- Email: [email protected]
Modeling of microstructure and constitutive relation during superplastic deformation by fuzzy-neural network
Journal of Materials Processing Technology 142 (2003) 197–202
Modeling of microstructure and constitutive relation during superplastic deformation by fuzzy-neural network Dunjun Chen∗ , Miaoquan Li, Shichun Wu Department of Materials Science and Engineering, Northwestern Polytechnical University, Xi’an 710072, PR China Received 9 January 2002; received in revised form 28 August 2002; accepted 14 February 2003
Abstract In this paper, an adaptive fuzzy-neural network model has been established to model the microstructure evolution and constitutive relation of 15 vol.% SiCp/LY12 aluminum composite during superplastic deformation. This network integrates the learning power of neural networks with fuzzy inference systems. During the training process of the network, the back-propagation learning algorithm is applied to optimally adjust the weight coefficients of the neural network and the parameters of the fuzzy membership functions. Then, the trained network is used to predict the microstructure evolution and constitutive relation of 15 vol.% SiCp/LY12 aluminum composite during superplastic deformation. The predicted results agree very well with the experimental data of the test samples. On the basis of the good prediction ability of the proposed fuzzy-neural network, the constitutive relation and microstructure of 15 vol.% SiCp/LY12 aluminum composite under various superplastic deformation conditions have also been calculated and analyzed. © 2003 Elsevier Science B.V. All rights reserved. Keywords: Fuzzy-neural network; Superplastic deformation; Microstructure evolution; Constitutive relation
1. Introduction Superplastic deformation has been a subject of scientific interest and industrial application for many years. The quality of products is mainly determined by the microstructure after superplastic deformation, so the microstructure evolution, including grain growth and cavity growth during superplastic deformation, has been playing a significant role in the research field of superplasticity. The factors affecting the microstructure of materials mainly include deformation temperature, strain rate and strain level, and how to optimally design and control those process variables is very important for effective control of the microstructure. On this account, many researchers [1–6] established the microstructure evolution models during superplastic deformation on the basis of experimental data, and these models described the relation between the microstructure and process variables by a specific mathematical expression. Similarly, the constitutive relations which reflect the deformation behavior of superplasticity were also described in the form of the curve diagram plotted by experimental results or the form of empiric mathematical equations [5–8]. ∗ Corresponding author. Present address: Department of Physics, Nanjing University, Nanjing 210093, PR China. E-mail address: [email protected] (D. Chen).
However, superplastic deformation is a complex dynamic process and process parameters present the non-linear relationship against the microstructure and deformation stress, so the microstructure evolution and the constitutive relation are difficult to be described accurately with a mathematical expression by the method of regression of experimental data. The past few years have witnessed the development of fuzzy-neural network technology. This integrates the excellent learning capability of neural networks with fuzzy inference systems for deriving the initial rules of a fuzzy system and tuning membership functions [9–14]. Furthermore, it does not require a mathematical description of the system. These advantages of fuzzy-neural networks make them suitable for a wide range of knowledge engineering and scientific applications, especially for treating non-linear systems and complex relational systems with unknown models [15,16]. In this paper, an adaptive fuzzy-neural network on the basis of an error back-propagation learning algorithm is applied to establish constitutive relation and correlate the relation between the microstructure and process parameters during superplastic deformation of 15 vol.% SiCp/LY12 aluminum composite. During the training process of the network, the process parameters of superplastic deformation are considered as input variables and grain size, volume fraction of cavities and true stress are taken as output variables of the
0924-0136/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0924-0136(03)00598-3
198
D. Chen et al. / Journal of Materials Processing Technology 142 (2003) 197–202
network. Then, the trained network is used to predict the microstructure evolution and constitutive relation of 15 vol.% SiCp/LY12 aluminum composite during superplastic deformation. By the comparison of the calculated results with the experimental data of the testing samples, it has been verified that this fuzzy-neural network model is capable of a good prediction.
