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Artificial Intelligence in numerical modeling of nano sized ceramic particulates reinforced metal matrix composites
Applied Mathematical Modelling 36 (2012) 5455–5465
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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Artificial Intelligence in numerical modeling of nano sized ceramic particulates reinforced metal matrix composites Mohsen Ostad Shabani ⇑, Ali Mazahery Department of Materials Science, Lenjan Branch, Islamic Azad University, Isfahan, Iran
a r t i c l e
i n f o
Article history: Received 10 June 2011 Received in revised form 18 December 2011 Accepted 21 December 2011 Available online 2 January 2012 Keywords: Algorithms Nano composite Abrasive
a b s t r a c t Artificial neural network models have the capacity to eliminate the need for expensive experimental investigation in various areas of manufacturing processes, including the casting methods. An understanding of the inter-relationships between input variables is essential for interpreting the sensitivity data and optimizing the design parameters. Aluminum is the best metal for producing metal matrix composites which are known as one of the most useful and high-tech composites in our world. Combining aluminum and nano Al2O3 particles will yield a material with high mechanical and tribological properties. In this investigation, the accuracy of various artificial neural network training algorithms in FEM modeling of Al2O3 nano particles reinforced A356 matrix composites has been studied. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Artificial neural network is a non-linear statistical analysis technique and is especially suitable for simulation of systems which are hard to be described by physical models. It provides a way of linking input data to output data using a set of nonlinear functions. Choosing the optimum architecture of the network is one of the challenging steps in artificial neural network (ANN) modeling. The term ‘architecture’ refers to the number of layers in the network and number of neurons in each layer. In order to design a neural network for a problem solution, a training algorithm is required. During the training process, the weights and biases in the network are adjusted to minimize the error and obtain a high-performance in the solution. At the end of the training and during the training error, mean absolute percentage error (MAPE) is computed between desired outputs and target outputs [1–6]. The finite element method is a powerful technique for solving differential or partial differential equations as well as integral equations. It is the most versatile numerical technique in modern engineering analysis and has been employed to study diverse problems in heat transfer, fluid mechanics, mechanical properties, rigid body dynamics, solid mechanics, electrical systems and many other fields [7–10]. Aluminum alloys are widely being used in the industry due to their excellent castability, good corrosion resistance and high strength-to-weight ratio [10–12]. However, their applications have often been restricted because conventional Al alloys are soft and notorious for their poor wear resistance. Wear can be defined as damage to a solid surface, generally involving progressive loss of material, due to relative motion between that surface and a contacting substance or substances. In many applications, metal matrix composites (MMCs) are subjected to sliding motion. In some cases this motion is intentional: for example, in an internal combustion engine piston or cylinder liner, or an automotive brake disk, or in the processing of material by machining, forging, or extrusion. It may also be unintentional, as in the fretting wear of a joint [13]. The wear resistance of aluminum alloys is improved as a consequence of the incorporation of ceramic fibers or particles which act as the ⇑ Corresponding author. Tel.: +98 912 563 6709; fax: +98 261 6201888. E-mail address: [email protected] (M.O. Shabani). 0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.12.059
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load bearing and abrasive member. Aluminum matrix composites (AMCs) represent a generation of engineering materials which combine metallic properties of matrix alloys (ductility and toughness) with ceramic properties of reinforcements (high strength and high modulus), leading to greater wear, strength in shear, compression and higher service-temperature capabilities [14–18]. In this study, nano Al2O3 particulates reinforced Al matrix composite have been processed using stir casting. Then, the experimental wear properties and the accuracy of various ANN training algorithms in FEM modeling of the composites have been investigated. 2. Experimental procedure Commercial casting Al–Si alloy and Al2O3 particles have been used in the present study. Initially, Al alloy was charged into the crucible, and heated to about 750 °C, which is above the liquidus temperature of the Al alloy. The composite was synthesized through solidification processing route using 0.5, 1, 1.5, 2, 2.5, . . . , 5 vol.% Al2O3 particles of size range 50 nm as reinforcement [18]. Composite slurry was step cast into the CO2-sand mould. 1 wt.% magnesium additive in powder form was also used as a wetting agent. Hardness measurements were carried out on a Brinell hardness testing machine (Eseway DVRB-M), using an indenter ball with 2.5 mm diameter at a load of 31.25 kg, and the mean values of at least five measurements conducted on different areas of each sample was considered in order to eliminate possible segregation effects and get a representative value of the matrix material hardness. The wear tests were performed at a sliding speed of 0.3 m/s under varying applied loads against case hardened steel disc using a pin-on-disk type test machine. The disk with a diameter of 50 mm and a thickness of 10 mm was made of the steel hardened up to 63 HRC and polished to very fine grade with surface roughness about 0.22 mm. Test specimens were cut and shaped in the form of pins having 6 mm in diameter and 25 mm in height. Before the abrasion tests, each specimen was polished to 0.5 lm. Fig. 1 shows Schematic diagram of the abrasion wear test. The pin-on-disk wear machine consists of the stationary pin pressed at the required load against the disk rotating at the defined speed. An AC motor ensures the stable running speeds of the disk. The testing machine is equipped with a set of measuring transducers. During the tests the friction force as well as coefficient of friction is measured continuously. The experiment was carried out at room temperature (21 °C, relative humidity 55%) with water as the lubricant. The samples were cleaned with acetone and weighed (up to an accuracy of 0.01 mg using microbalance) prior to and after each test. The temperature rise and friction force were recorded from the digital display interfaced with the wear test machine. Coefficient of friction was computed from the recorded frictional force and the applied load (i.e. the ratio of frictional force to the applied load). A set of three samples was tested in every experimental condition, and the average along with standard deviation for each set of three tests is measured. The wear tests were conducted up to the total sliding distance of 2000 m. The mass loss of the pin was used to study the effect of Al2O3 addition on the wear resistance of the composite materials under consideration. 2.1. Finite element method The finite element method (FEM) has undoubtedly become the most popular and powerful analytical tool for studying a wide range of engineering and physical problems. One of the important applications of FEM is the solidification. The basic concept is that the whole domain is divided into smaller elements of finite dimensions called ‘Finite Elements’. FEM method is used for discretization and to calculate the transient temperature field of quenching [1,19–22].
