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    Multilayered composite structure design optimisation using distributed parallel multi-objective evolutionary algorithms.pdf

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    Multilayered composite structure design optimisation using distributed parallel multi-objective evolutionary algorithms.pdf

    optimisationSpainlayertheareofto theirconsidered for each layer of multilayered composite laminates. Acombination of fibre types, thickness and orientation angles ofeach layer is computed by using a Finite Element Analysis FEAbased composite structure analysis tool COMPack [9] to calculatethe stiffness and strength parameters of a fibre-reinforced layer,straints.Unlikesingleobjectiveoptimisationproblems,thesolutionis a set of points known as Pareto optimal set. Solutions are com-pared to other solutions using the concept of Pareto dominance. Amulti-criteria optimisation problem can be formulated asMaximiseMinimisefixii 1; ...;NSubject to constraints gjx0 j 1; ...;Mhkx 6 0 k 1; ...;M⇑Corresponding author.E-mail addresses dsleecimne.upc.edu D.S. Lee, cmorillocimne.upc.edu C.Morillo, bugedacimne.upc.edu G. Bugeda, sergio.ollerupc.edu S. Oller,onatecimne.upc.edu E. Onate.Composite Structures 94 2012 1087–1096Contents lists available atComposite Structures1http//www.cimne.comofferversatilityincompositedesignduetothefactthatthestackingsequence of each orthotropic layer can take full advantage of thesuperior mechanical properties in terms of its strength, stiffness,and total weight. One of the goals in design optimisation for multi-layered composite structure is to increase its strength while lower-ing its weight with a given set of fibrous materials [2–5].This paper presents a research work on stacking sequencedesign optimisation for multilayered composite structure in a dis-cretised multi-objective approach using a parallel Multi-ObjectiveGenetic Algorithm MOGA [6–8]. For the stacking sequence designvariables, type of fibre, thickness and the orientation of fibres arefor composite structure. Section 4 conducts the multi-objective de-sign optimisation for multilayered composite structure. Section 5concludes the overall numerical results and shows the directionsfor future research avenue.2. Methodology2.1. Multi-objective optimisationOften,engineeringdesignproblemsrequireasimultaneousopti-misation of conflicting objectives and an associated number of con-1. IntroductionMulti-laminated composite structuresimportanttopicinthefieldsoffabricationmarine, and machine industries duedurability no corrosion – lower maintenancfire resistance, crash energy absorption,against cyclic loading no fatigue, reparabilitypair, etc. [1,2]. Multilayered fibre-reinforce0263-8223/ - see front matter C211 2011 Elsevier Ltd. Alldoi10.1016/j.compstruct.2011.10.009an ever-increasinglymechanical,aerospace,advantages such ase cost, survivabilityexcellent resistancerestoration and re-d material systems canand also the total weight of laminates. For the optimisation, aMOGA implementedin a Robust Multi-objective Optimisation Plat-form RMOP developed in CIMNE is used under the parallel/dis-tributed optimisation system and it is coupled to COMPack tofind the optimal combination of stacking sequences for multilay-ered composite plates which have lower weight, higher stiffnessand affordable total cost.The paper is organised as follows; the description of the meth-odology is given in Section 2. Section 3 describes the analysis toolDistributed optimisationEvolutionary algorithmsC211 2011 Elsevier Ltd. All rights reserved.Multilayered composite structure designmulti-objective evolutionary algorithmsD.S. Leea,⇑, C. Morillob, G. Bugedaa,b, S. Ollera,b, E. OnateaInternational Center for Numerical Methods in Engineering CIMNE, 08034 Barcelona,bUniversity Politcnica de Catalua, 08034 Barcelona, Spainarticle infoArticle historyAvailable online 17 October 2011KeywordsMultilayered composite structureFibre orientationStacking sequenceMulti-objective/multidisciplinary designabstractThis paper presents a researchite plate using a parallel/distributedmatic influence on the strengthmaterial systems offer versatilsequence of each orthotropicresults show that the optimalable cost when compared toof using a parallel optimisationjournal homepage www.elsevirights reserved.using distributed/parallela,b1work on stacking sequence design optimisation for multilayered compos-evolutionary algorithm. The stacking sequence of fibres has a dra-of multilayered composite plates. Multiple layers of fibre-reinforcedity in engineering material design due to the fact that the stackingcan offer full advantage of superior mechanical properties. Numericalcomposite structures have lower weight, higher stiffness and also afford-extreme and intermediate composite structures. In addition, the benefitssystem are also presented.SciVerse ScienceDirecter.com/locate/compstructwhere fi, gj, hkare, respectively, the objective functions, the equalityand the inequality constraints. N is