12-10
A SURFACE MODELING PARADIGM FOR ELECTROMAGNETIC
APPLICATIONS IN AEROSPACE STRUCTURES
RA4 J h a , SA Bokhari, V Sudhakar and Pi7 Mahapatra
D e p a r t m e n t of Aerospace Engineering
Indian I n s t i t u t e of Science
Bangalore, 560 012 India
ABSTRACT
A systernatic approach has been developed to model t h e s u r f a c e s e n c o u n t e r e d
in a e r o s p a c e engineering f o r EM applications.
The basis of this modeling is
t h e quadric canonical shapes which a r e t h e c o o r d i n a t e s u r f a c e s of t h e Eisenhart
C o o r d i n a t e Systems. The building blocks a r e visualized a s sections of quadric
cylinders and s u r f a c e s of revolution. These t r u n c a t e d quadrics can successfully
rnodel realistic a e r o s p a c e s t r u c t u r e s which a r e t e r m e d a s hybrid quadrics, of
which t h e s a t e l l i t e launch vehicles a r e illustrative exaniples.
Introduction
The shapes of aerospace s t r u c t u r e s a r e primarily governed by aerodynamic
considerations. The most c o m m o n body shape 15 cylindrical, with a conical
s t r u c t u r e in t h e front, t o effectively p e r f o r m t h e fluid-piercing function.
However, d u e t o payload and other logistical considerations, innumerable o t h e r
variants of t h e basic cone-cylinder conliguration a r e e n c o u n t e r e d i n p r a c t i c e .
Similarly, blunting t h e nose tip regulates t h e shock and h e a t transfer
c h a r a c t e r i s t i c s . A corribination of t h e s e r e q u i r e m e n t s may lead t o a shape
rescriibliiil; c . l o \ c l y ;iii < ' I l i l ) ~ o i d h i\ I< , i ' t i l r i i cir .i ~ ~ L ~ ~ .of~ ilr c~> voi > l lu t~i o i ~
i. ~ d
Similarly, itlost appendages t o t h e tliglit vehicles, like t h e wings and e x t e r n a l
control s u r f a c e s may b e closely approxiniated a s hybrid s u r f a c e s consisting of
s e g m e n t s with single curvatures. Such s e g m e n t s a r e typically derived from t h e
circular, elliptic and parabolic cylinders. I t is a p p a r e n t t h a t ;i vast majority
of a e r o s p a c e s t r u c t u r e s can be conveniently modeled ab coini,iilritlon> of simpler
quadric surfaces.
The work p r e s e n t e d in this paper c o n c e r n s t h e generalized solution of
s u r f a c e ray t r a c i n g problems associated with simple and hybrid s u r f a c e s of t h e
t y p e e n c o u n t e r e d in a e r o s p a c e vehicles and satellites. T h e primary motivation
f o r this work c o m e s froni t h e p r a c t i c a l n e e d t o develop a corriprchensive
t h e o r e t i c a l approach to t h e high frequency analysis of a n t e n n a c h a r a c t e r i s t i c s
for t h e a n t e n n a s l o c a t e d on r e a l i s t i c a e r o s p a c e s t r u c t u r e s .
Analytical Surface Generation and Surface Modeling
The quadrics most commonly encountered a m o n g t h e a e r o s p a c e shapes a r e t h e
quadric cylinders (QUACYLs) and t h e quadric s u r f a c e s of revolution (QUASORs).
The c o m p l e t e c l a s s of t h e quadric cylinders consist of circular, elliptic,
hyperbolic and parabolic cylinders. Similarly t h e right circular cone, s p h e r e ,
ellipsoid, arid tlic hyperboloid m d pir.hu1uid of rcvulutiuii coiistilutc tliv nolid e g e n e r a t e class of quiidric ~irr1dcc.s01 rc.volution. Tlicse Q L J A i Y L s and QOAX>l<s
a r e t h e c o o r d i n a t e s u r f a c e s of t h e a p p r o p r i a t e Eisenhart C o o r d i n a t e Systems [I].
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T h e Eisenhart S u r f a c e s (i.e., t h e c o i n p l e t e c l a s s of QUACYLs and QUAS3Rs)
forrri t h e basis of our s u r f a c e rrlodelirtg paradigiit. However, it tiiay be noted t h a t
t h e s e surface5 a r c geiierdlly iiilinite o r senti-infinite. Since, in p r a c t i c e , t h e
s u r f a c e s a r e but finite, t h e building blocks of t h e a e r o s p a c e shapes c a n be
considered a s t h e t r u n c a t e d s e c t i o n s of t h e s e quadrics and a r e identified a s tQUACYLs and t-QUASOlls.
Most a e r o s p a c e shapes a r e however q u i t e complex and c a n only be realized by
a combination of f i n i t e number of t h e s e t r u n c a t e d quadrics. Such s t r u c t u r e s in
our modeling have been identified as t h e hybrid quadrics (h-QUACYLr and hQUASORs). Figure I show s o m e 01 t h e illustrative e x a m p l e s of h-QUASORs.