Fig. 1. Schematic diagram of the abrasion wear test.
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The steps involved in the finite element analysis are as follows: The whole domain is divided into a finite number of sub-domains, which is called the discretization of the domain. Each sub domain is called an element. The collection of elements is called the finite-element mesh. From the mesh, a typical element is isolated and the variational formulation of the given problem over the typical element is constructed. An approximate solution of the variational problem is assumed and the element equations are made by substituting this solution in the above system. The element matrix, which is called stiffness matrix, is constructed by using the element interpolation functions. The algebraic equations so obtained are assembled by imposing the interelement continuity conditions. This yields a large number of algebraic equations known as the global finite element model, which governs the whole domain. The essential and natural boundary conditions are imposed on the assembled equations. The assembled equations so obtained can be solved by any of the numerical technique. The mathematical formulation of this solidification problem is given [1,10,22]:
qC
@Tðx; y; z; tÞ dfs ¼ K r2 Tðx; y; z; tÞ þ qL @t dt
ð1Þ
where q the density (kg/m3), C the specific heat (J/kg K), T the temperature (K), t the time (s), K is the thermal conductivity (W/m K), L is the latent heat (J/kg) and fs the local solid fraction. The fraction of solid in the mushy zone is estimated by the Scheil equation, which assumes perfect mixing in the liquid and no solid diffusion. With the liquidus and solidus having constant slopes, fs is then expressed as:
fS ¼ 1
Tf T T f T liq
1=ðk0 1Þ ð2Þ
where Tf is the melting temperature (K), TL the liquidus temperature (K), and k0 the partition coefficient. Then [22]:
ð2k0 Þ=ðk0 1Þ Tf T dfs 1 dT ¼ dt dt ðk0 1ÞðT f T liq Þ T f T liq
ð3Þ
The latent heat released during solidification of the remaining liquid of eutectic composition was taken into account by a device, which considers a temperature accumulation factor.
qC 0
@Tðx; y; z; tÞ dfs ¼ K r2 Tðx; y; z; tÞ þ qL @t dt
ð4Þ
where C0 can be considered as a pseudo-specific heat given by Shabani and Mazahery [22]:
dfs dT C M ¼ ð1 fs ÞC l þ fs C S
C0 ¼ CM L
ð5Þ ð6Þ
where the subscripts L, S and M refer to liquid, solid and mushy, respectively. The other properties such as thermal conductivity and density in the mushy zone are described similarly as the specific heat:
K M ¼ ð1 fs ÞK l þ fs K S
ð7Þ
qM ¼ ð1 fs Þql þ fs qS
ð8Þ
2.2. ANN algorithms There are various training algorithms used in neural network applications. It is hardly difficult to predict which of these learning will be the fastest one for any problem. Generally, it depends on some factors; the structure of the networks, in other words, the number of hidden layers, weights and biases in the network, aimed error at the learning, and application area, for instance, pattern recognition or classification or function approximation problem. Back-propagation algorithm (BP) is one of the most popular in engineering, the major steps of BP algorithm are listed as follows [19–28]:
Initial all weights (wij) to small random values Present an input and specify the desired output. Calculate outputs using the present value of wijs Find an error term, rj, for all the nodes. If vj, Oj and Ej stand for desired value of the jth output node, the computed value of the jth output node, and the computed value for the jth hidden layer node, then the error terms are calculated as; for outP put node j: rj = (vj Oj)Oj(1 Oj), for hidden layer node j: rj = Ej(1 Ej) krknjk, where k is over all nodes in the layer above node j.
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Adjust weights by wij(k + 1) = wij(k) + arjXi + n(wij(k) wij(k 1)), where (k + 1), (k), and (k 1) index next, present, and previous iteration, respectively. a is the learning rate. n is the momentum which determines the effect of past weight changes on the current direction of movement in weight space. It is used to filter out the high-frequency variations of the error surface. Present another input and go back to second step. All the training inputs are presented cyclically until weights stabilize. The neural network predictions were directly compared with the experimental obtained data to evaluate the learning performance. Also the ANNs can be judged by various parameters such as the mean square error (MSE), mean absolute error (MAE), normalized root mean square error (NSE) and mean absolute percentage error (MAPE) as follows [1,10]: N0 Q X 1 X jdn ðmÞ yn ðmÞj2 QN0 m¼1 n¼1 X x y MAE ¼ n ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P ðA FÞ2 NSE ¼ A2 " # N0 A F 1 X 100 MAPE ¼ N0 N¼1 A
MSE ¼
ð9Þ ð10Þ ð11Þ ð12Þ
where N0 is the number of output, Q the number of training sets, d desired output, and y the network output and where x = X X0 , X is the target output and X0 the mean of X, y = Y Y0 ; Y is the network output and Y0 the mean of Y. A and F represent the actual and forecasting values respectively. In this investigation the MAPE was used to evaluate the performance of model. Some of famous train algorithms are as follow [19–28]: Batch training with weight and bias learning rules: trains a network with weight and bias learning rules with batch updates. The weights and biases are updated at the end of an entire pass through the input data. BFGS quasi-Newton back propagation: is a network training function that updates weight and bias values according to the BFGS quasi-Newton method. BFGS quasi-Newton back propagation for use with NN model reference adaptive controller: is a network training function that updates weight and bias values according to the BFGS quasi-Newton method. Batch unsupervised weight/bias training: trains a network with weight and bias learning rules with batch updates. Weights and biases updates occur at the end of an entire pass through the input data. Cyclical order incremental update: trains a network with weight and bias learning rules with incremental updates after each presentation of an input. Inputs are presented in cyclic order. Powell–Beale conjugate gradient back propagation: is a network training function that updates weight and bias values according to the conjugate gradient back propagation with Powell–Beale restarts. Fletcher–Powell conjugate gradient back propagation: is a network training function that updates weight and bias values according to conjugate gradient back propagation with Fletcher–Reeves updates. Polak–Ribiére conjugate gradient back propagation: is a network training function that updates weight and bias values according to conjugate gradient back propagation with Polak–Ribiére updates. Gradient descent back propagation: is a network training function that updates weight and bias values according to gradient descent. Gradient descent with adaptive learning rule back propagation: is a network training function that updates weight and bias values according to gradient descent with adaptive learning rate. Gradient descent with momentum back propagation: is a network training function that updates weight and bias values according to gradient descent with momentum. Gradient descent with momentum and adaptive learning rule back propagation: is a network training function that updates weight and bias values according to gradient descent momentum and an adaptive learning rate. One step secant back propagation: is a network training function that updates weight and bias values according to the one-step secant method. Random order incremental training with learning functions: trains a network with weight and bias learning rules with incremental updates after each presentation of an input. Inputs are presented in random order. Resilient back propagation (Rprop): is a network training function that updates weight and bias values according to the resilient back propagation algorithm (Rprop). Sequential order incremental training with learning functions: rains a network with weight and bias learning rules with sequential updates. The sequence of inputs is presented to the network with updates occurring after each time step. Scaled conjugate gradient back propagation: is a network training function that updates weight and bias values according to the scaled conjugate gradient method. Levenberg–Marquardt back propagation: is a network training function that updates weight and bias values according to Levenberg–Marquardt optimization.
