Abstract
In this work, PA-6 nanocomposites containing different amounts of nanoclay (NC) were prepared using a corotating twin-screw extruder. In practice, it is hard task to identify the relationship between the extrusion process parameters and the tensile modulus of PA-6 nanocomposites by performing several experiments. One approach to map the relationship between the process parameters and the tensile modulus of PA-6 nanocomposites is the use of a non-linear system identification tool called the adaptive-neuro fuzzy inference system (ANFIS). In this study, to achieve high modeling accuracy and generalization capability, an efficient shuffled frog leaping (SFL) algorithm is proposed to learn all the parameters of the network. A multi-input single-output (MISO) ANFIS model is constructed and learned to predict the tensile modulus of PA-6 nanocomposites. The ANFIS model is constructed, trained and tested based on a collection of experimental data sets. Acceptable agreement has been observed between the experimental results and the predicted results by the proposed model. The statistical quality of the proposed model is significant due to its good correlation coefficient R2 values >0.98 between predicted values and experimental ones during the training and testing phase. Also, comparison results indicate the superior performance of the proposed scheme over the conventional reported methods due to its high approximation accuracy and good generalization capability.
1. Introduction
Polymer nanocomposites (PNCs) have emerged as a new and rapidly developing class of composite materials which have attracted considerable investment in research and development worldwide (1), (2), (3), (4). Nanocomposites are a class of composite materials where one of the constituents has at least one dimension in the range between 1 and 100 nm. One successful method to reduce polymeric materials deficiencies such as low heat distortion temperature, high moisture absorption, low mechanical properties, etc. is to add inorganic nanoparticles to them. Recently, various researches have been conducted in different fields of PNC which have shown substantial improvements in different properties compared with pure polymer and their conventional microcomposites, even at very low nanoclay (NC) content (5), (6), (7), (8), (9). To successfully produce polymer/NC nanocomposites with improved properties, dispersing inorganic fillers by inserting polymer chains or completely disrupting the layered structure is vital (10), (11).
On the one hand, the properties of nanocomposites are greatly influenced by the properties of the individual components (nanofiller and polymer), their size scale, and degree of mixing, morphology and interfacial characteristics. Several parameters such as NC content, screw speed (SS), melting temperature (MT) and feeding rate (FR) are influential in the extrusion process and can change the final properties of the prepared samples.
Polymer nanocomposites based on polyamide 6 (PA-6) have garnered much interest in recent years because of their readily tailored properties. The key point to enhance properties of PNCs such as PA-6/clay is delamination and dispersion of nanoscale layered clay fillers throughout the polymer matrix (12). Furthermore, increment of specific modulus, strength and heat distortion temperature under load as a factor of high reinforcement efficiency of 2 and 5 wt% can clearly depend on high surface to volume ratio of NC silicate layers in the PA6 matrix. However, platelet orientation and the nucleanation effect promote the crystal growth and degree of crystallinity of the nanocomposites. On the basis of current morphological features, the moisture resistance also improved (13), (14), (15), (16). The mentioned properties are the main importance of this special nanocomposite.
Predicting mechanical properties of nanocomposites is very important both in academia and industry, as it is a prerequisite for deeper understanding, analysis and drawing conclusions about the relationships among factors, microstructure and mechanical properties of these materials (17), (18), (19), (20).
Many methods have been used for modeling and analysis of a process that is affected by a set of factors. The basic method is the one factor at a time (OFAT) method which is considered to be less favorable compared to other methods because of requiring more runs for the same prediction accuracy. It can miss optimal settings of factors and can not estimate interactions. So, other different experimental and statistical design methods such as the response surface method (RSM) and regression analysis (RA) have attracted considerable attention in characterizing effects of some variables on the considered parameter (18), (21).
