1. Introduction
Titanium alloy TC18 (Ti-5Al-5Mo-5V-1Cr-1Fe) possesses the common excellent performance of both alpha phase and beta phase titanium alloy, such as high strength to weight ratio, high toughness, high hardness, high corrosion resistance, being non-magnetic and so on, with a new style alloy (alpha + beta). It has been widely used in aviation, biomedical, automotive fields, etc. [
1,
2]. However, there exist enormous challenges for traditional machining approaches due to its properties, including poor heat transfer performance, work-hardening and unstable chemical reaction, and deformation under high temperature conditions, which tends to cause serious tool wear, even much lower durability and shorter life than expectancy [
3,
4,
5]. Moreover, some researches showed that the quality of the machined surface of the work piece has significant influence on its mechanical properties, especially upon fatigue properties [
6,
7,
8]. To address these issues, some researchers used an improved genetic algorithm to optimize the milling parameters of TC18, and studied the forging process parameters of TC18 based upon the BP neural network, but it did not overcome the negative effect of the poor property of TC18 on the quality of the traditional processing technology, such as the heat affected zone, etc. [
9,
10]. Therefore, there is an urgent need for an advanced processing method to solve various problems that arise in the processing of TC18.
ASJ technology is one of the fastest growing and most advanced non-traditional processing technologies. It has the advantages of no thermal effects, no residual stress, good incision quality, high applicability of materials, being environment friendly and highly competitive in material processing [
11]. Wang studied the mechanism of kerf width and kerf angle formation during abrasive water jet machining [
12]. Azmir used the Taguchi experimental method and variance analysis to study the influence of processing parameters on the kerf angle upon the cutting of glass/epoxy composite laminate, and concluded that the type of abrasive is the most important controlling factor [
13]. Alberdi established a mathematical model based on pressure, the mass flow of the abrasive, target distance and transverse velocity processing parameters, which is used to predict the profile produced by AWJ cutting 1075-T651 [
14]. Feng used numerical simulations and experiments, concluding that the jet with the added polymer has better stability in air [
15]. Wang found that adding high polymer to the abrasive slurry to cut stainless steel would have better processing performance [
16]. Just as traditional machining relies on computer optimization control and an optimization of processing efficiency [
17,
18,
19], advanced computer algorithms can also be used to optimize the processing parameters of an abrasive water jet in order to obtain high-quality products. Azlan used an integrated system of SA and GA algorithms to optimize the parameters of the abrasive processing process [
20]. However, it depends on a great multivariate nonlinear regression model which is difficult to obtain. By considering the diameter of the focused nozzle and controllable process parameters such as work pressure, traverse speed and abrasive flow rate, Srinivasu modeled the artificial neural network to predict the depth of cut in the AWJ process, and also used a genetic algorithm to find out the optimal parameters combination [
21]. However, the accurate ANN (artificial neural networks) prediction modeling was constructed directly with enormous work and difficulty.
In order to study the effect of processing parameters on TC18, we used Taguchi’s orthogonal method to carry out the experiments. To optimize the machining process by ASJ for TC18, and at the same time taking into account the stability of the jet pressure in the experiment, we provided some new measures to analyze the experimental data. Firstly, a multivariate nonlinear regression model was established, and the reliability of the prediction model was verified by using a mathematical statistics formulae (MAPE, MSE and R
2) and some specific experimental data. Based on the verification results, the model was only used to determine the main influencing factors of the experiment, which indicated that Azlan’s method [
20] was not applicable here. We comprehensively utilized the good methods proposed by Azlan [
20] and Srinivasu [
21], meanwhile avoiding the restrictive conditions in [
20] that it must rely on a great nonlinear prediction model, and solved the difficult problem of directly establishing the neural network prediction model in [
21]. A back propagation artificial neural network (BP) prediction model, based on adaptive genetic algorithm (AGA) was established, and of which (BP-AGA) the validation was checked by using the same method as above. Then this study compared the multiple nonlinear regression method with the neural network method optimized by the adaptive genetic algorithm. It is found that BP-AGA is easier to model, offers better robustness and is more accurate. Finally, the BP-AGA and simulated annealing algorithm (SA) optimization technology were used to form a set of prediction systems, called integrated SA-BP-AGA. Through this integrated system, the best kerf angle and the parameters affecting the kerf angle were obtained. The study results can improve the performance for TC18 machining by ASJ.
4. Predictive Model of Kerf Angle Based on ANN-AGA
It can be seen from the above calculation results that the multivariate nonlinear regression is not good at prediction. Therefore, a new prediction model is established.
4.1. Methodology
During neural network training, network structure parameters and initial thresholds and weights determine the training duration and network quality of the network to a large extent. Due to the nature of the “black box” of neural networks, it leads to blindness in debugging and low training efficiency.
However, by using the AGA method to find the optimal initial training thresholds and weights of the neural network, the blindness of debugging is reduced to a certain extent, the efficiency of network training is greatly improved, and the quality of the network is indirectly improved. Finally, the test results are analyzed by the same statistical formulae (R2, MSE and MAPE).
An adaptive genetic algorithm optimizes the BP neural network flow as shown in
Figure 6.
After debugging the structure of neural network and random initial weights and thresholds, it is supplemented by the adaptive genetic algorithm to improve efficiency and quality. This genetic algorithm mainly includes chromosome coding, selection operation, mutation operation, crossover operation and fitness operation. Among them, the crossover ratio (pc) and the mutation ratio (pm) in the parameters of the genetic algorithm play very important roles in the performance of the algorithm. If the fixed pc and pm values are adopted, it is difficult to adapt to the change of population, and sometimes leads to the evolution of the past. In this paper, an adaptive algorithm based on Srinvivas is proposed. The pc and pm in the algorithm can change automatically with fitness values, which can maintain group diversity and ensure convergence, as shown in Equation (16) below.
where, pc
max, pc
min, pm
max and pm
min are the maximum, minimum crossover rate and mutation rate, respectively. f
max, f
avg are the maximum fitness value and the average fitness for each generation of population. f’ is the larger fitness value of the two individuals to cross, and f is the fitness value of the variant individuals.
Firstly, the chromosome was constructed by a binary encoding of the initial threshold and weight of the neural network, and the prediction error of the neural network was used as our fitness value. Then the individual difference was produced by chromosomal variation and cross, and the selection of the wheel was executed by the principle of survival of the fittest.
The optimal threshold and weight of the neural network were found. Finally, based on these parameters, the neural network was trained to get the best network model.
4.2. Neural Network Optimized by Adaptive Genetic Algorithm Based on Kerf Angle
It was finally determined that the structure of ANN is 4-11-1, that is, with 4 input nodes, 11 hidden nodes and 1 output nodes. The adaptive genetic algorithm parameters included, the population size is 24, the maximum cross rate is 0.7, the minimum cross rate is 0.1, the maximum mutation rate is 0.05, and the minimum mutation rate is 0.01. Similarly, the 22 sets of data selected above were used for training. A neural network prediction model with high accuracy was established by MATLAB 2016. The structure of the neural network is shown in
Figure 7. The regression performance of the neural network is shown in
Figure 8.
From
Figure 8, it can be found that the multivariate correlation coefficient R of training and testing is 0.97172 and 0.99783, respectively, with a high goodness of fit, listed in the
Table 4. Then, the remaining three groups are still used as checking groups, which are calculated by the statistical formulae R
2, MSE and MAPE. The results are listed into the
Table 5.
5. Comparison of the Two Analysis Methods
Currently, the main common methods for multivariate regression analysis are Forward, Forward, Backward and Stepwise. But in most cases, they are difficult to fit and are prone to multiple collinear troubles [
24]. Therefore, a large amount of time has to be used to perform various transformations on the data to obtain a higher coefficient of multiple correlation and determination.
However, using the artificial neural network to learn experimental data, and then predicting, only the optimization algorithm is needed to optimize the initial threshold and weight of the network, and then the simple network structure parameter adjustment can achieve the purpose.
From
Table 4 and
Table 5, the fitting quality and prediction performance of the regression model are not as good as the training quality and prediction performance of neural network model. Moreover, compared with the statistical analysis results of training quality and prediction performance, it can be seen that the neural network has better robustness and fault tolerance than the fitting regression method.
6. The Integrated SA-BP-AGA Optimization
Based on the trained neural network prediction model, a simulated annealing algorithm was used to find the optimal parameter combination of the minimum processing kerf angle. The integrated SA-BP-AGA of the above description is shown in
Figure 9.
The simulated annealing algorithm is a random search technique that is able to escape local optima using a probability function [
25]. SA is a relatively mature algorithm, widely used in VLSI (Very Large Scale Integration) design, image recognition and neural network computer research. It can be decomposed into three parts: Solution space, objective function and initial solution. Here, the solution space is composed of the ranges of various processing variables. The objective function is the AGA-optimized neural network, and the initial solution is set as the best parameter group that appears in the experiment, which is the 24th in the
Table 3. The optimal solutions of the MATLAB Optimization Toolbox is given in
Figure 10 on the base of these criteria as listed in
Table 6.
As shown in
Figure 10, the theoretically optimal solution was observed that the minimum kerf angle is 6.9425 × 10
−5. The set value of process parameters that lead to the minimum θ value are 31.5 mm/min for traverse speed, 0.7806 mm for standoff distance, 0.07841% for slurry concentration and 33.73 MPa for jet pressure.