A Neural Network (NN) and Response Surface Methodology (RSM) based prediction model for sintered aluminium performs

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1 A Neural Network (NN) and Response Surface Methodology (RSM) based prediction model for sintered aluminium performs Dr M. JOSEPH DAVIDSON Mech. Engg. Dept. NIT Warangal- Warangal India Dr N.SELVAKUMAR Mech. Engg. Dept., Mepco Schlenk Engineering College,Sivakasi - India nsk2966@yahoo.co.in Abstract The present work focuses on the two of the techniques namely Neural Network (NN) and Response Surface Methodology (RSM) for predicting the final density (FD) of sintered aluminium performs. In this work, the load, the aspect ratio and the initial preform density were taken as the input parameters and the response (output) variable measured was final density. Prediction of the response variable final density was obtained with the help of empirical relation between the response and the input variables using Response Surface Methodology s (RSM) Box-Behnken design of experiments (DOE) technique and also through Neural Network (NN). Predicted values of the response by both the techniques i.e. RSM and NN were compared with the experimental values and their closeness with the experimental values was determined. Moreover, it has been found that the final density of the preform increases with both the load applied and the initial preform density of the sintered preforms and the aspect ratio has only little effect on the densification. Keywords- Neural Network (NN); Response Surface Methodology (RSM); sintered aluminium; powder preforms; cold axial forming 1.Introduction Understanding the densification behavior of sintered aluminium preforms during forming is very important in achieving quality powder metallurgy (P/M) parts. For P/M materials there is a well-established dependence of strength upon density [1]. Due to the presence of voids in the aluminium preforms, the densification behaviour of the P/M product is considerably different from the conventional cast and wrought materials. For example, as porous materials are compressed, they spread in the transverse direction to a lesser degree than conventional materials. Also, the flow stress increases because of densification as well as cold working of the base metal, both of which occur during deformation. [2-3]. However the sintered preforms can be deformed again to a final size within reasonable dimensional tolerance to increase its density very close to the fully dense materials. [2]. Kuhn [3] has reported that the fundamental mechanical responses of porous metals namely densification, plastic flow and fracture can be determined through uniaxial compression tests of cylindrical specimens. During the compressive deformation of sintered powder metal preform, some metal flows on to the pores and subsequently there is a decrease in the bulk volume. As the metal fills the pores, simultaneously the density of the preforms increases [4]. Understanding the density distribution of powder compaction and the deformation behaviour of powder preforms during forming is very important in achieving quality P/M parts. Uneven density distribution gives rise to distortion of parts [2]. Abdul Rahman [5] has investigated the effect of relative density of the preform on the forming limit. Mamalis [6,7] has studied the porosity and micro defects on plastically deformed materials using yield criterion. P/M component manufactures are interested in predicting the final density of P/M components to fulfill the needs of the end user. However, estimating the final density prior to experimentation is very difficult. Recently modeling techniques are being used in the optimization and prediction of process parameters in the manufacturing areas. Neural networks have been used to study the deformation characteristics of sintered aluminium preforms [8,9]. Response surface methodology has been applied to access the bead quality of submerged arc welding of pipes [10]. However, not much work has been carried out to model the densification behavior of powder preforms when they are subjected to compressive loads. In this paper, the density attained by sintered aluminium preforms when they are subjected to cold axial forming operation is critically discussed. Axial compression tests were conducted on sintered aluminium preforms of aspect ratios 0.50, 0.75 and 1.00 having initial preform density values of 0.85, 0.885, and A mathematical model has been developed by incorporating the load, the aspect ratio and the initial preform density ISBN:

2 using Response Surface Methodology s Box Behnken Design of Experiments (D.O.E) technique. Using this model, the final density of the sintered aluminium powder metallurgical preforms can be ascertained within the ranges of investigation. The main and interaction effects of the process parameters have been studied by using analysis of variance (ANOVA) method. Also, a four layered back propagation Neural Network has been used to model the densification behavior of sintered aluminium preforms. The adequacy of both the developed models has been verified by conducting additional experiments. 2. Experimental work Atomized aluminium powder of high purity has been used in this experiment. The aluminium powder was used for preparing compacts of different height to diameter ratios (aspect ratios). The initial aspect ratios were 0.50,0.75 and 1.00.Compacts with initial aspect ratios given above were prepared from the previously mentioned powder by employing different compacting pressures in a universal testing machine so as to obtain various initial compaction densities of 85%, 88.5 % and 92% of the theoretical density. Molybdenum disulphide (MoS 2 ) was used to lubricate the die, the punch and the butt while preparing compacts. The sintering was carried 0 out in an electric muffle furnace at 520 ± 10 C for a holding period of one hour. The sintered aluminium preforms thus prepared were subjected to compressive loading as dictated by the design matrix developed through RSM s Box Behnken design. The final density (FD) attained by the preforms during each stage of load applied on the preforms was measured using Archimedes principle. MATLAB NN Toolbox has been used for the desigh, implementation and simulation of the neural network with feed forward back propagation algorithm. 3. NN training and validation A three-layered back propagation Neural Network has been used to model the densification behavior of sintered aluminium preforms as explained elsewhere [9]. Backpropagation neural networks are multi layer networks with the hidden layers of sigmoid transfer function and a linear output layer. The transfer function in the hidden layers should be differentiable and thus, either log sigmoid or tan sigmoid functions are typically used. In this study, the tan sigmoid transfer function, tansig is used for hidden layers and the purelin transfer function is used for the output layer. Purelin is a linear transfer function. They calculate a layers output from its net input. Each hidden layer and output layer is made of artificial neurons, which are interconnected through adaptive weights. The training function selected for the network is trainlm. Trainlm is a network training function that updates weight and bias values according to the Lavenberg-Marquardt algorithm. 3.1 Proposed NN Structure In this work, different combinations of layers and neurons were tried. Finally a four-layered network is selected which consists of an input layer, an output layer and two hidden layers. The input layer has three neurons corresponding to the load; the aspect ratio and the initial preform density. The hidden layers have thirty neurons each. The output layer contains one neuron corresponding to the response variable, final density (FD). The proposed network architecture is shown in Fig 1. The developed NN is trained several times until the number of neurons along with the initial weights and biases satisfies the error goal of 1e -5. Fig 1. Four layered back propagation NN topology 3.2 NN TRAINING The multiple data gathered through the above-explained experimental work have been used for training the NN model. Before training the network, the training data were normalized suitably. The training data were fed into the proposed network and after several iterations the network delivered a converged result in a lesser epoch (Fig 2). The weight values corresponding to the trained network were saved. These weight values were applied to another network having the same architecture as the trained network. Now the test data were fed into the network. Test data do not contain the data used for training the network. The output got from the network is the predicted value and is given in Table 4. It reveals that the model developed through NN can be used to predict the final density (FD) values. ISBN:

3 3. Finding the optimal set of experimental parameters that produce a maximum or minimum value of response. 4. Representing the direct and interactive effects of the process parameters through two and threedimensional plots. 6. Mathematical Model of final density 6.1 Response Equation for final density RSM s Box Behnken design consisting of 17 experiments was conducted for developing the mathematical model for final density. The input parameters and their levels chosen for this work are given in Table 1. The final density results for the 17 experiments are given in Table 2. Fig 2.The relationship between error value and number of epochs 4. Design of Experiments Design of experiments is a powerful analysis tool analyzing the influence of process variables over some specific variable, which is an unknown function of these process variables. It is the process of planning the experiments so that appropriate data can be analyzed by statistical methods, resulting in valid and objective conclusions. Statistical approval to experimental design is necessary if we wish to draw meaningful conclusions from the data. [11]. 5. Response Surface Methodology (RSM) RSM is a collection of statistical and mathematical methods that are useful for modeling and analyzing engineering problems. In this technique, the main objective is to optimize the response surface that is influenced by various process parameters. RSM also quantifies the relationship between the controllable input parameters and the obtained response surfaces [11-12]. The design procedure of RSM is as follows [10]. 1. Designing a series of experiments for adequate and reliable measurements of the response of interest. 2. Developing a mathematical model of the response surface with the best fittings. Std No S.No Parameter Low Level High Level 1 Load (LD) Aspect Ratio (ASPR) Initial Preform Density (IPD) Table 1 Input parameters and their levels Run NO Factor LD ASPR IPD FD Table 2 Experimental layout for the Box-Behnken design ISBN:

4 The response equation for final density (FD) so obtained is given by FD = * ( LD ) * ( ASPR) * ( IPD) * ( LD ) * ( ASPR ) * ( LD ) * (IPD) * (ASPR) *( IPD). 7. ANOVA and Respose Surface Graphs The analysis of variance (ANOVA) was applied to study the effect of the input parameters on the final density. The model summary statistics was done and it revealed that 2FI model is the best suggested model.so, for further analysis this model was used. ANOVA is commonly used to summarise the test for significance on individual model co-efficients.the value of "Prob > F" for the 2FI model is less than which indicates that the model terms are significant,.the Model F-value for the model is 39.56, which implies that the model is significant. There is only a 0.01% chance that a "Model F-Value" this large could occur due to noise. Std. Dev R Mean Adj. R C.V Pred. R PRESS Adeq Precision in reasonable agreement with the Adj. R 2 value. The R 2 value in this case is high and close to 1, which is desirable [13]. The adequacy of the model has also been investigated by the examination of residuals [11]. The residuals, which are the difference between the respective observed responses and the predicted responses, are examined using the normal probability plots of the residuals and the plot of the residuals verses the predicted response. If the model is adequate, the points on the normal probability plots of the residuals should form a straight line. On the other hand, the plots of the residuals verses the predicted response should be structure less, that is, they should contain no obvious pattern [13]. The normal probability plots of the residuals and the plots of the residuals verses the predicted responses for the final density values revealed that the residuals fall on a straight line implying that the errors are distributed normally. This implies that the model proposed is adequate and there is no reason to suspect any violation of the independence or constant variance assumptions [13]. Figures 7-8 give the 3D surface graphs for the final density. As the model is adequate these 3D surface plots can be used for estimating the final density values for any suitable combination of the input parameters namely the load, the aspect ratio and the initial preform density value. Table 3. Regression statistics Table 3 gives the regression statistics. The co-efficient of determination R 2 is used to decide whether a regression model is appropriate. The co-efficient of determination R 2 provides an exact match if it is 1 and if the residual increases R 2 decreases in the range from 1 to 0.As the number of variables increases, the residual decreases, so that the co-efficient of determination R 2, increases its value. So, to obtain a more precise regression model judgment, co-efficient of determination R 2 adjusted for the degrees of freedom Adj. R 2 is used. Adj. R 2 is used for comparing the residual per unit degree of freedom. Adequate precision compares the range of the predicted values at the design points to the average prediction error. It is a measure of the signal to noise ratio. Ratio greater than 4 indicates adequate model discrimination. In this particular case, it is , which is well above 4.So the model can be used to navigate the response space. Further, it is seen that the R 2 value is and the Adj. R 2 is The predicted R 2 value of is Fig 3. Normal probability plot of residuals for final density data. ISBN:

5 Fig 4. Contour plot in ASPR-LD plane at IPD=0.89 Fig 7. 3D surface graph for the final density (FD) at IPD=0.89, as ASPR and LD varies. Fig 5. Contour plot in IPD-LD plane at ASPR=0.75 Fig 8. 3D surface graph for the final density (FD) at ASPR=0.75, as IPD and LD varies. 8. Confirmation Test Fig 6. Contour plot in IPD-ASPR plane at LD=40 In order to verify the accuracy of the model developed, confirmation experiments were performed (Table 4). The test condition for the confirmation tests were so chosen that they be within the range of the levels defined previously. The predicted values and the associated experimental values were compared and the residuals and ISBN:

6 the percentage error were calculated. The error percentage is within permissible limits. So the response equation for the final density evolved through RSM can be used to successfully predict the final density values for any combination of the load, the aspect ratio and the initial preform density values within the range of the experimentation conducted. Again from Table 4 it is clear that NN is capable of predicting the final density values for any combination of the load, the aspect ratio and the initial preform density values within the range of the experimentation conducted. Moreover NN is capable of predicting the density values better than RSM within reasonable error value. Load (LD) ASPR IPD Final density (FD) Experim ental NN predic ted Error % Final density (FD) RSM predicte d Error % Table 4 Partial sample predicted data from the NN and RSM models. 9. Results and discussion A Box-Behnken design has been carried out to investigate the densification characteristics of sintered aluminium preforms. Table 4 reveals that both NN and RSM can be used for predicting the final density of the sintered aluminium preforms. A comparative study between the RSM and the NN revealed that the NN could predict the final density (FD) of sintered aluminium preform better than RSM within a few percent errors. Fig 4 is the contour plot of the final density in ASPR- LD plane at IPD=0.89.It reveals that aspect ratio has only little effect on the densification. Fig 5 is the contour plot of the final density in IPD- LD plane at ASPR=0.75. It reveals that the final density of the preform increases with both the load applied and the initial preform density of the sintered preforms. Fig 6 is the contour plot of the final density in IPD- ASPR plane at LD=40.It reveals that the aspect ratio has only little effect on the densification. Aspect ratio has little effect on the densification on the preforms with lower initial preform density values but it almost has no effect on the preforms with higher initial preform densities. Fig 7 is the 3D surface graph for final density at IPD=0.89. Figure 8 is the 3D surface graph for final density at ASPR=0.75. As the model is valid, these surface plots can be navigated for finding the desired final density for any suitable load, aspect ratio and initial preform density. 10 Conclusions In this paper, the densification behaviour of sintered aluminium preforms has been modeled using Neural Network and Response Surface Methodology. These approaches are used in an attempt to determine the final density attained by sintered aluminium preforms for various input parameters namely the load, the aspect ratio and the initial preform density values. It has been found that the load and the initial preform density values were significant parameters while the aspect ratio was relatively insignificant. Though the design and development of Neural Network is time consuming, it provides accurate results compared with RSM. REFERENCE 1. Smith, L.N.; Midha, P.S.; Graham, A.D. Simulation of metal powder compaction, for the development of a knowledge based powder metallurgy process advisor. Journal of Materials Processing Technology, 1998,79, Narayanasamy, R.; Ponalagusamy, R.; Subramanian, K.R. Generalised yield criteria of porous sintered powder metallurgy metals Journal of Materials Processing Technology, 2001,110(2), Kuhn, H.A.; Downey, C.L.; Deformation Characteristics and plasticity theory of sintered Powder materials. International Journal of Powder Mettallurgy.1971, 7(1), Narayanasamy, R.; Pandey, K. S. A study on the barrelling of sintered iron preforms during hot upset forging. Journal of Materials Processing Technology, 2000, 100 (1-3), ISBN:

7 5. Abdel-Rahman, M.; El-Sheikh, M. N. Workability in forging of powder metallurgy compacts. Journal of Materials Processing Technology, 1995,54, Mamalis, A.G.; Petrossian, G.L.; Manolakos, D.E. The effect of porosity and micro defects on plastically deformed porous materials. Journal of Materials Processing Technology, 1999,96, Mamalis, A.G.; Petrossian, G.L.; Manolakos, D.E. Limit design of porous sintered metal powder machine elements. Journal of Materials Processing Technology, 2001,98, Joseph Davidson, M. 2004,Simulation and analysis of deformation characteristics of sintered aluminium preforms during cold upset forming, M.E Thesis, Anna University, India. 9. Selvakumar, N.; Radha, P.; Narayanasamy, R.; Joseph Davidson, M. Prediction of deformation characteristics of sintered aluminium preforms using neural networks, 2004,12, weld bead quality in submerged arc welding of pipes. Journal of Materials Processing Technology, 1999, 88(1-3), 15, Montgomery,C. Design and Analysis of experiments, 4 th Edition, Wiley, New York, Jae-seob Kwak; Application of Taguchi and Response surface Methodologies for geometric error in surface grinding process. International Journal of machine Tool and manufacturing 2005 (45), Noordin, M.Y.; Venkatesh, V.C.; Sharif, S.; Elting, S.; Abdullah A. Application of Response Surface Methodology in describing the performance of coated carbide tools when turning AISI 1045 steel. Journal of Materials Processing Technology, 2004, 145, Gunaraj, V.; Murugan, N. Application of response surface methodology for predicting ISBN: