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1. INTRODUCTION

Composite pressure vessels due to lightweight and high performance are widely used in aerospace, automobile, petrochemical field for gas and liquidstorage1. In spite of their good performance, composite pressure vesselsmadebyfilament-windinghavecomplexityinanalyzingthegeometryand properties along the longitudinal axis,especiallyintheirdomesection2. Becauseasthewoundlayersareaddedonamandrel,thecurvilinearfiberpathleads to a continuous change in winding angleandthickness3,4. Moreover, the fiberpathdependsonthesurfacewherethefibersarewoundandtheshapeofthe surface is changing with already built-upfibersasthefilamentwindingprocess is carried out5. Especially the surface change during the winding process becomes relatively large becausethethicknessisthickernearthe polar opening than in the other dome parts. So it presents a strong challengetoexcellentdesignabouthow

to achieve optimization for multilayer composite structure with winding angles and thickness varies in thelongitudinalandthicknessdirectionsof composite pressure vessel6,7. In order to solve this problem, various intelligent optimization algorithms developed for composite pressure vessels8-11. The popular algorithms is genetic algorithm, which is originally proposed by Holland12 to simulate the evolution of biology, has been widely used to conduct the optimal design of composite laminated structures13. However, the local search ability is relatively weak in contrast with itsstrong global search ability. In addition, the genetic algorithm is confronted withtheproblemof‘‘prematurity”forsolving the multi-peak optimizationspace, which may provide the erroneous optimum results14.Asaresult,nobetterdesign method is suitable for composite pressure vessel under internal pressure. The reason may be conceived of that it is very hard to solve the low design efficiency that occurs when finite

element analysis and optimal design algorithms are used simultaneously.

In this paper, a new design method based on the clonal selection principle in the biological immunology is proposed to solve the minimum weight problem under the burst pressure constraint. The method consists of three main parts: the non-geodesic path algorithm, variable stiffness analysis algorithm and artificial immunealgorithm.AsoftwarecalledSimWind 1.0was developed on thebasis of this method to design the composite pressure vessel rapidly and efficiently. Finally, a representativecarbonfibercompositepressurevesselwithultrathinAl linerwasdesignedand fabricated by using this new design method and the software.

2. A NEW DESIGN METHOD FOR COMPOSITE PRESSURE VESSEL

2.1 The Non-geodesic Path AlgorithmAsmentionedinSection1,thewindingangle plays a very important role in the design of composite pressure vessel.

Optimal Design of Lightweight Composite Pressure Vessel by Using ArtificialImmuneAlgorithm

Weicheng Jiao, Yue Niu, Lifeng Hao, Fan Yang, Wenbo Liu, and Rongguo Wang*Center for Composite Materials and Structures, Harbin Institute of Technology

SUMMARYThisresearchaimstooptimizetheweightofcompositeoverwrappedpressurevessel(COPV)underinternalpressure. In the paper, a new design method based on the clonal selection principle in the biological immunology is proposed to solve the minimum weight problem under the burst pressure constraint. The method consists of threemainparts:thenon-geodesicpathalgorithm,variablestiffnessanalysisalgorithmandartificialimmunealgorithm.AsoftwarecalledSimWind1.0wasdevelopedonthebasisofthismethodinordertodesignthecompositepressurevessel rapidlyandefficiently.Finally,a representativecarbonfibercompositepressurevesselwithultrathinAllinerwasdesignedandfabricatedbyusingthisnewdesignmethodandthesoftware.

Keywords:Compositepressurevessel,Non-geodesicpathalgorithm,Variablestiffnessanalysisalgorithm,Artificialimmunealgorithm

*Correspondingauthor:wrg@hit.edu.cn

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Optimal Design of Lightweight Composite Pressure Vessel by Using Artificial Immune Algorithm

If the accurate winding angle distributions are not considered, the mechanical behavior of the wound part cannot be predicted precisely3.Thereforeitisrequiredtodeterminethewindinganglevariationinthethicknessdirection.

In general, the spiral winding angle a could be determined by the following Equation(1)15:

dαdx

=λ A2 sin2 α − r(x)r ''(x) cos2 α( ) − A2r '(x)sinα

r(x)A2 cosα (1)

Where xdenotestheaxialcoordinate,r(x) stands for the radial distance and it is the function of x. lstandsfortheslippagecoefficient.r’(x), r’’(x)arethefirstderivative and the second derivative of r(x), respectively. Adefinesas A = 1+ r '2 .

In order to calculate the winding angle and possible winding patterns, the algorithm shouldincludetwoconsiderationsasshowninEquation(1):First,theslippagelbetweenthefiberandthemandrelshouldbeconsidered.Itisaveryimportantparameter,whichrepresentsthefrictionalforcebetweenthefibertowsandthemandrelsurface,thebasisforfindingtheregionofpossiblewindingpatterns.Thedetails for determining the l value can be found in Ref.15. Second, windability shouldbeconsidered.Windabilitymeansaconsistentoverlapoffiberbandsonthe mandrel towards radial and circumferential directions.

So,bytheEquation(1),thefiberpathofthefirstplycanbecalculatedfromthegivenmandrelshape.Andthesecondplywaswoundonthetopsurfaceofthefirstply.Fromthesummedthicknessofpreviousplies,thefiberpathandthicknessofthenextplycouldbegenerated.Themandrelsurfacewasupdated.Thenthefiberpathofthesecondplycanbeobtained.Repeatthecalculationstepsuntilthewoundlayersequaltothegivennumberofplies.Finally,thewindinganglesfor each layer can be obtained. The whole steps of non-geodesic path algorithm for considering the winding angle change are shown in the Figure 1.

2.2 The Variable Stiffness Analysis AlgorithmFinite element analyses, which can handle the winding angle and thicknessvariation,arerequiredtopredicttheexactbehaviorofcompositepressurevessels.

Firstly,theexactshellthicknesspredictioniscriticalinfiniteelementanalysis.Becausewithoutknowingtheaccuratedomethickness,it’sdifficulttocreateanaccuratemodelandcalculatethestressesanddisplacementsprecisely.Buttheshellthickness,especiallydomethickness,variesasafunctionofdomeradiusandwindingpattern.Predictionoftheshellthicknessisnoteasy.Inthispaper,acubicsplinefunctionisusedtopredicttheshellthickness.Thecubicsplinefunctionwasexpressedasfollows:

t(ri ) = A× ri0 + B× ri

1 +C × ri2 + D × ri

3 (2)

Where t(ri)denotesthethicknessfordifferentradiusofcompositepressurevessel.A, B, C and Darecoefficient.HowtoobtaintheirvaluescanbefoundinRef.16.

Second, the variable stiffness analysis algorithm was established. The relationship betweenthestiffnessandshellthicknessfollowsasEquation(3):

σ skt k

σθkt k

τ sθk t k

k=1

#Plies

∑ = [Q]kεsεθγ sθ

k=1

# Layers

∑ (3)

Where the sks, s

kθ, tk

sθ, stands for stress

of k-th layer, tk is corresponding the resultsfromEquation(2),[Q]k is the stiffnessmatrixofk-th layer, εs, εθ

, gsθ,

stands for strain of k-th layer. Since the windinganglesandthicknesschangesduringthefilamentwindingprocess,the stiffness of the wound structure is not uniform. It can cause the large local deform, which will lead to the COPV failure earlier. In the variable stiffness analysis algorithm, the winding angle andthicknessvariationareconsidered.The lay-up wound structure can be optimaldesignedtomakethewholestrain small and uniformly. Finite element analyses based on the variable stiffness algorithm were performed by a commercialFEAcode,Ansys.Andthemaximumstraincriteriawereadopted.The stress and strain distributions of composite pressure vessels can be calculated for each structure design program.

2.3TheArtificialImmuneAlgorithmThe artificial immune system (AIS)defends against the invasion of the foreign pathogeny and protects the body through the antigen recognization, the cloning and hypermutation of B cells, the generation of memorycells and the death of antibody cells with low affinities. The AIScompares theobjective functionandconstraint conditions to the antigens, and compares the applicable results to the antibodies, and compares the applicable objective to the affinitybetween the antigen and antibody. The remarkable advantages of theAISoverthegeneticalgorithmlieinthe maintenance of diverse antibody cells and the presence of memory cellswith high affinities17.AndAISexhibitsfiveaspectofbehaviors17:(1)with the participation of the T-cells and antigen-presentingcells,theBcellsare

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inspired, cloned and decomposed to generate the antibody by recognizing theantigen;(2)thecloningoftheBcellsis accompanied by the hypermutation oftheBcells;(3)thecombinationofthe antibody and antigen represents theselectionprinciple;(4)theaffinitybetween the antibody and antigen indicatesthematchingextent;(5)thememory cells stem from the initial infection and those B cells withoutinspiration will perish.

In the new design method, the AIS Algorithm controls the overalldesign procedure. The initial antibody population is randomly generated in the form of binary bit string with suitable length. The non-geodesic path algorithm is applied to the selection of possiblefilamentwindingangles,andvariable stiffness analysis algorithm is applied to the calculation of the stress and strain distributions through finite element simulation. And theobjectivefunction,theburstpressure,andtheaffinitybetweentheantibodyandantigenarecalculated.Apenaltyfunction is proposed to deal with the constrained problem. The affinity function F for the minimum problem is written as:

f '(α, t) = f (α, t(ri ))+ β ×max[c ∗ PW − PB , 0],

F = 1f '(α, t)

(4)

Where β is a penalty factor. c is the stress ratio of the actual burst pressure PBtotheworkingpressurePW. Figure 2 showsaflowchartofthenewdesignmethod.

