[american institute of aeronautics and astronautics 47th aiaa/asme/asce/ahs/asc structures,...

Generating Curvilinear Fiber Paths

from Lamination Parameters Distribution

Shahriar Setoodeh∗, Adriana W. Blom †

Mostafa M. Abdalla ‡, Zafer Gurdal§

Aerospace Structures, Delft University of Technology

Kluyverweg 1, 2629 HS, Delft, The Netherlands

Contrary to the classical stacking sequence design of composite laminates with straight

fibers, each ply can be designed with curvilinear fiber paths resulting in variation of the

stiffness properties over the structure. Such laminates are often denoted as “variable-

stiffness panels” in the literature.

Instead of treating fiber orientation angles as spatial design variables, lamination para-

meters can be used. However, retrieving the actual stacking sequence requires additional

efforts at a post processing level. In this paper, we presume that the optimal distribution

of lamination parameters is already obtained for a particular design problem. Then we use

curve fitting techniques to obtain continuous fiber paths that result in a distribution of

the lamination parameters close to the optimal distribution in a least square sense while

satisfying the manufacturing curvature constraint.

The fiber orientation angle is expanded using a set of basis functions and unknown

coefficients. The unknown coefficients are then computed such that the assumed form rep-

resents the optimal distribution of the lamination parameters in a least square sense. The

key feature of such approach is that at this post-processing step, no more expensive finite

element analyses are needed and the curve fitting is performed using simple polynomial

and trigonometric function evaluations. The curve fitting problem is then solved using a

constrained nonlinear least square solver where maximum curvature is controlled using a

side constraint. Numerical results demonstrate the efficiency of the proposed formulation

for minimum compliance design problems. For the cantilever plate problem investigated,

the compliance of the approximate design is only than 2.5% larger than than the compliance

of optimal lamination parameters design. A methodology is also proposed to estimate the

thickness buildup due to the curved fiber paths.

I. Introduction

Contrary to the classical stacking sequence design of composite laminates with constant ply-angles overthe entire layer, each ply can be designed with spatially optimal fiber orientations resulting in variation of

the stiffness properties over the structure. Such laminates are often denoted as “variable-stiffness panels” inthe literature.1, 2 By allowing the stiffness properties of fiber-reinforced composites to vary from one point toanother, the design space is significantly expanded as compared to the traditional stacking sequence designproblem, therefore opening the possibility for improved designs for the same weight.

The minimum compliance design of variable-stiffness panels using fiber angles as continuous spatialdesign variables is well studied in the literature.3–5 Setoodeh et al. showed that the optimality criteria forthe minimum compliance design will reduce to the local minimization of the complementary strain energyat any point in the domain. Numerical studies show that substantial stiffness improvements can be gained

∗Postdoctoral Research Associate. Member, AIAA.†PhD Student, Member, AIAA‡Assistant Professor Member, AIAA.§Professor, Aerospace Structures Chair, Delft Technical University. Associate fellow, AIAA.

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American Institute of Aeronautics and Astronautics

47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Confere1 - 4 May 2006, Newport, Rhode Island

AIAA 2006-1875

Copyright © 2006 by Shahriar Setoodeh, Mostafa Abdalla, and Zafer Gurdal. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

only by reorientation of the orthotropic material in an optimal manner.5, 6 Such formulation, however, doesnot enforce the continuity of fiber paths and therefore designs might not be easily manufacturable.

Curvilinear fiber path design of composites, on the other hand, maintains the continuity of the fiber pathsand can improve structural properties such as stiffness and buckling load. Nagendra et al.7 implemented aglobal fiber path representation using linear combination of non-uniform rational B-Splines (NURBS) passingthrough a fixed number of control points for rectangular laminates. They studied optimal frequency andbuckling design of laminated composites plates subject to deformations, ply failure, and inter-laminar stressconstraints. The design variables were scalar multipliers of the different basis fiber paths. For a plate with ahole, Katz et al.8 used sequential linear programming to maximize the failure load based on the maximumstrain criterion. The plate was modeled by the finite element method, and sensitivities with respect tofiber orientation were calculated by variational methods. Results showed substantial improvements in loadcarrying capacity. Parnas et al.9 studied minimum weight design of composite laminates subject to theTsai-Hill failure criterion as a stress constraint by sequential quadratic programming. They constructed abi-cubic Bezier surface for layer thickness representation and cubic Bezier curves for fiber angles and usedcoordinates of the control points as design variables for reduced number of the design variables. Gurdalet al.1, 10, 11 studied curvilinear fiber paths design of rectangular panels in a series of publications. Thecurvilinear fiber paths in these publications were generated from a base curve that changes its orientationlinearly from one end of the panel to the other. In their formulation, they took into account constraints onthe radius of curvature of the fiber paths.