2. Fuzzy-neural network model In this section, an adaptive fuzzy-neural network is introduced to predict the microstructure and constitutive relation during superplastic deformation. A major characteristic of the network is that fuzzy rules and fuzzy membership functions can be automatically modified on-line, which is very fit to prediction and control of the complex systems. In addition, its learning algorithm is simplified on account of using multiplication algorithm instead of extremum operation. The structure of the fuzzy-neural network is shown in Fig. 1, x1 , . . . , xn represent input variables; Y is the output of the network; US, UM and UL represent membership functions; pi0 , . . . , pin are the weight coefficients of the middle layer of neural network; S and P represent sum and multiplication operations respectively; wi the weight value which denotes the degree of the rule; yi the outputs of the middle layer. The functions of the nodes in each layers of the network are described as follows: Layer 1. Layer 1 is called input variable layer. Each node in this layer only transport input values to the next layer. Layer 2. The number of fuzzy sets of each input variable and membership functions are designated in this
layer. Here, we divide typically each input variable into three sets: large (UL), medium (UM) and small (US) and use following Gaussian function to define all membership functions: (xj − aji )2 i µj = exp − (1) bji where aji and bji are the parameters of the membership functions which need to be optimized. Meanwhile, the value of membership function, which denotes the degree that an input value belongs to a fuzzy set is calculated in this layer. Layer 3. Each node in this layer represents one fuzzy rule, and the degrees of rules, i.e. the weight value wi are calculated in this layer. Fuzzy rules are described as follows: R1 : if x1 is UL, x2 is UL, . . . , xn is UL. Then y1 = p10 + p11 x1 + p12 x2 + · · · + p1n xn (2) w1 = µ11 µ12 · · · µ1n
(3)
Rm : if x1 is US, x2 is US, . . . , xn is US. Then m m m y m = pm 0 + p1 x1 + p2 x2 + · · · + pn xn
(4) m m wm = µm 1 µ2 · · · µn
(5)
where m is the number of rules, obviously m = 3n according to the array of fuzzy sets of input variables.
Fig. 1. The structure of the fuzzy-neural network.
D. Chen et al. / Journal of Materials Processing Technology 142 (2003) 197–202
Layer 4. Each node in this layer represents multiplication operation and the function of the node is y(4) = µi1 (x1 ) µi2 (x2 ) · · · µin (xn ) (pi0 + pi1 x1 + · · · + pin xn )
(6)
Layer 5. Each node in this layer represents sum operation and the function of the node is y(5) =
m i=1
[µi1 (x1 ) µi2 (x2 ) · · · µin (xn ) (pi0 + pi1 x1
+ · · · + pin xn )]
(7)
bji (k + 1) = bji (k) − β(yd − Y) i m i m i i y i=1 w − i=1 w y (xj − aji )2 wi × m
i 2 (bji )2 w i=1 (14) where η and β are the learning rate.
4. Experiments The 15 vol.% SiCp/LY12 aluminum composite was fabricated by spray atomization and codeposition. The chemical compositions of the LY12 matrix are: Cu 3.8–4.9; Mg 1.2–1.8; Mn 0.3–0.9 and the remaining Al, and the alloy
Layer 6. This layer is the output layer of the fuzzy-neural network. The output function of the network is m m i i [µi (x1 ) µi2 (x2 ) · · · µin (xn ) (pi0 + pi1 x1 + · · · + pin xn )] wy i=1 Y = m i = i=1 1 m i i i i=1 w i=1 [µ1 (x1 ) µ2 (x2 ) · · · µn (xn )] 3. Back-propagation learning of the network To optimally adjust the parameters of the membership functions and weight coefficients of neural network, an error back-propagation learning algorithm is presented by minimizing the error function E = 21 (yd − Y)2
(9)
where yd is the desired output and Y the current output. According to the back-propagation learning algorithm and the chain rule, we have pij (k + 1) = pij (k) − η
∂E ∂pij
(10)
199
(8)
was mixed with SiC particles having an average diameter of 10 m. After isothermal hot compression and isothermal hot forward extrusion (extrusion ratio: 10) of the composite, the uniaxial tension specimens were machined. The gage diameter and length of specimens were 5 mm × 15 mm. Tensile tests were carried out using a type CSS-1110C material-testing machine, at a constant cross-head speed both with and without the concurrent application of an external direct-current electric field from a commercial source. The measurement of microstructure under various process parameters and deformation extents was carried out on Leica LABOR-LUX 12MFS/ST microscopy, in which 20 visual fields at each location of each specimen were chosen to be measured. The microstructure variables, including the average diameter of grains and the volume fraction of cavities were taken as the average value of the three specimens.
