**3.1. Materials and porosity measurements**

186 Genetic Programming – New Approaches and Successful Applications

and mean pore diameter (mean flow pore diameter) [35].

that Equation 33 would give as low an absolute deviation as possible.

The evolutionary parameters for modelling by genetic programming were: population size 2000, maximum number of generations to be run 400, probability of reproduction 0.1, probability of crossover 0.8, probability of mutation 0.1, minimum depth for initial random organisms 2, maximum depth for initial random organisms 6, maximum depth of mutation fragment 6, and maximum permissible depth of organisms after crossover 17. The

**parameters** 

Three kinds of pores may be present in needle-punched nonwoven fabrics, namely, closed pores, open pores, and blind pores. The important pore structure characteristics of needlepunched nonwoven fabrics as filter media are the most constricted open pore diameter (smallest detected pore diameter), the largest pore diameter (bubble point pore diameter),

**3. The usage of genetic programming to predict woven fabric porosity** 

Porosity parameters based on an ideal geometrical model of porous structure give woven fabric constructor some useful information about porosity by developing a new product, but they are not in a good agreement with the experimental values. In order to balance the difference between the theoretical and experimental values of porosity parameters, genetic programming was used to develop models for predicting the following macro-porosity parameters of woven fabric: the area of macro-pore cross-section, macro-pore density, open porosity, and equivalent macro-pore diameter. The genetic programming is a variant of evolutionary algorithm methods described in many sources (e.g., [2, 3, 4]). The basic information on the evolutionary algorithms is given at the beginning of the section 4. We implemented Koza's variant of genetic programming [2]. In our research, the independent input variables (the set of terminals) were: yarn fineness *T* (tex), weave value *V*, fabric tightness *t* (%) and denting *D* (ends/dent in the reed). The set of terminals also included random floating-point numbers between –10 and +10. Variegated reed denting was treated as an average value of treads, dented in the individual reed dent. The dependent output variables were: area of macro-pore cross-section *Ap* (10-3 mm2), pore density *Np* (pores/cm2), and equivalent macro-pore diameter (µm). For all modelling, the initially set of functions included the basic mathematical operations of addition, subtraction, multiplication, and division. In the case of modelling the area of macro-pore cross-section and pore density the initially set of functions also included a power function, whereas the set of functions for modelling of equivalent macro-pore diameter included an exponential function. We then used the genetic programming system to evolve appropriate models consist of abovementioned sets of terminals and functions. Open porosity was calculated on the basis of predicted values of the area of macro-pore cross-section and macro-pore density and Equation 17. The equivalent macro-pore diameter was calculated on the basis of predicted values of the area of macro-pore cross-section using Equation 19. The fitness measure for modelling by genetic programming was exactly the same as defined by Equation 33 in section 4. The goal of the modelling was to find such a predictive model in a symbolic form, Our experiments involved woven fabrics made from staple yarns with two restrictions: first, only fabrics made from 100% cotton yarns (made by a combing and carding procedure on a ring spinning machine) were used in this research; second, fabrics were measured in the grey state to eliminate the influence of finishing processes. We believe that it is very hard, perhaps even impossible, to include all woven fabrics types to predict individual macroporosity parameters precisely enough, and so we focused our research on unfinished staple yarn cotton fabrics. We would like to show that genetic programming can be used to establish the many relations between woven fabric constructional parameters and particular fabric properties, and that the results are more useful for fabric engineering than ideal theoretical models. The cotton fabrics varied according to yarn fineness (14 tex, 25 tex, and 36 tex), weave type (weave value), fabric tightness (55% - 65%, 65% - 75%, 75% - 85%), and denting. The constructional parameters of woven fabric samples are collected in Table 1. They were woven on a Picanol weaving machine under the same technological conditions. The weave values of plain (0.904), twill (1.188), and satin (1.379) fabrics, as well as fabric tightness, were determined according to Kienbaum's setting theory [36].

We used an optical method to measure porosity parameters of woven fabrics, since it is the most accurate technique for macro-pores with diameters of more than 10 m. For each fabric specimen, we observed between 50 and 100 macro-pores using a Nikon SMZ-2T computeraided stereomicroscope with special software. We measured the following macro-porosity parameters: area of macro-pore cross-section, pore density, and equivalent macro-pore diameters.

