- Email: [email protected]
Effective parameters modeling in compression of an austenitic stainless steel using artificial neural network
Computational Materials Science 34 (2005) 335–341 www.elsevier.com/locate/commatsci
Effective parameters modeling in compression of an austenitic stainless steel using artificial neural network A. Bahrami a
a,*,1
, S.H. Mousavi Anijdan a, H.R. Madaah Hosseini a, A. Shafyei b, R. Narimani c
Department of Materials Science and Engineering, Sharif University of Technology (SUT), P.O. Box 11365-9466 Tehran, Iran b Department of Materials Engineering, Isfahan University of Technology (IUT), Isfahan, Iran c Department of Mechanical Engineering, Iran University of Science and Technology (IUST), Arak, Iran Received 29 September 2004; received in revised form 10 January 2005; accepted 11 January 2005
Abstract In this study, the prediction of flow stress in 304 stainless steel using artificial neural networks (ANN) has been investigated. Experimental data earlier deduced—by [S. Venugopal et al., Optimization of cold and warm workability in 304 stainless steel using instability maps, Metall. Trans. A 27A (1996) 126–199]—were collected to obtain training and test data. Temperature, strain-rate and strain were used as input layer, while the output was flow stress. The back propagation learning algorithm with three different variants and logistic sigmoid transfer function were used in the network. The results of this investigation shows that the R2 values for the test and training data set are about 0.9791 and 0.9871, respectively, and the smallest mean absolute error is 14.235. With these results, we believe that the ANN can be used for prediction of flow stress as an accurate method in 304 stainless steel. Ó 2005 Elsevier B.V. All rights reserved. PACS: 7.05.M; 84.35; 81.20.H Keywords: Artificial neural networks; 304 stainless steel; Back propagation; Flow stress; Temperature; Strain
1. Introduction * Corresponding author. Tel.: +98 21 686 5251; fax: +98 21 828 8074. E-mail address: [email protected] (A. Bahrami). 1 Present address: Department of Materials Science and Engineering, Azadi Avenue, Tehran, Iran.
As investigated by preceding researchers, the relationships between the temperature, strain-rate and strain with flow stress in 304 stainless steel is very complex. The most investigations on flow instability for this type of steel have been accomplished
0927-0256/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2005.01.006
336
A. Bahrami et al. / Computational Materials Science 34 (2005) 335–341
by some trial and error testing methods, which depends considerably on the designerÕs skills and experiences [1]. Semiatin et al. [2–4] evaluated phenomenological criterion based on the strain hardening and strain-rate sensivity. Also, Ziegler [5] evaluated continuum criterion based on the principle of maximum entropy. Ziegler proposed that a system undergoing large deformation shows instability when the strain-rate variations with strain-rate (_e) in the dissipation function (Dð_eÞ) satisfies the below relationship: dD Dðe_ Þ < d_e e_
ð1Þ
During deformation, power dissipation occurs through the temperature rise of the workpiece and microstructural changes (J co-content). The power partitioning between these two is divided by the strain-rate sensivity (m) of the flow stress (r) [1]. In a constant temperature and strain-rate, the J co-content was obtained as follows [6]: Z r m r_e ð2Þ J¼ e_ dr ¼ mþ1 0 In cold working, m is small and work hardening occurs. Prasad [7] applied the principle of reparability of the rate of entropy production into the conduction entropy and internal entropy. By considering the dissipation function corresponding to the microstructure changes (J co-content) a criterion for instability has been derived: nð_eÞ ¼
o ln ½m=ðmþÞ þm o ln e_
ð3Þ
Venugopal et al. have given an instability map. They suggested when nð_eÞ is negative, instability occurs. As can be seen, mechanical properties of 304 stainless steel is a complex function of temperature, strain-rate and strain and it is very difficult to develop a complete mathematical model to predict the flow stress of this steel. An engineering approach to predict the flow stress of engineering alloys is based on the utilization of artificial neural networks (ANNs). Neural networks consist of many computational elements, which operate in parallel and have been connected by several links with variable weights. These elements are typically
adopted during the learning process [8]. Recently, several models of neural networks have been proposed to predict the flow stress of metals [9,10]. The aim of this investigation is to develop a welltrained ANN model to predict the flow stress in compression tests for 304 stainless steel under cold and warm compression.