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Fig. 2. Flowchart of FEM–ANN scheme.
The Flowchart of FEM–ANN scheme is shown in Fig. 2. In the analysis of performance of various training algorithms, the same prepared learning and test set were used in the training processes of each learning algorithm. The performance analysis were done from the viewpoint of training duration, error minimization and prediction achievement. The neural network predictions were directly compared with the experimental obtained data to evaluate the learning performance. Finally, sensitivity data were used to further validate the model and to interrogate the model to try to gain a physical understanding of the predicted trends. The sensitivity factor (SF) measures the contribution of each input variable to the desired output, and it differs for each individual case in both the train and test data files. Therefore, the SF presented in this paper is an average of the contribution of an individual input variable towards the specified output. For every input
Fig. 3. Wear resistance of the nano composites: (a) Weight loss as a function of sliding distance, (b) effect of applied load on wear resistance.
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Fig. 4. Hardness as a function of nano Al2O3 particles volume fraction.
parameter, the percentage change in the output, as a result of the change in the input parameter, was calculated using the following relationship [7]:
"
# N0 %change in output 1 X 100 Relative sensitivity ¼ N0 j¼1 %change in input
ð13Þ
3. Results and discussion Sliding wear behavior of A356 alloys based composites containing Al2O3 particles were experimentally investigated using a pin-on-disk wear testing machine. Fig. 3(a) and (b) shows the weight loss as a function of sliding distance and applied load, respectively. It is noted that the weight loss of the composites is less than that of unreinforced alloy, increases with increase in sliding distance, and has a declining trend with increasing the particles volume fraction. In general, composites offer superior wear as compared to the alloy irrespective of applied load and sliding speed [29]. The hard dispersoids, present on the surface of the composite, act as protrusions, protect the matrix from severe contact with the counter surfaces [30–32], and thus resulting in less wear in composite as compared to that in the alloy. It is noted that the wear rate in all the samples increases marginally with applied load. The increase in the applied load leads to increase in the penetration of hard asperities of the counter surface to the softer pin surface, increase in micro cracking tendency of the subsurface and also increase in the deformation and fracture of asperities of the softer surface. The wear rate of the unreinforced alloy is found to be higher than that of the composites. This result is consistent with the rule that in general, materials with higher hardness have better wear
Fig. 5. The ANN architecture.
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and abrasive resistance (Fig. 4). The higher hardness of the composites could be attributed to the fact that Al2O3 particles act as obstacles to the motion of dislocation [33]. It is reported that an extensive mechanical mixing takes place between the aluminum matrix composite and the steel counterpart during sliding wear, and mechanically mixed layer (MML) containing elements from the two sliding counterparts are formed on the worn surface [34]. It was reported in the previous research that when the applied load induces stresses that exceed the fracture strength of carbide particles, the particles fracture and largely lose their effectiveness as load bearing components. The shear strains are transmitted to the matrix alloy and wear proceeds by a subsurface delamination process [35]. The coefficient of friction is related to the interaction of asperities between the counter surfaces, which in micro scale varies within specific range throughout the test period. This may result in fluctuation on friction coefficient within a narrow range in each of the material with sliding distance. In general the wear coefficient decreases with increase in applied load
Fig. 6. MAPE error values with, one hidden layer (a) and two hidden layers (b).
Fig. 7. The change of MAPE values for each training method for the first 10,000 s.
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before the specimen gets seizes. This is because the surface is covered with more stable, smoother and harder MML, which leads to the generation of fewer wear particles, the rise in temperature also more at higher applied load, which makes the material more plastic. This is also one cause of the lower wear particles generation. Some of the particles that are generated during sliding may also become compacted and lead to a further decrease in wear coefficient. The value of wear coefficient decreases with increase in sliding speed before the material seizes. This is also because of the greater plasticity due to higher frictional heating and formation of a more stable MML over the specimen surface. At seizure, the MML becomes unstable and fresh material is exposed to the counter surface. Because of the high temperature, the freshly exposed material becomes fused in localized region and adheres to the counter surface, leading to the generation of more wear particles or the transfer of more softer Al alloy matrix to the counter surface. This leads to a higher valued of wear coefficient during seizure. As the Al2O3 content increases, the degree of effective contact between the asperities of composite surface and counter surface decreases and thus the wear rate of composite reduces. The addition of Al2O3 also improves the hardness, strength
Fig. 8. Relative sensitivity as a function of percentage changes a: hardness, b: UTS, c: grain size, d: volume percentage of porosity and e: weight loss.