## **3.2. Predictive models of woven fabric porosity parameters**

Equations 29 and 30 present predictive models of the area of macro-pore cross-section Ap and macro-pore density Np, respectively [37]. Here *V* is the weave factor, *T* is the yarn linear density in tex, *t* is the fabric tightness in %, and *D* is the denting in ends per reed dent. The open porosity and equivalent diameter are calculated using Equations 17 and 19, respectively, where for Ap and Np the predicted values are taken into account. Because the model of the area of macro-pore cross-section is more complex, the functions f1, f2,…f10 are not presented here but are written in the appendix. When calculating the values of models, the following rules have to be taken into account: the protected division function returns to 1 if denominator is 0; otherwise, it returns to the normal quotient. The protected power function raises the absolute value of the first argument to the power specified by its second argument.

By a comparison of both GA models (Equations 29 and 30) with the theoretical ones (Equations 11-13 and 14), the complexity of GA models is obvious and derives from the factors involved in the models. Namely, factors involved in GA models don't ignore the irregularity of macro-pores cross-section area as well as the number of pores, due to the phenomenon of latticed pores in the case of staple yarns (which depends on the type of weave – factor V and fabric tightness – factor t) and the phenomenon of thread spacing irregularity (factor D), as theoretical models do. Theoretical model for the macro-pore crosssection area assumes that all macro-pores in woven structure have the same cross-section area regardless the type of used yarns, type of weave, fabric tightness and denting, whilst the theoretical model for the pore density assumes no reduction of the number of pores.

The Usage of Genetic Methods for Prediction of Fabric Porosity 189

<sup>9</sup> 95.024.1

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Figure 10 presents a comparison of the experimental, predicted, and theoretical values of macro-porosity parameters. Theoretical values of macro-pore density are calculated on the basis of an ideal model of porous structure using Equation 14. By calculation of the theoretical values of the area of macro-pore cross-section, open porosity, and the equivalent pore diameter, the circular, rectangular, and elliptical shape of macro-pore area are taken

Theoretical values of woven fabric porosity parameters deviate from experimental ones on average by 118.3% (min 8.8%, max 452.9%) for the area of the macro-pore with rectangular cross-section, 111.5% (min 14.5%, max 370.6%) for the area of the macro-pore with circular cross-section, 72.8% (min 0.2%, max 335.3%) for the area of the macro-pore with elliptical cross-section, 37.3% (min 0.0%, max 395.0%) for the macro-pore density, 232.6% (min 19.9%, max 1900.1%) for the open porosity of fabrics with rectangular pore cross-section, 221.0% (min 14.3%, max 1558.0%) for the open porosity of fabrics with circular pore cross-section, 166.3% (min 5.9%, max 1479.0%) for the open porosity of fabrics with elliptical pore cross-section, 43.7% (min 4.3%, max 135.1%) for the equivalent pore diameter where rectangular pore crosssection is taken into account, 43.7% (min 7.0%, max 116.9%) for the equivalent pore diameter where circular cross-section is taken into account, and 28.0% (min 0.1%, max 108.6%) for the

equivalent pore diameter where elliptical pore cross-section is taken into account.

The results of woven fabric porosity parameters determined with models based on genetic programming show very good agreement with experimental values (Figure 11) and justify the complexity of GA models. The predicted values deviate from experimental ones on average by 1.5% (min 0.0%, max 10.2%) for the area of the macro-pore cross-section, 2.0% (min 0.0%, max 8.0%) for the macro-pore density, 3.2% (min 0.0%, max 10.1%) for the open porosity, and 0.8% (min 0.0%, max 5.2%) for the equivalent macro-pore diameter. The

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**Table 1.** The constructional parameters of woven fabric samples

