neural network model of heat and fluid flow in gas metal arc fillet … · 2006. 8. 31. · neural...

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Neural network model of heat and fluid flow ingas metal arc fillet welding based on geneticalgorithm and conjugate gradient optimisation

A. Kumar*1 and T. DebRoy

Although numerical calculations of heat transfer and fluid flow can provide detailed insights into

welding processes and welded materials, these calculations are complex and unsuitable in

situations where rapid calculations are needed. A recourse is to train and validate a neural

network, using results from a well tested heat and fluid flow model to significantly expedite

calculations and ensure that the computed results conform to the basic laws of conservation of

mass, momentum and energy. Seven feedforward neural networks were developed for gas metal

arc (GMA) fillet welding, one each for predicting penetration, leg length, throat, weld pool length,

cooling time between 800uC and 500uC, maximum velocity and peak temperature in the weldpool. Each model considered 22 inputs that included all the welding variables, such as current,

voltage, welding speed, wire radius, wire feed rate, arc efficiency, arc radius, power distribution,

and material properties such as thermal conductivity, specific heat and temperature coefficient of

surface tension. The weights in the neural network models were calculated using the conjugate

gradient (CG) method and by a hybrid optimisation scheme involving the CG method and a

genetic algorithm (GA). The neural network produced by the hybrid optimisation model produced

better results than the networks based on the CG method with various sets of randomised initial

weights. The CG method alone was unable to find the best optimal weights for achieving low

errors. The hybrid optimisation scheme helped in finding optimal weights through a global search,

as evidenced by good agreement between all the outputs from the neural networks and the

corresponding results from the heat and fluid flow model.

Keywords: Neural network, Gas metal arc welding, Optimisation, Fillet welds

IntroductionSince the geometry, composition and structure of weldsare affected by the welding variables, these variables areoften adjusted by trial and error to achieve defect-free,structurally sound and reliable welds. However, thisapproach is time consuming and expensive, and does notalways provide optimum welds. Attempts have beenmade to obtain systematic correlations between weldingvariables and weld characteristics by the use of statisticalregression analysis,1 artificial neural networks1–13 andphenomenological modelling.14–24 In principle, regres-sion analysis can relate weld pool geometry to weldingvariables with the use of a large volume of experimentaldata. However, this approach is difficult for gas metalarc (GMA) fillet welding, due to complex interactionsbetween various physical processes, where each variable

affects the weld pool dimensions, cooling rate and otherparameters in a complex manner.17–20 In recent years,numerical models of heat transfer and fluid flow infusion welding have provided significant quantitativeinsights into both the welding processes and the weldedmaterials. The computed temperature and velocityfields, cooling rates, weld pool geometry for variousconcentrations of surface active elements, concentra-tions of oxygen, nitrogen and hydrogen, and formationof defects, have been studied quantitatively usingnumerical models. Although these models are recognisedas powerful tools for research, they are not extensivelyused in the welding industry, because the models arehighly complex, require specialised training to developand test, and consume a large amount of computer time.

The neural network models are capable of relatinginput variables such as welding process parameters andmaterial properties to weld characteristics such as theweld geometry1–5 and properties. Previous efforts tomodel the GMA fillet welding process using neuralnetworks were based on training the network withexperimental data. Since the volume of experimental

Department of Materials Science and Engineering, The PennsylvaniaState University, University Park, PA, USA

*Corresponding author, email [email protected]

� 2006 Institute of Materials, Minerals and MiningPublished by Maney on behalf of the InstituteReceived 12 August 2005; accepted 13 October 2005DOI 10.1179/174329306X84319 Science and Technology of Welding and Joining 2006 VOL 11 NO 1 106

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data required to train a neural network depends on thenumber of input and output variables, most previousworkers considered only few input parameters to keepthe necessary volume of experimental data tractable.1–4

For example, Kim et al.,1 Smartt and Johnson,2 Cooket al.3 and Li et al.4 developed neural network modelsof the GMA welding process in which the effects of pro-cess parameters such as welding speed, arc voltage andarc current were considered as inputs. Since the weldpool geometry depends on other welding variables, aswell as various material properties, the effects of manyof the welding variables and material properties cannotbe determined from the available neural networks.Furthermore, the output variables used in the previousneural networks were also limited. For example, theexisting neural network models do not provide anyinformation about some of the important parameters,such as the cooling rate and peak temperature. A reviewof previous work indicates that what is needed is aframework for rapid calculation of weld pool geometry,cooling rate and peak temperature in the weld pool forGMA welding of various materials.

A neural network trained with the results of anumerical heat transfer and fluid flow model cancorrelate various output variables, such as the weldpool geometry, cooling rate, liquid velocities and peaktemperatures, with all the major welding variables andmaterial properties. Furthermore, such correlationssatisfy the phenomenological laws. With the improve-ments in computational hardware in recent years, a largevolume of training and validation data can be generatedwith a well tested numerical heat transfer and fluid flowmodel in a realistic time frame. The weights in the neuralnetwork were calculated using a hybrid optimisationscheme involving the conjugate gradient (CG) methodand a genetic algorithm (GA). The neural networkproduced by the hybrid optimisation model producedbetter results than the networks based on the CGmethod. This paper seeks to document the problems,issues and lessons learnt in the development of a neuralnetwork model from the results of a heat transfer andfluid flow model that considers all the major inputvariables, including all important process variables andmaterial properties, and correlates them with outputvariables.

