Bending elastic stiffness constants of the tested anisotropic plate and the computed errors.
Keywords
1. Introduction
Nowadays, the demand and the necessity for the use of materials with specificcharacteristics have increased in many engineering fields. Due to this necessity of making new materials, composite materials have been an alternative, or maybe the unique option, to attempt a large number of design requirements suchas high strengthtoweight ratio,high resistance to mechanical shocks, chemical attacks, corrosion, and fatigue, that cannot be obtained only from the commonly used structural materials (metals, ceramics, polymers and wood). Because of this, their applications are present in the main industries such as aerospace, automotive, marine, and sportive
Thanks to their flexibility characteristicsthere are many combinations and arrangements and, consequently, constitutive properties, that are possible to be achieved. This particularity represents one of the main advantages of these materials. Nevertheless, some factors related to arrangements as the number of layers and the orientation of the fibers can introduce abehavior called anisotropy that, in the most of the cases, is not required. The anisotropy makes the structural analysis more complex due to increasing the number of independent variables, as for example, the number of bending and extensional elastic stiffness constants.
Recently, wide part of works presented by scientific literature whose goal is to identify constitutive parameters of materials (these being composites or not) is based in the called “inverse problems”. Experimental data such as geometry, resultant forces and strain (or stress) fields are used as input data, and the unknown variables are the required constitutive parameters. In general, the solution is basically associated to two methods: iterative (or also called indirect methods) and noniterative (or also called of direct methods). The first one is related to optimization problems where the design variables are constitutive parameters and the objective function represents, in general, a residue (or error) between experimental and numerical (generally obtained by finite elements) data. For the numerical simulations, it is considered structures that have the same geometrical characteristics and boundary conditions of the real ones. For each step, required parameters are checked out, and, the optimum represents the iteration whose residue (objective function) has its lowest value. Unlike indirect methods, the direct methods are ones where the required parameters are computed from the solution of constitutive equation(s) that are functions of these parameters.
In general, according to the type of experimental test, it is possible to separate the methods of elastic property identification in two categories: static (destructives and nondestructives) and dynamic methods (nondestructive), as shown in Fig.1. A large number of identification techniques that use data from these categories of tests have been proposed, especially ones dedicated to composite materials. It is possible to say that these techniques identify effective properties of the entire material. The way as each formulation is built, and, the adopted procedures and devices are the main differences among the many proposed methodologies.
Static tests with monotonic load are experimental tests that were more commonly used in the last years, and maybe the simplestones, for this material property identification. Despite the simplicity of these tests, some aspects render them less attractive than dynamic tests, such as the fact of requiring a number of samples with fiber orientations according to standard norms, for example, American Society for Testing and Materials (ASTM), which, in the most of cases, are not in accordance to the real characteristics of the required material. Furthermore, some variables difficult of controlling during the tests can contribute to worsen the experimental results, e.g., the presence of nonuniform stress fields near the ends of the sample from the clamped boundary conditions. For these reasons, dynamic tests have been considered an interesting alternative. In general, they are tests that combine experimental data with numerical methods, and allow the identification of elastic constants from only one unique sample or even from composite material part. Sample is usually thin plate (that reflect Kirchhoff’s hypotheses), cylindrical shell, or beam. In many cases, input data of the numerical methods are natural frequencies and/or mode shapes.
