORACIÓN DEL NÉCTAR

A n a tu ra l c a n d id a te fo r a n ite ra tiv e s o lu tio n of th e d is c re te rep resen tatio n of the MFIE is the N eu m an n expansion u sed b y H olliday (1985), an d H o llid ay et al (1987). H ow ever, alth o u g h the ex p an sio n h as b een used to form ally rep resen t the solution to the MFIE (Brown, 1982), th ere is no p ro o f th at the expansion, eith er of the MFIE or its discrete rep resen tatio n converges. The convergence or otherw ise o f th e discrete case cannot p ro v e the convergence or otherw ise of the contin u o u s case

an d vice versa. O n the o th er h a n d , the failure of the d iscrete case to converge w o u ld p ro v id e stro n g evidence th a t the convergence of the continuous case was n o t generally true. M oreover, if the convergence of the discrete case is unsure, it w ould be better in num erical w ork to replace it w ith an iteratio n w hose convergence w as certain. We have exam ined the convergence an d rate of convergence of the N e u m an n ex p an sio n ap p lied to the discrete rep resen tatio n of the MFIEs for G aussian, ro u g h surfaces. We have fo u n d th at w h e n the surface stru ctu re is of the sam e dim ensions as the electrom agnetic w avelength, the series diverges rapidly.

To solve the MFIE num erically, (11) is approxim ated w ith the discrete e q u atio n

1^1

(3.1)

2H ^(xjJ - J(xn) + ^ K (^n/^m ) J(^m)* m = 0

The m atrix K is b ounded, so for every J there is a positive constant a such that (Stakgold, 1979),

"N -1 N -1 2" 1/2 ■ N -1 1/2 I % K(xn,Xji^) J(xm) < a % ( J(^m) /

- n= 0 m = 0 _ _- m —0 _—

= I lig i I < a l IJI I. (3.2)

In the N eu m an n expansion the solution J(xj^) of the m atrix-equation (31) is the lim it of the sequence Jk, k = 0 ,1 ,..., «>, obtained from the iteration

N-1

Jk + l(^ n ) = 2H^(xn) - % K(xi^,Xm) Jk(xm ) m = 0

(3.3)

The iteration converges if a < 1. This is true for arbitrary Jq (Kreysig, p. 375,

1978). Furtherm ore,

T herefore, a n o rm of K less th an u n ity is a sufficient co n d itio n for convergence. The only algorithm ic m eth o d w e are aw are of to determ ine the no rm of K d irectly is to d eterm in e its sin g u lar values, (K is n o t H e rm itia n sy m m etric), (S ch illin g a n d Lee, 1988), (W ilk in so n a n d R heinsch, 1978). N um erically, this requires essentially the sam e effort as com puting the inverse of K; if it w as easy to determ ine a-priori the n o rm of K w e w o u ld n o t require an iterative solution to (31).

We n eed a step -b y -step m eth o d of id en tify in g divergence. A t each iteration w e substitute Jj^ in (3-3) to generate the quantity

N-1

2Hk(xn) = Jk(^n) **■ X Jk(^m)* (3*5) m=0

We then form the norm alized error

We will show th at satisfies the inequality

e k ^ a k e Q , (3-7)

an d if the iteratio n (3 5) is initialised b y setting Jq = 2H^, Eq satisfies th e

inequality

£q < a . (3-8)

The in eq u alities (3 7) an d (3-8) p ro v id e sufficient step -b y -step tests for divergence. If > Eq, or Eq > 1, then a > 1 an d the iteration diverges. The

inequalities (3*7) an d (3 8) are obtained as follows. From (3*6), (3-3) an d (3-5) w e have

i hV But, from (3 3) l l J k , i - J k N = I I K ( J ^ - J , . l ) l l < a I I T k - L l ' ' = a I I K O ^ . i - W I I ... S a k | | J j - J o l l , (310)

an d (3 7) then follows by reusing (3-9). We have from (3 9) and (3 3) that if

Jo = 2Hi s o ^ '. 'J r i q L L 21 i r f l l I l2H> + Jo+igol I 21 IHfl l I IKHM I I I l f II (311)

an d (3*8) follows using (3 2). From (3 9) it is also ap p aren t th at £|^ does n o t m easure the closeness of Jj, to the solution J(xn). H ow ever, ej^ 0 w hen Jj^ -> J(xn), and w e take the sm allness of £j^ to indicate th at Jj^ is close to the solution of (31)

