Kanat Profil Tasarımı İçin Genetik Algoritma Ve Artık Düzeltme Yöntemi

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İSTANBUL TECHNI CAL UNI VERSI TY  INSTI TUTE OF SCI ENCE AND TECHNOLOGY

M. Sc. Thesi s by Bül ent TUTKUN, B. Sc.

De part me nt : Space Engi neeri ng Progra mme: Space Engi neeri ng

MAY 2003

GENETI C ALGORI TH M AND RESI DUAL CORRECTI ON METHOD FOR I NVERS E DESI GN OF AI RFOI LS

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İSTANBUL TECHNI CAL UNI VERSI TY  INSTI TUTE OF SCI ENCE AND TECHNOLOGY

M. Sc. Thesi s by Bül ent TUTKUN, B. Sc.

(511001208)

Dat e of sub missi on : 5 May 2003 Dat e of defence exa mi nati on: 27 May 2003

Supervi sor ( Chai r man) : Prof. Dr. R. Alsan MERİ Ç Me mbers of t he Exa mi ni ng Co mmi ttee Prof. Dr. Sül ey man TOLUN

Prof. Dr. A. Rüste m ASLAN

MAY 2003

GENETI C ALGORI THM AND RESI DUAL CORRECTI ON METHOD FOR I NVERS E DESI GN OF AI RFOI LS

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İSTANBUL TEKNİ K ÜNİ VERSİ TESİ  FEN Bİ Lİ MLERİ ENSTİ TÜS Ü

GENETİ K ALGORİ TMA VE ARTI K DÜZELT ME YÖNTE Mİ YLE PROFİ L TERS TASARI MI

YÜKSEK Lİ SANS TEZİ Uzay Müh. Bül ent TUTKUN

(511001208)

MAYI S 2003

Tezi n Enstitüye Veril diği Tari h : 5 Mayı s 2003 Tezi n Savunul duğu Tari h : 27 Mayı s 2003

Tez Danı ş manı : Prof. Dr. R. Alsan MERİ Ç Di ğer Jüri Üyel eri Prof. Dr. Sül ey man TOLUN

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ACKNO WLEDGE MENTS

I woul d li ke t o t hank my super visor Pr of. Dr. R. Al san MERİ Ç f or his val uabl e support and encour age ment duri ng t hi s st udy.

Of course, I woul d al so l i ke t o t hank t o my f a mily. Their support i s al ways wi t h me whenever I need it.

And I woul d li ke t o send speci al t hanks t o a speci al person f or me.

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TABLE OF CONTENTS LI ST OF TABLES i v LI ST OF FI GURES v LI ST OF SYMBOLS vii ÖZET i x SUMMARY x 1. I NTRODUCTI ON 1 2. GENETI C ALGORI THMS 5

2. 1 Outli ne of a Genetic Al gorit hm 7

2. 2 Geneti c Operat ors 8

2. 3 Fit ness Assi gn ment 9

2. 4 Sel ecti on Met hods 9

3. B- SPLI NE CURVES 11

4. POTENTI AL FLOW AND PANEL METHOD 15

4. 2 Hess- Smit h Panel Met hod 16

5. I NVERS E DESI GN OF AI RFOI LS 27

5. 1 Inverse Desi gn with Geneti c Al gorit hm 27

5. 2 Inverse Desi gn with Resi dual Correcti on Algorit hm 28

6. RESULTS AND CONCLUSI ON 31

6. 1 Results of Inverse desi gn wit h Geneti c Al gorit hm 32

6. 2 Results of Inverse Desi gn wit h Resi dual Correcti on Al gorit hm 34

6. 3 Concl usi on 41

REFERENCES 42

APPENDI X A 44

APPENDI X B 60

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LI ST OF TABLES

Page nu mber Tabl e 2. 1 The correspondence of t er ms bet ween nat ural and artifi ci al

evol uti on 7

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LI ST OF FI GURES

Page Nu mber

Fi gure 1. 1 Defi niti on of an airfoil 1

Fi gure 1. 2 Aer odyna mi c force and mo me nt on t he body 2

Fi gure 1. 3 Aer odyna mi c force and its co mponent s 2

Fi gure 2. 1 Ge neti c al gorit hm fl owchart 7

Fi gure 3. 1 Eff ect s of var yi ng j ust one of t he poi nt s on B-spli nes 12

Fi gure 3. 2 Successi ve B-spli nes j oi ned t oget her 13

Fi gure 4. 1 Ai rf oil Anal ysi s No menclat ure for Panel Met hods 17

Fi gure 4. 2 Defi niti on of Nodes and Panel s 18

Fi gure 4. 3 Local Panel Coor di nat e Syst e m 20

Fi gure 4. 4 Ge o met ri c Int er pret ati on of Sour ce and Vort ex

Induced Vel ociti es 23

Fi gure 5. 1 Inverse desi gn wit h genetic al gorit hm fl ow chart 28 Fi gure 5. 2 Inverse desi gn wit h resi dual correcti on al gorit hm

fl ow chart 30

Fi gure 6. 1 Tar get geo met r y and pressure di stri buti on 31 Fi gure 6. 2 Res ults of geneti c al gorith m aft er generati on 200 32 Fi gure 6. 3 Res ults of geneti c al gorith m aft er generati on 1000 33 Fi gure 6. 4 Res ults of geneti c al gorith m aft er generati on 4000 34 Fi gure 6. 5 Res ults of resi dual correcti on al gorit hm aft er

iterati on 80 ( A, B, C=1) 35

Fi gure 6. 6 Res ults of resi dual correcti on al gorit hm aft er

iterati on 2931 ( A, B, C=3) 38

iterati on 789 ( A, B, C=10) 39

iterati on 9763 ( A, B, C=10) 40

Fi gure A. 1 Res ults of geneti c al gorith m aft er generati on 200 44 Fi gure A. 2 Res ults of geneti c al gorith m aft er generati on 400 45 Fi gure A. 3 Res ults of geneti c al gorith m aft er generati on 800 46 Fi gure A. 4 Res ults of geneti c al gorith m aft er generati on 1200 47 Fi gure A. 5 Res ults of geneti c al gorith m aft er generati on 1800 48 Fi gure A. 6 Res ults of geneti c al gorith m aft er generati on 2500 49 Fi gure A. 7 Res ults of geneti c al gorith m aft er generati on 3500 50

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Page Nu mber

Fi gure A. 8 Res ults of geneti c al gorith m aft er generati on 4000 51 Fi gure A. 9 Res ults of resi dual correcti on al gorit hm aft er

Fi gure A. 10 Res ults of resi dual correcti on al gorit hm aft er

iterati on 977 ( A, B, C = 1) 53

iterati on 316 ( A, B, C = 4) 54

iterati on 3907 ( A, B, C = 4) 55

iterati on 474 ( A, B, C = 6) 56

iterati on 5860 ( A, B, C = 6) 57

iterati on 1183 ( A, B, C = 15) 58

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LI ST OF SYMBOLS

L : Lift

D : Dr ag

N : Co mponent of R ( nor mal t o t he airfoil chor d)

A : Co mponent of R ( Tangenti al t o t he airfoil chor d)

M : Mo ment

R : Force



V : Free strea m vel ocit y

c : Chor d of airfoil

cl : Lift coeffi ci ent

cd : Dr ag Coeffi ci ent

cp : Pressure coeffi ci ent

Tcp : Tar get pressure coeffi cient

cm : Mo ment coeffi ci ent

p : Pressure

u : Vel ocit y i n hori zont al directi on

v : Vel ocit y i n verti cal directi on

sij

u , vsij : Vel ocit y co mponent s at t he mi dpoi nt of panel i i nduced by a s our ce

of unit strengt h at t he midpoi nt of panel j.

vij

u , v_vij : Vel ocit y co mponent s at t he mi dpoi nt of panel i i nduced by a vort ex of unit strengt h at t he midpoi nt of panel j.

