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ME 42104 : OPTIMIZATION TECHNIQUES

Lecture 1

G. N. Kotwal

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constraints on system’sbehaviourConstraints can be imposed on:

• equivalent stress• critical buckling load (local and global), can include

postbuckling c aracteristics• !requenc" o! vibrations (can be several)• cost• etc.

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o ce o es gnvariables

•

Design variables are selected to uniquely identify adesign. They have to be be mutually independent.

• Typical examples:

Area of cross section of bars in a truss structure Number of a specific steel section in a catalogue ofUB sections

oordinates of poles of B!splines defining the shape

of an aerofoil " etc.

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OPTIMIZATION PROBLEM

•A formal mathematical optimization problem : to find components of the vector x of designvariables :

#her e F( x ) is the objective function " gj( x ) are theconstraint functions " the last set of inequalityconditions defines the side constraints .

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Unc n!tr"#ne$ O%t#'"t# n%r ()e&!

S &e %t#'"t# n %r ()e&! $ n t #n* )*e "n+ c n!tr"#nt! "n$ c"n(e !t"te$ "!:

• #uc problems are called unconstrained optimizationproblems.

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%r ()e& .An %t#'"t# n %r ()e& c"n (e !t"te$ "! , )) -!.

#here $ is an n!dimensional vector called the design vector, /(X) istermed the objective Junction, and gj (X) and Ij (X) are known asinequality and equality constraints" respectively. The number ofvariables n and the number o constraints m and/or/! need not be relatedin any way" #he $roblem stated in %q. above is called a constrained

o$timi%ation $roblem"

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r ter a o system se ciency

riteria of structural efficiency are described by the ob&ective function F( x ). Typicalexamples:

cost#eightuse of resources 'fuel" etc.(stress concentrationetc.

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xamp e o a screteproblem

•

)ptimi*ation of a steel structure #here some of the members are described by +, designvariables. %ach design variable represents a number of a UB section from a catalogue of +,available sections.• )ne full structural analysis of each design ta-es + sec. on a computer.

• uestion: ho# much time #ould it ta-e to chec- all the combinations of cross!sections in

order to guarantee the optimum solution/

• Ans#er: 0+1 years

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/# cu)t+ #n ! )*#n "$#!crete %r ()e&

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M"t e&"t#c") M $e)#n E "&%)e!Ne- c n!u&er re!e"rc #n$#c"te! t "t %e %)e )# e t $r#n "( ut 0.5)#ter , ! $" "t " t#&e $ur#n t e !u&&er & nt !. T e ,"(r#c"t# nc !t , t e re$e!# ne$ ! $" c"n #! %r % rt# n") t t e !ur,"ce "re"

"n$ c"n (e e!t#&"te$ "t 1.0 %er !6u"re &eter , t e &"ter#")u!e$. A c#rcu)"r cr !! !ect# n #! t e & !t %)"u!#()e #*en currentt )#n "*"#)"()e , r &"nu,"cture. 7 r "e!t et#c re"! n! 8 t e e# t&u!t (e "t )e"!t t-#ce t e $#"&eter. Stu$#e! #n$#c"te t "t )$#n

c &, rt re6u#re! " $#"&eter (et-een 9 "n$ c&.

/e!# n *"r#"()e! : S#'e! : $8 8tA!!u&%t# n! : t #! !&")) 8 # n re$ #n t e c")cu)"t# n , * )u&e ,c"n.On)+ C+). Sur,"ce c n!#$ere$ , r * ). c")cu)"t# n!8 n t t e en$!.

T+%#c"))+ 8 c "n e #n %"r"&eter! -#)) c"u!e t e ! )ut# n t (erec &%ute$./e!# n %"r"&eter: C ;C !t < un#t "re" = 1 C. %er !6. c&. >

T e $e!# n ,unct# n! -#)) #nc)u$e : c &%ut"t# n , t e * )u&eenc) !e$ (+ t e c"n "n$ t e !ur,"ce "re" , t e c+)#n$r#c") !ect# n. * )u&e #n t e c"n ? @$2

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Se"rc Pr ce$ureN n)#ne"r Pr ()e&!:S )ut# n !e"rc e$ (+!e"rc < nu&. < #ter"t#*e;Ste%-#!e ") r#t &>&et $!.Pr %er #n#t#") ue!! #!nece!!"r+8 t re %er#ence.

ett#n c) !er t t e! )ut# n = C n*er enceIn#t#") ue!! D 0 = n nD,e"!#()e. t 1 (+#$ent#,+#n " !e"rc

$#rect# n. S) ;*ect r,r & 0 t 1>.1 #! (etter t "n 0 =

,e"!#()e (ut n nD%t#&").

In !"&e $#ret# n8 S) c"n#*e 1 "t BS)

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Complete algorit m decides wa"s o! !inding # and $

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C"nt#)e*er Be"&A c"nt#)e*er (e"& nee$! t (e $e!# ne$ t c"rr+#n " % #nt ) "$ 7 "tt e en$ , t e (e"& , )en t L. T e cr !! !ect# n , t e (e"&

-#II (e #n t e ! "%e , t e )etter I (e"& ! u)$ &eet %re!cr#(e$,"#)ure cr#ter#". T ere #! ")! " I#t n #t! $eFect# n. A (e"& ,n#&u& &"!! #! re6u#re$ t (e $e!# ne$. Ac8 Qc8 Ic ? C

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4G

S#'#n

%t#'"t# n Tr"ct rDtr"#)erc &(#n"t# n

)b&ective: toimprove the ride

characteristics

Design variables: properties of thesuspension system

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4

' n%t#'"t#

n 7)# t !#&u)"t r2inematic optimi*ation of a3te#art platform manipulatorfor a flight simulator.

