geometry package for generation of input data for a three-dimensional potential-flow program

8/14/2019 Geometry Package for Generation of Input Data for a Three-Dimensional Potential-Flow Program

1/70

..

N A S A Contractor Report 2962

A GeometryPackage forGeneration of InputDatafor a Three-DimensionalPotential-Flow Program

N. Douglas Halseyand John L. Hess

C O N T R A C T N A S 1 - 1 4 4 02J U N E 1 9 7 8


2/70

TECH LIBRARYKAFB, NU

NASA Contractor Report 2962

A GeometryPackage forGeneration of Input Datafor aThree-DimensionalPotential-Flow Program

N. Douglas Halsey and John L. HessMcDonneZZ DongZas CorporationLong Beach, California

Prepared fo rLangley Research Centerunder Contract NAS -14402

National Aeronauticsand SpaceAdministrationScientific and TechnicalInformation Office1978


3/70


4/70

.1 .2 .3.04.05.06.07.0

8 .0

9.0

1. TAB LE OF CONTENTSTable fContents . . . . . . . . . . . . . . . . . . . . . . . . . . i i iIndex of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . VSumnary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1In troduc t ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Nomenclature and Arrangement of Input Poi n t s . . . . . . . . . . . . 9Panel ingfsolated Components . . . . . . . . . . . . . . . . . . 1 37.1 General Fe atu resf the Paneling Method . . . . . . . . . . . . 1 37 .2 Dis t r ibu t ion of Po in t s Along N-lines . . . . . . . . . . . . . . 177.2 .1nput i s t r ibu t ion . na l te red . . . . . . . . . . . . . . 1 77.2.2npu tDistribution. Augmented i n Number . . . . . . . . . 187.2.3Constantncrementsn Arc Le ng th . . . . . . . . . . . . 207.2.4 Cosinepacing . . . . . . . . . . . . . . . . . . . . . . 207.2.5urvature-Dependent Distribution . . . . . . . . . . . . 227.2.6ser-Specifiedis tr ibution . . . . . . . . . . . . . . . 247 .3is t r ibu t ionf-l ines . . . . . . . . . . . . . . . . . . . . 247.3.1 The Planar-Section Mode . . . . . . . . . . . . . . . . . 257.3.2 The Arc-Length Mode . . . . . . . . . . . . . . . . . . . 31Ca lcu la t ionfn te r s ec t ion Curves . . . . . . . . . . . . . . . . . 328.1 General Featuresfhe Method . . . . . . . . . . . . . . . . . . 328 . 2 Res tr ic t ions and Limita t ions o f the Method . . . . . . . . . . . 328 .3D e t a i l s o f the Method of ol ut io n . . . . . . . . . . . . . . . 338.3.1 The I n i t i a l Search f o r "LineSegments nd Su rfa ce

8.3.2 The FinalSearch f o r "line Segments nd Su rfac eElements Containingnte rsec t ion o in ts . . . . . . . . . 34Elements Co ntainin g ntersectionPoin ts and th eApproximate Determination o f the In te r s ec t ion o in t s . . 358.3.3De rivation of a Mathematical Re pre sen tatio n of theSurf ace f anlement . . . . . . . . . . . . . . . . . . 378.3.4 Com putation of More Precise Values o f the Coordinatesof the In te r s ec t iono in t s . . . . . . . . . . . . . . . 428.3.5 es t Cases f o r the Intersection Method . . . . . . . . . 43Final Repaneling of Components. . . . . . . . . . . . . . . . . . . . 479.1 General Co nsid era tion s . . . . . . . . . . . . . . . . . . . . . 479.2 In te rs ec ti n g Components . . . . . . . . . . . . . . . . . . . . 489.3 Nonl i f t i n gntersected Components . . . . . . . . . . . . . . . 519.4 L i f t i n g Intersected Components . . . . . . . . . . . . . . . . . 54

i i i


5/70

10.0Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6111. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

i v


6/70

2.0 INDEX OF FIGURES1. Typical element distribution fo r a trapezoidal wing . . . . . . . . . . 32. Typical element distribution fo r a wing-fuselage case . . . . . . . . . 33. Surface elements used by NLR, Amsterdam, for an external store

configuration (1780 elements) . . . . . . . . . . . . . . . . . . . . . 44. Definition of frequently used terms . . . . . . . . . . . . . . . . . . 95. Illustration of numerical differentiation procedure . . . . . . . . . . 156. Comparison of curve-fit methods . . . . . . . . . . . . . . . . . . . . 167. Method of distributing points on an N-line - input distribution,augmented . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198. Point distribution on a supercritical wing section - nput distribution,

augmented . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199. Point distribution on a cylindrical fuselage section - onstantincrements in arc-length . . . . . . . . . . . . . . . . . . . . . . . 200. Method o f distributing points on an N-line - osine distribution . . . 211 . Point distribution on a supercritical wing section - osinedistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222. Point distribution on a supercritical wing section - curvature-dependent distribution . . . . . . . . . . . . . . . . . . . . . . . . 24. Point distribution on a section of a nonlifting component -curvature-dependent distribution . . . . . . . . . . . . . . . . . . . 244. Redistribution o f elements on a trapezoidal wing (cosine spacingchordwise, constant increments spanwise, planar-section mode) . . . . . 285. Redistribution o f elements on a supercritical wing (cosine spacingchordwise , constant increments spanwi e, planar-section mode ,mu1 tisection component option) . . . . . . . . . . . . . . . . . . . . 296. Redistribution of elements on a fuselage (constant increments around

circumference and in axial direction, planar-section mode, multi-secti on-component option) . . . . . . . . . . . . . . . . . . . . . . . 307. Comparison o f planar-section and arc-length modes of distributingN-1 ines - trut on a thick wing . . . . . . . . . . . . . . . . . . . 318. Intersection method test case - fntersection o f two circularcylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

V


7/70

19. Intersection method test case -intersection of two spheres . . . . . . 520 . Intersection method test case- intersection of two ellipsoids . . . . 521. Illustration of intersection method - thick wing intersecting acircular cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . 4622 . Final repaneling o f intersecting components -wing-fuselage case . . . 923. Final repaneling of intersecting components - wing-fuselage-tip-tankcase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5024. Final repaneling of nonlifting intersection components -wing-fuselagecase (zero-incidence ing, N-lines through every point on intersectioncurve) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 525 . Final repaneling of nonlifting intersected components- wing-fuselagecase (wing with10" incidence, N-lines through every point on inter-

section curve) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5526 . Final repaneling of nonlifting intersected components -wing-fuselagecase (zero-incidence wing, N-lines through every other point onintersection urve) . . . . . . . . . . . . . . . . . . . . . . . . . 5627 . Final repaneling of lifting intersected components -wing-pylon case. . 5728. Final repaneling of lifting intersected components -wing-pylon case(sparse element distribution on pylon) . . . . . . . . . . . . . . . . 829. Final repaneling of lifting intersected components- wing-pylon case(more points on pylon than surrounding region oning) . . . . . . . . 930. Final element distribution on a wing-fuselage-pylon case . . . . . . . 0

vi


8/70

3.0 SUMMARYT h e preparationofgeometricd a t a f o r i n p u t to th ree -d imens iona lpoten t ia l -

i s a very tedious, time-consuming (and therefore expensive) t a s k .re po rt de scr ibe s a geometry package th a t automates and simplifies thiso a l a r g e degree. Inpu t o the computer program for the geometry pack-

gecons is tsof a very sp ar se set of coord ina te da ta , o f ten w i t h an orderoffewer poin ts han required for the ac tua lpo ten t i a l - f lowca lcu la t ions .

such a s w i n g s , f u s e l a g e s ,e t c .a r e pane led au tom at ica l ly ,oneof severalpos s ib le element d i s t r i b u t i o n a l g o r i thms . Curves of inter-between components ar e ca lc ul at ed , u s i n g a hybr id c u r v e - f i t / r u r f a c e - f i t

roach . F i na l ly , n t e r sec t in g components a re epan e led so t h a ta d j a c e n teither side of the i n t e r s e c t i o nc u r v e s i n e u p i n a s a t i s f a c t o r y

nner f o r h ep o t e n t i a l - f l o wc a l c u l a t i o n s . The geometry package has beenthe NASA Langley version of the 3-D l i f t i n g po ten t i a l - f low

and i t i s p o s s i b l e o r u n many casescompletely (from i n p u t , throughw i t h o u t n t e r r u p t i o n .

se of this geometry package can s ignificantly reduce the time and expensei n making three-dimensionalpoten t ia l - f lowcalcu la t ions .


9/70

4.0 INTRODUCTIONW i t h theadvent of modern high-speed co m pu ter s, heaerodynamicis t

gained th e ab i l i t y t o s tu dy he e ffe c ts of conf ig ura t ion changes i n g r e a td e t a i l . Whereas befor e he m i g h t havebeen able t o e s t i m a t e h ee f f e c t s ofchanges i n such grossgeometricparameters a s sweep and aspec t r a t i o on thet o t a l l i f t of an i s o l a t e d wing, he no w can accu r a te ly ca lcu la te he e f fec t so f changes of q u i t e s m a l l d e t a i l s of t h e shape o f very complex configurations(wing-fuselage-nacelle-pyloncases , for example) on not o n ly l i f t b u t a l s o onspanwise and chordwise l o a d d i s t r i b u t i o n s , l o c a l pressures , f low angles ,e t c .This g a i n i n c o m p u t a t i o n a l a b i l i t y has made theprepara t ion of i n p u t d a t a f o rdesc r ib ingconfiguration geometry a very te di ou s, time-consuming ta sk . Generalthree-dimensional p o t e n t i a l - f l o w programs, such as he Hess program,( re fe rences 1 and 2 ) req uir e he geometry t o be i n p u t a s thecoord ina tes o fs e t s o f po in t s , which a r e grouped t o form quadri la teralsu r facee lemen ts .Many more elem ents are gene rally requ ired t o o b t a i n accu ra te f l ow s o l u t i o n st h a n t h e number required for adequa te p ic to r ia l ep re se n ta t ion . For example,thes imple rapezoidal wing of f i g u r e 1 , w h i ch can be represented f a i r l yaccu ra te ly by twenty or s o e lements ,requires on theorder o f two hundredelements t o o b t a i n a reasonable f l ow so lu ti on . More complex co nf ig ur at io ns ,such as hewing-fuselagecase o f f i g u r e 2 , m i g h t use nearly a thousand. Acase which was recently run by personnel o f N L R, Amsterdam, using 1780elements t o rep re sen t an e x t e r n a l - s t o r ec o n f i g u r a t i o n i s shown i n f i g u r e 3( re fe rence 3 ) .