* *

v ,

u : Local vel ociti es

q : Source strengt h

s : arc-l engt h coor di nat e

ri j : Dist ance from t he mi dpoi nt of panel i t o t he jt h node  : Shear stress

 : Angl e of att ack 

 : Free strea m densit y  q : V2 2 1   

 : pot enti al functi on 

 : Free strea m pot enti al functi on

s

 : Source di stri buti on pot enti al functi on

v

 : Vort ex di stri buti on pot enti al functi on  : strea m f uncti on

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 : Vort ex strengt h  : Panel angl e

nˆ : vect or nor mal t o t he panel

tˆ : vect or t angenti al t o t he panel

ri j : Dist ance from t he mi dpoi nt of panel i t o t he jt h node

ij

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GENETİ K ALGORİ TMA VE ARTI K DÜZELT ME YÖNTE Mİ YLE PROFİ L TERS TASARI MI

ÖZET

Ha va- uzay al anı nda t asarı mlanan kanat pr ofil şekill eri perf or mans ve i hti yaçl arı n karşılanması açısı ndan hayati r ol oynar. Bu nedenl e bu konuda bir çok çaba harcan makt adır. Yeni bir kanat pr ofil şekli t asarlarken, araştır macıl ar genelli kl e opti mizasyon veya t ers t asarı m t ekni kl eri ni kull anırlar. Opti mizasyonda t aşı ma, sürükl e me ve mo ment gi bi pr ofile ait bazı paramet rel er mi ni mize veya ma ksi mize edil meye çalışılır. Hal buki, t ers t asarı mda i se verilen bir para metre i çi n, ( bu genelli kl e bası nç dağılı mı dır) o para metreyi sağl ayan profil şekli bul un maya çalışılır.

Bu çalış mada, verilen hedef değerl eri sağl ayan bir pr ofil geo metrisi i ki f ar klı t ers tasarı m yönt e mi il e elde edil miştir. İ ki yönt emi n sahi p ol duğu al gorit mal ar da farklı dır. İl k yönt e m geneti k al gorit ma kull anmakt adır. Bu yönt e mde ayrı ca, şekil para metrel eri ni azalt mak i çi n B-spli ne eğrileri nden yararlanıl mıştır. Bu yönt e mi n

a macı 







 n 1 i 2 p 2 p ) (c ) Tc ( i

i değeri ni maksimi ze et mektir. Bur adaki Tcpi hedef bası nç dağılı mı ve

i p

c tasarlanan bası nç dağılı mıdır. Tasarl anan pr ofil geo met risi ni n anali zi nde, Smit h- Hess panel yönt e mi kull anıl mıştır.

İki nci t ers t asarı m yönt emi nde, artı k düzelt me al gorit ması kull anıl mıştır. Belli bir pr ofil geo metrisi ile başlayarak ( NACA 0012), her adı mda hesapl anan Y’leri n yar dı mı yl a hedef pr ofil geo metrisi ne ul aşılır. Y’l er

2 2 t 2 2 V V dx Y d C dx Y d B Y

A       di feransi yel denkl e mi kull anıl arak hesapl anırlar. İl k ol arak bu diferansi yel denkl e m sonl u farkl ar yakl aşı mı kull anıl arak ayrı kl aştırılır. Daha sonra, el de edil en üç-bant katsayıl ar matrisi Tho mas al gorit ması nı n yar dı mı yla çözül ür ve Y ’ler el de edilir. Bu yönt e mde biri ncisi gi bi anali z içi n Smit h- Hess panel yönt e mi ni kullanır.

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GENETI C ALGORI THM AND RESI DUAL CORRECTI ON METHOD FOR I NVERS E DESI GN OF AI RFOI LS

SUMMARY

In t he fi el d of aer ospace, desi gned airfoil shapes pl ay a cr uci al r ol e i n t er ms of perf or mance and meeti ng t he r equire ment s. So, ma ny eff orts are put on t hi s subj ect i n aer ospace. Whil e desi gni ng a ne w ai rfoil shape, researchers generall y use opti mizati on or i nverse desi gn t echni ques. I n opti mizati on, so me par a meters ( lift, dr ag, mo ment, et c.) of t he airfoil are tri ed t o be mi ni mi zed or maxi mi zed. Ho wever, i n i nverse desi gn, an ai rfoil shape i s desi gned for a gi ven par a met er (gener all y pr essure di stri buti on).

In t hi s wor k, t wo i nverse desi gn met hod wit h di fferent al gorit hms ar e used t o desi gn an airfoil geo met r y t hat fits t o gi ven t ar get val ues. First met hod utili zes a geneti c al gorit hm whi ch i s a search met hod. I n t he first met hod, al so B- spli ne cur ves ar e used t o decrease shape para met ers. Thi s met hod’s pur pose i s t o maxi mi ze t he

2 p n 1 i p Tc ) c ( i i  



 . Wher e i p

Tc is t he t ar get pr essure di stri buti on and i p

c is t he desi gned pr essure di stributi on. To anal yze t he desi gned airfoil geo met ries, Smit h-Hess panel met hod is used.

In second i nverse desi gn met hod, r esi dual correcti on al gorit hm i s utili zed. St arti ng wi t h an i niti al airfoil geomet r y ( NACA 0012), t arget airfoil geo met r y i s r eached wi t h t he hel p of Y ’s co mi ng from t he ₂ _t2 2 2 V V dx Y d C dx Y d B Y A       . First, t hi s differenti al equati on i s discritized wi t h fi nit e differences. Then obt ai ned t ri -di agonal coeffi ci ent mat ri x i s sol ved wi t h t he Tho mas Al gorit hm t o gi ve Y’s. Thi s met hod al so uses Smit h- Hess panel met hod t o anal yze t he airfoils.

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1. I NTRODUCTI ON

Wi t h t he advent of s uccessf ul po wer ed fli ght at t he t ur n of t he t wenti et h cent ur y, t he i mport ance of aer odynami cs r ose suddenl y. So, i nt erest gr e w i n t he underst andi ng of t he aer odyna mi c acti on of such lifti ng s urfaces as fi xed wi ngs on ai r pl anes and, l at er, rot ors on heli copt ers [1]. Consi der a wi ng as dra wn i n perspecti ve i n Fi gure (1. 1).

Fi gure 1. 1 Defi niti on of an airfoil

The wi ng ext ends i n t he y di recti on. The free strea m vel ocit y V i s par all el t o t he xz _ pl ane. Any secti on of t he wi ng cut by a pl ane parall el t o t he xz pl ane i s call ed an airfoil. The lift and mo ment s on t he airfoil are due mai nl y t o t he pr essur e di stri buti on.

The first Pat ent ed airfoil s hapes wer e devel oped by Hor ati o F. Philli ps i n 1884 [ 1]. Cl earl y, i n t he earl y days of po wer ed fli ght, airfoil desi gn was basi call y cust o mi zed and

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personali zed. Just mentioned above, t he aer odyna mi c f orces and mo me nt s on t he airfoil are due t o onl y t wo basi c sources:

a) Pressure di stri buti on over t he body surface b) Shear stress di stri buti on over t he body surface

The net effect of t he p and  di stri buti ons i nt egrated over t he co mpl et e body s urface is a result ant aer odyna mi c force R and mo ment M on t he body ( Fi gure 1. 2)

Fi gure 1. 2 Aer odyna mi c force and mo ment on t he body

Then t he result ant R can be split i nt o co mponent s. ( Fi gure 1. 3)

Fi gure 1. 3 Aer odyna mi c force and its co mponent s

The angl e of att ack is defi ned as t he angl e bet ween c and V . Fr o m geo me tri cal _ rel ati ons:

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        cos A sin N D sin A cos N L (1. 1)

As t hi s f or mul ati on s hows, L and D val ues of an airfoil are det er mi ned by di rectl y pr essure di stri buti on. If we use di mensi onl ess quantiti es, t hey are defi ned as foll o ws: Li ft coeffi ci ent; c q L c_l   (1. 2) Dr ag coeffi ci ent; c q D c_d   (1. 3)

Pr essure coeffi ci ent

    q p p c_p (1. 4) Mo me nt coeffi ci ent 2 m c q M c   (1. 5) wher e V2 2 1 q_  _ _ (1. 6)

Si nce lift and mo ment s co me fr o m t he pr essure di stri buti on on t he airfoil, t o cr eat e t he r equired lift or mo ment, airfoil geo met r y must f or m a s pecifi ed pr essur e di stri buti on. Ther ef ore, for many years r esearchers have st udi ed har d on airfoil desi gn t echni ques.

The aer odyna mi c desi gn of aircraft co mponent s i s oft en carri ed by means of one of t he foll owi ng four appr oaches:

1. ‘ ‘Cut and Tr y’ ’ anal ysis 2. Indirect Met hods

3. Opti mizati on Techni ques 4. Inverse Desi gn Techni ques

Thi s wor k i ncl udes mai nly i nverse desi gn and partl y opti mizati on.

