finite element method in engineering

3. Determination of Plane Temperature Distribution in solid using ANSYS workbench
insulated
insulated
Convective boundary
Convective boundary
Heat flux
30OC
Linear Quadrilateral Element
Serendipity Quadratic Quadrilateral Element
Given: - All dimensions are in mm
Material Selected :- Steel
Assume other necessary data.
Convection = 40 W/m2
.O
C
heat flux = 30 W/m2
A B
C D
E
F
Domain is A B C D E F
A-B Convective boundary
B-C incoming Heat flux boundary
C-D Convective boundary
D-E insulated boundary
E-F constant temperature boundary
F-A insulated boundary
Description Of Problem

4. INTRODUCTION

5. EXPERIMENTAL SETUP
• Experiments are conducted on METATECH (ECMAC) electrochemical machining equipment.
• KCl solution is chosen as electrolyte, as it has no passivation effect on the surface of the work piece.
• Rectangular block of 20 mm X 20 mm and 25 mm height made of EN 31 tool steel which is a high carbon alloy
steel with high degree of hardness, compressive strength and abrasion resistance is chosen as workpiece.
• In this study, the collection of experimental data adopts the face center cubic (FCC) design with three levels of
each of the four design factors. The process parameters selected for present investigation are electrolyte
concentration (X1), voltage (X2), feed rate (X3) and inter-electrode gap (X4).
• The design is generated using MINITAB 16.0 statistical package. Table 1 shows the factors and their levels in coded
and actual values.
• Surface roughness parameter is selected as response variables in the present study. Roughness measurement is
done using a stylus-type profilometer, Talysurf (Taylor Hobson, Surtronic 3 + ). Roughness measurements in the
transverse direction on the work pieces are repeated five times and average of five measurements of surface
roughness parameter values are recorded.

6. Design
factors
Unit Notation
LEVELS
-1 0 1
Electrolyte
concentration
[%] X1 15 20 25
Voltage [V] X2 8 10 12
Feed rate [mm/min] X3 0.1 0.2 0.3
Inter-
electrode gap
[mm] X4 0.2 0.25 0.3
Table 1. Experimental parameters and their levels

7. ARTIFICIAL NEURAL NETWORK
• Artificial neural networks (ANN) emulating the biological connections between neurons are known as soft
computing techniques. ANN can reproduce some functions of human behavior, which are formed by a finite
number of layers with different computing elements called neurons. In order to construct a network, the neurons
are interconnected. The organization of connections determines the type and objectives of the ANNs.
• The processing ability of the network is stored in the inter-unit connection strengths, or weights, which are turned
in the learning process. The training algorithm (or learning) is defined as a procedure that consists of adjusting the
weights and biases of a network that minimize selected function of error between the actual and desired outputs.
• In this study, the back-propagation learning algorithms likely Levenberg Marquardt (LM), gradient descent with
variable learning rate and momentum (GDX) and scaled conjugate gradient (SCG) are used to train the networks.
• Four process parameters viz. electrolyte concentration, voltage, feed rate and inter-electrode gap are considered.
The input layers of the neural network consist of four neurons whereas the output layer has a single neuron that
represents the predicted value of Ra .
• Some parameters (i.e. the number of training and testing data, learning rate, number of hidden layers and
processing function used) affect the accuracy, reliability and effectiveness of the neural network. It is seen that
the processing functions, tansig and logsig, produce almost the same performance in different problems. Hence,
only the tansig processing function and single hidden layer are used.
• A trial and error scheme is used to determine the appropriate number of hidden neurons and numbers of hidden
neurons are varied from 3 to 8. The goal of any training algorithm is to minimize the global error such as mean
squared error (MSE), mean absolute percentage error (MAPE) and correlation coefficient (R).

8. RESULTS AND DISCUSSION
• Artificial neural network models for predicting surface roughness have been developed using multi-layer feed
forward back propagation algorithm.
• To construct, four process parameters viz., electrolyte concentration, voltage, feed rate and inter-electrode
gap as the input neurons and corresponding surface roughness as the output neuron.
• Considering the FCC design, 31 experiments (datasets) are conducted and Ra values are recorded. For brevity
of the paper, the table presenting the experimental results is omitted. The data sets are randomly chosen for
training and testing to get rid of any bias. 22 datasets for training and 9 datasets for testing are used. The data
points are normalized so that the dataset ranges in -1 to +1.
• To construct the models, the hyperbolic tangent sigmoid function in the hidden layer and linear activation
function in the output layer is considered. One hidden layer is chosen in the study. The neurons in the hidden
layers are varied from 3 to 8 to find out the best neural network model for different training algorithms. To
train the network, three different training algorithm viz. Levenberg-Marquardt (L-M) algorithm, gradient
descent with variable learning rate and momentum (GDX) algorithm and scaled conjugate gradient (SCG)
algorithm are used.
• MATLAB 7.8 is used to develop the neural network, train and test the network. In this study, to select the best
network, minimum mean squared error (MSE), mean absolute percentage error (MAPE) and maximum
correlation coefficient (R) are considered.

