PREDICTION THE JOMINY CURVES BY MEANS OF NEURAL NETWORKS

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1 Tomislav Filetin, Dubravko Majetić, Irena Žmak Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Croatia PREDICTION THE JOMINY CURVES BY MEANS OF NEURAL NETWORKS ABSTRACT: Accurate prediction of hardenability based on the chemical composition is very important for steel production as well as for its users. An attempt has been made to establish a non-linear static discrete-time neuron model, the so-called Static Elementary Processor (SEP). Based on the SEP neurons, a Static Multi Layer Perceptron Neural Network is proposed to predict a Jominy hardness curve from chemical composition. To accelerate the convergence of proposed static error-back propagation learning algorithm, the momentum method is applied. The learning results are presented in terms that are insensitive to the learning data range and allow easy comparison with other learning algorithms, independent of machine architecture or simulator implementation. In the learning process datasets with heats are used comprising samples from 40 steel grades with different chemical composition. The mean error between measured and predicted hardness data and standard deviation for testing dataset (60 heats samples from 03 heats in question) is comparable with other published methods of prediction. The additional testing of three smaller groups Cr-steels; Cr-Ni-Mo (Ni-Cr-Mo) steels for hardening and tempering and Cr-Mo, Cr-Ni (Ni-Cr), Cr-Ni-Mo (Ni-Cr-Mo) steels for carburizing shows better accuracy then by testing with heterogeneous dataset. KEYWORDS: Steels, Jominy curve, Prediction of properties, Artificial neural network,. INTRODUCTION In order to predict a Jominy hardness curve many authors /,,3,4/ have established regression formulae or methods to calculate hardness at different Jominy distances, by means of statistical analysis of a great number of steel heats. Degree of accuracy when comparing between a measured and calculated or predicted hardness depends on the prediction methods and the source of measured Jominy data. The use of computational neural network (NN) as a method of artificial intelligence has rapidly increased over the past 7 years in different science and technology fields: chemical science, design of molecular structure and prediction of polymer properties /5/, prediction of weld deposits structures and properties as a function of a very large number of variables /6/, process control etc. After learning the basic relationships between input factors and output the NN method enable to generate the output variables. This method is very suitable for predicting materials properties in the case when some of the relevant influence factors are unknown, and for solving many complex phenomena for which physical models do not exist. This contribution is based on results of first attempt in application of SEP neurones. A Static Multi Layer Perceptron Neural Network is proposed to predict the Jominy hardness curve from chemical composition. In the learning and testing process datasets of 00 heats of 40 different steel grades are used. The intention of developing this approach is to establish a unique method for predicting Jominy hardness values for a wide range of chemical compositions (steel grades in question).. DESCRIPTION OF NEURAL NETWORK

2 Since artificial neural networks can effectively represent complex non-linear functions, they proved to be a very useful tool in prediction and identifying of highly non-linear systems. The neurone models most commonly applied are the Feed Forward Perceptron used in multi layer networks, and the Radial Basis Function neurone (RBF). Both networks are proved to be universal approximator of any static non-linear mapping. They are capable of identifying any non-linear unique state function to arbitrary desired accuracy. Several learning methods for feedforward neural networks have been proposed in literature. Most of these methods rely on the gradient methodology and involve the computation of partial derivatives, or sensitivity functions. In this sense, the well known error back propagation algorithm for feedforward neural network is used in adaptation of weights /8/, /9/. Theoretical works by several researchers, including /0/ and //, have proven that, even with one hidden layer, artificial neural networks can uniformly approximate any continuous function over a compact domain, provided the network has sufficient number of units, or neurones. Thus the network proposed in this study as plotted in Fig.. has three layers /9/. Each i-th neurone in the first, input layer has single input that represents the external input to the neural network. The second layer, which has no direct connections to the external world, is usually referred to as a hidden layer also consisting of static neurone presented by Fig.. Each j-th static neurone in hidden layer has an input from every neurone in the first layer, and one additional input with a fixed value of unity usually named as Bias. Each k-th neurone in the third, output layer, has input from every neurone in the second layer, and like the second layer, one additional input with fixed value of unity (Bias). The output of the third layer is the external output of the neural network. O O OK..... K Output layer..... J Hidden layer..... I Input layer U Static neuron model U UI Fig.. Static neural network The structure of a proposed static neuron model is plotted in Fig.. The non-linear activation function input and output at time instant (n) are given in () and () respectively: J = net ( n ) w j u j. () j= The non-linear continuous bipolar activation function is described in (): y ( n ) = γ( net( n )) =, () net( n ) + e

