Abstract
Purpose
This study aims to present a novel methodology for the evaluation of tribological properties of new nanocomposites with the A356 alloy matrix reinforced with aluminium oxide (Al2O3) nanoparticles.
Design/methodology/approach
Metal matrix nanocomposites (MMnCs) with varying amounts and sizes of Al2O3 particles were produced using a compocasting process. The influence of four factors, with different levels, on the wear rate, was analysed with the help of the design of experiments (DoE). A regression model was developed by using the response surface methodology (RSM) to establish a relationship between the observed factors and the wear rate. An artificial neural network was also applied to predict the value of wear rate. Adequacy of models was compared with experimental values. The extreme values of wear rate were determined with a genetic algorithm and particle swarm optimization using the RSM model.
Findings
The combination of optimization methods determined the values of the factors which provide the highest wear resistance, namely, reinforcement content of 0.44 wt.% Al2O3, sliding speed of 1 m/s, normal load of 100 N and particle size of 100 nm. Used methods proved as effective tools for modelling and predicting of the behaviour of aluminium matrix nanocomposites.
Originality/value
The specific combinations of the optimization methods has not been applied up to now in the investigation of MMnCs. In addition, using of small content of ceramic nanoparticles as reinforcement has been poorly investigated. It can be stated that the presented approach for testing and prediction of the wear rate of nanocomposites is a very good base for their future research.
Keywords
Citation
Stojanović, B., Gajević, S., Kostić, N., Miladinović, S. and Vencl, A. (2022), "Optimization of parameters that affect wear of A356/Al2O3 nanocomposites using RSM, ANN, GA and PSO methods", Industrial Lubrication and Tribology, Vol. 74 No. 3, pp. 350-359. https://doi.org/10.1108/ILT-07-2021-0262
Publisher
:Emerald Publishing Limited
Copyright © 2022, Emerald Publishing Limited
1. Introduction
Properties of aluminium metal matrix nanocomposites (MMnCs) depends highly on the distributions of nano-sized particles in the matrix (Bahmani et al., 2018; Varol et al., 2017; Velickovic et al., 2019b). A similar problem has been noticed at metal matrix composites (MMCs) where micro-sized reinforcements are used. Despite the improvement of tribological properties over the matrix alloy, the MMCs also had some shortcomings related primarily to their machinability, increased porosity and reduced ductility (Canakci et al., 2013). For this reason, over the past few years, the researchers are more focused on the investigation of MMnCs, which also could improve the tribological properties while maintaining good ductility.
Identifying the relationship between the involved factors and the observed output is greatly facilitated by applying optimization methods, as well as by using software tools to avoid a long-term process. Priyadarshi and Sharma (2016) applied the response surface methodology (RSM) to study the workability of aluminium nanocomposites. The influence of the type and amount of the reinforcement on the tangential cutting force was considered and the optimal conditions were obtained.
A combination of RSM and analysis of variance (ANOVA) was used for the optimization of the tribological characteristics of nanocomposites, tested under dry sliding conditions by Manivannan et al. (2018). The 6061 aluminium alloy matrix reinforced with aluminum oxide (Al2O3) and Gr particles were produced by the ultrasonic casting method. The factors influencing the wear and coefficient of friction were normal load, sliding speed and sliding distance. Ekka et al. (2015) applied the Taguchi method to test the effects of silicon carbide and Al2O3 nanoparticles amount, normal load, sliding speed and sliding distance on the behaviour of nanocomposites with 7075 aluminium alloy matrix, produced using the stir casting technique. Using the ANOVA analysis, they observed that the normal load and sliding distance had the greatest influence on wear. Prediction of the wear rate was performed using the artificial neural networks (ANN) and the regression model. The optimization methods for wear rate were successfully applied and it was established that the ANN model was more efficient than the regression model.
