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

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 evolution of prediction models (organisms) in their symbolic forms, it is necessary to define them in advance [38]. In this study, a quadratic polynominal equation with three variables was used as a prespecified model for the prediction of porosity parameters as follows:

$$Y = c\_1 + c\_2 m + c\_3 D + c\_4 T + c\_5 m^2 + c\_6 D^2 + c\_7 T^2 + c\_8 m D + c\_9 m T + c\_{10} D T + c\_{11} m D T \tag{31}$$

where, Y is the dependent output variable, *m* is the nonwoven fabric mass per unit area in g/m2, *D* is the nonwoven fabric thickness in mm, *T* is the fibre fineness in dtex, and c1…11 are constants. The main reasons for this selection were as follows: 1. a polynominal model is relatively simple, 2. for the problem studied we did not expect harmonic dependence of the output variables, 3. some preliminary modelling-runs with different types of prespecified models showed that the quadratic polynominal model provides very good selection in terms of prediction quality. In our research, the initial random population *P(t)* consisted of *N* prespecified models (Equation 31) where *N* is the population size. Of course, in our computer implementation of the GA, the population *P(t)* consisted only of the *N* sets of the real-valued vectors of model constants. The individual vector is equal to:

$$\mathfrak{c} = \{\mathfrak{c}\_{\prime}, \mathfrak{c}\_{\prime}, \cdots, \mathfrak{c}\_{\prime}\mathfrak{m}\} \tag{32}$$

The Usage of Genetic Methods for Prediction of Fabric Porosity 193

operation, one parental vector *c* was randomly selected. Then, the mutation took place in one randomly selected parental gene. During both the crossover and mutation processes, the numbers of crossover and mutational operations performed on parental vector(s), were randomly selected. The evolutionary parameters for modelling by genetic algorithms were: population size 300, maximum number of generations to be run 5000, probability of reproduction 0.1, probability of crossover 0.8 and probability of mutation 0.1. Tournament selection with a group size of 5 was used. For the purpose of the research 200 independent genetic algorithms runs were carried out. Only the best models are presented in the paper.

Bearing in mind the fact that nonwovens have very different structures and, thus, also porosity parameters due to their sequences when web-forming, bonding, as well as finishing methods, the nonwoven fabric samples were limited to one type of nonwoven fabrics – those needle-punched nonwoven fabrics made from a mixture of polyester and viscose staple fibres. Nonwoven multi-layered webs were first obtained from the same manufacturing process by subjecting the fibre mixtures to carding and then orienting the carded webs in a cross-direction by using a cross lapper to achieve web surface mass ranges of 100-150, 150-200, 250-300, and 300-350 g/m2, and a web volume mass range of 0.019-0.035 g/cm3. The webs were made from a mixture of polyester (PES) and viscose (VIS) staple fibres of different content, fineness, and lengths, as follows: samples 1–3 from a mixture of 87% VIS fibres (1.7 dtex linear density, 38 mm length) and 12.5% of PES fibres (4.4 dtex linear density, 50 mm length), samples 4–7 from a mixture of 60% VIS fibres (1.7 dtex linear density, 38 mm length) and 40% PES fibres (3.3 dtex linear density, 60 mm length), samples 8–11 from a mixture of 30% VIS fibres (3.3 dtex linear density, 50 mm length), 40% PES fibres type 1 (6.7 dtex linear density, 60 mm length) and 30% of PES fibres type 2 (4.4 dtex linear density, 50 mm length), samples 12–15 from a mixture of 70% PES fibres type 1 and 30% PES type 2. Multi-layered carded webs were further subjecting to pre-needling using needle-punching machine, under the following processing parameters of one-sided preneedle punching: stroke frequency 250/min; delivery speed 1.5 m/min; needling density 30/cm, depth of needle penetration 15 mm, and felting needles of 15x18x38x3 M222 G3017. The processing parameters of further two-sided needle-punching were as follows: stroke frequency 900/min; delivery speed 5.5 m/min; needling density 60/cm (30/cm upper and 30/cm lower), depth of upper and lower needle penetrations 12 mm, and felting needles of 15x18x32x3 M222 G3017. The webs were further processed through a pair of heated calendars at under 180 °C with different gaps between the rollers, in order to achieve further changes in fabric density and, consequently, in the porosity within the range of 80–92 %. The constructional parameters of the nonwoven fabric samples are collected in Table 2. All the nonwoven fabric samples were in a grey state to eliminate the influence of finishing treatments. The constructional parameters of the nonwoven fabric samples, e.g. the nonwoven fabric mass per unit area and thickness were measured according to ISO 9073-1 (Textiles – Test Methods for nonwovens – Part 1: Determination of mass per unit area) and ISO 9073-2 (Textiles – Test Methods for nonwovens – Part 2: Determination of thickness).