where ∂E ∂E ∂Y = = −(yd − Y) i ∂Y ∂pij ∂pj wi = −(yd − Y) m
i=1 w
m
i=1 w
m
i yi /
∂pij
i=1 w
i
xi i j
5. Application of network model
(11)
Hence, the p parameter now becomes pij (k + 1) =
pij (k) − η(yd − Y)wi , m i i i=1 w xj
x0i = 1, j = 1, 2, . . . , n
(12)
Similarly, we have aji (k + 1) = aji (k) − β(yd − Y) i m i m i i y i=1 w − i=1 w y 2(xj − aji )wi ×
m i 2 bji i=1 w (13)
In this section, we use the proposed fuzzy-neural network model to calculate the microstructure and constitutive relation during superplastic deformation of the 15 vol.% SiCp/LY12 aluminum composite. Here, the grain size, the volume fraction of cavities and true stress are taken as output variables of the network. The process parameters including deformation temperature, strain rate, electric field and true strain are treated as input variables, which are divided into three fuzzy sets: large (UL), medium (UM) and small (US). A fuzzy membership function is assigned for every set of input variables and Fig. 2 shows the initial membership functions for each input variable on the basis of input data of samples in Table 1. To demonstrate the potential of the proposed fuzzy-neural network, 32 data pairs from experimental results are provided in Table 1. Those data are partitioned into 20 set of input–output pairs as training samples, and the remaining
200
D. Chen et al. / Journal of Materials Processing Technology 142 (2003) 197–202
Fig. 2. The initial division of input spaces into three fuzzy regions and their corresponding membership functions of the division of (a) temperature, (b) strain rate, (c) electric field, and (d) true strain variables.
Table 1 Input–output pairs for training and test of fuzzy-neural network Type of samples
Number of pairs
Temperature, x1 (K)
Strain rate, x2 (×10−4 s−1 )
Electric field, x3 (kV/cm)
True strain, x4
Grain size, y1 (m)
Volume fraction, y2 (%)
True stress, y3 (MPa)
Training samples
01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20
753 753 753 753 763 763 763 763 763 773 773 773 773 773 773 783 783 783 783 783
5.6 2.2 11.3 2.2 5.6 2.2 11.3 2.2 5.6 3.3 3.3 3.3 3.3 3.3 3.3 5.6 2.2 11.3 2.2 11.3
1 3 2 0 1 3 2 0 1 0 0 0 2 2 2 1 3 2 0 3
0.93 0.6 0.2 0.8 0.93 0.6 0.2 1 0.8 0.3 1.2 1.5 0.3 1.2 1.6 0.93 0.6 0.2 1 0.4
15.1 15.6 15.5 15.5 16.2 16.6 16.4 16.2 16 16.4 16.2 15.6 15.3 14.8 14.3 18 18.2 18.2 18 18.4
6.7 4.8 1.9 6.6 6.3 4.3 1.5 6.9 6 2 8 15 1.5 5.5 15 5.8 3.7 0.9 6.4 1.8
8.2 5.3 19.1 5.1 6.8 4.4 18 3.8 5.5 8.8 5.2 6 7.8 2.5 5.1 4.3 2.2 16.3 2 15.4
Test samples
21 22 23 24 25 26 27 28 29 30 31 32
753 753 763 763 763 773 773 773 773 783 783 783
2.2 2.2 5.6 11.3 2.2 3.3 5.6 3.3 5.6 11.3 3.3 5.6
1 3 1 2 0 0 0 2 2 3 0 1
0.6 0.3 0.7 0.2 0.6 0.8 0.7 1.4 1.2 0.3 0.8 1
13.1 16.5 15.5 16.7 15.8 16.5 16 15 15.5 18.5 18 17.2
0.4 6.2 5.5 1.4 1 3 2.6 12 8.4 1.5 3.5 6.8
4 7.3 4.1 18.6 5.5 5 8.1 4.1 5.4 18 5.2 4.8
D. Chen et al. / Journal of Materials Processing Technology 142 (2003) 197–202
Fig. 3. The prediction error of test samples.
12 set of input–output pairs as test samples for validation. During the training process, learning rate η = 0.00001, β = 0.00001 and the initial values of pi0 , . . . , pi4 are taken to be zero. The training of network will be terminated when the system error, which denotes the value of the accumulative relative errors of outputs of 20 training samples, reaches 2%. Then this trained network is used as the final prediction system of the microstructure and constitutive relation of the material. The relative error of the true stress, the grain size and the volume fraction of cavities, between the experimental data from testing samples and the prediction results are plotted in Fig. 3. It can be seen that the test error value of the every test sample is not more than 10%. Meanwhile, the convergence rate of the proposed fuzzy-neural network is rather quick with the system error reaching 2% after iteration of 536,648 and 674 times for the three outputs, respectively. All of those demonstrate that the adaptive fuzzy-neural network model has merits of good prediction accuracy and quick learning rate.
