Optimization of Laminated Composite Structures

Using Sub-Laminate Based Optimization Concept

Andre Monicke ∗, Harri Katajisto †

Componeering Inc., Itamerenkatu 8, FI-00180 Helsinki, Finland

Petri Kere ‡, Juhani Koski §

Tampere University of Technology, Department of Mechanics and Design

P.O. Box 589, FI-33101 Tampere, Finland

In composite design, individual plies or groups of plies are frequently defined on specificapplication areas. Consequently, one part of the application area may belong to a differentzone than another part. Zones have unique lay-ups and related section data definitions.Modern finite element analysis (FEA) software provide tools for this so-called ply-basedmodeling where the pre-processing is handled very manufacturing oriented manner andwhere the input files for solvers are automatically converted to be internally solved with azone-based approach. Typically, several plies are introduced in the same application areas.Parts of this area specific stack of plies can be considered as a sub-laminate for whichthe laminate lay-up optimization problem is defined. In this paper, different sub-laminatebased lay-up formulation concepts are evaluated. Implementation and performance of thelaminate lay-up optimization problem formulations together with numerical results arereported.

I. Introduction

The design optimization of laminated composite structures is challenging due to the complex mechanicalbehavior of the structures and the number of design variables involved. A composite laminate is typically

formed from a number of fiber-reinforced layers having directional properties. Basically, for each layer of thelaminate, design variables are the choice of material system, thickness of the layer, and orientation of thelayer. In practice, laminate design is more constrained. The choice of material systems is almost limitless,but as a result of the conceptual design phase, only a few possible candidates are usually left.

For solid laminates the use of multiple material systems is beneficial in several applications, though theuse of a single material system is more common. Sandwich structures consist of two material systems inminimum. The thickness of reinforced layers is usually determined by the choice of material system andprocessing. However, in sandwich panels the thickness of the core material can be chosen quite freely. Thechoice of layer orientations is often constrained to 0, 90,+45, and −45 deg. Other off-axis directions maysubstitute ±45 deg. To avoid undesirable anisotropy of the structure, various constraints are often set forthe lay-up. For instance, symmetry with respect to the laminate midplane may be required and a balancedlay-up with an equal number of off-axis layers with minus and plus orientation is usually preferred. Due tomanufacturing, some regularity may be desired in the laminate lay-up. This can be achieved, for instance,

∗Engineering Consultant, M.Sc. (Eng.), Doctoral Student, Componeering Inc.†Senior Engineering Consultant, M.Sc. (Eng.), Componeering Inc.‡Docent, D.Sc. (Tech.), Tampere University of Technology. Member AIAA.§Professor, D.Sc. (Tech.), Tampere University of Technology

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by repeating the same sub-laminate a number of times. Some stacking sequences may be preferred againstothers.

For a laminate lay-up design, values of the design variable vector are typically restricted integers. Stochas-tic population-based search methods have been successfully used to solve complex discrete structural op-timization problems.1–7 Population-based search methods like genetic algorithms and particle swarm op-timization (PSO) techniques maintain and manipulate a population of solutions in the search for bettersolutions. Constraints may include sufficient reserve factor in the static failure analysis, sufficient reservefactor against shear buckling and non-existence of natural frequencies at specific frequency domain.

Typically, several plies are introduced in the same application areas. This area specific stack of pliescan be considered as a sub-laminate for which the laminate lay-up optimization problem is defined. Inthis paper, different lay-up formulation concepts considering practically all aspects of the laminate design,except material selection, namely layer orientations, layer multipliers and stacking sequence permutation, areevaluated. For laminate problems with many layers the pattern of the sub-laminate can be repeated a numberof times. With this approach the desired regularity for the laminate lay-ups is achieved. Computational costis considerably reduced due to the reduced design space. Still, the objective space is not much limited, whichhas been demonstrated in.6,7 In the presented approach, section data for individual zones is assembled fromthe associated sub-laminates and their respective order.