**4.1. Materials and porosity measurements** 

The absolute deviation *D*(*i*,*t*) of individual model *i* (organism) in generation time *t* was introduced as a fitness measure. It was defined as:

$$D(i, t) = \sum\_{j=1}^{n} \left| E(j) - P(i, j) \right| \tag{33}$$

where, *E*(*j*) is the experimental value for measurement *j*, *P*(*i*, *j*) is the predicted value returned by the individual model *i* for measurement *j*, and *n* is the maximum number of measurements. The goal of the optimisation task was to find such a predictive model (defined by Equation 31), that Equation 33 would give as low an absolute deviation as possible. Therefore, the aim was to find out appropriate real-valued constants in Equation 32. However, since it was unnecessary that the smallest values of the above equation also meant the smallest percentage deviation of this model, the average absolute percentage deviation of all measurements for individual model *i* was defined as:

$$\Delta(i) = \frac{D(i, t)}{|E(j)| \, | \, n \tag{34}$$

The Equation 33 was not used as a fitness measure for evaluating population, but only for finding the best organism within the population, after completing the run.

The altering of population *P(t)* was effected by reproduction, crossover, and mutation. For the crossover operation, two parental vectors, e.g., *c*1 and *c*2 were randomly selected. Then the crossover took place between two randomly-selected parental genes having the same index. Two offspring genes were created according to the extended intermediate crossover, as considered by Mühlenbeim and Schlierkamp-Voosen [40]. During the mutation operation, one parental vector *c* was randomly selected. Then, the mutation took place in one randomly selected parental gene. During both the crossover and mutation processes, the numbers of crossover and mutational operations performed on parental vector(s), were randomly selected. The evolutionary parameters for modelling by genetic algorithms were: population size 300, maximum number of generations to be run 5000, probability of reproduction 0.1, probability of crossover 0.8 and probability of mutation 0.1. Tournament selection with a group size of 5 was used. For the purpose of the research 200 independent genetic algorithms runs were carried out. Only the best models are presented in the paper.

#### **4.1. Materials and porosity measurements**

192 Genetic Programming – New Approaches and Successful Applications

evolution of prediction models (organisms) in their symbolic forms, it is necessary to define them in advance [38]. In this study, a quadratic polynominal equation with three variables was

where, Y is the dependent output variable, *m* is the nonwoven fabric mass per unit area in g/m2, *D* is the nonwoven fabric thickness in mm, *T* is the fibre fineness in dtex, and c1…11 are constants. The main reasons for this selection were as follows: 1. a polynominal model is relatively simple, 2. for the problem studied we did not expect harmonic dependence of the output variables, 3. some preliminary modelling-runs with different types of prespecified models showed that the quadratic polynominal model provides very good selection in terms of prediction quality. In our research, the initial random population *P(t)* consisted of *N* prespecified models (Equation 31) where *N* is the population size. Of course, in our computer implementation of the GA, the population *P(t)* consisted only of the *N* sets of the

The absolute deviation *D*(*i*,*t*) of individual model *i* (organism) in generation time *t* was

1 (,) () (,) *n*

where, *E*(*j*) is the experimental value for measurement *j*, *P*(*i*, *j*) is the predicted value returned by the individual model *i* for measurement *j*, and *n* is the maximum number of measurements. The goal of the optimisation task was to find such a predictive model (defined by Equation 31), that Equation 33 would give as low an absolute deviation as possible. Therefore, the aim was to find out appropriate real-valued constants in Equation 32. However, since it was unnecessary that the smallest values of the above equation also meant the smallest percentage deviation of this model, the average absolute percentage

> (,) ( ) 100% | ( )| *Dit <sup>i</sup> Ej n*

The Equation 33 was not used as a fitness measure for evaluating population, but only for

The altering of population *P(t)* was effected by reproduction, crossover, and mutation. For the crossover operation, two parental vectors, e.g., *c*1 and *c*2 were randomly selected. Then the crossover took place between two randomly-selected parental genes having the same index. Two offspring genes were created according to the extended intermediate crossover, as considered by Mühlenbeim and Schlierkamp-Voosen [40]. During the mutation

