Fuzzy Logic Method for Predicting the Effect of Main Fabric Parameters Influencing Drape Phenomenon

2.1 Materials

Sixty-three woven fabrics with twill and plain weaves are chosen for the preparation of samples in this study. Table 1 shows the maximum, minimum, and average standard deviation of fabric properties and drape parameters used under study. The fabric samples are conditioned in standard atmospheric conditions for at least 24 hours to allow them to attain a stable condition before evaluating the drape. The fabrics are 80% cotton and 20% cotton/polyester. The fabric weight is measured using French Standard NFG07-104 [19].

The method of measuring the bending rigidity is presented below.

2.1.1 Measurement of the bending rigidity

The method used to measure the bending (flexural) rigidity, B, of fabrics in this study is known as Cantilever test, and it is presented in Figure 1. It enables the measurement of the fabric bending rigidity in weft, warp, and skew directions. The test method consists of clamping a horizontal strip of each fabric at one end, slipping and allowing the rest of the strip to hang under its proper weight. When the tip of the strip reaches a plane inclined at 41.5° below the horizontal, the overhanging length is measured and marked (L).

Figure 1

Experimental device for the fabrics properties

The bending rigidity is calculated using the following equation:

(1)B(cg.cm)=0.01×W(gm2)×C3(cm)

where C = L/2 and L is the bending length in cm, W = the fabric surface mass in g/m², and B = the bending rigidity in cg · cm.

The above formula is used in the calculation of the bending rigidities B_warp, B_weft, and B_skew, respectively, in the warp, weft, and skew directions.

2.1.3 The measurement of DC, NN, FDI, and the DDR

Modifications in the system of Cusick drapemeter, as shown in Figure 2, are made by the addition of an achievable MATLAB application to calculate the DC (%) [20].

Figure 2

Experimental device for determination of drape parameters

Image analysis method is used to calculate the real surface of the drape to determine numerical DC. This method is based on the shift from the original color image to the grey level image and finally to the binary one. The method of image processing for the fabric drape measurement is characterized by precision, speed, and the ability to reach various measurements.

The DC can be calculated using the following formula:

(3)DC=drape surface (pixel)−π×(supported disc ray(cm)×c)2π×(total ray(cm)×c)2−π×(supported disc ray(cm)×c)2

where DC is the drape coefficient and c the correlation between the centimeter and the pixel.

The node number is an important parameter linked to the virtual appearance of clothing. The distribution of the different distances of all the points of the outer contour shows the presence of the minimums and the maximums as shown in Figure 3. The number of nodes is the number of maximums found in the curve representing the ray as a function of the angle of rotation.

Figure 3

Presentation of the angle of calculation of rays and the distribution of the rays throughout the drape

The DDR and the FDI are calculated using the equation formula submitted by [21] (Figure 4):

(4)DDR=R−radR−r

where r_ad is the rays mean value of fabric drape.

(5)FDI=rmax−rminR−r

where r_max is the maximum ray of draped sample and r_min is the minimum ray of draped sample.

Figure 4

The various dimensions used in the calculation of drape distance ratio (DDR%) and the folds depth index (FDI%)

2.2 Fuzzy logic method

The concept of Fuzzy logic comes from the observation that the Boolean variable taking only two values: true or false. Thus, several areas own reasoning is applying the Fuzzy logic of modeling, decision support, and the representation of human knowledge. Fuzzy logic allows the modeling of data imperfections and is to some extent similar to the flexibility of human reasoning.

By introducing the notion of degree of belonging into the verification of a condition, we allow one condition to be in another state than true or false. Fuzzy logic thus confers considerable flexibility on the reasoning that uses it, which makes it possible to take into account inaccuracies and uncertainties. One of the interests of Fuzzy logic in formalizing human reasoning is that rules are stated in natural language.

By its numerical aspects, it is opposed to modal logics. It is based on the mathematical theory of fuzzy sets by Zadeh [22], which presents an extension of the theory of classical sets to sets defined imprecisely.

To evaluate the performance of the obtained Fuzzy logic models, three statistical performance criteria, including correlation coefficient (R), mean absolute relative error (MARE), and root mean square error (RMSE), were used in this study. These parameters were calculated for testing (T) and training (tr) phases:

In Equations (6), (7), and (8), y^k_p is the k^th model predicted value, y_k is the observed value corresponding to the inputs that give, y_k is the mean of the observed values, and N is the number of data points. It has to be noticed that a smaller RMSE means a better model accuracy, whereas a higher R means the better the model performs in prediction.

3.1 Selection of inputs/outputs for the Fuzzy inference system

The Fuzzy logic system has a class of inputs and an output class. In Table 2 the minimum average and maximum values of some properties of fabrics are presented to be taken as inputs in the system. The input parameters are divided into two classes: the mechanical parameters that are bending rigidity in the skew, warp, and weft directions and the shear rigidity. Also, there is the construction parameter that is the basis weight. Statistical summary of output variables such as DC, node number, DDR, and the FDI are presented in Table 3.