Shabani et al. (2018), using a combination of the optimization methods, performed the factor optimization during the semi-solid processing of the nanocomposite with the A360 aluminium alloy matrix and TiC nanoparticles. Using an adaptive neuro-fuzzy inference system, the function objective was calculated and then minimized by using the particle swarm optimization (PSO). The applied methodology shows a reduced error between the experimental and predicted values, which confirms the validity of the mathematical model.
The research in this paper focuses on the application of the design of experiments (DoE) for wear tests and optimization methods for analysis and optimization of parameters that affect the wear of aluminium alloy-based nanocomposites. The RSM model has been applied to optimize and determine the level of factor values for obtaining the minimum and maximum wear rate. Likewise, the RSM model has been connected with a genetic algorithm (GA) and PSO to obtain the optimal conditions. A mathematical model has been developed using the RSM method and ANN. A back-propagation algorithm based on Levenberg–Marquardt (LM) is used to train the neural network. The obtained results are compared with the experimental results. A comparison of the obtained results has been made using different methods to validate the developed mathematical model.
2. Materials and applied methodology
2.1 Materials and wear testing
Nanocomposites were obtained using the compocasting method with mechanical alloying pre-processing, by varying the different amount of Al2O3 reinforcement particles (0.2, 0.3 and 0.5 wt.%) in the A356 aluminium alloy matrix (EN AlSi7Mg0.3). Sizes of the Al2O3 particles were approx. 30 and 100 nm. The chemical composition of the A356 alloy was (wt. %): Al-7.20Si-0.18Fe-0.02Cu-0.01Mn-0.25 Mg-0.01Zn-0.02Ni-0.11Ti. The equipment used for semi-solid processing (compocasting) is described in more detail elsewhere (Vencl et al., 2008), as well as the parameters of the production process (Veličković et al., 2019a).
Experimental research of the wear characteristics was carried out based on Taguchi orthogonal matrix L24. The experiments were performed on a tribometer with a block-on-disc contact geometry, under lubricated conditions. The gear oil (ISO VG 220, ISO L-CKC/CKD) was used for lubrication. The disc material was steel 42CrMo4, with a hardness of 50–55 HRC and the tested nanocomposites were block materials. The tests were carried out on the sliding distance of 1,000 m at the sliding speeds of 0.5 and 1 m/s and normal loads of 40 and 100 N. Wear scars on the blocks were measured after each test to calculate the wear volume.
DoE was used to study the wear rate of nanocomposites and input factors (reinforcement content, sliding speed, normal load and reinforcement size). The optimization methods, RSM, ANN, PSO and GA, were used to identify the significance of factors, determine the extreme values and predict the behaviour of materials.
2.2 Modelling using response surface methodology
The RSM is a compilation of scientific and numerical systems and uses a regression analysis technique, statistical analysis and designing experiments. The main objectives are:
determination of the relationship between quality characteristics and input level to understand the influence of factor changes on the response value; and
determination of the optimal system operating conditions or determination of the range of factors that meet the operational needs (Huang et al., 2016).
The most advanced applications of RSM are reflected in certain situations where several input variables potentially influence the response. The input variables are called factors and they are controlled by researchers. The exact shape of the objective function is not known, so a linear or quadratic polynomial equation is set up to establish the relationship between the factor (X1, X2, … Xk) and the response (у). A second-order model is generally used in the RSM:
2.3 Modelling using artificial neural network
ANN is a mathematical technique that mimics the operations of the human brain. It uses mutually connected nodes called neurons to transmit information. The ANNs have found their application in solving complex scientific and engineering problems because they shorten the time of the experiment and are used to predict the behaviour of the output based on the input factors. Each ANN consists of three layers: input layer, hidden layer and output layer. The number of neurons in the input layer is equal to the number of input factors, the number of output neurons is equal to the number of output factors, while the hidden layer can consist of several layers and the number of neurons in each hidden layer is flexible. The structure of a neural network can be presented as (Yusri et al., 2018):
ANNs are not programmed, but their training is performed, which means that it can take a long time before ANN is ready for use. Before the training starts, the user sets the input and output parameters and then the training is performed to get the output as close as possible to the set parameter. After training of the network, it is used for the prediction of the output for inputs that were not in the training data set. The ANNs are reliable for prediction within the trained data range, but they are not reliable for prediction beyond the trained data range (Canakci et al., 2012).