An alternative formulation for the variable stiffness design problem is to use lamination parameters asdesign variables instead of fiber angles. In general, the in-plane behavior of thin symmetric compositelaminates can be fully modeled using only four lamination parameters regardless of the actual number oflayers.12 Moreover, the laminate stiffness matrices are linear in terms of the lamination parameters. It is alsoknown that structural design problems such as minimum compliance and minimum fundamental frequencyare convex in the lamination parameters space.13, 14 Besides, it is well known that the feasible laminationparameters domain is also convex.15 These characteristics are very beneficial in the design of variable-stiffnesslaminates in the sense that they not only improve the computational efficiency, but also represent the mostgeneral lay-up configuration and thereby providing a theoretical lower bound on the optimal value of theobjective function.

In this paper, we seek a curvilinear fiber path design for optimal variable stiffness panels design usinglamination parameters. The fiber orientation angle is expanded using a set of basis functions and theunknown coefficients are computed such that this form represents the optimal distribution of the laminationparameters in a least square sense.

II. Variable-Stiffness Design Using Lamination Parameters

Lamination parameters as initially introduced by Tsai and Hahn16 represent the laminate lay-up config-uration in a compact form. In general, the in-plane stiffness behavior of symmetric composite laminates inthe classical lamination theory can be fully modeled using only four lamination parameters regardless of theactual number of layers.12 Moreover, the laminate stiffness matrices are linear in terms of the laminationparameters. Lamination parameters cannot be arbitrarily prescribed since the trigonometric functions en-tering their definition are related. The feasible domain for the in-plane lamination parameters is known tobe convex and defined by15

2V 21 (1 − V3) + 2V 2

2 (1 + V3) + V 23 + V 2

4 − 4V1V2V4 ≤ 1

V 21 + V 2

2 ≤ 1

−1 ≤ V3 ≤ 1.

(1)

where the in-plane lamination parameters for a laminate with thickness h are defined by

(V1, V2, V3, V4) =

1/2∫

−1/2

(cos 2θ(z), sin 2θ(z), cos 4θ(z), sin 4θ(z))dz, (2)

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in which z = z/h is the normalized z coordinate of the layers, and θ(z) is the fiber angle at z. The in-planelaminate stiffness matrix A is then a linear function of the lamination parameters. The feasible domain forbending lamination parameters (W1,W2,W3, and W4) is the same as the in-plane lamination parameters asdefined in Equation (1).

According to the definition of the lamination parameters in Equation (2), only two of the laminationparameters, namely V1 and V3, are required to fully model the in-plane stiffness of balanced symmetriclaminates while V2 = V4 = 0. This would simplify the above set of inequality constraints to

V3 ≥ 2V 21 − 1,

− 1 ≤ Vi ≤ 1 (i = 1, 3).

Instead of treating fiber orientation angles as spatial design variables, lamination parameters can be used.Such an approach limits the number of design variables to four for in-plane problems regardless of the actualnumber of layers. Besides, the design optimization is usually convex in the lamination parameters spacethereby simplifies the numerical optimization task significantly. However, retrieving the actual stackingsequence requires additional efforts at a post processing level. In the next section, we presume that theoptimal distribution of lamination parameters is already obtained for a particular design problem and thenwe use curve fitting techniques to obtain a continuous fiber orientation angle distribution that results in adistribution of the lamination parameters close to the optimal distribution in a least square sense.

III. Problem Formulation

Assume a rectangular domain as shown in Figure 1 for which the optimal distribution of the laminationparameters for a balanced symmetric variable-stiffness panel is already obtained. Such a distribution can befor design objectives such as minimum compliance13 or maximum fundamental frequency.14 We define thenormalized coordinates ξ and η as follows

ξ =2x − a

a, η =

2y − b

b, (3)

such that

− 1 ≤ ξ ≤ 1,−1 ≤ η ≤ 1. (4)

x

y

a

b

Figure 1. Rectangular domain and coordinate system

Assume a variable stiffness [±θ1 ± θ2 . . . ± θnq]s layup configuration with nh = 2nq being half number of

layers. We expand the fiber orientation angle for the k-th layer in the ξ-η plane by