There is a clear need t o automateas much a s poss ib le hep repa ra t ion ofthe i n p u t coord ina te s , b o t h i n orde r t o reduce the number o f po in t s npu t t othe programs and inorder t o relax some o f t h e r e s t r i c t i o n s on how thepoin tsmustbe di s t r i bu te d. The main d i f f i c u l t y i n d o i n g t h i s i s t h a t t h e r ea r e somany l o g i c a l l y d i f f e r e n tc a s e s t o cons ider . For example, a wing has d i f f e r e n tspacing equirements t h a n a fuse lage ; a fuse lag e has d i f fe ren t equire m entst h a n a nace l l e ,o r a pylon , e t c . Moreover, bodies o f similar type may requ i redifferent numbers and d i s t r i b u t i o n s o f points due t o theproximity o f o the rbodies . I f t h e r e a r e in tersections between bodies ,addi t iona l equirementsare imposed on the p o i n t d i s t r i b u t i o n s . A geometry package must be very2


10/70

Figure 1 . Typical element distribution for a trapezoidal wing.

Figure 2 . Typicalelement distribution or a wing-fuselage case.3


11/70

,F igure 3 . Surfacee lements usedby NLR, Amsterdam, f o r an e x t e r n a ls t o r ec o n f i g u ra t i o n (1 7 8 0 e l e m e n ts ) .v e r y f l e x i b l e t o a p p l y t o so many d i f fe re n t cases , bu t i t s h o u l d n o t b e sof l e x i b l e t h a t i t becomescumbersome t o use.

Th i s re p o r t d e s c r i b e s a g eo m e tr ypackaged ev elo pe d f o r u se w i t h t h ep o t e n t i a l - f l o w pro gram o f r e f e r e n c e s 1 and 2 and f o r e x t e n s i o n s o f h e m ethodwhich may replace i t i n h e u t u r e .S i n c e h e e q u i r e m e n t s o f f u tu r ep r o -grams a r e m o s t l y s p e c u l a t i v e a t h e p r e s e n t , c o m p a t i b i l i t y w i t h h e p r e s e n tprogram i s emphas ized.

The geometry package hasbeen i n co rp or a t ed n to he NASA-Langley ve rs iono f thepo te n t i a l - f l o w p rogram. Th i s p rog rama c c e p t s i n p u t e i t h e r n h eo r i g in a l n p u t o r m a td e s c r i b e d n e f e r e n c e 4 ( w i t h m i n o r m o d i f i c a t i o n s ) o ri n h e o r m a to f h e p r og ra md e sc ri be d i n r e f e r e n c e 5. With a smal l amounto f a d d i t i o n a l n p u t o c o n t r o l h e geo metry package, it i s p o s s ib l e o r u nanumber o f r e q u e n t l y o c c u r r i n g C ~ J ~ So m p l e t e l yw i th ou t human i n te rv en t i o n .For example, i s o la te d wings or b o d i e s may b e i n p u t o h e p ro gram w i t h aminimumnumber o f p o i n t s and th e geometry package wi a u g m e n t a n d re d i s t r i b u tet h e p o i n t s o h e numbera n d d i s t r i b u t i o n s p e c i f i e d b y h euser . Thesec o o rd i n a te s c an t h e n be p unched o n c a rd s f o r u s e w i t h a u to m a t i c p l o t t i n gp ro gra ms t o n s p e c t h e e s u l t s b e f o r e p r o c e e d i n g ,o r h e p o t e n t i a l - f l o w

4

_


12/70

program can proceed immediately to analyze the low. Options are provided toallow the ser to tailor the distributions to theeeds of his particularproblem, so it should also be possible to run cases having multiple non-intersecting wings or odies with no human intervention. More difficultcases, involving intersecting components,an also be treated by the geometrypackage. Some of the simpler cases involving only two components, suchs awing/fuselage case, can alsoe run completely without intervention,ut morecomplex cases almost certainlyhould be checked before proceeding with theflow calculation. Some cases cannot be completely handled by the geometrypackage, but in these cases, use of the geometry package to augmenthe pointdistributions and to calculate intersection curves still results in asignificant reduction in the effort required t o prepare the coordinate data.A n outline of the majoreatures of the geometry ackage is given below.o Significant reduction i n effort required to input a case to the

potential -flow program.o Two separate modes of program input available.o Complete compatibility with the potential-flow program -allowing many

cases to be run completely without interruption etween the geometrypackage calculations nd the potential-flow calculations,

o Paneling of isolated componentso Al lows very sparse input coordinate data.o Provides output coordinate data suitable for potential-flow

analysis.o Uses independent cubic curve-fits for interpolation in two

directions on surfaces (N-lines and "lines).o Provides six options for the point distributions on N-lines

and four options on "lines.o Allows all N-lines on a component to lie in parallel planes,

if desired.o Calculation f ntersection urves

o Requires a distinction between intersecting and intersectedcomponents.

5


13/70

o Intersecting components represented by their "lines only.o Intersected components represented by three-dimensional surface

fits.o Intersection curves defined by arrays of intersection points

between the "lines of th e ntersecting components and the fittedsurfaces of th e ntersected components.

o Repaneling of intersecting and intersected componentso Intersection curve made an N-line of the intersecting component.d Other N-lines on intersecting component shifted to restore a

smooth distribution.o Extra strip inside intersected component automatically generated,

if desired.o Intersected components repaneled t o insure that elements on

adjacent components line u p along the intersection curves.o Simpler repaneling options also provided.The remainder of this report documents he theory and operation of the

geometry package. Section 6 defines the geometric terms used i n the latersections and discusses the basic philosophy o f the geometric input to themethod. Section 7 describes the paneling of isolated, non-intersectingcomponents, including the options available, the applicability and limitationsof the options, user requirements, methods used, and sample results. Section8 covers the ethod o f calculating curves o f intersection between components,including the theory, restrictions, and verification cases. Finally,section 9 describes the methods that have been provided for repanelingcomponents after having calculated the curves of ntersection between them.

6


14/70

5.0 SYMBOLSA M a t r i xfu r f a c e - f i to e f f i c i e n t snl g e b r a i c form.A,B,C,D C o e f f i c i e n t so f h ee q u a t i o no f a p lane.A,B,C,D, A l g e b r a i c u r f a c e - f i t o e f f i c i e n t s .I , J , K , L ,EsFsGsH,M,N,O,P. . ~CdF

f

GiK

kRM

" l i n eNN-1 i e

P

L o c a l v a l u e o f h e c h o r d o f acomponent.S t r a i g h t - l i n e d i s t a n c e betw een a d j a c e n t p o i n t s o nacurve.A l s o u sed f o r h e d i s t a n c e r o m a p o i n t t o a p la n e.F u nc ti on s u sed t o e xp re ss a c u b i c c u r v e n te rm s o f h e u n c t i o n a lv alu es and f i r s t d e r i v a t i v e s a t t h e ends o f h e c u r v e o n l y .Dummy v a r i a b le used t o express heg e n er al o rm o f a f u n c t i o n u se dw i t hs e v e r a ld i f f e r e n t v a r i a b l e s .M a t r i x o f s u r f a c e - f i t c o e f f i c i e n t s n g e o m e t r i c f o rm.G en era l s u b s c r i p t u se d i n a v a r i e t y o f ways.Parameter used i n s p e c i f y i n g t h e d i s t r i b u t i o n o f N - l i n e s onacomponent.C u rv a tu re a t a p o i n t ona curve.S u b s c r i p t d e n o t i n g v a r i a b l e s a s s o c i a t e d w i t h a l i n e o r curve.M a t r i x o fc o ns ta n ts used i n c o n v e r t i n g s u r f a c e - f i t c o e f f i c i e n t sf romg e o m et ri c t o a l g e b r a i c fo rm .Curve onacomponen t, ge ne r a l l y un n i ng spanw ise on l i f t i n gcomponentsand i n t h e a x i a l d i r e c t i o n on n o n l i f t i n g components.T o t a l number o f d e f i n i n g p o i n t s on a c ur ve .Curve on acomponent, ge ne ra l ly un nin g c ho rd w is e on l i f t i n gcomponentsand i n h e c i r c u m f e r e n t i a l d i r e c t i o n on n o n l i f t i n gcomponents.T o t a l a r c e n g t h o f a c urv e.N or m a l i zed point number o f t h e p o i n t h a v i n g n d e x i.Arc l en g th a l o ng a cu rve .

7


15/70

so s lST

8

Arc lengths at beginning and end of a urve.Subscript denoting a surface.Superscript denoting the transpose f a-m atr ix.Parameters used in calculating surface fits.Coordinates o f a point in a Cartesian coordinate system. Alsosubscripts referring to these coordinates.Angle around a circle circumscribed about an airfoil section, usedi n determining the cosine point spacing distribution.Angle o f a curve at the defining point having index i.Angle of a straight-line segment of a curv e eginning at thedefining point having index i.

a


16/70

6.0 NOMENCLATURENDRRANGEMENT OF I N P U T O I N T SB e f o r e des c r i b i n g he geom et ry package i t s e l f , it i s n ece ssa ry t o

d iscuss heg e n e ra l scheme f o r n p u t t i n g p o i n t s and f o r o r d e r i n g h e e le m en tst h a t h e y o r m and t o d e f i n e some o f t he t e r m sw hic h a r e u se d f r e q u e n t l yt h ro u g h o u t h e e m a in d e r o f h e e p o r t . M o stp rocedures and d e f i n i t i o n s a r ei d e n t i c a l o h o s e d e s c r i b e d i n r e f e r e n c e s 1 and 2, b u t some ( f o r example,t h ed e f i n i t i o no f i g n o r e de l e m en ts " ) have been changed s l i g h t l y . R efe renceshou l d be made t o f i g u r e 4 t o c l a r i f y t h e d i s c u s s i o n w h ic h ol lo w s.

S T R I P O NN - L I N E S O N

M - L I N E S O NI N T E R S E C T E D

ELEMENTI N T E R S E C T E D

I N T E R S E C T I N GCOMPONENTV - N T E R S E C T I N GCQYPONENT

F i g u r e 4. D e f i n i t i o no f r e q u e n t l y usederms.

9


17/70

A complete configuration (such as a wing-fuselage-nacelle-pylon case)is assumed to be constructed of a number of components, ach of which is aset of associated points. Normally a single component is used to representa complete body (such as a wing or a fuselage), but any number of componentsper body is allowed. There are two ypes of components - onlifting andlifting. Nonlifting components, such as fuselages or other blunt-ended bodies,are represented by source distributions over their surfaces and hence have nocirculation. Lifting components, such as wings or other bodies with sharptrai ing edges , are represented by both surface source nd dipole distribu-tions. Circulation about any section of a lifting component is adjusted insuch a way as to satisfy the trailing-edge Kutta condition. Liftingcomponents also have associated dipole sheets which represent trailing ortexwakes. Points on the wakes must be input to the program, as well as pointson the bodies, and are considered to belong to the same components as theassociated body points.