Another strategy for modeling industrial process behavior uses soft computing approaches. Among them, the neural network (NN) is a well-known approach due to its ability to learn and generalize nonlinear functional relationships between input and output variables. It provides a flexible mechanism for modeling and prediction mechanical properties (22), (23), (24), (25), (26). The NN model is usually used when RA is unable to describe the relation. This method, which provides well-adjusted parameters, is definitely powerful in making an accurate predicting machine. This predicting machine looks at some experimental results data to train, the same as the RA model, but in spite of RA, no technical knowledge is gained from this model. It is because the NN model relates the predictors to the predicted variable via some special hierarchical calculations. Although, NN model predictions are very useful in most problems, it is not enough for technical exploration of the problem. However, NN-based models suffer from the following disadvantages: (i) the precision of the NN-based model depends on the selection of the NN structure, parameters of the neuron and a sufficient training data and training method. (ii) Design of the NN-based model for complex processes requires a large network and this problem may lead to overfitting and poor generalization performance, slow convergence and convergence to a local minimum. (iii) NNs have a block box structure and it is not possible to utilize user experiences in modeling tasks, also the NN-based model is not in a human readable form.
Another modeling tool that makes possible the application of user experiences for modeling task is the fuzzy system. Fuzzy modeling in the processing of PA-6 in a twin screw extrusion process is challenging due to the complexity of the process which consists of multiple inputs which are highly non-linear. Fuzzy modeling makes it possible to utilize experience, knowledge and large amounts of process data (27), (28). Its performance depends on determining the number and location of the fuzzy basis functions. Also, the development of fuzzy systems for complex problems with high input dimension leads to time consuming tasks which provide unnecessarily large and complex structure as a model.
One approach to overcome the mentioned difficulties and achieve better learning performance is the use of fuzzy systems in the network structure which is called the adaptive neuro-fuzzy inference system (ANFIS). ANFIS is a powerful tool in fuzzy modeling. It learns information about the given data sets in order to compute the membership functions that describe the associated fuzzy inference system.
Some meta-heuristic optimization techniques have been proposed to learn the ANFIS network (29), (30), (31). One of them is the genetic algorithm (GA) that can solve complex optimization problems because of its ability to handle both discrete and continuous variables and nonlinear objective functions without any gradient information. Despite its benefits, it requires a long processing time for a near optimum solution to evolve. Also, not all problems lend themselves well to a solution with GA. Furthermore, it suffers from premature convergence difficulty. The mentioned disadvantages lead to performance and effectiveness degradation of this algorithm. In order to reduce the processing time and improve the quality of solutions, another evolutionary algorithm, particle swarm optimization (PSO) was proposed for learning purposes. However, the PSO algorithm progresses slowly and is not able to adjust the velocity step size. So, it can not search at a finer grain. Furthermore, it fails to reach a global optimal point for multi model functions. To overcome the mentioned limitations, the SFL algorithm was developed that combines the advantages of the gene based memetic algorithm and the social behavior-based PSO algorithms.
In (31), three optimization algorithms, e.g. binary GA, PSO and discrete SFL algorithm were used to minimize the weights and costs of the composites under different applied loading conditions. The reported results propose the SFL algorithm as a proper optimization tool for optimizing engineering problem such as those related to composite materials.
In this work, in order to improve the function approximation accuracy and general capability of the ANFIS system, a self-tuning process that uses the SFL algorithm is proposed to adjust all linear and nonlinear parameters of the network. By minimizing a quadratic measure of the error derived from the output of the system, the design problem can be characterized by the proposed SFL formulation. The solution is directly obtained without any need for complicated computations. Moreover, an efficient method is expected to have good performance without requiring any derivatives or other auxiliary knowledge.
Tensile modulus is perhaps the most important mechanical property in PNCs. So, a variety of empirical models, including RSM and NN have been utilized to predict this property. For example, in (32), RSM and ANN models were developed for tensile modulus of PA-6/ nanocaly composites. Some works studied the application of NNs or fuzzy systems based on RSM for modeling and the prediction task (33), (34), (35). To our knowledge, no extensive study has been conducted using the RSM method and ANFIS for assessing the effects of the considered factors on the mechanical properties of PA-6/NC nanocomposites.