θ(k)(ξ, η) =

m−1∑

i=0

n−1∑

j=0

a(k)ij Li(ξ)Lj(η),

k =1, 2, . . . , nq

(5)

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where m and n are number of basis functions used in ξ and η directions respectively, a(k)ij are unknown

coefficients (design variables), Li(ξ) = Li(ξ)/Li is the normalized Lobatto polynomial such that |Li(ξ)| ≤ 1and L0,L1,. . . ,Lm are the Lobatto polynomials defined as

Li(ξ) =

ξ∫

−1

Pi−1(τ)dτ, i ≥ 2, (6)

where Pn(τ) is the Legendre polynomial. We note that

L0(ξ) = 1, L1(ξ) = ξ, L2(ξ) =1

2(ξ2 − 1), L3(ξ) =

ξ

2(ξ2 − 1)

L4(ξ) =1

8(1 − 6ξ2 + 5ξ4), L5(ξ) =

1

8ξ(3 − 10ξ2 + 7ξ4), L6(ξ) =

1

16(−1 + 15ξ2 − 35ξ4 + 21ξ6) . . .

L0 = 1, L1 = 1, L2 =1

2, L3 =

1

3√

3, L4 =

1

8, L5 =

1

35

√3

35(75 + 4

√30), L6 =

1

16, . . .

(7)

In the numerical implementations, however, the following recurrence relation has been used

Li(ξ) =1

i(ξPi−1(ξ) − Pi−2(ξ)). (8)

Now assume that the optimal distribution of the lamination parameters is already obtained using a finiteelement mesh with nn nodes. For simplicity, here we assume a balanced symmetric variable-stiffness panelwith V1 and V3 as nodal design variables, however, the proposed methodology can be trivially extendedto the general variable stiffness panels where all four lamination parameters are treated as nodal designvariables. For balanced symmetric variable stiffness panels, lamination parameters can be computed fromthe fiber orientation angle approximation of Equation (5) as

V1(ξ, η) =1

nq

nq∑

k=1

cos(2θ(k)(ξ, η)),

V3(ξ, η) =1

nq

nqs∑

k=1

cos(4θ(k)(ξ, η)).

(9)

Now we seek an approximate fiber orientation function that results in the closest distribution of thelamination parameters to their optimal distribution in a least square sense while satisfying the curvatureconstraint of the tow-placement machine. This is formulated as the following nonlinear least square problem

minX12

∑2nn

i=1 f2i (X)

subject to :

−θl ≤ θ(k)q ≤ θu

−κmax ≤ κ(k)q ≤ κmax

(k = 1, . . . , nq), (10)

in which X = {X(1),X(2), . . . ,X(nh)} is the vector of the design variables with nq × m × n entries and

X(k) = {a(k)00 , a

(k)01 , . . . , a

(k)0n , . . . , a

(k)m0, a

(k)m1, . . . , a

(k)mn} (11)

and (i = 1, . . . , 2nn)

fi(X) =

(Vj1 − V ∗

j1 ) i = 2j − 1 (j ∈ N)

(Vj3 − V ∗

j3) i = 2j

(12)

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Here V ∗

j1and V ∗

j3are the target optimal lamination parameters for Node j in the finite element mesh. The

bounds on the fiber orientation angles are denoted by θl and θu are set to −180◦ and 180◦ respectively. Themaximum allowed curvature κmax depends on the tow-placement machine used. For instance, for VIPERFPS-3000 of the Cincinnati Machine17 κmax = 1.57m−1 whereas for the same feeding rate for IngersollMachine Tools18 κmax = 3.333m−1. Curvature κ is computed for any nodal point on each layer using thefollowing relation

κ =dθ

ds=

∂θ

∂x

dx

ds+

∂θ

∂y

dy

ds

=2

Lx

∂θ

∂ξcos θ +

2

Ly

∂θ

∂ηsin θ

(13)

in which the derivative of the θ in the normalized ξ − η coordinate systems can be easily obtained usingEquation (5). For instance

∂θ

∂ξ=

m−1∑

i=0

n−1∑

j=0

a(k)ij L′

i(ξ)Lj(η), (14)

where L′

i(ξ) is computed using the following recurrence relation

L′

i(ξ) =1

i Li(Pi−1(ξ) + ξP ′

i−1(ξ) − P ′

i−2(ξ)), (15)

finally, P ′

i (ξ) is computed, in turn, using the following Legendre recurrence relation

P ′

i (ξ) = i Pi−1(ξ) + ξP ′

i−1(ξ). (16)