Numerical examples including a practical design problem of a drive shaft having two conflicting objectives,i.e., weight minimization with the maximum load bearing capacity, are used for the demonstration. Thestudy demonstrates the efficiency of the optimization concept to produce feasible high-performance solutions.

II. Laminate Lay-up Coding System

Frequently, for a fixed number of layers k = 1, 2, . . . , N and a fixed set Θ = {θ1, θ2, . . . , θl} of l allowedlayer orientations, a standard laminate code is encoded with a representing layer orientation identity

string.1,2, 6, 7 For instance, the standard laminate code [0/± 45/90]SE can be encoded as (1, 3, 4, 2, 2, 4, 3, 1),where the layer orientation identity design variables 1, 2, 3, and 4 refer to the layer orientations 0, 90,+45,and −45 deg, correspondingly, and SE stands for the symmetric even laminate structure.

For many applications with in-plane loading, the stacking sequence is not as important as the number oflayers in each orientation. Fixed layer thicknesses can be assumed, as it is often the case with predeterminedply selection. A laminate with layer orientations θ1, θ2, . . . , θl is simply stacked, i.e., all layers of the sameorientation are consecutive like in [(θ1)n1/(θ2)n2/ . . . /(θl)nl] where nl are multipliers for each layer orien-tation. For a 12-layer laminate using three layer orientations the constraint would be n1 + n2 + n3 = 12.Gurdal et al.2 remark that for a genetic search, a formulation of the constraint as n3 = 12 − n1 − n2 andn3 ≥ 0 is much more efficient than the first one. It is obvious since the design space of the second formulationis thirteen times smaller than the first one, while the number of feasible designs remains the same for bothvariants. However, if the number of layers for the whole laminate is not fixed, those kind of constraintscannot be enforced.

When the laminate is loaded with bending or out-of-plane shear forces, the stacking sequence plays avital role. One way of considering this aspect is to introduce a stacking sequence vector S, which deter-mines the locations of the specific layer orientations in the laminate. For a laminate with l allowed layerorientations, the stacking sequence vector includes l! permutations. For example, for a [θ1/θ2/θl] laminateS = {(1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1)}, where 1, 2, and 3 refer to the layer orientationsθ1, θ2, and θ3, respectively.

III. Elementary Laminate Based Optimization Concepts

III.A. Basic Principle Concept

The basic principle is to divide the number of layers N in the laminate into p stacks with q layers sothat p × q = N . The allowed number of different elementary laminates r is defined so that r ≤ p. Ply

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thickness t is fixed in the formulation but different materials are possible.The symmetric even laminate lay-up using the basic principle concept is formulated as follows. A laminate

`(~x) ∈ L where the design variables xi ∈ {1, 2, 3}, i = 1, . . . , n, with l = 3 correspond to the layer pairs02, 902,±45 deg, respectively, takes the form

`(~x) = [x1, x2, xi, . . . , xn]SE, xi ∈ {1, 2, 3} (1)

Using the parameters n, p, q, r, a laminate with n = 12 where the number of regular stacks is one, thenumber of layer pairs in the elementary laminate is 12, and the number of elementary laminates is one, isdefined as n = 12, p = 1, q = 12, r = 1.

III.B. Layer Angle Alphabet Concept

Layer angle alphabet concept utilizes elementary laminates and is based on the layer angle alphabet system.Multipliers m1 and m2 are introduced for the two elementary laminates, i.e., how many times particularelementary laminates occur in a row. The laminate is defined as

`(Θ, ~m) = [(θ1/θk/ . . . /θq)m1/(θq+1/ . . . /θ2q)m2]SE (2)

where the number of regular stacks in the laminate is defined as p = 2(m1 + m2).Applied to the formulation of the above described example with n = 12 and assumed that the number of

layers in both elementary laminates is the same with p = 4, q = 3, r = 2, l = 3, it is reasonable to constructthe lay-up as

`(~x) = [(x1, x2, x3)x4, (x5, x6, x7)x8]SE

wherexi ∈ {1, 2, 3}, i = 1, . . . , 3, 5, . . . , 70 ≤ x4 ≤ 3, x8 = 4− x4, x8 ≥ 0

(3)

Multipliers for elementary laminates are x4 and x8 with x8 as a support variable. The laminate structureconsists of two elementary laminates, which may have different lay-ups. The additional information requiredto build a lay-up is how many times the specific elementary laminate occurs in a row. This is specified bythe elementary laminate multipliers.