9. Algorithm Model
Training data Testing data
MSE MAPE R MSE MAPE R
L-M
4-3-1 0.00009157 0.353 0.99956 0.0749445 8.876 0.98458
4-4-1 0.00000205 0.046 0.9999 0.0445753 8.293 0.99038
4-5-1 0.00000043 0.024 1 0.0077101 3.746 0.99831
4-6-1 0.00000420 0.056 0.99998 0.03965559 7.799 0.99107
4-7-1 0.00000382 0.050 0.99997 0.07006014 10.390 0.98382
4-8-1 0.00204939 1.868 0.99996 0.02525059 7.302 0.98012
GDX
4-3-1 0.00048059 0.775 0.99854 0.038141 6.947 0.99106
4-4-1 0.00006021 0.158 0.9997 0.062704 8.209 0.98684
4-5-1 0.00000097 0.032 0.99999 0.011502 4.630 0.99727
4-6-1 0.00000158 0.048 0.99998 0.034425 7.514 0.99265
4-7-1 0.00000125 0.034 0.99991 0.027325 6.925 0.99457
4-8-1 0.00165456 0.990 0.9999 0.052778 7.794 0.99018
SCG
4-3-1 0.00094376 0.893 0.99509 0.09769107 8.527 0.98209
4-4-1 0.00000798 0.107 0.99995 0.051756 7.157 0.98901
4-5-1 0.00000113 0.037 0.99999 0.016579 4.492 0.99662
Comparison of network performances based on different training algorithms

10. Comparative study of experimental Ra and ANN predicted Ra data (a) for training set and (b) for testing set

11. • Performance of different networks trained and tested with LM, GDX and SCG algorithms are presented in Table
2. From the table, it is seen that architecture 4-5-1 trained using L-M algorithm has the minimum MSE and the
architecture is selected as the best performing network.
• The architecture also provides the maximum correlation coefficient (R) and minimum MAPE. Again, when
networks are trained using GDX algorithm, the architecture 4-5-1 gives the minimum MSE and is selected as
the best network. Again, architecture 4-5-1 provides the minimum MSE when the network is trained with SCG
algorithm.
• It is also seen from the comparative study of the performances among the best networks that the architecture
4-5- 1 trained with L-M algorithm is the best network based on the minimum MSE, minimum MAPE and
maximum correlation coefficient (R).
• It is also seen that the architecture has the minimum MAPE (0.024%) for training dataset. It implies that the
experimental values and ANN predicted values lie very close to each other.
• It is observed from the regression analysis that correlation coefficient (R) is 1 for training pattern. Correlation
coefficient 1 means the perfect correlation between the experimental and the predicted results. Comparative
study of the experimental Ra and ANN predicted Ra is presented in Fig. 1(a) and it is seen that ANN predicted
and experimental Ra are close to each other.
• Also the network is tested with a separate testing dataset and the comparison of experimental results and
predicted results is presented in Fig. 1(b). The correlation coefficient (R) is 0.998381 and it is a good correlation
between the predicted and experimental outputs.
• From the results, it is obvious that the network gives a good prediction of Ra in ECM of EN 31 tool steel and
the network has the generalization capability

12. CONCLUSIONS
• Neural network model for predicting surface roughness (Ra) in ECM of EN 31 tool steel work-piece is developed
using multi-layer feed forward back propagation algorithm.
• Electrolyte concentration, voltage, feed rate and inter-electrode gap are considered as the process parameters. The
design of experiment is done using FCC with 31 experimental run.
• From the experimental results it is seen that the architecture 4-5-1 trained with L-M algorithm gives the best
performance as compare to GDX and SCG algorithms for predicting Ra in ECM.
• The developed neural network model can predict surface roughness with about 96% accuracy. The network also
has a good generalization capability.
• Finally, it can be concluded that the developed ANN model is suitable for predicting Ra with high accuracy.

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