2. The data base The data used for this paper have been published by Venugopal et al. [1]. The composition of 304 stainless steel was C 0.08, S 0.005, P 0.035, Ni 10.3, Cr 18.6, Mn 1.7, Si 0.58, Mo 0.07 and balance Fe (all in weight percent), the temperature range was 20–600 °C, the strain-rate variation was from 0.001 s1 to 100 s1 and the range of strain was 0.1–0.5. They performed 360 compression tests on cylindrical specimens. Among these experiments, 300 data set were selected for training the ANN and the 60 remaining data set were used to validate the ability of the ANN model to predict the flow stress of 304 stainless steel. The ranges of variables, which have been used for this study, are given in Table 1.
3. Artificial neural networks 3.1. ANN principles ANNs, also called ‘‘neural networks’’, ‘‘parallel distributed processing’’ and ‘‘connectionist’’ models, developed out of the areas of artificial intelligence and cognitive science in their attempts to model the brain and its learning process. ANNs are collections of small individual interconnected
Table 1 The range of variables used to develop the neural network model for 304 stainless steel
Temperature Strain-rate Strain
Minimum value
Maximum value
Standard deviation
20 °C 0.001 S1 0.1
600 °C 100 S1 0.5
172.86 36.613 0.1414
A. Bahrami et al. / Computational Materials Science 34 (2005) 335–341
Input Layer
Hidden layer
Output layer
Temperature Strain rate
Flow stress
Strain
Fig. 1. Schematic diagram of multi-layer neural network.
processing units with weights associated with each connection. Fig. 1 shows the structure of ANN model with various layers. Learning is the first step necessary in inducing intelligence to neural networks. In learning, ANNs are ‘‘taught’’ by presenting sets of patterns to be ‘‘learned’’, and the network autonomously adjusts the connection weights among the processing units according to imposed learning rules and, thereby, obtains unique knowledge from the data. The learned neural network generates accurate outputs for the input data. For inputs that have not been experienced are partially damaged, distorted or mixed with noise, appropriate outputs are generated based on its internal knowledge stored in the connection weights. The basic concept of back propagation learning [11] is shown in Fig. 2, where first, the input patterns of each node are provided at the input layer. Then, this signal is converted and trans-
337
ferred to the hidden layer at each node, and finally, the signal generates outputs in the output layer. The output values are compared to the target values, and if there is a difference, the connection weights wji, wkj are adjusted in such a direction that the error is decreased. This adjustment is back propagated from the upper layers to the lower layers, which results in the adjustment of the connection weights in the lower layers. The back propagation learning algorithm has the following advantages. First, the ANN requires less expertise, knowledge and experimentation to determine the relationships between the inputs and outputs of non-linear systems. Second, it has been used successfully in similar applications, such as prediction [12–14]. 3.2. Learning procedure In this study the learning procedure was the back propagation algorithm. This ‘‘training’’ algorithm is performed step-by-step as follows: Step 1: The number of learning data is selected from the experimental. The ANN learns these data and determines the relationships between them. The data used here were collected from experiments of Venugopal et al. and are normalized between 0.5 and 0.5 to the following equation [15]:
Fig. 2. Schematic structure of the back propagation learning.
