Neural network-based expert systems for

Neural network-based expert systems for predictions of temperature distributions in electron beam welding process

Dhanunjaya Y. A. Reddy & Dilip Kumar Pratihar

The International Journal of Advanced Manufacturing Technology ISSN 0268-3768 Volume 55 Combined 5-8 Int J Adv Manuf Technol (2011) 55:535-548 DOI 10.1007/ s00170-010-3104-6

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Author's personal copy Int J Adv Manuf Technol (2011) 55:535–548 DOI 10.1007/s00170-010-3104-6

ORIGINAL ARTICLE

Neural network-based expert systems for predictions of temperature distributions in electron beam welding process Dhanunjaya Y. A. Reddy & Dilip Kumar Pratihar

Received: 12 June 2010 / Accepted: 30 November 2010 / Published online: 15 December 2010 # Springer-Verlag London Limited 2010

Abstract In the present paper, neural network-based expert systems have been developed for online predictions of temperature distributions on electron beam-welded plates. Finite element method is a popular tool to carry out this analysis. However, this analysis could be time consuming, and the obtained results might be dependent on a number of mesh parameters, namely shaping ratio, number of element divisions, and others. Thus, an expert system might be necessary for making online predictions of temperature distributions in welding after considering the said uncertainties. Neural network-based expert systems have been developed using the data collected through finite element analysis, and their performances are compared on some test cases. Once trained, the neural network-based expert systems could make the predictions in a fraction of a second. Keywords Expert system . Electron beam welding . Finite element analysis . Back-propagation neural network . Genetic-neural network

1 Introduction An expert system is a set of computer programs capable of determining input–output relationships of a process preferably in less computation time, and thus, making it suitable D. Y. A. Reddy : D. K. Pratihar (*) Soft Computing Laboratory, Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur 721302, India e-mail: [email protected] D. Y. A. Reddy e-mail: [email protected]

for online implementations. In order to develop knowledge base of an expert system, proper training is to be provided to it with the help of some input–output data obtained either analytically or through real experiments. For analytical estimation of the said data, a proper mathematical model is to be built in the form of differential equations, and so on. These differential equations can be solved in a number of ways, and finite element (FE) analysis is one of those methods. An FE analysis is a time-consuming approach, and its results depend on the selection of a number of parameters, such as type of elements, size of elements, their connectivity, and others. Thus, there is enough fuzziness (uncertainty) in FE analysis itself. An attempt was made by Subba Rao and Pratihar [1] to model the said fuzziness of FE analysis using fuzzy logic technique. Neural networkbased approaches can also be used for the development of the above expert system. In the present study, neural network-based expert systems have been developed for electron beam welding process using the data collected through its FE analysis. Electron beam (EB) welding has unique advantages over other traditional fusion welding methods due to highenergy density, deep penetration, large depth-to-width ratio, and small heat affected zone. During the EB welding process, the metal in the welded joint is heated to melt and locally vaporized by the high-energy density of the electron beam. For the whole process, the temperature distributions play a vital role. Rosenthal [2] studied the temperature distributions on an infinite sheet due to a moving point heat source considering the heat dissipation by conduction only. Several analytical 2-D or 3-D models of the electron beam thermal source had been developed by various researchers. Couedel et al. [3] established a 2-D analytical heat transfer model using a moving thermal source, considering the impact of source size and influence of the boundary on the