The minimization problem of Equation (10) is solved numerically using the constrained nonlinear pro-gramming code DFNLP of Schittkowski.19 The Jacobian of the objective function in Equation (12) iscomputed exactly by (i = 1, . . . , 2nn, j = 1, . . . , m × n × nq)

∂fi(X)

∂Xj=

− 2

nqLp(ξk)Lq(ηk) sin(2θ(l)(ξk, ηk)) i = 2k − 1(k ∈ N)

− 4nq

Lp(ξk)Lq(ηk) sin(4θ(l)(ξk, ηk)) i = 2k(17)

in which (ξk, ηk) are the coordinates of the k-th node in the finite element mesh and

l =

⌊j − 1

m × n

⌋+ 1,

p =

⌊s − 1

n

⌋,

q =(s − 1) mod n,

where ⌊i⌋ is the floor function of i and common residue of m and n is denoted by m mod n, and

s = (j − 1) mod (m × n) + 1.

The sensitivity of the fiber angle constraint can be readily obtained using Equation (14) while derivativeof the curvature constraint is computed from

∂κ

∂Xj=

2

LxLq(η)(L′

p(ξ) cos θ − ∂θ

∂ξLp(ξ) sin θ)

+2

LyLp(ξ)(L

′

q(η) sin θ +∂θ

∂ηLq(η) cos θ)

(18)

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The average residual norm r is used as a measure to examine the convergence of the present formulationfor an increasing number of basis functions m and n in Equation (5). The average residual norm is computedfrom

r =‖R‖√2nn

(19)

where the residual vector R = {f1(X), f2(X), . . . , f2nn(X)} is computed at the solution of Equation (10).

The minimization problem of Equation (10) is a general nonlinear one and therefore has potentially severallocal minima. To remedy this issue, for the lowest number of basis functions, i.e., 2 × 2, we solve Equation(10) for several randomly selected starting points and keep a candidate which results in the minimum valueof the average residual norm. Then this point is supplied to the next level approximation, e.g. 3 × 3, asa starting point by setting the new coefficients all to zero. This process can be repeated until the relativechange in the average residual norm is smaller than a given tolerance. Once an approximate fiber angledistribution is obtained, it has to be translated to fiber paths for tow placement machines. A methodologyis outlined in the Appendix based on the streamline theory to obtain curvilinear fiber paths as well asestimating thickness variations.

IV. Numerical Results

To demonstrate the performance of the presented methodology, a cantilever plate with aspect ratio of 3,as initially suggested by Pedersen,4 is considered here. The plate dimensions are a = 1.8m and b = 0.6mwith the boundary conditions and loadings as shown in Figure 2(a). The following material properties areused in the numerical simulations

E11 = 181.0 GPa, E22 = 10.3 GPa

G12 = 7.17 GPa, ν12 = 0.28.

Variable stiffness design of this problem for minimum compliance is studied by Setoodeh et al.13(noticethat the compliances reported in that paper are erroneously multiplied by a factor of two). The complianceis nondimensionalized as follows

C =E22hb3c

q20a5

, (20)

where a and b are the dimensions of the plate, q0 is the uniform distributed load, h is the laminate thickness,and c is the compliance in N · m.

The optimal distributions of the lamination parameters for a balanced symmetric variable stiffness panelare depicted in Figures 2(b) and 2(c) for a 31 × 11 finite element grid. The optimal compliance for thisexample is C∗ = 0.0374. Using the proposed methodology, we generate a curvilinear fiber path designwith maximum curvature of κmax = 1.57m−1 for nq = 1. In the first step, 50 random starting points aregenerated for a 2 × 2 expansion and then Equation (10) is solved. The candidate design with the lowestvalue of the residual norm is then used as an starting point for the next, i.e. 3 × 3, expansion by settingthe remaining coefficients to zero. This will guarantee the feasibility of the starting point in the new designspace. This process is repeated to a 10× 10 expansion. The convergence plot (average residual norm versusnumber of the basis functions) is given in Figure 3 for up to 10× 10 basis functions and the distributions ofthe lamination parameters based on the approximate fiber orientation angle are given in Figures 2(d) and2(e). The nondimensional compliance for the 10 × 10 fitting is C = 0.0433 which is 15.9% different fromthe target value. The approximate fiber orientation angles obtained for this example are depicted in Figures2(f) and 2(g). This has been achieved by setting nq = 1 in Equation (5), i.e., a [±θ]s, layup configuration.However, a general balanced symmetric lamination parameters (V1, V3) point is usually realizable by multipleequi-thickness layers.20 Therefore, by setting nq = 2, 3, . . . we are naturally expecting improved designs interms of the residual norm.