Each component consists of a set of points which can be connected i nsuch a way s to form a network of intersecting lines called N-lines and"lines. N-lines on lifting components are the lines running approximatelyin the chordwise direction. They divide a wing, for example, into a numberof distinct sections. "lines are the lines connecting corresponding pointson the N-lines. On nonl ifting components N-lines are generally thosesurrounding the major axis of the body (if it is possible to define an axisat all). In principle, the roles of the N-lines and "lines on nonliftingcomponents can be reversed without adversely affecting he execution o f theprogram, but in most portions of the geometry package, it is assumed that theN-lines play the role described above. The potential-flow program does notrequire that all N-lines on nonlifting components have the same number ofpoints. However since having variable numbers of points per N-line makes itimpossible to define "lines, most applications o f the geometry package requireeach N-line to have the same number of points as eac h ther N-line and each"line to have the same number of points as each other "line. N-lines maynot cross other N-lines (though they may touch at a oint); "lines also maynot cross other N-lines (though they may also touch at a oint).

10


18/70

Po i n t s m ustbe ordered so t h a t a l l p o i n t s on t h e i r s t N - l in e ( i n c l . y d i n gwake p o i n t s a f t e r t h e b ody p o i n t s f o r l i f t i n g com ponents) a r e n p u t con -s e c u t i v e l y , o l l o w e d b y a l l p o i n t s on t h e second N - l i n e a nd so on . The f i r s tN - l i ne npu t may be t he one a t e i t h e r ex trem e o f t h e componen t, bu t hec h o i c ed e t e r m i n e s h eo r d e ro f n p u to fp o i n t sa l o n g h eN - l i n e s . ng e n e r a l ,t h e o r d e r m ust be such t h a t h e n e g a t i v e o f h e c r o s s p r o d u c t o f h e v e c t o rf ro m one p o i n t o n an N - l i n e o h e n e x t p o i n t on t h e N - l i n e w i t h h e v e c t o rf r o m a p o i n t on t h e N - l i n e o h e c o r r e s p o n d i n g p o i n t on t h e n e x t N - l i n e r e s u l t si n a v ec to rw h ic h i s d i r e c t e d t o t h e e x t e r i o r o f t h e component. T h is e q ui re m e ntmay be s a t i s f i e d on a w ing, f o r example,by o r d e r i n g h e N - l i n e s r o m i pt o r o o t and o r d e r i n g p o i n t s on t h e N - l in e s f r o m t h e t r a i l i n g edge a lo n g h el o w e r s u r f a c e o h e e a d i n g edgeandback a lon g he u p p e rs u r f a c e o h et r a i l i n g edge. On a f u se l ag e h e e q u ir e m e nt i s s a t i s f i e d by a r r a ng i n g h eN - l i n e s f rom f r o n t t o backand th e po i n t s on each N - l i ne nc r ea s i ng c o u n t e r -c l o c k w i s e l o o k i n ga f t ) . The requ i rement i s a l s o s a t i s f i e d by r e ve r si ngb o t ht h e o r d e r o f h e N - l i n e s and t h e o r d e r o f h e p o i n t s on t h eN- l ines. Howeverp a r t s o f h e g eo me try p ackage r e q u i r e h a t N - l i n e s on fu se la ge s s t a r t a t t h ef r o n t o f h e body, o r more g e n e r a l l y , h a tN - l i ne s on n o n l i f t i n g componentss t a r t a t th e e nd fa r t h e s t r o m h e f i r s t " l i n e of any ot h e r components whichi n te r s e c t h e componen t.

The area between adjacentN- l i n es on a component i s des ignated a s t r i p .Each s t r i p on a l i f t i n g c om ponent haso n e c h a r a c t e r i s t i c v a l u e o f t h e d i p o l ed i s t r i b u t i o n and one l o c a t i o n w here t h eK u t t ac o n d i t i o n i s s a t i s f i e d . Onl i f t i n g components, it i s p o s s i b l e t o s p e c i f y t h a t a s t r i p havea d i p o l ed i s t r i b u t i o n b u t no s o u r c e d i s t r i b u t i o n s andno boundary co n di t io ns on anyo f t s e le me nts . Such a s t r i p ,c a l l e d an e x t r a s t r i p , s u s e f u l o r a v o i d i n gt h e a b r u p t e n d i n g o f a d i p o l es h e e t w h i c hw o u l d e s u l t i n a concent ra tedv o r t e x ) a l o n g h e c u r v e o f n t e r s e c t i o n o f tw o com ponentsand f o r c o n t r o l l i n gt h e b e h a v i o r o f h e d i p o l e s h e e t n e a r w i n g - t i p s when t h e p i e c e w i s e i n e a rv o r t i c i t y o p t i o n s used. R e f e r o e f e r e n c e 1 f o r more d e t a i l s .

The a r e a ( g e n e r a l l y q u a d r i l a t e r a l ) betw een ad j ac en t " l i nes on a s t r i pi s termed an e lement. Each e lement has one c o n tr o lp o in t where th e boundaryc o n d i t i o n s a r e s a t i s f i e d andone c h a r a c t e r i s t i c v a l u e o f h e s o u r c e

11


19/70

distribution. It is possible to designate some elements o be ignored elemen'ts.These elements do not have source distributions and no boundary conditionsare applied to hem. If the component is lifting, ignored elements do havedipole distributions, however. References 1 and 2 allow for ignored elementsto be defined only for lifting components,ut this restriction has beenlifted in the present program.


20/70

7.0 PANELINGOFSOLATEDOMPONENTS7.1 GeneralFeatures o f hePane l i ng M e thod

The f i r s t o p e r a t i o n p e rf orm e dby heg e o m et ry p acka ge i s t o p a ne l( i .e . , d i s t r i b u te t h e e l em en ts ) he componen ts as i so l a ted bod i es , w he the r orno t any o f t hem i n te r sec t .A l t h o u g h h e e s u l t in g e l e m e n td i s t r i b u t i o n s o ni n te r s ec t i n g componen ts may n o t be use fu l o r an a l yz i ng com p l e te con f i gu -r a t i o n s , h e y s e r v e a sa s t a r t i n g p o i n t o r d e t e r m i n i n g h e i n a l d i s t r i b u t i o n sas w e l l a s a l l o w i n g a c o n f i g u r a t i o n " b u i l d - u p " o be p e r f o rm e d i .e . h esu cc es s i v e ad d i t i on o f componen ts can be pe rfor med , i n o r de r o de te r m i nei n t e r f e r e n c ee f f e c t s ) . It i s assumed t h a t each omponent i s co m ple te l yin de pe nd en t o f a l l o t h e r s . S in ce a s i n g l e body may be omposed o f more th anonecomponent,however,and s i n c e h e c l o s e p r o x i m i t y o f a n o t h e r b od y maymodi fy hee l em en ts p a c in g e q u ir e m e n ts o f acomponent, it i s n o t a lw ays t r u et h a t n d i v i d u a l com ponents a r ec o m p l e t e l y n d e p e n d e n t .S u f f i c i e n t l e x i b i l i tyhasbeen d e sig n ed i n t o h e p a n e l i n g m ethod t o a l l o w p o i n t s t o bematchedwherecomponentsm eet m aking t h e i r " l i n e s c o n t i n u o u s ) and t o a l l o w h e u s e r os p e c i f y w ha te ve r d i s t r i b u t i o n hedeems ap p r o p r i a te o acc ou n t o r hep r o x i m i t y o f o t h e r com ponents.

The panel ingo f an i nd i v i d u a l com ponent i s accomp lished i n tw osteps.F i r s t t h e p o i n t s on t h e n i t i a l N - l i n e s a r e augmented i n numbera n d r e d i s t r i -bu tedaccord ing t o t h e numberand the s p a c i n g a l g o r i t h m s p e c i f i e d b y t h euse r .F o r h i s c a l c u l a t i o n , h e N - l i n e s m ust be r o u g h l yc ho rd w is e on l i f t i n gcomponentsand, if h e a t e r r e p a n e l i n g o f n t e r s e c t e d components i s r e q ui re d ,t h e N - l i n es m ust be r ou gh l y c i r c u m f e r e n t i a la b o u t an a x i s on n o n l i f t i n gcomponents . P o in ts n h e wake o f an N- l i ne o f a l i f t i n g component a r ed i s t r i b u t e d n d e p e n d e n t l yo f h e bo dy p o i n t s .P ro v i s i o n has been made t oa l l o w h e u s e r o n p u t d i f f e r e n t numbers o f p o i n t s on t h e n i t i a l N - l i n e so f acomponent. The pro ce ss o f r e d i s t r i b u t i n gp o i n t s a l o n g h eN- l i nes makesth e numbers o f p o i n t s equal , h u s a l l o w i n g M -l in e s o be formed. The seconds t e p i s t o augment and r e d i s t r i b u t e h e N - l i n e s a c c o r d i n g o h e number anda l g o r i t h m s p e c i f i e d b y t h euse r .Th i s i s done bya ug me ntin g and r e d i s t r i b u t i n gpo i n t s a l on g each " l i ne .

13


21/70

The p ro ce ss o f r e d i s t r i b u t i n g p o i n t s a l o n g e i t h e r N - l i n e s o r " l i n e sr e q u i r e s a m ethod o f n t e r p o l a t i n ga l o n gg e n e r a lcu rv es n space . S incet h e s e c u r v e s p a r t i c u la r l y h e N - l i n e s )a r e n o tu s u a l l y m o n o t o n i c i n e i t h e rx,y, o r Z c o o rd i n a te s , some o therparamete rmu st be useda s t h e n d e p e n d e n tv a r i a b l e o f n t e r p o l a t i o n . I n t h epr es en t method, thep a ra m e te r c ho se n i sth ea rc e n g tha l o n g h ep o l y g o n o r m e db yc o n n e c t i n gs t ra ight - l ines betweena d j a c e n tp o in t s. I n h e re ma in de r o f h i s e p o r t h e e r m a r c e n g t h "a l w a y s e f e r s o h i ss t r a i g h t - l i n ea p p r o x i m a t i o n . n t e r p o l a t i o n o r a p o i n ton a cuPve re qu i re s h r ee s e p a r a t e n t e r p o l a t i o n s , one f o r ea ch c o o r d in a t e ,Each s e p a r a t e n t e r p o l a t i o n i s accompl ished i n twos teps . I n t h e i r s t s tep ,n u m e r i c a l d i f f e r e n t i a t i o n o f h e dependent v a r i a b l e w i t h r e s p e c t o h e a r c -l e n g t h i s p e rf orm e d a t e ach d e f i n i n gp o i n t o n h ec u r v e . I n t h e second s t ep , h ev alu es o f h e u n c t i o n and i t s d e r i v a t i v e s a t t h e ends o f h e segm ents o f h ec u r v e a r e used t o d e r i v e h e c o e f f i c i e n t s o f c u b i c n t e r p o l a t i n g p o l y n o m i a l s .