201
Fig. 4. Curves of true stress vs. true strain.
Fig. 5. The effect of the electric field intensity on grain size.
during superplastic deformation, respectively. It can be observed that: (i) the electric field reduces the grain size during superplastic deformation, which provides the microstructure condition for the improvement of superplasticity; (ii) the
6. The calculation of constitutive relation and microstructure evolution On the basis of the present fuzzy-neural network model, the constitutive relation and microstructure of 15 vol.% SiCp/LY12 aluminum composite under various superplastic deformation conditions are worked out. Fig. 4 shows the computed and experimental true stress–true strain constitutive relation curve for 15 vol.% SiCp/LY12 aluminum composite under various superplastic deformation parameters. It can be seen that those curves are a combination of strain-hardening and strain-softening behavior, meanwhile, the results computed agree very well with the experimental data. Figs. 5 and 6 indicate the effect of the electric field intensity on grain size and volume fraction of cavities of 15 vol.% SiCp/LY12 aluminum composite
Fig. 6. The effect of the electric field intensity on volume fraction of cavities.
202
D. Chen et al. / Journal of Materials Processing Technology 142 (2003) 197–202
appropriate electric field can reduce the volume fraction of cavities during superplastic deformation; (iii) the computed results are in good accord with the experimental results. Similarly, we can obtain the relation between other process parameters and the microstructure of 15 vol.% SiCp/LY12 aluminum composite during superplastic deformation by this trained fuzzy-neural network. Here, they are not reiterated.
Acknowledgements The authors wish to express their gratitude to the National Natural Science Foundation of China and the Aeronautic Science Foundation of China for providing the funds of this investigation.
References 7. Conclusions This paper proposes a new adaptive fuzzy-neural network model for the microstructure evolution and constitutive relation during superplastic deformation. Its main features and advantages are: (i) it combines two technologies, namely fuzzy inference system and neural networks; (ii) fuzzy rules and fuzzy membership functions can be tuned optimally by using a back-propagation learning algorithm; (iii) its learning algorithm is simplified on account of using only sum and multiplication algorithms; (iv) it is also adapted to other engineering and scientific applications because its fuzzy membership functions and fuzzy rules can adapt and change according to new training data. The prediction results show good agreement with the experimental data, and also demonstrate that the fuzzy-neural network method has a good simulation precision. Therefore, we can quickly and conveniently select the optimum process parameters to achieve the desired results and understand the behavior of microstructure evolution during the superplastic deformation by means of the prediction of the fuzzy-neural network.
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
M.J. Stowell, Met. Sci. 17 (1983) 1–11. W. Beere, M.V. Speight, Met. Sci. 12 (4) (1978) 172–176. A.H. Choksih, Acta Metall. 35 (5) (1987) 1089–1101. M.B. Liu, S.C. Wu, J. Mater. Process. Technol. 41 (1) (1994) 115– 124. S.L. Semiatin, V. Seetharaman, I. Weiss, Mater. Sci. Eng. A 243 (1998) 1–24. V.M. Imayev, R.M. Imayev, G.A. Salishchev, Mater. Sci. Eng. A 208 (1996) 226–231. P.L. Charpentier, B.C. Stone, Metall. Trans. A 17 (12) (1986) 2227– 2237. A.H. Chokshi, A.K. Mukherjee, T.G. Langdon, Mater. Sci. Eng. R 10 (6) (1993) 237–274. S. Horikawa, T. Furuhashi, Y. Uchikawa, IEEE Trans. Neural Networks 3 (5) (1992) 801–806. J.S.R. Jang, IEEE Trans. Syst., Man Cybernet. 23 (3) (1993) 665– 685. N. Kasabov, J. Kim, M. Watts, A. Gray, Inform. Sci. 101 (3) (1997) 155–175. Y.C. Lee, C.H. Hwang, Y.P. Shih, IEEE Trans. Syst., Man Cybernet. 24 (5) (1994) 736–744. Y. Lin, IEEE Trans. Syst. 3 (2) (1995) 190–198. C.F. Juang, C.T. Lin, IEEE Trans. Syst. 6 (2) (1998) 12–32. J. Kim, N. Kasabov, Neural Networks 12 (1999) 1301–1319. C.T. Lin, C.S.G. Lee, IEEE Trans. Comput. 40 (12) (1991) 1320– 1366.
