DTIC ADA263113: An Iterative T-Matrix

8 Pages·0.74 MB·English

by  Defense Technical Information Center

*°" t" fogA Dl -A 263 113 IPONNMPAvGXl""EHV 011tI*,.lMl" v Form
___pvd
~
arft 0"i --W* &- PMPiit '1 "4MW"'0f*V-1 SAO M A. VV VA I2r1U 4 Xg'
1. Agoi t R*Pon Typo and~ Deas Covoerd
1943_Fi~nal Proceedings
4. Thie and Subtitle. S Funding Humbe,
An hterative T-Matlix Ca.uw
S(cid:127),(cid:127)g.,(cid:127)0"W" , 0601 153N
6. Auiho s()- 4111111M * e30
ELECTEE k,n 3!ýo
M. F. Werby and Guy V. Norton
APR 14 1993 Aixi':"'"W O5nN'
1?221183
___
7. Performing Organization Nae(i) and Address(**). m , Org ,,,zdton
Naval Research Laboratory GRepor Numbt,
Center for Environmental Acoustics PR 91 098o221
Stennis Space Center, MS 39529-5004
9. Sponforng/Monitlorng Agency Name(s) ael Address("). 10. Spon*orlofg1onhortng Agwc
Naval Research Laboratory Rpo Numbof
Basic Research Management Office
Stennis Space Center, MS 39529-5004 PR 91,098221
11. Supplemientary Notes.
Published in IMACS, Computational Acoustics.
12s. Ofltlbution/Avallability StatemrnL 12b. Dist, Ibutton C'W..
Approved for public release; distribution is unlimited.
13. Abstract (Maximum 200 words).
The T-matrix method, a technique invented by Peter Waterman, (1965, 1969), 1977, 1980) has proven to be valuable for
calculating classical scattering from bounded targets. The method is useful for acoustical, electromagnetic and elastic scattering
and is based on coupled boundary integral equations and partial wave expansions of relevant physical quaniltins Upon irun(al",n'
of the partial wave expansion, which are in principle infinite, a system of equations is derived that can be expres-ed in terms of ftir4n
matrices. This system in turn leads loan expression that maps the incident field onto the scattered field. The outcome is a powerful
algorithm that yields essentially exact results. The point at which the partial wave series is truncated is always an issue in using the
method. INto o few terms are Included, the results will be very inaccurate. Conversely. it many terms beyond that required for
convergence are included, at best the calculation will be needlessly time consuming and, at worst, can lead to numerical instability
due to numerical round-off errors. What would be ideal is a technique that leads to optimum truncation, that is, the point just beyond
which converqnnce is achieved. This type of technique can be obtained by a method based on starting with a "smalr t-matrix and
in which we build up the dimensions term by term via simple vector operations until convergence is achieved. The algorithm is
described and results are illustrated for somn examples
\ui93-07692
ij ,
S'
¾U
14. Sublect Terms.
Acoustic scattering, shallow water, waveguide propagation
1, Price Cod*.
17. Security Clesilflca(cid:127)lon 16. Security Clasilfication 19. Security Classification 20. Umltallon of Abstract.
of Report. of This Page. of Abstrsict.
Unclassified Unclassified Unclassified SAR
4iN 7Zi40-0-2WO-5W0 Standarr Form 7A ifA-v 2-- 9
Pivirusc by ANSI SMdZ tilU
2WS-IO
A
"
k,
ISSN 1067-0688
'oK
Volume 2,1 993
Section B
icalMo dellfing
~4S~it~e
ýan
iYthfftihia edI nera oM:Asoiatonfor Mathematicl ad Computer Modelling
'
-w~
iV.
S V '
.' .
~I
P5I 4w~ ,
THE ..EIGHTH, INTERNATIONAL ýCONFERENCE
44~4
~cll~ge -PArk,Marylandi Ui. S:A pfl19
Reproduced From
,
Best Available Copy t-i~A 4 ~~ '''
~ ~
N - V. A 4,-ývsIi4
- *-SMO.
V' Io
""t
A .~'-f '
Mi h m odell Jfl So. compufl,, VW.. 2p.F..P... 9L ,.