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and Young’s modulus of composite material as compared to the alloy. The high temperature strength, hardness and modulus of elasticity of composite also increase with increase in Al2O3 content. As a result, the sticking tendency due to softening of surface material with the counter surface reduces in case of composite as compared to the alloy. As a result, the seizure pressure of composite is noted to be significantly higher than that of the alloy. The MML, in composite, is thicker than that of alloy and it contains more amount of iron and Al oxide as compared to that in alloy. This makes the MML of composite stronger than that in alloy. Additionally, the MML in composite have the higher capability to withstand more frictional heating without localized adhesion. This is also one of the reasons to have higher seizure pressure and seizure temperature as compared to the alloy. This is exactly observed in all the alloys and composites investigated in the present study. Because of the lower degree of contact between the contact surfaces the temperature rise in case of composite is noted to be less as compared to the alloy. Additionally a large amount of energy is spent on scratching of counter surface by the Al2O3 particles and thus less energy is spent on heating of the specimen surface. Furthermore, the thermal conductivity of the thicker MML formed over the composite surface is less as compared to that formed over alloy surface. The specific heat of MML in composite is also expected to be less as compared to that in alloy as the MML in composite contains more iron as compared to that in alloy. These facts lead to less temperature rise on composite surface as compared to the alloy. The thickness and concentration of MML, and the effective contact between the counter surfaces decreases with increase in Al2O3 content which might be the prime factor for reduction in temperature rise with increase in Al2O3 content. It has been mentioned that the hard Al2O3 particles penetrate deep into the counter surface leading to formation of microchips from counter surface. As a result greater amount of frictional force required for sliding of composite over the counter surface. Additionally the temperature rise is also noted to be less in composite as compared to the alloy. This leads to reduction in slipping action in case of composite as compared to the alloy. As a result coefficient of friction in case of composite is noted to be more than the alloy. As the Al2O3 content increases number of Al2O3 particles penetrating to the counter surface increases and thus the coefficient of friction in composites increases with increase in Al2O3 content. The temperature rise is also noted to be decreased with increasing Al2O3 content. Thus the slipping action is also reduced with increasing Al2O3 content. This further leads to increase in coefficient of friction of composite with increase in Al2O3 content. Fig. 5 shows the ANN architecture used in this study. MAPE error values are computed in Fig. 6 and given at the end of training process for various neurons in the hidden layer. The obtained error values for different neurons number in the hidden layers and also hidden layers number were analyzed and given, graphically. This figure also gives information about the accuracy of seven famous training algorithms depending on the neurons number in the hidden layers and hidden layers number. It is evident that the least error value was obtained by using LM training algorithm with 1 hidden layer and 8 neurons. Newton back propagation with 1 hidden layer and 10 neurons in the hidden layers is the second training algorithm following the LM in terms of low error value. Higher errors were observed by others training algorithms. Although LM is popular in the artificial neural network domain (even if it is considered the first approach for an unseen MLP training task), it is not that popular in the metaheuristics field. The number of used hidden units determines the complexity of neural network. The accuracy of predicted values increases with increasing the number of hidden units. However, with increasing the number of hidden units, there may be an overriding in data.
Fig. 9. Comparison between the experimental and predicted values of weight loss.
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Fig. 10. Distribution of hardness with 3% Al2O3 in this model (vertical section).
In Fig. 7, the change of MAPE values for each training method is given for the first 10,000 s. The speed advantage of the LM method is evidently seen. Normally, it has the computational complexity, however it can give the results with much accuracy and in the less number of iterations than other methods. The computation time of the training algorithm is important in artificial neural network based applications. In this study, it was proven that the LM learning algorithm based on a non-linear least squares optimization technique is significantly efficient in the training of a neural network. In spite of the fact that the necessary computations for this algorithm is more than those of the back propagation algorithm, its convergence speed makes it more preferable algorithm in many neural network applications. MAPE is a good criterion to have information about learning performance [1]. The iterations were continued until the minimum MAPE error is obtained. In addition, success in the algorithms depends on parameters learning rate and momentum constant [35–41]. Faster algorithms such as Newton use standard numerical optimization techniques [1]. The Levenberg–Marquardt algorithm is an approximation to the Newton method used also for training artificial neural network. The Newton method approximates the error of the network with a second order expression, which contrasts to the back-propagation algorithm that does it with a first order expression. Fig. 8 shows the results of sensitivity analysis. Percentage changes (5, 10, 15 and 20) were selected in this study. For every input parameter, the percentage change in the output, as a result of the change in the input parameter, is observed. The volume percentage of Al2O3 and cooling rate are two the most sensitive parameters and temperature gradient has the smallest effect. Fig. 8 also shows that volume percentage of porosity and weight loss are two most sensitive parameters and grain size is the least sensitive one. In other words, Fig. 8 indicates the high dependence of porosity content and weight loss to cooling rate and volume fraction of nano particles. Fig. 9 shows the efficacy of the optimization scheme by comparing the ANN results with the experimental values. There is a convincing agreement between experimental values and predicted values for weight loss of nano composite using LM training algorithm. In order to exhibit some results of A356 composite, the distribution of hardness with 3% Al2O3 in this model is displayed in Fig. 10. 4. Conclusion Mechanical and tribological properties are considered to be related to cooling rate, temperature gradient and volume percentage of Al2O3. It is concluded that the hard nano particles resist against destruction action of abrasive and protect the surface, so with increasing its content, the wear resistance enhances. Additionally, the hard dispersoid makes the matrix alloy plastically constrained and improves the high temperature strength of the virgin alloy. It is noted that the LM neural network with 8 neurons in 1 hidden layers will be the fastest training algorithm and can present a very good performance for FEM modeling of nano composites behaviors. Sensitivity analysis showed that volume percentage of Al2O3, and cooling rate are two most sensitive parameters and temperature gradient has smallest effect. References [1] M.O. Shabani, A. Mazahery, Appl. Math. Model. 35 (2011) 5707–5713. [2] K. Elangovan, C. Sathiya Narayanan, R. Narayanasamy, Comput. Mater. Sci. 47 (2010) 1072–1078. [3] A. Bahmani, P. Davami, N. Varahram, Kovove Mater. 49 (2011) 253–264.
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Contents lists available at SciVerse ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Artificial Intelligence in numerical modeling of nano sized ceramic particulates reinforced metal matrix composites Mohsen Ostad Shabani ⇑, Ali Mazahery Department of Materials Science, Lenjan Branch, Islamic Azad University, Isfahan, Iran
a r t i c l e
i n f o
Article history: Received 10 June 2011 Received in revised form 18 December 2011 Accepted 21 December 2011 Available online 2 January 2012 Keywords: Algorithms Nano composite Abrasive
a b s t r a c t Artificial neural network models have the capacity to eliminate the need for expensive experimental investigation in various areas of manufacturing processes, including the casting methods. An understanding of the inter-relationships between input variables is essential for interpreting the sensitivity data and optimizing the design parameters. Aluminum is the best metal for producing metal matrix composites which are known as one of the most useful and high-tech composites in our world. Combining aluminum and nano Al2O3 particles will yield a material with high mechanical and tribological properties. In this investigation, the accuracy of various artificial neural network training algorithms in FEM modeling of Al2O3 nano particles reinforced A356 matrix composites has been studied. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Artificial neural network is a non-linear statistical analysis technique and is especially suitable for simulation of systems which are hard to be described by physical models. It provides a way of linking input data to output data using a set of nonlinear functions. Choosing the optimum architecture of the network is one of the challenging steps in artificial neural network (ANN) modeling. The term ‘architecture’ refers to the number of layers in the network and number of neurons in each layer. In order to design a neural network for a problem solution, a training algorithm is required. During the training process, the weights and biases in the network are adjusted to minimize the error and obtain a high-performance in the solution. At the end of the training and during the training error, mean absolute percentage error (MAPE) is computed between desired outputs and target outputs [1–6]. The finite element method is a powerful technique for solving differential or partial differential equations as well as integral equations. It is the most versatile numerical technique in modern engineering analysis and has been employed to study diverse problems in heat transfer, fluid mechanics, mechanical properties, rigid body dynamics, solid mechanics, electrical systems and many other fields [7–10]. Aluminum alloys are widely being used in the industry due to their excellent castability, good corrosion resistance and high strength-to-weight ratio [10–12]. However, their applications have often been restricted because conventional Al alloys are soft and notorious for their poor wear resistance. Wear can be defined as damage to a solid surface, generally involving progressive loss of material, due to relative motion between that surface and a contacting substance or substances. In many applications, metal matrix composites (MMCs) are subjected to sliding motion. In some cases this motion is intentional: for example, in an internal combustion engine piston or cylinder liner, or an automotive brake disk, or in the processing of material by machining, forging, or extrusion. It may also be unintentional, as in the fretting wear of a joint [13]. The wear resistance of aluminum alloys is improved as a consequence of the incorporation of ceramic fibers or particles which act as the ⇑ Corresponding author. Tel.: +98 912 563 6709; fax: +98 261 6201888. E-mail address: [email protected] (M.O. Shabani). 0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.12.059
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load bearing and abrasive member. Aluminum matrix composites (AMCs) represent a generation of engineering materials which combine metallic properties of matrix alloys (ductility and toughness) with ceramic properties of reinforcements (high strength and high modulus), leading to greater wear, strength in shear, compression and higher service-temperature capabilities [14–18]. In this study, nano Al2O3 particulates reinforced Al matrix composite have been processed using stir casting. Then, the experimental wear properties and the accuracy of various ANN training algorithms in FEM modeling of the composites have been investigated. 2. Experimental procedure Commercial casting Al–Si alloy and Al2O3 particles have been used in the present study. Initially, Al alloy was charged into the crucible, and heated to about 750 °C, which is above the liquidus temperature of the Al alloy. The composite was synthesized through solidification processing route using 0.5, 1, 1.5, 2, 2.5, . . . , 5 vol.% Al2O3 particles of size range 50 nm as reinforcement [18]. Composite slurry was step cast into the CO2-sand mould. 1 wt.% magnesium additive in powder form was also used as a wetting agent. Hardness measurements were carried out on a Brinell hardness testing machine (Eseway DVRB-M), using an indenter ball with 2.5 mm diameter at a load of 31.25 kg, and the mean values of at least five measurements conducted on different areas of each sample was considered in order to eliminate possible segregation effects and get a representative value of the matrix material hardness. The wear tests were performed at a sliding speed of 0.3 m/s under varying applied loads against case hardened steel disc using a pin-on-disk type test machine. The disk with a diameter of 50 mm and a thickness of 10 mm was made of the steel hardened up to 63 HRC and polished to very fine grade with surface roughness about 0.22 mm. Test specimens were cut and shaped in the form of pins having 6 mm in diameter and 25 mm in height. Before the abrasion tests, each specimen was polished to 0.5 lm. Fig. 1 shows Schematic diagram of the abrasion wear test. The pin-on-disk wear machine consists of the stationary pin pressed at the required load against the disk rotating at the defined speed. An AC motor ensures the stable running speeds of the disk. The testing machine is equipped with a set of measuring transducers. During the tests the friction force as well as coefficient of friction is measured continuously. The experiment was carried out at room temperature (21 °C, relative humidity 55%) with water as the lubricant. The samples were cleaned with acetone and weighed (up to an accuracy of 0.01 mg using microbalance) prior to and after each test. The temperature rise and friction force were recorded from the digital display interfaced with the wear test machine. Coefficient of friction was computed from the recorded frictional force and the applied load (i.e. the ratio of frictional force to the applied load). A set of three samples was tested in every experimental condition, and the average along with standard deviation for each set of three tests is measured. The wear tests were conducted up to the total sliding distance of 2000 m. The mass loss of the pin was used to study the effect of Al2O3 addition on the wear resistance of the composite materials under consideration. 2.1. Finite element method The finite element method (FEM) has undoubtedly become the most popular and powerful analytical tool for studying a wide range of engineering and physical problems. One of the important applications of FEM is the solidification. The basic concept is that the whole domain is divided into smaller elements of finite dimensions called ‘Finite Elements’. FEM method is used for discretization and to calculate the transient temperature field of quenching [1,19–22].