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constit ut es an i nverse desi gn pr obl e m. I n s ol uti on of t hi s pr obl e m, opti mizati on wi t h a geneti c al gorit hm i s used. That i s, i nverse pr obl e m i s transf or med i nt o an opti mizati on pr obl e m. Wi t h t he hel p of genetic al gorit hm, t he opti mum s et of 20 contr ol poi nt s i s f ound. Then t hi s set i s used t o for m t he airfoil shape by utili zi ng an al gorit hm f or dra wi ng a B- spli ne cur ve [2].

Ge neti c al gorit hms ar e search met hods used i n r ecent years. They di ffer i n concepti on fr o m ot her search met hods, i ncl udi ng t raditi onal opti mizati on met hods and ot her st ochasti c search met hods. The basi c di fference i s t hat whil e ot her met hods al ways pr ocess si ngl e poi nt s i n t he search space, geneti c al gorit hms mai nt ai n a popul ati on of pot enti al sol uti ons [3].

Shape opti mizati on based on geneti c al gorit hm [ 4], or based on evol uti onar y al gorit hms i n general, is a r el ati vel y young and pot enti al fi el d of r esear ch. The i nt erest t owar ds r esearchi ng evol uti onar y shape opti mizati on t echni ques appears t o be j ust st art ed t o gr ow, rat her t han reached a st abl e and mat ure st at e.

Currentl y t he most popul ar appli cati on ar ea of geneti c al gorit hms- based s hape opti mizati on see ms t o be t he s hape opti mizati on i n connecti on wi t h co mput ati onal fl ui d dyna mi cs ( CFD), especi all y aer odyna mi c shape opti mizati on i n t he fi el d of aircraft desi gn, for exa mpl e [5- 12].

Usi ng B- spli nes i n an opt i mizati on pr obl e m i s very hel pf ul i n a way t hat it lessens t he desi gn par a met ers. As a r esult of t hi s, cost of t he al gorit hm i s al so l essened. Thi s ki nd of appli cati on of B-spli nes may be seen i n literat ure, for exa mpl e [13, 14].

Anot her part of t hi s work i ncl udes an i nverse desi gn met hod i n whi ch a r esi dual correcti on al gorit hm i s used. Wit h t hi s al gorit hm, an airfoil shape t hat gi ves t he t ar get pr essure di stri buti on i s reached. I nverse desi gn met hod i s a ver y popul ar met hod i n aer ospace, for exa mpl e [15- 18].

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2. GENETI C ALGORI THMS

Ge neti c al gorit hms constit ut e a cl ass of search met hods especi all y suit ed for s ol vi ng co mpl ex opti mizati on pr obl e ms [ 3]. Search al gorit hms i n general consi st of syst e mati call y wal ki ng t hr ough t he search s pace of possi bl e sol uti ons until an accept abl e sol uti on i s found. Geneti c al gorit hms transpose t he noti ons of nat ural evol uti on t o t he worl d of co mput ers, and i mi tat e nat ural evol uti on. They wer e i niti all y i ntroduced by John Holl and [ 4] f or expl ai ni ng t he adapti ve processes of nat ural syst e ms and f or creati ng ne w artifi ci al syste ms t hat wor k on si mil ar bases. I n nat ure, ne w or gani s ms adapt ed t o t heir envir on ment devel op t hr ough evol uti on. Ge neti c al gorit hms evol ve s ol uti ons t o t he gi ven pr obl e m i n a si mil ar wa y. They mai nt ai n a coll ecti on of s ol uti ons---- a popul ati on of i ndi vi dual s----and s o perf or m a mul ti directi onal search. The i ndi vi dual s are r epresent ed by chr o mos o mes co mposed of genes. Geneti c al gorit hms operat e on t he chr o mos o mes, whi ch r epresent t he i nherit abl e pr operti es of t he i ndi vi dual s. By anal ogy wi t h Nat ure, t hr ough sel ecti on t he fit i ndi vi dual s---- potenti al sol uti ons t o t he opti mizati on pr obl em---- li ve t o repr oduce, and t he weak i ndi vi dual s, whi ch ar e not so fit, di e off. Ne w i ndi vi dual s are cr eat ed fr o m one or t wo par ent s by mut ati on and cr ossover, respecti vel y. They repl ace ol d i ndi vi dual s i n t he popul ati on and t hey are usuall y si mil ar t o t heir parent s. In ot her wor ds, i n a new gener ati on t here will be i ndi vi dual s t hat r esembl e t he fit i ndi vi dual s fr o m t he pr evi ous generati on. The i ndi vi dual s sur vi ve i f t hey are fitt ed t o t he gi ven envir on ment.

In t abl e 1 t he anal ogy of t er ms bet ween nat ure and artifi ci al evol uti onar y syst e ms i n general.

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Tabl e 2. 1 The correspondence of t er ms bet ween nat ural and artifi ci al evol uti on

Nat ure Evol uti onary co mput ation

Indi vi dual Sol uti on t o a pr obl e m Popul ati on Coll ecti on of sol uti on

Fit ness Qualit y of sol uti on

Chr o mos o me Repr esent ati on of a sol uti on

Ge ne Part of represent ati on of a sol uti on

Cr ossover Bi nar y search operat or

Mut ati on Unar y search operat or

Repr oducti on Reuse of sol uti ons

Sel ecti on Keepi ng good sub-sol utions

Evol uti on i s an e mer gent pr opert y of artifi ci al evol uti onar y s yst e ms. The co mput er i s onl y t ol d t o ( 1) mai nt ai n a popul ati on of s ol uti ons, ( 2) all ow t he fitt er i ndi vi dual s t o repr oduce, and ( 3) l et t he l ess fit i ndi vi dual s di e off. The ne w i ndi vi dual s i nherit t he pr operti es of t heir parents, and t he fitt er ones s urvi ve f or t he next generati on. The fi nal sol uti ons will be much bett er t han t heir ancest ors from t he first generati on.

Thi s evol uti on i s direct ed by fit ness. The evol utionar y search i s conduct ed t o war ds bett er r egi ons of t he search s pace on t he basi s of t he fit ness measure. Each sol uti on i n a popul ati on i s eval uated based on ho w we ll it sol ves t he gi ven pr obl e m. Correspondi ngl y, each me mber of t he popul ati on is assi gned a fit ness val ue. Geneti c al gorit hms use a separate search space and s ol uti on s pace. The search space i s t he space of coded s ol uti ons, i. e. genot ypes or chr omos o mes consi sti ng of genes. Mor e exactl y, a genot ype ma y consi st of several chro mos o mes, but i n mos t practi cal appli cati ons genot ypes are made of one chr o mos ome. The s ol uti on s pace i s t he s pace of act ual sol uti ons, i. e. phenot ypes. Any genot ype must be t ransf or med i nt o t he correspondi ng phenot ype bef ore its fit ness is eval uat ed.

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2. 1 The outli ne of a geneti c al gorit hm

Whe n s ol vi ng a pr obl e m usi ng geneti c al gorit hms, first a pr oper r epresent ati on and fit ness measur e must be desi gned. Many r epresentati ons are possi bl e, and wi ll wor k. So me ar e bett er t han t he ot hers, ho wever. Devi sing t he t er mi nati on crit erion s houl d be t he next st ep. The t ermi nati on crit eri on us ually all ows at most so me pr edefi ned nu mber of it erati ons and verifi es whet her an accept abl e sol uti on has been f ound. The geneti c al gorit hm t hen wor ks as foll ows (also shown i n Fi gure 2. 1):

Fi gure 2. 1 Geneti c al gorit hm fl owchart

1. The i niti al popul ati on i s fill ed wi t h i ndi vi dual s t hat are generall y creat ed at rando m. So meti mes, t he i ndi vi dual s i n t he i niti al popul ati on ar e t he s ol utions f ound by s o me met hod det er mi ned by t he pr obl e m domai n. I n t hi s case, t he scope of t he geneti c al gorit hm i s t o obtai n mor e accurat e sol utions.