P"Anpus Soeigla Pn**ed t. U S.A
NTIS CRA&I
IDTIC TAB U
c
I U~n"ou~t'cet7
AN JURATIVE T'-MA13G\X t U
hl, F. WERBY and GUY V. NORtOONN Ot'tb onf
I Avd~tbiti Codles
Naval Oceanographic and Atmosphen Re-s,,anh Lvw(cid:127)r-,,u N'--1" i
Theoreucal Acoustics, Stennis Space Center, NS 34529Q.5[ix Avai a(1 I
w\
ABSTRACT
The T-matrix method, a technique invented by Peter Waterman, (1965. 1969, 1971, 1977. t990 has prroven to be
vaJuable for calcuiaung classi.tl scattering from bounded targets. The method is useful for acoustical,
electromagnetic and elastic scattering and is based on coupled boundary integral equations and partial wave
expansions of relevant physical quantities. Upon truncation of the partial wave expansion, wh"hh are in pnnciple
infinite, a system of equations is derived that can be expressed in ferms of finite matrices This s)sitrm in turn leads
to an expression that maps the incident field onto the scattered field. The outco"e is a powerful algonthm that
yields essentially exact results. The point at which the partial wave senres is truncated is always an issue in using
the method. If too few terms wae included, the remsuts will be very maccurait Convusely. if many terms beyond
that required for convergence we included, at best the calculation wilt be needlessly time consuming and, at *woE,
can lead to numerical instability due to numencal round-off errors, What would be ideal is a technique that leads to
optimum truncation, that is. the point just beyond which convergence is achieved This type of technique can he
obtained by a method based on starting with a "small' T-mauix and to which we build up the dimensions term by
term via simple vector operations until convergence is achieved. The algonithm is described ad result(cid:127)s are
illustrated for some examples.
KEYWORDS
T-mao'ix: parti wave functions: convergence; acoustic. scattering.
INTRODUCTION
The appropriate number of basis functions must be used when generating a T-maurix-. if too few ae used, the T-
matrix will not converge (Waterman, 1965, 1969, 1971. 1977, 1980; Wall. 1980; Werby and Chin-Bing, 1985;
Werby, 1991). When generating a T-matrix for well-known shapes, a few rules of thumb usually pertain to the
number of basis functions to use, but not for arbitrary shapes. In the past if the T-matrx was insufficiently exact.
the T-matrix would have to be regenerated with more terms. This regeneration. however, did not guarantee that the
new T-matrix would converge. With this in mind, an iterative algorithm is created so that there will be no
guessing as to how many basis functions is needed to ensure convergence. This paper isd ivided into the following
sections. We will present the theory behind the procedure (starting with an nth order T-matrix). describe what
occurs at convergence, and show some numerical results.
THEORY
The iterative procedure is based on the method of bordering (faddeeva. 1959). In general, we start with a (ni n)
matrix T. Another matrix of order (n+l x n+l) can be constructed based on the original matrix T. It is thus
possible to build any order matrx by this procedure, since the expanded matrix T order (n+ I x n+I ) can be used as
the next staring point to obtain the matrix T of order (n+2 x n+2). Let us then begin with an nth order T-marnx
expressed ns
whi re Re mean; the regular part of Q.
792
The marix Q in general has the fullowing lom,
II
9Ace
)i the hasii (unction and k 1dich % owrC?'mr 'A< .Ul n .01 cll h
.0
Re .....k i J Rea d's
wiLh a Neumann txouMdary codUition or
ReQ,, k~ Re q, ~etd
-an
I
with a Dinchlet boundary condiuon. In general. if we exttand the T-matrix of owder In • i tt An Xdo(cid:127) of n- I -X
n* 1), Eq. (1) becomes
Tn L Q 1 1n 1 n
where f is a row veco and{ Z is a column vecor Wl %n+IA ÷| is scamla, Now is" !e xpresse as
where P is an (n x n) matrix. (cid:127) and i are vectors and b(cid:127)+ IA+ is a scalar. They have the folloA ng exrrssiorm;
P n.,n÷
ý- I n-,
nnI
Sbn+QC,nn+R
'=b n,, R+l,(cid:127) (cid:127) n+{.n ! - 4,.1 Q n;,nC n+ l.