Constructing the lay-up with p = 3, q = 4, r = 2, l = 3, defines the design variables as

`(~x) = [(x1, x2, x3, x4)x5, (x6, x7, x8, x9)x10]SE

wherexi ∈ {1, 2, 3}, i = 1, . . . , 4, 6, . . . , 90 ≤ x5 ≤ 3, x10 = 3− x5, x10 ≥ 0

(4)

III.C. Layer Multiplier Concept

Using the layer multiplier concept all layer orientations are simply stacked one upon the other in predefinedorder and the number of layers of a given orientation is defined by its multiplier nk as

`(Θ, ~n) = [(θ1)n1/(θk)nk/ . . . /(θq)nq]SE (5)

Let the stack permutation vector be defined as S = {(1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1)}.The layer multiplier concept couples the stacking sequence vectors s1 ∈ S and s2 ∈ S into the elementarylaminate concept. The two stacking sequence vectors can independently have the six permutations presentedabove while 1, 2, and 3 refer to the layer pairs 02, 902, and±45 deg, respectively. Layer pairs of the elementarylaminate can have different multipliers. The formulation of the concept with n = 12, p = 4, q = 3, r = 2, l = 3is presented as

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`(~x) = [((ζ1)xζ1, (η1)xη1, (ξ1)xξ1)x1, ((ζ2)xζ2, (η2)xη2, (ξ2)xξ2)x2]SE

wheresi = (ζi, ηi, ξi), i = 1, 2

xζi + xηi ≤ 3, xξi = 3− xζi − xηi

0 ≤ x1 ≤ 4, x2 = 4− x1

and~x = (ζ1, xζ1, η1, xη1, ξ1, xξ1, x1, ζ2, xζ2, η2, xη2, ξ2, xξ2, x2).

(6)

This paper concentrates on the influence of the stacking sequence vector on the optimization, as theeffective use of elementary laminates as building blocks for thick laminates is shown in.6,7

IV. Zone-Based Modeling vs. Ply-Based Modeling

In traditional finite element modeling, the composite structure is divided in zones for which laminatelay-ups are defined. Zone-based modeling of laminate structures has its limitations. In reality, there arelayers and sub-laminates that are shared by adjacent zones. In lay-up optimization of the complete struc-ture, the separate zones cannot be handled independently. The concepts of sub-laminates can be used toovercome these problems. A sub-laminate is a parameterized laminate that is handled independently andcan be used as a building block in other laminates. Lay-up optimization of a sub-laminate may reflect si-multaneously to several zones in the model. The elementary laminate concept is applicable to sub-laminatesas parameterization concept.

Figure 1. An example of a winding product that has two local patches of additional reinforcements at the ends ofthe shaft. The zones at the ends of the shaft are denoted by 1, 2 and 3. The continuous part of the winding productis represented with a single sub-laminate a whereas the two local patches of reinforcements are presented with sub-laminates b and c.

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In ply-based modeling the model is constructed from the manufacturing point of view from ply groups,which are here considered as sub-laminates. Sub-laminates are used as building blocks to assemble thelaminate cross sections for the simulation. This association is defined in the problem definition and itguarantees ply continuation in overlapping zones. Changes in the individual sub-laminates automaticallyreflect to the whole FE model. In practice, the system generates new section definitions for all zones of themodel. Updated section definition file replaces the original section definitions for the new design evaluation.The file is in FEA tool specific format.

