UDK 620.17:519.61/.64:669.018.95	ISSN 1580-2949
Original scientific article/Izvirni znanstveni članek	MTAEC9, 46(2)109(2012)
THE PERFORMANCE OF VARIOUS ARTIFICIAL NEURONS INTERCONNECTIONS IN THE MODELLING AND EXPERIMENTAL MANUFACTURING OF THE
COMPOSITES
PREDSTAVITEV RAZLIČNIH UMETNIH NEVRONSKIH POVEZAV PRI MODELIRANJU IN EKSPERIMENTALNI IZDELAVI
KOMPOZITOV
Mohsen Ostad Shabani, Ali Mazahery
Karaj Branch, Islamic Azad University, Karaj, Iran vahid_ostadshabany@yahoo.com
Prejem rokopisa - received: 2011-05-31; sprejem za objavo - accepted for publication: 2011-07-06
This study reports the performance of different artificial neural network (ANN) training algorithms in the prediction of mechanical properties. First, an experimental investigation was carried out on the mechanical behavior of an A356 composite reinforced with B4C particulates and then an ANN modeling was implemented in order to predict the mechanical properties, including the yield stress, UTS, hardness and elongation percentage. After the preparation of the training set, the neural network was trained using different training algorithms, hidden layers and the number of neurons in hidden layers. The test set was used to check the system accuracy for each training algorithm at the end of the learning. The results show that the Levenberg-Marquardt learning algorithm gave the best prediction for the yield stress, UTS, hardness and elongation percentage of the A356 composite reinforced with B4C particulates.
Keywords: composite, hardness, mechanical properties, ANN
V tem delu smo najprej opredelili mehanske lastnosti, vključno z mejo plastičnosti, natezno trdnostjo, trdoto in raztezkom kompozita A356, ojačenega z delci B4C, in nato uporabili kombinacijo umetne nevronske mreže in metode končnih elementov. Po pripravi treningpostavitve je bila nevronska mreža preizkušena z uporabo različnih algoritmov, skritih plasti in števila nevronov v skritih plasteh. Treningpostavitev je bila uporabljena za preverjanje natančnosti za vsak algoritem na koncu učenja. Rezultati kažejo, da da Levenberg-Marquardtov učni logaritem najboljšo napoved meje plastičnosti, natezne trdnosti, trdote in raztezka za kompozit A356, ki je ojačen z delci B4C.
Ključne besede: kompozit, trdota, mehanske lastnosti, ANN
processes where a complete understanding of the 1 INTRODUCTION	physical mechanisms is very difficult, or even impossible
Large quantities of castings are made each year from to acquire' as in the c,ase of materia7l properties where no
the aluminium alloy A356 (also known as Al-7Si-	satisfactory analytical model exists7-14.
0.3Mg). This alloy is one of the most popular alloys used	The aim of this study was investigate the
in industry due to its high fluidity and good "casta-	prediction performance of various training algorithms
bility
"1-5
using a neural network computer program for the
The addition of hard particles to a ductile metal	mecchanicfl properties of th^ A356 comPosite reinforced
matrix produces a material whose mechanical properties	with B4C Particulates. The results showed that the
Levenberg-Marquardt learning algorithms gave the best
are intermediate between the matrix alloy and the	result for this study.
ceramic reinforcement. The casting cooling rate, the	result for this study. reinforcement volume fraction, size, shape, and spatial
distribution are the most important parameters, playing a	2 EXPERIMENTAL role in the enhancement of the composite's mechanical
properties. A stronger adhesion at the particle/matrix	In this study, A356 was used as the matrix material
interface improves the load transfer, increasing the yield	and different volume fractions of B4C particles (1 % to
strength and stiffness, and delays the onset of	15 % B4C) with particle sizes ranging from 1 pm to 5 pm
particle/matrix de-cohesion6.	were used as the reinforcements.
An ANN is a logical structure with multi-processing	The melt-particle slurry was produced by a mecha-
elements, which are connected through interconnection	nical stirrer. Approximately 5 kg of A356 alloy was
weights. The knowledge is represented by the inter-	charged into the graphite crucible and heated up to a
connection weights, which are adjusted during the learn-	temperature above the alloy's melting point (750 °C).
ing phase. This technique is especially valuable in	The graphite stirrer, fixed on the mandrel of the drilling
machine, was introduced into the melt and positioned just below the surface of the melt. It was stirred at approximately 600 r/min and 750 °C. Then the step casting was poured into the CO2-sand mould.