Fig. 1. Schematic diagram of the abrasion wear test.
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The steps involved in the finite element analysis are as follows: The whole domain is divided into a finite number of sub-domains, which is called the discretization of the domain. Each sub domain is called an element. The collection of elements is called the finite-element mesh. From the mesh, a typical element is isolated and the variational formulation of the given problem over the typical element is constructed. An approximate solution of the variational problem is assumed and the element equations are made by substituting this solution in the above system. The element matrix, which is called stiffness matrix, is constructed by using the element interpolation functions. The algebraic equations so obtained are assembled by imposing the interelement continuity conditions. This yields a large number of algebraic equations known as the global finite element model, which governs the whole domain. The essential and natural boundary conditions are imposed on the assembled equations. The assembled equations so obtained can be solved by any of the numerical technique. The mathematical formulation of this solidification problem is given [1,10,22]:
qC
@Tðx; y; z; tÞ dfs ¼ K r2 Tðx; y; z; tÞ þ qL @t dt
ð1Þ
where q the density (kg/m3), C the specific heat (J/kg K), T the temperature (K), t the time (s), K is the thermal conductivity (W/m K), L is the latent heat (J/kg) and fs the local solid fraction. The fraction of solid in the mushy zone is estimated by the Scheil equation, which assumes perfect mixing in the liquid and no solid diffusion. With the liquidus and solidus having constant slopes, fs is then expressed as:
fS ¼ 1
Tf T T f T liq
1=ðk0 1Þ ð2Þ
where Tf is the melting temperature (K), TL the liquidus temperature (K), and k0 the partition coefficient. Then [22]:
ð2k0 Þ=ðk0 1Þ Tf T dfs 1 dT ¼ dt dt ðk0 1ÞðT f T liq Þ T f T liq
ð3Þ
The latent heat released during solidification of the remaining liquid of eutectic composition was taken into account by a device, which considers a temperature accumulation factor.
qC 0
@Tðx; y; z; tÞ dfs ¼ K r2 Tðx; y; z; tÞ þ qL @t dt
ð4Þ
where C0 can be considered as a pseudo-specific heat given by Shabani and Mazahery [22]:
dfs dT C M ¼ ð1 fs ÞC l þ fs C S
C0 ¼ CM L
ð5Þ ð6Þ
where the subscripts L, S and M refer to liquid, solid and mushy, respectively. The other properties such as thermal conductivity and density in the mushy zone are described similarly as the specific heat:
K M ¼ ð1 fs ÞK l þ fs K S
ð7Þ
qM ¼ ð1 fs Þql þ fs qS
ð8Þ
2.2. ANN algorithms There are various training algorithms used in neural network applications. It is hardly difficult to predict which of these learning will be the fastest one for any problem. Generally, it depends on some factors; the structure of the networks, in other words, the number of hidden layers, weights and biases in the network, aimed error at the learning, and application area, for instance, pattern recognition or classification or function approximation problem. Back-propagation algorithm (BP) is one of the most popular in engineering, the major steps of BP algorithm are listed as follows [19–28]:
Initial all weights (wij) to small random values Present an input and specify the desired output. Calculate outputs using the present value of wijs Find an error term, rj, for all the nodes. If vj, Oj and Ej stand for desired value of the jth output node, the computed value of the jth output node, and the computed value for the jth hidden layer node, then the error terms are calculated as; for outP put node j: rj = (vj Oj)Oj(1 Oj), for hidden layer node j: rj = Ej(1 Ej) krknjk, where k is over all nodes in the layer above node j.
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Adjust weights by wij(k + 1) = wij(k) + arjXi + n(wij(k) wij(k 1)), where (k + 1), (k), and (k 1) index next, present, and previous iteration, respectively. a is the learning rate. n is the momentum which determines the effect of past weight changes on the current direction of movement in weight space. It is used to filter out the high-frequency variations of the error surface. Present another input and go back to second step. All the training inputs are presented cyclically until weights stabilize. The neural network predictions were directly compared with the experimental obtained data to evaluate the learning performance. Also the ANNs can be judged by various parameters such as the mean square error (MSE), mean absolute error (MAE), normalized root mean square error (NSE) and mean absolute percentage error (MAPE) as follows [1,10]: N0 Q X 1 X jdn ðmÞ yn ðmÞj2 QN0 m¼1 n¼1 X x y MAE ¼ n ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P ðA FÞ2 NSE ¼ A2 " # N0 A F 1 X 100 MAPE ¼ N0 N¼1 A
MSE ¼
ð9Þ ð10Þ ð11Þ ð12Þ
where N0 is the number of output, Q the number of training sets, d desired output, and y the network output and where x = X X0 , X is the target output and X0 the mean of X, y = Y Y0 ; Y is the network output and Y0 the mean of Y. A and F represent the actual and forecasting values respectively. In this investigation the MAPE was used to evaluate the performance of model. Some of famous train algorithms are as follow [19–28]: Batch training with weight and bias learning rules: trains a network with weight and bias learning rules with batch updates. The weights and biases are updated at the end of an entire pass through the input data. BFGS quasi-Newton back propagation: is a network training function that updates weight and bias values according to the BFGS quasi-Newton method. BFGS quasi-Newton back propagation for use with NN model reference adaptive controller: is a network training function that updates weight and bias values according to the BFGS quasi-Newton method. Batch unsupervised weight/bias training: trains a network with weight and bias learning rules with batch updates. Weights and biases updates occur at the end of an entire pass through the input data. Cyclical order incremental update: trains a network with weight and bias learning rules with incremental updates after each presentation of an input. Inputs are presented in cyclic order. Powell–Beale conjugate gradient back propagation: is a network training function that updates weight and bias values according to the conjugate gradient back propagation with Powell–Beale restarts. Fletcher–Powell conjugate gradient back propagation: is a network training function that updates weight and bias values according to conjugate gradient back propagation with Fletcher–Reeves updates. Polak–Ribiére conjugate gradient back propagation: is a network training function that updates weight and bias values according to conjugate gradient back propagation with Polak–Ribiére updates. Gradient descent back propagation: is a network training function that updates weight and bias values according to gradient descent. Gradient descent with adaptive learning rule back propagation: is a network training function that updates weight and bias values according to gradient descent with adaptive learning rate. Gradient descent with momentum back propagation: is a network training function that updates weight and bias values according to gradient descent with momentum. Gradient descent with momentum and adaptive learning rule back propagation: is a network training function that updates weight and bias values according to gradient descent momentum and an adaptive learning rate. One step secant back propagation: is a network training function that updates weight and bias values according to the one-step secant method. Random order incremental training with learning functions: trains a network with weight and bias learning rules with incremental updates after each presentation of an input. Inputs are presented in random order. Resilient back propagation (Rprop): is a network training function that updates weight and bias values according to the resilient back propagation algorithm (Rprop). Sequential order incremental training with learning functions: rains a network with weight and bias learning rules with sequential updates. The sequence of inputs is presented to the network with updates occurring after each time step. Scaled conjugate gradient back propagation: is a network training function that updates weight and bias values according to the scaled conjugate gradient method. Levenberg–Marquardt back propagation: is a network training function that updates weight and bias values according to Levenberg–Marquardt optimization.