In this paper, the ANFIS model is developed to predict the tensile modulus of PA-6/NC nanocomposites based on the experimental data. The SFL algorithm is proposed to learn all the parameters of the ANFIS model to identify the relationship between the process parameters and the tensile modulus of PA-6 nanocomposites. It investigates the effect of vital formulation and processing parameters including NC content incorporated with SS, FR and melt temperature on the tensile modulus of the PA-6/NC nanocomposites. A multi-input single-output ANFIS model is constructed, trained and tested to model and predict tensile modulus. The SFL algorithm is used to learn all parameters of the network including fuzzy membership functions and hidden layer weights, during the training stage such that the mean square error between the model output and actual output is minimized. The results are compared with the Taguchi-optimized ANN and RSM models. Simulation and comparison results verify the effectiveness and merits of the proposed scheme.
2. Experimental
2.1. Material
PA-6 (BASF SE, Germany) (B5, with melt volume flow rate of 8 cm3/10 min at 275 ˚C, load: 5 kg) was purchased from BASF and used as received. Nanoclay (Nanofil9®, Southern Clay Products Inc., TX, USA, organically functionalized by stearyl benzyl dimethyl ammonium chloride with good adhesion to PA-6) was purchased from Southern Clay Products, a subsidiary of Rockwood Specialties.
2.2. Design of experiments
The design of experiment (DOE) using response surface method (RSM) was chosen for investigating the effects of different parameters on the mechanical properties. Among several methods of response surface methodology, the Box-Behnken method was employed to explore the effects of four key parameters on the tensile modulus of melt-blended nanocomposites. The appropriate RSM to conduct our experiments was the 27-trial set of the Box-Behnken method. Table 1 shows the levels of the factors, selected after initial screening runs. Detailed processing conditions are gathered in Table 2.
Obtained data set from experiments for modeling tensile modulus.
| Sample | Processing conditions | Obtained data | |||
|---|---|---|---|---|---|
| NC content (phr) | Melt temperature (°C) | Feeding rate (RPM) | Screw speed (RPM) | TM (MPa) | |
| S1 | 4 | 245 | 7 | 450 | 3351 |
| S2 | 4 | 235 | 11 | 600 | 3120 |
| S3 | 6 | 255 | 9 | 600 | 3498 |
| S4 | 4 | 245 | 7 | 750 | 3370 |
| S5 | 4 | 245 | 11 | 750 | 3477 |
| S6 | 4 | 245 | 9 | 600 | 3361 |
| S7 | 6 | 245 | 11 | 600 | 3985 |
| S8 | 2 | 245 | 9 | 750 | 3012 |
| S9 | 6 | 245 | 9 | 450 | 3843 |
| S10 | 2 | 245 | 7 | 600 | 3201 |
| S11 | 4 | 235 | 9 | 750 | 3625 |
| S12 | 4 | 235 | 9 | 450 | 3488 |
| S13 | 2 | 245 | 11 | 600 | 3200 |
| S14 | 4 | 235 | 7 | 600 | 3850 |
| S15 | 4 | 245 | 9 | 600 | 3659 |
| S16 | 4 | 245 | 9 | 600 | 3474 |
| S17 | 2 | 235 | 9 | 600 | 3293 |
| S18 | 6 | 235 | 9 | 600 | 3984 |
| S19 | 2 | 255 | 9 | 600 | 3350 |
| S20 | 4 | 255 | 11 | 600 | 3640 |
| S21 | 4 | 255 | 9 | 450 | 3492 |
| S22 | 4 | 255 | 9 | 750 | 3400 |
| S23 | 6 | 245 | 9 | 750 | 3458 |
| S24 | 4 | 245 | 11 | 450 | 3464 |
| S25 | 2 | 245 | 9 | 450 | 3210 |
| S26 | 4 | 255 | 7 | 600 | 3678 |
| S27 | 6 | 245 | 7 | 600 | 3938 |
2.3. Sample preparation and characterization
PA-6 pellets and NC powder, pre-dried at 90°C for 12 h, were tumble blended in dry conditions. Samples were prepared, based on material and processing conditions listed in Table 2, via melt intercalation in a lab scale co-rotating twin-screw extruder (ZSK25, L/D=40, Germany). Figure 1 illustrates the screw configuration used in this study; the screw had two high mixing zones equipped with kneading elements to enhance dispersive and distributive mixing in the system. After melt mixing, dry pelletized samples were injection molded (Engel Austria GmbH) into standard tensile bars (ASTM D-638) using a 3-ton Engel injection molding machine. Afterwards, the injection-molded specimens were sealed and placed in a vacuum desiccator for a minimum of 24 h prior to mechanical testing under dry conditions. Each tensile modulus value was obtained from averaging five specimen test results. The complete procedure of producing the considered nanocomposites has been presented in (36).