Microscopic examinations of the composites and matrix alloy were carried out using an optical microscope. The porosity measurements of the composites were obtained using Archimedes's method. Hardness and tensile tests were used to assess the mechanical behavior of the composites and the matrix alloy.
2.1 Prediction of cooling rate and temperature gradient with EEM
The numerical model is applied to simulate the solidification of binary alloys; the mathematical formulation of this solidification problem is given15:
dT(x, y, z, t) 2 pC-dt-= 2 T (x, y, z, t)+q (1)
where p/(kg/m3) is the density, K/(W/(m K)) is the thermal conductivity, C/(J/(kg K)) is the specific heat, q/(W/m3) is the rate of energy generation, T/K is the temperature, and t/s is the time.
The release of latent heat between the liquidus and solidus temperatures is expressed by:
q=p^^^	^2)
where L/(J/kg) is the latent heat and fs is 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:
\1/(k0 -1)
fs =1-
Tf - T
I Tf - Tliq
)
(3)
where Tf/K is the melting temperature, TLiq/K is the liquidus temperature and k0 is the partition coefficient. Then15:
öt
1
f Tf - T
(2 - k „)/(t 0 -1)
(k0 -1)( Tf - Tliq) ^ Tf - T,q J
öT öt
(4)
pC
,dT (x, y, z, t) ' dt
= KV2 T (x, y, z, t) + q
(5)
where C' can be considered as a pseudo-specific heat given by:
öfs
C' = CM-^öT C M = (1-fs)C 1 + fs C s
(6) (7)
where the subscripts L, S and M refer to liquid, solid and mushy, respectively. The other properties, such as the thermal conductivity and the density in the mushy zone, are described in a similar way to the specific heat:
Km = (1-fs)K1 + fs Ks pM = (1-fs)p1 + fs Ps
(8) (9)
The latent heat released during the solidification of the remaining liquid of eutectic composition was taken into account by a device that considers a temperature accumulation factor.
The finite-element method (FEM) was used for discretization. Based on the above transient-temperature model, the FEM method is used to calculate the transient temperature, cooling rate and temperature gradient (G).
2.2 Neural network training algorithms
There are various training algorithms used in neural network applications. However, it is difficult to predict which of these 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 during the learning, and application area, for instance, pattern recognition or classification or the function approximation problem. However, the data structure and the uniformity of the training set are also important factors that affect the system accuracy and performance. Some of the famous training algorithms are as follows7-14'16-26:
Resilient back propagation (Äprop): is a network training function that updates weight and bias values according to the Rprop algorithm.
Random order incremental training with learning functions: trains a network with weight and bias learning rules using incremental updates after each presentation of an input. Inputs are presented in a random order.
Gradient descent back propagation: is a network training function that updates weight and bias values according to the gradient descent.
BFGS quasi-Newton back propagation: is a network training function that updates weight and bias values according to the BFGS quasi-Newton method.
Bayesian regularization: is a network training function that updates the weight and bias values according to LM optimization. It minimizes a combination of squared errors and weights, and then determines the correct combination so as to produce a network that generalizes well. The process is called Bayesian regulari-zation.
In the analysis of the performance of various training algorithms, the same prepared learning and test set were used in the training processes of each learning algorithm. The performance analyses were made from the viewpoint of training duration, error minimization and prediction achievement. The neural network predictions were directly compared with the experimentally obtained data to evaluate the learning performance. The mean square error (MSE), which is a statistical and scientific
error-computation method, was used to analyze the error25.
3 RESULTS AND DISCUSSION
Microscopic examinations were carried out on the metal-matrix composite. Figure 1 shows that the B4C particles were distributed between the dendrite branches and were frequently clustered together, leaving the dendrite branches as particle-free regions in the material.
Figure 2 shows the variation of porosity with B4C content. It indicates that an increasing amount of porosity is observed with increasing the volume fraction
Figure 3: Variations of the hardness value of the samples as a function of the volume fraction of B4C
Slika 3: Spremembe trdote vzorcev v odvisnosti od volumenskega deleža B4C
Figure 1: Typical optical micrographs: a) the composite with the volume fraction of B4C 4 %, b) the composite with 13 % B4C Slika 1: Tipičen optični posnetek: a) kompozit z volumenskim deležem B4C 2 %, b) kompozit s 15 % B4C
Figure 2: Variations of porosity as a function of the volume fraction of B4C
Slika 2: Spremembe poroznosti v odvisnosti od volumenskega deleža B4C
of the composites. The porosity level increased, since the contact surface area was increased27-31.
