Tesis Metodo de Analisis Por Soldadura

Journal of Materials Processing Technology 142 (2003) 20–28

Delta ferrite prediction in stainless steel welds using neural networkanalysis and comparison with other prediction methods

M. Vasudevana,∗, A.K. Bhaduria, Baldev Raja, K. Prasad Raoba Metallurgy and Materials Group, Indira Gandhi Centre for Atomic Research, Kalpakkam, India

b Department of Metallurgy, Indian Institute of Technology, Chennai, India

Received 2 May 2002; received in revised form 11 December 2002; accepted 17 February 2003

Abstract

The ability to predict the delta ferrite content in stainless steel welds is important for many reasons. Depending on the service requirement,manufacturers and consumers often specify delta ferrite content as an alloy specification to ensure that weld contains a desired minimum ormaximum ferrite level. Recent research activities have been focused on studying the effect of various alloying elements on the delta ferritecontent and controlling delta ferrite content by modifying the weld metal compositions. Over the years, a number of methods includingconstitution diagrams, Function Fit model, Feed-forward Back-propagation neural network model have been put forward for predicting thedelta ferrite content in stainless steel welds. Among all the methods, neural network method was reported to be more accurate compared toother methods. A potential risk associated with neural network analysis is over-fitting of the training data. To avoid over-fitting, Mackayhas developed a Bayesian framework to control the complexity of the neural network. Main advantages of this method are that it providesmeaningful error-bars for the model predictions and also it is possible to identify automatically the input variables which are important inthe non-linear regression. In the present work, Bayesian neural network (BNN) model for prediction of delta ferrite content in stainlesssteel weld has been developed. The effect of varying concentration of the elements on the delta ferrite content has been quantified for Type309 austenitic stainless steel and the duplex stainless steel alloy 2205. The BNN model is found to be more accurate compared to that ofthe other methods for predicting delta ferrite content in stainless steel welds.© 2003 Elsevier Science B.V. All rights reserved.

Keywords:Neural network analysis; Delta ferrite content; Austenitic stainless steel; Duplex stainless steel

1. Introduction

The ability to estimate the delta ferrite content accuratelyhas proven very useful in predicting the various propertiesof austenitic SS welds. A minimum delta ferrite content isnecessary to ensure hot cracking resistance in these welds[1,2], while an upper limit on the delta ferrite content de-termines the propensity to embrittlement due to secondaryphases, e.g. sigma phase, etc., formed during elevated tem-perature service[3]. At cryogenic temperatures, the tough-ness of the austenitic SS welds is strongly influenced by thedelta ferrite content[4]. In duplex stainless steel weld metals,a lower ferrite limit is specified for stress corrosion crackingresistance while the upper limit is specified to ensure ade-quate ductility and toughness[5]. Hence, depending on theservice requirement a lower limit and/or an upper limit ondelta ferrite content is generally specified. During the selec-

∗ Corresponding author. Tel.:+91-4114-80232; fax:+91-4114-40381.E-mail address:[email protected] (M. Vasudevan).

tion of filler metal composition, the most accurate diagramto date WRC-1992 is used generally to estimate the�-ferritecontent[6]. The Creq and Nieq formulae used for generat-ing the WRC-1992 constitution diagram is given by Creq =Cr+Mo+0.7Nb and Nieq = Ni+35C+20N+0.25Cu. Thelimitation of these equations is that values of the coefficientsfor the different elements remain unchanged irrespective ofthe change in the base composition of the weld. However,the relative influence of each alloying addition given by theelemental coefficients in the Creq and Nieq expressions islikely to change over the full composition range. Further-more, these expressions ignore the interaction between theelements. Also, there are a number of alloying elements thathave not been considered in the WRC-1992 diagram. Ele-ments like Si, Ti, W have not been given due to consider-ations, though they are known to influence the delta ferritecontent. Hence, the delta ferrite content estimated using theWRC-1992 diagram would always be less accurate and maynever be close to the actual measured value. In the FunctionFit model [7] for estimating ferrite, the difference in freeenergy between the ferrite and the austenite was calculated

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M. Vasudevan et al. / Journal of Materials Processing Technology 142 (2003) 20–28 21

as a function of composition and this was related to ferritenumber (FN). The equation used in this model to determineFN is given below:

FN = A[1 + exp(B + C�G)]−1 (1)

whereA, B andC are the constants. The advantages of thissemi-empirical model over the WRC-1992 diagram includeits considering effect of other alloying elements and theease of extrapolation to higher Creq and Nieq values. ThisFunction Fit method can be used for a wide range of weldmetal compositions and owing to the analytical form of thismodel, the FN can be quantified easily. However, the ac-curacy of this method is not greater than the WRC-1992diagram. Vitek et al.[8,9] sought to overcome the majorlimitation of the constitution diagram and the Function Fitmethod of not taking into account the elemental interactions,by using neural networks for predicting ferrite in SS welds.The improvement in accuracy in predicting the delta ferritecontent by using neural networks, involving a feed-forwardnetwork with a back-propagation optimization scheme, hasbeen clearly brought in their study. The effect of various el-ements on the delta ferrite content for a few base composi-tions was examined by calculating the FN as a function ofcomposition. However, it was not possible in their analysisto directly interpret the elemental contributions to the finalFN. The prediction and measurement of ferrite in SS weldsremains of scientific interest due to limitations in all the cur-rent methods, and newer methods and constitution diagramsare continuously being proposed to predict the delta ferritecontent for a wider range of SS types. It was in this con-text that the development of a more accurate neural networkbased predictive tool for estimating the effect of various al-loying elements on the delta ferrite content for different SSwelds was taken up in this work.