the number of objective func-tions and x is an n – dimensional vector where its arguments arethe decision variables. For a minimisation problem, a vector x1issaid partially less than vector x2if8ifix16 fix2 and 9ifix1 6 fix2In this case the solution x1dominates the solution x2.As Genetic Algorithms GAs evaluate multiple populations ofpoints, they are capable of finding a number of solutions in a Paretoset. Pareto selection ranks the population and selects the non-dom-inated individuals for the Pareto fronts. A Genetic Algorithm thathas capabilities for multi-objective optimisation is termed Multi-Objective Genetic Algorithms MOGAs. Theory and applicationsof MOGAs can be found in Refs. [6–8].2.2. Robust Multi-objective Optimisation Platform RMOPRMOP is a computational intelligence framework which is acollection of population based algorithms including GeneticFig. 1. Topology of Robust Multi-objective Optimisation Platform RMOP.1088 D.S. Lee et al./Composite Structures 94 2012 1087–1096Fig. 2. Mechanism of COMPack.Table 3Fibre orientation angles.ID 12345678 9 101 12AngleC1760153045607590C015 C030 C045 C060 C075D.S. Lee et al./Composite Structures 94 2012 1087–1096 1089Fig. 3. Baseline multilayeredAlgorithm GA and Particle Swarm Optimisation PSO [6,10].Inthis paper, a GA searching method in RMOP is used denoted asRMOGA under the parallel/distributed optimisation system ifparallel, denoted as D-RMOGA. RMOGA uses a Pareto tournamentselection operator which ensures that the new individual is notdominated by any other solutions in the tournament.As shown in Fig. 1, RMOP consists of eight modules;C15 ELIU is an elite module for game strategies especially fordynamic Nash-Game. This module forces to link between Paretoand Nash Game to solve complex single and multi-objectivedesign problems.Table 2Layer thicknesses m.ID1234567Thickness 1 C2 10C042 C2 10C044 C2 10C046 C2 10C048 C2 10C041 C2Table 1Generic fibre material types. Note The density for matrix epoxy is 1.800e3 and 40 ofID 1 2 3Name Carbon Pan Carbon Pitch CarbonDensity kg/m3 1.825e3 2.025e3 1.600e3Cost US/kg 60 120 15Fig. 4. Boundary conditions for a composite plate.Fig. 5. Sample multilayered composite design.composite design.C15 EVAU is a module for evaluation and collecting results fromanalysis tools. It is also capable to handle other language-basedinterfaces.C15 IOPU is a module for handling input, output data and also plot-ting convergence history, initial population with/without buf-fer population, total populations, Pareto optimal front.8 9 10 110C031.2 C2 10C031.4 C2 10C031.6 C2 10C031.8 C2 10C032 C2 10C03this will be applied to each layer.456Rayon Glass Aramid Boron2.550e3 1.440e3 2.60e322.5 50 175Fig. 6. Pareto optimal front.C15 IRPU is an initial random population module.C15 MEAU is a module for allocating/dis-allocating memory for pop-ulation and it provide Parallel/Distributed optimisationenvironment.C15 NDOU is a module for computing Pareto-tournament, non-dom-inated sorting solutions from population.C15 RANU is a module for generating pseudo random numbermodule and Artificial Neural Network ANN.C15 SSOU is a searching module; selection, mutation, crossover forGA and also it produces velocity, positioning module for PSO.RMOP is easily coupled to any analysis tools such as Computa-tion Fluid Dynamic CFD, Finite Element Analysis FEA and/orComputer Aided Design CAD systems. In addition, it is capableto solve any engineering design application [11,12].3. Analysis of laminated composite structures3.1. Analysis of laminated composite structureThe analysis of composite structure by FEM based on the classi-cal mixture theory [13] follows the algorithm shown in Fig. 2,innon-linear problems with finite deformations for a multi-phasecomposite material.Each of those phases corresponds to a layer of the compositeplateandithasitsownconstitutivemodelindependentoftheotherphases. The algorithm starts in the reference configuration andthen, by means of stress transport operations, from the referentialto updated configuration ‘‘push-forward’’ resolves the constitu-tive equation for each of the phases that form the composite mate-rial [9,14–16]. Each of those phases can be isotropic or anisotropicand can present a different type of constitutive behaviour. Oncedetermined the stress state of each component it’s needed to findthe total stress of the composite material by Eq. 1, it allows alsofinding the internal forces in each point of the structure.rij CSijkleeklXnc1kcrijcXnc1kcCSijkleeklc1where eeklis the strain tensor for the composite and kcrepresents thevolumetric participation of component c in the composite i.e.kc dVc/dVo.Table 4Comparison of weight and displacement obtained by the baseline extreme andintermediate cases composite and the multi-objective optimal multilayeredcomposites.Type of composite Weight kg DisplacementmExtreme Case 1 lighter 0.792 0.01042Pareto member 1 0.919 16.0 0.00744 C028.6Intermediate composite 5.558 0.00286Pareto member 15compromised solution3.283 C041.0 0.00194 C032.2Extreme