9 pr'oc. of the 8th
Finjilly awn wntc 0q ais
ýAe
T Re j
_ .i .'..
bn.l~n+t bn 1.n÷-I
0 3 = - R e .l Q n I E lbl n~l,n.I R _ r h_n (cid:127),_l ,n_ (cid:127).l __n _ ..
Re i.+ Qn-'. Z,.. Re a,,.,,.,
0l4 = .- .
bn~lnl nl,n+l
Using this algorithm has several advantages. As already mentioned the algorithm takes the
guesswork out of choosing the number of basis functions. The T-macrx is conditioned as it is built: this zechnique
relaxes some of the complications associated with ill.conditioning. which is sometimes encountered for various
objects. Finally, inspection of the theory shows that the addition of additional columns and rows is an operation
involving either a matrix and a vector or two vectors, which if performed on a vector machine would be fast and
economical.
DISCUSSION OF CONVERGENCE
Partial wave series salutions of scattering and rearrangement collisions have long been used in quantum scattering.
There are two main reasons for doing so. First. ills much easier to solve for each of the partial wave soluvons.
which reduce from a three-dimensional problem to that of a seies of one-dimensional problems for central forces, or
in general for problems in the which angular contribution can be separated from the radial part. Second. convergence
is generally expected at some value of angular momentum due to the centrifugal repulsion effect. What this means
is. that for some value of ka where k is the wave number and a is some characteristic dimension of the object (the
radius for spheres, for example), at some point the centrifugal contribution, which occurs in a partial wave solution
(namely I(1+1)/r2). is greater than the component (ka)2. Thus the contribution from that centrifugal term is
repulsive and causes little overlap with the target when it dominates.
Figure I illustrates this point for several incident partial waves. As evident from the figure, when the partial wave
number increases, there is a resulting decrease in the overlap with the target. For suitably low values of I. the ka
component dominates, so the interaction region is significant. The question for what value of I the centrifugal
component will dominate has been answered for spheres by using the WKB approximation. It is generally agreed
that a value of 1=2 ka is large enough for the series to cut off. For spheroids the situation is more complicated,
since the system of partial wave solutions are coupled and since the overlap of the incident partial waves with the
surface displacement terms is rather complicated
For the spherical case them is mode coupling of the surface boundary terms with each partial wave, and this corrupts
the simple picture we have for spheres. Figure 2 illustrates the case of convergence for several spheroids with
varying aspect ratios that deviate between 2 and 18 for a kL.2 fixed at 6, where L is the object length and k the total
wavenumber (k=2WrA). We use a low accuracy T-matrix code so that numerical difficulties will be accentuated. For
low aspect ratio targets, convergence is maintained with increasing partial wave terms. As the number of partial
wave terms increase, more terms are required for convergence because of an overlap effect with the surface expansion
terms (Wery and Chin-Bing, 1985; Werby. 1991). Further, the point at which the answer is no longer accurate is
due to ill-conditioning that results from high aspect ratio targets which occurs earlier; thus, the "window of
convergence" narrows and makes the problem of convergence more urgent with increasing aspect ratio. The method
outlined in the previous section not only obviates the problem of convergence but also appears to condition the
matrix as well. Alternative methods exist that allow problems to be solved based on the coupled equations of
Art I Ii a I 'IM' itt l i
Waterntwn for clistic urgei~'s 4crbý and Gfee n. N83i asd It ~n tlc W IC X4 C1,:'Z a ;t-X1 r! 'A'
et al .1986) The problem of a target in A%a vC guide has tbeen diuAAed eenwtci's ji(cid:127) . r u IfCk. t ', ,
and W'erbt € I)'9
(cid:127)/5 10
12o
Fag. I, Compaitlhn of tit overlap wofh ttahreg et. foh seve-l incident
is a
parbtal rewvos.
At L.1*0 zL " 00 "L •1
foe
APLIATO TO RIGDARET
S 10 12 14 l1i II U Z2 24 21 2I
Fig. 2. Form function vs. mnatria order for various aspect ratio targets.