The goal of optimi*ation is todesign a manipulator #ithmaximum #or-space #hosecharacteristics are definedaccording to the manoeuvres ofan aircraft.

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S#'#n

%t#'"t# n 7)# t !#&u)"t r 'cont.(2inematic optimi*ation of a

3te#art platform manipulator for aflight simulator.

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S#'#n

%t#'"t# n 7)# t !#&u)"t r3ix design variables define the configuration of the platform.

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S#'#n %t#'"t# n%r ()e&!

St#r)#n en #ne

)b&ective: to improve thethermodynamic efficiency.

onstraint: po#er output.

Design variables: parameters ofthe engine.

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S "%e

%t#'"t# nO%t#'"t# n , " !%"nner

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S "%e

%t#'"t# nO%t#'"t# n , " !%"nner

A CA/ & $e) , " !tructure. M *e! , t e ( un$"r+ "re ")) -e$ "t t e #n$#c"te$ % #nt!

S "%

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S "%e%t#'"t#

nO%t#'"t# n , " !%"nner

4nitial and final designs. ourtesy of 5. 6assmusen

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S "%e %t#'"t# n%r ()e&!

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5

S "%e %t#'"t# n%r ()e&!

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5G

S "%e %t#'"t# n%r ()e&!

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5

O%t#'"t# n , "n "er , #) B!spline representation of the NA A ,,+7 aerofoil. The B!spline poles arenumbered from + to 78. Design variables: & and y coordinates of 77 B!spline poles ' ' 9

(.

E AMPLES: SHAPE OPTIMIZATION

;.A. ;right" .

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Pr ()e& $e n#t# n ;"er , #)8c nt.>

E AMPLES: SHAPE OPTIMIZATION

@roblem formulation:•)b&ective function 'to be minimi*ed(: drag coefficient at

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Re!u)t! ;"er , #)8 c nt.>E AMPLES: SHAPE OPTIMIZATION

6esults of

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Speci c Features of ShapeOptimization

CA/ & $e) ener"t# n #! $ ne nceO%t#'"t# n %r ce!! & $# e! t #! CA/ & $e) "n$ return!" *")#$ CA/ & $e) t "t nee$! t (e "n")+!e$ T e CA/ & $e) ")) -! , r t e u!e , "ut &"t#c t )!;&e! ener"t r8 "$"%t#*e 7E8 etc>

E "&%)e. L#n #n " 7E &e! $#rect)+ t %t#'"t# n c"n*# )"te t e ("!#c "!!u&%t# n! t e & $e) #! ("!e$ n:

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Pr ()e& $e n#t# n; %t#'"t# n , " ! e))>

A shell is described by a square reference plan. The mid!surface is described usingsquare patches. At the -eypoints the out!of!plane coordinate and its derivatives #ith respectto the in!plane coordinates have been specified.

& t

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r e& e n t n; %t#'"t# n , " ! e))8

c nt.> The geometry is assumed to be symmetric #ith respect to the diagonals. The designvariables are the out!of!plane coordinates of the -eypoints and the corresponding derivatives'+7 in total(. The out!of!plane coordinates of the corners are fixed. Also" the thic-ness of theshell is ta-en as a design variable. The shell is supported at its corner nodes" for #hich alldisplacement components are prescribed. The shell is loaded by a uniform out!of!plane load.

The optimi*ation problem is formulated as minimi*ation of the maximum displacement #hilethe volume remains belo# the specified limit.

Numerical studiy sho#ed that this optimi*ation problem has several local optima. T#odesigns corresponding to almost equally good optima are sho#n in the figures belo#.

P () & $ # #

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Pr ()e& $e n#t# n; %t#'"t# n , "

! e))8 c nt.>

Eirst design" normali*ed constraint equals ,.>>0

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Pr ()e& $e n#t# n; %t#'"t# n , " ! e))8

c nt.>

3econd design" normali*ed constraint equals ,.>>?

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9

Three!bay by four!bay by four!storey structure

Discrete variables arenumbers of sections froma catalogue

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9G

O%t#'"t# n, ,r nt -#n

, J3 J" u"rR"c#n7 r&u)" 1c"r

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O%t#'"t# n , ,r nt-#n , J3 J" u"r R"c#n7 r&u)" 1 c"r

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0

enet#cA) r#t &

Eront #ing of 505aguar 6acingEormula + car

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1

enet#cA) r#t &

3chematic layup of the compositestructure of the

#ing

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2

)ptimi*ation problem: minimi*e mass sub&ect to displacement constraints 'E4A andaerodynamics(6esult of optimi*ation by a genetic algorithm ' A(:)btained design #eight: .>8 2gBaseline design #eight: 8.7 2g4mprovement: .?F

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3

M"ter#") %t#'"t# n

%r ()e&;O. S# &un$8 TU ,/en&"r >

Design of a negative @oissonGs ratio material 'expands vertically #hen stretched

hori*ontally( using topology optimi*ation. Heft: base cell. entre: @eriodicmaterial composed of repeated base cells. 6ight: Test beam manufactured by

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ter %t & t n%r ()e& ;O. S# &un$8 TU ,

/en&"r >

Design of a material #ith negative thermal expansion. 4t is composed of t#omaterials #ith different thermal expansion coefficients + 9 + 'blue( and 7

9+, 'red( and voids. The effective thermal expansion coefficient is ,9 .+1.Heft: base cell. entre: thermal displacement of microstructure sub&ected toheating. 6ight: periodic material composed of repeated base cells.