O n e of t h e salient f e a t u r e s of our paradigm is t h e r e t e n t i o n of only convex
building blocks, and a n a t t e m p t t o describe t h e s u r f a c e s while retaining i t s
convexity. F r o m t h e r a y - t h e o r e t i c point of view t h i s has a d v a n t a g e t h a t on such
s u r f a c e s t h e s u r f a c e ray g e o m e t r i c p a r a m e t e r s c a n b e readily obtained.
Y e t a n o t h e r a d v a n t a g e of t h i s m o d e l i s r e a l i z a t i o n of t h e s h a p i n g
p a r a m e t e r s corresponding t o a l l t h e quadric in t h e Eisenhart C o o r d i n a t e System
in t e r m s of t h e c o o r d i n a t e p a r a m e t e r held C O s t a n t [2,31.
For example, t h e
general equation of t h e g e n e r a t i n g parabola x’ = by = (a2)y, of a paraboloid
o r a parabolic cylinder, incorporates t h e r e a l scalar c o n s t a n t a serving a s a
shaping p a r a m e t e r , by varying which parabolas of d i f f e r e n t sharpness/flatness
can be generated.
In s u m m a r y , t h e c o n c e p t of modeling with s e c t i o n s of generaiized quadrics
is versatile and c a n accurately describe various complex, practical and
realistic aerospace structures.
Although m o r e corriplex s u r f a c e s t h a n quadrics, such as, bicubics, have been
considered f o r c o m p u t e r modeling application 141 and found t o provide g r e a t e r
modeling flexibility because of t h e i r inherent ability t o minimize or e l i m i n a t e
slope discontinuities, t h e m a t h e m a t i c a l complexity of t h e s e s u r f a c e s results In
s e v e r e c o m p u t a t i o n a l loads f o r coiiiplex scenes.
I t thus appears that the
quadrics provide t h e best trade-off b e t w e e n t h e e l e m e n t simplicity and t h e model
siinplicity while describing complex scenes.
Su”ary
A s u r f a c e modeling paradigm has been systematically coithtructed (Fig. 2).
As a logical s t e p a f t e r r a y t r a c i n g over t h e canonical qu&ciric shapes (i.e.,
individual QUACYLs and QUASOKs), hybrid s t r u c t u r e s of p r a c t i c a l i n t e r e s t a r e
considered. Included in this c l a s s is t h e a i r c r a f t wing modeled P S a hybrid
QUACYL, for which t h e ray t r a c i n g p a r a m e t e r s have been obtained in closed form
[2]. F j g u r e 2 also r e p r e s e n t s t h e m a t h e m a t i c a l analysis s t r u c t u r e t h a t has been
formalized by t h e s e a u t h o r s [51. The paradigm also f a c i l i t a t e s t h e t r e a t m e n t of
t h e m o r e complex c a s e of t h e hybrid nondevelopable QUASORs such a s a s a t e l l i t e
launch vehicle modeled a s a CPOR-cylinder combinarion. T h e r a y g e o m e t r i c
p a r a m e t e r s h a v e been obtained in t h e quasi-analytical form.
228
HEi7ERENCES
[ I ] P. Moon and D.E. Spencer, Field Theory Handbook. Berlin: Springer-Verlag,
1971.
[2] 1I.M. J h a , S.A. Bokhari, V . Sudhakar and P.R. M a h a p a t r a , "Ray analysis on a
c l a s s of hybrid cylindrical a i r c r a f t wings", Electronics L e t t e r s (GB), vol.
24, pp. 21-22, Jan. 1988.
[3] R.M. J h a , S.A. Bokhari, V. Sudhakar and P.R. Mahapatra, "Ray analysis on a
class of nondevelopable s a t e l l i t e launch vehicles", Electronics L e t t e r s
(CB), vol. 24, pp. 22-23, Jan. 1988.
[4] G.Y. G a r d n e r , " S i m u l a t i o n of n a t u r a l s c e n e s u s i n g t e x t l u r e d q u a d r i c
surfaces", C o m p u t e r Graphics, Proceedings of SIGGRAPH 84, vol. IS, no. 3,
pp. 11-20, 1984.
151 R.M.
J h a , Surface Ray Tracing on Convex Quadrics with Applications to
Mutual Coupling between Antennas on Aerospace Bodies, Ph D Dissertation.
Sublnitted to D e p a r t m e n t of Aerospace Engineering, Indian I n s t i t u t e of
Science, Bangalore, India, Nov. 1988.
Fig. 1 S o m e e x a m p l e s of hybrid s u r f a c e s
of revolution (h-QUASOKs).
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I
AEROSPACE APPLICATIONS
Realistic or Idealized
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