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A. Bahrami et al. / Computational Materials Science 34 (2005) 335–341
ðx xmin Þ 0:5 xN ¼ ðxmax xmin Þ
ð4Þ
where xN is the normalized value of x, xmin and xmax are minimum and maximum values of flow stress. Step 2: The connection weights wji and wkj, which determine the state of the ANN, and bias values hj and hk are initialized with random values. Step 3: The normalized learning data are fed into the ANN. Step 4: From connection weight wji, output opi of the input layer node i and bias hj, the input value netpj for hidden layer node j is determined as X netpj ¼ wji opi þ hj ð5Þ i
In Eq. (5), the i subscript indicates a summation over all nodes in the layer in the direction of the hidden layer. The output opj of node j is determined as shown in Eq. (7) below using netpj and the sigmoid function shown in Fig. 3 and Eq. (6). 1 ð6Þ f ðxÞ ¼ 1 þ expðxÞ opj ¼ fj ðnetpj Þ
ð7Þ
Step 5: The input value netpk of the output layer node k is determined from output opj of the hidden layer nodes, connection weight wkj between the hidden layer node and the output layer node and bias (offset) hk of node k. Then, using netpk and the sigmoid function, output opk of node k is determined as
netpk ¼
X
wkj opj þ hk
ð8Þ
j
opk ¼ fk ðnetpk Þ
ð9Þ
In Eq. (8), the j subscript indicates a summation over all nodes in the layer in the direction of the output layer. Step 6: Using the difference between the desired output tpk and the actual output opk, the error term dpk is determined as dpk ¼ tpk opk f 0 netpk ð10Þ ¼ tpk opk 1 opk Step 7: The error term dpj of the hidden layer is computed as X dpj ¼ fj0 netpj dpk wkj
¼ opj 1 opj
k
X
dpk wkj
ð11Þ
k
In Eq. (11), the k subscript indicates a summation over all nodes in the layer in the direction of the hidden layer from the output layer. Step 8: The connection weight wkj between the hidden layer node and the output layer node using Eq. (12) and bias value hk of the hidden layer node using Eq. (13) are modified. The term a is called the learning rate, and b is called the momentum factor. Both of these constant terms have an effect on the speed and stability of the network. wðnewÞkj ¼ wðoldÞkj þ adpk opk þ bDwðoldÞkj
ð12Þ hðnewÞj ¼ hðoldÞk þ adpk opk þ bDhðoldÞk ð13Þ where
Fig. 3. Shape of logistic sigmoid function.
DwðoldÞ ¼ wðoldÞ wðolderÞ
ð14Þ
DhðoldÞ ¼ hðoldÞ hðolderÞ
ð15Þ
Step 9: The connection weight wji between the input layer node and the hidden layer node using Eq. (16) and bias value hj of the hidden layer node using Eq. (17) are modified.
A. Bahrami et al. / Computational Materials Science 34 (2005) 335–341
wðnewÞji ¼ wðoldÞji þ adpj opj þ bDwðoldÞji
35 30
ð16Þ
25
MAE
hðnewÞj ¼ hðoldÞj þ adpj opj þ bDhðoldÞj
20 15 10
ð17Þ
4. Results and discussion 4.1. Neural network results The ANN model with one hidden layer was used in this study. The inputs of the ANNs were temperature, strain-rate and strain and flow stress was considered as an output. In first step, training was performed for adjusting connection weights between input layer to hidden layer and between hidden layer and output layer until the network to produce outputs that were close enough to the desired outputs. Mean absolute error (MAE) of desired and predicted data was calculated from the following equation: . X MAE ¼ jx yj n ð18Þ where x = XX 0 ; X is the target output and X 0 is the mean of X, y = YY 0 ; Y is network output and Y 0 is the mean of Y and n is the number of data set. In this model, when MAE reached the minimum value, the weight and bias values encode the networkÕs state of knowledge. Training data for the development of the neural network was obtained from the published paper [1]. Fig. 4 shows
5 0 0
5
10
15
20
25
30
Hidden unit
Fig. 4. MAE for different hidden units.
1600
Predicted flow stress (MPa)
Step 10: Return to the procedures of Step 3 until all of the learning data have been learned. Step 11: If the number of iterations is less than a limit of iterations, repeat the procedures following Step 3. Step 12: Learning is stopped. Step 13: The input data of each measurement time interval are fed into the learned ANN, and the flow stress for each measurement time interval by ANN recalling is calculated.