The n u m e r i c a l d i f f e r e n t i a t i o n s done u s i n g a " w e ig h te d -a n gl e"approach( s e e f i g u r e 5 ) . I n h i s ap pro ach , t h ee n g t h sd i ) and ang les (emi) o fs t r a i g h t - l i n e a p p r o x i m a t i o n s o segments o f h e c u r v e a r e i r s t c a l c u l a t e d .The a n g l e a t h e m i d p o i n t o f e ach segm ent i s a ssum ed t o e q u a l h ea n gl e o ft h es t r a i g h t - l i n ea p p r o x i m a t i o n o h e segm ent. The a n g l e a t a ny o f h eg i v e np o i n t s on t h ec u r v e ( e i ) i s th e nd e t e r m i n e db y a k i n g an a v era ge o f h eang les o f ad ja ce n t segments , w e igh te db y h ed is ta n ce s t o h e m i d p o i n t s o f t h esegments,

(7.1 .l)A ng le s o f t h e f i r s t and l a s t p o i n t s on a c u rv ea red e te rm i n e d b ye x t ra p o l a t i o n ,assuming cons tantc u r v a t u r e betw een t h e i r s t two a n d ' l a s t tw o p o i n t s . Thed e r i v a t i v e s a t h e g i v e n p o i n t s a r e h e n found b y a k i n g h e a n g e n t s o f h ec a l c u l a t e da n g l e s .

I n o r d e r o a c i l i t a t e h e r e a t m e n t o f components h a v in gs e c t i o n sbounded by s t r a ig h t i n e s , such as a w ing w it h a k i n k n t s t r a i l i n g edgeo r a f u s e la g ew i t h a c y l i n d r i c a lm i d s e c t i o n , a m u l t i s e c t i o n com ponent o p t i o ni s p ro v id ed i n h en u m e r i c a ld i f f e r e n t i a t i o np r o c e d u r e .W i t h h i so p t i o n ,

14


22/70

SFigure 5. I l l u s t r a t i o n of numer ica l d i f ferent ia t ion procedure .thecurve i s d iv ided i n t o seve ra lsec t i ons . A t the ends o f th esec t ions ,ext rap ola t ion s are used t o determine heder iva t ives , just a s h e y a r e a t theends o f th ecomple t ecurve .Sect ionscons is t ing o f j u s t two poin t sa rerepresented by s t r a ig h t i n es . The s lopes o f t h e s es t r a i g h t i n e s th endetermine heder iva t ives a t t he ends of ad jacen t s ec t i ons .

The values o f the f u n c t i o n and i t s d e r i v a t i v e a t each end of a curvesegment const i tute f o u r pieces of in format ion w h i ch can be used t o determineth ec o e f f i c i e n t s of a cubic polynomial a p p r o x i m a t i n g thecurve. The form ofthe polynomial is

(7.1.2)where S i s t he ndependen tvar iable arc ength r a n g i n g from values Sot o S I , f represe nts he dependent var iab le (x , y, o r z c o o r d i n a t e s ) , andprimes denote d i f f er en t ia t io n . The form of i t s d e r i v a t i v e s

f ' (5) = f ' S o ) + (S - S o ) f"(S,) f ;s - S O ) * f ' I ' ( S O ) (7 .1 .3 )15


23/70

Given the valuesf f and f ' a t So and SI, he s imul taneousolu t ionof equatio ns 7.1.2) and (7 .1. 3)y ie ld s the valuesof f " ( S o ) and f " ' (So).Equation (7.1.2) can then be used t o determine the value of the unc t io n (S)a t any v a l ueof S w i t h i n the given curve segment.

T h i s cu rv e- fi t method does no t ns ure con tinu ity of the second deriv-a t iveo f hefunction and thus i s not a cubicsp l ine f i t i n the usual sense( re fe rence 6 ) . I t was chosen ra th er than a spline method because of ' i t sc o n s i s t e n t l ys u p e r i o r e s u l t s i n s e v e r a l e s tca se s . For example, f i gu re 6shows a comparison between this methodand a tr u ecub icspline method.The interpolated coordinates found by thepr ese nt method are cons iderab lyless wavy t h a n thoseca lcu la tedus ing hesp li n e method. Both methodsf a i l o rep re se n t he shape accu ra te ly i n th e a f t e r region of the body ,because of theh igher oca lcurva ture and theproximityof he ends of hecurve.Other comparisons involving a i r f o i l s andmore gen eral hap esalsoshowed smoother results f o r h e p re se nt method tha n hespline method, i n

+ INFL'TOINTSe INTERPOLATEDOINTSSPLINE METHOD)INTERPOLATEDOINTS (PRESENT METHOD)

Figure 6 . Comparison of curve-fit methods.16


24/70

s p i t e o f h e a l l e g e d p r o o f i n r e f e r e n c e 6 t h a t , i n genera l , sp l i ne p r oducest h e sm oothest o f a l l p o s s i b l e c u r v e i t s . The c o n t r a d i c t i o n o f h i s p r o o fmay p o s s i b l y be. due t o t h e u n u s u a l c h a r a c t e r o f h e n d e p e n d e n t v a r i a b l e (itb e i n g d e f i n e d a s a q u a n t i t y w h i c h v a r i e s s m o o t h l y a l o n g a nonsmooth curve)o r i t may b e d ue t o t h e p r e s e n t a t i o n of t h e r e s u l t s b y a g ra ph o f onei n t e r p o la t e d r e s u l t ( t h e z - c o o r d i n a te s ) as a f u n c t i o n o f o t h e r n t e r p o l a t e dr e s u l t s t h ex - c o o r d i n a t e s ) .

7 .2 D i s t r i b u t i o no fP o i n t sA l ong N - L i nesEach o f h e o p t i o n s o r d i s t r i b u t i n g p o i n t s a l o n g N - l i n e s ( e x c e p t h e

t r i v i a l o p t i o n o f e a v i n g h e n i t i a l d i s t r i b u t i o n unchanged) r e qu i r e s ana r r a y o f n o r m a l i z e d a r c e n g t h s w h i c h a p p l i e s o e v e r y N - l i n e o f h e com-ponen tunde rcons i de ra t i on . The fo rm u l as f o r t he sea r c - l e n g t hd i s t r i b u t i o n sa r eg iven be low.Given thes p e c i f i e dd i s t r i b u t i o n s , h e method c a l c u l a t e st h e d i s t r i b u t i o n oneach i n i t i a l N - l i n e and i n t e r p o l a t e s each c o o r d i n a t ei n de p e n d e n tl y t o d e t er m in e h ev a lu e s a t h e d e s i r e d o c a t i o n s . n somecases a l l p o i n t s onan N - l i n e c o i n c i d e ( a s a t h e end o f a p o i n t e d body, f o rexample). Then t h e methoddoes nota tt em p t t o n t e r p o l a t e ,b u ts i m p l yp r o v i d e s h esp ec i f i e d number o fp o i n t s o h a t N - l i n e . The f o l l o w i n go p t i o n sa r e a v a i l a b l e o r d i s t r i b u t i n g p o i n t s on N - l i ne s :

1 . I npu ti s t r i b u t i o n ,n a l t e r e d2. I n p u t i s t r i b u t i o n , augm ented i n number3. Cons tantnc r em en ts i n r ce n g t h4. Cons tantnc rem en ts on t h e upe r sc r i bed i r c l e cos i ne pac i ng )5. C u r v a t u r e - d e p e n d e n t d i s t r i b u t i o n6. U s e r - s p e c i f i e di s t r i b u t i o n

7 .2 .1 I n p u tD i s t r i b u t i o n ,U n a l t e r e dW i t h t h i so p t i o n h e m ethoddoesn o i n t e r p o l a t i o n . It should be used

whenever t h e n i t i a l d i s t r i b u t i o n a l r e a d y c o n t a i n s a s u f f i c i e n t number o fp r ope r l y spaced po i n t s . If t h e N - l i n e s o f h e component un de rcons i de r a t i ona r e t o be r e d i s t r i b u t e d , o r if t h e com ponent i s i n v o l v e d i n an i n t e r s e c t i o n w i t hano ther component, t he n he number of p o in ts n p ut mustbe t h e same on eachcomponent of heN - l i n e . I n a d d i t i o n , h e d i s t r i b u t i o no fp o i n t s on a d j a c e n t

17


25/70

N- l i n e ss ho uld be f a i r l y s i m i l a r so t h a t when c o r re s p o n d i n g p o i n t s o n N - l i n e sa r eco nn ec ted , he es u l t i n g l i ne s make smooth cu rves .T his i s h e o n l y .o p t i o n w h i c h may r e s u l t i n d i f f e r e n t numbers o f p o i n t s and d i f f e r e n td i s t r i b u t i o n s o f p o i n t s among t h eN - l i n e s o f acomponent. Al th ou gh noi n t e r p o l a t i o n s a r e r e q u i r e d , some c a l c u l a t i o n s a r e s t i l l n ece ssa ry t o d e t e r -m i n e h e d e r i v a t i v e s w i t h r e s p e c t o a r c e n g t h o f h e c o o r d i n a t e s a t eachp o in t on each N- l i ne . These a re needed i n a t e r c a l c u l a t i o n s o d e t e r m i n ei n t e r s e c t i o n c u r v e s .7 .2 .2 InputD is t r i b u t i o n , Augmented i n Number

T h i s o p t io n r e s u l t s n a d i s t r i b u t i o n w h i c h s s i m i l a r t o t h e n i t i a ld i s t r i b u t i o n ,b u t c o n t a i n s a d i f f e r e n t number o f p o i n t s . The i n i t i a l d i s t r i -b u t io n used i n h e c a l c u l a t i o n s h e d i s t r i b u t i o n o r h e i r s t N - l i n e onth e component, un less a l lp o i n t s c o i n c i d e on t h e i r s t N - l i n e . n h a t caset h e d i s t r i b u t i o n on t h e se con d N - l i n e s used. The m ethod w o rk sb y d e f i n i n ga "n o rm a l i z e d point number", pi , r a n g i n g r o m ze ro o one,

pi = (i l ) / ( N - 1) (7.2.1)where i i s t h e n d e xo f h ep o i n t a n d N i s h e o t a l number o fp o i n t s .A r rays o f n o r m a l i z e d p o i n t num ber a r e fo rm ed f o r b o t h n p u t and o u t p u t d i s -t r i b u t i o n s . The o u t p u ta r c - l e n g t hd i s t r i b u t i o n sd e t e r m i n e db y n t e r p o l a t i n gth e c u r v e o f n p u ta r c e n g t h v e r s u s n p u tn o r m a l i z e d po in t number t o t h eo u t p u tv a l u e so f h en o r m a l i z e dp oi nt number. Th isp ro ce du re i s l l u s t r a t e di n i g u r e 7. S u i t a b l e c ase s f o r h i s o p t i o n n c l u d e b o t h i f t i n g andnon-l i f t i n g t y p e com ponents. It i s e s p e c i a l l y u s e f u l o r p r e c i s e l y c o n t r o l 1 n gt h e d e s i r e d d i s t r i b u t i o n w i t h o u t h a v i n g o o a d a l a r g e number o f p o i n t s andw i t h o u th a v i n g o d e t e r m i n e i n a dva ncewhat the n u m e r i c a l v a l u e s o f h ea r c e n g t h sa r e .T y p i c a l e s u l t s o r a s e c t i o n o f a s u p e r c r i t i c a lw i n ga r es h o wn i n i g u re 8.