Now we investigate the effect of the mesh density, number of layers, and maximum curvature allowed bythe tow-placement machine on the residual norm and compliance of the final designs. First consider Table1 where the maximum allowed curvature is considered to be κmax = 1.57m−1. In this table, C∗ represents

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a

b

q0

(a) Geometry and loading

V1: -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

(b) V ∗1

V3: -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

(c) V ∗3

V3: -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

(d) V1

V3: -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

(e) V3

(f) +θ (g) −θ

Figure 2. Design of a cantilever (a/b = 3, 31 × 11 nodes, κmax = 1.57, nq = 1); (a) geometry and loading(b)-(c) optimal distribution of the lamination parameters, (d)-(e) approximated distribution of the laminationparameters, (f)-(g) approximated fiber angles.

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the optimal compliance obtained using lamination parameters, r is the average residual norm as defined inEquation (19), C is the compliance of the design with curvilinear fiber paths, and finally the difference ofthe two compliances in percentage is given in the last column. By increasing the mesh density, i.e. nn inEquation (10), r will increase for fixed number of terms in Equation (5). This was expected as for the samenumber of expansion terms (m × n), the same value of the residual norm cannot be achieved for a higherresolution of the approximate distribution. Also, by increasing half number of layers, i.e. nh, from to 2to 4 results in smaller values of r as well as the compliance. Table 2 gives the same set of results for themaximum allowed curvature of 3.333m−1 of Ingersoll Machine Tools.18 The add manufacturing capability,results in significant improvements over the previous results.

No. of Basis Functions

Ave

rag

eR

esid

ualN

orm

Non

dim

ensi

onal

Co

mp

lianc

e

20 40 60 80 100

0.2

0.3

0.4

0.5

0.6

0.7

0.01

0.03

0.05

0.07

rC

Figure 3. Convergence plot for the cantilever plate (31 × 11 nodes, nq = 1,κmax = 1.57m−1).

Table 1. Curvilinear fiber path designs with κmax = 1.57m−1 using a 10 × 10 expansion.

Grid C∗ r C %Difference

(a) nq = 1

19 × 7 0.0367 0.24096 0.0396 7.89

31 × 11 0.0374 0.28547 0.0433 15.90

(b) nq = 2

19 × 7 0.0367 0.12006 0.0372 1.45

31 × 11 0.0374 0.19671 0.0401 7.19

V. Discussions and Concluding Remarks

The optimal distribution of the lamination parameters provides the best possible variable stiffness designfrom a theoretical perspective. In this paper, we have proposed a methodology to generate curvilinear fiberpaths that replicate the optimal stiffness distribution in a least square sense. It was shown that laminationparameters distribution can be matched with a good accuracy and the stiffness of the curvilinear fiber pathdesigns can be less than 2% apart from the target design. The effect of the number of variable stiffness layers

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Table 2. Curvilinear fiber path designs with κmax = 3.333m−1 using a 10 × 10 expansion.

Grid C∗ r C %Difference

(a) nq = 1

19 × 7 0.0367 0.17671 0.0379 3.26

31 × 11 0.0374 0.21363 0.0389 4.00

(b) nq = 2

19 × 7 0.0367 0.11316 0.0372 1.40

31 × 11 0.0374 0.13132 0.0383 2.52

was also investigated. As expected, by increasing the number of layers, a better match to the target designcan be obtained.

One of the drawbacks of using lamination parameters as design variables is that local stress constraintscannot be incorporated into the design formulation since the actual layup configuration is not known. Usingthe proposed formulation, the local stress constraints can be accommodated in the fiber path generationstep. Since a given set of lamination parameters can be achieved using potentially many different layupconfigurations, one can seek a layup which satisfies the local stress constraints, yet matches the optimallamination parameters distribution is a least square sense. This can be easily done by using the midsurfacestrain filed of the optimal lamination parameters design and including stress constraints in Equation (10).

Acknowledgment

The first author would like to thank Prof. Klaus Schittkowski for providing him with the source code ofthe DFNLP solver.

References

1Gurdal, Z. and Olmedo, R., “In-Plane Response of Laminates with Spatially Varying Fibre Orientations: Variable StiffnessConcept,” AIAA Journal , Vol. 31, No. 4, 1993, pp. 751–758.

2Waldhart, C., Gurdal, Z., and Ribbens, C., “Analysis of Tow Placed, Parallel Fiber, Variable Stiffness Laminates,” 37thAIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Material Conference, Salt Lake City, UT, Apr.15-171996, pp. 2210–2220.