339
R2 = 0.9791
1400 1200 1000 800 600 400 200 0 0
200
400
600
800
1000
1200
1400
1600
Measured flow stress (MPa)
Fig. 5. Comparison of predicted and measured flow stress for testing data set.
the MAE for various hidden layers. According to this figure, the ANN with one hidden layer and 15 hidden units has the smallest error (14.235). Fig. 5 shows the predicted flow stress by ANN model versus measured value for testing set. The test data set is not involved in the model training process and was only used for the testing purpose. Therefore, the accuracy of an ANN model in prediction is clearly demonstrated by its performance on the testing data set. These results have been obtained from a network with 15 neurons in hidden layer and 1230 epoch. As it has been shown in Fig. 5, the correlation coefficient for the test data set is 0.9791. This means that a good correlation between predicted and measured data has been obtained. Fig. 6 shows the comparison of predicted and measured flow stress for all data set. The overall correlation coefficient for the all data set is 0.9871. This figure shows that the used network
Predicted flow stress (MPa)
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A. Bahrami et al. / Computational Materials Science 34 (2005) 335–341
1600 R2 = 0.9871
All data
1400 1200 1000 800 600 400 200 0 0
500
1000
1500
2000
Measured flow stress (MPa)
Fig. 6. Comparison of predicted and measured flow stress for all data set.
Fig. 7. Significance of input variables in flow stress.
could be capable for prediction with a minimum error.
strain-rate has a reverse effect on flow stress. Formerly Venugopal et al. [1] have been reported that strain-rate has a negative effect.
4.2. Sensivity analysis The sensivity test was carried out to determine the relative significance of each of the input parameters. In this analysis, a step-by-step method was carried on the trained ANN by varying each of the input parameter, one at a time, at a constant rate. Various constant rates (5, 10, 20) were selected in this study. For every input parameter, the percentage was changed in the output as a result of the change in the input parameter. The sensivity of each input parameter was calculated by the following equation: Sensivity level of X i ð%Þ N 1 X % change in output ¼ 100 N j¼1 % change in input j
ð19Þ
where N is the number of data sets used in this study to test the network, functionality. Results showed that the changes in output parameter were unaffected significantly by the rate of an input parameter when strain-rate was varied (5, 10, 20). Fig. 7 shows the sensitivity of variations for flow stress at each of the input variables. As can be seen, temperature is the most significant parameter. These results are very close to the pervious investigations, which reported that temperature is the most important parameter in flow stress of stainless steels [16,17]. This analysis showed that
5. Conclusions The aim of this paper was to use the neural networks for calculation of the flow stress in 304 stainless steel. An ANN model with one hidden layer and 15 neurons in the hidden layer is a benefit method for prediction of flow stress in 304 stainless steel, where temperature, strain-rate and strain are input parameters. Sensivity analysis showed that temperature is the most effective parameter, the effect of strain-rate is negative, and strain has the minimum effect on flow stress. These features enable us to use ANNs in stainless steel and will help people studying in this field. So, experimental studies can be reduced to a minimum at the places where the use of ANNs is appropriate. Also, the results showed that the ANN approach could be considered as an alternative and practical technique to evaluate the flow stress in 304 stainless steel.
References [1] S. Venugopal et al., Optimization of cold and warm workability in 304 stainless steel using instability maps, Metall. Trans. A 27A (1996) 126–199. [2] S.L. Semiatin, G.D. Lahoti, Deformation and unstable flow in hot forging of Ti–6Al–2Sn–4Zr–2Mo–0.1Si, Metall. Trans. A 12A (1981) 1705–1717, and 1719–1728.
A. Bahrami et al. / Computational Materials Science 34 (2005) 335–341 [3] S.L. Semiatin et al., Plastic instability and flow localization in shear at high rates of deformation, Acta Metall. 32 (1984) 1347–1354. [4] S.L. Semiatin et al., Effect of deformation heating and strain rate sensivity on flow localization during the torsion testing of 6061 aluminum, Acta Metall. 34 (1986) 167–176. [5] H. Ziegler, Progress in Solid Mechanics, 4, John Wiley and Sons, New York, NY, 1983, p. 93. [6] Y.V.R.K. Prasad et al., Modeling of dynamic material behavior in hot deformation: forging of Ti-6242, Metall. Trans. A 15A (1984) 1883–1892. [7] Y.V.R.K. Prasad, Recent advances in the science of mechanical processing, Indian J. Technol. 28 (1990) 435–441. [8] B.H.M. Sadeghi, A BP-neural network predictor model for plastic injection molding process, J. Mater. Process. Technol. 103 (2000) 411–416. [9] T. Cool, H.K.D.H. Bhadeshia, D.J.C. MacKay, The yield and ultimate tensile strength of steel welds, Mater. Sci. Eng. A 223 (1997) 186–200. [10] A. Bahrami, S. H. Mousavi Anijdan, et al., Prediction of mechanical properties of DP steels using Neural Network model, Revised in the Journal of Alloys and Compounds, 2004.