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Fig. 2. Flowchart of FEM–ANN scheme.
The Flowchart of FEM–ANN scheme is shown in Fig. 2. In the analysis of performance of various training algorithms, the same prepared learning and test set were used in the training processes of each learning algorithm. The performance analysis were done from the viewpoint of training duration, error minimization and prediction achievement. The neural network predictions were directly compared with the experimental obtained data to evaluate the learning performance. Finally, sensitivity data were used to further validate the model and to interrogate the model to try to gain a physical understanding of the predicted trends. The sensitivity factor (SF) measures the contribution of each input variable to the desired output, and it differs for each individual case in both the train and test data files. Therefore, the SF presented in this paper is an average of the contribution of an individual input variable towards the specified output. For every input
Fig. 3. Wear resistance of the nano composites: (a) Weight loss as a function of sliding distance, (b) effect of applied load on wear resistance.
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Fig. 4. Hardness as a function of nano Al2O3 particles volume fraction.
parameter, the percentage change in the output, as a result of the change in the input parameter, was calculated using the following relationship [7]:
"
# N0 %change in output 1 X 100 Relative sensitivity ¼ N0 j¼1 %change in input
ð13Þ
3. Results and discussion Sliding wear behavior of A356 alloys based composites containing Al2O3 particles were experimentally investigated using a pin-on-disk wear testing machine. Fig. 3(a) and (b) shows the weight loss as a function of sliding distance and applied load, respectively. It is noted that the weight loss of the composites is less than that of unreinforced alloy, increases with increase in sliding distance, and has a declining trend with increasing the particles volume fraction. In general, composites offer superior wear as compared to the alloy irrespective of applied load and sliding speed [29]. The hard dispersoids, present on the surface of the composite, act as protrusions, protect the matrix from severe contact with the counter surfaces [30–32], and thus resulting in less wear in composite as compared to that in the alloy. It is noted that the wear rate in all the samples increases marginally with applied load. The increase in the applied load leads to increase in the penetration of hard asperities of the counter surface to the softer pin surface, increase in micro cracking tendency of the subsurface and also increase in the deformation and fracture of asperities of the softer surface. The wear rate of the unreinforced alloy is found to be higher than that of the composites. This result is consistent with the rule that in general, materials with higher hardness have better wear
Fig. 5. The ANN architecture.
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and abrasive resistance (Fig. 4). The higher hardness of the composites could be attributed to the fact that Al2O3 particles act as obstacles to the motion of dislocation [33]. It is reported that an extensive mechanical mixing takes place between the aluminum matrix composite and the steel counterpart during sliding wear, and mechanically mixed layer (MML) containing elements from the two sliding counterparts are formed on the worn surface [34]. It was reported in the previous research that when the applied load induces stresses that exceed the fracture strength of carbide particles, the particles fracture and largely lose their effectiveness as load bearing components. The shear strains are transmitted to the matrix alloy and wear proceeds by a subsurface delamination process [35]. The coefficient of friction is related to the interaction of asperities between the counter surfaces, which in micro scale varies within specific range throughout the test period. This may result in fluctuation on friction coefficient within a narrow range in each of the material with sliding distance. In general the wear coefficient decreases with increase in applied load
Fig. 6. MAPE error values with, one hidden layer (a) and two hidden layers (b).
Fig. 7. The change of MAPE values for each training method for the first 10,000 s.
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before the specimen gets seizes. This is because the surface is covered with more stable, smoother and harder MML, which leads to the generation of fewer wear particles, the rise in temperature also more at higher applied load, which makes the material more plastic. This is also one cause of the lower wear particles generation. Some of the particles that are generated during sliding may also become compacted and lead to a further decrease in wear coefficient. The value of wear coefficient decreases with increase in sliding speed before the material seizes. This is also because of the greater plasticity due to higher frictional heating and formation of a more stable MML over the specimen surface. At seizure, the MML becomes unstable and fresh material is exposed to the counter surface. Because of the high temperature, the freshly exposed material becomes fused in localized region and adheres to the counter surface, leading to the generation of more wear particles or the transfer of more softer Al alloy matrix to the counter surface. This leads to a higher valued of wear coefficient during seizure. As the Al2O3 content increases, the degree of effective contact between the asperities of composite surface and counter surface decreases and thus the wear rate of composite reduces. The addition of Al2O3 also improves the hardness, strength
Fig. 8. Relative sensitivity as a function of percentage changes a: hardness, b: UTS, c: grain size, d: volume percentage of porosity and e: weight loss.