As an example, a winding product that has two local patches of additional reinforcements at the ends ofthe shaft, is illustrated in Figure 1. The zones at the ends of the shaft are denoted by 1, 2 and 3. The partof the structure that involves the continuous winding product is represented with a single sub-laminate awhereas the two local patches of reinforcements are presented with separate sub-laminates b and c.

The concept guarantees that all designs generated by the optimization procedure can be manufactured.Basically, the concept provides the same automated conversion from ply-based model to zone-based modelas the current FEA tools. Still, majority of expert users rely on zone-based modeling. This approachsupports the more familiar modeling technique, where users have more transparent control over the system.Multiple, similar type of sub-laminate definitions can be managed simultaneously even if they are consideredindependently in the simulation. This facilitates the management of large number of input variables, sub-laminates and zones. Optionally, the sub-laminate designs can directly be feed to the pre-processor ifply-based tools are used. In that case the related tool makes the conversion to zone-based manner.

V. Simulation and Optimization Environment

Several applications are combined in the computational environment as illustrated in Figure 2, namelythe optimization and design environment software modeFRONTIER,11 the composite analysis software

ESAComp,12 and the finite element analysis code Elmer.8,9, 13

Figure 2. Schematic view of the simulation and optimization environment.

ESAComp role in the process is to provide material database, definition of lay-up optimization problemsfor sub-laminates, definition of the association between zones and sub-laminates, submission of optimizationproblems for modeFRONTIER, laminate level analyses, laminate and material data definitions in Elmerspecific format, and element failure analyses. ModeFRONTIER provides multi-objective optimization capa-

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bilities, process integration environment, and visualization of results.Laminate lay-up formulation concepts are introduced and reviewed with respect to the example cases. In

these studies, first the laminate lay-up is divided into two zones, which consist only of one sub-laminate. Inthis work, the amount of layers for the lay-ups is not fixed, as the weight of the structure is to be minimizedwhile simultaneously maximizing the reserve factor, i.e., minimizing the inverse reserve factor. Consequently,the optimal number of layers is determined.

This presentation also deals with the simulation and optimization of a carbon fiber-reinforced plastic(CFRP) driveshaft under combined loads. The simulation is performed with linear static using Reissner-Mindlin-Von Karman type shell facet model.8,9 The shell facet model was implemented in ESAComp usingElmer open source computational tool for multiphysics problems.

VI. Multiobjective Optimization Problem

Amultiobjective optimization problem of the form

min~x∈S

~z (~x ) = min~x∈S

{z1(~x ), z2(~x ), . . . , zm(~x )} (7)

is considered.10 The feasible set S is defined by constraints as S = {~x | ~g(~x ) ≤ 0, ~h(~x ) = 0}. The componentszi : S → R, i = 1, 2, . . . ,m of the vector objective function are called criteria and they represent the designobjectives by which the performance of the design point is measured. The image of the feasible set in thecriterion space is Λ = {~z ∈ Rm | ~z = ~z(~x ), ~x ∈ S}.

Definition 1. A solution ~x ∗ is Pareto optimal for the problem 1 if and only if there exists no ~x ∈ S suchthat zi(~x ) ≤ zi(~x ∗) for all i = 1, 2, . . . ,m and zi(~x ) < zi(~x ∗) for at least one i = 1, 2, . . . ,m. The points~z ∗ = ~z(~x ∗) ∈ Λ in the criterion space are called the minimal points.

Definition 2. A solution ~x ∗ is weakly Pareto optimal for the problem 1 if there does not exist another~x ∈ S such that zi(~x ) < zi(~x ∗) for all i = 1, 2, . . . ,m. The corresponding points ~z ∗ = ~z(~x ∗) ∈ Λ in thecriterion space are called the weakly minimal points.

VII. Results

VII.A. Cylindrical Shells

The model considers a complete cylinder with ply properties including engineering constants and firstfailure (FF) stresses and strains represented in Table 1.