2.4 Optimization using genetic algorithm
The GA is an optimization method that is widely used to solve numerous problems (Elsayed et al., 2014). The method is based on the principles of natural selection and belongs to the group of population-based metaheuristic methods. The working principle involves the creation of a population that represents a set of potential solutions. The solution converges to the optimum by an iterative process of creating new generations in which better individuals survive. The initial population was created from randomly selected values from the allowed domain.
The basic phases of GA are selection, crossover and mutation. By applying these procedures, the new individuals are obtained by the initial values, while the solution converges to the optimum. The optimal solution is the value of the best individual in the population. Crossover is a phase of an algorithm that represents the process of combining two individuals (as parents), where new individuals are obtained and they retain the characteristics of both parents. A mutation is a process of a complete change of an individual, where a completely new individual is obtained. The mutation process is very important, primarily to avoid premature convergence. The selection, crossover and mutation processes are repeated throughout each generation until the stopping criterion is met. The method is also used to solve the problem of multi-criteria optimization (Long, 2014).
2.5 Optimization using particle swarm optimization technique
The PSO method belongs to the group of swarm intelligent-based algorithms (Shin and Kita, 2014; Mahmoodabadi, 2014). The basic characteristic of this method is searching all over the allowed domain. The algorithm is also good because it contains only one phase. The working principle is based on the so-called particle acceleration, the distance of the particle position from the best value of the given particle (best – xp,i) and the position of the globally best particle (global best – xg,i). The position of a particle at a given moment is a potential solution. Only the best position is adopted and it is transmitted through the iterative optimization process. Each new solution depends on these two components, velocity and position. The position is defined with xi, while the velocity is represented by νi. The new value is obtained by the following equation:
The total size of a population is calculated from:
3. Results and discussion
3.1 Microhardness results
The hardness of nanocomposites was measured by the Vickers method with an applied load of 5 N, according to the ASTM E384 standard. The loading time was 15 s. Figure 1 shows a diagram of the average hardness values at room temperature.
An increase in hardness is observed with the addition of reinforcements from 0.2 to 0.5 wt.% with the size of 100 nm. Nanocomposites with the reinforcement size of approx. 30 nm show a decrease in hardness with the addition of 0.2 and 0.3 wt.%, while an improvement was recorded with the addition of 0.5 wt.%. The increase in microhardness for nanocomposites with 0.5 wt.% Al2O3, regardless of the reinforcement size, can be attributed to the applied compocasting process, i.e. better distribution of nanoparticles in the matrix. The nanoparticles were used to prevent the movement of the dislocation. The forced movement of the indenter increases the compression of the particles near the indenter, and this leads to an increase in the hardness of the nanocomposites. In addition, the increase in hardness is a consequence of the good adhesive bond between the ceramic nanoparticles and the metal matrix. The improvement in the hardness of nanocomposites compared to the matrix alloy has been proven by others researchers (Karbalaei Akbari et al., 2013; Ashok Kumar et al., 2018).
3.2 Wear testing results
The experiments were performed by the projection matrix to obtain the wear data in a controlled manner. The influence of factors on the wear rate was monitored by four input factors. The considered factors and their levels are shown in Table 1.
The experiment plan was based on the orthogonal matrix L24 with corresponding responses, i.e. the wear rates which are shown in Table 2. Analyzing nanocomposites with different sizes of the Al2O3 reinforcement in the A356 alloy matrix, it can be noticed that the improvement in the wear resistance of the nanocomposite with the particle size of 100 nm is higher than the nanocomposites with an particle size of approx. 30 nm at any reinforcement amount.