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and Young’s modulus of composite material as compared to the alloy. The high temperature strength, hardness and modulus of elasticity of composite also increase with increase in Al2O3 content. As a result, the sticking tendency due to softening of surface material with the counter surface reduces in case of composite as compared to the alloy. As a result, the seizure pressure of composite is noted to be significantly higher than that of the alloy. The MML, in composite, is thicker than that of alloy and it contains more amount of iron and Al oxide as compared to that in alloy. This makes the MML of composite stronger than that in alloy. Additionally, the MML in composite have the higher capability to withstand more frictional heating without localized adhesion. This is also one of the reasons to have higher seizure pressure and seizure temperature as compared to the alloy. This is exactly observed in all the alloys and composites investigated in the present study. Because of the lower degree of contact between the contact surfaces the temperature rise in case of composite is noted to be less as compared to the alloy. Additionally a large amount of energy is spent on scratching of counter surface by the Al2O3 particles and thus less energy is spent on heating of the specimen surface. Furthermore, the thermal conductivity of the thicker MML formed over the composite surface is less as compared to that formed over alloy surface. The specific heat of MML in composite is also expected to be less as compared to that in alloy as the MML in composite contains more iron as compared to that in alloy. These facts lead to less temperature rise on composite surface as compared to the alloy. The thickness and concentration of MML, and the effective contact between the counter surfaces decreases with increase in Al2O3 content which might be the prime factor for reduction in temperature rise with increase in Al2O3 content. It has been mentioned that the hard Al2O3 particles penetrate deep into the counter surface leading to formation of microchips from counter surface. As a result greater amount of frictional force required for sliding of composite over the counter surface. Additionally the temperature rise is also noted to be less in composite as compared to the alloy. This leads to reduction in slipping action in case of composite as compared to the alloy. As a result coefficient of friction in case of composite is noted to be more than the alloy. As the Al2O3 content increases number of Al2O3 particles penetrating to the counter surface increases and thus the coefficient of friction in composites increases with increase in Al2O3 content. The temperature rise is also noted to be decreased with increasing Al2O3 content. Thus the slipping action is also reduced with increasing Al2O3 content. This further leads to increase in coefficient of friction of composite with increase in Al2O3 content. Fig. 5 shows the ANN architecture used in this study. MAPE error values are computed in Fig. 6 and given at the end of training process for various neurons in the hidden layer. The obtained error values for different neurons number in the hidden layers and also hidden layers number were analyzed and given, graphically. This figure also gives information about the accuracy of seven famous training algorithms depending on the neurons number in the hidden layers and hidden layers number. It is evident that the least error value was obtained by using LM training algorithm with 1 hidden layer and 8 neurons. Newton back propagation with 1 hidden layer and 10 neurons in the hidden layers is the second training algorithm following the LM in terms of low error value. Higher errors were observed by others training algorithms. Although LM is popular in the artificial neural network domain (even if it is considered the first approach for an unseen MLP training task), it is not that popular in the metaheuristics field. The number of used hidden units determines the complexity of neural network. The accuracy of predicted values increases with increasing the number of hidden units. However, with increasing the number of hidden units, there may be an overriding in data.
Fig. 9. Comparison between the experimental and predicted values of weight loss.
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Fig. 10. Distribution of hardness with 3% Al2O3 in this model (vertical section).
In Fig. 7, the change of MAPE values for each training method is given for the first 10,000 s. The speed advantage of the LM method is evidently seen. Normally, it has the computational complexity, however it can give the results with much accuracy and in the less number of iterations than other methods. The computation time of the training algorithm is important in artificial neural network based applications. In this study, it was proven that the LM learning algorithm based on a non-linear least squares optimization technique is significantly efficient in the training of a neural network. In spite of the fact that the necessary computations for this algorithm is more than those of the back propagation algorithm, its convergence speed makes it more preferable algorithm in many neural network applications. MAPE is a good criterion to have information about learning performance [1]. The iterations were continued until the minimum MAPE error is obtained. In addition, success in the algorithms depends on parameters learning rate and momentum constant [35–41]. Faster algorithms such as Newton use standard numerical optimization techniques [1]. The Levenberg–Marquardt algorithm is an approximation to the Newton method used also for training artificial neural network. The Newton method approximates the error of the network with a second order expression, which contrasts to the back-propagation algorithm that does it with a first order expression. Fig. 8 shows the results of sensitivity analysis. Percentage changes (5, 10, 15 and 20) were selected in this study. For every input parameter, the percentage change in the output, as a result of the change in the input parameter, is observed. The volume percentage of Al2O3 and cooling rate are two the most sensitive parameters and temperature gradient has the smallest effect. Fig. 8 also shows that volume percentage of porosity and weight loss are two most sensitive parameters and grain size is the least sensitive one. In other words, Fig. 8 indicates the high dependence of porosity content and weight loss to cooling rate and volume fraction of nano particles. Fig. 9 shows the efficacy of the optimization scheme by comparing the ANN results with the experimental values. There is a convincing agreement between experimental values and predicted values for weight loss of nano composite using LM training algorithm. In order to exhibit some results of A356 composite, the distribution of hardness with 3% Al2O3 in this model is displayed in Fig. 10. 4. Conclusion Mechanical and tribological properties are considered to be related to cooling rate, temperature gradient and volume percentage of Al2O3. It is concluded that the hard nano particles resist against destruction action of abrasive and protect the surface, so with increasing its content, the wear resistance enhances. Additionally, the hard dispersoid makes the matrix alloy plastically constrained and improves the high temperature strength of the virgin alloy. It is noted that the LM neural network with 8 neurons in 1 hidden layers will be the fastest training algorithm and can present a very good performance for FEM modeling of nano composites behaviors. Sensitivity analysis showed that volume percentage of Al2O3, and cooling rate are two most sensitive parameters and temperature gradient has smallest effect. References [1] M.O. Shabani, A. Mazahery, Appl. Math. Model. 35 (2011) 5707–5713. [2] K. Elangovan, C. Sathiya Narayanan, R. Narayanasamy, Comput. Mater. Sci. 47 (2010) 1072–1078. [3] A. Bahmani, P. Davami, N. Varahram, Kovove Mater. 49 (2011) 253–264.
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