Engineering constants FF stresses [MPa] FF strains [%]

E1 [GPa] 139.3 Xt 1950 Xε,t 1.3999

E2 = E3 [GPa] 11.1 Xc 1480 Xε,c 1.0625

G12 = G31 [GPa] 6.0 Yt = Zt 48 Yε,t = Zε,t 0.4324

G23 [GPa] 3.964 Yc = Zc 200 Xε,c = Xε,c 1.8018

ν12 = ν13 0.3 S 79 Sε 1.3167

ν23 0.4 R 79 Rε 1.3167

tply [mm] 0.125 Q 79 Qε 1.9928

Table 1. Mechanical properties of the unidirectional AS4 CFRP plies used for the cylndrical shells.

The laminate coordinate system x-axis coincides with the undeformed cylinder longitudinal axis. E1

is the elastic modulus of the unidirectionally reinforced layer along the reinforcing fibers, E2 is its elastic

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modulus in the transverse direction along the shell plane. The laminate coordinate system x-axis is used asa base from which each layer is rotated to the layer coordinate system.

The inside of the cylinder is considered as the geometric reference surface. In all the examples bothcylinder ends of the FE-model are constrained to remain plane and circular to maintain the initial radiuswhile the axial load is transferred to nodal forces.

The geometry of the cylindrical shell is defined as shown in Figure 3. The length and the diametre ofthe FE-model of the CFRP cylinder are 1000 mm and 300 mm, respectively. The cylinder is clamped andcircumference constrained at x = 0 mm, and free and circumference constrained at x = 1000 mm.

(a) Displacements of the cylindrical shell. (b) The cylindrical shell geometry.

Figure 3. Schematic view of the cylindrical shell geometry.

The multiobjective optimization problem of the form

min~x∈S

~z (~x ) = min~x∈S

[W (~x )

1/RF (~x )

](8)

S = {~x | g(~x ) =1/RF (~x )(1/RF )a

− 1 ≤ 0, `j(~x) ∈ Lj , j = 1, 2} (9)

is considered, where W (~x ) is the weight measure of the structure, j = 1, 2 are the two allowable sub-laminateconfigurations each in its own zone as x = 0− 500 mm and x = 500− 1000 mm, respectively, and (1/RF )a

denotes the maximum allowed inverse reserve factor for each sub-laminate.One sub-laminate is based on the layer multiplier concept, the other is based on the layer multiplier

concept with additional stacking sequence vector. Layers are put in stacks of two so that the optimizedlaminates are automatically balanced. Table 2 describes the load cases LC1-LC3 applied for the cylindricalshell. In LC2 and LC3 the laminates are by definition symmetric even.

LC1 LC2 LC3

Fx(x = L) [kN] 1000 0 0

Mx(x = L) [kNm] 0 70 10

P [kPa] 0 0 -500

(1/RF )a 1.0 1.0 0.25

Table 2. Load cases of the cylindrical shell.

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Figure 4. The shell loads.

The two sub-laminates for the simple formula-tion without stacking sequence vector consists of 3parts. Part 1 can have 0 to 4 pairs of 0 deg lay-ers, i.e., x1 ∈ {0, 2, . . . , 8}. Part 2 also allows 0 to4 pairs of layers but the orientation is to be opti-mized chosen from 15 deg to 75 deg in steps of 15deg, so that x2, x3 ∈ {0, 1, . . . , 4}. Pairs are alwaysput as ±θ. Part 3 has again a fixed orientation 90deg and between 0 to 4 pairs of layers are allowed,i.e., x4 ∈ {0, 2, . . . ., 8}. Additionally the sum of thelayers has to be greater than 0.

Figure 5. The shell coordinate system and the shell loads.

For the second sub-laminate the variables x5 − x8 are used respectively. The first sub-laminate is thebuilding block for the left half of the cylinder, the second sub-laminate the one for the right half. Actually,here the sub-laminate equals the laminate for the zones.