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[11] D.E. Rumelhart, J.L. Mcclelland, in: Parrallel Distributed Processing, Vols. 1 and 2, MIT Press, Cambridge, 1986. [12] S.H. Mousavi Anijdan, A. Bahrami, et al., Prediction of mechanical properties of forged stainless steel DIN 1.4962 using Artificial Neural Network, in: Proceeding of 6th Symposium of Iron and Steel, Bafgh, Yazd, Iran, 2004, pp. 305–359. [13] S. Calcaterra et al., Prediction of mechanical properties in spheroidal cast iron by neural networks, J. Mater. Process. Technol. 104 (2000) 74–80. [14] W.T. Chien, C.Y. Chou, The predictive model for machinability of 304 stainless steel, J. Mater. Process. Technol. 118 (2001) 442–447. [15] M.A. Yescas, H.K.D.H. Bhadeshia, D.J.C. Mackay, Estimation of the amount of retained austenite in austempered ductile iron using neural network, Mater. Sci. Eng. A 311 (2001) 162–173. [16] S.S. Heeker, M.G. Stout, K.P. Staudhammer, J.L. Smith, Room-temperature plastic flow and strain-induced martensitic transformation in 1.2 wt% Ni metastable austenitic stainless steel sheets, Metall. Trans. A 13A (1982) 619– 626. [17] T. Angel, Formation of martensite in austenitic stainless steels, J. Iron Steel Inst. 177 (1954) 165–172.
Effective parameters modeling in compression of an austenitic stainless steel using artificial neural network A. Bahrami a
a,*,1
, S.H. Mousavi Anijdan a, H.R. Madaah Hosseini a, A. Shafyei b, R. Narimani c
Department of Materials Science and Engineering, Sharif University of Technology (SUT), P.O. Box 11365-9466 Tehran, Iran b Department of Materials Engineering, Isfahan University of Technology (IUT), Isfahan, Iran c Department of Mechanical Engineering, Iran University of Science and Technology (IUST), Arak, Iran Received 29 September 2004; received in revised form 10 January 2005; accepted 11 January 2005
Abstract In this study, the prediction of flow stress in 304 stainless steel using artificial neural networks (ANN) has been investigated. Experimental data earlier deduced—by [S. Venugopal et al., Optimization of cold and warm workability in 304 stainless steel using instability maps, Metall. Trans. A 27A (1996) 126–199]—were collected to obtain training and test data. Temperature, strain-rate and strain were used as input layer, while the output was flow stress. The back propagation learning algorithm with three different variants and logistic sigmoid transfer function were used in the network. The results of this investigation shows that the R2 values for the test and training data set are about 0.9791 and 0.9871, respectively, and the smallest mean absolute error is 14.235. With these results, we believe that the ANN can be used for prediction of flow stress as an accurate method in 304 stainless steel. Ó 2005 Elsevier B.V. All rights reserved. PACS: 7.05.M; 84.35; 81.20.H Keywords: Artificial neural networks; 304 stainless steel; Back propagation; Flow stress; Temperature; Strain
1. Introduction * Corresponding author. Tel.: +98 21 686 5251; fax: +98 21 828 8074. E-mail address: [email protected] (A. Bahrami). 1 Present address: Department of Materials Science and Engineering, Azadi Avenue, Tehran, Iran.