• L is the length and D is the diameter of the target. The region
)i Ther ar clse oft.resfrbipen.tt.r.w al een arrows represepnrtb est he 'windioe,w soofr caonn vehragre ncsec."attrers Tedont suprtbd
(cid:127) APPLICATION TO RIGID TARGETS
,(cid:127) There are two classes of targets for impenetrable problems. i.e., soft and hard scatierers. They do not support body
resonances; therefore, we examine acoustic quantities appropriate for such non-resonant targets as circumferentially
diffracted, or creeping, waves. These waves arise when scattering end-on from a spheroid in which one the return
signal is observed at the origin of the signal, There are two competing mechanisms. One arises from specular
scattering (geometric) and the other arises from the creeping waves. The result is a coherent effect in which the two
Wjleas i ada at So mlPeo i lmh
a Mii
umn~g (he dunenmU~f Ahrn th Ing (0a
Mor P,Olleu nmer, 'X(U t~j
C:1
C(b
200'o
Fig 3. Backatero
b8to1; and c) aI"zridj~(e of f1.,
"Biseatic angula r di90b~ 0 ~ ~1i5 0.''
Wit) relative 1 soumwn toe of a
4b rati01 Of 301 011: me oreb' menOasct
nethartw tho O90bd5je -e- CShU5fr"ve at ~PC Oethf eltIssy s cat1tU a fSro ( e th ob~je wca eeoxa -m-ednee ia.~ 1ý`9Y1 it spheirno sipda ocfe tr smCI. 7 .(dent~~
1
a~re Y prx~ mtos~a p tu ISs in the ine)~)C in Fig.4n-n n i
rwar sappiny- In allrurs.~ freq. ltOehh hIgh00whc
~Z~t~gdiaec
cfly is Siffil MIY region" where
s' Cienllty hihthat wave
Proc. of the 8th ('cm
90" 900
•-4 7 -2S 1800 00 3 -26 f
\(cid:127) __ ..- (b)
270* 270"
Fig. 4. a) Bistatic scattering from 30 to I aspect ratio rigid spheroid
end on; b) broadside incidence for klJ2=200.
SUMMARY
We have described an iterative method that produces an (n+m x n+m) matrix from an (n x n) matrix. With the u(cid:127)e
of an ilustration we described what must occur in order to produce convergence. Finally, we presented examples of
scattering rcsults achieved using this technique.
"REFERENCES
Faddeeva, V. N, (1959). ComputationalM ethods ofrLinear Algebra. Dover Press, New York.
Norton, G. V. and M. F. Werby (1991). A numerical technique to describe acoustical scattering and propagauon
from an object in a waveguide, submitted to J. Appl. Physics.
Wall, D. N. (1980). Methods of overcoming numerical instabilities associated with the T-matrix method. In:
Acoustic, Electromagnetic.a nd Elastic Wave Scattering-Focw on the T-matrnx Approach (V. K. and V
V. Varadan, eds.), p. 269. Pergamon Press, New York.
Waterman, P. C. (1965). Matrix formulation of electromagnetic scattering. Proc. IEEE, 51 (3). p. 802.
Waterman, P. C. (1969). New foundation of acoustic scattering. J. Acoust. Soc. Am-, 4.5, p. 1417.
Waterman, P. C. (1971). Symmetry, unitarity, and geometry in electromagnetic scattering, Phys. Rev. D3, p. 825
Waterman, P. C. (1977). Matrix theory of elastic wave scattering 11: a new conservation law, J. Acoust. Soc Am.
63 (6), p. 1320.
Waterman, P. C. (1980). Survey of T.matrix methods. In: Acoustic, Electromagnetic, and Elastic Wave Scattering-
Focus on the T-matrix Approach (V.K. and V.V. Varadan, eds.), p. 61. Pergamon Press, New York.
Werby, M. F. and L. R. Green (1983). An extended unitary approach for acoustical scattering from elastic shells
immersed in fluids, J. Acoust. Soc. Am.. JA (2). p. 625.
Werby, M. F. and S. Chin-Bing (1985). Numerical techniques and their use in extension of T-matrix and null-field
approaches to scattering, Int. J. Comp. Math. Appls., 11 (7/8), p. 717.
Werby, M. F., G. Tango and L. H. Green (1986). Eigenvalue and matrix transform methods in the solution of
acoustical scattering problems from submerged objects. In: Proceedings, Ist IMACS Symposium on
Computational Acoustics, (D. Leem R. Steinberg, and M. Schultz, eds.) Vol. 11,p . 608. North-Holland.
Werby, M. F. (1991). A coupled higher order T-matrix, Accepted for publication in I. Acoust. Soc. Am.