Screw configuration.
3. ANFIS for modeling tensile modulus of PA-6 nanocomposites
This section explains the ANFIS construction and learning for modeling tensile modulus of PA-6 nanocomposites. First, a brief description of ANFIS is provided and then learning and modeling tasks are described.
3.1. Description of ANFIS
The adaptive neuro-fuzzy inference system is a fuzzy inference system implemented in the framework of an adaptive NN. It is a powerful modeling tool that can represent an accurate model by combining the inference property of fuzzy systems and adaptation and the learning capabilities of NNs. Such a system can be trained with no need for the expert knowledge which is usually required for the design of standard fuzzy logic systems.
The ANFIS architecture of the first order TSK fuzzy inference system with two inputs x and y and one output f is shown in Figure 2. For this system, a common rule set with two fuzzy if-then rules has the following form (37), (38):
Anfis architecture (37).
where p1, q1, r1, p2, q2, r2 are the consequence parameters to be determined during the training stage and A1, A2, B1 and B2 are the premise parameters. Figure 2 shows that ANFIS structure consists of a five layers and each layer has different nodes.
In the ANFIS architecture in Figure 2, a circle indicates a fixed node whereas a square indicates an adaptive node. The functionality of the nodes in each layer can be summarized as follows (37):
Layer 1: In this layer, the fuzzification process takes place. Its nodes are adaptive and its output forms the membership values of the premise part as follows
Layer 2: The nodes in this layer are fixed. The output of its nodes computes the firing strength of the fuzzy rule as follows
Layer 3: The nodes of this layer are fixed. They perform a normalization of the firing strength from the previous layer as
Layer 4: Every node in this layer is an adaptive node. The output of each node is calculated as
where pi, qi and ri are consequent parameters.
Layer 5: The single node in this layer computes the overall output as the summation of all incoming signals from layer 4:
As mentioned above, ANFIS architecture has two adaptive layers (i.e. layer 1 and 4). Layer 1 incorporates adjustable parameters that are called premise parameters and are related to the input membership function. Also, layer 4 has three adjustable parameters (pi, qi and ri) called consequent parameters. Thus, an important issue in ANFIS modeling is to tune all the adjustable parameters including the premise and consequence parameters to make the output of the ANFIS matches the training data.
3.2. Shuffled frog leaping algorithm for learning of all parameters of ANFIS
After determining the ANFIS structure and initialization, its parameters must be tuned. In order to avoid trial and error, a self-tuning process which employees the SFL algorithm is proposed for network learning. The SFL algorithm is a memetic meta-heuristic method that is derived from a virtual population of frogs in which the individual frog represents a set of feasible solutions (39). Each frog is distributed in a different subset of the whole population referred to as a memeplex. The different memeplexes are considered as a different culture of frogs that are located at different places in the solution space. Each culture of frogs performs simultaneously an independent deep local search using a PSO-like method. To ensure global exploration, after a deterministic number of memeplex evolution steps, information is passed between memeplexes in a shuffling process. Shuffling improves frog ideas’ quality after being infected by the frogs from different memeplexes. Moreover, to improve information, random virtual frogs are generated and substituted in the population if the local search cannot find better solutions. After this, local search and shuffling processes (global relocation) continue until defined convergence criteria are satisfied. The design procedure of the SFL approach is described as follows (39):
Step 1. The initial population of “N” frog is randomly generated within the feasible space Ω i.e. P={X1, X2, …, XN} denotes initial population. In this step, to solve S-dimensional problems, the position of the “ith” frog is represented as Xi=(xi1, xi2, …, xiD). Also, the suitable fitness function is defined to evaluate the position of each frog.