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thermal field. Nguyen et al. [4] gave an analytical solution for a double-ellipsoidal power density moving heat source in a semi-infinite body. Wei and Shian [5] proposed an approximate 3-D heat conduction model by satisfying interfacial energy and momentum balances at the keyhole cavity. He and DebRoy [6] established a transient, 3-D numerical heat transfer and fluid flow model based on the solution of the equations of conservation of mass, momentum, and energy to calculate the temperature and velocity fields in the weld pool. Some other investigators (Miyazaki and Giedt [7], Baeva et al. [8], Ho and Wei [9]) also reported various shapes of thermal source, including elliptical, cylindrical, and conical cavity. FE analysis had been carried out on welding of large structures by Brown and Song [10, 11]. Tian et al. [12] developed a thermal model and studied keyhole formation mechanism in EB welding using Gaussian input heat flux. Some attempts were also made to optimize weld volume and establish input–output relationships. Gunaraj and Murugan [13] minimized weld volume in submerged arc welding process using quasi-Newton method. Vasudevan et al. [14] utilized a genetic algorithm (GA) [15] to achieve the target bead geometry in tungsten inert gas welding by optimizing the process parameters. Correia et al. [16] adopted an approach, where a GA was used as a tool to decide near-optimal settings of a gas metal arc welding process. The search for the near-optimal settings was carried out step by step with the help of a GA predicting the next experiment based on the previous and without using the knowledge of modeling equations among the inputs and outputs of the process. The GA was able to locate near-optimal conditions, with a relatively small number of experiments. Nagesh and Datta [17] developed a back-propagation neural network (BPNN) to establish the relationships between the process parameters and weld bead geometric parameters, in a shielded metal arc welding process. However, EB welding is slightly different from the conventional welding process as discussed earlier. Recently, some studies have been reported on optimization of weld bead geometry [18], bead profile predictions [19], and determination of input–output relationships [20] of EB welding process.

Table 1 Thermal and mechanical properties of Al alloy

T (K) 298 373 473 573 673 773

The present study deals with design and development of a neural network-based expert system capable of predicting temperature distributions on EB-welded plates, online. The training data necessary for developing the ES have been collected through FE analysis by varying the mesh parameters, namely shaping ratio and number of element divisions along the y direction. The neural networks are updated using back-propagation (BP) algorithm and GA, separately. The performances of the trained neural networkbased expert systems have been tested with the help of some other data related to the said process. The remaining part of this paper is organized as follows: Section 2 describes the method of data collection for the present study. The working principles of BP neural network and genetic-neural system are explained in Section 3. Results are stated and discussed in Section 4. Some concluding remarks are made in Section 5.

2 Data collection The aim of the present study is to determine temperature profiles in EB-welded plates. Differential equations related to heat transfer in welding are solved using FE analysis and the data related to temperature profiles are obtained for various combinations of inputs of the FE analysis tool, as discussed below. 2.1 Heat transfer governing equations Let us assume that an electron beam moves in the positive x direction of a large aluminum (Al) alloy plate during the welding. The governing equation for conduction mode of heat transfer can be written as follows: rCp




 @T @ @T @ @T @ @T @ ðΔH Þ ¼ k þ k þ k r @t @x @x @y @y @z @z @x

ð1Þ where x, y, z are the Cartesian coordinates; ρ is the density of the plate material; Cp is the specific heat; k is the thermal conductivity of the plate material; T is the temperature; H=fL L, where L is the latent heat of fusion (389 kJ/kg) and

1 (W/mK)

α (*10−6 K)

C (J/kgK)

E (GPa)

σ0.2 (MPa)

ρ (g/cm3)

180 188.4 180 184.2 184.2 184.2

23.2 23.2 24.3 25 25 25

1,089 1,089 1,172 1,298 1,298 1,298

47 40.8 40 40.2 38.6 32.5

120 107 80 50 25.7 17.3

2.74 2.74 2.74 2.74 2.74 2.74

v 0.33 0.33 0.33 0.33 0.33 0.33

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2500

2.3 Finite element model y=0

Temperature (C)

2000

1500

1000

y=0.8mm y=0.1.7mm

500

y=2.5mm 0 0

2

4 6 Time(s)x0.1

8

10

Fig. 1 Temperature profiles at different y values

fL is assumed to vary linearly with temperature in the mushy zone. The heat input of the electron beam is assumed to have a Gaussian distribution as follows: 0

3h P 3r2 q1 ðx; yÞ ¼ expð Þ; . . . Πa2 a2

ð2Þ

where η΄ is the electron beam energy absorption efficiency of the molten material, which has been kept equal to 0.7; P is the welding power; a represents the radius of the electron qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi beam; r ¼ ðx  vt0 Þ2 þ y2 , where v is the welding speed 0 and t denotes time. During the welding process, the heat supplied by the electron beam is conducted to the edges of the workpiece. Radiation losses will occur, when there are significant temperature differences between the workpiece and environment. The amount of radiation heat transfer can be expressed as follows:   4 q ¼ "l Ts4  T1 ð3Þ

