Prediction of the Sintering Shrinkage of Glass-Alumina Functionally Graded Materials by a BP Artificial Neural Network

The shrinkage of the glass-alumina functionally graded materials (G-A FGMs) as a function of sintering temperature, layers, and the alumina content was predicted by a back propagation artificial neural network (BP-ANN). The BP-ANN was composed of an input layer, a hidden layer, and an output layer. 21 sets of experimental data were trained, in which the temperature, layers, and the alumina content as input parameters whereas the shrinkage as the output parameter. 5 sets of experimental data were used to identify the accuracy of the BP-ANN. From the prediction, selection of the hidden layer neurons is essential for the convergence of the BP-ANN. The minimum predicted errors less than 6.6% are obtained with 8 neurons. Comparison of the predicted shrinkage shows that the increase of layers or alumina content is beneficial to the increase of the shrinkage and expansion resistance for the G-A FGMs.


Introduction
Functionally graded materials (FGMs) are newly developed composites for the application in high-tech fields, especially in those limiting circumstances as at super-high temperatures or with intensive thermal shocks [1].The spatial gradient variation of the properties and the macro functions of the FGMs can be optimized by the gradient variation of compositions as well [2].It is now a general opinion that the properties of FGMs differ from those of homogeneous materials due to the gradient variation of compositions [3].
Glass-alumina functionally graded materials (G-A FGMs) are essential FGMs as the stress can be redistributed which can effectively inhibit the formation of Hertzian cone cracks owing to the gradual variation of the Young's modulus [4].Additionally, the sliding contact friction properties can also be improved [5].Cannillo and coworkers prepared G-A FGMs with CaO-ZrO 2 -SiO 2 glass and polycrystalline alumina as raw materials and the influence of temperature and time on the percolation depth is discussed in detail [6].A mathematical model has been established according to the results from SEM and XRD-EDS as well [7].In addition, the influence of microstructure on fracture has been simulated [8].Recently, we prepared G-A FGMs by a novel freeform fabrication method proposed by our research group with Na 2 O-CaO-SiO 2 glass and polycrystalline alumina as raw materials [9,10].The sintering properties of the G-A FGMs with differed layers were investigated and an integrated liquidphase sintering model concerning both the viscous flow of the glass and the contribution of the activated alumina grain boundaries for G-A FGMs with 4 layers was established [11,12].As observed in the previous research [11], the sintering shrinkage of the G-A FGMs vary with temperatures, layers, and the Al 2 O 3 content.However, the mechanism of the sintering shrinkage can not be accurately determined as the variation is very complex and no regular relations are exhibited.
Artificial neural networks (ANNs) have become a useful tool for analysis of nonlinear systems because of their specific features as non-linearity, adaptability (i.e.learning from input parameters), generalization, and model independence (a prior model is not need) [13].It is generally accepted that ANNs provide more accurate predictive capabilities than methods based on traditional linear or non-linear statistical regression [14] and the superiority of ANNs over regression techniques increases as the dimensionality and/or non-linearity of the problem increases [15].ANNs have been found to outperform regression techniques in the prediction of both structural and functional ceramic material properties [16][17][18][19][20]. Additionally, the prediction of dielectric properties of organic materials has been attempted [21] and, again, ANNs have been found to be superior.
In this article, a back-propagation artificial neural network (BP-ANN) is applied to investigate the sintering shrinkage of G-A FGMs.The network is trained using the experimental data to learn the functional relationships between inputs and outputs and subsequently the trained network is used to predict the unknown results to ensure the reliability of the trained network.

BP Network Establishment 2.1 Basis of BP-ANN
BP-ANN is the most typical and useful feed forward neural network with backpropagation of errors, where the information is transmitted in a forward direction only [22].Besides the input and output layer neurons (also as nodes), the network has one or more hidden layers with neurons.The input signals transmit from the input neurons to the final output neurons via the hidden layer neurons successively and the power of the neural computation comes from the interconnection between adjacent layers.An individual neuron consists of weighted inputs, a combination function, an activation function and one output and there is no linking between the neurons in one layer.The outputs of one layer are connected to the inputs of the next layer to form the network topology.The performance of the network is determined by the form of the activation function, the training algorithm and by the network architecture.
Fig. 1 Scheme of the 3-layer BP neural network Fig. 1 shows the three-layer BP-ANN used for this prediction, where the number of neurons in the input and output layers is determined by the number of independent and dependent variables, respectively.The number of hidden layer neurons is determined by the complexity of the problem and is often obtained by the trial-and-error approach although evolutionary computing techniques such as genetic algorithms [23] have been used to determine the optimal network architecture.

