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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 =U4−U3cos4θ

Q22 =U1−U2cos2θ+U3cos4θ

Q16 =1

2U2sin2θ+U3sin4θ

Q26 =1

2U2sin2θ−U3sin4θ

Q66 =U5−U3cos4θ

(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=t(Γ0+ Γ1V1A+ Γ2V2A+ Γ3V3A+ Γ4V4A)(4a)

B=t2

4(Γ1V1B+ Γ2V2B+ Γ3V3B+ Γ4V4B)(4b)

D=t3

12(Γ0+ 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

0−U20

0 0 0

(5b)

Γ2=

0 0 U2

2

0 0 U2

2

U2

2

U2

20

(5c)

Γ3=

U3−U30

−U3U30

0 0 0

(5d)

Γ4=

0 0 U3

0 0 −U3

U3−U30

(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)3−1≤ViD ≤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 :

1−V1D+1

4(V1A+ 1)3≥0

V1D−1

4(V1A+ 1)3+ 1 ≥0

1−V3D+1

4(V3A+ 1)3≥0

V3D−1

4(V3A+ 1)3+ 1 ≥0

var. :−1≤V1A≤+1

−1≤V3A≤+1

−1≤V1D≤+1

−1≤V3D≤+1

(9)

25th ABCM International Congress of Mechanical Engineering (COBEM 2019)

October 20-25, 2019, Uberlândia, MG, Brazil

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 ≤t≤20mm

(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

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

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.