Mathematical model

Heat transfer and fluid flow model of GMA filletwelding to generate databaseThe datasets for training, validation and testing of theneural network were generated by using a well testedheat transfer and fluid flow model that solves theequations of conservation of mass, momentum andenergy in three dimensions. Because of the complexity ofGMA fillet welding, the following simplifying assump-tions15,16,18,20 are made to make the computational worktractable:

(i) the thermophysical properties needed for calcu-lations such as the thermal conductivity andspecific heat of the workpiece material areassumed to be constant for simplicity

(ii) the liquid metal flow is assumed to beincompressible and Newtonian. The effect ofturbulent flow in the weld pool is taken into

account through the use of the enhancedthermal conductivity and viscosity of the liquidmetal15,16

(iii) the heat transported from the filler metaldroplets is taken into account using a timeaveraged volumetric heat source14,15,17,25

(iv) both the heat and current flux from the arc areassumed to have a Gaussian distribution at theweld top surface.14,21 For example, heat flux isdefined by

qr~gIV

ps2jexp {

dr2

r20

!

where qr is the heat flux, g is the arc efficiency, Iis the current, V is the voltage, d is the arcdistribution factor, r is the radial distance fromthe arc location, and r0 is the radius of the arc.The distributions of heat flux and currentdensity are assumed to be unaffected by thedeformation of the weld pool top surface.14,15

The thermofluid model takes into account the complexfillet joint shape, the deformation of the weld pool topsurface, heat transfer by the hot metal droplets and theaddition of the filler metal from the consumableelectrode.15,16,18,24 The calculations require the use of adeformable, curvilinear grid system for accurate calcula-tion of heat transfer and fluid flow. Therefore, thegoverning equations are transformed from the Cartesianto the curvilinear coordinate system.14,15 The discretisedequations for enthalpy, three components of velocityand pressure are formulated by integrating the corre-sponding governing equations over all the interiorcontrol volumes in the computational domain.26 Theseequations were solved to obtain the temperature andvelocity fields using the modified SIMPLE algorithm.26

Subsequently, the free surface profile of the weld poolwas calculated based on the temperature field obtainedin the previous step. After the solution of the free surfaceprofile, the z locations of grids were adjusted to fit thesurface profile, and the temperature and velocity fieldswere then recalculated in the fitted grid system. Thecalculation procedure was repeated until temperatureand velocity fields converged. More details of thenumerical model of heat transfer and fluid flow ofGMA fillet welding are available in the literature14–20,24

and are not repeated here. The original fillet geometryis transformed into a rectangular domain of length450 mm, width 108 mm and depth 18 mm by using anappropriate coordinate transformation.14,15 A 72666647 grid system with finer grids near the heat source wasused for maximum resolution of variables.

Neural network modelAll the 22 input variables and their range of values usedto develop the neural network are listed in Table 1.While most of these variables are easily understood, thefollowing comments may be of interest. The inputvariables such as arc efficiency, arc power distributionfactor and arc radius determine how heat is absorbed atvarious locations from the arc.21 The droplet effi-ciency15–18 is defined as the ratio of the total sensibleheat input due to metal droplets Qt and the total heatinput IV, i.e. gd5Qt/(IV), where I is arc current (A)and V is voltage (V). Since temperature independentthermophysical properties of the solid alloy are used in

Kumar and DebRoy Neural network model of heat and fluid flow

Science and Technology of Welding and Joining 2006 VOL 11 NO 1 107

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the model, a question arises as to how to select theirvalues. Since the heat flow in the solid region near theweld pool affects both the size and shape of the weldpool as well as the temperature field in the entireworkpiece, it is appropriate to use thermophysicalproperties at a temperature closer to the melting pointthan to the ambient temperature. The effective thermalconductivity and viscosity are used to representenhanced heat and momentum transfer within the weldpool because of the fluctuating components of velocitiesin a strong recirculating flow confined in a small weldpool. These variables represent system properties, andtheir values, determined by reverse modelling, areavailable in the literature17–20 for GMA fillet weldingfor various heat input values. The effects of surfaceactive elements have not been rigorously studied in thedevelopment of the neural network, to simplify calcula-tions. Therefore, the results are valid for low concentra-tions of surface active elements.

All the important output variables from the modelincluded in the neural network are listed in Table 2. Thethree output variables describing the weld cross-sectionare penetration, leg length and throat. These are shownin Fig. 1. In addition, the length of the weld pool isconsidered as an output of the neural network, as shownin Table 2. Other outputs of the neural network modelinclude the peak temperature in the weld pool,maximum velocity in the weld pool and cooling timebetween 800uC and 500uC. The cooling time wascalculated on the workpiece surface along the weldingdirection.

Figure 2 shows the structure of each neural network,which contains an input layer, a hidden layer and anoutput layer. The input layer comprises all of the 22input variables, which are connected to neurones in thehidden layer through the weights assigned for each link.The number of neurones in the hidden layer is found byoptimising the network. In mathematical terms, wedescribe the output from a neurone y at each node(represented by circles in Fig. 2) as follows27–29

y~fXNi~0

wixi

!(1)

Table 2 Root mean square error for different output variables

Output variable Training data Validation data Testing data

Leg length, mm 0.01 0.02 0.14Actual throat, mm 0.02 0.18 0.29Penetration, mm 0.02 0.08 0.11Length of the weld pool, mm 0.16 0.18 0.19Peak temperature in the weld pool, K 4.27 4.58 4.82Cooling time between 800uC and 500uC, s 0.04 0.06 0.06Maximum velocity in the weld pool, mm s21 2.27 2.53 1.85

1 During the GMA fillet welding process, the weld geo-

metry is commonly specified by three quantities that

affect the joint properties: penetration, throat and leg

length

Table 1 The range of input variables used in the generation of training, validation and testing datasets

Variables Minimum Maximum Mean

Arc current, A 200.0 410.0 326.3Arc voltage, V 25.0 42.0 33.8Welding speed, mm s21 4.2 8.5 6.4Wire feed rate, mm s21 120.0 290.0 199.6Arc efficiency 0.4 0.7 0.5Arc radius, mm 4.0 6.5 5.1Arc distribution factor 0.5 3 1.4Droplet efficiency 0.1 0.2 0.13Contact tube to workpiece distance, mm 17.5 30.0 23.4Wire radius, mm 0.5 0.9 0.7Effective thermal conductivity, J m21 s21 K21 83.6 543.4 298.5Effective viscosity, kg m21 s21 2.061022 21.061022 7.961022

Density, kg m23 7000.0 8500.0 7742.1Solidus temperature, K 1690.0 1790.0 1741.7Liquidus temperature, K 1745.0 1815.0 1784.6Enthalpy of solid at melting point, kJ kg21 731.5 1149.5 1002.4Enthalpy of liquid at melting point, kJ kg21 1045.0 1463.0 1280.2Specific heat of solid, J kg21 K21 543.4 794.2 677.0Specific heat of liquid, J kg21 K21 689.7 919.6 789.7Thermal conductivity of solid, J m21 s21 K21 14.6 40.5 26.9Coefficient of thermal expansion, 1/K 0.0 1.761025 9.161026

dc/dT of alloy without any surface active element, N m21 K21 25.561024 22.561024 24.261024



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where x1, x2, x3, …, xN are the input signals to theneurone and w1, w2, w3, …,wN are the synaptic weightsthat embody the non-linear relationships betweenthe input and the output variables. The combinationof a fixed input x051 and an extra input weight w0accounts for the bias input. The activation function,denoted by f, captures the non-linear interactions ofvarious welding variables such as the arc current,voltage, wire feed rate, welding speed and materialproperties such as weld geometry, cooling rate and peaktemperature in the weld pool during GMA fillet welding.The following hyperbolic tangent function (which is asymmetrical sigmoid function) is used as the activationfunction

y~tanh aXNi~0

wixi

!(2)

where a is the slope parameter of the sigmoid function.By varying the parameter a, we can obtain sigmoidfunctions of different slopes.27–29 An increase in thevalue of a increases the slope of the activation functionand vice versa. A very high value of the slope makes thecurve close to a step function, while a low value retardsthe convergence rate. Based on the findings of previousstudies, a value of 1.5 was used to achieve rapidconvergence.29,30 Furthermore, the use of the tanhfunction in equation (2) as the activation function helpsin keeping the problem reasonably well conditioned. Anattractive feature of the hyperbolic tangent function isthat its derivative, given by f9512f2, does not increasecomputational volume significantly.27–29

To find the weights W, a modified back propagationalgorithm31 is used for the training of these neuralnetworks.27–29 The algorithm tries to minimise the

objective function, i.e. the least square error betweenthe predicted and the target outputs, and is given by

E~1

2

Xp

d (p)o �y(p)o� �2

(3)

where p represents the number of training datasets and orepresents the number of output nodes, which is 1 inthis work. The desired outputs of the network, such asweld penetration, leg length, throat, cooling rate andpeak temperature, are dependent on input weldingconditions, material properties and network parameterssuch as the weights. The working procedure of the backpropagation algorithm is explained in Appendix A. Thebasic or the original back propagation algorithm adjuststhe weights in the steepest descent direction (negative ofthe gradient).27–29,31 This is the direction in which theerror decreases most rapidly. Since this algorithmrequires a learning rate parameter to determine theextent to which the weights change in an iteration, i.e.the step sizes, its performance depends on the choice ofthe value of the learning rate.27–29,31 A slight modifica-tion of the back propagation algorithm includes amomentum term. The momentum term ensures that theprevious changes in the weights are considered indetermining the current direction of changes of weights.Although there is some guidance for the selection ofthese parameters, they are more oriented towardsspecific problems such as pattern recognition, and theirperformance varies with the type of problem.27–29,31

Owing to difficulty in the selection of the learning rateparameter and momentum term, the original backpropagation algorithm was modified by replacing thesteepest gradient method with the CG method foroptimising the weights.31

2 Architecture of the neural network model used in this work. The input layer comprises 22 variables, which are con-

nected to a hidden layer. The output of the network is either penetration, leg length, throat, weld pool length, cooling

time between 800uC and 500uC, maximum velocity or peak temperature in the weld pool



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Modified back propagation algorithmIn the CG method,17,18,20,27,31 the weights are updatedafter each iteration, based on the error calculatedusing equation (3) for the entire training dataset. Forminimising the objective function, the method requirescalculation of the step size or change in weights and thesearch direction. The step size is calculated at eachiteration by using Brent’s algorithm,27–29,31 whereasthe search direction is calculated by conjugating theprevious direction with the current gradient of theobjective function. Brent’s iterative line search algo-rithm19,24 utilises the parabolic interpolation andGolden section search method27–29,31 to locate the lineminima. At the first iteration, since there is no previousdirection, the search is performed in the direction ofthe steepest descent. The directions of search atsubsequent iterations are calculated by conjugating theprevious direction with the current gradient. Theseconjugate directions are actually calculated on theassumption that the error surface is quadratic.27–29

However, if the algorithm discovers that the currentline search direction is not downhill, it simply calculatesthe line of steepest descent and restarts the search inthat direction. Once a point close to a minimum isfound, the quadratic assumption holds true and theminimum can be located very quickly using the Goldensection search method.27–29,31 The steps involved inthe calculation are as follows.22,23,28,31 A line searchis performed to determine the optimal distance tomove along the current search direction. If we let pkdenote the direction vector at iteration k of thealgorithm, then the weights are updated using thefollowing rule27–29,31

