2020 Volume 61 Issue 1 Pages 176-180
Except in the case of martensitic transformation during quenching and age-hardening, the mechanical properties (tensile strength and hardness) of many metallic materials are often determined by its chemical composition. If mechanical properties can be predicted from the chemical composition of molten metal before casting, it can contribute to the stabilization of quality and the reduction of the testing process of tensile strength and hardness. In the case of gray cast iron, mechanical properties are often discussed with five main elements (C, Si, Mn, P and S). Multiple regression shows low performance in terms of correlation coefficient. Therefore, trace elements other than the five main elements should be considered since the influence of trace elements on mechanical properties is mostly nonlinear, making it difficult to analyze by multiple regression. Given that deep neural network (DNN) can take nonlinear cases into consideration, we investigated whether mechanical properties can be predicted from chemical compositions including trace elements, and obtained the following findings. For comparison, we also analyzed mechanical properties by multilayer perceptron (MLP) and multiple regression (MR). As a result, the prediction accuracy of DNN, MLP and MR improved by the consideration of not only the five main elements but also 18 other elements including trace elements. Prediction error of tensile strength analyzed by DNN was less than half of MR. Increasing the number of layers and the number of nodes in DNN improved the prediction accuracy of mechanical properties, demonstrating the effectiveness of DNN.
This Paper was Originally Published in Japanese in J. JFS 91 (2019) 253–257.
For many metallic materials, such as cast iron and steel, excluding phase transition and precipitation, mechanical properties are often determined by chemical components. Most metallic materials consist of a large number of elements including trace elements, and it is difficult to predict the mechanical properties including all of them. In gray cast iron, graphite crystallizes as flakes in the cast iron metallographic structure from the chemical components of molten cast iron. The structure of gray cast iron changes because of various parameters, such as chemical components of molten metal when casting, casting temperature, and inoculation. In terms of the influence of cast iron components, C, Si, Mn, P, and S are major elements that constitute cast iron, and their ratio is an important factor that determines the mechanical properties. Specifically, carbon and silicon, the basis of graphite crystallization, have a major influence on the tensile strength and hardness of cast iron. However, cast iron contains trace elements in addition to major elements, and mechanical properties change depending on the contents of these trace elements. The relationship between most of the elements and mechanical properties is nonlinear;1) thus, sufficient accuracy for prediction cannot be achieved with a conventional statistics method such as multiple regression (MR) analysis.
The mechanical properties of cast iron are known to differ based on the metallographic micro-structure, even if the chemical components are identical.2–5) Many previous studies have shown that inoculation and graphite morphology influence the mechanical properties, but the present study targeted gray cast iron without inoculation in order to improve the accuracy of the mechanical property prediction with chemical components as the input parameter. Therefore, the present work focuses on 13 trace elements and predicts the mechanical properties from 13 trace elements and five major elements: a total of 18 species of chemical components. A previous study showed the influence of several combinations of elements on mechanical properties,6) but it did not make predictions based on the 18 elements.
In the present study, a deep neural network (DNN) was used to consider the nonlinear relationship between variables. In this manner, tensile strength and the Brinell hardness of gray cast iron are predicted based on the chemical compositions including the trace elements. To compare the accuracy of prediction, MR and multilayer perceptron (MLP) were used.
In the experiment, the tensile strength and Brinell hardness of gray cast iron were predicted with 3,315 cases of experimental data. The experimental data were obtained from casting Kb-type knockoff specimens before pouring a mold for products. For melting, a 5t-low-frequency induction furnace was used, molten metal casting into a furan-resin self-hardening mold. The type of graphite was type D.
2.1 Sample materials 2.1.1 Sample compositionsFC150, FC200, FC250, FC300, and FC350 as JIS (Japanese Industrial Standard) were the targets, and their chemical compositions were measured through an emission spectrochemical analysis of the molten metal before pouring. Table 1 shows the target range of chemical compositions for each element.
The casting size of the round bar was φ25 × 300 mm, and the shape of the round bar before machining the parallel portion of the tensile strength test was φ20 × 300 mm.
2.1.3 Test methods for mechanical properties(1) Tensile test
The tensile test specimens was prepared from the Kb-type knockoff specimens. The tensile test specimens were prepared according to No. 8 in JIS Z 2241, and the rate of stress increase was 10 N/mm2·s−1.
(2) Hardness test
The grip of the test specimen was cut off after the tensile test, and the Brinell hardness of this cross-section was measured with a load of 3,000 kgf.
2.1.4 Examination of compounds in the specimensFigure 1 shows the results of X-ray diffraction measurements for the extraction residue of FC300. It only shows graphite and cementite. Because carbonitride has not been confirmed, its influence on mechanical properties can be omitted. Some trace elements likely solve to cementite. Because cast iron has higher sulfur content compared with typical steel materials, MnS is easily generated. However, the hardness of MnS was approximately 95 on the Brinell scale,7) and, on the scanning electron microscopy map (Fig. 2), the area ratio was low. It was assumed that there was no influence on mechanical properties.
XRD result of extracted residues of FC300 specimen.
Element (C, Si, Mn, S, Fe) mapping of FC300 specimen by SEM, EDX.
The contents of 18 constituent elements of cast iron were used for input variables. The elements examined were five major elements (C, Si, Mn, P, and S) and 13 trace elements (Cu, Ni, Cr, Mo, Ti, V, Al, Sn, Mg, B, Sb, Zn, and N). Predictions were made in two ways: considering the 18 elements to measure the influence of trace elements and considering only the five major elements. There are more elements that constitute cast iron other than those considered in the present study, but the above 18 elements were selected as these are likely to have an influence on the properties of the product and results based on past studies.
2.3 Analytical methods 2.3.1 Deep neural network (DNN)A DNN is a neural network with multiple hidden layers. The learning of this network is called deep learning. The structure of a DNN is shown in Fig. 3. Figure 3 shows that X1 is the input layer, X2, X3, and X4 are hidden layers, and X5 is the output layer. The circles inside the layers are nodes. The input layer is the variable used for prediction, consisting of 18 nodes corresponding to the amount of each element in the chemical components. The output layer is the variable to be predicted. A model each was created for tensile strength and the Brinell hardness that predicts one mechanical property from 18 input variables. Hidden layers are those between the input layer and the output layer. Each node of the hidden layers is expressed as a mixture of input variables and is converted to an activation function. For the activation function, the rectified linear function (ReLU),8) and sigmoid function are used. This calculation is expressed with the following equation.
| \begin{equation} y = f\left(\sum_{i = 1}^{n}w_{i}x_{i} + b \right) \end{equation} | (1) |
| \begin{equation} X_{\text{i}+1} = f(W_{i}X_{i} + B_{i}) \end{equation} | (2) |
Structure of deep neural network.
Predictions of mechanical properties by the DNN are divided into two processes to evaluate the performance of the model: learning and testing. Generally, with machine learning, the data used for learning have fit with extremely high precision; thus, to evaluate whether the model is effective, different data must be used for learning and testing. In the model learning, the DNN parameters are learned based on the learning data with chemical component values as input variables and with mechanical properties as the target of prediction. Based on this learned model, mechanical properties of test data are predicted. By comparing the prediction results and actual data of mechanical properties, model performance can be evaluated.
2.4 Evaluation methodsIn the evaluation, the mean absolute error between the actual value and predicted value was measured by 10-fold cross-validation. The prediction of mechanical properties is better when the mean absolute error is smaller. The mean absolute error is obtained with the following equation.
| \begin{equation} \mathrm{MAE} = \frac{1}{n} \sum_{i = 1}^{n}| e_{i} | \end{equation} | (3) |
As the model parameters, six hidden layers and 100 nodes per layer were used in the DNN. For the activation function, ReLU8) was used. The learning method for the parameters was adaptive moment estimation (Adam).9) For the initial value of the parameters, the initial value reported by He et al.10) was used. The output of each layer was normalized through batch normalization (BN).11) For the multilayer perceptron, one hidden layer with 400 nodes was used. For the activation function, the sigmoid function was employed, and, for the learning method, the stochastic gradient descent method was used. MR does not take interaction of variables into consideration. The value of each variable in the experimental data was normalized between 0 and 1 via min–max normalization.
Figures 4 and 5 show predicted results of the tensile strength and Brinell hardness for each method, respectively. Figure 4 shows that the mean absolute error for the tensile strength when 18 elements were used as input variables was 5.12 for the DNN, 12.37 for the MLP, and 12.31 for MR. When five elements were used as input variables, the mean absolute error was 6.94 for the DNN, 13.82 for the MLP, and 13.76 for MR. Figure 5 shows that the mean absolute error for Brinell hardness when 18 elements were used as input variables was 4.18 for the DNN, 7.80 for the MLP, and 7.78 for MR. When five elements were used as input variables, it was 5.61 for the DNN, 8.98 for the MLP, and 8.96 for MR. Mean absolute errors for the tensile strength and Brinell hardness were both the smallest with the DNN. In the case of all methods, by considering 18 elements including trace elements instead of using only the five major elements, the prediction error was reduced. The amount of reduction in the mean absolute error when increasing the input variables from five elements to 18 was the greatest for the DNN. Based on this result, the prediction that only uses major elements is insufficient, and trace elements should be considered in order to predict the mechanical properties of gray cast iron accurately.
Comparison of mean absolute error of tensile strength after prediction.
Comparison of mean absolute error of Brinell hardness after prediction.
Figures 6 and 7 show scatter plots of the actual values and predicted values for tensile strength. They show that the prediction accuracy with the DNN (Fig. 6) was high for the whole range of actual values. Meanwhile, the prediction accuracy with MR (Fig. 7) was low for a range where actual values of tensile strength were low. For MR, this is likely because the regression equation has a high correlation with actual values from 250 to 350 MPa, where the number of data points is large, and this equation leads to higher errors when the actual values are small. With the DNN, because there are multiple hidden layers, mechanical properties can be predicted with high accuracy by capturing the relationship between variables.
Scatter plot of predicted tensile strength by deep neural network using 18 elements and measured tensile strength value.
Scatter plot of predicted tensile strength by multiple regression using 18 elements and measured tensile strength value.
From the effects of the method of learning multiple hidden layers (Fig. 8), the prediction error became 12.78 with one hidden layer and 31.81 with six hidden layers. Although the number of the hidden layer increased, the prediction accuracy worsened. This is because the MLP does not use a method to learn multiple hidden layers. If one of the learning methods of MLP was changed for the learning method of the DNN, the error was 10.43 with ReLU, 10.63 with BN, and 5.79 with Adam. Using a learning method for multiple hidden layers reduced the prediction error.
Effects of learning method for mean absolute error of tensile strength in neural network.
Figures 9 and 10 show the influence of the numbers of hidden layers and nodes on tensile strength and Brinell hardness, respectively. For both tensile strength and Brinell hardness, as the layer became deeper, the prediction error decreased. As the number of nodes of each layer increased—50, 100, and 200—prediction error decreased. Compared with the 100 nodes used for comparison with other methods, 200 nodes had less prediction error. However, if the optimal numbers of layers and nodes are selected from the test data, the prediction ability of the DNN is overestimated; thus, the number of nodes was set at 100. When the number of layers is increased to seven or more, the prediction error might become smaller, but, to balance with the calculation cost, up to six layers were examined.
Change in mean absolute error of tensile strength according to number of hidden layers and number of nodes in each layer of deep neural network.
Change in mean absolute error of Brinell hardness according to number of hidden layers and number of nodes in each layer of deep neural network.
In this study, it was shown that mechanical properties could be predicted with high accuracy by using a DNN. Using multiple hidden layers increased the prediction accuracy. Specifically, in the prediction of tensile strength, accuracy doubled compared with MR. To improve the prediction accuracy further, parameters at the time of casting that have an influence on mechanical properties (for example, casting temperature) need to be considered. The molten metal for the experimental data used for the present study was not inoculated. Thus, inoculated data are necessary to examine the influence of inoculation. With a DNN, sufficient data make highly accurate prediction possible, but it is unclear whether a DNN would work on data with mixed information of multiple graphite shapes and inoculation conditions. Therefore, future examination is necessary.
In the present study, the prediction accuracy of a DNN was evaluated through comparison with an MLP and a MR to improve the accuracy of mechanical property prediction for gray cast iron. A chemical composition of 18 elements including trace elements and five major elements (C, Si, Mn, P, and S) was used as input parameters, as well as the predicted tensile strength and Brinell hardness of uninoculated gray cast iron. In this manner, the influence of the numbers of hidden layers and nodes on the mean absolute error was examined. The results are as follows.