3 where u J = represents a threshold unit, also called Bias. u J = BIAS w J u u w w net γ y u 3 w 3 u J w J Fig.. Static neuron model Learning algorithm for optimal parameters The goal of the learning algorithm is to adjust the neural network parameters (the weights) based on a given set of input and desired output pairs (supervised learning) and to determine the optimal parameter set that minimises a performance index E as follows: N E = ( Od ( n ) O( n )), (3) n= where N is the training set size, and the error is the signal defined as difference between the desired response Od ( n) and the actual neurone response O( n). This error, which is calculated at the output layer, is propagated back to the input layer through the static neurons in hidden layer. The result is a well-known error-back propagation learning algorithm. The adjustment of weights occurs for each input-output data pair (pattern or stochastic learning procedure). The linear activation function given in (4) is a chosen transfer (activation) function for static neurones in output layer: O ( n ) = γ ( net ( n )) net ( n ), (4) k k k = k where k =,,..., K is the number of neural network outputs. To determine the optimal network parameters that minimise the performance index E, a gradient method can be applied. Iteratively, the optimal parameters (the weights coefficients) are approximated by moving in the direction of steepest descent /8/, /9/: ϑnew = ϑold + ϑ, (5) E ϑ = η E = η, (6) ϑ where η is a user-selected positive learning constant (learning rate). The choice of the learning constant depends strongly on the class of the learning problem and on the network architecture. The learning rate values ranging from 0-3 to 0 have been reported throughout the technical literature as successful for many computational back-propagation experiments. For large constants, the learning speed can be drastically increased; however, the

4 learning may not be exact, with tendencies to overshoot, or it may be never stabilised at any minimum. Finally, a measure of performance must be specified. All error measures will be reported using nondimensional error index NRMS, Normalised Root Mean Square error, given in (7). Normalised means that the root mean square is divided by the standard deviation of the target data (σ dn ) //. Thus the resulting error index, or index of accuracy is insensitive to the dynamic range of the learning data, and allows easy comparison with other learning algorithms, independent of machine architecture or simulator implementation. Training started with random weights values between - and +, and the networks were trained with η = and user-selected positive momentum constant α = NRMS = N n= ( O O d n N σ d n n ). (7) 3. RESULTS OF PREDICTION The dataset for learning can be compiled from any source of chemical analysis data and measured Jominy hardness data. Data from a single source should produce the best results. Our dataset for learning and testing is derived from a single source and contains very heterogeneous groups of nonboron constructional steel grades for hardening and tempering and for carburizing (40 steel grades - 03 heats): - Unalloyed steels - Cr - Cr-Mo - Cr-V - Cr-Mo-V - Cr-Ni (Ni-Cr) - Cr-Ni-Mo (Ni-Cr-Mo) - Mn-Cr - Mo-Cr. It has been aimed as much as possible to find a generally applicable approach for predicting the Jominy hardness from large range of chemical compositions. The ranges of chemical composition of the heats in question are: 0,-0,70 %C; 0,-,4 %Si; 0,-, %Mn; 0,4-,96 %Cr; 0,4-,76 %Ni; 0,08-0,3 %V; 0,0-0,35 %Mo; 0,00-0,34 %Cu; 0,00-0,34 %Al. The main problem in this investigation was the selection of representative dataset of samples for learning. The dataset we used for training contains heats (about 60 % of all steel heats in question). Input data are the results of chemical analysis of melt (wt. % of 9 chemical elements - C, Si, Mn, Cr, Ni, Mo, Cu, Al and V) and output data are the hardnesses at J-distances:.5, 3, 5, 7, 9,, 3, 5, 0, 5, 30, 40 and 50 mm.