In this simulation, the element type Solid 226 has been considered for the thermomechanical-coupled field analysis. The element has 20 nodes, with up to 5 degrees of freedom per node. Structural capabilities are elastic only and include large deflection and stress stiffening. Thermoelectric capabilities include Seebeck, Peltier, Thomson effects, and Joule heating effects. In addition to thermal expansion, structural thermal capabilities include the Piezocaloric effect in dynamic analyses. The Coriolis effect is available for analyses with structural degrees of freedom. The heat fluxes are calculated and tabulated as array parameters into ANSYS [21] based on the coordinates and further applied on their corresponding surface. In the present EB welding simulation, mapped meshing is used. The mapped meshing concept is valid only in two and three-dimensional problems (no line elements). The solid modal entities (areas and volumes) meshed with this option use quadrilateral area elements or hexahedral (brick) volume elements. In this simulation, number of divisions along the y-axis is varied from 40 to 150 in steps of ten, and for each deviation, spacing ratio is varied from 1 to 5 with an increment of 1 for each step. 2.4 Temperature profiles Electron beam welding is assumed to be carried out at y=0 and along x direction, and the temperature values are determined at y=0, 0.8, 1.7, and 2.5 mm using ANSYS [21] package of FE analysis. The temperature profiles are plotted for the above mentioned values of y by varying the mesh parameters, namely shaping ratio from 1 to 5 and

where ε is the heat emissivity; 1 is the Stefan–Boltzman constant; Ts and T∞ are the surface and ambient temperatures, respectively. 2.2 Material properties Considering the fact that the temperature gradient is quite large around the welding zone and material properties change considerably, temperature-dependent thermal properties are used to increase the accuracy of the heat transfer solution. Thermal and mechanical properties of the Al alloy considered in this study are listed in Table 1 [12], where T represents the temperature; 1 is thermal conductivity; α denotes the coefficient of thermal expansion; C indicates the heat of fusion; E is the Young’s modulus; σ0.2 represents the stress distribution; ρ denotes the density and v is Poisson’s ratio.

Fig. 2 Structure of the neural network

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Fig. 3 Flowchart of the GA–NN system

number of element divisions along the y direction from 40 to 150. Figure 1 shows the temperature profiles. In the present study, the temperature profiles are split into two, namely pre- and post-maximum temperature curves. The profiles before and after the maximum temperature values are fitted to third order polynomial functions as y ¼ a0 þ a1 x þ a2 x2 þ a3 x3 , where a0, a1, a2, and a3 are the coefficients. It is evident that each profile has a unique set of a0, a1, a2, and a3. Using this data set of size equal to 60, the neural networks are trained considering the mesh parameters as inputs, and a0, a1, a2, and a3 as the outputs.

3 Neural networks A neural network consisting of three layers (namely input, hidden and output) of neurons has been considered, as shown in Fig. 2. As there is no guarantee that a neural network with multiple hidden layers will always perform better than that with a single hidden layer, only one hidden layer has been used in the network for simplicity. Two and four neurons are used in the input and output layers to

represent the process parameters and responses, respectively. A parametric study is conducted to obtain the appropriate number of neurons in the hidden layer. A linear transfer function y=x is used in the neurons of input layer, as it is a general practice. The log-sigmoid and tan-sigmoid bx bx transfer functions of the forms y ¼ 1þ"1ax and y ¼ ""bx " þ"bx are utilized for the neurons of hidden and output layers, respectively, where a and b are the coefficients of transfer functions, and x is the input of a neuron. The outputs of the neural network, such as α0, α1, α2, and α3 could be either positive or negative. Therefore, a tan-sigmoid transfer function has been used in the neurons of output layer. The hidden neurons might have either log-sigmoid or tan-sigmoid or any other non-linear transfer functions. Fig. 4 Results of BPNN parametric study. a Average absolute percentb deviation in prediction vs. number of hidden neurons. b Average absolute percent deviation in prediction vs. bias value. c Average absolute percent deviation in prediction vs. momentum constant. d Average absolute percent deviation in prediction vs. learning rate. e Average absolute percent deviation in prediction vs. a. f Average absolute percent deviation in prediction vs. b1. g Average absolute percent deviation in prediction vs. b2. h Average absolute percent deviation in prediction vs. b3. i Average absolute percent deviation in prediction vs. b4