2. Each i ndi vi dual i n t he current popul ati on is evaluat ed usi ng t he fit ness measure.

3. If t he t er mi nati on crit eri on is met, t he best sol uti on is ret ur ned.

4. Fr o m t he current popul ati on, i ndi vi dual s are sel ect ed based on t he pr evi ousl y co mput ed fit ness val ues. A ne w popul ati on i s f or med by appl yi ng t he geneti c

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operat ors (repr oducti on, cr ossover, mut ati on) t o t hese i ndi vi dual s. The sel ect ed i ndi vi dual s are call ed par ent s and t he r esulti ng i ndi vi dual s offspri ng. I mpl e ment ati ons of geneti c al gorit hms di ffer i n t he way of constr ucti ng t he ne w popul ati on. So me i mpl eme nt ati ons ext end t he current popul ati on by addi ng t he ne w i ndi vi dual s and t hen cr eat e t he ne w popul ati on by o mitti ng t he l east fit indi vi dual s. Ot her i mpl e ment ati ons creat e a separat e popul ation of ne w i ndi vi dual s by appl yi ng t he geneti c operat ors. Mor eover, t here are geneti c al gorit hms t hat do not use generati ons at all, but conti nuous repl ace ment.

5. Acti ons st arti ng fr o m st ep 2 ar e r epeat ed until the t er mi nati on crit eri on i s satisfi ed. An it erati on is call ed generati on.

2. 2 Geneti c operat ors

In each generati on, t he geneti c operat ors are applied t o sel ect ed i ndi vi duals fr o m t he current popul ati on i n order t o cr eat e a ne w popul ati on. Generall y, t he t hr ee mai n geneti c operat ors of r eproducti on, cr ossover and mut ati on ar e e mpl oyed. By usi ng different pr obabiliti es f or appl yi ng t hese oper at ors, t he s peed of conver gence can be contr oll ed. Cr ossover and mut ati on oper at ors must be caref ull y desi gned, si nce t heir choi ce hi ghl y contri but es t o t he perf or mance of t he whol e geneti c al gorit hm.

Repr oducti on: A part of t he ne w popul ati on can be cr eat ed by si mpl y copyi ng wi t hout change sel ect ed i ndi vi dual s fr o m t he present popul ati on. Thi s gi ves t he possi bilit y of sur vi val for already devel oped fit soluti ons.

Cr ossover: Ne w i ndi vi dual s are generall y cr eated as offspri ng of t wo parent s ( as such, cr ossover bei ng a bi nar y oper at or). One or mor e s o-call ed cr ossover poi nt s ar e sel ect ed ( usuall y at r andom) wi t hi n t he chr o mos ome of each parent, at t he sa me pl ace i n each. The part s deli mited by t he cr ossover poi nts are t hen i nt erchanged bet ween t he parent s.

Mut ati on: A ne w i ndi vidual i s creat ed by maki ng modi fi cati ons t o one sel ect ed i ndi vi dual. The modi fi cati ons can consi st of changi ng one or mor e val ues i n t he represent ati on or i n addi ng/ del eti ng parts of t he r epresent ati on. I n geneti c al gorit h ms mut ati on i s a sour ce of vari abilit y, and i s applied i n additi on t o cr ossover and

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repr oducti on. At different st ages of evol uti on, one may use di fferent mut ati on operat ors. At t he begi nni ng mut ati on operat ors r esulti ng i n bi gger j u mps i n t he sear ch space mi ght be pr eferred. Lat er on, when t he s ol uti on i s cl ose by, a mut ati on oper at or leadi ng t o sli ght er shifts in t he search space coul d be favor ed.

2. 3 Fit ness assi gn me nt

The pr obabilit y of sur vival of any i ndi vi dual i s det er mi ned by it s fit ness: t hr ough evol uti on t he fitt er i ndi vi dual s overt ake t he l ess fit ones. I n or der t o evol ve good sol uti ons, t he fit ness assi gned t o a sol uti on must directl y reflect its ‘ goodness’, i. e. t he fit ness f uncti on must i ndi cat e ho w well a sol uti on f ulfills t he r equire ment s of t he gi ven pr obl e m. Fit ness assi gn ment can be perf or med i n several different ways:

 We defi ne a fit ness f uncti on and i ncor por at e it i n t he geneti c al gorit h m. Wh en eval uati ng any i ndi vi dual, t his fit ness functi on is co mput ed for t he i ndi vi dual.  Fit ness eval uati on i s perfor med by dedi cat ed separat e anal ysis soft war e. I n

such cases eval uati on can be ti me- consu mi ng, t hus sl owi ng do wn t he whol e evol uti onar y al gorit hm.

 So meti mes t here i s no expli cit fit ness f uncti on, but a hu man eval uat or assigns a fit ness val ue t o t he sol uti ons present ed t o hi m/ her.

 Fit ness can be assi gned by co mpari ng t he i ndi vi dual s i n t he current popul ati on.

2. 4 Sel ecti on met hods

Onl y sel ect ed i ndi vi duals of a popul ati on are all owe d t o have offspri ng. Sel ecti on i s based on fit ness: i ndi vi dual s wit h bett er fit ness val ues are pi cked mor e frequentl y t han i ndi vi dual s wit h wor se fit ness val ues. The most co mmonl y used sel ecti on sche mes:

Fit ness- proporti onal selecti on: When usi ng t hi s sel ecti on met hod, a sol uti on has a pr obabilit y of sel ecti on di rectl y pr oporti onal t o it s fit ness. The mechani s m t hat all ows fit ness pr oporti onal sel ecti on i s si mil ar t o a r oul ett e wheel t hat i s partiti oned i nt o sli ces. Each i ndi vi dual has a share dir ectl y pr oporti onal t o it s fit ness. When t he

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roul ett e wheel i s r ot at ed, an i ndi vi dual has a chance of bei ng sel ect ed correspondi ng t o its share.

Ranked sel ecti on: The probl e m of fit ness- pr oporti onal sel ecti on i s t hat it i s directl y based on fit ness. I n most cases, we cannot defi ne an accur at e measur e of goodness of a s ol uti on, so t he assi gned fit ness val ue does not express exactl y t he qualit y of a sol uti on. Still, an i ndi vidual wit h bett er fit ness val ue i s a bett er i ndi vi dual. I n r ank based sel ecti on, t he i ndi vi dual s are or dered accor di ng t o t heir fit ness. The i ndi vi dual s are t hen sel ect ed wit h a probabilit y based on so me li near functi on of t heir rank.

Tour na ment sel ecti on: I n t our na ment sel ecti on, a set of n i ndi vi dual s ar e chosen fr o m t he popul ati on at r ando m. Then t he best of t he pool i s sel ect ed. For n =1, t he met hod is equi val ent t o r ando m s el ecti on. The hi gher i s t he val ue of n, t he mor e di rect ed t he sel ecti on is t owar ds bett er i ndi vi dual s.

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3. B- SPLI NE CURVES

For mul a for a cubi c B-spli ne i n t er ms of para met ric equati ons whose para met er is u. Gi ven t he poi nt s pi (xi,yi), i0,1,....,n, t he cubi c B-spli ne f or t he i nt er val

is , 1 n ,..., 2 , 1 i ), p , p ( i i1        2 1 k k i k i(u) b p B , where (3. 1) 6 ) u 1 ( b 3 1    , 3 2 u 2 u b 2 3 0    , (3. 2) 6 1 2 u 2 u 2 u b 2 3 1    , 6 u b 3 2  , 0u1

pi refers t o t he poi nt ( xi , yi ); it i s a t wo- co mponent vect or. The coeffi ci ents, t he bk’s, ser ve as a basi s and do not change as we move from one set of poi nt s t o t he next. Obser ve t hat t hey can be consi dered wei ghti ng f act ors appli ed t o t he coordi nat es of a set of f our poi nt s. The wei ght ed su m, as u vari es from 0 t o 1, generat es t he B- spli ne cur ve.

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2 i 3 1 i 2 3 i 2 3 1 i 3 i x u 6 1 x ) 1 u 3 u 3 u 3 ( 6 1 x ) 4 u 6 u 3 ( 6 1 x ) u 1 ( 6 1 ) u ( x               (3. 3) 2 i 3 1 i 2 3 i 2 3 1 i 3 i y u 6 1 y ) 1 u 3 u 3 u 3 ( 6 1 y ) 4 u 6 u 3 ( 6 1 y ) u 1 ( 6 1 ) u ( y              

Not e t he not ati on her e: xi ( u) and yi ( u) are f uncti ons of u and xi , yi are co mponent s of t he poi nt p. The u- cubi cs act as wei ghti ng f actors on t he coor di nat es of t he f our successi ve poi nt s t o generat e t he cur ve. For exampl e, at u = 0, t he wei ght s appli ed are 1/ 6, 2/ 3, 1/ 6 and 0. At u =1, t hey ar e 0, 1/ 6, 2/ 3, and 1/ 6. These val ues var y t hr oughout t he i nt er val fro m u = 0 t o u = 1.