Wkz1~WkzDWk~Wk�gkLELW

� �0

~Wkzgkpk (4)

where gk is the learning rate or search step sizecalculated at each iteration by using Brent’s method.32,33

For the first iteration, the initial direction p0 is the sameas the steepest descent direction (negative of thegradient)

p0~� g0~�LELW

� �0

(5)

where g is the gradient of error with respect to the weightof the link connecting any two consecutive layers. Thenthe next search direction is determined so that it isconjugate to previous search directions. The generalprocedure for determining the new search direction is tocombine the new steepest descent direction with theprevious search direction22,23

pk~� gkzbkpk�1 (6)

The various versions of CG are distinguished by themanner in which the constant bk is computed. For thePolak–Ribiére update, the constant bk is computedfrom17,18,20,22,23,32 the following equation

bk~gk�gk�1ð ÞTgkgTk�1gk�1

(7)

where T refers to transposition of the matrix, i.e. rowschanged to columns. This is the product of the previouschange in the gradient, with the current gradient dividedby the norm squared of the previous gradient. The

gradient between the hidden node h and the output nodeo is given by27–29,31

g~goh~LELwoh

~�Xp

d(p)o y(p)h (8)

d(p)o ~ d(p)o �y(p)o � f 0 x(p)o

� �(9)

�where y

(p)h and y

(p)o represent the output at hidden and

output nodes, respectively, and x(p)o is the input. For theinput-to-hidden connections, the gradient betweenhidden node h and input node i is given by27–29,31

ghi~LELwhi

~�Xp

d(p)h y(p)i (10)

d(p)h ~f

0½x(p)h �Xo

d(p)o woh (11)

The value of output nodes o is 1 in equations (8), (9) and(11), since only one output is produced by each of theneural networks. In the gradient descent algorithms,calculations are started at some point on the errorfunction defined over the weights, and an attempt ismade to move to the global minimum of the function.The gradient based methods can easily get trapped inlocal minima. Stochastic optimisation techniques arecapable of finding the global minima and avoiding localminima.34 Therefore, a genetic algorithm (GA)34 is usedalong with the CG method to find the optimal globalweights in the present work. A parent centric recombi-nation (PCX) operator based generalised generation gap(G3) GA model24,35–38 was used in the present study.The PCX operator is a steady state, elite preserving,scalable and computationally fast population alterationmodel.35 This model was chosen because it has beenshown to have a higher convergence rate on standardtest functions as compared to other evolutionaryalgorithms and classic optimisation algorithms, includ-ing other real parameter GAs with the unimodal normaldistribution crossover (UNDX) and the simplex cross-over (SPX) operators, the correlated self-adaptiveevolution strategy, the covariance matrix adaptationevolution strategy (CMA-ES), the differential evolutiontechnique, and the quasi-Newton method.35 Detaileddescription of this model is available in the litera-ture24,35–38 and is not included here. The variousterms used to describe GA are explained in Table 3.Specific application of GA for determination of theoptimised weights of the neural network is given inAppendix B.

Calculation procedure

Number of hidden layers in the networkA large database is required for training, validation andtesting of the neural network. The number of trainingdatasets should be more than the number of weightsconnecting different nodes. The number of hidden layersin a neural network depends on the type of the problemand the relationships between the input and the outputvariables. Theoretically, any continuous variation ofoutput with respect to input can be represented by asingle hidden layer.39,40 Two hidden layers are neededwhen the relationships between the input and the outputvariables are discontinuous.39,40 The use of more than



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the optimal number of hidden layers in the network mayresult in undesirable overfitting of the data.27–30,39,40 Asingle hidden layer was used, since the outputs arecontinuous in nature in GMA fillet welding. For a singlehidden layer network, the number of weights is given bythe following

Number of weights in the network

~(niz1)nhz(nhz1)no (12)

where ni is the number of input variables, i.e. 22 in thepresent work, nh is the number of nodes in the hiddenlayer, and no is the number of output variables, i.e. 1.Since the number of weights increases with the increasein the number of nodes, an optimal number of nodes isrequired.

Database generation based on design ofexperimentsA database for training of the neural networks wasgenerated on the basis of the design of experiments tocapture the effects of all the welding parameters andmaterial properties.41 The original L81 (3

40) orthogonalarray contains 81 rows and 40 columns of variables,each having three levels of values for capturing theinteractions among all 40 variables.41 However, in thecase of GMA fillet welding, we have 22 input variables.In the L81 (3

40) array, only 22 columns are used and theremaining 18 remain vacant. Since only three levels ofweld process parameters are not capable of capturingthe complex interactions among variables, the array ismodified to L81 (9

66316) based on linear graph theory,41

as explained in Appendix C. In the modified array, thesix most important variables, current, voltage, weldingspeed, wire feed rate, effective thermal conductivity andeffective viscosity, are assigned nine levels, and theremaining 16 variables are assigned three values each.The rest of the variables, such as the efficiency, arcradius, arc power distribution factor, contact tube toworkpiece distance (CTWD) and material properties,such as density, specific heat of the solid, specific heat ofthe liquid and latent heat, are kept at three levels. Thisprocedure increases the degrees of freedom and helps tocapture the effects of the variables, which have a largeinfluence on the weld geometry and cooling rate. TwoL81 (9

66316) arrays for each of the Fe 1005, Fe 1045and A36 steels were used. For each steel, we used twodifferent sets of nine levels of values of current, voltage,welding speed, wire feed rate, effective thermal con-ductivity and effective viscosity to capture the inter-actions. Similarly, material properties and otherparameters such as arc efficiency, arc radius, arc power

distribution factor and CTWD are different in eacharray. Therefore, we have 18 levels of all the inputvariables in the form of six L81 (9

66316) arrays, or 486datasets for training. For validation and testing of theneural network, an additional 50 and 25 differentdatasets were generated using three-dimensional heattransfer and a fluid flow model. These datasets forvalidation and testing were generated randomly byselecting the values of variables that are different fromthe training dataset. The ranges of all the 22 inputvariables used for the generation of datasets are shownin Table 1. The different levels or values of the variablesare decided based on their sensitivity to weld geometry.In the database, variables such as arc current, arcvoltage, welding speed, wire feed rate, effective thermalconductivity and effective viscosity, which have a majorinfluence on weld geometry, are taken at many levelscompared to the other remaining variables, as shown inFig. 3. The material properties in the database wereselected around their corresponding values for Fe 1005steel, Fe 1045 steel and A36 steel.