3Hyer, M. W. and Charette, R. F., “Use of Curvilinear Fiber Format in Composite Structure Design,” 30th Structures,Structural Dynamics, and Material Conference, AIAA, Mobile, AL, April 1989, pp. 1011–1015.

4Pedersen, P., “Examples of Density, Orientation, and Shape-Optimal 2D-Design for Stiffness and/or Strength withOrthotropic Materials,” Structural Optimization, Vol. 26, No. 1-2, 2003, pp. 37–49.

5Setoodeh, S., Gurdal, Z., and Watson, L. T., “Design of Variable-Stiffness Composite Layers Using Cellular Automata,”Computer Methods in Applied Mechanics and Engineering, Vol. 37, No. 4-5, 2006, pp. 301–309.

6Setoodeh, S., Abdalla, M. M., and Gurdal, Z., “Combined Topology and Fiber Path Design of Composite Layers UsingCellular Automata,” Structural and Multidisciplinary Optimization, Vol. 30, No. 6, 2005, pp. 413–421.

7Nagendra, S., Kodiyalam, S., and Davis, J. E., “Optimization of Tow Fiber Paths for Composite Design,” Proceedings ofthe AIAA/ASME/ASCE/AHS/ASC 36th SDM Conference, New Orleans, LA, April 10-13 1995, pp. 1031–1041.

8Katz, Y., Haftka, R. T., and Altus, E., “Optimization of Fiber Directions for Increasing the Failure Load of a Plate witha Hole,” ASC Technical Conference, Vol. 4, 1989, pp. 62–71.

9Parnas, L., Oral, S., and Ceyhan, U., “Optimum Design of Composite Structures with Curved Fiber Courses,” CompositeScience and Technology , Vol. 63, 2003, pp. 1071–1082.

10Gurdal, Z. and Tatting, B., “Cellular Automata for Design of Truss Structures with Linear and Nonlinear Response,”Proceedings of the 41st AIAA/ASME/ASCE/AHS Structures, Structural Mechanics and Materials Conference, AIAA Paper,Atlanta, GA, 2000, pp. 2000–1580.

11Tatting, B. and Gurdal, Z., “Cellular Automata for Design of Two-Dimensional Continuum Structures,” Proceedings ofthe 8th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, AIAA Paper, Long Beach,CA, 2000, pp. 2000–4832.

12Gurdal, Z., Haftka, R. T., and Hajela, P., Design and Optimization of Laminated Composite Materials, John Wiley &Sons Inc., 1999.

13Setoodeh, S., Abdalla, M. M., and Gurdal, Z., “Design of Variable-Stiffness Laminates Using Lamination Parameters,”Composites Part B: Engineering , Vol. 195, No. 9-12, 2006, pp. 836–851.

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14Abdalla, M. M., Setoodeh, S., and Gurdal, Z., “Design of Variable Stiffness Composite Panels for Maximum FundamentalFrequency using Lamination Parameters,” European Conference on Spacecraft Structures, Materials & Mechanical Testing ,Noordwijk, The Netherlands, May 2005.

15Hammer, V. B., Bendsøe, M. P., Lipton, R., and Pedersen, P., “Parametrization in Laminate Design for OptimalCompliance,” International Journal of Solids and Structures, Vol. 34, No. 4, 1997, pp. 415–434.

16Tsai, S. W. and Hahn, H. T., Introduction of Composite Materials, Technomic, Lancaster, 1980.17Evans, D. O., Vaniglia, M. M., and Hopkins, P. C., “Fiber Placement Process Study,” 34th International SAMPE

Symposium and Exhibition, May 1989, pp. 1822–1833.18Moruzzi, M., Oldani, T., Tatting, B. F., Gurdal, Z., and Blom, A. W., “Tailoring of Composite Layup through Tow-

Placement Manufacturing Techniques,” Proceedings of the SAMPE 2006 Conference, Long Beach, CA, May 2006.19Schittkowski, K., DFNLP20: A Fortran Implementation of an SQP-Gauss-Newton Algorithm, April, 2005.20Setoodeh, S., Abdalla, M. M., Gurdal, Z., and Tatting, B., “Design of Variable-Stiffness Composite Laminates For Maxi-

mum In-Plane Stiffness Using Lamination Parameters,” 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics& Materials Conference, Austin, TX, April 18-21 2005.