18


26/70

+.++ I N P U TO I N T SW T E :L LO I N T S A R ELOCATEDTQUAL INCREMENTS +O U TP U T P O I N T S I N N O RM A LI ZE D P O I N TW B E R .

+. + . .+

+

++0.0 .0.0 0.2 0.4 0 .6 0.8 1 .o

NORMALIZED POINT NUMBER - [(i 1 ) / ( N - ) ]Figure 7 . Method o f distributing points on an N-line - nput distributionaugmented.

+ I N P U TO I N T SN = 15) o OUTPUT POINTSN = 25)

Figure 8 . Point distribution on a supercritical wing section - nput distribu-tion, augmented.19


27/70

7.2.3 Constant Increments in Arc LengthThis option results in points distributed uniformly according to the

formulaSi/P = (i - l ) / ( N - 1 ) (7.2.2)

where S is the arc length measured from the first point and P is thetotal arc length around the perimeter of th e N-line. It should be used forsmooth bodies which do not have large variations i n curvature (for example,the cylindrical fuselage section illustrated i n figure 9. It should usuallynot be used f or wing sections, since, for a reasonable number of points, therewould be too few points to accurately define the shap e near the leading edgeor other high-curvature regions and too coar se spacing at the trailing edgefor the utta condition to be accurately applied.

Figure 9. Point distribution on a cylindrical fuselage section - onstantincrements in arc length.7.2.4 Cosine Spacing

Cosine spacing is a commonly-used distribution, dating rom some of theclassical two-dimensional theories (reference 7). Its name derives i n anobvious way from the usual formula fo r the spacing

20


28/70

Xi/C - 2 C 1 + c o s [ ( i - l ) a / (N - 1)]> (7.2.3)w here i s h e cho rd ,w h ic h i s assumed t o be p a r a l l e l o h ex - a x i s . T h es i g n i f i c a n c e o f h i s method,as i l l u s t r a t e d n f i g u r e 10, i s t h a t p o i n t s a r echosen h av ing hex - coo r d i na tesco r r espond i ng oe q u a l n c r e m e n t s i n ang l ea ro un d a c i r c l e c i r c u m s c r i b e d a b o u t h e N - l i n e w i t h e a d i n g and t r a i l i n g edgest o u c h i n g h e c i r c l e .T h i s e s u l t s i n v e r y i n es p a c i n gn e a r e a d i n g andt r a i l i n g edgesand coarsespac ing i n reg ion s away f rom hese edges . It i s au s e f u l d i s t r ib u t i o n o p t i o n o r w i n g s e c t i o n s b u t w o u l d p r o b a b l y n o t becomnonlyused f o r n o n l i f t i n g com ponents.

I EOUAL INCREMENTS OF

F i g u r e 10. M ethod o f d i s t r i b u t i n g p o i n t s on an N - l i n e - o s i n e d i s t r i b u t i o n .Eq uat io n (7.2.3) i s n o t d i r e c t l y u s e f u l o r d e t e r m i n i n g h e s p e c i f i e d

a r c - l e n g t h d i s t r i b u t i o n s w h i c h h e m ethod o f t h i s g eo me try package r e q u i r e s .Todo t h i s , h e w i n g s e c t i o n i s f i r s t s ca le d t o u n i t chord, t r a n s l a t e d op u t t h e o r i g i n o f t h e c o o r d i n a t e system a t h e e a d i n g edge, and r o t a t e d t omake thec h o r dp a r a 1 e 1 o h ex-a x i s . The ang le ( 6 ) about thesuper sc r i bedc i r c l e i s t h e n c a l c u l a t e d o r each p o i n t on t h e N - li n e b y s o lv in g h e e q u a t i o n

X i / C = 7 (1 + cos B ) (7.2.4)

21


29/70

using the transformed values of th e -coordinates. The arc-length distributioncorresponding to th e input points is also calculated. For both the angl e andarc-length calculations, only the first N-line on a compone nt is used (unlessits length is zero, in which cas e the second one is used). Given the valuesof the rc-length distribution and the angle istribution for the nputcoordinates, and the desired values of the angl e istribution for the outputcoordinates (uniform increments), the arc-length distribution for the outputcoor dina tes is determined by interpolation.

Results of this option for the same upercritical wing section usedpreviously are shown in figure 1 1 .

Figure 1 1 . Point distribution on a supercritical wing section -c os in edistribution.

7.2.5 Curvature - ependent DistributionThe accuracy and computational efficiency of th e otential-flow method

are both improved by using fine spacing in regions of high curvature andcoarser spacing in regions of low curvature. This option produces a pointdistribution which depends on the curvatur e istribution along the firstN-line o f the component (or the second if the length of the first is zero).If the component is of lifting type, the point distribution is also mad e tobe a function of the proximity t o the trailing and leading edges in order toallo w the trailing-edge Kutta condition to be accurately satisfied and tokeep the variation o f the spacing smooth in the leading-edge region where thecurvature varies rapidly.

22


30/70

The following relationship between the curvaturend the spacing incrementsis employed :

Asi = (1 - A S ~ ~ ~ / A S ~ ~ ~ ) ( ~i/kmax) + AS i n/Asmax'Asmax 7.2.5)where Asi is the ith increment in normalized arc length between adjacentpoints, ki is the absolute value of the curvaturet the center of the thsegment, kmax is the maximum absolute value of the curvaturen th e N-line,and Asmi n and Asmax are, respectively, the minimum and maximum allowableincrements in arc length. In this application, the ratio A s m i n / ~ s m a x isspecified to be 0.25.

To implement this relationship requires n iterative procedure. Firstthe arc-length and curvature distributions are calculated. Then, if thecomponent is of lifting type, the variable whichlays the role of thecurvature in equation (7.2.5) is modified to make ts value at the trailingedge equal to its leading-edge value nd to make it vary linearly from theleading and trailing-edge values to thealues at 7.5 percent o f the N-line'sperimeter on either side of the leading nd trailing-edge points. Thismodification insures that oints will be closely spaced near the trailingedge and that the spacing ill not vary abruptly near the eading and trailingedges. To start the iterations, estimates of the values of bsmax and allthe values of AS ^ are required. These are a1 1 taken to be equal to the totallength o f the curve divided by the number of segments it contains. Given theestimated values o f Asmax and AS^, the curvature at the center f eachsegment (modified as described above) s determined by interpolation of thecurvature versus arc-length relationship and used in equation (7.2.5) toupdate the estimated alues of AS^. These values are then scaled to maketheir sum equal to the total length of the N-line and searched to determinethe value of AS,,,^^. These updated values of AS^ and ~s~~~ are then usedin the next iteration and the process is repeated a maximum f five times oruntil the required scale factor or the lengths of the increments issufficiently close to nity.

Results of this optare shown in figure 12.shown in figure 13.

ion for the supercritical wing section used prevResults for a section of a nonlifting component

iouslyare

23


31/70

Figure 12. Point distribution on a supercritical wing section-curvature-dependent distribution.

Figure 13. Point distribution on a section of a nonlifting component-curvature-dependent distribution.

7.2.6 User-Specified DistributionFor N-lines o f unusual shape or when components are located very close

together, it is possible that none of the options described above will producean adequate point distribution. To account for this possibility, provisionhas been made for the user to specify (as input data) values o f the normalizedarc lengths, which are used directly to determine he augmented coordinates.

7.3 Distribution of N-linesAfter the points on the initial N-lines have been redistributed, each

N-line on a component has the same number f points distributed in a similarmanner. Connecting corresponding points on a1 1 N-1 ines generates a set o f 1 inesdesignated "lines. Since the distribution of points is the same on each N-line,the "lines are smooth and have fairly small curvature. The process o f24


32/70

augment ing and redistributing the N-lines i s accompllshed by augmenting andredistributing poin tsa long these "lines. For theseca lcu la t ions , i t isreq uire d hat each "l in e have a tota l eng th g rea te r han ze ro .

There are ouro p t i o n sa v a i l a b l e o r the d i s t r i b u t i o n Of "lin es. They ar e:1 . Input d i s t r i b u t i o n , una1 te re d2 . Input d is t r ibu t ion , augmented i n number3 . Constantncrements4. User - spec i f i edi s t r ibu t ion

The descript ions of a1 1 four options are completely analogous t o the descr ip-t io ns of the co r re spond ing op t ions fo rd i s t r ibu t ion o f points on N-linesdescribed i n sec t ion 7.2.

There are twomodes of op era tio nof th is portion of the geometry package.The f i r s t (and most common)mode i s des igna ted he p lanar-sec t i on mode and thesecond i s termed thearc-length mode i n thediscuss ions which follow .7.3.1 The P1 anar-Section Mode

The usual way o f i n p u t t i n g da ta o hepotentia l-f low program i s to maketheN-lines on a component planarcu tsper pen dic ula r o some ax is of th ecomponent. For nonlif t ing components , t h i s convention is just a conveniencet o th euser. For l i f t i n g components, however, thee lemen tsare approximatedby trapezoids, so t h a t i f th e N-1 in es ar e notpara1 le1 , he accuracy of repre-sentation of the component by i t s elements is no tas good.