Normalising inputs and outputsThe values of the input and output variables varysignificantly. The vastly different scales of inputs andbias values lead to ill-conditioning of the problem.27–29

Whereas large inputs cause ill-conditioning by leading tovery small weights, large outputs do so by leading tovery large weights.27–29 To eliminate the ill-conditioningproblem, the data were normalised using the followingformula29

x0~2x� xminxmax�xmin

� ��1 (13)

where x is the original value of the variable, x9 is thenormalised value, and xmin and xmax represent theminimum and maximum values of the variable inthe whole dataset. Equation (13) normalises the datain the range21 to 1. The range of values of all input andoutput parameters from 21 to z1 implies that thestandard deviation cannot exceed 1, and its symmetryabout zero means that the mean will typically berelatively small. Furthermore, its maximum derivativeis also 1.5, so that back propagated errors will be neithermagnified nor attenuated more than necessary.29

Selection of initial weightsIn the back propagation algorithm, the magnitude of theerror propagated backwards through the network isproportional to the value of the weights. If all theweights are the same, the back propagated errors will be

Table 3 Terminology used in genetic algorithm

Biological termsEquivalent neural network variables and representationin genetic algorithm

Genes: units containing hereditary information In the form of weights of the network variablesw1, w2, …, wn, e.g. w1520.10; w250.17; wn50.26

Chromosome/individual: a number of genesfolded together

A set of values of weights taken together,i.e. (20.10, 0.17, …, 0.26)

Population: collection of many chromosomes/individuals Collection of multiple sets of weights:(20.10, 0.17, …, 0.26), (0.15, 0.27, …, 0.24), (0.33, 20.14, …, 0.43)

Parents: chromosomes/individuals participating increating new individuals (or offspring)

Parents, e.g. (20.10, 0.17, …, 0.26), (0.15, 0.27, …, 0.24)

Fitness value: value of fitness function determines if achromosome/individual survives or dies

Objective function or fitness function:calculated for each set of input variables using equation (3)



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the same, and consequently all of the weights will beupdated by the same amount.27–29 To avoid this sym-metry problem, the initial weights of the network wereselected randomly. Furthermore, to avoid the prematuresaturation of the network, the initial values of theweights were distributed inside a small range of values,i.e. in the interval 20.5 to 0.5. When the weights aresmall, the neurones operate in the linear regions of theactivation function, and consequently the activationfunction does not saturate.

The calculation starts with the selection of the numberof nodes in the hidden layer. The total number ofweights in the network is calculated on the basis of thenumber of nodes in the hidden layer. The weights arethen initialised randomly in the interval 20.5 to 0.5, asdescribed in the previous section. In the next step, amodified back propagation algorithm is used to mini-mise the error on the training dataset. The weightscalculated by the CG method are stored as one possibleset of weights. This process is repeated 10 times withdifferent randomly selected initial weights for fixedvalues of nodes in the hidden layer. All of these 10 sets ofweights are provided as input to the GA. The final aimof the GA is to find the weights in the network through asystematic global search that will give the least errorbetween the neural network prediction and heat transferand fluid flow calculations. The flowchart of thecalculation scheme is presented in Fig. 4. The con-vergence is based on the error in training and valida-tion data. When the error during validation starts to

increase, the calculation is stopped to avoid overfitting,even if the error in the training dataset decreases withiteration.

Results and discussionThe number of nodes in the hidden layer was varied toget an optimum number of nodes that resulted in aminimum mean square error (MSE), as shown in Fig. 5.The MSE is defined as follows

MSE~2:E

p~

1

p

Xp

d(p)o �y(p)o� �2

(14)

where E is the objective function represented byequation (3). Figure 5a shows that log(MSE) decreasesalmost linearly with increase in the number of hiddennodes. The results are shown for penetration as theoutput variable. Other output variables also showed thesame trend. These runs were conducted using the CGmethod with 10 different randomly selected initial sets ofweights in the neural network to avoid any local optimalsolution. Figure 5b shows a similar trend for thevariation in log(MSE) for leg length. The log(MSE)for penetration and leg length becomes almost constantfor more than 19 hidden nodes in the network. Figure 6shows that most of the runs for 19 hidden nodes gavevery small values of error within 15 000 iterations.However, the convergence rate depended on the choiceof initial set of weights. This is the main difficulty inusing the CG method alone to find the optimal weights.