Modeling of microstructure and constitutive relation during superplastic deformation by fuzzy-neural network Dunjun Chen∗ , Miaoquan Li, Shichun Wu Department of Materials Science and Engineering, Northwestern Polytechnical University, Xi’an 710072, PR China Received 9 January 2002; received in revised form 28 August 2002; accepted 14 February 2003
Abstract In this paper, an adaptive fuzzy-neural network model has been established to model the microstructure evolution and constitutive relation of 15 vol.% SiCp/LY12 aluminum composite during superplastic deformation. This network integrates the learning power of neural networks with fuzzy inference systems. During the training process of the network, the back-propagation learning algorithm is applied to optimally adjust the weight coefficients of the neural network and the parameters of the fuzzy membership functions. Then, the trained network is used to predict the microstructure evolution and constitutive relation of 15 vol.% SiCp/LY12 aluminum composite during superplastic deformation. The predicted results agree very well with the experimental data of the test samples. On the basis of the good prediction ability of the proposed fuzzy-neural network, the constitutive relation and microstructure of 15 vol.% SiCp/LY12 aluminum composite under various superplastic deformation conditions have also been calculated and analyzed. © 2003 Elsevier Science B.V. All rights reserved. Keywords: Fuzzy-neural network; Superplastic deformation; Microstructure evolution; Constitutive relation
1. Introduction Superplastic deformation has been a subject of scientific interest and industrial application for many years. The quality of products is mainly determined by the microstructure after superplastic deformation, so the microstructure evolution, including grain growth and cavity growth during superplastic deformation, has been playing a significant role in the research field of superplasticity. The factors affecting the microstructure of materials mainly include deformation temperature, strain rate and strain level, and how to optimally design and control those process variables is very important for effective control of the microstructure. On this account, many researchers [1–6] established the microstructure evolution models during superplastic deformation on the basis of experimental data, and these models described the relation between the microstructure and process variables by a specific mathematical expression. Similarly, the constitutive relations which reflect the deformation behavior of superplasticity were also described in the form of the curve diagram plotted by experimental results or the form of empiric mathematical equations [5–8]. ∗ Corresponding author. Present address: Department of Physics, Nanjing University, Nanjing 210093, PR China. E-mail address: [email protected] (D. Chen).
However, superplastic deformation is a complex dynamic process and process parameters present the non-linear relationship against the microstructure and deformation stress, so the microstructure evolution and the constitutive relation are difficult to be described accurately with a mathematical expression by the method of regression of experimental data. The past few years have witnessed the development of fuzzy-neural network technology. This integrates the excellent learning capability of neural networks with fuzzy inference systems for deriving the initial rules of a fuzzy system and tuning membership functions [9–14]. Furthermore, it does not require a mathematical description of the system. These advantages of fuzzy-neural networks make them suitable for a wide range of knowledge engineering and scientific applications, especially for treating non-linear systems and complex relational systems with unknown models [15,16]. In this paper, an adaptive fuzzy-neural network on the basis of an error back-propagation learning algorithm is applied to establish constitutive relation and correlate the relation between the microstructure and process parameters during superplastic deformation of 15 vol.% SiCp/LY12 aluminum composite. During the training process of the network, the process parameters of superplastic deformation are considered as input variables and grain size, volume fraction of cavities and true stress are taken as output variables of the
0924-0136/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0924-0136(03)00598-3
198
D. Chen et al. / Journal of Materials Processing Technology 142 (2003) 197–202
network. Then, the trained network is used to predict the microstructure evolution and constitutive relation of 15 vol.% SiCp/LY12 aluminum composite during superplastic deformation. By the comparison of the calculated results with the experimental data of the testing samples, it has been verified that this fuzzy-neural network model is capable of a good prediction.