*j Dit Ej Pi j* 

2 22 *Y c c m c D c T c m c D c T c mD c mT c DT c mDT* 1 2 3 4 5 6 7 8 9 10 11 (31)

c = (*c*1 , *c*2 , · · ·, *c*11) (32)

(33)

(34)

used as a prespecified model for the prediction of porosity parameters as follows:

real-valued vectors of model constants. The individual vector is equal to:

deviation of all measurements for individual model *i* was defined as:

finding the best organism within the population, after completing the run.

introduced as a fitness measure. It was defined as:

Bearing in mind the fact that nonwovens have very different structures and, thus, also porosity parameters due to their sequences when web-forming, bonding, as well as finishing methods, the nonwoven fabric samples were limited to one type of nonwoven fabrics – those needle-punched nonwoven fabrics made from a mixture of polyester and viscose staple fibres. Nonwoven multi-layered webs were first obtained from the same manufacturing process by subjecting the fibre mixtures to carding and then orienting the carded webs in a cross-direction by using a cross lapper to achieve web surface mass ranges of 100-150, 150-200, 250-300, and 300-350 g/m2, and a web volume mass range of 0.019-0.035 g/cm3. The webs were made from a mixture of polyester (PES) and viscose (VIS) staple fibres of different content, fineness, and lengths, as follows: samples 1–3 from a mixture of 87% VIS fibres (1.7 dtex linear density, 38 mm length) and 12.5% of PES fibres (4.4 dtex linear density, 50 mm length), samples 4–7 from a mixture of 60% VIS fibres (1.7 dtex linear density, 38 mm length) and 40% PES fibres (3.3 dtex linear density, 60 mm length), samples 8–11 from a mixture of 30% VIS fibres (3.3 dtex linear density, 50 mm length), 40% PES fibres type 1 (6.7 dtex linear density, 60 mm length) and 30% of PES fibres type 2 (4.4 dtex linear density, 50 mm length), samples 12–15 from a mixture of 70% PES fibres type 1 and 30% PES type 2. Multi-layered carded webs were further subjecting to pre-needling using needle-punching machine, under the following processing parameters of one-sided preneedle punching: stroke frequency 250/min; delivery speed 1.5 m/min; needling density 30/cm, depth of needle penetration 15 mm, and felting needles of 15x18x38x3 M222 G3017. The processing parameters of further two-sided needle-punching were as follows: stroke frequency 900/min; delivery speed 5.5 m/min; needling density 60/cm (30/cm upper and 30/cm lower), depth of upper and lower needle penetrations 12 mm, and felting needles of 15x18x32x3 M222 G3017. The webs were further processed through a pair of heated calendars at under 180 °C with different gaps between the rollers, in order to achieve further changes in fabric density and, consequently, in the porosity within the range of 80–92 %. The constructional parameters of the nonwoven fabric samples are collected in Table 2. All the nonwoven fabric samples were in a grey state to eliminate the influence of finishing treatments. The constructional parameters of the nonwoven fabric samples, e.g. the nonwoven fabric mass per unit area and thickness were measured according to ISO 9073-1 (Textiles – Test Methods for nonwovens – Part 1: Determination of mass per unit area) and ISO 9073-2 (Textiles – Test Methods for nonwovens – Part 2: Determination of thickness).


**Table 2.** The constructional parameters of nonwoven fabric samples

The porosity parameters of the nonwoven fabric samples were measured using the Pascal 140 computer aided mercury intrusion porosimeter, which measures pores' diameters between 3.8 - 120 µm, and operates under low pressure. The mercury intrusion technique is based on the principle that non-wetting liquid (mercury) coming in contact with a solid porous material can not be spontaneously absorbed by the pores of the solid itself because of the surface tension, but can be forced by applying external pressure. The required pressure depends on the pore-size and this relationship is commonly known as the Washburn equation [9]:

$$P = \frac{-2 \cdot \gamma \cdot \cos \theta}{r} \tag{35}$$

The Usage of Genetic Methods for Prediction of Fabric Porosity 195

3 2 2 2

(36)

(37)

porosity (%). The volume of penetrated mercury is directly the measure of the sample's pore volume expressed as a specific pore volume in mm3/g, and is obtained by means of a capacitive reading system. The average pore diameter is evaluated at 50% of the cumulative

Equations 36 and 37 present predictive models of the total porosity ε and mean pore diameter dp, respectively. Here *T* is the fibre fineness in dtex, *m* is the nonwoven fabric mass

150.1 8.61 10 3.21 10 66.04 1.09 10 28.74

103.12 0.39 6.01 0.73 2.12 10 30.77 3.79

Figure 12 presents a comparison of the experimental, predicted and theoretical values of porosity parameters, e.g. total porosity and mean pore diameter. The theoretical values of total porosity and mean pore diameters were calculated using Equation 22 and 27-28,

**Total porosity (%)**

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

experimental predicted theoretical

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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

experimental predicted theoretical

*<sup>p</sup> d mDT m D T*

*T mD mT DT mDT*

2 2 3 2 2

*m DT m D*

3

**4.2. Predictive models of nonwoven fabric porosity parameters** 

per unit area in g/m2, and *D* is the nonwoven fabric thickness in mm.