Many authors have proposed to identifythe elastic constants by iterative procedures adopting RayleighRitz and finite element as numerical methods. The difference among the identification techniques based on these iterative procedures is basically in the way as the optimization problem is formulated, for example, the type of adopted search method to find the minimal, the boundary conditions, the geometric characteristics of the samples, the type of anisotropy of the test material, the type of experimental devices, and the numerical method used to compute the mode shapes (or operational modes) with their respective frequencies (Deobald& Gibson, 1988; Pedersen &Frederiksen, 1992; Lai & Lau, 1993; Ayorinde& Gibson, 1995; Rikards&Chate, 1998; Ayorinde& Yu, 1999, 2005; Rikards et al., 1999; Bledzki et al., 1999; Hwang & Chang, 2000; Araujo et al., 2000; Chakraborty&Mukhopadhyay, 2000; Rikards et al., 2001; Lauwagie et al., 2003; Lauwagie et al., 2004; Lee &Kam, 2006; Cugnoni et al., 2007; Bruno et al., 2008; Pagnotta&Stigliano, 2008; Diveyev&Butiter, 2008a, 2008b).
In works that do not use iterative process, natural frequencies and mode shapes, or operational frequencies and modes, are input data of an algorithm based on the differential equation that governs the transversal vibration of sample in a specific direction and under specific boundary conditions (Gibson, 2000; Alfano&Pagnotta, 2007). In this methodology, it can be included the use of Virtual Fields Method, VFM (Grédia, 1996, 2004; Grédiac& Paris, 1996; Grédiac&Pierron, 1998, 2006; Pierron et al., 2000, 2007; Pierron&Grédiac, 2000; Grédiac et al., 1999a, 1999b, 2001, 2006; Giraudeau&Pierron, 2003, 2005; Chalal et al., 2006; Toussaint et al., 2006; Avril&Pierron, 2007; Pierron et al., 2007; Avril et al., 2008; Giraudeau et al., 2006). For VFM, weighting functions are the called virtual fields. Due to sensitivity to experimental errors and the presence of noise during the dynamic testing, it was proposed the use of specific virtual fields named “special virtual fields” (Grédiac et al., 2002a, 2002b, 2003). In order to decrease the noise contribution, the use of more accurate experimental modal analysis techniques or the application of some signal smoothing (or filtering) technique is mandatory.
The majority of works identifies only the bending stiffness matrix or directly the engineering elastic constants. However, the extensional elastic stiffness matrix is also needed to model composite materials under multiaxial loads. In general, these stiffness matrices are independent. The extensional stiffness matrix relates the inplane resultant forces to the midplane strains, and, the bending stiffness matrix relates the resultant moments to the plate curvatures. In a laminate composite, if only the stacking sequence of layers is changed, the bending matrix is changed but the extensional matrix remains the same. In other words, different laminates can have different bending stiffness matrices and the same extensional stiffness matrix. It is not possible to obtain the extensional matrix from the bending matrix. It will be possible only if the stacking sequence of layers and their thickness are known and, also, if the material is the same for all lamina.
Sometimes, it is more convenient to use effective laminate engineering constants rather than the laminate stiffness. These effective laminated engineering constants may be easily obtained from the extensional elastic constants. However, due to difficulties on experimental inplane modal analysis, such as the necessity of using specific devices to measure inplane displacements and to excite high frequencies, the identification of extensional elastic stiffness constants using modal testing is less attractive. The main challenge to perform inplane vibration testing is the excitation and measurement ofonly inplane and not outofplane vibration modes. Today there are some new techniques that are suitable for this kind of problems, for example, the excitation by piezoelectric (PZT) and measurements by digital image correlation.
In this chapter, a review about the VFM applied to compute bending elastic stiffness constants proposed by Grédiac& Paris, 1996 is presented. Furthermore, a formulation based on the VFM is proposed in order to identify the extensional elastic stiffness matrix ofKirchhoff’s thin plates. The linear system of equations that provides the required elastic constants is obtained from differential equations that govern the forced vibration of anisotropic, symmetric and nondamped plates under inplane loads. The common procedures to find the weak form (or integral form) of these equations are applied here. The correct choice of weighting functions (which are the virtual fields) and mode shapes representsa key characteristic to the accuracy of the results. Numerical simulations using anisotropic, orthotropic, quasiisotropic plates are carried out to demonstratethe accuracy of the methodology.
2. Identification of elastic constants using VFM
2.1. Review of the Virtual Fields Method  VFM
The VFM has been developed for extracting constitutive parameters from fullfield measurements and it is associated to problems of identification of parameters from constitutive equations. Two cases are clearly distinguished: constitutive equations depending linearly on the constitutive parameters and nonlinear constitutive equations. The type of constitutive equations is chosen
Mathematically, the VFM is based on the principle of virtual work and can be written as:
where
where
It is possible to see in Eq. (3) that each virtual field originates a new equation involving the constitutive parameters. The VFM relies on this important property. It is a method based on setting virtual fields that provide a set of equations. This set of equations is used to extract the required unknown constitutive parameters. The correct choice of the virtual fields that combine to actual fields in Eq. (3) is the key issue of the method. Their number and their type depend on the nature of
2.2. Review of the identification method of bending elastic stiffness matrix
The method proposed by Grédiac&Paris,1996, consists of obtaining elastic constants based on the partial differential equation that governs the transversal vibration of an anisotropic thin plate (Kirchhoff’s plate). This equation is given by:
where
After some mathematical manipulations in Eq. (4), Grédiac& Paris,1996 obtained a linear system in which the unknown variables are the elastic constants. Briefly, the sequence of operations is as follows: (a) multiply both sides of Eq. (4) by an arbitrary weighting function; (b) integrate twice by parts along the plate domain; (c) eliminate the boundary integrals by applying the freeedge boundary conditions; (d) decompose the displacement function
where indices
Eq. (5) can be represented in matrix form as:
where, considering
2.3. Identification method of the extensional elastic stiffness matrix
In the general case of composite laminates, each lamina is assumed to have orthotropic material properties. After the assembly, the behavior can be anisotropic due to the interaction of different laminas. Considering a plate under plane state of stress and using Hooke’s generalized law, stresses can be integrated over its thickness yielding the following forcedeformation equations:
where N and M are vectors that contain normal forces and resultant moments, respectively, A is the extensional elastic stiffness matrix, B is the coupling elastic stiffness matrix (B is a null matrix in the case of a symmetric laminate), D is the bending elastic stiffness matrix, ε and κ are vectors that contain middle plane linear strains and rotations, respectively. Considering a symmetrical (B = [0]) and fully anisotropic laminate under freeedge inplane vibration (the plate is not under bending) and using the equilibrium relations, the following equations can be written:
where
where
where
where
where
where
Multiplying Eq. (17) by a weighting function
Multiplying Eq. (19) by a weighting function
Substituting Eqs. (20) and (21) into Eq. (15), and reorganizing the terms, one can write:
Now, if freeedge boundary conditions are considered, boundary integrals of Eq. (22) vanish. Considering that the plate is vibrating, functions
where
Now, if the same previous mathematical procedures used in Eq. (10) are used in Eq. (11), one obtains:
2.3.1. Choice of the weighting functions
Eqs. (25) and (26) are theoretically valid for isotropic, orthotropic, or anisotropic plates, provided that the laminate is symmetrical. As it can be seen, the function
W(x, y) = x^{2}, which applied to Eqs. (25) and (26) provides the following integral equations:
W(x, y) = y^{2}, which applied to Eqs. (25) and (26) provides the following integral equations:
W(x, y) = 2xy, which applied to Eqs. (25) and (26) provides the following integral equations:
Defining:
Eqs. (34)(38) can be written as:
or, in matrix form:
Eq. (39) can also be rewritten in a compact form as:
where, considering
from where the extensional elastic constants
3. Results and comments
A commercial finite element code (ANSYS 11.0) was used to give particular mode shapes and their corresponding natural frequencies from both inplane and outofplane numerical modal analysis. Element SHELL99 was used and plates under freeedge boundary conditions were considered.
To exemplify the method proposed by Grédiac and Paris (1996), it was used an anisotropic plate with dimensions 0.450 x 0.350 x 0.0021 m and density 1500 kg/m^{3}. It was used a laminate with 8 plies, [0 45 90 135]_{S}, and the following engineering elastic constants by ply:
Table 1 shows the bending elastic stiffness constants computed using the engineering constants and the classical theory of laminates, and it also shows the errors computed after applying the identification method. As can be observed, the technique is able to find very satisfactory results when it is used the correct modes. The problem of this technique is the high sensitivity to noise presence because of secondorder derivatives. More results and comments about this method can be found in Grédiac& Paris, 1996.
In order to verify the accuracy of the extensional elastic stiffness identification method, it was used six graphite/polymer symmetric laminated plates (Table 2): a fully anisotropic with all
As can be seen in Table 2, the constants
Bending elastic constants  N x mm  Errors (%) 