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a

b

c

d

e

Number of hidden neurons = 7

f

Number of hidden neurons = 7

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g

h Number of hidden neurons = 7

i

Number of hidden neurons = 7

Number of hidden neurons = 7

Fig. 4 (continued)

However, hidden neurons with log-sigmoid transfer have been found to yield some interesting results during preliminary runs of the network. The neural network has been trained using BP algorithm and GA, separately.

differences between the network-predicted values and actual values (target values) of the responses. The meansquared error E is determined as follows: L X P 2 1 X 1 Tij  Oij ; L  P i¼1 j¼1 2

3.1 Back-propagation neural network

E¼

A neural network (NN) has been trained using a BP algorithm and following a batch mode of supervised learning [22]. The network is iteratively trained to reduce the mean-squared error. The error has been calculated as the

where Tij and Oij represent the target and predicted values of the responses, respectively; L indicates the number of training cases and P represents the number of outputs of the network. In the present study, a batch mode of training has been used.

ð4Þ

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Table 2 Results of BPNN parametric study y values, mm

Maximum temperature

Parameters of BPNN No. of hidden neurons

Bias (b)

Learning rate (η)

Momentum constant (α΄)

Transfer function coefficients Hidden layer a

0 0.8 1.7 2.5

Pre Post Pre Post Pre Post Pre

7 8 5 7 6 8 5

0.0004 0.0002 0.00065 0.0009 0.00085 0.00075 0.00045

0.15 0.30 0.25 0.40 0.20 0.15 0.45

0.15 0.45 0.30 0.20 0.20 0.25 0.30

4.75 3.25 3.75 4.00 3.50 3.25 3.50

Output layer b1

b2

b3

b4

3.25 2.75 3.25 2.00 4.25 2.75 3.75

3.75 2.25 4.00 2.74 3.50 3.75 4.25

4.50 4.25 4.25 3.50 4.00 4.25 5.25

5.25 5.00 5.25 4.25 3.75 3.50 3.75

Fig. 5 Results of GANN parametric study. a Fitness vs. pm. b Fitness vs. population size. c Fitness vs. maximum number of generations

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Table 3 Results of GANN parametric study

y values, mm

Maximum temperature

Parameters of GANN Pm

0 0.8 1.7 2.5

Pre Post Pre Post Pre Post Pre

To update the weights of the NN, a BP algorithm with a 0 momentum term a has been utilized, as given below: ΔWjk ðtÞ ¼ h

@E ðtÞ þ a0 ΔWjk ðt  1Þ; @Wjk

where η indicates the learning rate; α΄ represents the momentum constant; t denotes the iteration number; @E and @W can be determined using the chain rule of jk differentiation like the following: @E @E @Yk @Uk ¼   ; @Wjk @Yk @Uk @Wjk where Yk and Uk represent the output and input, respectively, of kth neuron lying on the output layer. The NN parameters, such as number of neurons in the hidden layer, learning rate, momentum constant, transfer function coefficients, and bias value are determined with the help of a parametric study. The number of neurons to be present in the hidden layer is varied first, keeping the other parameters fixed at their respective mid-values. Once the number of hidden neurons is decided (corresponding to which, the performance of the network is found to be good), it has been kept unaltered during the study and the second parameter is allowed to vary following the same procedure, and so on. Thus, the optimal/near-optimal values of the above parameters have been determined in stages. A batch mode of training has been provided to the NN with the help of 60 known cases. The performance of the trained neural networks has been tested for ten cases. 3.2 Genetic-neural network The schematic view of the developed genetic-neural network (GANN) approach [22] used in the present work is shown in Fig. 3. The weights, coefficients of activation functions, and bias values are supplied by the GA string, whereas the NN computes the expected outputs. The number of neurons of the hidden layer has been kept the same as that of the

0.001233 0.001533 0.002133 0.001833 0.002433 0.002133 0.000933

Population size

Maximum no. of generations

250 230 190 210 180 270 240

450 380 420 390 440 410 380

BPNN. A typical GA string used in the present work is shown below:

0001……010101110…….010101010…….10010 Network weights Transfer function coefficients