2. Fuzzy-neural network model In this section, an adaptive fuzzy-neural network is introduced to predict the microstructure and constitutive relation during superplastic deformation. A major characteristic of the network is that fuzzy rules and fuzzy membership functions can be automatically modified on-line, which is very fit to prediction and control of the complex systems. In addition, its learning algorithm is simplified on account of using multiplication algorithm instead of extremum operation. The structure of the fuzzy-neural network is shown in Fig. 1, x1 , . . . , xn represent input variables; Y is the output of the network; US, UM and UL represent membership functions; pi0 , . . . , pin are the weight coefficients of the middle layer of neural network; S and P represent sum and multiplication operations respectively; wi the weight value which denotes the degree of the rule; yi the outputs of the middle layer. The functions of the nodes in each layers of the network are described as follows: Layer 1. Layer 1 is called input variable layer. Each node in this layer only transport input values to the next layer. Layer 2. The number of fuzzy sets of each input variable and membership functions are designated in this
layer. Here, we divide typically each input variable into three sets: large (UL), medium (UM) and small (US) and use following Gaussian function to define all membership functions: (xj − aji )2 i µj = exp − (1) bji where aji and bji are the parameters of the membership functions which need to be optimized. Meanwhile, the value of membership function, which denotes the degree that an input value belongs to a fuzzy set is calculated in this layer. Layer 3. Each node in this layer represents one fuzzy rule, and the degrees of rules, i.e. the weight value wi are calculated in this layer. Fuzzy rules are described as follows: R1 : if x1 is UL, x2 is UL, . . . , xn is UL. Then y1 = p10 + p11 x1 + p12 x2 + · · · + p1n xn (2) w1 = µ11 µ12 · · · µ1n
(3)
Rm : if x1 is US, x2 is US, . . . , xn is US. Then m m m y m = pm 0 + p1 x1 + p2 x2 + · · · + pn xn
(4) m m wm = µm 1 µ2 · · · µn
(5)
where m is the number of rules, obviously m = 3n according to the array of fuzzy sets of input variables.
Fig. 1. The structure of the fuzzy-neural network.
D. Chen et al. / Journal of Materials Processing Technology 142 (2003) 197–202
Layer 4. Each node in this layer represents multiplication operation and the function of the node is y(4) = µi1 (x1 ) µi2 (x2 ) · · · µin (xn ) (pi0 + pi1 x1 + · · · + pin xn )
(6)
Layer 5. Each node in this layer represents sum operation and the function of the node is y(5) =
m i=1
[µi1 (x1 ) µi2 (x2 ) · · · µin (xn ) (pi0 + pi1 x1
+ · · · + pin xn )]
(7)
bji (k + 1) = bji (k) − β(yd − Y) i m i m i i y i=1 w − i=1 w y (xj − aji )2 wi × m
i 2 (bji )2 w i=1 (14) where η and β are the learning rate.
4. Experiments The 15 vol.% SiCp/LY12 aluminum composite was fabricated by spray atomization and codeposition. The chemical compositions of the LY12 matrix are: Cu 3.8–4.9; Mg 1.2–1.8; Mn 0.3–0.9 and the remaining Al, and the alloy
Layer 6. This layer is the output layer of the fuzzy-neural network. The output function of the network is m m i i [µi (x1 ) µi2 (x2 ) · · · µin (xn ) (pi0 + pi1 x1 + · · · + pin xn )] wy i=1 Y = m i = i=1 1 m i i i i=1 w i=1 [µ1 (x1 ) µ2 (x2 ) · · · µn (xn )] 3. Back-propagation learning of the network To optimally adjust the parameters of the membership functions and weight coefficients of neural network, an error back-propagation learning algorithm is presented by minimizing the error function E = 21 (yd − Y)2
(9)
where yd is the desired output and Y the current output. According to the back-propagation learning algorithm and the chain rule, we have pij (k + 1) = pij (k) − η
∂E ∂pij
(10)
199
(8)
was mixed with SiC particles having an average diameter of 10 m. After isothermal hot compression and isothermal hot forward extrusion (extrusion ratio: 10) of the composite, the uniaxial tension specimens were machined. The gage diameter and length of specimens were 5 mm × 15 mm. Tensile tests were carried out using a type CSS-1110C material-testing machine, at a constant cross-head speed both with and without the concurrent application of an external direct-current electric field from a commercial source. The measurement of microstructure under various process parameters and deformation extents was carried out on Leica LABOR-LUX 12MFS/ST microscopy, in which 20 visual fields at each location of each specimen were chosen to be measured. The microstructure variables, including the average diameter of grains and the volume fraction of cavities were taken as the average value of the three specimens.