64363.9  0.02 

24155.8  0.04 

8875.1  0.02 

10032.7  1.22 

6019.6  0.63 

6019.6  0.64 
The key point of this technique of identification is related with the correct choice of mode shapes together to the weighting functions (virtual fields). The correct mode shapes are called here by “suitable modes” and the correct combination between these modes and the weighting functions are called by “suitable combinations”. The identification of the suitable modes is not difficult, as it will be shown in the next topics. But, the suitable combinations are more difficult because they depend on the type and the geometry of the material. Fortunately, there are some aspects that help finding the best choice. Unlike the bending stiffness identification method originally proposed, for this method there are a lot of modes and suitable combinations that give satisfactory results.
10^{8} [N/m] 
Aniso [90 0 0 45]_{S} 
Ortho I [90 0 90 0]_{S} 
Ortho II [0]_{4S} 
Ortho III [30]_{4S} 
Ortho IV [30 30]_{2S} 
Quasiiso [90 45 0 45]_{S} 

2.7865  2.5186  4.6724  2.7840  2.7840  1.9776 

0.3610  0.0905  0.0905  0.9020  0.9020  0.6315 

0.2692  0  0  1.4012  0  0 

1.7096  2.5186  0.3648  0.6301  0.6301  1.9776 

0.2692  0  0  0.4641  0  0 

0.4025  0.1320  0.1320  0.9436  0.9436  0.6730 
Table 3 shows some errors computed for the anisotropic plate, rectangular and square. It was considered only the first fifteen inplane modes shapes. Anisotropic plates, in general, give very satisfactory results using the combinations among suitable modes. This factor can be justified by the fact of these combinations be hardly involved with all required extensional elastic constants
The numerical contribution of each mode to the computation of a specific constant cannot be jeopardized by numerical contribution of another mode during the solution of the system given by Eq. (40).The suitable modes are those that when associated with weighting functions do not null or give very low values for integrals of the right (K matrix) and/or left (C matrix) hand sides of Eq. (40). The suitable combinations are one composed by suitable modes and that give more accurate results. In the majority of the cases, combinations using a higher number of suitable modes can be suitable combinations. According to Table 3 is possible to see that using combinations with only two suitable modes very satisfactory results can be obtained. Satisfactory results would also be obtained using combinations with any modes since the number of suitable modes among all used modes is higher than nonsuitable modes. But the accuracy of these results cannot be guaranteed for all combinations.
In general, for the orthotropic and isotropic materials is more difficult to find the suitable combinations when it is compared to fully anisotropic materials. It is necessary to take care to correctly identifying the combinations that give the best results. In these types of materials not all combinations are among suitable modes that can be considered as being suitable combinations. According to values found to terms of the K and C matrices, Eq. (40), and using combinations among suitable modes, it is possible to see the following types of systems:
Anisotropic rectangular plate – suitable modes: 2, 3, 6, 9, 10, 13, and 14  
Suitable combinations 
Errors (%)  







2369  2.20  1.95  1.56  1.89  1.46  0.78 
236  0.04  0.20  0.21  0.93  0.79  0.97 
239  0.37  2.63  0.27  0.78  0.93  0.28 
269  0.30  2.46  0.17  1.45  1.03  0.02 
369  0.17  1.77  0.11  0.97  1.09  0.21 
23  0.00  0.32  0.24  0.78  0.53  1.08 
69  0.01  2.06  0.11  1.62  1.50  1.51 
Anisotropic square plate – suitable modes: 2, 3, 6, 9, 10, 13, and 14  
Suitable combinations 
Errors (%)  







2369  3.73  3.41  2.23  3.21  1.88  1.06 
236  0.20  0.62  0.23  0.86  0.09  1.08 
239  0.41  3.86  0.54  1.36  0.77  0.50 
269  0.35  3.43  0.44  2.00  1.61  0.13 
369  0.24  2.59  0.14  1.66  0.84  0.30 
23  0.04  0.06  0.17  1.02  0.97  1.49 
69  0.08  2.66  0.25  2.70  1.08  1.93 
where
Table 4 shows errors computed to some suitable combinations for these orthotropic plates. As can be seen, very satisfactory results can be obtained using correct combinations of modes. For this type of orthotropy, it can be more difficult to compute an accurate value for constant
Ortho I rectangular plate – suitable modes: 2, 3, 6, 8, 11, and 14  
Suitable combinations 
Errors (%)  Differences (N/m)  







23681114  0.14  3.80  0.12  1.22  2.7 x 10^{4}  0.8 x 10^{4} 
2368  0.11  2.90  0.24  0.74  2.5 x 10^{4}  0.6 x 10^{4} 
36811  0.17  1.58  0.31  1.67  1.4 x 10^{4}  0.5 x 10^{4} 
2811  0.17  0.29  0.23  1.46  1.5 x 10^{4}  0.6 x 10^{4} 
6811  0.18  0.89  0.34  1.91  4.8 x 10^{4}  1.0 x 10^{4} 
38  0.02  5.75  0.91  0.13  1.4 x 10^{4}  8.4 x 10^{3} 
811  0.25  0.42  0.40  2.17  2.6 x 10^{4}  1.1 x 10^{4} 
Ortho I square plate – suitable modes: 2, 3, 6, 7, 11, and 12  
Suitable combinations 
Errors (%)  Differences (N/m)  