3 Range of input variables in the database used for training, validation and testing of the network. The normalised

value of the variables was obtained using equation (13) and corresponding minimum and maximum values listed in

Table 1. Various combinations of these 22 input variables were generated using the modified L81 (966316) array in

the database



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In order to solve this problem and obtain globallyoptimised weights, the weights obtained by using the CGmethod for 10 different runs were included in the inputto a GA based optimisation model. Since the GArequires a population of at least 100 individuals (ordifferent sets of weights)24,34,35 to start the calculations,

5 a Variation of log(MSE) for penetration with number of

hidden nodes for training data by using the CG optimi-

sation method after 50 000 iterations. Dots represent

log(MSE) obtained in 10 different runs, which were

taken to examine the effect of initial weights on the

final converged solution. The box whisker plot shows

the variation and the mean of the log(MSE) for differ-

ent numbers of hidden nodes. Seventy-five percent of

the data are inside the shaded box. b Variation of

log(MSE) for leg length with number of hidden nodes

for training data by using the CG optimisation method

6 Variation of log(MSE) for penetration training data with

number of iterations for 19 hidden nodes by using the

CG optimisation method

4 Flowchart of the modified back propagation algorithm

using the hybrid optimisation model after coupling of

the generalised generation gap (G3) genetic algorithm

with the conjugate gradient method



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the rest of the 90 sets of weights were generatedrandomly in the interval 20.5 to 0.5.Figure 7a shows the log(MSE) for penetration with

the best optimised set of weights for various numbers ofhidden nodes with the hybrid optimisation scheme. Forthe first 19 hidden nodes, the error decreased continu-ously with the increase in number of nodes, and thenstarted to increase slightly with the increase in nodes.The lower value of MSE may also be due to overfittingof the network. The performance of the network wastested using the validation and testing datasets. Thenetwork was trained using only the training data. Thevalidation data were randomly generated independentlyof the training data. During training, if the networklearns the effects of input variables on the output, theMSE on the validation set improves with the iterations.However, if the network minimises the MSE on thetraining data by overfitting, the MSE on the validationdataset will increase. This behaviour indicates that theinteractions between different input variables of thetraining dataset are accurately modelled for the trainingdata but not for all possible values of input variables.

This behaviour also means that the performance of thenetwork may vary significantly for training, validationand testing datasets. To avoid overtraining (or over-fitting), the training was stopped after some iterationswhen the performance of the network for the validationdata was optimal. The testing data were used to checkthe overall performance of the network.

Figure 7a shows that 19 hidden nodes providethe lowest value of log(MSE) for penetration for thetraining, validation and testing data. Similarly, theminimum value of log(MSE) for leg length was foundfor 19 hidden nodes, as shown in Fig. 7b. Since a neuralnetwork with 19 hidden nodes showed low errors for allother output variables, 19 hidden nodes were selectedfor all the variables for simplicity. Figure 8 shows thevariation of log(MSE) in penetration and leg length withiterations using the hybrid method for 19 nodes in thehidden layer. Initially, the error for both training andvalidation data decreases with iterations. However, oncethe network gets the optimal weights, the error becomesalmost constant. The calculation was stopped when theerror for validation data started to increase with changein weights or iterations. Furthermore, the number ofiterations depended on the output variables and was notthe same for all variables. For example, for leg lengthand penetration, the calculations were stopped after42 000 and 48 000 iterations, respectively, based on theresults in Fig. 8. Table 2 shows the root mean square(rms) error for all the output variables. The rms errors inthe training data for penetration, leg length and peaktemperature in the weld pool were 0.02 mm, 0.01 mmand 4.27 K, respectively. These rms errors were quitesmall in comparison to the magnitude of these outputvariables. The rms errors in training data for otheroutputs were also very small. However, the rms errorswere higher for both the validation and testing data.Leg length showed a good match for all three datasets.This is because leg length depends mainly on the heatinput and is not significantly influenced by the impinge-ment of droplets.17–19 Furthermore, the rms error for

8 Variation of averaged log(MSE) in penetration and leg

length for all the 100 individual members in the popu-

lation by using the hybrid optimisation method for 19

nodes in the hidden layer. The calculation was

stopped when the error on the validation data started

to increase

7 Variation of log(MSE) in a penetration and b leg length

for training, validation and testing data by using the

hybrid optimisation scheme after 50 000 iterations; the

19 nodes in the hidden layer provide less error on all

three datasets



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penetration was large because its calculation involvescomplex interactions between various welding processparameters such as arc current, welding speed and wirefeed rate. The rms error in testing data was generallyhigher because the testing data were selected near theextreme range of input variables, with the exception ofmaximum velocity, for which the errors in the testingdata are smaller. To further check the accuracy of theresult predicted by the neural network, the relativeerrors were calculated as follows

(Relative % error)i~100|d i�yid i

� �(15)

where di is the desired value of the penetration for the ithdata point, and yi is the calculated value of thepenetration by the neural network for the same dataset.The results are presented as typical frequency v. relativepercentage error for penetration in Fig. 9. This error hasa classic Gaussian distribution, with the centre aroundzero. Furthermore, more than 90% of the data has anerror of less than 0.8% in the training dataset. This lowpercentage error indicates the accuracy of the neuralnetwork.

Various runs were conducted using the hybridoptimisation method with different randomly selectedinitial sets of weights in the population to obtain theoptimised values of all weights. The hybrid method gaveeither a smaller or the same error as the CG method.The computations took approximately 6 h for 19 hiddennodes on a 3.06 GHz Intel P4 CPU with 512 MbPC2700 DDR-SDRAM memory. Figure 10 comparesthe relative percentage error for training, validation andtesting data for different best sets of optimal weightsobtained using the CG method alone and the hybridmethod involving CG and GA. The results show that thebest set of weights obtained using the hybrid methodprovides less error than the best sets obtained using theCG method. Furthermore, the relative percentage errorwith the CG method varied significantly with the initialvalues of the weights. As a result, the CG method aloneis not capable of finding the best optimal network. Therelative percentage errors in training data with the CG

method in runs 4, 6 and 8 were low. However, all ofthese runs resulted in a large relative percentage error invalidation and testing data (Fig. 10b and c). Use of theCG method to determine the weights produced sub-optimal solutions. With use of the optimal weights

9 Frequency plot showing number of training datasets

of penetration lying in different ranges of relative per-

centage error

10 Comparison of relative percentage error in penetra-

tion for a training data, b validation data, and c test-

ing data, calculated by taking 10 different runs of the

CG method alone in the neural network and the

hybrid optimisation method. These box whisker plots

show that the hybrid optimisation method always pro-

duced less error than the CG method



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produced by the hybrid method, the average relativepercentage errors in training data, validation data andtesting data were 0.5%, 2.4% and 3.8%, respectively, forthe predicted value of the penetration. Table 4 showsthat the average relative percentage errors in trainingdata for other output variables were less than 0.5 %,except for maximum velocity, which had a maximumerror of 0.7%.

Computed values of penetration, actual throat, leglength, length of weld pool, peak temperature, coolingtime and maximum velocity in the weld pool obtainedfrom the neural network model were compared withtheir corresponding values obtained from the heattransfer and fluid flow model to examine the accuracyof the neural network. Figure 11 shows all the outputdata obtained using different sets of welding variablesused during the training, validation and testing of thenetwork. All the neural networks of GMA fillet weldcomprised only one hidden layer containing 19 nodes foreach of the seven output variables, i.e. penetration,actual throat, leg length, length of weld pool, peak

temperature, cooling time and maximum velocity in theweld pool. Figure 11 shows the predicted values ofoutputs from the neural network and the correspondingvalues calculated using the heat transfer and fluid flowmodel. The plots show that all points lie on or very closeto the diagonal line, and the results obtained from theneural network agree well with the values calculatedusing the heat transfer and fluid flow model. Table 4shows the average absolute value of the relativepercentage error in all the output variables for training,validation and testing data. The maximum average valueof the relative percentage error in all outputs was 1.1, 2.4and 6.0 in the training, validation and testing data,respectively. The results indicate that the neural networkcan be used for simulations with predetermined goodaccuracy.

Summary and conclusionsA set of multiple feedforward neural networks wasdeveloped for GMA fillet welding to calculate penetration,

11 Comparison of output variables, i.e. a penetration, b leg length, c throat, d length of weld pool, e peak temperature

in the weld pool, f cooling time between 800uC and 500uC, and g maximum velocity Umax in the weld pool, calculatedfrom the heat transfer and fluid flow model (x-axis) with corresponding values predicted by the neural network

model of GMA fillet weld. The diagonal lines in each plot show that, ideally, all the points should lie on this line.

The training data, validation data and test data comprise 486, 50 and 25 datasets, respectively



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leg length, throat, weld pool length, cooling timebetween 800uC and 500uC, maximum velocity and peaktemperature in the weld pool. These neural networkmodels used 22 input variables, including weldingprocess parameters and material properties. A hybridoptimisation scheme involving a CG and a GA forfinding global optimal weights resulted in low errors intraining, validation and testing data. The resultsobtained using the hybrid scheme were better than thoseobtained with the CG method alone. For each outputvariable, a separate neural network model was devel-oped. This approach provided superior results andgreater flexibility than one neural network for all theoutput variables. These neural network models canreplace the output of complex heat transfer and fluidflow models with significant computational economy.

Appendix A

Back propagation algorithm

The back propagation training consists of two passes ofcomputation, a forward pass and a backward pass.27–31

In the forward pass, an input vector (i.e. set of weldingvariables) is applied to the sensory nodes of the network.The signals from the input layer propagate to the unitsin the first layer, and each unit produces an outputaccording to equation (2). The outputs of these units arepropagated to units in subsequent layers, and thisprocess continues until the signals reach the outputlayer, where the actual response of the network (i.e. weldgeometry parameters such as penetration, throat, leglength and cooling time) to the input vector is obtained.During the forward pass, the synaptic weights of thenetwork are fixed. During the backward pass, on theother hand, the synaptic weights are all adjusted inaccordance with an error signal, which is propagatedbackwards through the network. The basic idea is thatthe objective function (given by equation (3)), which is adiscrepancy between the desired solution and thepredicted values, has a particular surface over the weightspace, and therefore an iterative process such as the CGmethod can be used for its minimisation. In short, thebasic back propagation algorithm is as follows:27–29,31

(i) initialise the input layer

(ii) propagate signals forward

(iii) calculate the error in the output layer

(iv) back propagate the error

(v) update the weights.

Appendix BPCX based G3 genetic algorithmThe steps involved in calculating the optimal values ofweights are as follows:

(i) a population is a collection of many individuals,and each individual represents a set of randomlychosen values of all the weights. A parent refersto an individual in the current population. Thebest parent is the individual that has the bestfitness, i.e. gives the minimum value of theobjective function, defined by equation (3), inthe entire population. The best parent and twoother randomly selected parents are chosenfrom the population

(ii) from the three chosen parents, two offspringor new individuals are generated, using arecombination scheme. PCX based G3 modelsare known to converge rapidly when threeparents and two offspring are selected. Arecombination scheme is a process for creatingnew individuals from the parents. The recombi-nation scheme (step (ii)) used in the presentmodel is based on the PCX operator24,34–38

(iii) two new parents are randomly chosen from thecurrent population of the individuals

(iv) a subpopulation of four individuals thatincludes the two randomly chosen parents instep (iii) and two new offspring generated instep (ii) is formed

(v) the two best solutions, i.e. the solutions havingthe least values of the objective function, arechosen from the subpopulation of four mem-bers created in step (iv). These two individualsreplace the two parents randomly chosen in step(iii).

The calculations are repeated from step (i) again untilconvergence is achieved, as shown in Fig. 12. Therecombination scheme (step (ii)) used in the presentmodel is based on the PCX operator. A brief descriptionof this operator, tailored to the present problem, is asfollows.

The first three parents, i.e. (w01,w02,w

03, . . . ,w

0n),

(w11,w12,w

13, . . . ,w

1n), (w

21,w

22,w

23, . . . ,w

2n), are randomly

selected from the current population. Here, the sub-scripts represent the number of weights in the network,and the superscripts denote the parent identificationnumber. The mean vector or centroid

g!~ w01zw

11zw

21

3,w02zw

12zw

22

3,w03zw

13zw

23

3, � � � ,w

0nzw

1nzw

2n

3

� �

of the three chosen parents is computed.To create an offspring, one of the parents,

say x!(p)~(w01,w02,w03, . . .w0n), is chosen randomly. The

direction vector d!(p)

~ x!(p)� g! is next calculated fromthe selected parents to the mean vector or centroid.Thereafter, from each of the other two parents, i.e.

(w11,w12,w

13, . . .w

1n), and (w

21,w

22,w

23, . . .w

2n), perpendicular

Table 4 Average absolute value of the relative percentage error for different output variables

Output variable Training data Validation data Testing data

Leg length 0.1 0.1 1.2Actual throat 0.1 1.0 1.8Penetration 0.5 2.4 3.8Length of the weld pool 0.4 0.4 0.5Peak temperature in the weld pool 0.1 0.2 0.2Cooling time between 800uC tand 500uC 0.2 0.2 0.7Maximum velocity in the weld pool 0.7 0.8 0.8



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distances Di to the direction vector d!(p)

are computed

and their average D is found. Finally, the offspring, i.e.

y!~ w01,w0

2,w0

3, . . .w0

n

� �, is created as follows24,34–38

y!~ x!(p)zwf d!(p)��

��z Xni~1,i=p

wgD h!(i)

where h!(i)

are the orthonormal bases that span the

subspace perpendicular to d!(p)

, and wf and wg arerandomly calculated zero mean normally distributedvariables.

Appendix C

The original L81 (340) orthogonal array was modified to

L81 (966316) on the basis of linear graph theory. The

columns in the L81 (340) orthogonal array have two

degrees of freedom.41 Since any variable in the modifiedL81 (9

66316) array with nine levels requires eight degreesof freedom,41 it was necessary to determine theappropriate four columns from the L81 (3

40) orthogonalarray to which to assign this variable. The procedure forconverting the standard orthogonal array to a multilevelL81 (9

66316) array includes the following steps.Step 1: Establish the standard L81 (3

40) orthogonalarray.

Step 2: Choose a suitable linear graph. For example, alinear graph joining four columns in the L81 (3

40)orthogonal array is shown in Fig. 13a. Based on thisgraph, four columns, i.e. columns 6, 18, 28 and 35, wereselected from the standard L81 (3

40) orthogonal array asshown in Table 5.

Step 3: The linear graph was modified with differentcombinations of 1s, 2s and 3s from columns 6, 18 and 28without having to consider the 35th column, as shown inTable 6. In Table 6, only a few rows of the array areshown, to save space.

Step 4: Columns 6, 18, 28 and 35 in the original L81(340) orthogonal array were replaced with the modifiednine level column.

The process was repeated with the other linear graphsshown in Fig. 13 to achieve six columns, each with ninelevels of values, in the original orthogonal array. Thefirst few rows of the resulting multilevel L81 (9

66316)array are shown in Table 6.

12 The working principle of the GA based on the generalised generation gap (G3) model and using the PCX operator

13 Linear graphs of L81 (340) orthogonal array for col-

umns a 6, 18, 28 and 35, b 1, 8, 9 and 10, c 2, 14, 17

and 20, d 3, 23, 27 and 31, e 4, 32, 37 and 39, and f

5, 15, 24 and 33, to capture interactions among

variables



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Acknowledgements

This research was supported by a grant from the USDepartment of Energy, Energy Programs Office underDE-FC07-011D14204. Mr Amit Kumar gratefullyacknowledges the award of a Fellowship from theAmerican Welding Society. The authors thank Dr J. M.Vitek of Oak Ridge National Laboratory and Mr S.Mishra of Penn State for many helpful comments. Wethank Ms Marie Quintana of Lincoln Electric and Dr S.S. Babu of Oak Ridge National Laboratory for theirinterest in the work.

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Table 5 Columns of L81 (340) orthogonal array selected

on the basis of the linear graph shown inFig. 13a

Column case 6 18 28 Modified 35

1 1 1 1 1 12 1 2 2 2 23 1 3 3 3 34 2 1 2 4 35 2 2 3 5 16 2 3 1 6 27 3 1 3 7 28 3 2 1 8 39 3 3 2 9 1

Table 6 First nine rows of the modified multilevel L81(966316) orthogonal array

Variable 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

Case 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1Case 2 1 2 2 2 2 2 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2Case 3 1 3 3 3 3 3 1 1 1 1 3 3 3 3 3 3 3 3 3 3 3 3Case 4 2 1 2 3 4 4 2 2 2 2 1 1 1 1 2 2 2 2 3 3 3 3Case 5 2 2 3 1 5 5 2 2 2 2 2 2 2 2 3 3 3 3 1 1 1 1Case 6 2 3 1 2 6 6 2 2 2 2 3 3 3 3 1 1 1 1 2 2 2 2Case 7 3 1 3 2 7 7 3 3 3 3 1 1 1 1 3 3 3 3 2 2 2 2Case 8 3 2 1 3 8 8 3 3 3 3 2 2 2 2 1 1 1 1 3 3 3 3Case 9 3 3 2 1 9 9 3 3 3 3 3 3 3 3 2 2 2 2 1 1 1 1


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