Step 2. In this step, the performance of each frog is computed based on its position, and then the frogs are sorted in a descending order according to their fitness. Finally, the entire population is divided into m memeplexes, each of which consisting of n frogs, i.e. N=n×m. In this process, the first frog goes to the first memeplex, the second frog goes to the second memeplex, frog m goes to the mth memeplex, and frog m + 1 back to the first memeplex, and so on.
Step 3. The position of the ith frog which is denoted by Di is adjusted according to the difference between the worst fitness frog (Xw) and the best fitness frog (Xb) by the following function:
where rand () generates a random number between 0 and 1.
Step 4. During memeplex evolution, the worst frog Xw leaps toward the best frog Xb. According to the original frog leaping rule, the position of the worst frog is updated as follows:
where Dmax is the maximum allowed change of frog’s position in one jump.
Step 5. If this repositioning process in step 4 produces a frog with better fitness, it replaces the worst frog, otherwise, the calculation in steps 3 and 4 are repeated with respect to the global best frog (Xg), (i.e. Xg replaces Xb). If no improvement becomes possible in this case, then a new frog is randomly generated to replace the worst frog. This evolution process is continued for a specific number of iterations (39), (40).
To learn all the parameters of ANFIS using the SFL algorithm, it is assumed that there are L input–output pairs (x(l), y(l), y(l)), l=1, 2, …, L. Now, the objective is to develop the neuro-fuzzy basis functions expansion such that the error between the ANFIS output
The obtained parameters by the SFL algorithm are used during modeling and prediction tasks.
3.3. Modeling tensile modulus of PA-6 nanocomposites using ANFIS
In this section, ANFIS is constructed to model the relation between the NC content, melt temperature (MT), SS and FR as inputs and the tensile modulus (M) as output. In this regard, 16 fuzzy rules that model the data behavior are defined to construct a multi input-single output (MISO) model that has four inputs and one output. Once the type and number of membership functions are determined, a Sugeno type fuzzy inference system is initially constructed to design the ANFIS model. An error is realized from the difference between the desired output and the ANFIS output. Then, SFL-based learning algorithm is applied to find the optimum values of the premise and consequence parameters, respectively, in such a way to minimize the error between the input and output pairs. The process of training is repeated until the desired error criterion is achieved. Figure 3 shows the constructed ANFIS structure.
Proposed ANFIS structure.
4. Results and discussion
The tensile modulus of PA-6 nanocomposites is predicted using the ANFIS model learned by the SFL algorithm. The multi-input single-output ANFIS model consisting of four input variables including NC, MT, FR and SS at three levels that are chosen based on the primary experiment was developed to predict the tensile modulus as an output variable. The Sugeno-type fuzzy inference system was generated by using 16 fuzzy rules that describe the relationship between input and output variables. After network construction, all of its linear and nonlinear parameters including weights of the network and parameters of the membership functions should be learned, online. The objective is to learn the ANFIS structure for modeling and predicting the tensile modulus based on the training data set.
As a first input variable, NC content has been taken at three levels of 2, 4 and 6 phr. As a second variable, MT has been changed from 235 to 245 and 255°C. FR has been varied from 7 to 9 and 11 rpm. As a last one, the levels of SS are set to 450 to 600 and 750 rpm. In this work, 27 samples based on the Box-Behnken design of experiments were produced using a twin-screw extruder and processing conditions reported in Table 1. Seventeen data points are determined as a training data set and six data points, which are different from the training data, are chosen as a test data set to test the proposed model.
In fuzzy modeling, input membership functions can be described by different shapes including triangular, sigmoidal, bell-shaped, etc. In this work, Gaussian type fuzzy membership functions were used to describe each input variable where the width and location of them will be optimized, automatically. The performance of the developed model was evaluated by the R2 and MAPE factors.