0.46 0.16 1.13 3.79 10

0.89 0.24 0.20 32.16 0.10

*mD mt DT mDT*

volume of mercury.

respectively.

2

**Figure 12.** Results of nonwoven fabric porosity parameters

where, P is the applied pressure, ϒ is the surface tension of mercury, θ is the contact-angle and r is the capillary radius. The distribution of pore size, as well as the total porosity and the specific pore volume can be obtained from the relationship between the pressure necessary for penetration (the pore dimension) and the volume of the penetrated mercury (pore volume). There are certain main assumptions necessary when applying the Washburn equation: the pores are assumed to be of cylindrical shape and the sample is pressure stable.

Each nonwoven sample of known weight was placed in the dilatometer, then the air around the sample was evacuated and finally the dilatometer was filled with mercury by increasing the pressure up to the reference level. The volume and pressure measurements' data were transferred into the computer programme and the following data were detectable or calculated: the specific pore volume (mm3/g), the average pore diameter (µm) and the total porosity (%). The volume of penetrated mercury is directly the measure of the sample's pore volume expressed as a specific pore volume in mm3/g, and is obtained by means of a capacitive reading system. The average pore diameter is evaluated at 50% of the cumulative volume of mercury.

#### **4.2. Predictive models of nonwoven fabric porosity parameters**

194 Genetic Programming – New Approaches and Successful Applications

**Table 2.** The constructional parameters of nonwoven fabric samples

and this relationship is commonly known as the Washburn equation [9]:

Fabric mass per unit area *m*, g/m2

1 2.0 143 1.202 2 2.0 142 0.941 3 2.0 142 0.576 4 2.3 173 1.509 5 2.3 201 1.558 6 2.3 171 0.941 7 2.3 200 1.071 8 5.0 259 1.360 9 5.0 259 1.261 10 5.0 279 1.182 11 5.0 274 1.112 12 6.0 298 1.400 13 6.0 304 1.266 14 6.0 352 1.347 15 6.0 343 1.235

The porosity parameters of the nonwoven fabric samples were measured using the Pascal 140 computer aided mercury intrusion porosimeter, which measures pores' diameters between 3.8 - 120 µm, and operates under low pressure. The mercury intrusion technique is based on the principle that non-wetting liquid (mercury) coming in contact with a solid porous material can not be spontaneously absorbed by the pores of the solid itself because of the surface tension, but can be forced by applying external pressure. The required pressure depends on the pore-size

> 2 cos *<sup>P</sup> r*

where, P is the applied pressure, ϒ is the surface tension of mercury, θ is the contact-angle and r is the capillary radius. The distribution of pore size, as well as the total porosity and the specific pore volume can be obtained from the relationship between the pressure necessary for penetration (the pore dimension) and the volume of the penetrated mercury (pore volume). There are certain main assumptions necessary when applying the Washburn equation: the pores are assumed to be of cylindrical shape and the sample is pressure stable.

Each nonwoven sample of known weight was placed in the dilatometer, then the air around the sample was evacuated and finally the dilatometer was filled with mercury by increasing the pressure up to the reference level. The volume and pressure measurements' data were transferred into the computer programme and the following data were detectable or calculated: the specific pore volume (mm3/g), the average pore diameter (µm) and the total

(35)

Fabric thickness *D*, mm

*T*, dtex

Ref. Average fibre fineness

Equations 36 and 37 present predictive models of the total porosity ε and mean pore diameter dp, respectively. Here *T* is the fibre fineness in dtex, *m* is the nonwoven fabric mass per unit area in g/m2, and *D* is the nonwoven fabric thickness in mm.

$$\begin{aligned} \varepsilon &= 150.1 - 8.61 \cdot 10^{-2} m + 3.21 \cdot 10^{-2} D - 66.04 \cdot T - 1.09 \cdot 10^{-3} m^2 - 28.74 \cdot D^2 + \\ &+ 0.89 \cdot T^2 + 0.24 \cdot m \cdot D + 0.20 \cdot m \cdot T + 32.16 \cdot D \cdot T - 0.10 \cdot m \cdot D \cdot T \\\\ d\_p &= 103.12 - 0.39 \cdot m + 6.01 \cdot D - 0.73 \cdot T - 2.12 \cdot 10^{-3} m^2 - 30.77 \cdot D^2 - 3.79 \cdot T^2 + \\ &+ 0.46 \cdot m \cdot D + 0.16 \cdot m \cdot t + 1.13 \cdot D \cdot T - 3.79 \cdot 10^{-3} \cdot m \cdot D \cdot T \end{aligned} \tag{37}$$

Figure 12 presents a comparison of the experimental, predicted and theoretical values of porosity parameters, e.g. total porosity and mean pore diameter. The theoretical values of total porosity and mean pore diameters were calculated using Equation 22 and 27-28, respectively.

**Figure 12.** Results of nonwoven fabric porosity parameters

In Figure 12, the theoretical values of total porosity and mean pore diameter as well as predicted values of pore diameter are linked with lines while samples (1-3, 4-7, 8-11, and 12- 15) are arranged regarding their decreased porosity. The results show that nonwovens with similar porous structure and lower porosity also have lower pore diameter. The experimental values of total porosity are for some samples not in a good agreement with theoretical ones, while samples which should have the highest porosity actually have the lowest (samples No. 1, 8, and 12). The reason may lie in fact, that these samples contain more closed pores which are not detectable with mercury porosimetry.

The Usage of Genetic Methods for Prediction of Fabric Porosity 197

3 2

*Vt T TT D <sup>D</sup>*

4.74 ( )

*t t*

tightness, denting), image analysis as testing method of porosity measurements, and genetic programming, and 2. total porosity and mean pore diameter of nonwoven fabrics based on the constructional parameters of nonwoven fabrics (fibre linear density, fabric mass per unit area, fabric thickness), mercury intrusion porosimetry as testing method of porosity measurements, and genetic algorithm, were developed. Open porosity and equivalent pore diameter of woven fabric were also predicted using values calculated on the basis of predictive models of the area of macro-pore cross-section and pore density, and known mathematical relationships. All proposed predictive models were created very precisely and could serve as guidelines for woven/nonwoven engineering in order to develop fabrics with

In general, for prediction of porosity parameters of woven or nonwoven samples both modelling tools can be used, e.g. GA and GP. Usually, GP method is used for more difficult problems. Our purpose was to show usability and effectiveness of both methods. By woven fabric modelling, the range of porosity parameters' measurements was substantial larger with more input variables when compared to the nonwoven fabrics (and this means more difficult problem), so the GP was used as modelling tool. By GP modelling, the models are developed in their symbolic forms, thus more precise models are developed in regard to the GA modelling, where only coefficients of prespecified models are defined. At the same time, for GP modelling more measurements data are desired for better model accuracy, while by GA modelling good results are achieved by lower number of measurements (in our case 27 measurements were available for woven fabrics and only 15 for nonwoven fabrics). The advantage of GP modelling is its excellent prediction accuracy, while its disadvantage is the complexity of the developed models. In general, by GA modelling, the developed models

the desired porosity parameters.

are simple but less accurate.

Polona Dobnik Dubrovski

*Faculty of Mechanical Engineering, Slovenia* 

*Faculty of Mechanical Engineering, Slovenia* 

*Department of Textile Materials and Design, University of Maribor,* 

1 2

*T T <sup>f</sup> x D*

6.43 12.856 35.3 , <sup>2</sup> <sup>2</sup>

*V Dt T t <sup>D</sup>*

*Department of Mechanical Engineering,University of Maribor,* 

**Author details** 

Miran Brezočnik

**Appendix** 

The results show that the theoretical values of porosity parameters deviate from experimental ones on average by 8.0% (min 0.0%, max 15.4%) for total porosity and by 19.7% (min 2.9, max 57.3%) for pore diameter, whilst the predicted values, calculated using Equations 36-37, are in better agreement with the experimental ones. The mean predicted error is: 1.1% (from 0.0% to 4.4%) for the total porosity and 1.9% (from 0.0% to 12.4%) for the average pore diameter. The correlation coefficients between the predicted and experimental values are 0.9024 and 0.8492 for the total porosity and the average pore diameter, respectively. Scatter plots of the experimental and predicted values for porosity parameters, are depicted in Figure 13.

**Figure 13.** Scatter plots of experimental and predicted porosity parameters using GA models