Neural network operation
The following steps illustrate the network operation.
(1) Input some data 1 2 ( , , , ) ( , , , ) to the input layer and the output layer respectively, where k is the number of the training dataset, n the number of input neurons and p is the number of output neurons.
(2) Random values are assigned to the weight w ij (i=1, 2, …, n; j=1, 2, …, p) connecting the input layer and the hidden layer, and the weight v jt (j=1, 2, …, p; t=1, 2, …, q) connecting the hidden layer and the output layer, where i, j, and t are the numbers of the input, the hidden, and the output neurons being calculated respectively.
(3) Calculate the input s j of the hidden layer neurons using the input data 1 2 ( , , , ) , the connecting weight w ij , and the bias θ j .Then calculate the output b j of the hidden layer neurons by applying a Sigmoid activation function (1) for s j , ( ) (4) Calculate the output L t of the output layer neurons using the output b j of the hidden layer neurons, the connecting weight v jt , and the bias γ t .Then we can get the response C t of the output layer neurons by the activation function (2), , where t=1, 2,…, q; ( ) (5) Calculate the general error k t d of the output layer neurons using the experimental data 1 2 ( , , , ) , where t=1, 2,…,q (6) Calculate the general error k j e of the hidden layer neurons using v jt , k t d , and b j by Where t=1, 2,…, q; j=1, 2,…, p; and α (0<α<1) is the learning rate and controls the adjustments to the weights/biases.
The network's performance is measured after each epoch has been completed and is determined by a mean square error (MSE) function given by [24] ( ) Where y is the output predicted by the network, t the experimentally measured output, and N is the number of records in the dataset.The training process corresponds to an iterative decrease in the error function and continues until a predetermined value E is reached, when training is halted.

Network architecture design 2.3.1 Number of hidden layer neurons
The number of the optimized hidden layer neurons can be determined by the trial-anderror approach as no accurate theory and method for choosing that has been determined.Assuming that the 3-layer network contains n input neurons and q output neurons, we used the following equation to calculate s given by s n q a = + + (8) Where s is the number of the hidden layer neurons, and a is a constant between 0 and 10.In this article, n is 3 and q is 1, thus the ranges of the number of the hidden layer neurons can be determined between 2 and 12.In this investigation, the number is chosen from 3 to 12 to investigate the accuracy of the network.

Weights, biases, and the learning rate initialization
Weight and bias values initialization have a great effect on the error convergence of the network [20].If choosing appropriately, it will be beneficial to the network convergence.The range of the weight values initialization is usually from -1 to 1 as it is sensitive for Sigmoid function which is favorable to increasing the learning efficiency.In this investigation, according to our previous trial-and-error method, the range of the weight values initialization is determined as 0.1~1 and a faster network convergence is obtained.The learning rate is determined as 0.5.

Materials Datasets
According to the previous investigation [11], temperature, number of layers of the FGMs and alumina content are chosen as input parameters and the shrinkage rate of the G-A FGMs as the output parameter.26 samples have been produced by the proposed method [9][10][11] and the relative shrinkage along the gradient direction is measured.Among the data, 21 samples were selected for the ANN training process, while the remaining 5 samples were used to verify the generalization capability of ANN.Tab.I shows the high and low levels of effective variables on shrinkage properties of the G-A FGMs.
The activation functions for both the hidden and the output layers are a Sigmoid function which is sensitive when the output values range from 0 to 1.0.Therefore, the preprocessing of the raw data is conducted before training, which is favorable for the network learning efficiency.In this investigation, processing of the input and output data is conducted according to equation ( 9) by min max min Where x max and x min are the maximum and the minimum of the input or output values respectively.