Using the formulation without stacking sequence vector the 0 deg layers are always going to be first inthe final sub-laminate appearing as many times as the multiplier defines. Part 2 defines the orientationand amount of layers coming after part 1 in the sub-laminate and part 3 the ones afterwards. When thestacking sequence is used, two variables x9, x10 ∈ {0, 1, . . . , 5} are added to represent the stacking sequencepermutations.

Loadcases LC1-LC3 have been calculated with the same design of experiments (DOE) size, i.e., random,80 designs, and 70 generations with multi-objective genetic algorithm MOGAII provided by modeFRON-TIER. In Tables 3-5 Pareto designs of example runs are set out with their ID, i.e., the design number, theinverse reserve factor, and the weight measure based on the number of layers used in the sub-laminate. Table3 contains also the lay-ups corresponding to the Pareto optimal designs.

As expected for the axial loadcase (LC1) the minimum weight design which fulfills the constraints iscomposed of only 0 deg layers. Here the minimum weight feasible solution has 3 pairs of 0 deg layers ineach sub-laminate as [(02)3]. For both formulations, with and without stacking sequence vector, equallywell performing designs were found. In the calculated runs it was not obvious that the formulation withstacking sequence vector would necessarily need more designs to reach the results of similar quality to theother formulation. In fact, it often found slightly better performing designs, as the maximum number of0 deg layers was reached, and with the stacking sequence vector the less efficient ±θ deg pairs could beshifted to the inside of the structure. However, one can see in Table 3 from the existence of two differentsub-laminates that the optimum design for the weight measures 24 and 20 were not yet reached. Other runsprovided those as well, but at later generations.

For the second load case LC2 with only torsion, similar observations can be made. The same designswhere found for the same weight measure values. As expected, the lightest design which fulfills the constraintscontains (+45,−45) layer pairs, such that the the lay-up in each sub-laminate is [(+45/− 45)3]SE.

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Without stacking sequence With stacking sequence

ID 1/RF W Lay-up ID 1/RF W Lay-up

863 0.346 28 [08, (+15,−15)3] 878 0.306 28 [(+15,−15)4, 06]

1896 0.398 24 [08, (+15,−15)2] 872 0.324 24 [(+15,−15)2, 08], [(+15,−15)3, 06]

1452 0.470 20 [06, (+15,−15)2] 1025 0.374 20 [(+15,−15)2, 06], [(+15,−15)1, 08]

921 0.519 16 [08] 479 0.446 16 [08]

1143 0.686 12 [06] 447 0.519 12 [06]

Table 3. A subset of Pareto optimal solutions using simple sub-laminate concept with and without stacking sequencevector for the cylindrical shell with LC1.

Without stacking sequence With stacking sequence

ID 1/RF W ID 1/RF W

1999 0,4315 44 3989 0,4151 48

2237 0,4430 42 5260 0,4272 44

3417 0,4431 40 3515 0,4401 40

2471 0,4554 38 2544 0,4538 38

4146 0,4555 36 3206 0,4538 36

1641 0,4687 34 3339 0,4685 34

3423 0,4688 32 2264 0,4686 32

1313 0,4842 30 3338 0,4842 30

834 0,4843 28 4075 0,4843 28

487 0,5010 26 1005 0,5010 26

713 0,5011 24 688 0,5011 24

1578 0,5193 22 1089 0,5193 22

1167 0,5193 20 922 0,5193 20

1109 0,5421 18 1118 0,5421 18

4146 0,5421 16 772 0,5421 16

5120 0,7165 12 2046 0,7165 12

Table 4. A subset of Pareto optimal solutions using simple sub-laminate concept with and without stacking sequencevector for the cylindrical shell with LC2.