A potential risk associated with neural network analy-sis is over-fitting of the training data. To avoid over-fitting,Mackay[10] developed a Bayesian framework to control thecomplexity of the neural network, with its main advantagesbeing that it provides meaningful error-bars for predictionsand also enables identifying the input variables that are im-portant in the non-linear regression. Hence, in the presentstudy, Bayesian neural network (BNN) analysis was appliedto develop a generalized model for FN prediction in stainlesssteel welds and the effect of variations in the concentrationof the elements on the FN for 309 stainless steel and duplexstainless steel base compositions were also quantified.

2. Database

As the aim of the present work was to model for theFN as a function of chemical composition, the databaseof 924 datasets for shielded metal arc (SMA) weld com-positions and delta ferrite contents, representing the com-mon 300-series SS weld compositions (viz., 308, 308L, 309,309L, 316, 316L, etc.) and duplex stainless steels used for

Table 1Range, mean and standard deviations of the composition variables (input)and the FN (output)

Elements Minimumvalue

Maximumvalue

Mean Standarddeviation

C 0.0000 0.2000 0.0401 0.0219Mn 0.3500 12.6700 1.8805 1.7897Si 0.0300 6.4600 0.5255 0.3477Cr 1.0500 32.0000 20.5135 2.7615Ni 4.6100 33.5000 11.3062 2.5628Mo 0.0100 10.7000 1.4186 1.6363N 0.0100 2.1300 0.0889 0.1415Nb 0.0000 0.8800 0.0287 0.0982Ti 0.0000 0.3300 0.0200 0.0284Cu 0.0000 6.1800 0.1405 0.4368V 0.0000 0.2300 0.0373 0.0404Co 0.0000 0.6900 0.0300 0.0463Fe 45.5990 72.5150 63.9412 4.3320FN 0.0000 98.0000 12.0358 17.3052

generating the WRC-1992 diagram was used[11]. For thedatasets in which the composition values for elements suchas Nb, Ti, V, Cu and Co were not available, their values wereassumed zero.Table 1shows the range, mean and standarddeviation of the each composition variable (input) and theFN (output). This simply gives the idea of the range coveredand cannot be used to define the range of applicability of theneural network model as the input variables are expected tointeract in neural network analysis. In BNN analysis, sizeof the error-bars define the range of useful applicability ofthe trained network.

3. BNN analysis

The networks employed consist of 13 input nodesxi rep-resenting the 13 composition variables, a number of hiddennodeshi and one outputy. The schematic structure of thenetwork is shown inFig. 1. The single output represents theFN. Both the input and output variables were normalizedwithin the range±0.5 as follows:

xN = x − xmin

xmax − xmin− 0.5 (2)

wherexN is the normalized value ofx, which has maximumand minimum values given byxmax andxmin. Eighty differ-ent neural network models were created using the datasets,with the number of the hidden units varying from 1 to 16 andwith five different sets of random seeds used to initiate thenetwork for a given number of hidden units. All these mod-els were trained on a training dataset that consisted of a ran-dom selection of half the datasets (i.e. 462 datasets), whilethe remaining half formed the test dataset that was used toexamine how the model generalizes with unseen data.

For calculating the outputs from the inputs, the linearfunctions of the inputsxj are multiplied by the weightswij

and operated by the following hyperbolic tangent transfer

22 M. Vasudevan et al. / Journal of Materials Processing Technology 142 (2003) 20–28

Fig. 1. Schematic diagram of the network structure showing the input nodes, hidden units and the output node.

function so that each input contributes to every hidden unit,whereN is the total number of input variables:

hi = tanh

N∑

j

w(1)ij xj + θ

(1)i

(3)

Here the bias is designated asθ and is analogous to theconstant in linear regression. The transfer from the hiddenunits to the output is linear, and is given by:

y =N∑i

w(2)i hi + θ(2) (4)

The outputy is therefore a non-linear function ofxj, with thefunction usually selected for flexibility being the hyperbolictangent. Thus, the network is completely described if thenumber of input nodes, output nodes and the hidden unitsare known along with all the weightswij and biasesθi.These weightswij are determined by training the networkand involves minimization of a regularized sum of squarederrors.