As investigated by preceding researchers, the relationships between the temperature, strain-rate and strain with flow stress in 304 stainless steel is very complex. The most investigations on flow instability for this type of steel have been accomplished
0927-0256/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2005.01.006
336
A. Bahrami et al. / Computational Materials Science 34 (2005) 335–341
by some trial and error testing methods, which depends considerably on the designerÕs skills and experiences [1]. Semiatin et al. [2–4] evaluated phenomenological criterion based on the strain hardening and strain-rate sensivity. Also, Ziegler [5] evaluated continuum criterion based on the principle of maximum entropy. Ziegler proposed that a system undergoing large deformation shows instability when the strain-rate variations with strain-rate (_e) in the dissipation function (Dð_eÞ) satisfies the below relationship: dD Dðe_ Þ < d_e e_
ð1Þ
During deformation, power dissipation occurs through the temperature rise of the workpiece and microstructural changes (J co-content). The power partitioning between these two is divided by the strain-rate sensivity (m) of the flow stress (r) [1]. In a constant temperature and strain-rate, the J co-content was obtained as follows [6]: Z r m r_e ð2Þ J¼ e_ dr ¼ mþ1 0 In cold working, m is small and work hardening occurs. Prasad [7] applied the principle of reparability of the rate of entropy production into the conduction entropy and internal entropy. By considering the dissipation function corresponding to the microstructure changes (J co-content) a criterion for instability has been derived: nð_eÞ ¼
o ln ½m=ðmþÞ þm o ln e_
ð3Þ
Venugopal et al. have given an instability map. They suggested when nð_eÞ is negative, instability occurs. As can be seen, mechanical properties of 304 stainless steel is a complex function of temperature, strain-rate and strain and it is very difficult to develop a complete mathematical model to predict the flow stress of this steel. An engineering approach to predict the flow stress of engineering alloys is based on the utilization of artificial neural networks (ANNs). Neural networks consist of many computational elements, which operate in parallel and have been connected by several links with variable weights. These elements are typically
adopted during the learning process [8]. Recently, several models of neural networks have been proposed to predict the flow stress of metals [9,10]. The aim of this investigation is to develop a welltrained ANN model to predict the flow stress in compression tests for 304 stainless steel under cold and warm compression.
2. The data base The data used for this paper have been published by Venugopal et al. [1]. The composition of 304 stainless steel was C 0.08, S 0.005, P 0.035, Ni 10.3, Cr 18.6, Mn 1.7, Si 0.58, Mo 0.07 and balance Fe (all in weight percent), the temperature range was 20–600 °C, the strain-rate variation was from 0.001 s1 to 100 s1 and the range of strain was 0.1–0.5. They performed 360 compression tests on cylindrical specimens. Among these experiments, 300 data set were selected for training the ANN and the 60 remaining data set were used to validate the ability of the ANN model to predict the flow stress of 304 stainless steel. The ranges of variables, which have been used for this study, are given in Table 1.
3. Artificial neural networks 3.1. ANN principles ANNs, also called ‘‘neural networks’’, ‘‘parallel distributed processing’’ and ‘‘connectionist’’ models, developed out of the areas of artificial intelligence and cognitive science in their attempts to model the brain and its learning process. ANNs are collections of small individual interconnected
Table 1 The range of variables used to develop the neural network model for 304 stainless steel
Temperature Strain-rate Strain
Minimum value
Maximum value
Standard deviation
20 °C 0.001 S1 0.1
600 °C 100 S1 0.5
172.86 36.613 0.1414
A. Bahrami et al. / Computational Materials Science 34 (2005) 335–341
Input Layer
Hidden layer
Output layer
Temperature Strain rate
Flow stress
Strain
Fig. 1. Schematic diagram of multi-layer neural network.