Figure 3 displays the results of the hardness tests. The hardness of the MMCs increases with the volume fraction of particulates in the alloy matrix. The higher hardness of the composites could be attributed to the fact that the B4C particles act as obstacles to the motion of dislocations32-36. Figure 4 shows the typical stress-strain curves obtained from uniaxial tension tests. The considerable increase in strain-hardening observed during the plastic deformation of composites is rationalized by the resistance of the hard reinforcing particles to the slip behavior of the Al matrix. The elongation to fracture of the composite materials was found to be very low, and no necking phenomenon was observed before fracture. On the other hand, the elongation to fracture of the un-reinforced Al alloy was about 15 %.
The input and output data set of the model is illustrated schematically in Figure 5. In Figure 6, the obtained MSE values for training data were given for each training algorithm. The obtained error values for
Figure 4: Stress-strain curves for volume fractions Al/ 3 % B4C (B), Al/ 7 % B4C (C), Al/ 10 % B4C (D), Al/ 12 % B4C (M) and Al/ 15 % B4C (N)
Slika 4: Odvisnosti napetost - deformacija za volumenske deleže Al/ 3 % B4C (B), Al/ 7 % B4C (C), Al/ 10 % B4C (D), Al/ 12 % B4C (M) in Al/ 15 % B4C (N)
Figure 5: Schematic representation of the neural network architecture Slika 5: Shematičen prikaz nevronske arhitekture
different numbers of neurons in the hidden layers and the number of hidden layers were analyzed and presented graphically. This figure also gives information about the accuracy of five famous training algorithms depending on the number of neurons in the hidden layers and the number of hidden layers. It is evident from this figure that the smallest error value was obtained by using the Levenberg-Marquardt training algorithm with two hidden layers and eight neurons (MSE = 6.4). BEGS quasi-Newton back propagation with three hidden layers and nine neurons in the hidden layers follows the
Figure 7: Comparison between the experimental and predicted values: a) elongation percentage, b) UTS
Slika 7: Primerjava med eksperimentalnimi in predvidenimi vrednostmi za: a) raztezek in b) natezno trdnost
Figure 6: Evaluation of the training performance of the networks for different training algorithms according to the MSE values with: a) one hidden layer, b) two hidden layers, c) three hidden layers and d) four hidden layers
Slika 6: Ocena parametrov treninga mreže za različne treningalgoritme po MSE-vrednostih za: a) eno skrito plast, b) dve skriti plasti, c) tri skrite plasti in d) štiri skrite plasti
Levenberg-Marquardt algorithm (MSE = 8.1), and thirdly the gradient descent back propagation including four hidden layers and six neurons in the hidden layers has clearly much more error than the previous two cases (MSE = 14.4). The most error was obtained from the Resilient back-propagation training algorithm and the Random order incremental training with learning functions. The Levenberg-Marquardt training algorithm was found to be the fastest training algorithm; however, it requires more memory with the same error convergence bound compared to the training methods25. MSE is a good criterion to have information about learning performance. The iterations were continued until it is decided that the minimum MSE error is obtained.
Figure 7 shows the efficacy of the optimization scheme by comparing the ANN results with the experimental values. There is a convincing agreement between the experimental values and the predicted values for UTS and the elongation percentage of the A356 composite reinforced with B4C particulates for the Levenberg-Marquardt training algorithm.
4 CONCLUSION
1)	The mechanical properties modeling was developed to predict the hardness, yield stress, ultimate tensile strength and elongation percentage.
2)	The effect of various training algorithms on the prediction of the mechanical properties of the fabricated A356 composite reinforced with B4C particulates was investigated. The prediction of the ANN model was found to be in good agreement with the experimental data.
3)	According to the results, the Levenberg-Marquardt learning algorithm gave the best prediction for hardness, yield stress, ultimate tensile strength and elongation percentage for the A356 composite. It is believed that an ANN with two hidden layers and eight neurons (MSE = 6.4) gave an accurate prediction of the mechanical properties of the fabricated A356 composite reinforced with B4C particulates.
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