Tab. I Training Datasets
No. Temperature

Network training and prediction
The number of hidden layer neurons has a dramatic effect on network convergence.The network does not converge even after 2000 epochs of training using the Levenberg-Marquardt algorithm when the neurons are 3 and 4. As shown in Fig. 2 and Fig. 3, the errors after 2000 epochs are 0.00443883 and 0.00101319, respectively.
Rapid convergence of the network is achieved when the number of neurons is between 5 and 12.However, no regular effect is observed.Fig. 4 shows the MSE and the training epochs for different neurons.From the figure, the training epochs do not decrease with the increased number of neurons.When the number of neurons is 6 or 8, more rapid convergence rate is achieved while the MSE does not reach the minimum.In contrast, the value of MSE generally decreases with the increased number of neurons.It needs to be addressed that the value of MSE and the training epochs for convergence increase to a maximum when the number of neurons is 11, showing that more training epochs are needed even the value of MSE is larger.It is essential that the relative prediction errors vary as the number of neurons changes.Tab.II gives the relative errors between the experimental and predicted shrinkage of 95G+90G FGMs with different neurons.The overall error reduces as the number of neurons increases and the error is less than 6.02% when the number is 8.At 675°C, the predicted values differ from the experiment ones and entirely larger errors are observed when the network has 9 or more neurons.Tab.III gives the relative errors between the experimental and predicted shrinkage of 90G+85G FGMs with different numbers of neurons.From the Tab., no regular increase or decrease of the predicted errors with increased number of neurons are exhibited.For 7 neurons, the predicted results differ greatly from the experimental ones.But effective prediction is realized for 8 neurons.With the increased number of neurons, the overall prediction is effective except that at 675°C.Tab.IV gives the relative errors between experimental results and predicted shrinkage of 95G+90G+85G FGMs with different neurons.Effective prediction for all numbers of neurons is realized compared with the 95G+90G FGMs and the 90G+85G FGMs, in which more accurate prediction is realized for 5, 8 and 11 neurons.In general, the predicted errors for all FGMs are relatively small for 8 neurons and the maximum error is less than 6.6%.Predicted relative errors are shown in Fig. 5.It needs to be addressed that the difference between the predicted and experimental results at 675°C for all FGMs is still being investigated; however, no reasonable results have been obtained.According to the analysis made in section 4.1, 8 hidden layer neurons are adopted to predict the sintering shrinkage of G-A FGMs with different compositions.Fig. 6 shows the temperature dependent shrinkage of the G-A FGMs with different layers and alumina content.From the figure, greater shrinkage of all G-A FGMs from 650°C to 690°C is exhibited.From 725°C to 800°C, relatively slow increase is exhibited for both 95G+90G+85G and 90G+85G FGMs.However, a relative expansion for 95G+90G FGMs is exhibited from 725°C, that is, the shrinkage starts to decrease, and reaches the minimum at about 750°C, after which it begins to increase steadily again.The shrinkage of both the 95G+90G FGMs and the 90G+85G FGMs increases apparently while that of the 95G+90G+85G FGMs increases gently from 800°C.In general, better shrinkage and expansion resistance properties are confirmed in FGMs from 700°C to 800°C.The shrinkage of the three-layer FGMs increases gently until 875°C, indicating that increased layers and alumina content are beneficial to the shrinkage and expansion resistance, which is in accordance with the experimental results confirmed by previous investigations [11].

Conclusions
The shrinkage of glass-alumina functionally graded materials as a function of temperature, layers, and the alumina content was predicted by a back propagation artificial neural network. 2 conclusions are drawn from the prediction as follows: (1) Selection of the hidden layer neurons has a great effect on the network convergence.The network does not converge after 2000 epochs of training trained by the Levenberg-Marquardt method for 3 or 4 neurons.With 8 hidden layer neurons, the optimal predicted results are obtained.And the predicted maximum errors of the G-A FGMs with different compositions are less than 6.6%.
(2) BP-ANN prediction shows that better shrinkage and expansion resistance properties are confirmed in G-A FGMs from 700°C to 800°C.In addition, the increase of layers and alumina content is beneficial to the shrinkage and expansion resistance.

Fig. 2 a
Fig. 2 a) Training error curve of the hidden layer with 3 neurons Fig. 2 b) Training error curve of the hidden layer with 3 neurons (for online use only)

Fig. 3
Fig. 3 Training error curve of the hidden layer with 4 neurons

Fig. 4 Fig. 4
Fig. 4 Training error curve of the hidden layer with 4 neurons (for online use only)

Fig. 5 4 . 2
Fig. 5 Relative errors of the prediction of the G-A FGMs with different compositions

Fig. 6
Fig. 6 Effect of temperature on the relative shrinkage rate of the G-A FGMs with different alumina content