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This is an author-deposited version published in: https://sam.ensam.eu Handle ID: .http://hdl.handle.net/10985/11438

To cite this version :

Marco MONTEMURRO, Anita CATAPANO - On the effective integration of manufacturability constraints within the multi-scale methodology for designing variable angle-tow laminates - Composite Structures - Vol. 161, p.145-159 - 2017

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within the multi-sale methodology for designing variable

angle-tow laminates

Maro Montemurro

∗

Arts etMétiers ParisTeh,I2M CNRSUMR 5295,F-33400 Talene, Frane.

Anita Catapano

Bordeaux INP,UniversitédeBordeaux,I2MCNRS UMR 5295,F-33400 Talene, Frane

Author for Correspondene:

MaroMontemurro, PhD,

Arts etMétiersParisTeh,

I2M CNRSUMR5295,

F-33400Talene,Frane,

tel: +3355 68 45422,

fax: +3354 0006 964,

e.mail: maro.montemurroensam.eu,

maro.montemurrou-bordeaux1.fr

∗

Correspondingauthor.

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In this work a multi-sale two-level(MS2L) optimisation strategy for optimising

VATompositesispresented. IntheframeworkoftheMS2Lmethodology,thedesign

problemissplitandsolvedintotwosteps. Attherststepthegoalistodeterminethe

optimumdistributionof the laminatestiness properties overthe struture (maro-

sopisale),whiletheseondstepaimsatretrievingtheoptimumbres-pathineah

layermeetingalltherequirementsprovidedbytheproblemathand(mesosopisale).

TheMS2Lstrategyhasbeenimprovedinordertointegratealltypesofrequirements

(mehanial, manufaturability, geometri,et.) within the rst-levelproblem. The

proposed approah relies on: a) the polar formalism fordesribing thebehaviour of

theVATlaminate,b)theiso-geometrisurfaesfordesribingthespatialvariationof

boththelaminatestiness properties(maro-sale)andthelayersbres-path(meso-

sale)and) anhybridoptimisationtool(geneti andgradient-basedalgorithms)to

performthesolutionsearh. TheeetivenessoftheMS2Lstrategyisproventhrough

anumerialexampleonthemaximisation oftherstbuklingfator ofaVATplate

subjettobothmehanialandmanufaturabilityonstraints.

Keywords:

VAT Laminates; Optimisation; Bukling; Geneti Algorithms; B-splines; Composite Ma-

terials.

1 Introdution

Anisotropi materials, suh as bres-reinfored omposite materials, are extensively used

inmanyindustrialeldsthankstotheir mehanial performanes: highstiness-to-weight

and strength-to-weight ratiosthat lead to a substantial weight saving when ompared to

metalli alloys. In addition, the reent development of new manufaturing tehniques of

omposite strutures, e.g. automated bre plaement (AFP) mahines, allows for going

beyond the lassial design rules, thus leading the designer to nd innovative and more

eient solutions than the lassial straight bre ongurations. The use of the AFP

tehnology brought to the emergene of a new lass of omposite materials: the variable

angle tow (VAT) omposites, [1, 2℄. A modern AFP mahine allows the bre (i.e. the

tow) to be plaed along a urvilinear path within the onstitutive lamina thus implying

a point-wise variation of the material properties (stiness, strength, et.). Of ourse,

this tehnology enables the designer to take advantage of the diretional properties of

ompositesinthemosteetiveway. Theinterestofusingvariablestiness(VS)laminates

is onsiderably inreased during the last years: in the meantime some works on the a

posteriori haraterisation of the elasti response of suh materials have gained a lot of

attention from the sienti ommunity of omposites materials, [3, 4℄. Although the

utilisation of VAT laminates onsiderably inreases the omplexity of the design proess

(mainlydue to the large numberof design variables involved withinthe problem), on the

otherhanditleadsthedesignertooneivenon-onventionalsolutions haraterisedeither

by a onsiderable weight saving or enhaned mehanial properties when ompared to

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that an be ahieved interms of mehanial performanes (stiness, bukling behaviour,

et.) byusingaVSplateinwhiheahplyisharaterisedbyaurvilinearpathofthetow

(i.e. a VATonguration)insteadof theonventional straightbreformat ispresentedin

[2℄. Theauthors make use of a sensitivityanalysis and a gradient-based searh tehnique

to determine theoptimal breorientation ina given number ofregions oftheplate. This

work proved that a onsiderable inrement of the bukling load of the struture an be

obtained whenemploying a VATsolutionfor thelayeredplate.

The omplexity of the design proess of a VAT laminated struture is mainly due to

twointrinsi properties ofVATomposites,i.e. theheterogeneity andtheanisotropythat

intervene at dierent sales of the problem and that vary point-wise over the struture.

Moreover,afurtherdiultyisduetothefatthatsuhaproblemisintrinsiallyamulti-

sale design problem: in the most general ase, the designer should take into aount,

withinthe samedesign proess, thefullset ofdesign variables (geometrial and material)

governing the behaviour of the struture at eah harateristi sale (miro-meso-maro).

Up to now no general rules and methods exist for theoptimum design ofVATlaminates.

Onlyfew workson this topi an befound inliterature, andall of them always make use

ofsome simplifyinghypothesesandrules to getasolution. An exhaustive reviewfousing

on onstant and variable stiness design of omposite laminates is presented in [9, 10℄.

In [11℄ the rst natural frequeny of VS omposite panels is maximised by onsidering

on the one hand the lamination parameters and the lassial laminate theory (CLT) for

the desription of the loal stiness properties of the struture and, on the other hand,

a generalised reiproal approximation algorithm for the resolution of the optimisation

problem. This approah is limited to the determination of the stiness properties of an

equivalent homogeneous plate, sine the lay-up design phase is not at all onsidered. In

[12℄theleast-weight design problemof VAT laminatessubmitted to onstraints inluding

thestrengthandthe radius ofurvatureisonsidered. Thedesign variablesarethelayers

thikness and bres angles whih are represented by bi-ubi Bezier surfaes and ubi

Bezier urves, respetively. A sequential quadrati programming method is used to solve

theoptimisation problem.

A two-level strategy wasemployed in[13℄to design a VATlaminated plate: this work

represents the rst attempt of applying a multi-sale numerial strategy whih aims at

determining, at the rst level, theoptimum distribution of the stiness properties of the

struture(intermsofthelaminationparametersofthelaminate),whileattheseondlevel

theoptimumpath (ineahonstitutive layer)mathingloallythelaminationparameters

resultingfromtherststep. However,themajordrawbakofthisworkatuallywasinthe

determination of the urvilinear bres-path of eah layer: the resulting path was dison-

tinuousbeause theauthors had not foreseenanumerial strategy able to simultaneously

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on the other hand the optimum distribution of lamination parameters provided by the

rststep ofthe proedure. In[14℄the two-levelapproahwas abandoned andtheauthors

stated the problem bydiretly onsidering thebres path ineah ply asdesign variables.

However, as in[11℄, this approah always leads to a disontinuous bres-pathand, unlike

thestrategy proposedin [11℄, itleads alsoto the emergene of a new issue: theresulting

optimisationproblemwashighlynon-onvexsineitwasformulateddiretlyinthespaeof

thelayersorientations (whihvaryloallyovertheplate). Aordingly,in[14℄theauthors

onlude that suh an issue an be potentially remedied by formulating in a proper way

the design problem of VAT laminates in the framework of the two-level strategy and by

trying to overome the issueof the ontinuity of the bres-path diretlyin the rst level

ofthestrategy wherethe designvariablesarethelaminate mehanial properties(e.g. the

laminationparameters asinthetheoretial framework of[13℄).

Another issueoftenaddressedbyresearhesonVATlaminatesonerns thetowplae-

ment tehnology whih ould introdue several dierenes (i.e. imperfetions) between

thenumerial model of the VAT omposite and thereal struture tailored withthe AFP

proess, if the design methodology does not take into aount the manufaturability re-

quirements. Tothispurposein[15℄anissuelinkedtotheAFPtehnology isaddressed: the

overlap oftow-plaed ourses thatinreases theply thikness (the build-up phenomenon)

thus aeting the strutural response and the surfae quality of thelaminate. The work

of Blom et al. [15℄ presents a method for designing omposite plies with varying bres

angles. The bre angle distribution per ply is given while, using a streamline analogy,

theoptimal distributions of bres oursesisdetermined for minimisingthemaximumply

thikness or maximising thesurfae smoothness. An improved researh on this topi has

been developed in[16℄where analgorithmispresentedto optimisethebrespathinorder

toensuremanufaturability. Afurtherworkfousingonthedevelopment and/orimprove-

ment of manufaturing tehniques for tailoring VAT laminates in order to minimise the

imperfetions indued bythefabriation proess is presented in[17℄. Theontinuous tow

shearing(CTS)tehnique,utilisingthe abilitytosheardrytows,isproposedasanalterna-

tivetehniqueto the well-known AFPproess. Later, theworkpresentedin[17℄hasbeen

improvedthrough theintrodutionofa omputer-aidedmodellingtool[18℄whih anre-

ateaurate niteelement models reetingthe bretrajetories and thiknessvariations

ofVATomposites manufatured usingthe CTS tehnique.

Tooveromethepreviousrestritions,reentlytheauthorspresentedin[19℄arstwork

foused on the generalisation and extension of the multi-sale bi-level (MS2L) proedure

for the optimum design of omposite strutures (initially introdued in [20, 21℄) to the

ase of VAT omposites. Suh a MS2L proedure has already been employed by the

authors andtheir o-workersinthepast for thedesignand optimisation of severallasses

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isharaterisedontheonehandbytherefusalofsimplifyinghypothesesandlassialrules

usually employed in the framework of the design proess of laminates, and on the other

handbyaproperandompletemathematialformalisationoftheoptimumdesignproblem

ateahharateristisale(meso-maro). TheMS2Lstrategyreliesontheuseofthepolar

formalism(initially introdued byVerhery [28℄, and later extendedto theaseof higher-

ordertheoriesbyMontemurro[29,30,31℄)for thedesriptionoftheanisotropibehaviour

of theomposite. The real advantage inusing the Verhery's polarmethod is inthe fat

thattheelasti responseofthestrutureatthemaro-saleisdesribed intermsoftensor

invariants,theso-alledpolarparameters havingapreisephysialmeaning(whihislinked

totheelastisymmetriesofthematerial)[32℄. OntheotherhandtheMS2Lstrategyrelies

ontheuseofa partiulargenetialgorithm (GA)ableto dealwitha speiallassofhuge-

size optimisation problems (from hundreds to thousands of design variables) dened over

a domain of variabledimension, i.e. optimisation problems involving a variable number

ofdesign variables [25℄.

As far as onerns the problem of designing VAT omposites in [19℄several modia-

tionshavebeenintrodued inthetheoretialandnumerialframeworkoftheMS2Ldesign

proedure at both rst and seond levels. At the rst level (laminate marosopi sale)

of the proedure, where the VAT laminate is modelled as an equivalent homogeneous

anisotropi plate whose mehanial behaviour is desribed in terms of polar parameters

(whih vary loally over the struture) the major modiations onerned: 1) the utilisa-

tion of higher-order theories (First-order Shear Deformation Theory (FSDT) framework

[29,30℄)fortakingintoaounttheinueneofthetransverseshearstinessontheoverall

mehanial responseof VAT omposites;2) the utilisation ofB-spline surfaesfor obtain-

ing a ontinuous point-wise variation of the laminate polar parameters. Regarding the

seond-level problem (laminate mesosopi sale, i.e. the ply level) the main modia-

tions were: 1) the utilisation of B-spline surfaes for obtaining a ontinuous point-wise

variationofthe bres-pathwithineah ply;2) apropermathematial formalisationofthe

manufaturability onstraints linked to the AFPproess intheframework of theB-spline

representation. Allofthesemodiationsimplyseveraladvantagesfortheresolution ofthe

related optimisation problems (both at rst and seond level of the strategy) as detailed

in[19℄. However, the major drawbak of the approah proposedin [19℄is inthefat that

themanufaturabilityonstraintshave been introdued onlywithin theseond stepofthe

design proedure. Atually, when using suh an approah, there is no warranty that the

optimisation algorithm ould nd an optimum bres path able to meet on the one hand

the optimum distribution of the laminate polar parameters resulting from the rst step

and onthe other handthemanufaturabilityonstraintsimposedduringthe seondstep.

To overome suh an issue, in the present work the theoretial formulation of the

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have been improved: in partiular the manufaturability onstraints are now integrated

withinthe rst-level problem(thestrutural optimisation)whiletheseond-level problem

(the lay-up design) is now formulated as an unonstrained minimisation problem as all

therequirements (geometrial, tehnologial, mehanial, et.) are satisedsine therst

stepof the MS2Lstrategy. The paperisorganised asfollows: thedesign problemand the

MS2Lstrategy aredisussed inSetion 2. Themathematial formulation oftherst-level

problem is detailed in Setion 3, while the mathematial statement of the seond-level

problem(the lay-up design) ispresented inSetion 4. A onise desriptionof theFinite

Element(FE)modeloftheVATlaminatedplateisgiveninSetion5,whilethenumerial

results of the optimisation proedure areshownin Setion 6. Finally, Setion 7 ends the

paperwithsome onluding remarks.

2 A new design paradigm for VAT laminates

2.1 Desription of the problem

The optimisation strategy presented in this study is applied to a VAT laminated plate

omposedofaxednumberofplies,henethetotalthiknessoftheplateisxeda priori.

The bres-tow ismade of arbon-epoxy pre-pregstrips whose elasti properties are listed

inTable 1.

ConerningthemehanialbehaviouroftheVATplate,furtherdetailshavetobeadded

inorderto learly dene the theoretial framework ofthis work:

•

^the^geometryôf^the^laminated^strutureând^theâpplied^Boundary^Conditions^(BCs)

areknownand xed;

•

^the^VÂT^plate îsômposedôfîdential^plies^(i.e. ^same ^materialând ^thikness);

•

^the^material^behaviour^is ^linear^elasti;

•

^the^VA^T^plate^isquasi-homogeneousandfullyorthotropi [22,23,27,19℄point-wise,

i.e. theseproperties apply loallyineah point of thestruture;

•

ât ^the ^maro-sale ^(i.e. ^the ^sale ôf ^the ^struture) ^the êlasti ^response ôf ^the ^VÂT

plateisdesribedinthetheoretialframeworkoftheFSDTandthestinessmatries

of the plate are expressedin terms of the laminate polar parameters [29, 30℄whih

onstitute alsothedesign variables oftheVATplateat themarosopi sale.

Asfar asonerns the mesosopi sale of the VATlaminate (i.e. that oftheonstitutive

ply) nosimplifying hypotheses aremadeon therestof thedesign parameters ofthelami-

natedplate,i.e. thedesignvariablesofthestak,namelythelayerpositionandorientation

angle(whih variespoint-wisefor eahlayer).

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The main goal of the design strategy is the maximisation of the rst bukling load of a

VATplate subjet to

•

feasibility onstraints onthemehanial parameters (i.e. thelaminate polar param-

eters)governing the behaviourof thestrutureat themarosopi sale;

•

manufaturability onstraints on the loalradius of thetow (i.e. theloal steering)

due totheonsidered AFPtehnology.

The optimisation proedure is artiulated into thefollowing two distint (butlinked) op-

timisation problems.