3.3 Modelling and optimization using response surface methodology
RSM is used to analyse the experimental results and provides quantitative measurements of possible interactions between factors, which are difficult to obtain by other techniques. It consists of a group of mathematical and statistical approaches in which the considered response depends on several important variables, and the objective of this method is to optimize and develop the corresponding mathematical model for the observed response (Manivannan et al., 2018). The mathematical model for the wear rate is represented as:
The results of the regression analysis are shown in Table 3. When R2 is close to the unit, this indicates a good correlation between the experimental and predicted values. Since the value of R2 was 0.95, it can be concluded that the proposed model of response surfaces is adequate for expressing the wear rate.
The validity of the developed quadratic model and statistical significance of regression coefficients were determined using the ANOVA. The statistical analysis of the experimental results of the wear rate of the nanocomposites is performed using the ANOVA at a 95% confidence level (i.e. a significance level of 0.05). ANOVA analysis is used to determine the effect of each factor on the wear rate and the results are given in Table 4. This table contains the degrees of freedom, the sum of the squares, the mean sum of the squares, the F-statistical value and the probability values. Generally, the statistical significance for the output/response is determined using p-values and then using the F-statistical values of ANOVA. The sources (factors) with the F-value higher than the value F from Table 4 and the value p lower than the significance level (here 0.05) are considered to have a significant influence on the wear rate (response).
It is noted that all observed factors influence the wear rate because their p-value is lower than 0.05. Likewise, at the 95% confidence level, it is noticed that there is also the influence of interactions B*C, B*D and C*D, which should not be neglected in the research of the wear characteristics of the nanocomposites. The assumption of normality is checked using a normal probability plot of the residuals (Figure 2).
Based on Figure 2, it can be concluded that the proposed model of the wear rate follows the normal distribution. Two-dimensional contour plots (Figure 3) are used to analyse the influence of factor interaction on the wear rate, where two factors are varied simultaneously, while two other factors remain at a constant central level. In Figure 3, the green light colour is for a minimum value, the green is for the average value and the dark green is for the highest value for the wear rate [Figures 3(a) and 3(d)]. The blue colour, in Figures 3(b) and 3(c), shows the lowest value of wear rate in the plot, while dark green shows the highest wear value. In general, the diagrams indicate the connection and significance of the factors and outputs. It can be concluded that the minimum wear is achieved [Figure 3(a)] by the particle size of 100 nm with a reinforcement content of more than 0.35 wt.%.
The behaviour of the wear rate depending on the reinforcement content and the normal load can be seen from the plot shown in Figure 3(b), where it is observed that the dependence is in the form of a parabola with the increased reinforcement content. A great number of studies prove an improvement in mechanical and tribological characteristics of nanocomposites, reinforced with ceramic particles, compared to the matrix alloy (Muley et al., 2015; Han et al., 2017; Surendran et al., 2017).
3.4 Worn surface analysis
The wear scars on the tested materials are the result of direct contact with the counter-body elements of the tribometer (steel disc). The wear tests are followed by the analysis of the obtained wear scars to define the dominant wear mechanism. The morphology and composition of the worn surfaces were analysed by scanning electron microscopy (SEM) and energy dispersive spectroscopy (EDS). The worn surfaces of all nanocomposites had a similar appearance. As a representative, the worn surface of the nanocomposite A356/Al2O3 with an average particle size of 100 nm is shown in Figure 4.
Analyzing the nanocomposites with different average sizes of the Al2O3 reinforcement, it can be noticed that the improvement in the wear resistance of the nanocomposite with the particle size of 100 nm is higher than the nanocomposites with the particle size of approx. 30 nm at any reinforcement content. According to Figure 4, it can be concluded that abrasion is the dominant wear mechanism for all tested nanocomposites, with adhesion as a secondary wear mechanism. The nanocomposite with 0.5 wt.% and particle size of 100 nm shows the highest wear resistance. By analyzing the results, it is concluded that the wear rate of all tested nanocomposites increases with the increase of the normal load and sliding speed. The influence of the sliding speed on the wear rate is a consequence of the mixed lubrication regime, while the influence of the normal load is expected in accordance with other researches. The conclusion for the mixed lubrication regime is based on the values of the coefficient of friction. The presence of the transferred material from the steel disc to the tested nanocomposites was observed during SEM analysis and confirmed by the EDS analysis (Figure 5).