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Without stacking sequence With stacking sequence

ID 1/RF W ID 1/RF W

3828 0,0872 32 2756 0,0872 32

2180 0,0950 30 1332 0,0950 30

1184 0,0951 28 1005 0,0951 28

1599 0,1049 26 968 0,1049 26

1372 0,1049 24 1358 0,1049 24

2281 0,1153 22 1523 0,1153 22

2699 0,1153 20 1024 0,1153 20

1438 0,1461 18 715 0,1461 18

2311 0,1461 16 1866 0,1461 16

3699 0,1962 14 2164 0,1962 14

3624 0,1962 12 2245 0,1962 12

Table 5. A subset of Pareto optimal solutions using simple sub-laminate concept with and without stacking sequencevector for the cylindrical shell with LC3.

Figure 6. Criterion space for the cylindrical shell with LC3 optimized using the simple sub-laminate concept withstacking sequence vector, the same run as shown in Table 5. Pareto optimal solutions are marked with bold squares.

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Although load case LC3 is based on the combined load, it is still not demanding for a complex lay-up forthe optimal solution. With both formulations the minimum weight feasible design was found to be 3 pairs of(+60,−60) layers in each sub-laminate as [(+60/−60)3]SE. As in LC2 there is not obvious benefit to gain byusing the formulation with stacking sequence, on the other hand, the number of designs for a Pareto optimalsolutions of similar quality does not differ clearly. Judging from the runs of all three load cases LC1-LC3the initial DOE seemed to have a bigger effect on the amount of designs needed than the used formulation.

VII.B. The Drive Shaft

For the drive shaft the length and the diametre of the FE-model are 1000 mm and 100 mm, respectively. Atx = 1000 mm rotations are fixed, a displacement constraint is applied and the circumference is constrained.The shell is optimized using three sub-laminates sharing different areas. The zones are as x = {0− 50, 50−100, 100− 900, 900− 950, 950− 1000} mm as shown in Figure 1.

The multiobjective optimization problem of the form

min~x∈S

~z (~x ) = min~x∈S

[W (~x )

1/RF (~x )

](10)

S = {~x | g(~x ) =1/RF (~x )(1/RF )a

− 1 ≤ 0, `j(~x) ∈ Lj , j = 1, 2, 3} (11)

is considered, where j = 1, 2, 3 are the three allowable sub-laminate configurations sharing the zones asdefined above, 1/RF is the inverse reserve factor and W the mass index based on the amount of layers in asub-laminate and area covered on the drive shaft.

The load case includes axial force Fx = −20 kN, torsion Mx = 1000 Nm, and displacement boundaryconditions uy(x = L) = 5 mm. As in the examples before each sub-laminate consists of 3 parts to beoptimized. Part 1 can have an orientation from 30 deg to 60 deg in steps of 5 deg and part 3 the same range,but in steps of 15 deg, i.e., x1 ∈ {0, 1, . . . , 6} and x4 ∈ {0, 1, 2}. All parts have a multiplier between 0 and5 as x2, x3, x5 ∈ {0, 1, . . . , 5}. Additionally depending on the formulation 1 variable for each sub-laminateis introduced as x6 ∈ {0, 1, . . . , 5}. Altogether that allows 2 × 1013 designs. Of course, for the inner sub-laminate a minimum of one layer constraint is applied. The problem was solved with MOGAII algorithm,running for 100 generations with 50 designs random DOE.

The minimum weight design is illustrated in Figure 8. Due to allowing 30 deg to 60 deg layers in thebeginning and/or the end of the laminate, the both formulations can create designs of the same quality.Compared to the load cases LC1-LC3, the actual one shows a more difficult border shape in the criterionspace to explore the Pareto optimal solutions as illustrated in Figure 7. From the charts presented there onecannot yet generalize a preference for a certain formulation, as the Pareto designs are found equally well.Both formulations show that they are able to explore the border of the criterion space, but in a single runlack to cover the whole front. Generally the bigger DOE size would decrease that effect.

Figure 8 shows the lay-ups of the minimum weight design found and the inverse reserve factor plot forthe drive shaft built out of those layups. It is clear that the most critical points are the ends of the shaftand the transitions from one zone to another. Higher loads or forced displacements increase that effect andlead to the need of more reinforcements in the zones near the ends.