5 The second open problem is the normalising of input and output data. For normalising the possible minimum and maximum (range of data) in testing dataset and the predictive influence of each input data on outputs has to be known. The difference between the predicted and measured Jominy hardness, for different heats at each Jominy distance, was calculated and standard deviation of errors was determined for each heat and each J-distance. It is obvious that accuracy of the results depends on the steel type and J-distance. The results of testing with learning dataset on learning dataset show (Table ) that the mean difference between measured and learned data as well as standard deviations are relatively small. That points to good consistency of dataset for learning in relation to all expected heats for testing. Table: Mean errors and standard deviation of errors for hardenability results of testing with learning data on learning dataset J-distance:, HRC 0,0035-0,03 0,0 0,037 0,033 0,034 0,047 0,073 0,03-0,038-0,074-0,04-0,037 HRC for entire dataset = 0,0045 σ,060 0,999,94,98,74,57,474,43,5,73,808,785,956 σ for entire dataset =,63 N = NRMS = 0,3 The results of prediction for 60 samples with different chemical composition using learning dataset with heats show that the mean differences between measured and predicted hardness and standard deviations are small and comparable with other methods /3,4,5/ (Table ). The C and Crsteels have a greater mean difference and standard deviation than steel grades with higher hardenability, particularly at J-distances where hardness sheer falls down. Table : Mean errors and standard deviation of errors for hardenability results of predicting with 60 different heats J-distance:, HRC 0,8 0,7 0,7-0,59-0,87-0,033-0,05 0,06-0,96-0,9-0,099 0,054-0,0 HRC for entire dataset = -0,045 σ 0,97 0,879,50,96,470,60,706,539,384,485,848,950 3,084 σ for entire dataset =,35 N = 60 NRMS = 0,7 Figure 3 illustrates the differences between the measured and predicted Jominy curves for two steels: 30 NiCrMo and 0 MnCr 5.

6 NiCrMo Hardness, HRC MnCr 5 Fig. 3 Comparison between measured and predicted Jominy curves with NN of two steels: 30 NiCrMo and 0 MnCr 5 Besides, the accuracy of prediction based on relatively similar chemical composition dataset of defined steel group was estimated (Table 3). Hardenability data are predicted for three datasets: - Cr-steels for hardening and tempering - Cr-Ni-Mo (Ni-Cr-Mo) steels for hardening and tempering 3 - Cr-Mo, Cr-Ni (Ni-Cr) and Cr-Ni-Mo (Ni-Cr-Mo) steels for carburizing, with learning dataset of heats The results are shown in table 4, 5 and 6. measured predicted Table 3: The limits of chemical composition of tested steel groups Wt. % Cr-steels Cr-Ni-Mo for hardening and tempering Cr-Mo, Cr-Ni, Cr-Ni-Mo for carburizing C 0,33-0,45 0,7-0,43 0,-0,8 Si 0,3-0,37 0,9-0,8 0,7-0,3 Mn 0,60-0,85 0,36-0,8 0,3-,09 Cr 0,87-,7 0,45-,96 0,47-,9 Ni 0,09-0,5 0,46-,89 0,09-,74 Mo 0,0-0,06 0,6-0,43 0,0-0,8 Cu 0,4-0,34 0,08-0,9 0,5-0,34 Al 0,004-0,043 0,008-0,03 0,003-0,03 Table 4: The results of predicting with different heats of Cr-steels Jominy distance, mm J-distance:, HRC 0,663 0,458-0,4-0,444-0,76-0,398-0,9 0,036 0,574, 3,38,74 -,35 HRC for entire dataset = 0,55 σ 0,67 0,38,00,79,35,9,70,3,49,36 0,75,9 3, σ for entire dataset =,5 N = 6 NRMS = 0,06