In Table 4, the selected classes are three in number (low, medium, and high). The choice is made according to the following classification of results found by the Fuzzy-C-mean (FCM) method [23].

3.2 Choice of membership functions

This study section is devoted to the choice of the membership function for the different properties. It is conducted according to the results found in the Fuzzy classification made by Hamdi et al. [23]. The choice of interval data and the upper and lower limits of each class in every parameter are performed according to the results presented in Table 1.

3.2.1 Membership function for input variables

The membership functions for input variables are presented in Figure 5. They are the bending rigidities, shear stiffness, and surface density. It is remarkable that the selected classes are three in number and that the shape used is trapezoidal. The coordinates of each main item belonging to a form with a specific class are presented in Table 1. We note that the presentation of the membership function of each class has its own specificity. In fact, the first class is characterized by three points: the first is the minimum experimental value and the level of the highest membership, the second point is the minimum experimental value of the second class, and the third point corresponds to the maximum experimental value in this class with a zero-affiliation degree. The second class is characterized by four points.

Figure 5

Membership function for input variables: (a) B_skew, (b) B_warp, (c) B_weft, (d) shear rigidity, and (e) surface mass

The first is the minimum experimental value in this class. The second is the maximum experimental value in the first class. The third point is the minimum experimental value in the third class and the fourth point is the maximum experimental value in the second class and having a zero-affiliation degree. Finally, the third class is characterized by three points. The first is the minimum experimental value in this class and has a degree equal to zero. The second point is the maximum experimental value in the second class and the third point represents the maximum value in the third class.

3.2.2 Membership function for output variables

In Figure 6, the membership function for output variables is presented through the DC, NN, DDR, and FDI. The same procedure is followed for the submission of membership functions as cited in the foregoing and according to the data mentioned in Table 4.

Figure 6

Membership function for output variables: (a) drape coefficient (DC), (b) number of node number (NN), (c) draped distance ratio (DDR), and (d) foldss depth index depth (FDI)

3.3 List of rules used

In Table 5, a list of rules linking the input and output variables is presented. Intuitively, it seems that the input variables as such are approximately appreciated by the brain, thus corresponding to the degree of verification of condition of Fuzzy logic. A comprehensive system of rules based on the Fuzzy logic helping to make decisions is called Fuzzy inference system. This list is based on the expertise of some industrialists, professionals, and researchers who are experts in the field of textile and clothing and the experimental tests performed in this work. In this table, the following codes are used:

• “L”: This code stands for the term “Low.”

• “M”: This code stands for the term “Medium.”

• ““H””: This code stands for the term ““High.””

3.4 General structure of the Fuzzy system

The executed Fuzzy calculation system is characterized by the following parameters:

• The type of Fuzzy system = ‘Mamdani’.

• Number of inputs = 5.

• Number of outputs = 4.

• Rules number = 23.

• The method used in the procedure is “And” = ‘min’.

• The method used in the procedure is “Or” = ‘max’.

• The method used in the phase is fuzzification = ‘centroid’.

• Figure 7 shows the general structure of the fuzzy system.

3.5 Fuzzy response surfaces

Fuzzy logic is a method based on the principle of membership degrees. In this context and depending on the rules that link inputs and outputs to the membership of each property of functions with upper and lower limits of each class and each variable, one can draw the evolution of each output depending on the selected inputs. Figure 8 presents the Fuzzy response surfaces of variations of drape parameters according to the bending rigidities in the warp and skew directions.

Figure 7

The general structure of fuzzy system

Figure 8

The Fuzzy surfaces responses from drape parameters according to bending rigidities in the skew and warp direction

According to Figure 7, we can conclude that DC increases with B_skew and B_warp. The number of node increases when B_skew and B_warp decreases. The DDR increases when B_skew and B_warp decreases. The FDI increases with B_skew and B_warp.

3.6 Evaluating the results of calculating drape parameters

To validate the results found by the realized computing system, conducting a test step is necessary. Figure 9 shows the correlation between the experimental values and the values calculated by the Fuzzy system. This analysis is performed to evaluate the effectiveness and reliability of the method based on Fuzzy logic of calculating drape parameters (DC, NN, DDR, and FDI).

Figure 9

The correlation between the experimental values of drape parameters and those calculated by the Ffuzzy system: (a) drape coefficient (DC), (b) number of node number (NN), (c) drape distance ratio (DDR), and (d) folds depth index (FDI).

Throughout this figure, a high correlation values appears. Indeed, the correlation coefficient values of DC, NN, DDR, and FDI are as follows: R_DC = 0.943, R_NN = 0.936, R_DDR = 0.969, and R_FDI = 0.946. From these results, it is possible to conclude that the computing system is considered as efficient to a great extent and that it has reached a success rate. At the same time and according to Table 6, the values of the average of the root mean square error (RMSE) obtained are acceptable because they are less compared to the standard deviation, which proves the acceptance limit of this error for all the characteristics of drape studied. In the footsteps of what has just been mentioned, the mean absolute relative errors (MARE) are very low. From these results, we can conclude that our computing system is considered representative and significant.