Bias value

Five bits are used to represent each parameter in a GA string. The real values of the parameters are supplied to the neural network to make it ready before passing the training data. The mean-squared error has been calculated utilizing Eq. 4 and used as the fitness value of the GA string. The GA operators, such as tournament selection, uniform crossover, and bitwise mutation are utilized to modify the solutions. The number of hidden neurons has been maintained the same with that of the BPNN, whereas a GA is used to optimize the neural network parameters, namely connecting weights, transfer function coefficients, and bias value. The appropriate GA parameters, such as mutation probability, population size, and maximum number of generations have been obtained through a parametric study. To carry out the said study, the suitable ranges for variation of mutation probability, population size, and number of generations are decided first, and then the first parameter, namely mutation probability is varied keeping the other two parameters fixed at their respective mid-values. The value of mutation probability, thus obtained, is kept unaltered, and the second parameter, namely population size is varied after keeping the third parameter fixed at its mid-value. This method is continued to decide optimal/near-optimal values of the said parameters. It is to be noted that a uniform crossover with probability equal to 0.5 has been utilized in this study. The performance of the optimized network has been tested on the same set of ten cases.

4 Results and discussion The maximum temperature and its profile have been predicted for different values of y using both the BPNN as well as GANN as discussed below.

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Fig. 6 Temperature profile of the first test case

4.1 Predictions of temperature profiles at y=0

Fig. 8 Percent deviation in prediction of maximum temperature for the post-maximum temperature curve at y=0

The performance of BPNN depends on the quality and quantity of data used in training. It is also dependent on its architecture, connecting weights, learning rate, momentum constant, coefficients of transfer functions, and bias value. To determine an optimal set of the above parameters, a study has been carried out by varying one parameter at a time and keeping the other parameters unaltered. Figure 4 shows the results of the parametric study conducted to determine the optimal network. In the first stage, the

number of neurons in the hidden layer is varied in the range of two to eight, keeping the other parameters, viz. learning rate η, momentum constant α΄, coefficient of transfer function of the hidden neurons a, coefficient of transfer function of the output neurons: b1, b2, b3, and b4, and bias value fixed to 0.5, 0.5, 3.0, 3.0, 3.0, 3.0, 3.0, and 0.0005, respectively. It is interesting to note that the NN with seven neurons lying in its hidden layer has showed the best performance in terms of percent deviation in predictions of the temperature coefficients. Thus, in the second stage and onwards, the number of hidden neurons is kept fixed at seven. In a similar way, the optimal/near-optimal values of η, α΄, a, b1, b2, b3, b4, and bias values are determined in stages. As the

Fig. 7 Percent deviation in prediction of maximum temperature for the pre-maximum temperature curve at y=0

Fig. 9 Average absolute percent deviation in prediction of the temperature profiles for the pre-maximum temperature zone at y=0

Results of both BPNN and GANN have been stated and discussed below. 4.1.1 Results of BPNN

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Fig. 10 Average absolute percent deviation in prediction of the temperature profiles for the post-maximum temperature zone at y=0

Fig. 12 Percent deviation in prediction of maximum temperature for the post-maximum temperature curve at y=0.8 mm

Bias value=0.0004 optimal parameters are determined in stages (but not simultaneously), there is no guarantee that the obtained network will be globally optimal but it could be a nearoptimal one. The parameters of the optimal network are found to be as follows: Number of neurons in the hidden layer=7 Learning rate η=0.15 Momentum constant, α΄=0.15 Coefficient of transfer function of the hidden neurons, a=4.75 Coefficient of transfer function of the output neurons: b1, b2, b3, and b4 = 3.25, 3.75, 4.50, and 5.25, respectively

Fig. 11 Percent deviation in prediction of maximum temperature for the pre-maximum temperature curve at y=0.8 mm

The similar study has been carried out for various pre- and post-maximum temperature curves at different values of y, and the results have been shown in Table 2. It is important to mention that for y=2.5 mm, no significant change in temperature distribution is observed in post-maximum temperature region. Thus, the temperature after reaching its maximum value is found to remain almost constant. 4.1.2 Results of GANN In the proposed genetic-neural system (GANN), the number of hidden neurons has been kept fixed to seven (as found to

Fig. 13 Average absolute percent deviation in prediction of the temperature profile for the pre-maximum temperature zone at y=0.8 mm

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Fig. 14 Average absolute percent deviation in prediction of the temperature profile for the post-maximum temperature zone at y=0.8 mm

be optimal in the previous approach on BPNN). The other parameters, like the coefficients of transfer functions of the neurons lying on the hidden and output layers and bias value are varied within their respective ranges, during optimization (training). As the performance of a GA depends on its parameters, such as crossover probability (pc), mutation probability (pm), population size and maximum number of generations, an extensive study is conducted to determine the optimal set of GA parameters. A uniform crossover scheme with a probability of 0.5 is used and the optimal values of other GA parameters are decided through a careful study. The probability of mutation pm is varied in the range of

Fig. 15 Percent deviation in prediction of maximum temperature for the pre-maximum temperature curve at y=1.7 mm

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Fig. 16 Percent deviation in prediction of maximum temperature for the post-maximum temperature curve at y=1.7 mm

0.1/L to 1.0/L (where L indicates the string length), after keeping the other parameters, namely population size and maximum number of generations fixed to their respective mid-values. The optimal value of pm is, thus, determined. The similar procedure is adopted for obtaining the optimized values of population size and maximum number of generations, one after another. Results of the GA parametric study are shown in Fig. 5. The following parameters are found to yield the best performance of the GA. Mutation probability, pm =0.001233 Population size=250 Maximum number of generations=450

Fig. 17 Average absolute percent deviation in prediction of the temperature profiles for the pre-maximum temperature zone at y=1.7 mm

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y ¼ 1; 030:55 þ 19; 769:36x  93; 718:39x2 þ 160; 820:06x3 . . .

ð6Þ y ¼ 1; 057:40 þ 20; 233:91x  96; 029:14x2 þ 164; 224:89x3 . . .

ð7Þ For this test case, the post-maximum temperature curves are represented by Eqs. 8, 9, and 10 as obtained by the ANSYS, BPNN, and GANN, respectively. y ¼ 36; 821:88  179; 566:80x þ 298; 386:91x2  164; 985:38x3 . . .

ð8Þ

y ¼ 37; 208:29  181; 653:69x þ 302; 076:19x2  167; 123:70x3 . . .

ð9Þ

Fig. 18 Average absolute percent deviation in prediction of the temperature profiles for the post-maximum temperature zone at y=1.7 mm

y ¼ 37; 985:48  185; 877x þ 309; 572:19x2  171; 481:6x3 . . .

The similar study is conducted for various pre- and postmaximum temperature curves at different values of y, and the results are shown in Table 3. The performances of trained neural networks have been tested for ten cases. A unique set of the coefficients: a0, a1, a2, and a3 have been obtained for each test case using the ANSYS FE package, BPNN, and GANN. Equations 5, 6, and 7 have been obtained through the ANSYS, BPNN, and GANN, respectively, as the pre-maximum temperature curves for a test case, as given below. y ¼ 1; 045:89 þ 20; 031:95x  95; 030:80x2 þ 162; 749:91x3 . . .

ð10Þ The corresponding temperature profiles determined by the above three approaches are shown in Fig. 6. The results obtained through the BPNN and GANN are compared with the ANSYS results. The values of percent deviation in prediction of maximum temperature and average absolute percent deviation in predictions of the profile are calculated. Figures 7 and 8 show the values of percent deviation in predictions of maximum temperature for the pre- and post-maximum temperature curves,

ð5Þ

Fig. 19 Percent deviation in prediction of maximum temperature for the pre-maximum temperature zone at y=2.5 mm

Fig. 20 Average absolute percent deviation in prediction of the temperature profiles for the pre-maximum temperature zone at y=2.5 mm