23671112  0.09  5.81  0.05  1.89  2.7 x 10^{4}  1.6 x 10^{4} 
23611  0.10  5.09  0.06  1.92  4.4 x 10^{4}  8.7 x 10^{4} 
36712  0.38  2.33  0.08  2.92  6.4 x 10^{4}  1.7 x 10^{4} 
2611  0.13  5.47  0.11  2.51  1.0 x 10^{5}  1.8 x 10^{4} 
2712  0.22  5.37  0.04  1.95  1.1 x 10^{5}  0.5 x 10^{5} 
211  0.02  9.04  0.18  0.56  1.3 x 10^{4}  1.9 x 10^{4} 
611  0.18  5.63  0.30  3.46  1.1 x 10^{5}  2.7 x 10^{4} 
Table 5 shows the errors computed to some suitable combinations for these orthotropic plates. Using correct combinations very satisfactory results can be obtained. Similar to ortho I plate, for this type of orthotropy, constant
Ortho II rectangular plate – suitable modes: 2, 3, 5, 9, 10, 11, 14, and 15  
Suitable combinations 
Errors (%)  Differences (N/m)  







235  2.77  6.06  0.08  0.38  2.4 x 10^{4}  0.5 x 10^{4} 
111415  1.14  2.15  0.92  1.15  2.3 x 10^{5}  0.1 x 10^{5} 
23  2.36  0.57  0.19  0.72  1.7 x 10^{4}  0.1 x 10^{4} 
210  2.10  0.82  0.60  2.42  2.0 x 10^{5}  0.4 x 10^{5} 
214  2.04  5.22  0.37  1.12  3.9 x 10^{4}  1.2 x 10^{4} 
1114  1.64  2.14  0.87  0.98  1.8 x 10^{5}  0.1 x 10^{4} 
Ortho II square plate – suitable modes: 2, 4, 5, 9, 10, 12, 13, and 14  
Suitable combinations 
Errors (%)  Differences (N/m)  







245  4.13  8.63  0.39  0.45  3.5 x 10^{4}  0.1 x 10^{4} 
2414  2.43  0.78  0.72  2.89  4.2 x 10^{4}  1.1 x 10^{4} 
24  3.35  0.10  0.21  1.25  3.2 x 10^{4}  0.1 x 10^{4} 
214  1.63  1.59  1.30  4.72  1.1 x 10^{5}  0.3 x 10^{5} 
514  2.67  13.54  2.21  2.82  5.6 x 10^{4}  1.0 x 10^{4} 
914  2.12  62.05  0.85  3.89  1.1 x 10^{5}  0.2 x 10^{5} 
For these plates, rectangular and square the computed matrices K and C, Eq.(40), are full matrices, similar ones of the anisotropic plates. Thus, the majority of combinations among suitable modes are suitable combinations. Combinations that are not suitable present high errors for all constants, what, consequently, make them easy to be identified. Table 6 shows the errors computed to some suitable combinations. Using correct combinations very satisfactory results can be obtained.
Ortho III rectangular plate – suitable modes: 2, 3, 6, 8, 10, 12, and 13  
Suitable combinations 
Errors (%)  







3612  0.25  0.51  0.10  1.05  0.69  0.99 
81012  4.26  5.42  3.67  3.43  4.60  5.62 
23  0.85  1.72  1.22  1.39  0.10  0.96 
36  0.63  0.32  0.04  0.22  0.53  0.14 
612  0.71  1.06  0.05  1.39  1.19  1.30 
1013  4.42  2.05  0.71  4.07  5.21  0.63 
Ortho III square plate – suitable modes: 2, 4, 6, 8, 9, 12, and 13  
Suitable combinations 
Errors (%)  Differences (N/m)  