The Windows-based program for this method was developed with the C++ languageandused,calledSimWind1.0as shown in Figure 3.Andthissoftwareapplication was programmed to enable all the design processes to feed the resultsbacktoeachother.

3. RESULTS

A representative composite pressurevesseliscomposedofa5A03Aluminium

Figure 1. non-geodesic path algorithm

Figure 2. Flowchart of optimal design procedure for composite pressure vessels

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Optimal Design of Lightweight Composite Pressure Vessel by Using Artificial Immune Algorithm

linerwiththicknessisabout0.8mmandT700 carbon fiber reinforced HIT-1epoxyresinwaschosenforfabricatingcomposite layers.

The constant parameters in the optimization analysis of composite pressure vessels are chosen as: the maximumoperatinginnerpressureis

5MPa,theburstpressureislargerthan10MPa,thecapacityofthecompositepressurevesselsV=780L, the innerradius and length of the cylinder is 760mmandthenumberofcompositelayers is 6.

The basic design conditions are summarized as:

1. Themaximumworking pressureis5MPaandtheburstpressureislargerthan10MPa.

2. Theyieldofthelinerisprohibited.

3. Theweightreductionisthemostimportant goal of this design.

MinimizeW = f (α, t(ri )) (5)

where W is the weight, a, t(ri) are the same parameters as mentioned before, further more a∈[a0, an],t(ri)∈[t0, tn].a0, ancouldbeobtainedfromEquation(1) and t0, tn can be calculated from Equation(2).

The aluminum liner is just only forair tightness, not bear loading. The carbon fiber/epoxy composite bearall pressure loading and is considered to be linear-elastic and transversely isotropic. The material parameters are listed in Table 1.

At last, through the non-geodesicalgorithm and variable stiffness analysis algorithm, the possible winding angle is calculated a∈ [-3.86°,29.74°],andthethicknessofcomposite layer is calculated t∈[0.8,16]mm.thestressratioisc =2.35,thestring length of bits for each variable is10,thepenaltyfactorisβ=0.2,thestring length in the clonal selection is 10andthehypermutationprobabilityis0.1.

Table 2 shows the optimization results of the weight, burst pressure and affinity of each antibody for16 antibodies and 100 iterations.By comparison, the results for theantibody7with thehighest affinity0.129 among all antibodies areoptimum.

The composite pressure vessel product is shown in the Figure 4. It is very light. Apersoncanliftiteasily.

Figure 3. Windows-based program, SimWind 1.0

Table 1. Material properties of T700/ HIT-1 epoxyE1 E2 G12 G23 v12 v23 Xt Yt Density161.3GPa 8.8GPa 5.3GPa 2.7GPa 0.33 0.45 2300MPa 30MPa 1550Kg/m3

Table 2. The optimization results for 16 antibodies after 100 iterationsAntibody number Weight (Kg) Burst pressure (MPa) Affinity1 37.7 11.32 0.1192 36.5 10.96 0.1233 39.0 11.71 0.1154 36.5 10.97 0.1235 38.1 11.43 0.1186 37.6 11.29 0.1197 35.0 10.51 0.1298 36.4 10.93 0.1239 39.1 11.74 0.11510 35.3 10.54 0.12811 35.4 10.51 0.12812 38.8 11.65 0.11613 38.2 11.48 0.11714 36.1 10.86 0.12415 36.4 10.94 0.12316 39.0 11.70 0.115

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4. CONCLUSIONS

In this research, the optimal design of filament wound composite pressurevessel under internal pressure was performed.Anewdesignmethodhasbeen proposed. It includes the non-geodesic path algorithm, variable stiffness analysis algorithm and artificialimmunealgorithms.Finally,the new design method was applied to a representative filament woundcomposite pressure vessel. The results show that the lightweight COPV could bring weight savings of up to 25%~30%comparedtotraditionalone.Consequently,thenewdesignmethodisefficientandsuccessful.

ACKNOWLEDGEMENTS

The authors gratefully acknowledgethefinancialsupportfromtheMajorState Basic Research DevelopmentProgram of China (973 Program,No.2011CB605605), SpecializedResearch Fund for the DoctoralProgram of Higher Education (No.20122302120033)andResearchFundofNationalKeyLaboratoryofScienceandTechnologyonAdvancedComposites in Special Environments (9140C490107130C4901).

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Figure 4. the lightweight composite pressure vessel

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Optimal Design of Lightweight Composite Pressure Vessel by Using Artificial Immune Algorithm

328 Polymers & Polymer Composites, Vol. 22, No. 3, 2014

Weicheng Jiao, Yue Niu, Lifeng Hao, Fan Yang, Wenbo Liu, and Rongguo Wang