In fig. 3-1 w e show the norm alized error £j^ generated b y the N eu m an n ex p an sio n a t each iteratio n , for a v ertically p o larize d , electrom agnetic w a v e (2-3), n o rm ally in cid en t o n a G aussian ro u g h surface. The figure

show s fo u r cases w ith the sam e co rrelatio n -len g th of 0-4X, b u t w ith d iffe re n t RMS h e ig h ts. The RMS slo p e of the surface is given by arctan(V 2a/Ç ); and (A) illustrates a RMS slope of 20°, (B) a RMS slope of 25°, (C) a RMS slope of 35°, and (D) a RMS slope of 45°.

1 0 .0 -q o L_ L_ <D “O <D N “Ô

E

0.01

30

20

10

0

Fig. 31. The convergence of the Neumann expansion. The graph shows the normalized error with the number of iterations k. The correlation-length is 04 wavelengths and (A) the RMS slope is 20°; (B) the RMS slope is 25°; (C) the RMS slope is 35°; and (D) the RMS slope is 45°.

T hree curves, (B), (C) an d (D) clearly diverge. W ith a < 1 one of the cases, curve (B), satisfies the inequality (3-8), b u t fails at the second iteration to satisfy the in eq u ality (3 7), the rem ain in g tw o fail to satisfy b o th the in eq u a lity (3-7) an d (3-8). O ne of the cases curve (A) does ap p aren tly converge. M oreover the convergence is rapid; the norm alized erro r is less th an 0.01 w ith in 13 iterations.

T he RMS slo p e is clearly a factor in d e te rm in in g w h e th e r th e expansion diverges, b u t we have found that the rate of divergence depends u p o n the surface correlation-length too. In fig. 3-2 w e show four cases w ith a correlation-length of 0-8X. The RMS slope of curves (A) - (D) in fig. 3-2 is the sam e as curves (A) - (D) in fig. 3 1 .

10.0-q

0.01

0

10

20

30

number o f iterations

Fig. 3 2. The convergence of the Neumann expansion. The graph shows the normalized error with the number of iterations k. The correlation-length is 0*8 wavelengths and (A) the RMS slope is 20®; (B) the RMS slope is 25®; (C) the RMS slope is 35®; and (D) the RMS slope is 45®

In fig. 3*2 tw o of the curves, (C) and(D) clearly diverge, b u t a t a rate w hich is m arg in ally slo w er th a n curves (C) a n d (D) in fig. 3-1. The a p p a re n t convergence of curve (A) in fig. 3*2 is rap id , an d m arginally faster th an curve (A) in fig. 3 1 . The expansion also apparently converges in fig. 3-2(B), w hereas in fig. 3-1(B) it does not. H ow ever, since fig. 3*2(B) m arginally fails at the second step to satisfy the inequality (3-7) w ith a < 1, w e suspect th at it

w ould diverge if w e took sufficient iterations.

W e h av e sh o w n th at w h e n the surface stru c tu re is o f the sam e dim ensions as the electrom agnetic w avelength, the N e u m an n series m ay d iv erg e ra p id ly . The re stric tio n n o rm ally p lac ed o n th e N e u m a n n ex p an sio n , i.e. g / X « 1 and a /Ç « 1, are lim itations th a t p e rm it us to ignore all b u t the first tw o term s. In ou r w ork here, w e have concentrated on cases w here the RMS h eig h t an d correlation len g th are o f the sam e order, an d of the sam e order of the electrom agnetic w av elen g th , because w e anticipated th at this w o u ld be a region of the p aram eter space w here th e N e u m a n n ex p an sio n m ay hav e difficulties co n v erg in g . W e hav e fo u n d th a t the exp an sio n m ay p ro v id e a rap id n u m erical so lu tio n for sm all values of a /^ .a n d a. To the extent th at the num erical representation is a good approxim ation to the MFIE (1-1), w e also consider th at our results p ro v id e stro n g evidence th at the N e u m an n ex p an sio n can n o t be u sed w ith o u t qualification to p ro v id e a form al solution to th e ro u g h surface MFIE.

3*2 The conjugate-gradient method, and avoiding rounding errors