246  2.62  3.94  3.18  0.59  2.02  3.23 
248  1.20  2.40  1.64  1.27  0.11  1.19 
24  2.35  1.11  0.02  5.46  4.26  2.90 
26  0.76  2.76  3.52  0.01  0.53  2.90 
213  1.75  2.75  1.51  1.92  0.33  1.25 
1213  3.08  2.41  2.63  0.53  1.85  2.54 
Table 7 shows errors computed to some suitable combinations. Using correct combinations, very satisfactory results can be obtained.
Ortho IV rectangular plate – suitable modes: 2, 4, 5, 8, 11, 12, and 14  
Suitable combinations 
Errors (%)  Differences (N/m)  







245  1.52  2.15  1.28  1.54  1.7 x 10^{4}  0.6 x 10^{4} 
248  1.50  2.09  0.83  1.14  1.4 x 10^{4}  0.9 x 10^{4} 
51112  0.27  0.09  0.04  1.11  1.2 x 10^{4}  1.9 x 10^{4} 
24  1.18  1.83  1.02  2.87  1.2 x 10^{4}  0.4 x 10^{4} 
25  0.49  1.01  1.01  1.41  1.8 x 10^{4}  0.7 x 10^{4} 
511  0.90  0.37  0.04  1.05  3.1 x 10^{4}  0.2 x 10^{4} 
Ortho IV square plate – suitable modes: 1, 4, 6, 9, 11, 12, and 14  
Suitable combinations 
Errors (%)  Differences (N/m)  







1612  0.16  0.30  0.13  3.07  1.7 x 10^{4}  2.5 x 10^{4} 
6912  0.18  0.35  0.30  2,68  1.3 x 10^{4}  4.8 x 10^{4} 
91214  0.18  1.87  0.40  0.68  6.2 x 10^{4}  8.1 x 10^{4} 
112  0.59  0.26  0.74  2.79  2.0 x 10^{4}  2.9 x 10^{4} 
411  0.94  1.55  2.77  0.23  2.2 x 10^{4}  0.8 x 10^{4} 
1214  0.26  2.17  0.19  0.90  9.7 x 10^{4}  9.4 x 10^{4} 
Table 8 shows errors computed to some suitable combinations. Using correct combinations, very satisfactory results can be obtained.
Quasiiso rectangular plate – suitable modes: 1, 4, 6, 8, 11, 12, and 14  
Suitable combinations 
Errors (%)  Differences (N/m)  







146  0.60  1.47  0.19  0.19  9.1 x 10^{3}  4.7 x 10^{3} 
148  0.09  0.80  0.11  0.96  7.4 x 10^{3}  9.5 x 10^{3} 
1411  0.33  2.04  0.26  0.08  1.4 x 10^{3}  8.1 x 10^{3} 
14  0.25  0.54  0.29  1.93  5.7 x 10^{4}  7.1 x 10^{4} 
16  0.28  0.91  1.77  0.00  1.0 x 10^{4}  0.1 x 10^{4} 
48  0.73  0.39  0.05  0.95  1.1 x 10^{4}  0.3 x 10^{4} 
Quasiiso square plate – suitable modes: 2, 3, 7, 8, 10, and 11  
Suitable combinations 
Errors (%)  Differences (N/m)  







237  0.37  1.33  0.35  0.91  4.7 x 10^{5}  4.7 x 10^{5} 
238  0.32  1.36  0.33  0.97  4.4 x 10^{5}  4.4 x 10^{5} 
21011  0.07  3.29  0.45  0.81  2.3 x 10^{5}  1.1 x 10^{5} 
23  0.33  0.79  0.34  2.68  2.2 x 10^{4}  1.3 x 10^{4} 
37  0.56  1.73  0.43  0.80  6.1 x 10^{5}  5.2 x 10^{5} 
811  0.89  3.05  0.06  1.34  1.1 x 10^{5}  1.2 x 10^{5} 
Figs. 3 to 14 show the fifteen first inplane mode shapes to all the analyzed plates: rectangular and square geometry.
4. Conclusions
The identification of elastic properties using VFM has shown to be a very efficient technique since the correct combinations among mode shapes and weighing functions are used. This factor is the key point to find the correct results. However, the identification of these suitable combinations is not so simple in some situations, mainly to the extensional elastic stiffness identification method. Fortunately, there are some characteristics that can help to find such combinations, as it was shown here. A great advantage of this method is related to the large number of possibilities to make combinations able to give very satisfactory results.
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