Case 2 heavier 22.8 0.00036Pareto member 40 11.023 C051.6 0.00026 C028.01090 D.S. Lee et al./Composite Structures 94 2012 1087–1096Fig. 7. Histogram for weight top and displacement bottom obtained by the extreme, intermediate cases and Pareto members 1, 15 and 40 obtained by D-RMOGA.Later, it’s possible to verify the balance between those forcesand the external applied forces. Details of this theory includingextension to the large deformation cases and its numericalimplementation can be found in Ref. [15].3.2. Analysis tools COMPackCOMPack in an analysis kit designed by Quantech and CIMNEable to use the necessary tools in order to create the finite elementmodel to perform structural simulations of composite materialstructures [8,14–16]. COMPack is able to determine the structuralproperties such as elastic behaviour, ultimate tensile and compres-sion strength and damage level of a composite material. One of theprincipal benefits of COMPack it’s the capability of working withthe constitutive model of the composite material in detail. To doso, it takes in account all mechanical and physical properties,amount and orientation of each of its forming fibre and matrixmaterials, and follows a FEM procedure to solve the structuralproblem.4. Stacking sequence design optimisation for multilayeredcomposite structure4.1. Problem formulationThe problem considers a multi-objective composite stacking se-quence design optimisation to find lighter and stiffer multilayeredcomposite structures. Fig. 3 shows the baseline composite consist-ing of symmetric non-balanced laminates A and B. The symmetriclaminate A has five layers; each layer contains 60 of fibre and 40two of its sides and a constant punctual force is applied in the cen-tral position.4.2. Design variablesThe candidate multilayered composite consists of five layerswith five fibrous materials, five thicknesses and five orientationangles, as shown in Fig. 5. Design variables are limited to six ortho-tropic materials, eleven thicknesses, and twelve orientation anglesas shown in Tables 1–3. It can be predicted that the lighter multi-Fig. 9. Fitness 3; total cost US vs. Fitness 1; weight kg.Fig. 10. 3D view of Pareto front obtained by D-RMOGA.Table 6D.S. Lee et al./Composite Structures 94 2012 1087–1096 1091of epoxy. The boundary conditions for the structural simulation areshown in Fig. 4 where a quadrilateral plate is simply supported inTable 5Comparison of fibre stacking sequences and orientations obtained in multi-objectivedesign optimisation. Note ID numbers for material, thicknesses and orientations canbe found in Tables 1–3.Composite StackingsequenceThicknesses OrientationsExtreme Case 1 5–5–5–5–5 1–1–1–1–1 0.0005 m 1–1–1–1–1Pareto member 1 5–1–6–2–5 1–1–1–1–1 0.0005 m 4–6–4–7–4Intermediate Case 2–3–4–5–6 4–4–4–4–4 0.003 m 1–1–1–1–1Pareto member 15 1–3–1–1–5 1–2–2–5–4 0.0019 m 3–3–7–5–4Extreme Case 2 6–6–6–6–6 11–11–11–11–11 0.01 m 1–1–1–1–1Pareto member 40 1–5–3–5–1 9–9–9–8–2 0.0082 m 5–11–2–4–1Fig. 8. Fitness 2; displacement m vs. Fitness 1; weight kg.Comparison of multilayered composite weight, displacement, and total cost obtainedby the baseline extreme and intermediate cases composite and the multidisciplinaryoptimal multilayered composites.Type of composite WeightkgDisplacementmTotal costUSExtreme Case 1 lighter 0.792 0.01042 39.6Pareto member 1 0.8709.80.00735C029.543.1 8.8Intermediate composite 5.558 0.00286 471.74Pareto member 17compromised solution4.184C025.00.00203C029.0221.85C053.0Extreme Case 2 heavier 22.8 0.00036 3990.0Pareto member 40 12.489C045.20.00023C036.1542.16C086.4layered composite will be a combination of five Aramid MaterialID – 5 with lower thickness Thickness ID – 1 and 0C176 orientationangles, denoted as Extreme Case 1, while the heaviest multilayeredcomposite will be a combination of five Boron Material ID – 6with higher thickness Thickness ID – 11 and 0C176 orientationangles, denoted as Extreme Case 2. One intermediate multilayeredcomposite can be a combination of Material ID 1–2–3–4–6 withthickness ID – 4 0.6 mm and 0C176 orientation angles. The detailsof orthotropic material can be found in Ref. [17].4.3. Multi-objective multilayered composite structure designoptimisation4.3.1. Problem definitionThe problem considers a multi-objective multilayered compos-ite structure design optimisation using D-RMOGA with paralleland RMOGA without parallel. The fitness functions are to mini-mise the weight of the multilayered composite while minimisingits maximum displacement as shown in the following equations1092 D.S. Lee et al./Composite Structures 94 2012 1087–1096Fig. 11. Histogram for weight top, displacement middle and total cost bottom obtainedD-RMOGA.by the extreme, intermediate cases and Pareto members 1, 17 and 40 obtained byf1 minWCompositeXni0WLayeri2f2 mindTotalffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDx2Dy2Dz2q3where WComposite, dTotalrepresent the weight and displacement ofthe multilayered composite, respectively.4.

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