processing units with weights associated with each connection. Fig. 1 shows the structure of ANN model with various layers. Learning is the first step necessary in inducing intelligence to neural networks. In learning, ANNs are ‘‘taught’’ by presenting sets of patterns to be ‘‘learned’’, and the network autonomously adjusts the connection weights among the processing units according to imposed learning rules and, thereby, obtains unique knowledge from the data. The learned neural network generates accurate outputs for the input data. For inputs that have not been experienced are partially damaged, distorted or mixed with noise, appropriate outputs are generated based on its internal knowledge stored in the connection weights. The basic concept of back propagation learning [11] is shown in Fig. 2, where first, the input patterns of each node are provided at the input layer. Then, this signal is converted and trans-
337
ferred to the hidden layer at each node, and finally, the signal generates outputs in the output layer. The output values are compared to the target values, and if there is a difference, the connection weights wji, wkj are adjusted in such a direction that the error is decreased. This adjustment is back propagated from the upper layers to the lower layers, which results in the adjustment of the connection weights in the lower layers. The back propagation learning algorithm has the following advantages. First, the ANN requires less expertise, knowledge and experimentation to determine the relationships between the inputs and outputs of non-linear systems. Second, it has been used successfully in similar applications, such as prediction [12–14]. 3.2. Learning procedure In this study the learning procedure was the back propagation algorithm. This ‘‘training’’ algorithm is performed step-by-step as follows: Step 1: The number of learning data is selected from the experimental. The ANN learns these data and determines the relationships between them. The data used here were collected from experiments of Venugopal et al. and are normalized between 0.5 and 0.5 to the following equation [15]:
Fig. 2. Schematic structure of the back propagation learning.
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A. Bahrami et al. / Computational Materials Science 34 (2005) 335–341
ðx xmin Þ 0:5 xN ¼ ðxmax xmin Þ
ð4Þ
where xN is the normalized value of x, xmin and xmax are minimum and maximum values of flow stress. Step 2: The connection weights wji and wkj, which determine the state of the ANN, and bias values hj and hk are initialized with random values. Step 3: The normalized learning data are fed into the ANN. Step 4: From connection weight wji, output opi of the input layer node i and bias hj, the input value netpj for hidden layer node j is determined as X netpj ¼ wji opi þ hj ð5Þ i
In Eq. (5), the i subscript indicates a summation over all nodes in the layer in the direction of the hidden layer. The output opj of node j is determined as shown in Eq. (7) below using netpj and the sigmoid function shown in Fig. 3 and Eq. (6). 1 ð6Þ f ðxÞ ¼ 1 þ expðxÞ opj ¼ fj ðnetpj Þ
ð7Þ
Step 5: The input value netpk of the output layer node k is determined from output opj of the hidden layer nodes, connection weight wkj between the hidden layer node and the output layer node and bias (offset) hk of node k. Then, using netpk and the sigmoid function, output opk of node k is determined as
netpk ¼
X
wkj opj þ hk
ð8Þ
j
opk ¼ fk ðnetpk Þ
ð9Þ
In Eq. (8), the j subscript indicates a summation over all nodes in the layer in the direction of the output layer. Step 6: Using the difference between the desired output tpk and the actual output opk, the error term dpk is determined as dpk ¼ tpk opk f 0 netpk ð10Þ ¼ tpk opk 1 opk Step 7: The error term dpj of the hidden layer is computed as X dpj ¼ fj0 netpj dpk wkj
¼ opj 1 opj
k
X
dpk wkj
ð11Þ
k
In Eq. (11), the k subscript indicates a summation over all nodes in the layer in the direction of the hidden layer from the output layer. Step 8: The connection weight wkj between the hidden layer node and the output layer node using Eq. (12) and bias value hk of the hidden layer node using Eq. (13) are modified. The term a is called the learning rate, and b is called the momentum factor. Both of these constant terms have an effect on the speed and stability of the network. wðnewÞkj ¼ wðoldÞkj þ adpk opk þ bDwðoldÞkj
ð12Þ hðnewÞj ¼ hðoldÞk þ adpk opk þ bDhðoldÞk ð13Þ where
Fig. 3. Shape of logistic sigmoid function.
DwðoldÞ ¼ wðoldÞ wðolderÞ
ð14Þ
DhðoldÞ ¼ hðoldÞ hðolderÞ
ð15Þ
Step 9: The connection weight wji between the input layer node and the hidden layer node using Eq. (16) and bias value hj of the hidden layer node using Eq. (17) are modified.