1. First-level problem. The aim of this phase is the determination of the optimum

distribution of the mehanial design variables of the VAT struture in order to

minimise theonsidered objetive funtion and to meet, simultaneously, thefull set

ofoptimisation onstraints provided bythe problem at hand. At thislevelthe VAT

omposite plate is modelled as an equivalent homogeneous anisotropi ontinuum

whose behaviour is desribed in terms of laminate polar parameters, whih vary

point-wise over the struture and onstitute the design variables of the rst-level

problem. Moreover, thanks to a proper formulation of the optimisation problem

(detailed inSetion 3), at this stage the designer an add further requirements, e.g.

manufaturability onstraints, strength and damage riteria, et. by ating diretly

onthe resulting distribution of thelaminatepolar parameters.

2. Seond-level problem. The purpose of this design phase is the determination of

the optimum lay-up of the laminate omposing the struture meeting the optimum

distribution of the polar parameters provided by the rst level of the strategy. At

thisstage,thedesignvariablesarethelayerorientationangles whih varypoint-wise

ineahply (namely thebres-path).

It is noteworthy that the integration of the manufaturability onstraints within the

rst-level problem impliessome important onsequenes on theformulation as well ason

theresolution tehnique of the seond-levelproblem thatwill be detailedinSetion 4.

3 Mathematial formulation of the rst-level problem

Asdisussed in [19℄, in orderto apply theMS2L numerial optimisation strategy [23,27℄

totheaseofVATompositessomemajor modiationshavebeen introdued. Regarding

therst-levelproblem thesemodiationsfouson:

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•

^theutilisationofhigher-ordertheories(inthisasetheFSDT)fortakingintoaount

the inuene of the transverse shear stiness on the overall mehanial response of

theVATlaminate;

•

^theutilisation ofB-spline surfaesfor expressingthevariationof thelaminate polar

parameters overthe struture.

The rst point represents a very important step forward in the MS2L strategy when

applied to every kind of omposite struture (lassial or VAT) as it allows to properly

design thinaswell asmoderately thik plates.

The seond modiation leads to important onsequenes, too. Suh onsequenes

onstitute justasmanyadvantages forthe resolution oftherelatedoptimisation problem.

Firstly, the utilisation of iso-geometri surfaes leads to a onsiderable redution in the

numberof mehanial design variables (atthemaro-sale), i.e. thepolarparameters de-

nedineahpointoftheontrolnet oftheB-splinesurfae. Seondly,thankstothestrong

onvex hull property of theB-splineblending funtionstheoptimisation onstraints ofthe

problem, related to the speiations of the onsidered appliation, an be imposed only

ontheontrol pointsofthenet: iftheyaresatised onsuhpointsthey areautomatially

metoverthe wholedomain.

However, in the present work, in order to integrate themanufaturability onstraints

within the mathematial formulationof the rst-level problemfurther modiations have

been introdued:

•

ônly^the^variationôfône^polar^parameter^(namely^the^polarângle,âs^disussedⁱⁿ^the

next subsetion)is desribed interms of iso-geometri surfae over the plate,while

theothers arekeptonstant;

•

^thanks ^to ^the ^previous assumption, the tehnologial onstraint on the minimum

admissible radius of urvature of the tow an be diretly imposed on the B-spline

surfaedesribing the variation ofthe polarangle.

Theselastaspetsareofparamountimportane: duetothefuntionalrelationshipexisting

betweenthepolarangleandthepliesorientationangles[29,30℄,whetherthemanufatura-

bility onstraint is satised bythe optimumdistribution of thepolarangle overtheplate

it will be automatially met also by the B-spline surfaes desribing the variation of the

brespath ineah layerasdisussed inSetion 4.

As previously stated the goal of the rst level of the strategy is the maximisation of

the bukling load of the VAT laminate by simultaneously satisfying the feasibility and

manufaturability onstraints on the distribution of the laminate polar parameters over

theplate. All ofthese aspetsare detailedinthe following subsetion.

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In the framework of the FSDT theory [33℄ the onstitutive law of the laminated plate

(expressedwithin the global frameof thelaminate

R = {0; x, y, z}

⁾ ^an^be ^stated^as:





 {N}

{M}







=





[A] [B]

[B] [D]









 {ε 0 } {χ 0 }







,

⁽¹⁾

{F} = [H] {γ 0 } ,

⁽²⁾

where

[A]

^,

[B]

^and

[D]

^are^the^membrane, membrane/bending oupling and bendingsti- ness matries of the laminate, while

[H]

^is ^the out-of-plane shear stiness matrix.

{N}

^,

{M}

^and

{F}

âre^the ^vetorsôf ^membrane ^fores, ^bending ^momentsând ^shear ^fores ^per

unit length, respetively, whilst

{ε 0 }

^,

{χ 0 }

^and

{γ 0 }

âre ^the ^vetors ôf în-plane ^strains,

urvatures and out-of-planeshear strains ofthelaminate middle plane,respetively,[33℄.

Inordertoanalysetheelastiresponseofthemultilayerplatethebestpratieonsists

inintroduing the laminate homogenisedstiness matriesdened as:

[A ∗ ] = 1

h [A] ,

[B ∗ ] = 2

h 2 [B] ,

[D ∗ ] = 12

h 3 [D] ,

[H ∗ ] =



 

  1

h [H] (basic) , 12

5h [H] (modified) .

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where

h

^is^the ^total ^thikness ^of^the^laminated ^plate.

Asdisussedin[29,30℄,intheframeworkofthepolarformalismitispossibletoexpress

theCartesian omponents of these matries interms of their elasti invariants: it an be

proven that, also in the FSDT theoretial framework, in the ase of a fully orthotropi,

quasi-homogeneous laminate the overall number of independent mehanial design vari-

ables desribing its mehanial response redues to only three, i.e. the anisotropi polar

parameters

R A 0K ∗

^and

R A 1 ∗

^and^the^polar^angle

Φ A 1 ∗

^(this^lastrepresentingtheorientation of themain orthotropy axis) of the homogenised membrane stiness matrix

[A ∗ ]

^. ^F^or ^more

details on the polar formalism and its appliation in theontext of the FSDT thereader

isaddressed to[29,30,32℄.

For a VAT omposite, in the most general ase, all of the three independent polar

parameters an vary point-wise over the struture [19℄. However, the aim of this work

is to improve the MS2L proedure in order to integrate the manufaturability onstraint

on the loal steering (i.e. the loal radius of urvature of the tow) sine the rst step

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angle

Φ A 1 ∗

ân ^vâry ôver ^the ^struture ^(and ^suh â ^variation îs êxpressed ^by ^means ôf â

B-splinesurfae)while

R A 0K ∗

^and

R A 1 ∗

âre^keptônstant. În^partiular,ⁱⁿ^themathematial framework of the B-spline surfaes the variation of the laminate polar angle

Φ A 1 ∗

^an ^be

expressedas:

Φ A 1 ∗ (ξ, γ) =

n p

X

i=0 m p

X

j=0

N i,p (ξ) N j,q (γ ) Φ A 1 ∗ (i,j) .

⁽⁴⁾

Eq. (4)fullydesribesaB-splinesurfaeofdegrees

p

^and

q

^along^the^parametri^oordinates

ξ

^and

γ

^,respetively,asdepited inFig.1.

The dimensionless oordinates

ξ

^and

γ

^an ^be arbitrarily dened: a natural hoie onsistsoflinking them to the Cartesianoordinatesof thelaminated plate,

ξ = x

a , γ = y

b ,

⁽⁵⁾

where

a

^and

b

âre ^the ^lengths ôf ^the ^plate êdges âlong

x

^and

y

^axes, respetively. In Eq.(4)

Φ A 1 ∗ (i,j)

⁽

i = 0, · · · , n p

^,

j = 0, · · · , m p

⁾ îs ^the ^value ôf ^the ^laminate ^polar ângle ât

thegeneriontrolpoint (thesetof

(n p + 1) × (m p + 1)

^ontrol ^points^forms ^the^so-alled

ontrol network),while

N i,p (ξ)

^and

N j,q (γ)

^are^the

p th

^-degree^and

q th

^-degree^B-spline^basis

funtions(along

ξ

^and

γ

^diretions,respetively)denedonthenon-periodi,non-uniform knotvetors:

Ξ =







0, · · · , 0

| {z }

p+1

, Ξ p+1 , · · · , Ξ r − p −1 , 1, · · · , 1

| {z }

p+1





 ,

Γ =







0, · · · , 0

| {z }

q+1

, Γ q+1 , · · · , Γ s − q −1 , 1, · · · , 1

| {z }

q+1





 .