The plana r sec tion mode of op era tio n re dis tr i bu te s th e po in ts on the"lines i n such a way that a l l N-lines, exceptperhaps the f i r s t and l a s t oneson a component, are p a r a l l e l . T h e f i r s t and la s t N-l ines g enera l ly define theplanform view of he component, which is notrequ ired o becomposed onlyofs t r a i g h t l i n e s . The method does not a l ter these N-lines d u r i n g this por t iono fthe c a l c u l a t i o n s . The d i s t r i b u t i o n sdesc r ib ed above gen e ra l ly r e f e r o h ed i s tances between the p a r a l l e lpl an es , normalized by th edi sta nc es between th ef i r s t and l a s t N- l ine s . A t the edqesof the component, when th e f i r s t and l a s tN-l inesareno tp lanar , the d is t r ib u t io ns re f e r o d i s t an ces between the f i r s t

25


33/70

p o i n t on t h e N - l i n e u n d e rc on sid er at io n and t h e f i r s t p o i n t on t h e f i r s t N - l i n eo f t h e component, re so l v e d n he d i r e c t i o n p e r p e n d i c u l a r o h e p a r a l l e l p l a n e sand norma l i zedb y h ed is ta n ce between t h e f i r s t p o i n t s on t h e f i r s t and l a s tN - l i n e so f he component. The o r i e n ta t i o n o f h e p l a n e s s d e f i n e d b y s p e c i f y -i n g ( as i n p u t d a t a ) h e d i r e c t i o n c o s i n e s o f a ve c t o rp e r p e n d i c u l a r o h e p l a n e s .F o r t h i s c a l c u l a t i o n , h e sense o f h e no rm al v e c t o r sn o t m p o r t a n t . However,i f i n t e r s e c t i o n c u r v e s a r e ca lcu la te d and th e components aresubsequen t ly epan-e l e d , h e n h ev e c t or m ust p o i n t n h e d i r e c t i o n o f i n c r e a s i n gN - l i n e s . If t h ed i r e c t i o n c o s i n e s a r e n o t s p e c i f i e d and t h e p l a n a r s e c t i o n mode i s u sed , d e fa u l tv a lu e s o f ( 1.0 , 0.0, 0.0) f o r n o n l i f t i n g com ponentsand (0.0, -1 .O, 0.0) f o r lift-i n g compon ents a re assumed. When t h ed e f a u l tv a l u e sar e used, N - l in es must bei n p u t from f r o n t t o ba ck on n o n l i f t i n g com ponents and f ro m t i p t o r o o t o f l i f t i n gcomponents.

The f i r s t s te p i n d i s t r i b u t i n g t h e p o i n t s on t h e " l i n e s i s t o d e te r m in et h e e q u a t i o n o f h e p l a n e s r e p r e s e n t i n g h e N - l i n e s and t h e n t e r p o l a t i n g p o l y -nom ia l s ep rese n t i ng he l i ne segments . The equa t i ons o f h e p lanes o f t h eN-1 ine 2 ar e o f form

Ax + By t Cz t Di = 0 (7.3.1)where A, B, and C arequa l toh e x, y, and z d i r e c t i o n o s i n e s ft h e v e c to r n orma l t o h e p lanes (andhence ar e he same f o r a1 1 p lanes on th ecomponent). The valu es o f Di ( d i f f e r e n t o r each p l a n e )a r eg i v e n by

Di = Dl + Ki (D 2 - Dl ) (7.3.2)where Dl and D2 a r e h e o e f f i c i e n t so fh ep l a n e sp a s s i n g h r o u g h h ef i r s t and l a s t p o i n t s , r e s p e c t i v e l y , on t h e f i r s t " l i n e o f t h e com ponent andKi i s t h ev a l u eo f h ed i s t r i b u t i o np a r a m e t e r o r h eN - l i n e u n d e rcons ide r -a t i o n . Dl i s d e t e r m i n e dro mh ee l a t i o n s h i p

(7.3.3)where xl, y, , and z1 a r e h e o o r d i n a t e s o f t h e i r s tp o i n t on t h e i r s t" l i n e . De i s d e te rm in ed i n a s i m i l a r manner, u s i n g h e a s tp o i n t on t h ef i r s t " l i n e . The i n t e rp o l a t in gp o l y n o m i a l s o r h e l i n e segm ents a r ed e te rm i n e db y h ec u rv e - f i tp ro c e d u red e s cr ib e d i n s e c t i o n 7.1.26


34/70

The second step i n d i s t r i b u t i n g points on the " l ines i s to determine which"line segments in te rs ec t which plane s . For th is step, i t i s assumed that" l i n ecurva ture i s smal l enough so ha t s t ra ig h t - l in e approximat ions to he " l ine seg-ments may be used.

Then, knowingwhich "line segments intersect which planes andknowing theequations of the "line segments and theplanes , hecoord ina tes ofeach inter-sec t ionpo in t can becomputed. Too this fo r any po in t ,s u b s t i t u t e h ee x p r e s -s i o n s o rx , y , zas a func t ion of a rc engtha long he"li ne segment (eachhaving the form of equa t ion(7 .1 .2)) n to heequa t ionof heplane(7 .3 .1) . Thisgives a s inglecubicequa t ion w i t h S i , thearc engtha long he"line segmentt o h e n t e r s e c t i o np o i n t ,a s h eonly unknown. This i s solved by Newton'smethod. Using the i n t e r s e c t i o npo in t o f the p l a n e and thes t ra igh t - l ine app rox -imation to he " l in e segment to de te rm ine he s ta r t ingpo in t, he method ty pi -c a l l y r e q u i r e s o n l y h r e e o r fou r te ra t ion s o converge to an adequa teso lu t ion .

Fig ure s 14 and 15 show the op views of two typical wings b o t h beforeand after r e d i s t r i b u t i n g h e N-lines. Since the wing i n f i g u r e 1 4 is t r ape -zo ida l i n planform and has a linear twist d i s t r i b u t i o n ,o n l y h e N - l i n e s a tth e t i p and rootare needed i n t h e n i t i a l g e o m e tr i c e p r es e nt a ti o n .S incet h e n i t i a l r e p r e s e n t a t i o n s o f t h e t i p and roo ts ec t ionsa rea l readyp lana rand all" l i n e sa r e s t r a i g h t l i nes , the pl an ar -se cti on mode nd th earc - lengthmode (described i n th enex t ec t ion ) produce ide n t i c a l e su l t s . A more gen-er al ca se , fo r which the twomodes ofoperationshouldgivev e r y d i f f e r e n tr e s u l t s is shown i n f igu re15. In this case , he"l ines on the e n t i r eo u t -boa rd ha l f o f the span a re s t ra ig h t l i ne s , b u t they a r e curved on the nbo ardport ion.Therefore , hem ulti sect ion component op tiondescribed i n section 7.1i s used ,a l lowing he inearpor t ionof he geometry t o be represented us ingonly tw o N-lines. Figure 16 shows the op view of a fuselagebefore and a f t e rr e d i s t r i b u t i n g the N-lines . This fuselage has a cylindricalmidsec t ion w i t hsem i-e l l ipso ida l ec t ion s ore and a f t . Again, themultisection componentopt iona l lows he inearport io n o be represented u s i n g only two N-lines. Ina l l t h r e e o f t h e s e c a s e s , the redistributed N-1 ines a re equa l ly spaced .

27


35/70

"" .- . _ . _ " ." ._ . . - __ _. . . . . . . _. - . . -._

INITIAL ELEMENTDISTRIBUTION

ELEMENT DISTRIBUTIONAFTERREPANELING

Figure 14 . R e d is t r ib u t io n o f elements on a trapezoidalwing cosinespacingcho rdwi se, cons tant ncre men ts spanwi se, planar- secti'on mode).

28


36/70

INITIAL ELEMENTDISTRIBUTION

ELEMENT DISTRIBUTIONAFTER REPANELING

Figure 15. Redistribution o f elements on a supercritical wing (cosine spacingchordwise, constant increments spanwise, planar-section mode,mu1 tisection component ption).

29


37/70

w0

I N I T I A L E L E M E N T D I S T R I B U T I O N

ELEMENTDISTRIBUTIONAFTERREPANELING

Figure 16. Redistribution o f elements on a fuselage (constant increments around circumferenc e and inaxial direction, planar-section mode, multisection component option).


38/70

7.3.2 The Arc-Length ModeIn ome ca se s he N -lin e a t the edge of a component maybe so highlynonplanar

t h a t a st r ip bounded by this N-line andan adjacentplanar N-line would varys t rong ly i n w i d t h , r e s u l t i n g i n a reas oosparsely covered w i t h elements . Anexample of this i s the under-wing pylon shown i n f i g u r e1 7 ( a ) . In this figurethe h icknessof he wing has been exaggerated toh e l p l l u s t r a t e h epo in t . I nsuch cases hear c- le ng th mode of d i s t r i b u t i n g N-lines on the component should beused. In this mode thespec i f i eddis t r ibut ionparamete r s e f e r t o f r a c t i o n so ft h e o t a la r c eng ths o f t he l i n e s ,rat he r ha n o normalized dis ta nce s betweenpla ne s. The po in ts on a l l l i n e s o f a component are d i s t r i b u t e d i n a s i m i l a rmanner, so t ha t he N- l ines i n the v i c i n i t y o f theno np lan ar edge of th e compo-nent curve smoothly around to c re a te a more uniform d is t r ib u t i o n of the elem ents ,a s l l u s t r a t e d i n f i g u r e1 7 ( b ) . In this mode of ope ra tio n, he method of re d is -t r i b u t i n g pointsalong M-lines i s completelyanalogous to h e method ofredis-t r i b u t i n g pointsalongN-l ines.

(a )lanar-Section Mode. ( b ) Arc-Length Mode.Figure 17 . Comparison of pla na r-s ec tio n and arc -le ng th modes o f d i s t r i b u t i o n ofN-lines - s trut on a thick wing.

31


39/70

8.0 CALCULATION O F INTERSECTION CURVES8.1 General Fe atu res of th e Method

The second major operation performed by the geometry package i s the cal cu-la t ionof hecu rve s of in te rs e ct io n between body components. For this calcu-l a t i o n i t i s assumed th at he "l in es of one of the components (de sign ated heint ers ec t in g component) p ierce the sur face of the o th er component (des ign atedthe nte rse cte d component) . The f i n a ls o l u t i o n is a se t of i n t e r sec t i onp o i n t s ,one for each augmented M-line on the in te rs ec t in g component. In general , h emethod ca lc ula te s he n terse c t ion p o i n t o f a curve and a surface and hencerequi res a sur face- f i t method as well as a cu rv e- f i t method. The su r f ac e- f i tmethod uses he heory of par am etriccubicsur facepatchesoriginated byCoons( reference 8 ) and extensively developed and applied by a number of otheri n v e s t i g a t o r s r e f e r e n c e 9 , for example) . The curve- f i t method i s the sameone used i n thepan el ing of is ol at ed componentsand described i n sec t ion 7.1.

8 .2 Restr ict ions and Limitat ionsof the MethodThe intersection method i s designed to handlecases which oc cu rf requent ly

i n a i r c r a f t a p p l i c a t i o n s , su ch a swin g-fu sela ge nte rsec t ion s, wing-pyloni n t e r s e c t i o n s ,e t c . t i s notdesigned fo r complex cas es i n which the i n t e r -sec t ioncurve is discont inuousorfo rc a se s n v o lv i ng f i l l e t s o r sm ooth t r a n s i -t ions fromonecomponent to h en e x t. Use of th e method i s l i m i t e d oc a s e s

I i n which the o l low ing es t r ic t ionsapp ly :1.

2.

3.