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respectively, at y=0. Moreover, Figs. 9 and 10 display the values of average absolute percent deviation in predictions of the temperature curves at y=0 for all the test cases. The values of average absolute percentage of deviation in predictions of the temperature profiles for the pre- and post-maximum temperature curves are found to be equal to 0.812% and 0.833%, respectively, by the BPNN; whereas, these values are seen to be equal to 0.785% and 0.628%, respectively, by the GANN, after considering all the test cases. Thus, GANN has performed better than the BPNN. It may be due to a more robust and exhaustive search carried out by the GA in comparison with the BP algorithm. 4.2 Predictions of temperature profiles at y=0.8 mm A parametric study is carried out as discussed above for each of these two approaches. The optimal values of the parameters for the BPNN and GANN are found to be as displayed in Tables 2 and 3, respectively. The trained neural network has been tested for ten cases as mentioned above. A unique set of the coefficients: a0, a1, a2, and a3 have been obtained for each test case. Results of the BPNN and GANN are compared with those of the ANSYS. Figures 11 and 12 show the values of percent deviation in predictions of maximum temperature for the pre- and post-maximum temperature curves, respectively, at y=0.8. Figures 13 and 14 display the values of average absolute percent deviation in predictions of the temperature curves. The BPNN has yielded the values of average absolute percent deviation in predictions for the pre- and postmaximum temperature zones as 6.94% and 6.35%, respectively, whereas the respective values are obtained by the GANN as 5.56% and 3.77%, respectively, after considering all the test cases. Thus, GANN has outperformed the BPNN and the reason behind it has been explained earlier. 4.3 Predictions of temperature profiles at y=1.7 mm At y=1.7 mm, the temperature profiles have been predicted using the BPNN and GANN, whose optimized parameters are shown in Tables 2 and 3, respectively. Figures 15 and 16 display the values of percent deviation in predictions of maximum temperature for the pre- and post-maximum temperature curves, respectively, as obtained by the BPNN and GANN. The values of average absolute percent deviation in predictions of the temperature profiles for the pre- and post- maximum temperature zones are shown in Figs. 17 and 18, respectively, as determined by the BPNN and GANN for ten test cases. The average of ten such values has been calculated for both the BPNN- and GANN-predicted results. For the premaximum temperature zone, it is found to be equal to 7.504% and 6.213% for the BPNN and GANN, respective-

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ly; whereas, for the post-maximum temperature zone, it has been obtained as 4.207% and 3.573% by the BPNN and GANN, respectively. Thus, the performance of GANN is seen to be better than that of BPNN, and it has happened so, due to the reason mentioned earlier. 4.4 Predictions of temperature profiles at y=2.5 mm At y=2.5 mm, no significant change in temperature has been observed in the post-maximum temperature zone. Thus, results related to pre-maximum temperature zone have been reported here. Figures 19 and 20 display the values of percent deviation in prediction of maximum temperature and those of average absolute percent deviation in predictions of the temperature profiles, as obtained by the BPNN and GANN. Considering all the test cases, the values of average absolute percent deviation in predictions of temperature profiles have been calculated and are found to be equal to 7.521% and 5.419% for the BPNN and GANN, respectively. It is clearly evident from the above results that GANN has yielded better results compared to BPNN. It might have happened due to the reasons that the solutions of BP algorithm may get stuck at the local minima, whereas the chance of GA solutions for getting trapped into local minima is less. 4.5 Comparison of computation time Time taken for each of ten test cases has been determined to predict the temperature profiles using the ANSYS package, BPNN and GANN. The ANSYS package has taken an average of 2.9 h as CPU time to predict the temperature profile, whereas optimized BPNN and GANN are found to take 0.002 s only for the same. Thus, both the trained BPNN and GANN can be used to predict the temperature profiles, online.

5 Conclusions In this work, predictive tools for determining temperature profiles of EB welding have been developed. The meshing parameters, namely number of element divisions and spacing ratio of finite element analysis are varied in the model to obtain different temperature profiles. Neural network models have been developed for the predictions of pre- and post-maximum temperature profiles. The following conclusions have been made form this study: &

Both the neural network-based expert systems are able to predict the temperature profiles in EB welding accurately. However, GANN has outperformed the

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BPNN in terms of average absolute percent deviation in predictions of temperature profiles. It has happened so, as a GA-based search is used in the former approach, in place of a gradient-based search of the latter approach. Being a gradient-based search, the BP algorithm has a tendency to get stuck at the local minima, whereas a more exhaustive search is carried out by the GA in GANN approach. Thus, the chance of GA solution for being trapped into the local minima is less. &

Both the trained BPNN and GANN approaches are seen to predict the temperature profiles in a fraction of a second, whereas the ANSYS package is found to take about 3 h time to obtain the said temperature profiles in a P-IV PC. Thus, neural network-based expert system proposed in this paper, may be an efficient tool for making online predictions of temperature profiles in EB welding.

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