The Usage of Genetic Methods for Prediction of Fabric Porosity 189

$$\begin{split} A\_{p} &= \frac{1}{V^{2}t^{2}} \Biggl( f\_{1} + \frac{t}{f\_{2} - f\_{3}} \Bigg) \cdot \Bigg( f\_{8} \left( f\_{9} + f\_{10} \right) + TV \left( f\_{4} + f\_{7} + \frac{f\_{5}f\_{6}}{t(T+t)} \right) \Bigg) \\\\ N\_{F} &= \frac{(D+V)(D+D) \left( t + \frac{9+t}{T-V} \right)}{2.93 + T} - \frac{D \left( t + \left( T + V - t + \frac{Vt}{D} \right)^{-7} + \frac{-1.4 + 2D + 0.95DV^{-1.7} + \frac{9+D+t}{-6.6 + T}}{T + \frac{t}{D+V}} \right)}{(D+T) \left( 1.055 - 0.8816V + \frac{D}{V \sqrt{\frac{T}{V^{2}}}} \right)} \\\\ D \Bigg( \frac{D\left( 1.05 - V - \frac{6.3}{(-1+D)DT^{\frac{1}{2}}} \right)}{DT - t} + \frac{T - \frac{t}{D + V^{2}}}{T + \frac{9.5 + T}{D - T + V}} \Bigg) \\\\ \frac{1.055 - V + \frac{3.56 - 5.187^{\circ}(D+T)}{V(1.05 + V + T)}}{1.055 - V + \frac{3.56 - 5.187^{\circ}(D+T)}{V(1.05 + V + T)}} \end{split} \tag{30}$$

188 Genetic Programming – New Approaches and Successful Applications

Ref. Yarn linear density *T*,

tex

**Table 1.** The constructional parameters of woven fabric samples

By a comparison of both GA models (Equations 29 and 30) with the theoretical ones (Equations 11-13 and 14), the complexity of GA models is obvious and derives from the factors involved in the models. Namely, factors involved in GA models don't ignore the irregularity of macro-pores cross-section area as well as the number of pores, due to the phenomenon of latticed pores in the case of staple yarns (which depends on the type of weave – factor V and fabric tightness – factor t) and the phenomenon of thread spacing irregularity (factor D), as theoretical models do. Theoretical model for the macro-pore crosssection area assumes that all macro-pores in woven structure have the same cross-section area regardless the type of used yarns, type of weave, fabric tightness and denting, whilst the theoretical model for the pore density assumes no reduction of the number of pores.

> Weave value *V*

Fabric tightness t, %

Denting *D,*  ends/reed dent

> Figure 10 presents a comparison of the experimental, predicted, and theoretical values of macro-porosity parameters. Theoretical values of macro-pore density are calculated on the basis of an ideal model of porous structure using Equation 14. By calculation of the theoretical values of the area of macro-pore cross-section, open porosity, and the equivalent pore diameter, the circular, rectangular, and elliptical shape of macro-pore area are taken into account.

> Theoretical values of woven fabric porosity parameters deviate from experimental ones on average by 118.3% (min 8.8%, max 452.9%) for the area of the macro-pore with rectangular cross-section, 111.5% (min 14.5%, max 370.6%) for the area of the macro-pore with circular cross-section, 72.8% (min 0.2%, max 335.3%) for the area of the macro-pore with elliptical cross-section, 37.3% (min 0.0%, max 395.0%) for the macro-pore density, 232.6% (min 19.9%, max 1900.1%) for the open porosity of fabrics with rectangular pore cross-section, 221.0% (min 14.3%, max 1558.0%) for the open porosity of fabrics with circular pore cross-section, 166.3% (min 5.9%, max 1479.0%) for the open porosity of fabrics with elliptical pore cross-section, 43.7% (min 4.3%, max 135.1%) for the equivalent pore diameter where rectangular pore crosssection is taken into account, 43.7% (min 7.0%, max 116.9%) for the equivalent pore diameter where circular cross-section is taken into account, and 28.0% (min 0.1%, max 108.6%) for the equivalent pore diameter where elliptical pore cross-section is taken into account.