A d i s t i n c t i o n c a n be made between in tersec t ing and in tersec tedcomponents.A component can in te rs ec t on ly one oth er componentand can be in te r -se cte d by only one ot he r component. If a body intersects o r i sintersected by severalbodies , i t must b e divided up i n tos e v e r a lcomponents. A si n g le component an in te r s e c t one component ndbeintersected by another component, however.The "line s on th e n te rs e ct in g component pie rce the sur face of theint ers ec ted component a t a sh arpang le .Sur facesa ren o t a n g e n twhere they meet.

32


40/70

4.

5.

6.

Therequi res

The fines on the intersecting component extend sufficiently farinto the interior of the intersected component to allow a planarrepresentation o f the elements on th e intersected component to beused in the process of searching for the elements which re inter-sected by the M-1 ines.The intersecting component as at least one N-line which lies entirelyin the interior of th e intersected component.No line on the intersecting component intersects he intersectedcomponent more than nce.

8.3 Details of the Method of Solutionmethod of calculation of the intersection curve between two componentsthat one component be assigned the role of the intersecting component

I !

and the other component be assigned the role of the intersected component.This is done by the user in the input data to the geometry package. The inter-section curve is then defined by the set of intersection points between thelines on the intersecting component and the elements on the intersectedcomponent. In order to define this curve at a sufficient number of points,every line on the intersecting component (after redistributing and augmentingthe points on the N-lines) is used. On the intersected component, in principleeither the original input elements or the elements after augmenting oints onjust the N-lines, or the elements after augmenting points on both N-lines andlines could be used for defining surface patches which are intersected by thelines of the intersecting component. Since parametric cubic surface patchesare used, however, it is not necessary to use the large number f elements thatexist after augmenting points on both N-lines and lines. Since the inputpoints may vary in number from one -line to the next, and since the istribu-tions may also be dissimilar, use of the original input elements could resultin surface patches having much more extreme curvature and hence less accuracy)than use of either of the other twoets of elements. Therefore, the elementsafter augmenting points on the N-lines but before augmenting points on theM-lines are used fo r the formation of surface patches.

One possible method f or calculating intersectionssurface might operate as follows: Derive mathematical

of an M-1 ine and aformulas to represent

33


41/70

each segment of the "line and each element of the sur face and f i n d the r o o t s o ftheequat ionsobta inedby.equat ingcoordinatevalu es on th e segments and the e l e -ments . I f a pa r t icu larequ at io n has no re a l r o o t s ,o r i f i t does have real rootsb u t th e po in t s hey represen t fa l lou t s ide he bounds of thee lement or the seg-ment, then heelement and the segment have no intersection p o i n t s . I f he equa-t ion does have realro ot s and thepo in ts hey epresen t do f a l l w i t h i n th e boundsof both theelement and the s u r f a c e , h e n h e r o o t sa r e h e n t e r s e c t i o n p o i n t s .When a l l such equ ati on s havebeen solved , he ntersect io npointsaregrouped ,somehow, to det erm ine he nte rsec t ionc u rv es . Such a methodwould have tos o l v ea l ar g e number of eq uatio ns a number equal to h e number of "lin e segmentsmultiplied by the numberof surfaceel em en ts ) and many of the se equations wouldfa i l o have so lu t ion s . I t would , t he re fo re , be very uneconomical t o use.

A more economical approach,adopted i n the present method, f i r s t comparesmaximum and m i n i m u m coordinateva lues on the "line segments and t he su r f aceelements and el im ina tes f rom considerat ion hosecom bination s which obviouslycontain no intersect ionpoints. Thenan app rox im ate nter sec t ion method,w i t h planarelem ent eprese ntat ions, i s used to determine which " l ine segmentsand sur faceelementsconta in n tersec t ionpo ints and to get approximatevaluesof the n tersec t ionp o i n tco ord inate s . Then, mathematical rep res en tat ion sa reobta inedonlyfor hoseelementsconta ining ntersec t ionpoints , and t h e s e a reused, together w i t h t he r epresen t a t i ons o f he M-1 ine segments, to ca lcu la t ethe n t e r sec t i onpo in t s more accura t e ly .Deta i l s of this approach are givenbel ow.8.3.1 The I n i t ia l Search fo r "LineSegments nd Su rfa ce Elements C o n t a i n i n gI n t e r s e c t i o n P o i n t s

The ba sic nfo rm atio n needed to conduct the i n i t i a l stages of thesearchin c l udes th e maximum and minimum values of each coordinate on each element o fth e nt er se ct ed component, on each s t r ip of th e n te rs e ct e d component, on eachsegment of the n te rsec t ing " l ine , and on the ent i r e n te rsec t ing " l ine .Because i t i s assumed that he ntersect ing "l in es extend some di s ta nc e nt othe bodyand a pla na relem ent eprese ntation canbe used, i t i s l e g i t im a t e oassume th at th e maximum and m i n i m u m values onan element must occur a t oneof i t s corners . A quick comparison of thecoordinatesof hecornerpoints

34


42/70

t h u s determines the extrem a. Extreme va lu es on a str ip a redetermined'by com-paring heextremevalues o f each of i t s elem en ts. Values on th e"line segmentsar e determined more acc ura tely , assuming cu bic rela t ion ship s between the coord-ina t eva lue s and thes t r a igh t - l i nea r c e n g t h sa l o n g the "line (equat ions 7 .1 .2)and (7 .1 .3 )). Equation equation 7.1.2) toz e r o s e p a r a t e l y o rx,y, and z) andsolvinggiv es two va lue so farc- lengtha long he" l ine( foreachcoordinate) .The re al so lu t io ns which fa1 1 w i t h i n the bounds of he segment a r e used to ca l cu-la te coord ina t e va lue s which a re compared, along w i t h thev a l ue s a t t h e en ds o fth e segm ent, to'd ete rm in e he extreme va lu es . Maximum and m i n i m u m values on theen t i r e in te rs ec t in g " l in e ar e determined by comparing the extreme values of eachof i t s segments.

Having determined the extreme values of thecoopdinates on s t r ip s andelements of the nt ers ec ted componentand on segments and the en t i re ty of thei n t e r s e c t i n g" l i n e , the searchingprocedurebegins by comparing the extremecoordinatevalues on the f i r s t strip of he nte rsec ted component w i t h th eva lues o r the entire "line. If the rangeof any of the coordinatevalueson the s t r ip f a i l s t o in clu de a t l e a s t p a r t of th erange of the values of thesame coordinate on the "line, then no intersect ion can possiblyoccur on th atstrip. Successive strips are compared w i t h the" l ine u n t i l one i s foundwhich may possiblyconta in he n tersec t ionpo in t. Then each element on th es t r ip i s compared w i t h th e" l ine u n t i l a pos s ib le n terse c ted e lement i sfound. Then each segment on th e nt er se ct in g" l ine is compared w i t h th ee l e -ment u n t i l a poss ib l e n t e r se c t i ng segment i s found .8.3.2 The FinalSearch f o r "LineSegments nd Su rfac e Elements Co ntainin gIn t e r sec t ionPo in ts and th e Approximate Determination o f the I n t e r -sec t i onPoin t s

Having found an element on the intersected componentand a segmen t ofthe i n t e r s e c t i n g "line which s h a re , a t l e a s t p a r t i a l l y , the r anges o f t he i rx, y, and z-coo rdin ates , a more care fu l check on whether or no t hey inter-s e c t i s made. If the element i s n o ta l r eady r i angu la r , i t is div ided n totwo tr iangles by drawing one of the diagonalsacross the quadr i l a t e r a lr eg ion .Each element i s t h e n represented bywo pl an ar , r ia ng ul ar subelements . Thep lan a r coe f f i c i en t s o f each subelement are de termined, s ta r t ing f rom thegenera l form of the equat ion o f a plane

35


43/70

A x + By + Cz + D = 0 (8.3.1)b y f i r s td i v i d i n g h r o u g h b y D ( i l lu s t r a t i n g h a t h e r ea r eo n l y h r e eindependent unknowns i n equat ion 8 .3 .1 ) ) and re a r r an g in g o g e t

A/Dx + B/Dy + C/Dz = -1 (8.3.2)and t h e n s u b s t i t u t i n g h e c o o r d i n a t e s o f h e c o r n e r p o i n t s o f h e subelem enti n t o e q u a t i o n (8.3 .2 ) t o o b t a i n a t h i r d -o r d e rsys tem o f l i nea r e q u a t i o n s w h i c hi s e a s i l y s o l v e d o r A/D, B/D, and C/D . A , B, and C ar e hen oundbym u l t i p l y i n gb y any a r b i t r a r i l y chosen ( f in i t e ,n o n z e r o )v a l u eo f D .

A check i s th en made t o see if h e ends o f t h e " l i n e segment l i e ono p p o s i t es i d e so f h ep l a n e .T h i s i s done b yc o m p u t i n g h ed i r ec tedd i s tancef r o m e ach p o i n t o h e p l a n e , u s i n g t he . o rm u la

dA2+ B2 + C 2 (8.3.3)where d i s h ed i s ta n ce and (xl , y1, z1 ) a r e h ec o o r d i n a t e so f h ep o i n t .If t h ev a l u eo f d has th e same s ig n o rb o th end po in ts , he segment doesn o t n t e r s e c t h ep l a n e o f t h e r i a n g u l a r s ub ele me nt. I f t h i s i s t r u e o rbothsubelements on thee lement , heelemen t and th e " l in e segment do n o ti n t e r s e c t and t h enex t segm ent i s cons ide r ed.