3.5 Artificial neural networks results
An ANN with architecture 4–5–1 was modelled, which means that this network has 4 neurons in the input layer (considered influential factors), 5 in the hidden layer and one neuron in the output layer (wear rate) with log-sigmoid transfer function (LOGSIG). The feed-forward back propagation neural network and the LM algorithm (TRAINLM) are widely used for the development of neural networks (Varol et al., 2013).
After the modelling and training of the network, the regression coefficient for training, validation and testing was obtained, as well as the overall regression coefficient of the trained network (Figure 6). The total regression coefficient is close to 1 (R = 0.99246), which means that the correlation of the experimental data and the network output is good. Based on ANN analysis it can be concluded that this network can be used, with high reliability, to predict the wear rate.
Based on the developed mathematical model, using RSM and ANN analysis, it is possible to predict the wear rate of the nanocomposites within the limits of the experiment. Figure 7 shows the wear rate values obtained by the experiment, by the prediction using ANN and by prediction using the RSM model. The negative values of wear rate are obtained by using the developed mathematical model, although this phenomenon is not physically possible. These values can be explained by the approximation of the real model in given test conditions. The negative values that occur are close to zero, so they can be considered as the relevant minimum values of wear rate. Based on this comparative presentation of the results, it can be concluded that there is a satisfactory correlation between the obtained results.
3.6 Results obtained from genetic algorithm and particle swarm optimization
The optimization problem formulation was performed based on the response surface model presented in equation (7), which was the objective function. The identified factors represent the optimization variables. The wear rate depends on the four mutually dependent variables that were figured in this function. The objective function was minimized following the ranges on the variables: 0.2 ≤ A ≤ 0.5, 0.25 ≤ B ≤ 1, 40 ≤ C ≤ 100, 30 ≤ D ≤ 100. GA was used for optimization in the first case. The modulation algorithms were oriented to find the optimum value. The population size was set to 50, while the maximum number of generations was 200. Crossover probability was 0.85, while mutation probability was 0.01. The 30 consecutive repetitions of simulations were performed per each GA and PSO method, to confirm the optimal value by each repetition.
The PSO algorithm was used for the optimization in the second case. The optimization conditions were set to achieve the optimum value for 30 consecutive repetitions. For the operation of the PSO algorithm, the number of iterations was defined as 200 and the number of particles as 50. With the use of equation (5) intensity of particles were calculated as: ωmax = 0.9 and ωmin = 0.4.
The optimization was performed for the RSM model, as the objective function (Table 5). In addition, ANN prediction was performed and these values were compared with the optimal ones (Table 6). All predictions indicate that the wear rate is the lowest for the same values of variables, but the wear rate values differ depending on whether the optimization is performed based on RSM or the prediction by using ANN. The comparison of the maximum wear rates (Table 6) was performed for GA, PSO, ANN and experimental data. The deviations of the obtained values from the experimental data were also presented.
4. Conclusions
The motivation of this research was to develop a new nanocomposite with an aluminium alloy matrix, as well as to test its characteristics and to determine the potential direction of the future development.
The wear rate values range from 2.64 × 10−7 to 3.40 × 10−4 mm3/m. Nanocomposites with a particle size of 100 nm showed higher wear resistance than nanocomposites with particles of 30 nm. Improvements in the microhardness of nanocomposites with 0.5 wt.% particles over the matrix alloy were approx. 4.8 and 2.4% for particle size of 100 and 30 nm, respectively.
The ANOVA analysis of the influential factors confirmed that the wear rate was influenced by sliding speed, normal load and reinforcement size, while the reinforcement content influenced wear rate to a lower extent. The lowest wear would be obtained if the nanocomposite has Al2O3 particle size of 60 nm with content above 0.35 wt.%.
An ANN model was developed based on the experimental results to predict wear rate in the limits of the testing parameters. Predicted values were compared to experimental results and there were certain deviations, but the trained network corresponded to the behaviour of the considered phenomenon. Likewise, a mathematical model (RSM model) was developed that defined the wear resistance of the nanocomposite depending on the influencing factors. The results obtained by the RSM model were verified by comparison with the experimental values with the coincidence of 87.03%.
By combining the GA and PSO optimization methods, it was possible to determine the extreme values of the wear rate. The highest wear resistance was for reinforcement content of 0.44 wt.%, sliding speed of 1 m/s, normal load of 100 N and reinforcement size of 100 nm, and the worst was for reinforcement content of 0.20 wt.%, sliding speed of 0.25 m/s, normal load of 100 N and reinforcement size of 30 nm. For the highest wear rate error of ANN prediction was 0%, while GA-RSM and PSO-RSM had 12.65% error compared to experimental results.
Using the combination of mentioned methods, it is possible to predict the behaviour of new material from the aspect of wear resistance. This procedure greatly shortens the testing time and expands the spectrum of conclusions when testing and developing new nanocomposites.
Figures
Input factors and their levels
| Factors | Unit | Level I | Level II | Level III |
|---|---|---|---|---|
| (A) Reinforcement content | wt.% | 0.2 | 0.3 | 0.5 |
| (B) Sliding speed | m/s | 0.25 | 1 | – |
| (C) Normal load | N | 40 | 100 | – |
| (D) Reinforcement size | nm | 30 | 100 | – |
Experimental results
| No. | Reinforcement content (wt.%) | Sliding speed (m/s) | Normal load (N) | Reinforcement size (nm) | Wear rate × 10–4 (mm3/m) |
|---|---|---|---|---|---|
| 1. | 0.2 | 0.25 | 40 | 30 | 1.088 |
| 2. | 0.2 | 0.25 | 40 | 100 | 0.506 |
| 3. | 0.2 | 0.25 | 100 | 30 | 3.404 |
| 4. | 0.2 | 0.25 | 100 | 100 | 1.123 |
| 5. | 0.2 | 1.00 | 40 | 30 | 0.533 |
| 6. | 0.2 | 1.00 | 40 | 100 | 0.320 |
| 7. | 0.2 | 1.00 | 100 | 30 | 0.844 |
| 8. | 0.2 | 1.00 | 100 | 100 | 0.711 |
| 9. | 0.3 | 0.25 | 40 | 30 | 0.970 |
| 10. | 0.3 | 0.25 | 40 | 100 | 0.450 |
| 11. | 0.3 | 0.25 | 100 | 30 | 2.530 |
| 12. | 0.3 | 0.25 | 100 | 100 | 0.783 |
| 13. | 0.3 | 1.00 | 40 | 30 | 0.228 |
| 14. | 0.3 | 1.00 | 40 | 100 | 0.036 |
| 15. | 0.3 | 1.00 | 100 | 30 | 0.623 |
| 16. | 0.3 | 1.00 | 100 | 100 | 0.095 |
| 17. | 0.5 | 0.25 | 40 | 30 | 0.914 |
| 18. | 0.5 | 0.25 | 40 | 100 | 0.345 |
| 19. | 0.5 | 0.25 | 100 | 30 | 2.165 |
| 20. | 0.5 | 0.25 | 100 | 100 | 0.689 |
| 21. | 0.5 | 1.00 | 40 | 30 | 0.089 |
| 22. | 0.5 | 1.00 | 40 | 100 | 0.003 |
| 23. | 0.5 | 1.00 | 100 | 30 | 0.542 |