A. Bahrami et al. / Computational Materials Science 34 (2005) 335–341
wðnewÞji ¼ wðoldÞji þ adpj opj þ bDwðoldÞji
35 30
ð16Þ
25
MAE
hðnewÞj ¼ hðoldÞj þ adpj opj þ bDhðoldÞj
20 15 10
ð17Þ
4. Results and discussion 4.1. Neural network results The ANN model with one hidden layer was used in this study. The inputs of the ANNs were temperature, strain-rate and strain and flow stress was considered as an output. In first step, training was performed for adjusting connection weights between input layer to hidden layer and between hidden layer and output layer until the network to produce outputs that were close enough to the desired outputs. Mean absolute error (MAE) of desired and predicted data was calculated from the following equation: . X MAE ¼ jx yj n ð18Þ where x = XX 0 ; X is the target output and X 0 is the mean of X, y = YY 0 ; Y is network output and Y 0 is the mean of Y and n is the number of data set. In this model, when MAE reached the minimum value, the weight and bias values encode the networkÕs state of knowledge. Training data for the development of the neural network was obtained from the published paper [1]. Fig. 4 shows
5 0 0
5
10
15
20
25
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Fig. 4. MAE for different hidden units.
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Step 10: Return to the procedures of Step 3 until all of the learning data have been learned. Step 11: If the number of iterations is less than a limit of iterations, repeat the procedures following Step 3. Step 12: Learning is stopped. Step 13: The input data of each measurement time interval are fed into the learned ANN, and the flow stress for each measurement time interval by ANN recalling is calculated.
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R2 = 0.9791
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Measured flow stress (MPa)
Fig. 5. Comparison of predicted and measured flow stress for testing data set.
the MAE for various hidden layers. According to this figure, the ANN with one hidden layer and 15 hidden units has the smallest error (14.235). Fig. 5 shows the predicted flow stress by ANN model versus measured value for testing set. The test data set is not involved in the model training process and was only used for the testing purpose. Therefore, the accuracy of an ANN model in prediction is clearly demonstrated by its performance on the testing data set. These results have been obtained from a network with 15 neurons in hidden layer and 1230 epoch. As it has been shown in Fig. 5, the correlation coefficient for the test data set is 0.9791. This means that a good correlation between predicted and measured data has been obtained. Fig. 6 shows the comparison of predicted and measured flow stress for all data set. The overall correlation coefficient for the all data set is 0.9871. This figure shows that the used network
Predicted flow stress (MPa)
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All data
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Fig. 6. Comparison of predicted and measured flow stress for all data set.
Fig. 7. Significance of input variables in flow stress.
could be capable for prediction with a minimum error.
strain-rate has a reverse effect on flow stress. Formerly Venugopal et al. [1] have been reported that strain-rate has a negative effect.
4.2. Sensivity analysis The sensivity test was carried out to determine the relative significance of each of the input parameters. In this analysis, a step-by-step method was carried on the trained ANN by varying each of the input parameter, one at a time, at a constant rate. Various constant rates (5, 10, 20) were selected in this study. For every input parameter, the percentage was changed in the output as a result of the change in the input parameter. The sensivity of each input parameter was calculated by the following equation: Sensivity level of X i ð%Þ N 1 X % change in output ¼ 100 N j¼1 % change in input j
ð19Þ
where N is the number of data sets used in this study to test the network, functionality. Results showed that the changes in output parameter were unaffected significantly by the rate of an input parameter when strain-rate was varied (5, 10, 20). Fig. 7 shows the sensitivity of variations for flow stress at each of the input variables. As can be seen, temperature is the most significant parameter. These results are very close to the pervious investigations, which reported that temperature is the most important parameter in flow stress of stainless steels [16,17]. This analysis showed that
5. Conclusions The aim of this paper was to use the neural networks for calculation of the flow stress in 304 stainless steel. An ANN model with one hidden layer and 15 neurons in the hidden layer is a benefit method for prediction of flow stress in 304 stainless steel, where temperature, strain-rate and strain are input parameters. Sensivity analysis showed that temperature is the most effective parameter, the effect of strain-rate is negative, and strain has the minimum effect on flow stress. These features enable us to use ANNs in stainless steel and will help people studying in this field. So, experimental studies can be reduced to a minimum at the places where the use of ANNs is appropriate. Also, the results showed that the ANN approach could be considered as an alternative and practical technique to evaluate the flow stress in 304 stainless steel.
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