APPENDIX: Manufacturability of the Variable-Stiffness Panels

The approximate fiber angle distribution θ(x, y) of the variable-stiffness plate has to be translated into aformat that can be used for production with a tow-placement machine. A tow-placement machine constructslaminate plies by laying down courses of material that follow predefined paths. Each course consists of up to24 individual tows with a total width of 3”. Each of these tows can be fed independently to allow for in-planesteering, and they can also be cut and restarted independently, thereby enabling thickness control within aply. If, however, the course width is kept constant (i.e. no tows are dropped) and the fiber paths are notparallel, a ply will exhibit gaps and/or overlaps. In the next section we will first propose a methodology usingstreamline analogy to translate the optimum fiber angle distribution into continuous fiber paths that serveas centerlines for the individual courses. The distance between individual centerlines will not be constantand if no gaps are allowed, overlaps between courses will occur. Since we assumed constant thickness in thelamination parameters formulation overlaps are not allowed and therefore tows need to be dropped. However,if there are too many courses overlapping each other, dropping tows might not be sufficient anymore. Inorder to predict if a constant thickness ply is feasible or not, an estimate of the amount of overlap is needed.This is accomplished by first assuming that each centerline is covered by a course with a constant coursewidth, so that thickness will be built up due to overlapping courses. If the amount of overlap within thepanel does not exceed a certain number (e.g. eight times the constant thickness), it is assumed that constantthickness can be achieved by dropping tows. After the derivation of the continuous fiber paths, a methodwill be introduced to estimate the thickness distribution before dropping tows.

A. Development of Continuous Fiber Paths

In order to translate the known fiber angle distribution to continuous fiber paths we assume that the contin-uous fiber paths can be represented by streamlines, so that the value of the stream function, Ψ, is constantalong a fiber. If a rectangular coordinate system with x− and y−coordinates is used and the distance alonga fiber is defined as s, the mathematical expression for the stream function is given by:

Ψ(x, y) = C and dΨ = 0

where dΨ =∂Ψ

∂x

dx

ds+

∂Ψ

∂y

dy

ds

= Ψ,x cos θ(x, y) + Ψ,y sin θ(x, y)

(21)

where x = x(s) and y = y(s). Taking into account the boundary conditions for Ψ, this partial differentialequation can be solved for Ψ(x, y) either analytically or numerically. In order to avoid conflicts at theboundaries of the domain, boundary conditions are only given at the inflow boundaries:

Ψ(x, y) = Ψ∗(x, y) if t · N ≤ 0

where t = cos θ(x, y) i + sin θ(x, y) j

and N is the normal to the surface boundary, pointing outward

(22)

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The conventions in this definition are shown in Figure 4(a). In order to solve the differential Equation21, boundary conditions are needed. We have chosen the boundary conditions to be such that the change ofthe flow function normal to the fiber direction is constant along the inflow boundary:

|Ψ∗

,n| = Constant

where Ψ,n = Ψ,xdx

dn+ Ψ,y

dy

dn

= −Ψ,x sin θ(x, y) + Ψ,y cos θ(x, y)

(23)

Combining equations 21 and 23, the partial derivatives at the boundary can be found:

[cos θ sin θ

− sin θ cos θ

]{Ψ∗

,x

Ψ∗

,y

}=

{0

Ψ∗

,n

}⇒

{Ψ∗

,x

Ψ∗

,y

}= Ψ∗

,n

{sin θ

− cos θ

}(24)

Besides the derivatives at each inflow boundary, one value for Ψ∗ is needed in order to obtain all boundaryconditions. Which point is known per inflow boundary depends on the connection with neighboring outflowregions. This is explained below and illustrated by Figure 4(b).

N = [ 0, 1]

N = [ 1, 0]

N = [ 0,−1]

N = [−1, 0]

t

t = [ cos θ, sin θ ]

t

n

n = [− sin θ, cos θ ]

n

(a) Conventions

type I type IIa type I

type Itype IIatype IIb

τ

ττ

Inflow boundary

Outflow boundary

(b) Boundary definitions

Figure 4. Definitions for the derivation of continuous fiber paths

In this figure the inflow regions are indicated by continuous lines, while outflow regions are represented bydashed lines. Transitions from inflow to outflow regions and vice versa are indicated by small circles, whichare assigned a type. Type I transitions always occur at a discontinuity of the surface boundary. A veryimportant characteristic of a Type I is that the neighboring outflow region always depends on the adjoininginflow region at that transition. That is at this transition the streamlines that exit at the outflow regionoriginate from the neighboring inflow region. Therefore the value of the flow function can not be knownat this transition. Type II transitions always occur along the continuous boundary and they can connectan inflow region with either an outflow region that depends on the adjoining inflow region (Type IIa) or anoutflow region that is independent of the neighboring inflow region (Type IIb). In the latter case, the valueof Ψ∗ at the transition (xt, yt) is known and this value will serve as a starting point to determine Ψ∗ at thisinflow boundary. Type IIa on the other hand can not provide a starting value. Mathematically these twotransitions are described as:

if τ · t ≥ 0 ⇒ Type IIa and Ψ∗

in(xt, yt) ≡ Ψ∗

out(xt, yt)

if τ · t < 0 ⇒ Type IIb(25)

In these equations τ represents a vector that is tangent to the surface boundary and points from theoutflow to the inflow region (as shown in Figure 4(b)). Finally, the sign of Ψ∗

,n needs to be switched at everytransition and if the value at (0, 0) is chosen to be zero, all boundary conditions are set. However, if there ismore than one inflow region, the boundary conditions themselves depend on the solution of the differentialEquation (21) and then they will have to be found implicitly. For that purpose a numerical scheme with aregular grid was set up. At the inner domain and at the outflow boundaries, Equation (21) is used, wherethe derivatives are calculated according to the scheme in table 3.

The reason for using both backward and forward differences is to avoid gridpoints that do not belong tothe domain. The boundary conditions are also included in the numerical scheme, where the relation between

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Table 3. Differentiation scheme

Ψ,x Ψ,y

cos θ ≥ 0 backward -

cos θ < 0 forward -

sin θ ≥ 0 - backward

cos θ < 0 - forward

the points on the boundary is found by integrating Equation (24). In this equation, the sign of Ψ,n is alwaysthe opposite of the Ψ,n of the neighboring outflow boundaries. Using all this information a numerical solutionscheme can be set up. Once the numerical scheme is set up, it can be solved for both the stream functionand the boundary conditions. By plotting making a contourplot of the stream function, the continuous fiberpaths are found, because these are represented by the contourlines of the stream function. The exact valueof Ψ,n determines the distance between the individual fiber paths.

B. Prediction of Thickness Distribution using Tow-Placement

The continuous fiber paths that are obtained in section A are assumed to serve as centerlines for the tow-placement machine. If each path is covered by a course with a constant width and no gaps between coursesis allowed, overlaps will occur and thickness will be built up. The amount of thickness buildup can beestimated by assuming an infinitely small course width, which results in a continuous thickness distribution,although in reality discrete overlap regions will occur due to the finite head width. In the case of infinitelysmall courses, the thickness buildup will be proportional to one over the distance |d| between two fibers, i.e.:

t(x, y) ∝ 1

|d| (26)

Since |d| is proportional to 1/Ψ,n(x, y) the thickness will be proportional to the derivative of the streamfunction with respect to the normal direction:

t(x, y) ∝ 1

|d| ∝1

1/Ψ,n(x, y)∝ dΨ

dn(x, y) (27)

If no gaps are desired, the constant in Equation (23) should be divided by the minimum of t, so that

min(t) becomes 1. As soon as the fiber orientation θ(x, y) is not constant, overlaps will occur, although forthe design using lamination parameters the thickness was assumed constant. When the thickness buildupis not too severe (e.g. tmax/tmin < 8), constant thickness can be achieved by cutting and restarting towswith the tow-placement machine. Furthermore it would be more efficient for manufacturing purposes to uselarger course widths, although then some deviations from the ideal fiber orientation will occur and a trade-offbetween production time and performance will become necessary. Even if the predicted thickness buildup islarger, constant thickness can still be achieved by not (completely) using all the continuous paths as coursecenterlines. Of course the obtained thickness distribution depends on the assumed boundary conditions forthe stream function and therefore it can be changed by changing the boundary conditions. Fiber paths alongwith thickness distributions for a [±θ1/ ± θ2] layp is depicted in Figure 5.

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(a) +θ1 (b) +θ2

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

(c) +t1

1 1.5 2 2.5 3 3.5 4 4.5 5

(d) +t2

(e) −θ1 (f) −θ2

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

(g) −t1

1 1.5 2 2.5 3 3.5

(h) −t2

(i) ±θ1 (j) ±θ2

Figure 5. Fiber paths and thickness estimations for a [±θ1/ ± θ2]s layup.

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[american institute of aeronautics and astronautics 47th aiaa/asme/asce/ahs/asc structures,...

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