No w we can exa mi ne t wo B- spli nes det er mi ned fr o m a set of exactl y four poi nt s. Fi gure 3. 1a and 3. 1b s how t he effect of var yi ng j ust one of t he poi nt s. As you woul d expect, when p2 i s moved up war d and t o t he l eft, t he cur ve t ends t o f oll ow; i n f act, it is pull ed t o t he opposit e si de of p1. You may be s ur prised t o see t hat t he cur ve i s never ver y cl ose t o t he t wo i nt er medi at e poi nt s, though it begi ns and ends at positi ons so me what adj acent. It wi ll be hel pf ul t o t hi nk of t he cur ve generat ed fr o m t he defi ni ng equati on f or B1 as associ at ed wit h a cur ve t hat goes fr o m near p1 t o p2. It i s al so hel pf ul t o re me mber t hat poi nt s p0, p1, p2, and p3 are used t o get B1.

Fi gure 3. 1 Effect s of varyi ng j ust one of t he poi nts on B-spli nes

Because a set of f our poi nt s i s r equired t o generate onl y a porti on of t he B- spli ne, t hat associ at ed wi t h t he t wo i nner poi nt s, we must consi der ho w t o get t he B- spli ne f or

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mor e t han f our poi nt s as well as ho w t o ext end t he cur ve i nt o t he r egi on out si de of t he mi ddl e pair. For t hi s we can mar ch al ong one poi nt at a ti me, f or mi ng ne w s et s of four. We abandon t he first of t he ol d set when we add t he ne w one.

The conditi ons t hat we wa nt t o i mpose on t he B- spli ne ar e: conti nuit y of t he cur ve and it s first and second deri vati ves. It t ur ns out t hat t he equati ons f or t he wei ghti ng fact ors (t he u- pol yno mi als, t he bk) are such t hat t hese r equire ment s are met. Fi gur e 3. 2 shows how t hree successi ve parts of a B-spli ne mi ght l ook.

Fi gure 3. 2 Successi ve B-spli nes j oi ned t oget her

We can su mmari ze t he properti es of B-spli nes as foll ows:

1. B- spli nes are pi eced t oget her so t hey agree at t heir j oi nt s i n t hree ways: a. 6 p p 4 p ) 0 ( B B i i 1 i 2 1 i i        b. 2 p p ) 0 ( B ) 1 ( B ' i i 2 1 i ' i       c. B_i''(1)B_i''_₁(0)p_i 2p_i_₁ p_i_₂

2. The porti on of t he cur ve det er mi ned by each gr oup of four poi nt s is wit hi n t he convex hull of t hese poi nt s.

No w we consi der ho w t o gener at e t he ends of t he j oi ned B- spli ne. If we have poi nt s from p0 t o pn, we already can construct B-spli nes B1 t hr ough Bn- 2. We need B0 and

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Bn- 1. Our pr obl e m i s t hat, usi ng t he pr ocedur e al ready defi ned, we woul d need additi onal poi nt s out si de t he do mai n of t he gi ven poi nt s.

Fi rst, we can add mor e poi nt s wit hout artifi ci alit y by maki ng t he added poi nt s coi nci de wit h t he gi ven extre me poi nt s. If we add not j ust a si ngl e fi ctitious poi nt at each end of t he set, but t wo at each end, we wi ll fi nd t hat t he ne w cur ves not onl y j oi n pr operl y wit h t he porti ons already made, but st art and end at t he extreme poi nt s as we want ed. I n s u mma ry, we add fi ctiti ous poi nt s p- 2, p- 1, pn+1, and pn +2 , wi t h t he first t wo i denti cal wit h p0 and t he l ast t wo i denti cal wit h pn.

The mat ri x for mul ati on for cubi c B-spli ne is:





6 p M u p p p p 0 1 4 1 0 3 0 3 0 3 6 3 1 3 3 1 1 u u u 6 1 ) u ( B b T 2 i 1 i i 1 i 2 3 i                                  (3. 4)

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4. POTENTI AL FLOW AND PANEL METHOD

4. 1 Governi ng Equati ons

Conser vati on of mass (conti nuit y equati on): 0

 

 V (4. 1)

For an irrot ati onal fl ow: 

 

V (4. 2)

Ther ef ore, f or a fl ow t hat i s bot h i nco mpr essi bl e and i rr ot ati onal equati on 1 and 2 can be co mbi ned t o yi el d 0 ) (    t hen 0 2  (4. 3)

Equati on 3 is Lapl ace’s equati on

For a t wo di mensi onal i nco mpr essi bl e fl ow, a strea m f uncti on can be defi ned s uch t hat y u     (4. 4) x v      (4. 5)

The conti nuit y equati on, V0, expressed i n cart esi an coor di nat es, is 0 x          y v u V (4. 6)

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0 x y y x x y y x 2 2                                    (4. 7)

Si nce t he fl ow is irrot ati onal: 0 y u x v       (4. 8)

Substit uti ng equati on (4.4) and (4. 5) i nt o equati on (4. 8): 0 y y x x _                       t hen 0 y x 2 2 2 2         (4. 9) Thi s i s also Lapl ace’s equati on. So, t he stream f uncti on al so satisfies Lapl ace’s equati on.

4. 2 Hess- Smit h Panel Met hod

Ther e are many choi ces as t o ho w t o f or mul at e a panel met hod (si ngul arity s ol uti ons, vari ati on wi t hi n a panel, si ngul arit y strengt h and di stri buti on, et c.) The si mpl est and first trul y pr acti cal met hod was due t o Hess and S mit h [ 19]. It i s based on a di stri buti on of sources and vorti ces on t he surface of t he geo met r y. In t heir met hod:

v s       _ (4. 10)

wher e,  i s t he t ot al pot enti al f uncti on and its t hree co mponent s are t he pot enti al s correspondi ng t o t he free strea m, t he s our ce di stributi on, and t he vort ex di stri buti on. These l ast t wo di stri buti ons have pot enti all y l ocally var yi ng strengt hs q(s) and (s), wher e s i s an ar c-l engt h coor di nat e whi ch s pans t he co mpl et e surface of t he airf oil i n any way you want.

The pot enti als creat ed by t he di stri buti on of sources/ si nks and vorti ces are gi ven by:



_   lnrds 2 ) s ( q s



      ds 2 ) s ( v (4. 11)

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wher e t he vari ous quantities are defi ned i n t he Fi gur e bel ow:

Fi gure 4. 1 Airf oil Anal ysis No mencl at ure for Panel Met hods

Noti ce t hat i n t hese f or mul ae, t he i nt egrati on i s t o be carri ed out al ong t he co mpl et e surface of t he airfoil. Usi ng t he super positi on pri nci pl e, any s uch di stributi on of sources/si nks and vortices satisfies Lapl ace’s equati on, but we will need t o fi nd conditi ons f or q(s) and (s) such t hat t he fl ow t angency boundar y conditi on and t he Kutt a conditi on are satisfied.

Noti ce t hat we have multi pl e opti ons. In t heor y, we coul d:

 Us e t he sour ce strengt h di stri buti on t o satisfy fl ow t angency and t he vortex di stri buti on t o satisfy t he Kutt a conditi on.

 Us e ar bitrar y co mbi nati ons of bot h sources/si nks and vorti ces t o satisfy bot h boundar y conditi ons si mult aneousl y.

Hess and Smit h made t he foll owi ng vali d si mplifi cati on:

Take t he vort ex strengt h t o be const ant over t he whol e airfoil and use t he Kutt a conditi on t o fi x its val ue, whil e all owi ng t he s ource strengt h t o var y fr om panel t o panel so t hat, t oget her wi t h t he const ant vort ex di stri buti on, t he fl ow t angency boundar y conditi on is satisfi ed ever ywher e.