Simulation results are shown in Figures 4–7. Figure 4 (A) and (B) shows the actual and predicted values of tensile modulus at the training and testing phase, respectively. A very good accuracy is observed at the training and testing phase. Also, the predicted values versus the actual experimental ones at both phases are depicted in Figure 5. As can be seen from Figure 5, R2 factor is found as R2>0.98 for each phase.
Actual and predicted tensile modulus using ANFIS: (A) training data and (B) testing data.
Correlation between experimental and ANFIS predicted values for tensile modulus: (A) using training data, (B) using testing data.
Correlation between experimental and ANN-predicted values of tensile modulus: (A) using training data, (B) using testing data (32).
Tensile modulus surfaces versus NC content and melt temperature: (A) proposed scheme, (B) Taguchi-optimized ANN, (C) RSM method.
In order to show the effectiveness of the proposed scheme, the results were compared by the Taguchi-optimized ANN and RSM models. The model was applied to the training and testing data set and the results were presented in Figure 6. To highlight the superior performance of the proposed scheme, R2 and MAPE values for each method have been reported in Table 2. Reported statistical results indicate the superior performance of the proposed scheme in the sense of the good agreement between experimental and predicted values by the proposed ANFIS model. Therefore, it seems reasonable to conclude that the proposed model can be a suitable tool for predicting the tensile modulus of the PA-6 nanocomposites.
Figures 7–9 show the effect of the input factors on tensile modulus. The results are reported for the proposed scheme, the Taguchi-optimized ANN (32) and RSM (35) models. From Figure 7, the reported results show the nonlinear map between the NC content and MT as input variables and TM as an output variable. However, the predicted output by the proposed scheme and the ANN model is more similar than the RSM model to the experimental results. For example, from Table 1, sample S3, NC=6 and MT=255 results in TM=3498, while the obtained result from the RSM model is TM=3600. Also, it can be seen from Figure 8 that high NC content and high FR result in a high tensile modulus, while comparison of the predicted results with the experimental results shows that the proposed scheme provides an accurate prediction. For example, from Table 1, sample S7, FR=11 and NC=6 results in TM=3985. From Figure 8, only the proposed model predicts this value while the considered ANN and RSM predict 3800 and 3830, respectively. Figure 9 shows the nonlinear behavior between the SS and NC content as input variables and TM as an output variable. Although the general behavior obtained from each scheme is similar the obtained behavior by the proposed scheme predicts certain features that are not obtained from the Taguchi-optimized ANN and RSM results. As the proposed model integrates the properties of inference property of the fuzzy systems, the learning ability of NNs and optimization capability of evolutionary algorithm for modeling purposes. Owning to the mentioned properties, the proposed model uses expert knowledge to describe the nonlinear process in the term of fuzzy rules and then invokes the SFL algorithm as an evolutionary algorithm to learn all the parameters of the networks.
Tensile modulus surfaces versus NC content and feeding rate: (A) proposed scheme, (B) Taguchi-optimized ANN, (C) RSM method.
Tensile modulus surfaces versus NC content and screw rate: (A) proposed scheme, (B) Taguchi-optimized ANN, (C) RSM method.
In summary, the reported results verify the superiority of the proposed scheme over ANN and RSM models in terms of prediction accuracy. They show that the proposed scheme has better ability to capture the input-output relation based on the training data set; thus, it can capture the nonlinear behavior of the tensile modulus and provide a more precise representation of the actual response-factor relationship.
5. Conclusion
The multi-input single-output ANFIS structure was developed to predict the tensile modulus of PA-6 nanocomposite samples. The Sugeno-type fuzzy inference system was constructed based on the expert knowledge; then the SFL algorithm was invoked to learn all linear and nonlinear parameters of the network. The performance of the developed model was evaluated during the training and testing phases. Good agreement was observed between the predicted and experimental results. To highlight the superior performance of the proposed scheme, the obtained results were compared by the Taguchi-optimized ANN and RSM model. The reported results verify the superior performance of the proposed model than other methods. So, it can be concluded that the proposed ANFIS model, which combines inference property of fuzzy systems, learning ability of NNs and optimization capability of evolutionary algorithms, is a proper model for accurate prediction of tensile modulus of PA6 nanocomposite samples.
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