VIII. Conclusions

Sub-laminate based lay-up optimization formulation concepts are considered. One sub-laminate is basedon the layer multiplier concept, the other is based on the layer multiplier concept with additional stack-

ing sequence vector. Four different example cases are used to illustrate the behavior of the concepts toparametrize composite lay-ups in the computational environment aimed at multicriterion design optimiza-tion of composite structures. The presented concepts performed equally well. Due to the stacking sequencevector, the design space increased by factor 6 for each sub-laminate. Yet the amount of designs needed to get

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(a) Run a with stacking sequence. (b) Run b with stacking sequence.

(c) Run c without stacking sequence. (d) Run d without stacking sequence.

Figure 7. Examples of criterion space for different optimization runs of the drive shaft.

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(a) Lay-ups for the three zones.

(b) Inverse reserve factor plot.

Figure 8. The minimum weight design found for the drive shaft problem.

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a subset of Pareto optimal solutions did not increase accordingly. Instead at the given DOE size, the set ofinitial designs had a more drastic effect on the average number of designs needed. Further studies using thestacking sequence vector in more demanding loadcases involving buckling or combined load case includingnatural frequency analysis should be carried out to expose the potential performance gain provided by theformulations.

Acknowledgments

The third author is employed by National Board of Patents and Registration of Finland.

References

1Le Riche, R. and Haftka, R. T., “Improved Genetic Algorithm for Minimum Thickness Composite Laminate Design”,Composites Engineering, Vol. 5, pp. 143–161, 1995.

2Gurdal, Z., Haftka, R. T., Hajela, P., Design and Optimization of Laminated Composite Materials, Wiley, 1999.3Autio, M., “Determining the Real Lay-up of a Laminate Corresponding to Optimal Lamination Parameters by Genetic

Search”, Structural and Multidisciplinary Optimization, Vol. 20, 2000, pp. 301–310.4Jalkanen, J. and Koski, J., “Heuristic Methods in Space Frame Optimization”, Proc. 46th

AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Austin, TX, USA, 18–21April, 2005. Paper AIAA-2005-1900.

5Kathiravan, R. and Ganguli, R., “Strength Design of Composite Beam Using Gradient and Particle Swarm Optimization”,Composite Structures, Vol. 81, 2007, pp. 471–479.

6Bassanese, A., Ozen, M., Monicke, A., Katajisto, H., Palantera, M., Spagnolo, M., Micchetti, F., Perillo, M., “ProcessIntegration and Multi-Objective Design Optimization as New Design Methodologies for Composite Structures”, Proc. SAMPE’08, Long Beach, CA, USA, 18–22 May, 2008.

7Monicke, A., Katajisto, H., Kere, P., Palantera, M., Perillo, M., “Engineering Oriented Formulation for Laminate Lay-upOptimization”, Journal of Structural Mechanics, Vol. 41, 2008, pp. 137–151.

8Kere, P. and Lyly, M., “On Post-Buckling Analysis and Experimental Correlation of Cylindrical Composite Shells withReissner-Mindlin-Von Karman Type Facet Model”, Computers & Structures, Vol. 86, 2008, pp. 1006–1013.

9Kere, P. and Lyly, M., “Reissner-Mindlin-Von Karman Type Shell Facet Model for Buckling Simulation of ImperfectCylindrical Composite Shells”, Mechanics of Advanced Materials and Structures, Vol. 18, 2011, pp. 115–124. To appear.

10Miettinen, K., Nonlinear Multiobjective Optimization, Norwell: Kluwer Academic, 1999.11modeFRONTIER, modeFRONTIER web site at http://www.esteco.it.12ESAComp - Software for Analysis and Design of Composites, Release 4.1, http://www.componeering.com ESAComp

web site.13Elmer Open Source Finite Element Software for Multiphysical Problems, Release 5.4, http://www.csc.fi/elmer Elmer

web site.

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