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Al t er nati ves t o t hi s choi ce ar e possi bl e and r esult in di fferent t ypes of panel met hods. As k i f you want t o kno w mor e about t he m. Usi ng t he panel deco mpositi on fr o m t he fi gure bel ow,

Fi gure 4. 2 Defi niti on of Nodes and Panel s

we can ‘ ‘discreti ze’ ’ Equati on (10) i n t he foll owi ng way:

ds 2 r ln 2 ) s ( q ) sin y cos x ( V N 1 j panel_j

 

_      _           (4. 12)

Si nce Equati on ( 4. 12) i nvol ves i nt egrati ons over each di scret e panel on t he s urface of t he airfoil, we must some ho w par a met eri ze t he vari ati on of s our ce and vort ex strengt h wi t hi n each of t he panel s. Si nce t he vortex strengt h was consi dered t o be a const ant, we onl y need worr y about t he sour ce strengt h di stri buti on wi t hi n each panel.

Thi s i s t he maj or appr oxi mati on of t he panel met hod. Ho wever, you can see ho w t he i mport ance of t hi s appr oxi mati on shoul d decrease as t he nu mber of panel s,

 

N (of course t hi s will i ncrease t he cost of t he comput ati on consi der abl y, s o t here are mor e effi ci ent alternati ves.)

If we t ake t he si mpl est possi bl e appr oxi mati on, t hat i s, t o t ake t he s our ce strengt h t o be const ant on each of t he panel s:

i q ) s ( q  on panel I, i = 1, ……, N

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Ther ef ore, we have N + 1 unkno wns t o sol ve f or i n our pr obl e m: t he N panel s our ce strengt hs q_i and t he const ant vort ex strengt h . Consequentl y, we wi ll need N + 1 i ndependent equati ons whi ch can be obt ai ned by f or mul ati ng t he fl ow t angency boundar y conditi on at each of t he N panel s, and by enf orci ng t he Kutta condit i on di scussed pr evi ousl y. The s ol uti on of t he pr obl e m wi ll require t he i nversi on of a mat ri x of si ze ( N +1) x ( N +1).

The fi nal questi on t hat r e mai ns i s: wher e shoul d we i mpose t he fl ow t angency boundar y conditi on? The foll owi ng opti ons are avail abl e:

 The nodes of t he surface paneli zati on.

 The poi nt s on t he surface of t he act ual airfoil, halfwa y bet ween each adj acent pair of nodes.

 The poi nts l ocat ed at t he mi dpoi nt of each of t he panel s.

We wi ll see i n a mome nt t hat t he vel ociti es are i nfi nit e at t he nodes of our paneli zati on, whi ch makes t he m a poor choi ce for boundar y conditi on i mpositi on.

The second opti on is reasonabl e, but rat her diffi cult t o i mpl e ment i n practi ce.

The l ast opti on i s t he one Hess and S mit h chose. Al t hough it suffers fr om a sli ght alt erati on of t he s urface geo met r y, it i s easy t o i mpl e ment and yi el ds f airl y accur at e results f or a r easonabl e nu mber of panel s. Thi s l ocati on i s al so used f or t he i mpositi on of t he Kutt a conditi on ( on t he l ast panel s on upper and l ower surfaces of t he airfoil, assu mi ng t hat t heir mi dpoi nt s r e mai n at equal di st ances fr o m t he t raili ng edge as t he nu mber of panel s is i ncreased).

If we want t o i mpl e ment t he met hod; consi der t he it h panel t o be l ocat ed bet ween t he it h and (i + 1)t h nodes, wit h its ori ent ati on t o t he x-axi s gi ven by

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i i 1 i i l y y sin    i i 1 i i l x x cos    (4. 13)

wher e li i s t he l engt h of t he panel under consi derati on. The nor mal and t angenti al vect ors t o t his panel, are t hen gi ven by

j i nˆ_i sin_iˆcos_iˆ (4. 14) j i t cos ˆ sin ˆ ˆ_i _ __i _ __i

The t angenti al vect or i s ori ent ed i n t he di recti on from node i t o node i +1, whil e t he nor mal vect or, if t he airfoil is traversed cl ock wi se, poi nt s i nt o t he fl ui d.

Fi gure 4. 3 Local Panel Coor di nat e Syst e m

Furt her mor e, t he coor di nat es of t he mi dpoi nt of t he panel are gi ven by

2 x x x i i 1 i    2 y y y i i 1 i    (4. 15)

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) y , x ( v v ) y , x ( u u i i i i i i  

The fl ow t angency boundar y conditi on can t hen be si mpl y writt en as (un)0, or, for each panel

0 cos v sin ui i  i i   for i = 1, ……, N (4. 16)

whil e t he Kuttt a conditi on is si mpl y gi ven by

N N N N 1 1 1

1cos v sin u cos v sin

u       

(4. 17)

wher e t he negati ve si gns are due t o t he f act t hat t he t angenti al vect ors at t he first and last panel s have nearl y opposit e directi ons.

No w, t he vel ocit y at t he mi dpoi nt of each panel can be co mput ed by s uperpositi on of t he contri buti ons of all sour ces and vorti ces l ocated at t he mi dpoi nt of ever y panel (i ncl udi ng itself). Si nce t he vel ocit y i nduced by t he sour ce or vort ex on a panel i s pr oporti onal t o t he s our ce or vort ex strengt h i n t hat panel, q_iand  can be pulled out of t he i nt egral i n Equati on ( 4. 12) t o yi el d         N 1 j N 1 j vij sij j i V cos q u u u (4. 18)         N 1 j N 1 j vij sij j i V sin q v v v

wher e usij , vsij are t he vel ocit y co mponent s at t he mi dpoi nt of panel i i nduced by a sour ce of unit strengt h at t he mi dpoi nt of panel j. A si mil ar i nt er pret ation can be found f or uvij , vvij. I n a coor di nat e syst e m t angent ial and nor mal t o t he panel, we can perf or m t he i nt egral s i n Equati on ( 4. 12) by noti ci ng t hat t he l ocal vel ocit y

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co mponent s can be expanded i nt o absol ut e ones accor di ng t o t he f oll owi ng transf or mati on:

j * j * sin v cos u u    (4. 19) j * j * cos v sin u v   

No w, t he l ocal vel ocit y co mponent s at t he mi dpoi nt of t he it h panel due t o a unit -strengt h source di stri bution on t hi s jt h panel can be writt en as

dt y ) t x ( t x 2 1 u j l 0 * 2 *2 * * sij       (4. 20) dt y ) t x ( y 2 1 v j l 0 * 2 *2 * * sij     

wher e (x*,y*)are t he coor di nat es of t he mi dpoi nt of panel i i n t he l ocal coor di nat e syst e m of panel j. Carr ying out t he i nt egral s i n Equati on (4. 20) we fi nd t hat





t lj 0 t 2 1 2 * 2 * * sij ln (x t) y 2 1 u        (4. 21) j l t 0 t * * 1 * sij t x y tan 2 1 v      

These r esults have a si mpl e geo met ri c i nt er pret ation t hat can be di scer ned by l ooki ng at t he fi gure bel ow. One can say t hat

ij 1 ij * sij r r ln 2 1 u     (4. 22)         2 2 v* l 0 ij sij

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Fi gure 4. 4 Geo met ri c Inter pret ati on of Sour ce and Vort ex Induced Vel ocities

rij i s t he di st ance fr o m t he mi dpoi nt of panel i t o t he jt h node, whil e ij i s t he angl e

subt ended by t he jt h panel at t he mi dpoi nt of panel i. Noti ce t hat u 0

*

sii  , but t he

val ue of v*_siiis not so cl ear. Whe n t he poi nt of i nt erest appr oaches t he mi dpoi nt of t he panel fr o m t he out si de of t he airfoil, t hi s angl e _ii . Ho wever, when t he mi dpoi nt of t he panel i s appr oached fr o m t he i nsi de of t he airfoil, _ii . Si nce we are i nt erest ed i n t he flo w out si de of t he airfoil onl y, we will al wayst ake _ii .

