Machine-learning assisted compositional optimization of 2xxx series aluminum alloys towards tensile strength

High-strength 2xxx series aluminum alloys (Al-Cu system) have been favored by the aerospace and railway transportation industries. Traditionally, developing new materials with targeted properties is guided by extensive experiments and expert experience, causing the development process to be dismayingly slow and expensive. Here, a Kriging model-based efficient global optimization(EGO) lgorithm is applied to search for new 2xxx series aluminum alloys with high tensile strength in a huge search space. After four iterations, the alloy’s ultimate tensile strength increased by 60 MPa, which is higher than that of the best alloy in the initial data set. This study demonstrates the feasibility of using machine-learning to search for 2xxx alloys with good mechanical performance.


Introduction
2xxx series aluminum alloys (Al-Cu system) have been widely used as structural materials in the aerospace industry mainly due to their low density, high strength, and good formability [1,2]. With the rapid development of high-speed trains, 2xxx alloys is used as substitutes for steel and iron structural components in high-speed trains for achieving light-weighting purposes [2]. However, for some high load-bearing applications in highspeed trains (e.g., the axle box body and connecting rod), inadequate tensile strength at room temperature (RT) limits the use of 2xxx alloys. Therefore, it is necessary for 2xxx alloys to further improvement of the RT tensile strength and thus get more opportunities in new application areas.
Optimizing alloy composition is one of the most effective and common methods to improve the tensile strength of metals [3][4][5][6][7]. 2xxx alloys are precipitation-strengthened aluminum alloys and usually contain copper (Cu), magnesium (Mg), manganese (Mn) and titanium (Ti) alloying elements. The literature shows that adding yttrium (Y) can significantly improve the microstructure and mechanical properties of the alloy. Wang et al added different amounts of Y to 3 × 04 aluminum alloy. The results show that when the content of Y is 0.2 wt.%, the large skeleton eutectic in the alloy disappears and thick striped second phase becomes thin, leading to the best mechanical properties of the alloy [8]. Li et al added 0.10 wt.% Y to 2519 aluminum alloy and obtained finer θ-Phase and AlCuY metal compounds with high-temperature stability [9]. Wan et al studied the effect of adding Y on the microstructure of recycled Al-7Si-0.3Mg-1.0Fe casting alloy, and found that Y not only reduced the length and volume fraction of β-Fe phase but also refined the primary aluminum dendrite [10]. Based on this, we introduced trace Y in the alloy composition design process [8][9][10]. Relevant studies have shown that machinelearning can establish a quantitative relationship between material composition and performance [4,5,[11][12][13]. In a complete machine learning system, the three keywords 'data, model and performance' can be used to summarize their internal relationship. They play interconnected roles: data is the basis, performance is the result, and the model is the key. It is crucial to successfully select an appropriate model to establish the relationship between material composition and properties and thus to predict the properties of alloys. Reddy Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
et al successfully established a machine-learning model between 'composition + heat treatment' and performance prediction by combining a neural network model and genetic algorithm [11]. Ozerdem et al also established BP neural network model to predict the yield strength, tensile strength and elongation of Cu alloy [12]. Wen et al also proposed a performance-oriented material design strategy. By using machine-learning model and correction function, they successfully searched for alloys with high hardness in Al-Co-Cr-Cu-Fe-Ni dataset [13]. At present, many models are applied in the field of machine learning: neural network model, SVM support vector machine model, regression model and Kriging model. However, the neural network model and other models have some shortcomings, such as the probability of the neural model falling into local optimization and the inaccuracy of the iteration resultthat depends on the random initial weight value. More importantly' it is difficult to find a suitable kernel function for nonlinear problems for the SVM support vector machine model. Besides, it is not sensitive to existing data. In order to overcome the above shortcomings, the Kriging model is adopted in this work.
In addition, the research of machine learning methods on 2xxx series aluminum alloy is less, compared with the successful application of machine-learning methods to improve the strength of other alloys. In this work, our primary purpose is to prove the possibility of improving the strength of the 2xxx aluminum alloys by applying machine-learning. Therefore, the composition of 2xxx aluminum alloys is optimized by an efficient global optimization (EGO) algorithm based on the Kriging model.

Materials and methods
We adopt the design idea of 'building data set → cross-validation → constructing Kriging model → iterative optimization' to optimize the composition of 2xxx aluminum alloy, as shown in figure 1. A kriging model has constructed on the DACE [14] (Design and Analysis of Computer Experiments) toolbox, in which we choose zero-order polynomial and Gaussian functions as regression and correlation functions [15]. After the validation and establishment of an effective Kriging model, the optimization can be directly carried out according to the output results of the model. However, the problem lies in: (1) If the best value of the model output is directly selected, a large number of initial sample points are required to obtain the global best value, which is not desirable from the perspective of efficiency and economy. (2) If the Kriging model is updated only according to the optimal value of the model output, it is easy to fall into the local optimum because the predicted error of the model is not taken into account. We choose an efficient global optimization algorithm (EGO algorithm) based on Kriging to solve this problem. The adaptive point adding function in the EGO algorithm called 'Expected improvement (EI)' is mainly proposed for the problem of falling into local optimization. Suppose g max = max (g1, g2,K, g m ) is the maximum (optimal) value in the sample [16], then the improvement value at any point u is: In the formula, Φ(·) and j(·) are the cumulative distribution function and probability density function of standard normal distribution, respectively.
The predicted value and predicted error of the EI function to the model are monotonically increasing. When the predicted value is large, ] is relatively large, bringing the algorithm search to the maximum. When the predicted error is large, the algorithm searches for the location with a large predicted error, thus improving the global optimization of the algorithm. By maximizing the addition of EI function points, the accuracy of the Kriging model can be improved and local optimization can be avoided [16]. The EGO algorithm cleverly uses EI value to balance the global and local search and therefore to reasonably determine new sampling points [16]. In the iterative optimization process, optimizing the parameter θ is also crucial to constructing the kriging model. But in DACE, the pattern search method (Hooke and Jeeves method) is simply used to optimize θ. Therefore, the genetic algorithm is used to optimize θ in this experiment. Every time we build or rebuild a kriging model, θ is optimized beforehand. respectively, according to the composition of several classic 2xxx aluminum alloys. Considering that the Y element also has an obvious effect on improving the strength of 2xxx aluminum alloy, it is decided to add Y element (0.0-0.3wt.%) when designing the composition of the alloy. 32 alloys components were designed for melting and heat treatment through Latin hypercube distribution. Finally, the tensile strength was measured, and the 'composition-tensile strength' data set was established.
(2): Based on the above data set, conduct 'cross-validation' to evaluate the model.
Where y represents the predicted UTS, y max represents the maximum value of UTS in the training data set, σ is the predicted uncertainty expressed by the root mean square error, Φ and j are the standard normal cumulative distribution and probability density function, respectively.
(5): Judge whether the EI function value meets the convergence condition of the algorithm, that is, whether the EI function value is less than 1% of the current maximum tensile strength value. If the algorithm conditions are met, the optimization will stop. If the convergence condition of the algorithm is not met, add the three newly generated sample points to the training sample set and repeat step (3) ∼ (5).
In a 7.5kw resistance furnace, 99.99% pure Al and pure Mg, Al-50Cu, Al-10Mn, Al-10Ti, Al-10Y master alloys (wt.%) are melted in a graphite crucible. After all the alloy was melted, it was stirred for 3 min to promote melt composition and kept at 720°C for 15 min. Then, the molten metal was cast in a metal mold coated with zinc oxide on the surface at 200°C. After solution treatment at 500°C for 1 h, the ingot was water quenched and followed by immediate aging treatment (160°C/15 h).
Phase constitution was determined by using an x-ray diffractometer (XRD; Empyrean) with Cu Kα radiation in the 2-theta span of 10°-90°. Specimens for scanning electron microscope (SEM; JSM 7800 F) analysis were unetched. The gauge length of the tensile specimens was designed as 8 mm and the stain rate was set as 1 × 10 −3 s −1 . The tensile direction was the same as the extrusion direction, and 3 samples were tested at each condition and then their average value was adopted.

Model evaluation
After obtaining the data set, the current model needs to be validated and then evaluate its feasibility. Figure 2(a) and (b) are the cross-validation results of the initial data set. As shown in figure 2(a), most of the dots are on or on both sides of the 45°line, but the predicted values of some sample dots are quite different from the actual values (in the red circle). From only figure 2(a), it is not shown directly and effectively whether the model's accuracy meets the requirements of machine-learning. Consequently, we calculated the standard residual of the model according to formula 3-1, and the result is shown in figure 2(b). This shows that all the sample points are within the confidence interval of the model prediction [−3, +3] [14]. Therefore, the accuracy of the model can be considered to be reasonable and reliable. Among them, G (u i ) is the actual value of the sample point u , i G u î ( ) and s u î ( )are the predicted value and standard deviation of the model constructed at the point u i after removing the point u , i respectively.  Table 1 shows the composition and tensile strength of the new alloy in the iterative optimization process. In the four iterations, except for alloy 1-2, alloy 1-3, and alloy 3-2, the tensile strength of the other 9 alloys was greater than 307 MPa (the optimal value in the data set). As shown in figure 2(e), with the increase of iteration times, the tensile strength of the alloy generally shows an increasing trend. It is worth noting that the tensile strength of alloys 4-1, 4-2, and 4-3 increased significantly in the fourth iteration. The tensile strength of alloys 4-1 and 4-3 exceeds 350 MPa, and the tensile strength of alloy 4-3 reaches 367 MPa, which is 60 MPa higher than the optimal value of the data set.

Reverse predicted of the model
The trained machine-learning model can predict the tensile strength value of any alloy in the composition space and give the predicted error. After four iterations, cross-validation continues to evaluate the accuracy of the model. As shown in figures 2(c) and (d), the accuracy of the model is still reliable. Based on this, we can consider using the model output results to study the influence of alloy composition changes on the tensile strength value, which provide a reference for the design of 2xxx series aluminum alloys. As shown in figures 3(a) and (b), adjusting the content of Cu and Mg has a more significant impact on the strength of the alloy, and the effect of Cu is much more effective than that of Mg. When the Cu content is 4.0 ∼ 4.5 wt.%, the alloy has a relatively high tensile strength, and the Mg can be changed in a wide range at this time.  Figure 4 show the SEM images and XRD patterns of the 0-9 and 4-3 alloys. As shown in figures 4(a) and (b), the morphology of 0-9 alloy shows two phases: continuous phase and massive phase distributed along the grain boundary. After EDS and XRD analysis, the continuous phase distributed along the grain boundary is the Al 2 Cu phase, and the massive phase is the Al 20 Ti 2 Y phase. Figures 4(c) and (d) show that the second phase of the 4-3 alloy displays a discontinuous shapeand no massive second phase. According to EDS and XRD, we found that there are also two second phases in 4-3 alloys: Al 2 Cu and Al 2 CuMg. In figure 4(e), it can see that the Al 2 CuMg phase exhibits network distribution and the Al 2 Cu phase grows along with the Al 2 CuMg phase. The formation of network Al 2 CuMg phase is mainly due to two eutectic reactions during solidification, L(liquid)→α(Al)+θ(Al 2 Cu)+S(Al 2 CuMg) and L(liquid)→α(Al)+θ(Al 2 Cu) [17,18]. According to the Al-Cu-Mg ternary diagram, after nucleation and growth of α(Al), the ternary eutectic first forms from the liquid phase (α + θ + S) [19]. When there are not enough residual Mg atoms to form ternary eutectic, binary eutectic reaction occurs: L(liquid)→α(Al)+θ(Al 2 Cu). Therefore, the θ (Al 2 Cu) phase appears in the matrix in the form of free eutectic and then grows in combination with ternary eutectic to form a massive phase [20,21].

Conclusions
This research shows the potential of machine-learning methods in developing new 2xxx aluminum alloys with high mechanical properties. Based on six commercial 2xxx series aluminum alloys, we successfully discovered some new alloy compositions. The following conclusions can be drawn from this research: (1)The EGO algorithm can accelerate the composition design of 2xxx aluminum alloy. After 4 iterations, the tensile strength of alloy 4-3 reaches 367 MPa and optimal alloy compositions is obtained. (3)After four iterations, the composition and morphology of the second phase changed significantly, the massive Al 20 Ti 2 Y phase disappeared, and the network Al 2 CuMg phase appeared in the Al 2 Cu phase.