When an " l i n e segment which does cr o ss h e p la n e o f oneof th e subelementsi s found, t h e p o i n t o f i n te rs e c t i o n o f the segment and the p la n e i s fo un dus ingthe m ethod desc r i bed i n sec t i on 7.3.1. Fo r h i sc a l c u l a t i o n , h e l i n e seg-m en t i s a g a in e p r e se n t edby a cub i cp o l y n o m i a l .G i v e n h e n t e r s e c t i o np o i n t ,it i s n e xt n e ce s sa ry t o c he ckw hethe r o r n o t it f a 1 1s w i t h i n t h e t r i a n g u l a rr e g i o n o f h e subelement. To o th is , he p r esen t m e thodchecks th a t each s id eo f h e r i a n g l e i e s on t h e same s i d e o f b o t h h e n t e r s e c t i o n p o i n t and t h eoppos i te apex o f t h e r i a ng l e . To de te r m i neha t tw o po i n t s , A and B y l i eo n t h e same s i d e o f a l i n e r o m p o i n t s 1 t o 2, the method tak es hec r ossp r o d u c t fh e e c t o r sr o m A t o 1 and from A t o 2 and ther ossp r o d u c t o f t h ee c t o r sro m B t o 1 and from B t o 2 and checks hethero r n o t h e d o t p r o d u c t o f h e two r e s u l t i n g v e c t o r s s p o s i t i v e .36


44/70

I f he p o i n t does not f a l l w i t h i n the bounds o f e i t h e r t r i a n g u l a r subele-ment, thene xt segment of th e "l in e s checked. I f none o f the segmentsin t e r sec t s hee l emen t , henext element i s checked i n the abovemanner. Ifno element on th ecur ren t str ip con ta ins he n t e r sec t i on p o i n t , t h e e n t i r eprocedure , nc lud ing he i n i t i a l s ea rchdescr ibed i n sec t ion 8 .3 .1 , i srepeated for t he nex t st r ip , and so on, u n t i 1 e i ther an e lement o f t h e i n t e r -sec ted body and a segment of th e in te rse c t in g " l in e which con ta in he n ter -s e c t i o n p o i n t a r e f o u n d , o r al l po ss ib le combinat ions of elements an d M-linesegments areexhaus ted. f an approximate in ter se c t i on p o i n t is found i nthis manner, i t i s u s e d a s a s t a r t i n g p o i n t for th e te ra t i v e method des-cr ib ed below, t o determinea more pre c i se n te rse c t io n p o i n t . I f t h e methodf a i l s t o f i n d an approximate in te rs ec t io n p o i n t , ca l cu l a t i onscon t inue ,s t a r t i n g w i t h thenex t" l ine on th e nt er se ct in g component. n th isc a s e ,execution o f the computer program i s t ermina ted a f t e r he n t e r se c t i on ca l cu -l a t i o n ss i n c e h e f i n a lrep an el in g method req ui re s he n te rse c t io n curve t obe u n i n t e r rup t ed and compl e t e l y def i ned .8.3.3 Derivat ion of a Mathematical Representation of theSurface of anElement

In orde r t o compute a more ac cu ra te n ter se ct i on po in t , i t i s necessaryt o obta in an equation f o r th esur f ace of the element .This i s done u s i n g thetheory of param et r ic ubic ur facesdescr ibed i n references 8 and 9. Thef o l l o w i n g di scuss ion i s a sumnary of re levantpor t ions o f t he heory .

Requiredgeomet r ic quant i t i es a t each corner p o i n t o f the-e le me nt undercons idera t ion nclude he o l lowing (as well as he or responding y and zVal ues ) :

where t he ubs c r ip t s nd i ca t ed i f f e r e n t i a t i o n a n d u and w ar e h e param-e t e r s upon which thes u r f a c e f i t i s based. On the boundary curves of anelement , u r e p r e s e n t s a f r a c t i o n of t h es t r a i g h t - l i n ear c en g th betweenth e two M-1 ine boundary curves and w r e p r e s e n t s a f r a c t i o n of t h es t r a i g h t -1 n e a r c 1ength between the two N-1 ine boundary curves. On t h e i n t e r i o r o fan element, u and w r e p r e s e n tq u a n t i i e s analogous to hosedefined abov'e

37


45/70

~

b u t , o r e le me nt s w i t h compound c u r v a t u r e , h e p h y s i c a l n t e r p r e t a t i o n o f t h ep ara me te rs i s e s s o bvio us. u a t a c o r n e r n d i c a t e s h ed e r i v a t i v eo f xw i t h r e s p e c t o s t r a ig h t - l in e a r c e n g t h a l o n g h e N - l in e p a s s i n g h r o u g h h ec o r n e r , n o r m a l i z e d b y h e s t r a i g h t - l i n e d i s t a n c e b etw ee n a d j a c e n t p o i n t son th e N-1 in e . xw i n d i c a t e s h ed e r i v a t i v ew i t h e s p e c t os t r a i g h t - l i n ea r c e n g t ha l o n g h e " l i nep a s s i n g h r o u g h h e c o r n e r ,n o r m a l i z e db y h es t r a i g h t - l i n ed i s t a n c e b etw ee n a d j a c e n tpo in t s on th e l i n e . xuw andwui n d i c a t ec r o s sd e r i v a t i v e te rm s i n h e d i r e c t i o n so f h e l i n e and N - l i n e .The f i r s t d e r i v a t i v e t er ms n o tn o r m a l i z e d )w e r e p r e v i o u s l y e q u i r e d o r h ec a l c u l a t i o n s o r e d i s t r i b u t e and augment p o i n t s a l o n g h e " l i n e s and N - l i n e s .They need o n l y be n o r m a l i z e d b y h e p r o p e r e l e m e n t s i d e e n g t h o be a p p l i c -ab le e re . The c ros s er i va t i veer ms xuw and xwu are bta ine d y numer-i c a l l y d i f f e r e n t i a t i n g t h e un no rm alize d f i r s t d e r i v a t i v e term sa l on g he l i nes and N - l i n es , esp ec t i v e l y , and thenn o r m a l i z i n gb y h ep r o d u c to f h e e n g t h so f h ea d j a c e n te l e m e n ts i de s . The tw o c r o ss - de r i va t i veva lues a t each co r ne ra r e henave r aged ,s ince hee q u a ti o n u se d t o r e p r e s e n tth e s u r f a c e m p l ie s h a t h e y a r e e q u a l .These geomet r i cq u a n t i i e s , o r x, y , and z a n d f o r e a c h c o r n e ro f h ee lem en t, co ns t i t u te he s o - c a l l e d " p a r a m e t r i cc u b i cp a t c h c o e f f i c i e n t s i ngeomet r i c fo rm." The c o e f f i c ie n ts canbe ar ranged in m a t r i x o r m as f o l lo w s(shown o n ly or h e x c o o r d i n a t e s ) :

I x10 xll xwlo xwC G X l = x X X "uoo uo l uwoo uwolX

I xu lo x u l l X XuwluwlL(8.3.5)

I n h i sm a t r i x h e u and w s u b s c r i p t sa g a i n e p r e s e n tde r i va t i ve s as bove.The 0 and 1 s u b s c r i p t s o g e t h e r n d i c a t e h e o r n e ro f he e l em ent e i ngc o ns id e re d , h e f i r s t n d i c a t i n g t h e N-1 i n e and t h e second i n d i c a t i n g t h e" l i ne . The 0 i n d i c a t e she lower -numbered l i n e on the lement and th e 1i n d i c a t e s h eh i g h e r - n u m b e r e d i n e .S i m i l a rm a t r i c e sa r ea l s o o r m e d o r h ey and z-coordinatee r m s .38


46/70

T h e matrixo f c o e f f i c i e n t s i n geometric form, [G,], and the correspondingand z mat r i ces on t a in a l l the information needed to derive a l g e b r a i c

expres s ions fo r the coo rdina tes a t any poin t on the sur f ace of the element i nterms of the paramet r icvar iab les . These ex pre ssio nsare of the fol lowing form(again shown only for the x coord ina t es ) :

x ( u , w ) = w3 3Axu + Bxu 2 + C X u + D X ]+ w2 3E xu + Fxu2 + G,u + Hxl+ w [Ixu 3 + Jxu2 + Kxu + Lx]+ [Mxu 3 + Nxu 2 + PX u + Px]

The coef f ic ients o f equ ation 8.3.6) canbe"matrix o f c o e f f i c i e n t s i n a l g e b r a i c form,"

p x Bx cxE x Fx GxIX J x K xCAX 1 =

L M X Nx b,To show how themat r ix of c o e f f i c i e n t s

(8.3.6)

grouped t o form th e so-ca l l ed[A,] a s fo l lo ws :

IIX (8.3.7)i n geometric form may e co nvertedi n t o themat r ix o f c o e f f i c i e n t s i n a l g e b r a i c form, i t i s f i r s t necessary t o s t u d yth ep r o p e r t i e s of equa t ion 8 .3 .6 ) u r the r . On the boundaries of the elementone of th epa ramet r i c va r i ab l es i s cons t an t , e i t he rze ro o rone, and the o t h e rv a r i e s from zero t o one. For de f in i t en ess , assume th a t hev a r i a b l e w i sequal to er o . Equation (8.3.6) theneduces t o a cubic quat ion w i t h u asth eonlyndependent var iable . On the oppo s i t e i de o f the element, w i sequal t o one and equation (8.3.6) reduces toano the rcubicequat ion,againw i t h u a s the only ndependentvar iable. T h e en t i r e u r f ace o f t he e lementcan be considered to be a c o l l e c t i o n o f cubiccurves w i t h w constant and uva ria bl e. Each of the secur ve s has th e form ofeq uat ion 7.1 .2), which anbeconverted to h e form

f ( u ) = F 1 ( u ) f ( 0 ) + F 2 ( u ) f ( l ) + F 3 ( u ) f ' ( 0 ) + F 4 ( u ) f ' ( l ) (8.3.8)39


47/70

b y s o l v i n g o r h e secondand t h i r d d e r i v a t i v e te rm s i n (7.1.2) as de scr ibe d i nse ct io n 7.1. When t h i s s done, the erm s F1 (u),F2(u),F3(u),. and F4 (u )areg i v e n b y

F1(u) 2~ - ~ + 1F2(u) = 02u + 3uF3(u) = u3 - 2u + u

3 2F4(u) = u -

where

CMI =-2 +11t 3 -21

0 + 1 00 0 0

I n m a t r i xn o t a t i o n ,e q u a t i o n (8.3.8) becomesF(u ) = [F] - [ f 0 ) f ( 1 ) f ' ( 0 ) f'(l)lT

(8.3.9)

(8.3.10)

(8.3.11 )

(8.3.12)= [ u 3 2 u 1 1 [M [ f ( O ) f ( 1 ) f ' ( 0 ) f ' ( 1 ) I T

where T i n d i c a te s h a t h e r a n s p o s e o f t h em a t r i x s t o be tak en . Now, us in g(8.3.12), the qu at io ns for x(u) and xw(u) on the boundaryurves w = 0and w = 1 can be w r i t t e n as3 2X(U,O) = [u u u 1 1 [MI . [xoo xl0 xu IT

00 xu lo

X w ( U , l ) = [U u u 13 [MI 9 [X3 2 X X01 wllwolu w l l IT40


48/70

or more compactly(8.3.14)

Equation (8.3.12) can a l so be used t o derive th e expressionx ( u , w ) = [w 3 2 W 11 [MI [ X ( U , O ) X(u,1) X w ( U , o ) x w ( u , l ) l T (8.3.15)

or equ iva l en t ly

Combining (8.3.1.4) and (8.3.16)givesx ( U , w ) = [u-? U 2 u 11 [MI [G,] [MIT [w w w 132 T

Sinceequat ion (8.3.6) canbe rearranged and w ri t te n asX ( U , W ) = [u3 u 2 u !I * LAXI [ W w w 1 I2 T

(8.3.17)

(8.3.18)the matrixo f c o e f f i c i e n t s i n geometric form i s converted t o thematr ixofc o e f f i c i e n t s i n algebraic form by the

geometry package for generation of input data for a three-dimensional potential-flow program

Documents