Si mil arl y, f or t he vel ocity fi el d i nduced by t he vort ex on panel j at t he mi dpoi nt of panel i we can si mpl y see t hat

       



2 dt y ) t x ( y 2 1 u ij l 0 2 * 2 * * * vij j (4. 23) ij 1 ij l 0 2 * 2 * * * vij r r ln 2 1 dt y ) t x ( t x 2 1 v j         



and fi nall y, t he fl ow t angency boundar y conditi on, usi ng Equati on ( 4. 18), and undoi ng t he l ocal coor dinat e transf or mati on of Equati on (4. 19) can be written as



     N 1 j i 1 iN j ijq A b A (4. 24)

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wher e ) cos cos sin (sin v ) cos sin sin (cos u cos v sin u A i j i j * sij i j i j * sij i sij i sij ij                   (4. 25) whi ch yi el ds ij j i ij 1 ij j i ij cos( ) r r ln ) sin( A 2         (4. 26)

Si mil arl y for t he vort ex strengt h coeffi ci ent



           N 1 j ij j i ij 1 ij j i 1 iN sin( ) r r ln ) cos( A 2 (4. 27)

The ri ght hand si de of t his matri x equati on is gi ven by

) sin(

V

bi   i  (4. 28)

The fl ow t angency boundar y conditi on gi ves us N equati ons. We need an additi onal one pr ovi ded by t he Kut ta conditi on i n or der t o obt ai n a s yst e m t hat can be s ol ved. Accor di ng t o Equati on (4. 17)



        N 1 j 1 N 1 N , 1 N j j , 1 N q A b A (4. 29)

Aft er si mil ar mani pul ati ons we fi nd t hat



           N , 1 k kj 1 kj j k kj j k j , 1 N r r ln ) cos( ) sin( A 2 (4. 30) kj j k N , 1 k N 1 j kj 1 kj j k 1 N , 1 N cos( ) r r ln ) sin( A 2 

 

          

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) cos( V ) cos( V bN1  1   N  (4. 31)

wher e t he s u m



_k_₁_,_Nare carri ed out onl y over t he first and l ast panel s, and not t he range

 

1,N . These vari ous expr essi ons set up a matri x pr obl e m of t he ki nd

b Ax

wher e t he mat ri x A i s of si ze ( N + 1) x( N + 1). Thi s syst e m can be s ket ched as foll ows:                                                                      1 N N i 1 N i 1 1 N , 1 N N , 1 N i , 1 N 1 , 1 N 1 N , N NN Ni 1 N 1 N , i iN ii 1 i 1 N , 1 N 1 i 1 11 b b : b : b q : q : q A A ... A ... A A A ... A ... A : : : : A A ... A ... A : : : : A A ... A ... A (4. 32)

Noti ce t hat t he cost of i nversi on of a f ull mat ri x s uch as t hi s one i s O ( N + 1)3 , s o t hat, as t he nu mber of panel s i ncreases wit hout bounds, t he cost of sol vi ng t he panel pr obl e m i ncreases rapi dly.

Fi nall y, once you have sol ved t he syst e m f or t he unkno wns of t he pr obl em, it i s easy t o construct t he t angential vel ocit y at t he mi dpoi nt of each panel accor di ng t o t he foll owi ng for mul a

                                                N 1 j ij i j ij 1 ij j i ij 1 ij j i ij j i N 1 j j i ti ) cos( r r ln ) sin( 2 r r ln ) cos( ) sin( 2 q ) cos( V V (4. 33)

And kno wi ng t he t angenti al vel ocit y co mponent, we can co mput e t he pr essur e coeffi ci ent ( no appr oxi mati on si nce Vni = 0) at t he mi dpoi nt of each panel accor di ng t o t he foll owi ng for mul a

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2 2 ti i i p V V 1 ) y , x ( C   

from whi ch t he f orce and mo ment coeffi ci ent s can be co mput ed assu mi ng t hat t hi s val ue of Cp is const ant over each panel and by perfor mi ng t he di scret e sum.

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5. I NVERS E DESI GN OF AI RFOI LS

In i nverse desi gn, pur pose i s t o fi nd a pr oper airfoil shape, whi ch gi ves t he pr e-specifi ed pr essure di stri buti on. I n t hi s wor k t wo met hods used t o i mpl e ment i nverse desi gn. First met hod used i s based on a geneti c al gorit hm. Second met hod i s based on a resi dual correcti on al gorit hm.

5. 1 Inverse desi gn wit h Ge neti c Al gorit hm

In t hi s met hod, Fortran code of a geneti c al gorit hm writt en by Davi d L. Carr oll ( Uni versit y of Illi noi s) i s used . I nverse desi gn probl e m i s sol ved as an opti mizati on pr obl e m such t hat, t he val ue of t he bel ow equati on is maxi mi zed.

2 p n 1 i p Tc ) c ( Funcval i i   



 (5. 1)

So an i nverse pr obl e m may t ransf or m i nt o an opti mizati on pr obl e m wi t h t hi s for mul ati on. Thi s geneti c al gorit hm code i s co mpi led t oget her wit h S mit h-Hess Panel Met hod Code and B- spl i ne Cur ve Gener at or Code. Fl ow chart of t hi s code i s as foll ow:

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Fi gure 5. 1 Inverse desi gn wit h geneti c al gorit hm flo w chart

5. 2 Inverse Desi gn wit h Resi dual Correcti on Al gorit hm

In t hi s met hod, a r esi dual correcti on met hod i s used [ 17]. Correspondi ng differenti al equati on: 2 2 t 2 2 V V dx Y d C dx Y d B Y A       (5. 2)

A, B, C are ar bitrar y const ant s det er mi ni ng t he rate of change of t he airfoil.

If we use fi nit e differences t o di scriti ze t he equation, appr oxi mati on of dx Y d on t he upper surface: i 1 i i 1 i x x Y Y x Y           _{(5. 3)} Input Tcp

Popul ati on generati on

Ai rf oil shape generati on wi t h B-spli ne

Co mput ati on of cp’s wit h panel met hod

Co mpari son of cp and Tcp

Ma xi mu m generati on number is reached Yes Exit No i-1 i +1 i

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Appr oxi mati on of dx Y d on t he l ower surface: 1 i i 1 i i x x Y Y x Y           (5. 4)

Then appr oxi mati on of 2 2 dx Y d  on t he upper surface:                           2 x x x x Y Y x x Y Y x Y 1 i 1 i 1 i i 1 i i i 1 i i 1 i 2 2 (5. 5)

A si mil ar equati on occurs on t he l ower surface. At t he l eadi ng edges and t he t raili ng edges Y i s accept ed as zero. Wi t h t hese appr oxi mati ons, we can writ e f ollo wi ng for mul ati on:

2 i 2 t 1 i 1 i 1 i i 1 i i i 1 i i 1 i i 1 i i 1 i i V V x x x x Y Y x x Y Y C 2 x x Y Y B Y A i                            (5. 6)

Then t he coeffi ci ent s of the syst e m K_i Y_i ₁ L_i Y_i M_i Y_i ₁ V_t2 V_i

i        _ _ are: ) x x )( x x ( C 2 x x B K 1 i 1 i i 1 i i 1 i i                            1 i i 1 i 1 i 1 i i i 1 i i x x 1 x x 1 x x C 2 x x B A L (5. 7) i +1 i i-1 i-1 i i +1

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) x x )( x x ( C 2 M 1 i 1 i 1 i i i       

As a r esult of t hi s f or mul ati on, we can constit ut e t ri-di agonal Nx N coeffi cient mat ri x. To s ol ve t hi s syst e m Thomas al gorit hm i s used. So we can have a Yfor each poi nt on t he airfoil. Then, usi ng t his Ywe can fi nd ne w Y val ue for each poi nt, such t hat:

Y Y

Ynew  old  (5. 8)

A, B, and C const ant s det er mi ne t he sensiti vit y of Y; t he bi gger const ant s l ead t o s mall er Yval ues. Thi s al gorit hm code i s al so co mpil ed t oget her wit h S mit h- Hess Panel Met hod Code. Fl ow chart is as foll ows:

Fi gure 5. 2 Inverse desi gn wit h resi dual correcti on al gorit hm fl ow chart Input Vt and st arti ng airfoil shape

Cal cul ati on of Y_i’s

Obt ai ni ng Ynew Yold Y

Co mput ati on of ne w V wit h panel met od

St oppi ng crit eri a is satisfied? Yes Exit

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6. RESULTS AND CONCLUSI ON

To t est pr evi ousl y mentioned t wo met hods, a t est case i s utili zed. Eppl er 361 airfoil t hat i s desi gned f or r ot orcrafts i s used f or t est case. Fl ow ar ound Eppl er 361 airfoil wi t h 5- degree angl e of att ack i s anal yzed. As a r esult of t hi s anal ysi s, pr essur e di stri buti on and vel ocit y di stri buti on on t he airfoil ar e obt ai ned. Bel ow fi gur es s ho w Eppl er 261 airfoil geo metry and pressure distri buti on on t he airfoil respectivel y.

