Literature Review
3.4 High temperature compression test
3.5.2 Artificial neural network modeling of the flow stress
The flow stress of all alloys was predicted by artificial neural network (ANN) modeling. In order to obtain better prediction of flow stress (σ) during isothermal hot deformation of the investigated alloys, artificial neural network (ANN) modeling was carried out. This was based on the concept that
( , , ). T
(2.15)The deformation flow stress flow stress (σ) can be predicted by artificial neural network, which is a data driven model, if sufficient number of data sets is available. A typical artificial neural network (ANN) architecture consists of three layers, viz. an input layer consisting of three neurons representing the external parameters of strain (ε), strain rate ( ̇) and temperature (T) a hidden layer containing n numbers of neurons and an output layer with one neuron representing flow stress. Appropriate values for
the weight and bias of the network architecture for each alloy was arrived at by training the network using several data sets.
The artificial neural network modeling was carried out by the multiple layer perception (MLP) feed forward back propagation network. The input layer consisting of three neurons ( ̇, ε, and T) and the flow stress (σ), in the output layer formed the data sets for training the network. The ‘Neural Network’ tool box available with the MATLAB (R2010a) software package was used for the present modeling. Training of the neural network was done using the artificial neural network tool kit of MATLAB software, using ‘TRAINLM’ function. ‘TRAINLM’ is a network training function that updates weights and bias values in a back propagation algorithm according to Levenberg–Marquardt optimization. Levenberg–Marquardt algorithm is a highly efficient method for solving non-linear optimization problems [40]. Single layer hidden neurons were used in the network architecture. The number of neurons in the hidden layer, the transfer functions at the input-to-hidden layer and hidden-to-output layer were optimized by trial and error method during the network training and testing stages. The mean square error (MSE) during the training and testing was determined for each trial. The network architecture was finally frozen based on the minimum mean square error value obtained during both the training and testing stages.
During the modeling, strain (ε), temperature (T) and ln( ̇) values were transformed to lie within the range of 0.1 and 0.9. When the neural network is trained by minimizing the sum-squared error, the order of the error is independent of the magnitude of the target value. Therefore, it is likely that percentage error may be more in the case of target values of lower magnitude. In order that the percentage error in prediction is more or less uniform for low and high values of flow stress (σ), normalized values of ln(σ) were taken as the output layer of the network. The output was then operated by exponential function to get the values of flow stress (σ).
Justification for this can be given as follows [40]: Let us consider that for a particular flow stress value of*, the predicted value is given by ln( * e) where e is the error in prediction. The corresponding predicted flow stress (σ) value is then obtained by taking the exponential of the ln( * e) term. Therefore, the percentage error in prediction is expressed as [40]:
exp(ln * ) *
exp( ) 1
*e e
.
(3.3)
Since this percentage error is a function solely of e and is independent of*, the error is expected to be fairly uniform irrespective of the target value.
Modeling of flow stress (σ) for various combinations of strain (ε), strain rate ( ̇) and temperature (T) was carried out by artificial neural network. For each alloy, a total number of 150 input-output data sets (for combination of 5 true strain rates, 5 deformation temperatures and 6 true strain values considered) were obtained from the compression tests. It was decided to use 78 data sets for the network training, 48 data sets for the testing and 24 data sets from the remaining, for validation purpose. The validation data sets were not used earlier for the testing or training purposes. Both training and testing of the network was carried our independently. A number of numerical trials were carried out with single hidden layer neural network. The
“tansig”, “logsig” and “purelin” transfer functions were tried with while simultaneously varying the number of neurons in the hidden layer, in order to arrive at the best network architecture and processing function for each of the investigated alloys. The root mean square (RMS) functional error used as a measure of performance, can be expressed as
2 f
err 2
( *)
RMS ,
n
(3.4)where σ is the experimental flow stress value and σ* is the predicted flow stress value.
Both training and testing errors were calculated separately. Effective error for training and testing data is given by:
Effective error = max. of [RMSferr of training data, RMSferrof testing data]. (3.5) The sum squared training error goal for flow stress (σ) was fixed at 0.00001. After a number of trials with various initial weights and biases, the best neural network architecture was frozen for which (i) functional root mean square (RMS) error was minimum, (ii) minimum number of data sets has a deviation error of 5 %, (iii) maximum deviation during testing and training is within 10 %,
Predictability of the trained artificial neural network (ANN) model was verified viz. employing standard statistical parameters such as coefficient of determination (R2). It is expressed as
,
,
,
,
2 1
( )2
1 ( 1) 2
( )2
2 1
SSresidual R
SStotal
SSresidual in Ei Pi
SStotal n
n Ei E i
n
s
s
(3.5)
where s = standard deviation, Ei = Experimental value, Pi = predicted value, n = total number of data and ̅ are the average values of Ei. The simulation results are presented and discussed in section 4.4.