| 24. | 0.5 | 1.00 | 100 | 100 | 0.013 |
Regression coefficients
| Term | Coef | SE coef | T-value | p-value |
|---|---|---|---|---|
| Constant | 0.612 | 0.102 | 5.99 | 0.000 |
| A | −0.2357 | 0.0623 | –3.78 | 0.003 |
| B | −0.4535 | 0.0513 | –8.84 | 0.000 |
| C | 0.3251 | 0.0513 | 6.34 | 0.000 |
| D | −0.3652 | 0.0513 | –7.12 | 0.000 |
| A*A | 0.219 | 0.124 | 1.77 | 0.102 |
| A*B | 0.0169 | 0.0612 | 0.28 | 0.787 |
| A*C | −0.0884 | 0.0612 | –1.45 | 0.174 |
| A*D | 0.0338 | 0.0612 | 0.55 | 0.590 |
| B*C | −0.2001 | 0.0509 | –3.93 | 0.002 |
| B*D | 0.2289 | 0.0509 | 4.50 | 0.001 |
| C*D | −0.1888 | 0.0509 | –3.71 | 0.003 |
R2 = 95.31%, R2(adj) = 91.00%, R2(pred) = 80.36%
Analysis of variance for wear rate
| Source | DF | Adj SS | Adj MS | F-value | p-value |
|---|---|---|---|---|---|
| Model | 11 | 15.1253 | 1.37503 | 22.15 | 0.000 |
| Linear | 4 | 11.3764 | 2.84409 | 45.82 | 0.000 |
| A | 1 | 0.8892 | 0.88917 | 14.33 | 0.003 |
| B | 1 | 4.8496 | 4.84962 | 78.13 | 0.000 |
| C | 1 | 2.4925 | 2.49250 | 40.16 | 0.000 |
| D | 1 | 3.1451 | 3.14509 | 50.67 | 0.000 |
| Square | 1 | 0.1948 | 0.19477 | 3.14 | 0.102 |
| A*A | 1 | 0.1948 | 0.19477 | 3.14 | 0.102 |
| 2-way interaction | 6 | 3.2269 | 0.53782 | 8.66 | 0.001 |
| A*B | 1 | 0.0048 | 0.00476 | 0.08 | 0.787 |
| A*C | 1 | 0.1298 | 0.12977 | 2.09 | 0.174 |
| A*D | 1 | 0.0190 | 0.01900 | 0.31 | 0.590 |
| B*C | 1 | 0.9605 | 0.96051 | 15.47 | 0.002 |
| B*D | 1 | 1.2575 | 1.25754 | 20.26 | 0.001 |
| C*D | 1 | 0.8554 | 0.85535 | 13.78 | 0.003 |
| Error | 12 | 0.7448 | 0.06207 | ||
| Total | 23 | 15.8701 |
Results of predicting the minimum value of wear rate using PSO, GA and ANN
| Optimization technique | Reinforcement content (wt.%) | Sliding speed (m/s) | Normal load (N) | Reinforcement size (nm) | Min. fitness function value – wear rate × 10–4 (mm3/m) |
|---|---|---|---|---|---|
| GA-RSM | 0.44 | 1 | 100 | 100 | –0.132 |
| PSO-RSM | 0.44 | 1 | 100 | 100 | –0.132 |
| ANN | 0.44 | 1 | 100 | 100 | 0.033 |
Comparison of results for maximum wear rate values
| Optimization technique | Reinforcement content (wt.%) | Sliding speed (m/s) | Normal load (N) | Reinforcement size (nm) | Max. fitness function value – wear rate × 10–4 (mm3/m) | Error (%) |
|---|---|---|---|---|---|---|
| Experimental | 0.20 | 0.25 | 100 | 30 | 3.40 | 0 |
| GA-RSM | 0.20 | 0.25 | 100 | 30 | 2.97 | 12.65 |
| PSO-RSM | 0.20 | 0.25 | 100 | 30 | 2.97 | 12.65 |
| ANN | 0.20 | 0.25 | 100 | 30 | 3.40 | 0 |
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Acknowledgements
This work has been performed as a part of activities within the projects TR 35021 and 451-03-9/2021-14/200105, supported by the Republic of Serbia, Ministry of Education, Science and Technological Development, and its financial help is gratefully acknowledged.