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It is noteworthy that the dimensions of the knot-vetors

Ξ

and

Γ

are

r + 1

^and

s + 1

^,

respetively,with:

r = n p + p + 1 ,

s = m p + q + 1 .

⁽⁷⁾

For a deeper insightin thematter thereader isaddressedto [34℄.

Thisnewformulationofthemehanialdesignvariablesoftherst-levelproblemallows

thedesignertoimposethemanufaturabilityonstraintsdiretlyonthespatialdistribution

of the polar angle

Φ A 1 ∗

^: ^if ^this requirement is met at this stage of the strategy it will be automatiallymetbytheoptimumlay-upresulting fromtheseond-levelproblemwithout

imposing further onstraints, as detailed in Setion 4. Furthermore, this formulation is

haraterisedby thefollowing advantages:

•

^theûseôfâniso-geometrisurfaefor desribingthevariationof

Φ A 1 ∗

^over^the^stru-

tureimpliesthatthethree independent polarparameters have nodisontinuityover

the plate(

R A 0K ∗

^and

R A 1 ∗

^being^onstant);

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•

^thanks^to ^the^B-spline representation thelaminate polar angle must be determined

solely on eah point of the ontrol net, implying in this way a signiant redution

inthe numberofdesign variables involved withintherst-level problem.

Therefore,the optimisation variables oftheprobleman begrouped into thevetor:

x = n

Φ A 1 ∗ (0,0) , · · · , Φ A 1 ∗ (n p ,m p ) , R A 0K ∗ , R A 1 ∗ o

.

⁽⁸⁾

Thetotal numberof design variables is heneequal to

2 + (n p + 1) × (m p + 1)

^.

Thanks to the utilisation of the B-splinesurfae for representing the spatial variation

of

Φ A 1 ∗

^,^the tehnologial onstraint on theminimumadmissibleradius ofurvatureofthe towan be stated inastraightforward way:

g 1 ( x ) = r adm − r min ≤ 0 ,

⁽⁹⁾

where

r adm

îs^the^minimumâdmissible ^radiusôfûrvatureôf^the^tow^whose ^vâlue^depends

upon the AFP proess, while

r min

îs ^the ^loal ^least ^radius ôf ûrvature ôf ^the ^B-spline

surfae desribing thevariation of

Φ A 1 ∗

^.

r min

^is^dened ^as:

r min = min

(x,y) r(x, y) , r(x, y) = t · ∇Φ A 1 ∗ −1

, x ∈ [0, a] , y ∈ [0, b] .

(10)

InEq. (10)

t

isthe loaltangent vetorof theangular eld

Φ 1 A ∗ (x, y)

^,^while

∇Φ A 1 ∗

^is^the

gradient ofthe orthotropy orientation withrespetto oordinates

(x, y)

^,^namely

t =

cos Φ A 1 ∗ , sin Φ A 1 ∗ ,

∇Φ A 1 ∗ = 1

a

∂Φ A 1 ∗

∂ξ , 1 b

∂Φ A 1 ∗

∂γ

.

(11)

Inaddition,intheformulationoftheoptimisationproblemfortherstlevelofthestrat-

egy,thegeometriandfeasibilityonstraintsonthepolarparameters (whiharisefromthe

ombinationofthelayersorientationsandpositionswithinthestak)mustalsobeonsid-

ered. Theseonstraints ensure thatthe optimumvalues ofthe polar parameters resulting

fromtherststeporrespondtoafeasiblelaminatethatwillbedesignedduringtheseond

stepof the optimisation strategy,see [35℄. Sine thelaminate is quasi-homogeneous, suh

onstraints an be written onlyfor matrix

[A ∗ ]

^as^follows:



 



 



−R 0 ≤ R A 0K ∗ ≤ R 0 , 0 ≤ R A 1 ∗ ≤ R 1 , 2

R A 1 ∗ R 1

2

− 1 − R A 0K ∗ R 0 ≤ 0 .

(12)

(13)

therelevant optimisation variables, i.e. on

R A 0K ∗

^and

R A 1 ∗

^. ^Hene, ^the ^resulting feasibility onstraint on thelaminate polar parameters is:

g 2 ( x ) = 2 R A 1 ∗

R 1 2

− 1 − R 0K A ∗

R 0 ≤ 0 .

⁽¹³⁾

For a wide disussionupon thelaminate feasibilityand geometrial boundsas well as

ontheimportane ofthequasi-homogeneity assumption thereader is addressedto [35℄.

3.2 Mathematial statement of the problem

Therst-levelproblemfousesonthedenitionoftheoptimaldistributionofthelaminate

polar parameters. Inthis bakground,thesolutionof thestrutural optimisation problem

is searhed for an orthotropi and quasi-homogeneous (point-wise) plate subjet to given

BCs.

Therefore theoptimisation probleman beformulated asfollows:

min

x − λ ( x )

subjetto:

g i ( x ) ≤ 0 , (i = 1, 2)

(14)

where

λ

^is^the ^rst ^bukling ^fator^of ^the^laminated ^struture.

3.3 Numerial strategy

Problem(14) isa non-linear, non-onvex problem interms ofthe mehanial design vari-

ables. Itsnon-linearityandnon-onvexityisduetothenatureoftheobjetivefuntion,the

rst bukling fator,thatis anon-onvexfuntion interms oftheorthotropy orientation.

In addition, the omplexity of suh a problem is also due to the feasibility and manu-

faturability onstraints imposed on the polar parameters of the plate, see Eqs. (9) and

(13). We reall that the overall number of design variables and optimisation onstraints

for problem(14) is

2 + (n p + 1) × (m p + 1)

^and^two,respetively.

For the resolution of problem (14) a hybrid optimisation tool, omposed of the GA

BIANCA [25℄ interfaed with the MATLAB fminon algorithm [36℄, oupled with a FE

model of the plate (used for numerial alulation of the rst bukling load) has been

developed,seeFig. 2.

TheGABIANCAwasalreadysuessfullyappliedtosolvedierentkindsofreal-world

engineering problems, e.g. [23, 27℄. As shown in Fig. 2, the optimisation proedure for

the rst-level problem is split intwo phases. During the rst phase theGA BIANCA is

interfaed with the FE model of the VAT plate: for eah individual at eah generation,

a FE-based bukling analysis is arried out for the evaluation of the rst bukling load

(14)

BIANCAandelaboratedbyanANSYSParametriDesignLanguage(APDL)marowhih

generatestheB-splinesurfaerepresentingthedistributionofthepolarangle

Φ A 1 ∗

^over^the

VAT plate,inorder to alulate therst bukling loadof thestruture together withthe

minimum radius of urvature of the angular eld. At the end of the FE analysis, the

GAelaborates theresults provided bythe FEmodel (intermsofobjetive and onstraint

funtions)inorder to exeutethe geneti operations. These operations arerepeated until

theGAmeetsthe user-denedonvergene riterion.

The generi individual of the GA represents a potential solution for the problem at

hand. The genotype of the individual for problem (14) is illustrated in Fig. 3 and it

is haraterised on the one hand by a modular part omposed of

(n p + 1) × (m p + 1)

hromosomeswith only one geneodingthe valueof thepolar angle

Φ A 1 ∗ (i,j)

^while^on ^the

other hand by a xed hromosome with only two genes eah one oding the remaining

polar parameters, i.e.