> The results of woven fabric porosity parameters determined with models based on genetic programming show very good agreement with experimental values (Figure 11) and justify the complexity of GA models. The predicted values deviate from experimental ones on average by 1.5% (min 0.0%, max 10.2%) for the area of the macro-pore cross-section, 2.0% (min 0.0%, max 8.0%) for the macro-pore density, 3.2% (min 0.0%, max 10.1%) for the open porosity, and 0.8% (min 0.0%, max 5.2%) for the equivalent macro-pore diameter. The

correlation coefficients between the predicted and experimental values are 0.9999, 0.9989, 0.9941, and 0.9997 for the area of macro-pore cross-section, macro-pore density, open porosity, and equivalent diameter, respectively.

The Usage of Genetic Methods for Prediction of Fabric Porosity 191

**Macropore density (pores cm-2 )**

0 500 1000 1500 2000 2500

**Experimental values**

**Equivalent diameter (10-6 m)**

0 50 100 150 200 250 300 350 400 450 500

**Experimental values**

R² = 0,9989

**Predicted values**

**Figure 11.** Scatter plots of experimental and predicted porosity parameters using GP models

**4. The usage of genetic algorithm to predict nonwoven fabric porosity** 

In this research, the genetic algorithm was used for definition of predictive models of nonwoven fabric porosity parameters. Since needle-punched nonwoven fabrics have completely different porous structure when compared to woven fabrics, it is inappropriate to focus on open porosity through the prediction of the area of macro-pore cross-section and macro-pore density. The most valuable porosity parameters for needle-punched nonwoven porous structure characterisations are total porosity and mean pore diameter, and those parameters were the subjects of our research. Since the basic steps in evolutionary computation are well-known, only a brief description follows. Firstly, the initial population *P(t)* of the random organisms (solutions) is generated. The variable *t* represents the generation time. The next step is the evaluation of population *P(t)* according to the fitness measure. Altering the population *P(t)* by genetic operations follows. The genetic operations alter one or more parental organism(s); thus, creating their offspring. The evaluation and alteration of population takes place until the termination criterion has been fulfilled. This can be the specified maximum number of generations or a sufficient quality of solutions [38]. More comprehensive information on evolutionary computation can be found in [39].

The independent input variables were fibre fineness - T (dtex), nonwoven fabric area mass - m (g/m2), and nonwoven fabric thickness - D (mm). The dependent output variables were mean pore diameter dp (µm) and total porosity ε (%). Since the GA approach is unsuitable for the

R² = 0,9997

**Predicted values**

R² = 0,9999

R² = 0,9941

0 20 40 60 80 100 120 140 160

**Experimental values**

**Open porosity (%)**

0 5 10 15 20 25 30 35

**Experimental values**

**Area of macropore cross-section (10-6 m)**

0,0 20,0 40,0 60,0 80,0 100,0 120,0 140,0 160,0

0,0 5,0 10,0 15,0 20,0 25,0 30,0 35,0

**parameters** 

**Predicted values**

**Predicted values**

**Figure 10.** Results of woven fabric porosity parameters

The models are based on image analysis technique and assumption that woven samples are transparent. The boundary limits for the validity of the models are as follows: 1. the minimal values for yarn linear density, weave factor and fabric tightness, are 14 tex, 0.904, and 55%, respectively, 2. the maximal values for yarn linear density, weave factor and fabric tightness are 36 tex, 1.379, and 85%, respectively.

porosity, and equivalent diameter, respectively.

are 36 tex, 1.379, and 85%, respectively.

**Figure 10.** Results of woven fabric porosity parameters

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correlation coefficients between the predicted and experimental values are 0.9999, 0.9989, 0.9941, and 0.9997 for the area of macro-pore cross-section, macro-pore density, open

**Area of macropore cross-section (10-3 mm2)**

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

experimental predicted theoretical

**Open Porosity (%)**

experimental predicted theoretical-rectangular

The models are based on image analysis technique and assumption that woven samples are transparent. The boundary limits for the validity of the models are as follows: 1. the minimal values for yarn linear density, weave factor and fabric tightness, are 14 tex, 0.904, and 55%, respectively, 2. the maximal values for yarn linear density, weave factor and fabric tightness

theoretical-elliptical theoretical-circular

experimental predicted theoretical-rectangular

**Macropore density (pores cm-2)**

theoretical-elliptical theoretical-circualr

**Figure 11.** Scatter plots of experimental and predicted porosity parameters using GP models
