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Composite panel optimization using lamination parameters and inverse distance weighting interpolation

Authors:
COB-2019-1745
COMPOSITE PANEL OPTIMIZATION USING LAMINATION
PARAMETERS AND INVERSE DISTANCE WEIGHTING
INTERPOLATION
Victor N. Capacia
Jose A. Hernandes
Solid Mechanics and Structures, ITA - Av. Heitor Villa Lobos, 1850
victorncapacia@hotmail.com; hernades@ita.br
Saullo G. P. Castro
Delft University of Technology, Department of Aerospace Structures and Materials, Kluyverweg 1, 2629 HS Delft, The Netherlands
S.G.P.Castro@tudelft.nl
Abstract. Since the middle of last century, the use of composite structures has increased in the aerospace industry. The
process of defining efficient structure laminate is still a challenge even with the processing power of modern computers.
At the same time, the competitive aeronautical industry demands an efficient way for defining the ideal structure stiffness
associated with specific requirements.
Throughout the past decades, the scientific academy evolved its knowledge and strategy to design orthotropic laminates.
The formulation and use of lamination parameters in optimizations has shown to be a good alternative to local minimum
problems.
In this work, the use of lamination parameters is used to define the stiffness matrix of specially orthotropic laminates.
Equally, an interpolation strategy is implemented as a manner to reduce the number of design variables within the process
of an structural optimization.
Keywords: Structural analysis, laminates, optimization
1. INTRODUCTION
The application of composites in civilian or military aircraft followed the typical stages that every new technology
goes through during its implementation. At the beginning, limited application on secondary structure minimized risk
and improved understanding by collecting data from tests and experience. This limited usage was followed by wider
applications, first in smaller aircraft, capitalizing on the experience gained earlier.
The structural optimization of composite material is one of the main themes of this growing technology. However,
the optimization task of such complex material is no easy task. The most common parametrization when optimizing
composite material is the use of lamination angles gradients in order to design the laminate. However, the problem
becomes poorly conditioned, ending up in local optimum instead of the final solution of the problem. In the decade of
1960, Tsai and Pagano (1968) first developed an elegant way to express the stiffness transformation of an orthotropic
material from one coordinate system to a generic one. By use of various trigonometric identities between sin and cos,
the transformed reduced stiffness could be written as a function of some constants, called the invariant properties of an
orthotropic lamina.
The lamination parameters were initially introduced by Hahn and Tsai (1980) and represent a non-dimensional through
the thickness integration of the layer angles. For layup optimization problems, using lamination parameters, reduces
significantly the number of design variables in comparison to using ply orientation angles and thickness, which may result
in non-convex optimization problem related to periodic functions and discrete number of plies or constrained design
space (HAMMER et al., 1997a). In the recent past, the academic community noticed its potential to be used as design
variables in laminates optimization problems. Moreover, only eight lamination parameters are necessary to describe a
general laminate, independently of its total thickness, and its domain is always within the range of [-1,1]. Fukunaga and
Vanderplaats (1991) optimized the buckling load of a cylindrical shell making use of a symmetric and balanced laminate.
Also considered the coupling effects of bend-twisting negligible, making the laminate especially orthotropic and reducing
the necessary lamination parameters from eight to four. In their work, the optimization behaved well and converged
always for the same optimum point.
Victor N Capacia, Jose A. Hernandes and Saullo G. P. Castro
Composite panel optimization using lamination parameters and inverse distance weighting interpolation
Despite the fact of being a good alternative, the use of lamination parameters also has its difficulties. For example,
not all combination of lamination parameters can be associated with a real laminate. To transform the orientations of a
real laminate into lamination parameters is an easy task, however to start from lamination parameters and end up with
the orientations of a laminate is a non-linear problem and it can have one, many or no solutions. There is a relation
between the lamination parameters in order for then to represent a real laminate. This relation is called feasible region
and many authors along the last years tried to develop a simple manner of representing this relation. Grenestedt and
Gudmundson (1993) used a variational approach to implicitly determine the feasible region of orthotropic symmetric
laminates. Furthermore, Grenestedt and Gudmundson (1993) derived explicit expressions between certain sets of the
in-plane and out-of-plane lamination parameters and additionally proved that the feasible region was necessarily convex
(Bloomfield et al., 2009).
In this work, the design variables will be interpolated using the inverse distance weighting (IDW) Method, as a manner
of reducing the total number of design variables an interpolation approach is investigated in order to find a satisfactory
balance between computational cost and weight reduction. The parametrization method is called inverse distance weighted
interpolation and it can be applied in any type of geometry. The optimization is performed in a two-step approach as a
manner to, first, maximize the loads multiplier of a linear buckling problem and, afterwards, minimize the wing total
weight. The case study is performed for two different numbers and distributions of design variables. A commercial
software called MSC NASTRAN is used in order to model and analyze the wing structure.
2. OPTIMIZATION FORMULATION
Lamination parameters were first introduced only as a simpler and elegant manner of expressing the stiffness transfor-
mation of a general laminate. However, its potential to be used as design variables was identified by the scientific academy
and it has been used in many recent studies in the area.
2.1 Lamination parameters
Tsai and Pagano (1968), formulated an elegant way of representing laminate stiffness, in terms of orthotropic layer
invariants and trigonometric functions:
Q11 =U1+U2cos2θ+U3cos4θ
Q12 =U4U3cos4θ
Q22 =U1U2cos2θ+U3cos4θ
Q16 =1
2U2sin2θ+U3sin4θ
Q26 =1
2U2sin2θU3sin4θ
Q66 =U5U3cos4θ
(1)
Where Qij is the reduced transformed stiffness matrix of the laminate and Uiare the invariants properties of an
orthotropic lamina. The laminate invariants are the following:
U1=3Q11 + 3Q22 + 2Q12 + 4Q66
8
U2=Q11 Q22
2
U3=Q11 +Q22 2Q12 4Q66
8
U4=Q11 +Q22 + 6Q12 4Q66
8
U5=Q11 +Q22 2Q12 + 4Q66
8
(2)
Where, Qij are the reduced stiffnesses of the lamina.
In the integration through the thickness of a laminate with only one material type, the terms Ui for i=1,2...5, become
constant and the following so called lamination parameters are defined:
V1(A,B,D) = Z1
2
1
2
cos(2θ)[1, z, z2]dz (3a)
25th ABCM International Congress of Mechanical Engineering (COBEM 2019)
October 20-25, 2019, Uberlândia, MG, Brazil
V2(A,B,D) = Z1
2
1
2
sin(2θ)[1, z, z2]dz (3b)
V3(A,B,D) = Z1
2
1
2
cos(4θ)[1, z, z2]dz (3c)
V4(A,B,D) = Z1
2
1
2
sin(4θ)[1, z, z2]dz (3d)
Where, zis the normalized through the thickness coordinate of the layers. Therefore, matrices A, B, D can be rewritten
as
A=t0+ Γ1V1A+ Γ2V2A+ Γ3V3A+ Γ4V4A)(4a)
B=t2
41V1B+ Γ2V2B+ Γ3V3B+ Γ4V4B)(4b)
D=t3
120+ Γ1V1D+ Γ2V2D+ Γ3V3D+ Γ4V4D)(4c)
Where, tis the laminate thickness. The following matrices of invariants are used:
Γ0=
U1U40
U4U10
0 0 U5
(5a)
Γ1=
U20 0
0U20
0 0 0
(5b)
Γ2=
0 0 U2
2
0 0 U2
2
U2
2
U2
20
(5c)
Γ3=
U3U30
U3U30
0 0 0
(5d)
Γ4=
0 0 U3
0 0 U3
U3U30
(5e)
Lamination parameters are interesting option of design variables for optimization, since a generic laminate with any
number of layers can be fully described with only 12 lamination parameters. This number can be reduced to 8, in case of
a symmetric laminate constituted of the same material. In this case, because of the symmetry of the stiffness matrix and
thickness, all the bending-extension coupling (B matrix) is reduced to 0. Likewise, when defining a balanced laminate
all the coupling between shear and extension cease to exist, eliminating terms A16 and A26. Thereafter, parameters
V2A and V4A also become zero. Finally, for simplicity, the coupling terms of the bending-twisting, D16 and D26 are
going to be neglected in the current work; then the symmetric and balanced laminate can be regarded as an especially
orthotropic laminate (FUKUNAGA et al., 1991). Therefore, the laminate stiffness can be represented by only 4 lamination
parameters.
Lamination parameters are defined to be nondimensional quantities that represent the stiffness of a laminate and, as
trigonometric equations, are restricted in the interval between -1 and 1. However, not all combinations among lamination
parameters results in physically viable laminates. The formulation of the relationship between lamination parameters
has been developed by several authors in an incremental fashion to define a feasible region for special laminates. In
the especially orthotropic laminated plates and shells that eliminate all coupling terms, the stiffness characteristics are
governed by two in-plane and two out-of-plane lamination parameters (DIACONU et al., 2002). The first to define a
Victor N Capacia, Jose A. Hernandes and Saullo G. P. Castro
Composite panel optimization using lamination parameters and inverse distance weighting interpolation
fundamental relationship between two in-plane or two out-of-plane lamination parameters was Miki (1982) and Miki
(1985). Yet, he did not described a relationship between in-plane and out-of-plane parameters. Later, Fukunaga and
Sekine (1992) described a feasible region for a symmetric laminate with extension-shear coupling or bending-twisting
coupling. This type of laminate is governed by four in-plane and four out-of-plane lamination parameters. However, a
relationship between in-plane and out-of-plane parameters was not described.
The lamination parameters have been used successfully as design variables in the optimization problems for vibration
by Fukunaga et al. (1994), buckling by Fukunaga et al. (1995) and topological design by Hammer et al. (1997) (DIA-
CONU et al., 2002). Next, Grenestedt and Gudmundson (1993) used a variational approach to implicitly determine the
feasible region of orthotropic symmetric laminates. Furthermore, Grenestedt and Gudmundson (1993) derived explicit
expressions between certain sets of the in-plane and out-of-plane lamination parameters and additionally proved that the
feasible region was necessarily convex (Bloomfield et al., 2009). For an especially orthotropic laminate, Grenestedt and
Gudmundson (1993) formula for the coupling of in-plane region with out-of-plane region becomes:
1
4(ViA + 1)31ViD 1
4(ViA 1)3+ 1 (6)
2.2 Inverse Distance Weighting
In the inverse distance weighting method, each sample point is pondered during interpolation according with its
distance from the unknown point. The farther a sample point gets from the interpolated one, less influence it has over it.
Figure 1. Influence zone
Figure 1 shows an example of an interpolated point in the center of the circle. In a scale from blue to red it is shown
qualitatively the measured point influence over the interpolated one. A weighting coefficient is applied to sample points
values in order to account for its distance and consequently its influence over the interpolated point. The closer the sample
point is to the interpolated one, the greater the coefficient will be, and closer in value both points will be. As the coefficient
decreases, the value of the unknown point deviates the value of the nearest observational point. The weighting function
can be seen in Eq. (7).
wi=1
d(x, xi)p(7)
Where d is the distance between x (interpolated point) and xi (known point). At the same time, the inverse distance
weighting interpolation method has some disadvantages. The quality of the interpolation result can decrease, if the
distribution of sample data points is uneven. Besides that, maximum and minimum values in the interpolated surface can
only occur at sample data points. This often results in small peaks around the sample data points and it is the main reason
to strategically choose the sample points location. The interpolating function for a value u at a point x on samples xi =
u(xi) for i=1,2,..n can be seen in Eq. (8).
u(x) = Pn
i=1 wi(x)ui
Pn
i=1 wi(x)(8)
In the case of having a big interpolating area, it is possible to restrict the area that is going to be included in the
summation of Eq. (8). The shape of the neighborhood restricts how far and where to look for the measured values to
25th ABCM International Congress of Mechanical Engineering (COBEM 2019)
October 20-25, 2019, Uberlândia, MG, Brazil
be used in the prediction. Other neighborhood parameters restrict the locations that will be used within that shape. As
the structure of the investigated wing presents no directional influence over the thickness and lamination parameters,
all data points are going to be considered equally in all interpolations. In this work, the search neighborhood is going
to be considered a circle with half wing span radius. It is also possible to divide the search neighborhood into sectors
and consider only the maximum value of each sector. No sectioning restriction is going to be applied in the current
optimization.
3. Wing structure
The conceptual aircraft used by Castro (2009) is called PRIME 900 and was a preliminary project of the 10th EM-
BRAER’s specialization program. The current composite wing holds twenty five ribs and three spars. In Tab. 1 it is
possible to visualize the main dimensions of the wing.
Table 1. Prime 900 Wing Dimensions
Variable Value
Root chord 7.164 m
Kink chord 4.229 m
Tip chord 1.480 m
Dihedral Angle 5.0o
Sweep Angle 38.5o
Kink Span 3.684 m
Half Span 12.874 m
Front wing box height 0.727 m
Rear wing box height 0.650 m
The current wing holds three spars and twenty five ribs. No stringers were used in the current wing. Castro (2010)
also optimized wing rib geometry and its position. In the current work his optimized geometry is used in order to optimize
its composite structure stiffness. The wing can be visualized in Fig. 3. The maximum takeoff weight of this aircraft is
100,000lbs. Consequently 50,000 lbs will be supported by each wing. In the current model two load cases will be used,
a 3G positive (667 kN) and a 1G negative (222 kN). Also, for simplification a trapezoidal load distribution will be used
instead of an elliptic. The pressure applied is a result of the total load divided by the upper or lower skin area.
Figure 2. Representation of wing loading simplification, Lemos (2013)
Each color depicted in Fig. 3 corresponds to an elastic region of the finite element model, where thickness and
material properties will be optimized. As discussed previously, the elastic region of the current wing needs four lamination
parameters to be defined. In this work, using the idea of the IDW, a few property regions on upper and lower skin will
be chosen to have its lamination parameters used as design variables. All the other regions will have their lamination
parameters interpolated by the IDW equations.
Also, the thicknesses will be optimized independently for all elastic regions. Having in mind the need of maximizing
the buckling efficiency at the same time as reducing the total weight of the wing, the optimization will be divided into two
steps. Each step will have its own design variables and constraints as well as, its specific design objective.
4. Two-step approach
At the first optimization step the stiffness matrix of the wing constitutive material will be enhanced in order to maxi-
mize its buckling eigenvalue. Using lamination parameters as design variables in a few property regions it is possible to
interpolate all the other regions stiffness matrices using a few design variables to optimize the entire wing structure. In
Victor N Capacia, Jose A. Hernandes and Saullo G. P. Castro
Composite panel optimization using lamination parameters and inverse distance weighting interpolation
Figure 3. Property regions
this manner, the focus is perform a computationally cheap optimization step maximizing the wing buckling eigenvalue
while constraining the laminate feasible region. Two different model configuration with two different number of lamina-
tion parameter design variables will be optimized in this first step. The main idea of testing two different models is to
understand the influence of the number of design variables and its distribution through each skin on a big structure linear
buckling load multiplier, using a reduced number of design variables.
Figure 4. Lamination parameters design variables distribution
Spars 1 and 2 will also have its structure optimized in the same manner. A few property regions will be chosen to have
its lamination parameters optimized and the remaining regions will have lamination parameters interpolated. Spar 3 will
not be optimized. All the ribs structures will also have its lamination parameters optimized, however, having in mind its
small dimensions, nothing will be interpolated. In Fig. 4 it is possible to visualize two models that will be verified. In
the figure, regions with red color represent design variable region. While grey color illustrates interpolating regions. Case
1 represents only three independent property regions on each skin chordwise and also three span-wise, with a total of 9
independent regions on each skin. Three regions on spar 1 and spar 2 and one region per rib. Resulting in a sum of 49
regions. In case 2, four independent property regions are used chordwise on each skin. No modification on ribs and spars
design regions were made from case 1. Also, case 2 adds two rolls of independent regions span wise, summing a total of
71 design regions on the entire wing, of which, 20 of them in each skin.
In a summarized way, the first step optimization problem statement holds Eq. (9) form.
Max λ
s.a :
1V1D+1
4(V1A+ 1)30
V1D1
4(V1A+ 1)3+ 1 0
1V3D+1
4(V3A+ 1)30
V3D1
4(V3A+ 1)3+ 1 0
var. :1V1A+1
1V3A+1
1V1D+1
1V3D+1
(9)
25th ABCM International Congress of Mechanical Engineering (COBEM 2019)
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As a manner of translating the step 1 optimization result in the maximization of the buckling load multiplier, a second
step optimization is going to be performed. The focus is to take advantage of the first optimization and reduce the wing
weight through a sizing step. The design objective is the minimization of weight while restraining wing elements from
buckling. Plate elements thicknesses are going to be used as design variables for this step. Differently than the previous
approach no interpolation is going to be used and all property regions within the wing are going to receive its own design
variable. The main objective at this stage is to investigate if the step 1 approach is sufficiently capable of defining feasible
laminates for a wide structure while reducing the numbers of design variables to maintain a computationally cheap and
efficient optimization. The final weight is also going to be a good manner of comparing the current work results with
previous works performed by several authors with the same wing geometry and loads. A total of 143 thicknesses design
variables are implemented at this stage. They are divided as:
52 design variables for the upper skin;
52 design variables for the lower skin;
25 design variables for the ribs (1 for each rib);
14 design variables for the spars (7 for each spar);
The problem statement of the second step optimization is as follows:
M in W eight
s.a :
1
λ<1
var. : 3mm t20mm
(10)
5. Runs and analysis
This section aims to expose all results obtained in a resumed manner and debate some conclusions that were taken
from it. After this preview a best solution case is going to be chosen in order to deeply expose its own results in the
next subsections. For both model configuration a total of 4 optimization runs were performed. For each design variable
configuration, the optimization was investigated for the power function owning the values of p = 1 and p = 2. Table 2
holds the resume of results.
Table 2. Step 1 results summary
Model 1 Model 2
Power function 1 2 1 2
Design Cycles 36 42 26 38
λinitial 2.16 2.16 2.16 2.16
λfinal 3.44 3.40 3.38 3.41
According to Tab. 2, it is easy to see that Model 1, using p value of 1 was the one with the best final value. These results
are counter intuitive, having in mind that it is common to think that the more design variables a problem have, the best the
result. However, what recent optimization publications shows, is that, typically, the optimization performance in finding
the optimum solution increases until an optimum point of number of design variables, after which, it only decreases
with the increase of design variables. Quadros (2017), with his Lagrange interpolation faced the same conclusion. More
variables trends to add more non-linearities, which, in turn, can add more local minimum to the objective function when
the problem is not convex by its nature. And even if the increase of number of design variable does not affects convexity,
it can change the objective function curvature, also decreasing convergence capabilities of the solver. Furthermore, a
problem with more non-linearities is hard to converge, either in gradient-based algorithm or not. The results also shows
that, in this case, Model 1, with less design variables was capable of adapting better to the current buckling problem and,
despite converging faster, Model 2 was not able to hold the best results. Besides that, it was noticed a small difference
between the results for the interpolation power functions, p = 1 and p = 2. Further analysis on this fact is going to be
drawn in the next subsection.
Focusing on performing a deeper investigation of the influence of the power function in the interpolation method, a
Python algorithm was wrote in order to be able to see the interpolation function behavior. Figure 5 shows the interpolation
function acting on the upper panel, upon the result for lamination parameter V1A for Model 1 with p = 1.
Victor N Capacia, Jose A. Hernandes and Saullo G. P. Castro
Composite panel optimization using lamination parameters and inverse distance weighting interpolation
Figure 5. Contour of the interpolation function of V1Aresults at upper panel of Model 1 with p= 1
Figure 6. Contour of the interpolation function of V1Aresults at upper panel of Model 1 with p= 2
Yet, in the current model, properties are interpolated based on the location of a center element. As a consequence,
small variations of the power function do not have a big influence on the model interpolating results. As an example of
that, Fig.7 shows how Fig. 5 interpolating function really appears in the model results.
Figure 7. Contour of real V1Aresults for the upper panel of Model 1 with p= 1
The above figure shows the exact result that Model 1 holds for p = 1. With the change of p = 1 to p = 2 would
have a tiny influence on the final result for the same values. Yet, Model 1 with power function of 1 was able to find a
load multiplier 1% bigger than when using a power function of 2. It is easy to conclude that an equilibrium between
number and location of design variables has to be found, as well as, its combination with the power function value for the
algorithm to find the optimum solution. At this stage, in Step 2, a sizing optimization is performed in order to quantify
how much the found lamination parameters could reduce the total weight of the wing. Table 3 resumes the final weight
obtained for each group of structure, as well as, the total wing weight for Model 1 with interpolating power function of 1.
Table 4 resumes the current optimization results. It is easy to conclude that Step 2 is much more computationally
expensive than Step 1. With a total of 1127 design cycles with an elapsed time of 6 hours.
Table 5 resumes the final weight obtained for each group of structure, as well as, the total wing weight for Model 2
with interpolating power function of 1.
Is possible to perceive a reduction in both upper and lower panel weights. That can be explained by the increase in
the total number of design variables in each panel from 9 to 20. However, the optimizer seemed not to found such good
25th ABCM International Congress of Mechanical Engineering (COBEM 2019)
October 20-25, 2019, Uberlândia, MG, Brazil
Table 3. Components weight
Component Weight [kg]
Upper skin 319.73
Lower skin 203.17
Ribs 133.94
Spar 1 23.97
Spar 2 17.11
Spar3 4.46
Total 702.38
Table 4. Optimization summary for Model 1, Step 2, with p=1
Model 1
Power function 1
Design Cycles 1127
Initial weight 1801.1kg
Final weight 702.38
Table 5. Component weight
Component Weight [kg]
Upper skin 315.10
Lower skin 198.34
Ribs 148.76
Spar 1 26.22
Spar 2 19.12
Spar3 4.46
Total 712.0
results for the ribs and spars. In comparison to configuration 1 both total spar and ribs weight increased.
6. Conclusion
In this work, an interpolation parametrization was presented for a linear buckling design optimization of a composite
wing structure through lamination parameters theory. The stated approach focused on investigating the potential of reduc-
ing the number of design variables while maximizing the critical buckling load and, at the same time, reducing the total
weight of a complex aeronautical structure. By the use of the proposed method, it was possible to increase the buckling
load of a composite wing up to 159.26%, starting with a quasi-isotropic laminate. It was also possible to reduce its weight
to almost 38.54% of its initial total weight. The analysis was extended to a second configuration case where more panel
design variables were included. It was concluded that, contrary to what was expected, more design variables added more
complexity to the optimization model. Typically, the optimization performance in finding the optimum solution increases
until an optimum number of design variables, after which, it only decreases with the increase of design variables. More
variables tends to add more non-linearities, which, in turn, can change the objective function curvature, or create local
minima, thereby, decreasing convergence capabilities of the solver.
Also, for each panel the behavior of the interpolating function was evaluated. Two types of power function values were
used on the inverse distance weighting interpolation method, p = 1 and p = 2. The interpolating surface became smoother
with the increase of p value up to a point where local values generates steps in its surface. With higher values of p, the
interpolated points get to be more influenced by local values and suffer less influence of the further points. However, the
current model interpolates properties, not elements. The consequence is that a few elements gets the interpolated results
based on the location of a center element. Hence, small variations of the power function does not have a big influence
on the model interpolating results. Yet, the configuration holding a p value of 1 showed the best results. The strategy to
use lamination parameters as design variables, had the purpose of obtaining an objective function with better behavior
and more convex. Yet, local minima were identified such that different starting points led to distinct optimum solution.
Having in mind the focus of understanding the behavior of the optimization algorithm, it was decided to start always
from the same point in which V1A = V3A = V1D = V3D = 0. This combination represents a quasi-isotropic laminate
and, at this manner, a fair comparison between the model could take place. All design variables respected the Grenested
Victor N Capacia, Jose A. Hernandes and Saullo G. P. Castro
Composite panel optimization using lamination parameters and inverse distance weighting interpolation
and Gudmunson (2013) feasible region imposed. Also, the fact that the feasible region equation results in a continuous
domain, all the interpolated lamination parameters also necessarily respected the constraint.
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8. RESPONSIBILITY NOTICE
The authors are the only responsible for the printed material included in this paper.
... More complex fiber paths can be modeled using the Lagrange polynomial scheme proposed by [18,30] for rectangular domains and by Capacia et al. [31,32] using an arbitrary distribution of control points. Prescribed fiber angles at the M×N reference points placed over the plane, as shown in Fig. (1b), determine the polynomial coefficients of the fiber path. ...
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