R A 0K ∗

^and

R A 1 ∗

^. ^Due ^to ^the ^strong ^non-onvex ^nature ^of ^problem

(14), the aim of thegeneti alulationis to provide a potential sub-optimal point inthe

design spae whih onstitutes the initial guess for the subsequent phase, i.e. the loal

optimisation, wherethe fminon gradient-based algorithm isinterfaed withthesame FE

modelof the VATplate.

4 Mathematial formulation of the seond-level problem

The seond-level problem onerns the lay-up design of the VAT laminated plate. The

goal of this problem is the determination of at leastone staking sequene satisfying the

optimumspatialdistributionofthepolarparameters overthestrutureresulting fromthe

rstlevelofthestrategyandhavingtheelastisymmetriesimposedtothelaminatewithin

theformulation ofthe rst-level problem,i.e. quasi-homogeneity andorthotropy.

In the aseof a VATsolution the bres-pathvariespoint-wiseineveryply omposing

thelaminate. Hene a properdesription of thebres-path isneessary to formulate and

solve the seond-level problem of the MS2L strategy. To this purpose, the point-wise

variationof the bres orientation (in eah ply) is desribed through theuse of a B-spline

surfae. As explained in [19℄ the use of B-spline surfaes allows, as in the ase of the

rst-levelproblem, to redue thetotal number ofdesign variables.

Nevertheless, unlike the approah proposed in [19℄, the improved formulation of the

rst-levelproblemalongwiththeintegrationofthemanufaturabilityonstraintsdesribed

inSetion3leadtosomemajormodiations(andsimpliations)alsofortheseond-level

problem. Inpartiular, asin[19℄thebres-pathinthe generiply isrepresented through

aB-spline surfaeas:

δ k (ξ, γ) =

n p

P

i=0 m p

P

j=0

N i,p (ξ) N j,q (γ) δ k (i,j) with k = 1, · · · , n ,

⁽¹⁵⁾

(15)

where

δ k (i,j)

istheorientation angle at thegeneri ontrol pointfor the k-thlayer. More-

over,this B-spline surfae isharaterisedbythe sameparameters (withtheexeption of

thevalueoftheorientationineahontrol point)asthoseutilised forthedenitionofthe

B-splinesurfae ofthepolarangle

Φ A 1 ∗ (ξ, γ)

^,^see ^Eqs. ^(4)-(7).

As a onsequene of both the new formalisation of the rst-level problem and the

assumptionsmadeon the spatialdistributionof the polar parametersof thelaminate,the

valueofthe bre orientation angleineahontrol point (foreah layer)an bealulated

asfollows:

δ k (i,j) = Φ A 1 ∗ (i,j) + δ k (ξ 0 , γ 0 ) − Φ 1 A ∗ (ξ 0 , γ 0 ) ,

⁽¹⁶⁾

where

δ k (ξ 0 , γ 0 )

^and

Φ 1 A ∗ (ξ 0 , γ 0 )

^are^the^bresorientation angleforthek-thlayerandthe loalorthotropyorientation ofthe laminate,respetively,alulated atthearbitrarypoint

(ξ 0 , γ 0 )

^. Âs ît^learly âppears ^from Êq. ⁽¹⁶⁾ ^the ônly ûnknowns âre ^the ^layer orientation angles

δ k (ξ 0 , γ 0 )

^. ^Therefore,ⁱⁿ^this^bakground^(and^unlike^the^approah^proposedⁱⁿ^[19℄)

thedesignvariablesoftheseond-levelproblemarenotthelayersorientationanglesineah

point of the ontrol net, rather the value of the orientation angle at the arbitrary point

(ξ 0 , γ 0 )

^for^eah ^ply^,^namely

δ k (ξ 0 , γ 0 )

^. ^This^fatîmplies^someônsequenesôf^paramount

importane whih onstitutes as many advantages in the formalisation and resolution of

theseond-levelproblem.

Firstly,whenomparedtotheapproahpresentedin[19℄thenumberofdesignvariables

oftheseond-levelproblemisdrastiallyredued andpassesfrom

n × (n p + 1) × (m p + 1)

to only

n

^,^i.e. ^the ^pliesorientation angles dened for thearbitrarypair

(ξ 0 , γ 0 )

^.

Seondly, the seond-level problem an now be formulated as an unonstrained min-

imisation problem(beause the manufaturability onstraintsare integrated into therst

stageoftheMS2Lstrategy)andsolvedsolelyinone(arbitrary)pointoftheVATlaminated

plate:

δ k min (ξ 0 ,γ 0 ) I (δ k (ξ 0 , γ 0 )) k = 1, · · · , n .

⁽¹⁷⁾

InEq. (17)

I(δ k (ξ 0 , γ 0 ))

îs^the ôverall ôbjetive ^funtion^whih îs^dened âs:

I (δ k (ξ 0 , γ 0 )) = X 6

i=1

f i (δ k (ξ 0 , γ 0 )) .

⁽¹⁸⁾

where

f i (δ k (ξ 0 , γ 0 ))

^are ^quadrati ^funtions ⁱⁿ ^the ^spae ^of ^polar parameters, eah one representing a requirement to be satised. For theproblem at hand the partial objetive

(16)

f 1 (δ k (ξ 0 , γ 0 )) =

Φ A 0 ∗ (δ k (ξ 0 , γ 0 )) − Φ A 1 ∗ (δ k (ξ 0 , γ 0 ))

π/4 − K A ∗ (opt)

2 ,

f 2 (δ k (ξ 0 , γ 0 )) =

"

R A 0 ∗ (δ k (ξ 0 , γ 0 )) − R 0 A ∗ (opt) R 0

# 2

,

f 3 (δ k (ξ 0 , γ 0 )) =

"

R A 1 ∗ (δ k (ξ 0 , γ 0 )) − R 1 A ∗ (opt) R 1

# 2

,

f 4 (δ k (ξ 0 , γ 0 )) =

"

Φ A 1 ∗ (δ k (ξ 0 , γ 0 )) − Φ A 1 ∗ (opt) (ξ 0 , γ 0 ) π/4

# 2

,

f 5 (δ k (ξ 0 , γ 0 )) =

|| [C] (δ k (ξ 0 , γ 0 ))||

|| [Q] ||

2

, f 6 (δ k (ξ 0 , γ 0 )) =

|| [B ∗ ] (δ k (ξ 0 , γ 0 ))||

|| [Q] ||

2

,

(19)

where

K A ∗ (opt) = (

1 if R A 0K ∗ (opt) < 0 ,

0 otherwise .

⁽²⁰⁾

In Eq. (19)

f 1 (δ k (ξ 0 , γ 0 ))

^represents ^the ^elasti requirement on the orthotropy of the laminate having thepresribed shape,

f 2 (δ k (ξ 0 , γ 0 ))

^,

f 3 (δ k (ξ 0 , γ 0 ))

^and

f 4 (δ k (ξ 0 , γ 0 ))

^are

therequirementsrelatedtothepresribedvaluesoftheoptimalpolarparametersresulting

fromtherst-levelproblem, while

f 5 (δ k (ξ 0 , γ 0 ))

^and

f 6 (δ k (ξ 0 , γ 0 ))

^are^linked^to^the^quasi-

homogeneityondition. Formoredetailsonthemeaningofthepartialobjetivefuntions,

on the elasti symmetries of the laminate in the framework of the FSDT and on the

symbols appearing in Eq. (19), the reader is addressed to [29, 30℄.

I (δ k (ξ 0 , γ 0 ))

^is ^a

positivesemi-denite onvexfuntioninthespaeoflaminatepolarparameters, sineitis

dened as a sum of onvexfuntions, seeEqs. (18)-(19). Nevertheless, suh a funtion is

highlynon-onvexinthe spaeofpliesorientationsbeausethelaminatepolarparameters

dependuponirular funtionsofthelayersorientationangles,see[29,30℄. Moreover,one

of the advantages of suh a formulation onsists inthe fat that the absolute minima of

I(δ k (ξ 0 , γ 0 ))

âre^knownâ^priori^sine^theyâre^the ^zeroesôf^this^funtion. ^Fôr^more^details

aboutthe nature oftheseond-levelproblemsee [25,26,21℄.

Conerning thenumerial strategyfor solvingproblem(17)theGABIANCAhasbeen

employed to nd a solution also for the seond-level problem. In the most general ase,

thegenotype ofthe individualisomposedof

n

^hromosomes^(one^for êah^ply), êahône

haraterisedby asinglegene odingthelayerorientation angle.

5 Finite element model of the VAT laminate

Inordertodetermine theurrentvalueoftheobjetivefuntion(therst buklingfator)

and that of the optimisation onstraints of problem (14) a lassial eigenvalue bukling

(17)

alulation, dierent ongurations of the VAT plate requires the reation of an ad-ho

input le for the FE model that has to be interfaed with the hybrid (GA + gradient-

basedalgorithms) optimisation tool.

The FEmodelof the VAT laminated plate(see Fig. 4) employed during therst step

of theMS2L strategy,is built within the ANSYS environment andis made ofSHELL281

elementsbased onthe Reissner-Mindlin kinematimodel,having 8nodesand sixDegrees

OfFreedom (DOFs) pernode. The mesh size ishosen afterpreliminary meshsensitivity

analyseson theonvergene of thevalueof the rst bukling load for a given setof BCs.

It was observed that a mesh having 2482 DOFs issuient to properly evaluate therst

buklingloadof thestruture.

It is noteworthy that the B-Spline mathematial formalism hasbeen implemented by

the authors into the ANSYS environment by using the APDL [37℄ for reating a set of

appropriate maros that were integrated within the FEmodel of theVATplate. At this

stage,the plateismodelled asanequivalenthomogeneousanisotropiplatewhose stiness

matries(

[A ∗ ]

^,

[B ∗ ]

^,

[D ∗ ]

^and

[H ∗ ]

⁾^vary point-wise, i.e. for eah element disretisingthe real struture. In partiular, in order to properly dene, for every element of the VAT

plate,theorretvalueofitsstinesspropertiesthefollowingstrategyhasbeen employed:

1. for a given set of the laminate polar parameters dened in eah ontrol point (i.e.

thedesign variables passed from the optimisation tool to theFE modelof the VAT

plate),build theorrespondingB-splinesurfaes;

2. disretisethe plateinto

N e

^elements;

3. x the element index

i

^: ^for ^the

i

^-th êlement ^retrieve ^the ^Cartesian ôordinates ôf

its entroid, i.e.

(x i e , y i e )

^and ^alulate ^the orresponding dimensionless oordinates

(ξ e i , γ i e )

^aording^to ^Eq. ^(5);

4. alulatethelaminate polarparameters (andhenetheCartesianomponentsofthe

stiness matries of the laminate) for

(ξ e i , γ e i )

^and ^assign ^the ^material ^properties ^to

the element

i

^;

5. repeatpoints

3

^and

4

^for êahêlement ôf ^the^plate.

Finally,the linearbukling analysisisperformedusingtheBCsdepited inFig.4and

listedinTable2.

6 Studied ases and results

In this setion a meaningful numerial example is onsidered in order to prove the ee-

tiveness of the MS2L strategy for the optimum design of VAT laminates. Asdepited in

(18)

N y /N x = 0.5

^. ^The ^plate ^has^a ^square ^geometry ^with ^side ^length

a = b = 254

^mm ^and ^is

madeof

n = 24

^plies^whose ^material^properties^are^those ^listedⁱⁿ^T^able^1. ^Conerning^the

rst-levelproblem, theparameters deningthe B-splinesurfaewhihdesribe thespatial

distribution of the polar angle over the VAT plate are set as:

n p = m p = 4

^(hene ^ve

ontrol points along eah diretion),

p = q = 2

^(degrees ^of ^the ^blending ^funtions ^along

eahdiretion). Moreover,the B-splineisdened overthefollowing uniform knot-vetors:

Ξ =

0, 0, 0, 1 3 , 2 3 , 1, 1, 1 , Γ =

0, 0, 0, 1 3 , 2 3 , 1, 1, 1 .

(21)

Aordingly,for therst-level problemtheoverallnumberof design variablesand optimi-

sation onstraints is

27

^and ^two, respetively. The mehanial design variables together withtheirnature andboundsfor the rst-levelproblemarelisted inTable3. Inaddition,

thereferenevaluefor theminimumadmissible radius ofurvatureof thetow, i.e.

r adm

^is

setequal to

80

^mm.

Conerning the seond-level problem, the parameters dening the B-spline surfae

whihdesribesthepoint-wisevariationofthebreorientation angle(foreahply)arethe

same asthoseemployed during therststep ofthe strategy. However, inorderto further

simplify the problem of retrieving an optimum stak, the searh spae for problem (17)

hasbeen restrited to apartiular lass ofquasi-homogeneous laminates: thequasi-trivial

(QT)stakingsequenes whihonstituteexatsolutions withrespetto therequirements

of quasi-homogeneity, i.e. funtions

f 5 (δ k (ξ 0 , γ 0 ))

^and

f 6 (δ k (ξ 0 , γ 0 ))

ⁱⁿ ^Eq. ^(19). ^QT

solutions anbe foundfor laminateswithidential plies(samematerial andthikness)by

ating only on the position of the layers within the stak. Indeed, the potential of QT

staks is in the fat that they onstitute exat solutions (in terms of quasi-homogeneity

ondition) regardless to the value of the orientation angle assigned to eah layer: in this

way the orientations represent free parameters whih an be optimised to full further

elasti requirements. The proedure for searhing QT staks is (oneptually) quite sim-

ple: it sues to x the number of layers

n

^and ^the ^num^ber ^of ^saturated ^groups

n g

^(i.e.

the number of possible dierent orientation angles within the stak) and look for all the

permutationsofthepositionofeahgroupmeetingthequasi-homogeneityondition. More

detailsonthistopianbefoundin[38℄. Nevertheless,asdisussedin[38℄,theproblemof

determiningQT staks gives riseto a huge numberofsolutions: thenumberofQT staks

rapidlyinreasesalongwith

n

^. ^To^this^purpose^a^database^of^QT ^staks^has^been ^built^for

dierent ombinations of

n

^and

n g

^.

For theproblemathand,andforeahonsideredase, thenumber ofsaturatedgroups

n g

^has^been^xed^a^priori. ^Let^be

n sol

^the^numberôf^QT^staks^forâ^partiularômbination

of

n

^and

n g

^. Êah^solutionôlleted ^within^the^databaseîs ûniquelyîdentied ^byân

ID sol

(i.e. aninteger) whihvariesintherange

[1, n sol ]

^. ^Therefore^the

ID sol

^represents^a^further

(19)

design variable along with the

n g

orientation angles. As stated previously, the solution searh is performedthrough theGA BIANCA; in thease of QT staks the struture of

the individual's genotype is simple beause it is omposed of a single hromosome with

n g + 1

^genes: ^the ^rst ^one ^odes ^the ^variable

ID sol

^whilst ^the ^remaining ^genes ^ode ^the

orientation angles assoiated to every group whih are ontinuous variables in the range

[-90

◦

,90

◦

℄.

Regarding the settingof the genetiparameters for theGABIANCAutilised for both

rst and seond-level problems they are listed in Table 4. Moreover, onerning the

onstraint-handling tehnique for the rst-level problem the Automati Dynami Penal-

ization (ADP) method has been employed, see [39℄. For more details on the numerial

tehniquesdevelopedwithinthe newversionofBIANCAand themeaning ofthevalues of

thedierent parameters tuning theGAthe reader isaddressed to [25,21℄.

Asfarasonerns thefminon optimisation toolemployedfor theloalsolutionsearh

attheendofthe rststep, thenumerial algorithmhosento arryout thealulations is

theative-set methodwithnon-linear onstraints. For more detailson thegradient-based

approahes implemented into MATLAB,the reader isaddressedto [36℄.

Before starting the multi-sale optimisation proess a referene struture must be de-

ned inorder to establish a referene value for the rst bukling fator of theplate. The

referenestrutureisstill asquareplate ofside

a = b = 254

^mm^omposed^of

24

^unidire-

tionalbre-reinfored laminae whose material properties are those listed inTable 1. The

stakingsequeneofthereferenesolutionis

[0/−45/0/45/90/45/0/−45/90/45/90/−45] s

^.

Thehoie ofthereferenesolutionhasbeen orientedtowardsasymmetriquasi-isotropi

stak, of ommon use in real-world engineering appliations, whih onstitutes a good

ompromisebetween weight and stinessrequirements(in termsofbuklingload): suha

ongurationisharaterisedbyabuklingfator

λ ref = 81.525

^when

N x = 1

^N/mm^and

N y = 0.5

^N/mm.

6.1 Case 1: design without manufaturability onstraints

Problem(14)hasbeenrstlysolvedtodesignaVATplatewhihisnotsubjettomanufa-

turabilityonstraints. Inthisasetheonlyoptimisationonstraintisthatonthefeasibility

ofthepolarparameters

R 0K A ∗

^and

R 1 A ∗

^,^see^Eq. ^(13). ^Conerning^the^rst-level^problem,

theoptimumdistributionofthelaminatepolarangle

Φ 1 A ∗

^over^the^VAT^plate^isillustrated in Fig. 5, while the optimum value of the polar parameters

R 0K A ∗

^and

R 1 A ∗

^as ^well ^as

thatof

Φ 1 A ∗

^for êahôntrol ^point âre^listed ⁱⁿ^Tâbles ⁵ ând^6,respetively.

Regarding thesolutionofthe seond-levelproblem,thenumber ofsaturatedgroups

n g

is set equal to three. In the ase of a laminate omposed of 24 plies and three saturated

groups the overall number of QT staks is

n sol = 26

^. ^Assigning ^the ^salars ^zero, ^one

and two to the rst,seond and third saturated group, respetively,the optimal staking

(20)

[0 1 0 1 2 0 2 0 2 2 0 0 0 0 1 1 0 1 0 1 2 0 2 0] .

⁽²²⁾

Thismeans thatwithin the optimumstaklayers

•

^1,^3,^6,^8,^11-14, ^17, ^19,²²^and ²⁴ ^belong ^to ^the^saturated^group ^0, ^i.e. ^they^share

thesameorientation

δ 0 (ξ, γ)

^;

•

^2,^4, ^15, ^16, ¹⁸ ^and ²⁰ ^belong ^to ^the ^saturated ^group ¹ haraterised by the loal

orientation angle

δ 1 (ξ, γ)

^;

•

^5,^7,^9,^10,²¹^and²³ ^belong^to^the^saturated^group ² ^sharing ^theorientationangle

δ 2 (ξ, γ)

^.

SinetheorientationangleforeverysaturatedgroupisdesribedbymeansofaB-spline

surfae, it sues to list the value of the orientation in eah point of the ontrol net to

uniquelydenetheB-splinesurfae. Tothispurposetheoptimumvaluesoftheorientation

angleineahpointoftheontrolnetforeverysaturatedgrouparereportedinTables8-10,

whilst anillustration ofthe optimumbres-path (foreah group)isdepited in Fig. 6.

It is noteworthy that the optimal solution found at the end of the MS2L design pro-

edureis haraterised bya buklingfator of

145.74

^whih ^is^about

78%

^higher ^than ^the

refereneounterpartand,inthemeantime, satisesthefeasibilityonstraintonthepolar

parameters.

6.2 Case 2: design inluding manufaturability onstraints

In this setion the problem of theoptimum design of a VAT laminated plate is solved in

themost general ase,i.e. bytakinginto aount alsothemanufaturabilityonstrainton

theminimum radius of urvature ofthe tow imposed by theAFPfabriation proess, see

Eq. (9). Conerning the rst-level problem, the optimum value of the polar parameters

R 0K A ∗

^and

R 1 A ∗

âs ^well âs ^that ôf

Φ 0 A ∗

^for êah ôntrol ^point âre ^reported ⁱⁿ^Tables ⁵

and7,respetively,whilst theoptimumdistribution ofthelaminate polarangle

Φ 0 A ∗

^over

theVATplate isillustrated inFig. 7.

Asshown inTable 5,for thisseond ase apartiular solution hasbeen obtained: the

optimum value of the polar parameter

R 1 A ∗

îs êqual ^to ^zero ⁱⁿ êah ^point ôf ^the ^VÂT

omposite plate whih means that loally the struture is haraterised by the square

elasti symmetry, see [31℄. In this ase, the main diretion of orthotropy is represented

(point-wise)bythespatial distributionof the polar angle

Φ 0 A ∗

^. ^The relationshipbetween thetwo polar angles,whih representsa tensor invariant too,is:

Φ 0 A ∗ − Φ 1 A ∗ = K A ∗ π

4 ,

⁽²³⁾

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too. Moreover, inthis ase

K A ∗ = 1

^beause

R 0K A ∗ < 0

^,^see^Table^5.

As far as onerns the solution of the seond-level problem, a number of saturated

groups

n g = 4

^has ^been ^hosen. În ^the âse ôf â ^laminate ômposedôf ²⁴ ^plies ând ^four

saturatedgroups onlyone QT stakexistshaving theform:

[0 1 2 3 2 3 1 3 0 2 0 1 0 1 3 1 2 0 2 3 2 3 0 1] ,

⁽²⁴⁾

wherean integerbetween zeroand threehasbeen assignedto eahsaturatedgroup. How-

ever, in performing the solution searh for problem (17), the GA provided an optimum

stakwith onlytwo saturatedgroups having theform:

[0 0 2 2 2 2 0 2 0 2 0 0 0 0 2 0 2 0 2 2 2 2 0 0] ,

⁽²⁵⁾

whih means that groups

0

^and

1

âre îdential âs^well âs ^groups

2

^and

3

^. ^The ^optimum

valuesof the orientation angles ineah point ofthe ontrol net for every saturated group

are reported in Tables 11 and 12, whilst an illustration of the optimum bres-path (for

eahgroup) isdepited inFig. 8.

It is noteworthy that the optimal solution found at the end of the MS2L design pro-

edureis haraterised bya buklingfator of

136.09

^whih ^is^about

67%

^higher ^than ^the

refereneounterpartand,inthe meantime, satisesboth thefeasibilityonstraint onthe

polar parameters and the manufaturability onstraint on the minimum (loal) radius of

urvature ofthetow.

6.3 General disussion of results

FromaarefulanalysisoftheoptimumongurationoftheVATlaminatedplateprovided

bytheMS2L proedure, itis possible to deduethefollowing fats.

•

^The^point-wise^variationôf ^the^laminate^polarângles ^resulting ^from ^the^rst ^step ôf

thestrategy istotallyasymmetri. Symmetrisolutions are,ofourse,possible: itis

suienttoimposethesymmetryonditiondiretlyonthevaluesofthepolarangles

at the points of the ontrol network of the B-spline surfaes. However, in order to

state and solve the optimisation problem inthemost general ase, suh a ondition

hasnot been imposedinthis study.

•

^When^looking ^at ^the^optimum ^spatialdistribution of thelaminate polar parameters

(Tables5-7),oneannotiethatthelaminateanbeharaterisedeitherbyanordi-

naryorthotropy shapewith

K A ∗ = 0

^(ase¹⁾ âs^well âs^byâ ^square^symmetry^with

R A 1 ∗ = 0

^(ase^2). ^Indeed,ⁱⁿ^the^rst^ase^(where^themanufaturabilityonstraint is not onsidered) the spatial distribution of

Φ 1 A ∗

^over ^the^plate ^is haraterisedbya

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turewhihstien"thestruturebyinreasingthebuklingfator. Conversely,when

lookingatthesolutionofase2(wheretherequirement ontheloalsteeringistaken

intoaount)thevariation of

Φ 0 A ∗

îs^smoother^than ^thatôf^the^previous âseⁱⁿôr-

derto omplywith the manufaturability onstraint. Moreover the solution of ase

2isloatedontheboundary ofthe elastidomainofthelaminate polarparameters,

seeTable3. Aordingly,when searhing for an optimumstaksolution of problem

(17) the GA found a point-wise QT symmetri ross-ply stak (indeed the angular

dierene between the two saturated groups is equal to

90 ◦

^for ^eah ^point ^of ^the

plate,seeTables 11 and12) whose mainaxis oforthotropy(i.e.