7 The standard deviations of errors for steels with 0,-0,4 % C and 0,8-, % Cr at different J- distances resulted from the NN method are comparable with the same from published regression equations /4,5/ and Database Method /6/ (Fig. 4). The greatest standard deviation derived from NN method occurs near inflection point of the Jominy curve. Standard Deviation of Error σ (HRC) Regression Equations /5/ 3.0 Database Calculation /6/ NN Method Jominy distance, mm Fig. 4 Comparison of standard deviation of error for hardenability predictions performed using neural network (NN) and published Database Method /6/ as well as regression derived equations /5/ using steels with 0,-0,4 % C and 0,8-, % Cr. Table 5: The results of predicting with different heats of Cr-Ni-Mo (Ni-Cr-Mo) steels for hardening and tempering J-distance:, HRC -0,08-0,39-0,08-0,7 -,089 -,039 -,066-0,95-0,96 -,36 -,396 -,7 -,46 HRC for entire dataset = -0,859 σ 0,87 0,6 0,65,0,33,49,83,00,9,95,56,78,54 σ for entire dataset =,5 N = 3 NRMS = 0,64 Table 6: The results of predicting with different heats of Cr-Mo, Cr-Ni (Ni-Cr) and Cr-Ni-Mo (Ni-Cr-Mo) steels for carburizing J-distance:, HRC -0,459-0,388-0,45-0,375 0,38 0,404 0,47 0,765 0,75 0,377 0,346 0,09 0,05 HRC for entire dataset = 0,3 σ 0,83 0,87,06,77,,6,9,88,87,89,96,,0 σ for entire dataset =,8 N = 9 NRMS = 0,6 The mean standard deviation of errors for all of the three groups is smaller than from prediction with the group of 60 different steels. Fig. 5 shows the comparison of standard deviation of errors at different distances from quenched end of the Jominy probe for three tested steel groups.

8 Standard Deviation of Error σ (HRC) Cr steels (N=6) Jominy distance, mm Cr-Ni-Mo steels for hardening and tempering (N=3) Cr-Mo, Cr-Ni, Cr-Ni-Mo steels for carburizing (N=9) Fig. 5 Comparison of standard deviation of error for hardenability predictions using neural network for three groups of steels For the time being the number of heats for testing with NN method is too small for more accurate comparison of these three methods and for drawing final conclusions. 4. CONCLUSION The application of the neural network method for predicting the Jominy hardenability curve, as shown by presented testing, encourage further investigation. The accuracy of prediction depends on the accuracy (standard deviation) of measured data. The measured data should reflect real relations between chemical composition and Jominy hardness. This preliminary experience shows following evident benefits of application of the NN for predicting the Jominy hardenability in a steel production: - accurate prediction of hardenability for each new heat in the production based on own learned dataset (from own metallurgical history), - possibility of optimizing the chemical composition of steel for required hardenability, - avoiding the Jominy testing after the production. To achieve better accuracy in application of NN, additional activities are needed in following directions:

9 a) selecting the optimum size of learning dataset with the representative data for expected heats in question. The learning datasets have to be as wide as possible and should contain enough different shapes of Jominy curves; b) testing the other approaches in normalising of data; c) testing with a greater number of heats from different sources and comparison of those results; d) application of other NN algorithms (e.g., Radial Basis Function Neural Networks - RBF). Acknowledgements The authors wish to thank dr. F. Grešovnik from Željezarna Ravne Slovenia for the help in collecting the hardenability data and to Ministry of Science and Technology of Republic of Croatia for financial support of this research within the project titled Computerised simulation and materials development. 4. REFERENCES // B. G. Sumpter, D. W. Noid, Proceedings from ANTEC 95, p // H. Bhadeshia, Materials World, Nov. 996, p /3/ E. Just, Met. Prog., Vol 96, 969, p. 87- /4/ H. Gulden K. Kriger, D. Lepper, A. Lubben, H. Rohloff, P. Schuler, V. Schuler and H.J. Wieland, Stahl und Eisen, Vol 09 (No. ), 989, p. 3-7 /5/ H. Gulden K. Kriger, D. Lepper, A. Lubben, H. Rohloff, P. Schuler, V. Schuler and H.J. Wieland, Stahl und Eisen, Vol (No. 7), 99, p. 0-0 /6/ W.T. Cook, P.F. Morris, L. Woollard, J. of Materials Engineering and Performance, (997)6, p /7/ J. S. Kirkaldy, S.E. Feldman, J. Heat Treat. (989)Vol 7, p /8/ J.M. Zurada, Artificial Neural Systems, W.P. Company, USA, (99). /9/ D.Majetic,V.Kecman,Technical Report TR93-YUSA-0, MIT, Cambridge, USA, (993), p /0/ G.Cybenko, Mathematics of Control, Signals, and Systems, Vol., (989), p // K.Funahashi, Neural Networks, Vol., (989), p // A.Lapedes,R.Farber,Technical Report, Los Alamos National Laboratory, Los Alamos, New Mexico, (987)