Eppler 361 -0.1 -0.05 0 0.05 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x/c y /c x/c versus Cp -2 -1,5 -1 -0,5 0 0,5 1 1,5 0 0,2 0,4 0,6 0,8 1 x/c Cp Target Cp

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6. 1 Res ults of Inverse Desi gn wit h Geneti c Al gorit hm

In i nverse desi gn wi t h geneti c al gorit hm, r esults of pr essure di stri buti on and airf oil geo met r y obt ai ned fr o m di fferent it erati on nu mber ar e pr esent ed wi t h t he t ar get pr essure di stri buti on and airfoil geo met r y.

For generati on nu mber 200, results may be seen bel ow: x/c versus Cp -2 -1.5 -1 -0.5 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 x/c Cp Target Cp New Cp x/c versus y/c -0.1 -0.05 0 0.05 0.1 0 0.2 0.4 0.6 0.8 1 x/c y /c Target Airfoil New Airfoil

Fi gure 6. 2 Results of geneti c al gorit hm aft er generati on 200

For generati on nu mber 200, it i s seen t hat results i s not so good and t her e ar e re mar kabl e differences i n pr essure di stri buti ons and, of course, airfoil geo met r y. Whi l e it erati on nu mber i ncreases, results obt ai ned st art t o r ese mbl e t o t arget val ues. For exa mpl e i n generati on nu mber 1000, results may be seen bel ow:

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x/c versus Cp -2 -1.5 -1 -0.5 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 x/c Cp Target Cp New Cp x/c versus y/c -0.1 -0.05 0 0.05 0.1 0 0.2 0.4 0.6 0.8 1 x/c y /c Target Airfoil New Airfoil

Al t hough r esults see m better wit h r espect t o generati on nu mber 200, di fferences fr o m target val ues ar e not negli gi bl e. So i f we conti nue t o it erat e, we can co me up wi t h good r esults. As an exampl e, obt ai ned r esults i n generati on nu mber 4000 ma y be seen bel ow:

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x/c versus Cp -2 -1,5 -1 -0,5 0 0,5 1 1,5 0 0,2 0,4 0,6 0,8 1 X/C Cp Target Cp New Cp x/c versus y/c -0,1 -0,05 0 0,05 0,1 0 0,2 0,4 0,6 0,8 1 x/c y /c Target Airfoil New Airfoil

As it i s seen i n t he fi gures above, at t he generati on nu mber 4000, t ar get pr essur e di stri buti ons and airfoil geo met r y are obt ai ned wit h sli ght differences. Mor e r esult s from di fferent generati on nu mbers are avail abl e i n t he appendi x. At first, l ooki ng at generati on nu mber 4000, cost of t hi s co mput ation may be r egar ded as hi gh. But, si nce t he ti me consu med by t he al gorit hm f or one generati on i s t oo l ow, actuall y it i s not so costl y.

6. 2 Res ults of Inverse Desi gn wit h Resi dual Correcti on Al gorit hm

Whi l e i mpl e menti ng t hi s met hod, different A, B, and C const ant val ues are used t o see t he effect s of t hese const ant s on r esults and iterati on nu mber. I n addition, f or t hi s met hod t wo different criteri a are defi ned. These are:

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a)



Tc c



1.5 2 n 1 i p p_i  _i 



 and b)



Tc c



1 2 n 1 i p p_i  _i 





Whe n t hese crit eri a are satisfi ed, al gorit hm st ops. For exa mpl e, for t he case of A = 1, B = 1, C = 1, obt ai ned results f or crit eri a ( a), aft er t he it erati on nu mber 80 ar e as foll ows: x/c versus Cp -2 -1.5 -1 -0.5 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 x/c Cp Target Cp Comp. Cp x/c versus y/c -0.1 -0.05 0 0.05 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x/c y /c eppler 361 Comp Airfoil

Fi gure 6. 5 Results of resi dual correcti on al gorit hm aft er it erati on 80 ( A,B, C=1)

If we appl y t he second crit eri a wit h t he sa me A, B and C val ues, aft er 977 it erati on we co me up wit h:

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x/c versus Cp -2 -1.5 -1 -0.5 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 x/c Cp Target Cp Comp. Cp x/c versus y/c -0.1 -0.05 0 0.05 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x/c y /c eppler 361 Comp Airfoil

Fi gure 6. 6 Results of resi dual correcti on al gorit hm aft er iterati on 977 ( A,B, C=1)

It i s seen t hat, f or t hi s crit eri a, differences occurred about t he l eadi ng edge i n first crit eri a mostl y di sappeared. So we can r each nearl y t he sa me pr essure distri buti on wi t h t ar get pressure di stributi on.

If we t ake t he A = 3, B =3, C = 3 f or t he sa me t wo crit eri a, pressure di stri buti on, airfoil geo met r y and it erati on nu mber at whi ch t he crit eri a i s satisfi ed are pr esent ed bel ow.

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x/c versus Cp -2 -1.5 -1 -0.5 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 x/c Cp Target Cp Comp. Cp x/c versus y/c -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x/c y /c eppler 361 Comp Airfoil

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x/c versus Cp -2 -1.5 -1 -0.5 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 x/c Cp Target Cp Comp. Cp x/c versus y/c -0.1 -0.05 0 0.05 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x/c y /c eppler 361 Comp Airfoil

Fi gure 6. 8 Results of resi dual correcti on al gorit hm aft er iterati on 2931 ( A, B, C=3)

As expect ed, iterati on numbers get s l ar ger, whil e there is an i ncrease i n A, B, C. To see t hi s effect mor e evi dentl y , it may be benefi ci al t o see t he r esults of t he case A = 10, B = 10, C = 10. For t he first crit eri a, aft er 789 it erati on, obt ai ned results are:

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Fi gure 6. 10 Results of resi dual correcti on al gorit hm aft er iterati on 9763 ( A, B, C=10)

By usi ng t hese t hree cases, we can see t he effect s of A, B, C const ant s on t he required it erati on t o satisfy t he crit eri a. Foll owi ng tabl e sho ws t he trend.

Tabl e 6. 1 Co mpari son of results accor di ng t o constant s and crit eri a

Val ues of A, B, C Re qui red iterati on nu mber for t he fi rst criteri a

Re qui red iterati on nu mber for t he second criteri a

1 80 977

3 237 2931

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6. 3 Concl usi on

Bot h i nverse desi gn met hods gi ve accept abl e r esults. Met hod wi t h genetic al gorit h m creat es st arti ng generati ons r ando ml y. Ther ef ore, alt hough obt ai ned r esults ar e i n t he accept abl e li mit s, so me differences fr o m t ar get val ues t ake att enti on. But on t he ot her hand, r ando mness of t he geneti c al gorit hm al so creat es an advant age. That i s, si nce geneti c al gorit hm has r ando mness, accur acy of t he al gorit hm does not change t oo much dependi ng on positi on on t he airfoil. And t hi s f eat ure of t he al gorit h m ma y creat e an advant age i n t he case of mor e co mpl ex geo met ri es.

The met hod wi t h r esi dual correcti on al gorit hm s ee ms r el ati vel y better t han t he met hod wi t h geneti c al gorit hm. For t hi s t est case, because r esi dual correcti on al gorit hm st arts it erati on wi t h a cert ai n airfoil ( NACA 0012), it reaches better r esults. But si nce, i n t hi s al gorith m, t ar get val ues are r eached by usi ng Y’s, sensitivit y of

Y

 is i mport ant f or obt ai ni ng t ar get val ues. So al gorit hm accur acy vari es accor di ng t o positi on on t he airfoil.

For exa mpl e, alt hough obt ai ned r esults f or ot her part of t he airfoil fit ver y well t o t he target val ues, al gorit hm accur acy does not sho w t he sa me t rend i n t he l eadi ng edge. In l eadi ng edge, r e mar kabl e di fference i s seen. To correct t hi s sit uati on, we must decrease val ue of crit eri a and t hi s l eads t o an i ncrease i n it erati on nu mber.

In additi on t o t hese, i n t he case of mor e co mpl ex pr obl e ms, first met hod i s used t o creat e an i niti al airfoil geo met r y for t he second met hod.

(54)

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