Φ 0 A ∗

⁾ ^varies^loally

to maximise the bukling fator and to meet, at the same time, the tehnologial

onstraint.

•

Ûnlike ^the ^vast ^majority ôf ^works ^reported ⁱⁿ ^literature ^[40℄, ^the ôptimum ^bres-

path for eah ply is very general. In the framework of the proposed approah, the

point-wise variation of the bres-path inevery lamina does not follow simple linear

orparaboli variations (withrespettolaminate globalframe),ratheritisdesribed

by a general B-spline surfae, see Eq. (15). This fat, together with the very gen-

eralformulationofthedesignproblemof VATlaminates, allows thedesignertond

an optimum stak meeting all the requirements (i.e. elasti and manufaturability

onstraints) whihareintegrateddiretlyintherststepofthestrategy (i.e. at the

marosopisale), seeEq. (14),withouttheneed ofafurtherpost-proessingtreat-

ment to simplify the trajetory of the tows in order to omply with theonstraints

imposed bythe AFPproess.

•

^Finally^,^the ôptimum^bres-path ^(forêah^layer)^foundât ^theênd ôf^the^seond ^step

ofthe MS2Lproedure doesnot need of a further stepfor thereonstrution of the

CADmodelbeausethevariationofthebres-pathisdesribedbyaB-splinesurfae

whihisfullyompatible withseveralstandard leformats(IGES,STLandSTEP),

allowing in this way a rapid exhange of information among the CAD tool and the

software ofthe AFPproess.

7 Conlusions

Inthiswork animprovedversionof themulti-sale two-leveldesign/optimisationmethod-

ology of VATompositestrutures (initiallypresented in[19℄)has been introdued. This

designparadigmessentiallyreliesontheutilisationofaMS2Loptimisationproedurehar-

aterised byseveralfeaturesthatmakeitan original,eetiveand generalmethodfor the

multi-saledesign ofVATstrutures. Inthepresentworkthisstrategy hasbeenemployed

to deal with theproblem of the maximisation of thebukling fator of a VAT multilayer

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thedesignproess isnot submitted torestritions: anyparameterharaterising theVAT

omposite(at eah sale)isan optimisation variable. Thisallows thedesigner to lookfor

a true global minimum, hard to be obtained otherwise. On theother hand, both the for-

mulationofthedesignproblemandtheMS2Loptimisation strategyhavebeen generalised

and improved inorder to be applied to theproblemof designinga VSomposite.

In the framework of the MS2L design methodology several modiations have been

introdued for both rst and seond level problems. As disussed in [19℄, regarding the

rst-levelproblem the mainmodiations are:

•

^the^use ^of higher-order theories (introdued asresult of [29,30,31℄) for taking into

aount the inuene of the transverse shear stiness on theoverall mehanial re-

sponseofVATomposites;

•

^the utilisation of B-spline surfaes for desribing the distribution of the laminate

polarparameters overthestruture whihleadsto aontinuouspoint-wisevariation

ofthe laminate stiness matries.

Theseaspets leadto someimportant advantagesfor theresolution of therelatedoptimi-

sation problem. Inthemost general asetheutilisation ofB-splinesurfaesfor desribing

the point-wise variation of the polar parameters leads to a onsiderable redution in the

number of design variables (the polar parameters have to be dened solely ineah point

of the ontrol network of theB-spline surfae). Inthis bakground, thanks to the strong

onvex-hull property of the B-spline blending funtions, the optimisation onstraints of

the problem (related to the speiations of the onsidered appliation) an be imposed

only on the ontrol points of the network: if they are satised on suh points they are

automatiallymet overthewholedomain.

In this study a new formalisation of the rst-level problem is presented: unlike the

approah proposed in [19℄, only the polar angle

Φ A 1 ∗

^vâries ôver ^the ^struture ^(ad îts

spatialdistribution isalways desribed through aB-splinesurfae), whilst theanisotropi

moduli

R A 0K ∗

^and

R A 1 ∗

ôf ^the ^VÂT^laminate âre^keptônstant. ^This ^fat^gives^rise ^to ^some

important advantages:

•

^the feasibility onstraints on the laminate polar parameters an be stated only one

timeforthewholeVATlaminate (andnotineahpointoftheontrol netasin[19℄);

•

^the ^new formulation of the mehanial design variables allows for integrating the

manufaturability requirements linked to the AFP proesssine the rst step of the

strategy asoptimisationonstraintsonthespatialdistributionofthepolarangle

Φ A 1 ∗

^.

The integration of the tehnologial onstraints sine the rst step of the MS2L strategy

togetherwiththe newformalisationof therst-level problemimply someonsequenes of

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laminate):

•

^the^bres-path ôfêah ^layerân ^be ^desribed ^throughâ ^B-spline ^surfae^having ^the

sameparameters(numberofontrolpoints,degreealongeahdiretion,omponents

ofthe knotvetors)of the B-splinesurfae employed for

Φ A 1 ∗

^during^the^rst^step ^of

the proedure; moreover thevalueof the orientation angle ateah ontrolpoint (for

eahlayer) an be gotthrougha straightforward analytial relationship;

•

^the seond-level problem an be formulated and solved only in an arbitrary point

of the VAT laminate thus implying a signiant redution of the number of design

variables;

•

^themathematial formulation of the seond-level problemis onsiderably simplied

anditanbestated intheformofan unonstrainedminimisationproblem(beause

manufaturability onstraints arealreadyinluded within therst-levelproblem).

Allofthesemodiationsallowtogobeyondthemainrestritionsharaterisingthedesign

ativitiesand researh studieson VATomposites thatan befound inliterature.

Finally,theeetiveness oftheimprovedversionoftheMS2Lstrategyhasbeenproven

throughameaningfulnumerialexample. Theoptimisationtoolallowstondanoptimum

VAT laminate haraterised bya signiant inrement of the rst bukling fator (about

78%

^when manufaturability onstraints are not involved and about

67%

^when ^they ^are

integrated within the design problem) when ompared to a referene lassial solution

(omposedofstraight-bresunidiretional plies).

Conerning the perspetives of this work, there are still some theoretial, numerial

and tehnial aspetsthat need to be deeplyinvestigated and developed inorder to make

theproposedapproahaverygeneralandomprehensivestrategyabletoprovidesolutions

that are both eient (true optimal ongurations) and manufaturable. Of ourse, this

ation passes through a real understanding of thepotential and thetehnologial restri-

tions linked to the AFP proess. Currently, only the tehnologial onstraint on the tow

steering has been integrated in the MS2L strategy. A step forward an be realised by

properlyformalisingandinluding into thedesign problemotherkindsof manufaturabil-

ity onstraints: tows gap and overlapping, the variation of thebre volume fration due

to imperfetions, et. Moreover, from a numerial point of view, the designer ould be

interestedinoptimisingalsothenumberofdesign variables(i.e. thenumberofparameters

tuningtheshapeoftheB-splinesurfaes)involved intoboth levelsoftheMS2Lproedure:

this point an be easily taken into aount by exploiting the original features of the GA

BIANCA. Finally, further modiations may also be onsidered in theformulation of the

design problem depending on the nature of the onsidered appliation, e.g. by inluding

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proess,osts, et.

Researh isongoing onall oftheprevious aspets.

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