where ∂E ∂E ∂Y = = −(yd − Y) i ∂Y ∂pij ∂pj wi = −(yd − Y) m
i=1 w
m
i=1 w
m
i yi /
∂pij
i=1 w
i
xi i j
5. Application of network model
(11)
Hence, the p parameter now becomes pij (k + 1) =
pij (k) − η(yd − Y)wi , m i i i=1 w xj
x0i = 1, j = 1, 2, . . . , n
(12)
Similarly, we have aji (k + 1) = aji (k) − β(yd − Y) i m i m i i y i=1 w − i=1 w y 2(xj − aji )wi ×
m i 2 bji i=1 w (13)
In this section, we use the proposed fuzzy-neural network model to calculate the microstructure and constitutive relation during superplastic deformation of the 15 vol.% SiCp/LY12 aluminum composite. Here, the grain size, the volume fraction of cavities and true stress are taken as output variables of the network. The process parameters including deformation temperature, strain rate, electric field and true strain are treated as input variables, which are divided into three fuzzy sets: large (UL), medium (UM) and small (US). A fuzzy membership function is assigned for every set of input variables and Fig. 2 shows the initial membership functions for each input variable on the basis of input data of samples in Table 1. To demonstrate the potential of the proposed fuzzy-neural network, 32 data pairs from experimental results are provided in Table 1. Those data are partitioned into 20 set of input–output pairs as training samples, and the remaining
200
D. Chen et al. / Journal of Materials Processing Technology 142 (2003) 197–202
Fig. 2. The initial division of input spaces into three fuzzy regions and their corresponding membership functions of the division of (a) temperature, (b) strain rate, (c) electric field, and (d) true strain variables.
Table 1 Input–output pairs for training and test of fuzzy-neural network Type of samples
Number of pairs
Temperature, x1 (K)
Strain rate, x2 (×10−4 s−1 )
Electric field, x3 (kV/cm)
True strain, x4
Grain size, y1 (m)
Volume fraction, y2 (%)
True stress, y3 (MPa)
Training samples
01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20
753 753 753 753 763 763 763 763 763 773 773 773 773 773 773 783 783 783 783 783
5.6 2.2 11.3 2.2 5.6 2.2 11.3 2.2 5.6 3.3 3.3 3.3 3.3 3.3 3.3 5.6 2.2 11.3 2.2 11.3
1 3 2 0 1 3 2 0 1 0 0 0 2 2 2 1 3 2 0 3
0.93 0.6 0.2 0.8 0.93 0.6 0.2 1 0.8 0.3 1.2 1.5 0.3 1.2 1.6 0.93 0.6 0.2 1 0.4
15.1 15.6 15.5 15.5 16.2 16.6 16.4 16.2 16 16.4 16.2 15.6 15.3 14.8 14.3 18 18.2 18.2 18 18.4
6.7 4.8 1.9 6.6 6.3 4.3 1.5 6.9 6 2 8 15 1.5 5.5 15 5.8 3.7 0.9 6.4 1.8
8.2 5.3 19.1 5.1 6.8 4.4 18 3.8 5.5 8.8 5.2 6 7.8 2.5 5.1 4.3 2.2 16.3 2 15.4
Test samples
21 22 23 24 25 26 27 28 29 30 31 32
753 753 763 763 763 773 773 773 773 783 783 783
2.2 2.2 5.6 11.3 2.2 3.3 5.6 3.3 5.6 11.3 3.3 5.6
1 3 1 2 0 0 0 2 2 3 0 1
0.6 0.3 0.7 0.2 0.6 0.8 0.7 1.4 1.2 0.3 0.8 1
13.1 16.5 15.5 16.7 15.8 16.5 16 15 15.5 18.5 18 17.2
0.4 6.2 5.5 1.4 1 3 2.6 12 8.4 1.5 3.5 6.8
4 7.3 4.1 18.6 5.5 5 8.1 4.1 5.4 18 5.2 4.8
D. Chen et al. / Journal of Materials Processing Technology 142 (2003) 197–202
Fig. 3. The prediction error of test samples.
12 set of input–output pairs as test samples for validation. During the training process, learning rate η = 0.00001, β = 0.00001 and the initial values of pi0 , . . . , pi4 are taken to be zero. The training of network will be terminated when the system error, which denotes the value of the accumulative relative errors of outputs of 20 training samples, reaches 2%. Then this trained network is used as the final prediction system of the microstructure and constitutive relation of the material. The relative error of the true stress, the grain size and the volume fraction of cavities, between the experimental data from testing samples and the prediction results are plotted in Fig. 3. It can be seen that the test error value of the every test sample is not more than 10%. Meanwhile, the convergence rate of the proposed fuzzy-neural network is rather quick with the system error reaching 2% after iteration of 536,648 and 674 times for the three outputs, respectively. All of those demonstrate that the adaptive fuzzy-neural network model has merits of good prediction accuracy and quick learning rate.
201
Fig. 4. Curves of true stress vs. true strain.
Fig. 5. The effect of the electric field intensity on grain size.
during superplastic deformation, respectively. It can be observed that: (i) the electric field reduces the grain size during superplastic deformation, which provides the microstructure condition for the improvement of superplasticity; (ii) the
6. The calculation of constitutive relation and microstructure evolution On the basis of the present fuzzy-neural network model, the constitutive relation and microstructure of 15 vol.% SiCp/LY12 aluminum composite under various superplastic deformation conditions are worked out. Fig. 4 shows the computed and experimental true stress–true strain constitutive relation curve for 15 vol.% SiCp/LY12 aluminum composite under various superplastic deformation parameters. It can be seen that those curves are a combination of strain-hardening and strain-softening behavior, meanwhile, the results computed agree very well with the experimental data. Figs. 5 and 6 indicate the effect of the electric field intensity on grain size and volume fraction of cavities of 15 vol.% SiCp/LY12 aluminum composite
Fig. 6. The effect of the electric field intensity on volume fraction of cavities.
202
D. Chen et al. / Journal of Materials Processing Technology 142 (2003) 197–202
appropriate electric field can reduce the volume fraction of cavities during superplastic deformation; (iii) the computed results are in good accord with the experimental results. Similarly, we can obtain the relation between other process parameters and the microstructure of 15 vol.% SiCp/LY12 aluminum composite during superplastic deformation by this trained fuzzy-neural network. Here, they are not reiterated.
Acknowledgements The authors wish to express their gratitude to the National Natural Science Foundation of China and the Aeronautic Science Foundation of China for providing the funds of this investigation.
References 7. Conclusions This paper proposes a new adaptive fuzzy-neural network model for the microstructure evolution and constitutive relation during superplastic deformation. Its main features and advantages are: (i) it combines two technologies, namely fuzzy inference system and neural networks; (ii) fuzzy rules and fuzzy membership functions can be tuned optimally by using a back-propagation learning algorithm; (iii) its learning algorithm is simplified on account of using only sum and multiplication algorithms; (iv) it is also adapted to other engineering and scientific applications because its fuzzy membership functions and fuzzy rules can adapt and change according to new training data. The prediction results show good agreement with the experimental data, and also demonstrate that the fuzzy-neural network method has a good simulation precision. Therefore, we can quickly and conveniently select the optimum process parameters to achieve the desired results and understand the behavior of microstructure evolution during the superplastic deformation by means of the prediction of the fuzzy-neural network.
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
M.J. Stowell, Met. Sci. 17 (1983) 1–11. W. Beere, M.V. Speight, Met. Sci. 12 (4) (1978) 172–176. A.H. Choksih, Acta Metall. 35 (5) (1987) 1089–1101. M.B. Liu, S.C. Wu, J. Mater. Process. Technol. 41 (1) (1994) 115– 124. S.L. Semiatin, V. Seetharaman, I. Weiss, Mater. Sci. Eng. A 243 (1998) 1–24. V.M. Imayev, R.M. Imayev, G.A. Salishchev, Mater. Sci. Eng. A 208 (1996) 226–231. P.L. Charpentier, B.C. Stone, Metall. Trans. A 17 (12) (1986) 2227– 2237. A.H. Chokshi, A.K. Mukherjee, T.G. Langdon, Mater. Sci. Eng. R 10 (6) (1993) 237–274. S. Horikawa, T. Furuhashi, Y. Uchikawa, IEEE Trans. Neural Networks 3 (5) (1992) 801–806. J.S.R. Jang, IEEE Trans. Syst., Man Cybernet. 23 (3) (1993) 665– 685. N. Kasabov, J. Kim, M. Watts, A. Gray, Inform. Sci. 101 (3) (1997) 155–175. Y.C. Lee, C.H. Hwang, Y.P. Shih, IEEE Trans. Syst., Man Cybernet. 24 (5) (1994) 736–744. Y. Lin, IEEE Trans. Syst. 3 (2) (1995) 190–198. C.F. Juang, C.T. Lin, IEEE Trans. Syst. 6 (2) (1998) 12–32. J. Kim, N. Kasabov, Neural Networks 12 (1999) 1301–1319. C.T. Lin, C.S.G. Lee, IEEE Trans. Comput. 40 (12) (1991) 1320– 1366.