The BNN analysis of Mackay[10] allows the calcula-tion of error-bars with two components—one representingthe perceived level of noise(σv) in the output and the otherindicating the uncertainty in the data fitting. This latter com-ponent of the error-bars emanating from the Bayesian frame-work allows the relative probabilities of the models withdifferent complexity to be assessed. This enables estimationof quantitative error-bars, which vary with the position in theinput space depending on the uncertainty in fitting the func-tion in that space. Hence, instead of calculating a unique setof weights, a probability distribution of weights is used todefine the uncertainty in fitting. Therefore, these error-barsbecome large when data are sparse or locally noisy. In thiscontext, a very useful measure of the error is the logarithmof the predictive error (LPE) given by the following:

LPE =∑n

1

2

[(t(n) − y(n))2

σ(n)2

y

+ log(2πσny )1/2

](5)

wheret is the target for the set of inputsx, while y the cor-responding network output.σy is related to the uncertaintyof fitting for the set of inputsx. By using LPE, the penalty

for making a wild prediction is reduced if that predictionis accompanied by an appropriately large error-bar, with alarger value of the LPE implying a better model. Further, bythis method it is also possible to automatically identify theinput variables that are significant in influencing the outputvariable, as the input variables that are less significant aredown-weighted in the regression analysis.

3.1. Over-fitting problem

In BNN analysis, two solutions are implemented whichcontribute to avoid over-fitting. The first is contained in thealgorithm due to MacKay[12]: the complexity parametersα andβ are inferred from the data, therefore allowing auto-matic control of the model complexity. The second residesin the training method. The database is equally divided intoa training set and a testing set. To build a model, about 80networks are trained with different number of hidden unitsand seeds, using the training set; they are then used to makepredictions on the unseen testing set and are ranked by LPE.

3.2. Committee model

The networks with different number of hidden units willgive different predictions. But predictions will also dependon the initial guess made for the probability distributionof the weights (the prior). Optimum predictions are oftenmade using more than one model, by building a committee.The predictiony of a committee of networks is the averageprediction of its members, and the associated error-bar iscalculated according toEq. (6):

y = 1

L

∑l

y(l)

σ2 = 1

L

∑l

σ(l)2

y + 1

L

∑l

(y(l) − y)2 (6)

whereL is the number of networks in a committee. Notethat we now consider the predictions for a given single set ofinputs and that exponentl refers to the model used to producethe corresponding predictiony(l). In practice, an increasingnumber of networks are included in a committee and their


performances are compared on the testing set. Most often,the error is minimum when the committee contains morethan one model. The selected models are then retrained onthe full database.

4. Results and discussions

The characteristics of the BNN model on FN is discussedin detail elsewhere[13]. The comparison between the pre-dicted and measured FN values for the committee of mod-els (38 models in the committee) is shown inFig. 2 for thecomplete dataset. There was excellent agreement betweenthe measured and the predicted FN values. The correlationcoefficient was determined to be 0.98025.

4.1. Significance of the individual elements on the FN

Fig. 3 indicates the significanceσw of each of the inputvariables as perceived by the first five neural network mod-els in the committee. Theσw value represents the extent towhich a particular input explains the variation in the out-put, as for a partial correlation coefficient in linear regres-

Fig. 2. Comparison between the predicted and measured FN values forthe entire dataset using the committee of models.

Fig. 3. Perceived significanceσw values of the first five FN models foreach input.

sion analysis. It is observed fromFig. 3 that the elementsMn and Nb are not significant in influencing the FN. Theobservation of Mn not having a significant influence on theFN for the 300-series austenitic SS is in agreement with thereported results that the variation in Mn from 1 to 12 wt.%had almost no effect on the as-deposited FN[14]. However,the element Nb, which was found in the present study tohave an insignificant effect on the FN is included in the Creqformula used in the WRC-1992 diagram. As expected, Crand Ni were found to be the main elements influencing theFN. As per the present model, the elements that influencethe FN in order of significance are: Mo> N > V > Ti >

Cu > Co > Si > C > Fe. However, some of these elementsnamely V, Ti, Co and Si are not included in the Nieq andCreq formulae used in the WRC-1992 diagram.

4.2. Comparison of accuracy of present model with otherexisting methods

Analysis of the error distributions (measured FN–predictedFN) for the present BNN model shows that the absoluteerror was<2.5 for most of the datasets used in the training,while for the FNN-1999 model the absolute error was<3for about 80% of the dataset used in training. It is impor-tant to note here that in the present BNN model, the entiredatasets were used for retraining the committee of models,while in the FNN-1999 model only 90% of the datasetswere used for training. Further, the error distributions forthe present BNN model is symmetrical about zero (Fig. 4)implying good fitting of the model to the datasets. Also,the “tail” of the error distributions is less compared to theother methods.Table 2shows the comparisons of the errorsfor the BNN model and the FNN-1999 model. Compari-son of the quantified error distributions with those of theFNN-1999 model shows that the present BNN model issuperior. It has been reported that the FNN-1999 model ismore accurate compared to the WRC-1992 diagram andFunction Fit model. Hence, the root mean square (RMS)error between the measured and predicted FN values forthe present BNN model and the other three methods werecompared and it was found that the present BNN model

Fig. 4. Error distributions (experimental FN–predicted FN) for the com-plete dataset for BNN model.


has the lowest error among the four methods, BNN modelshowing an improvement of 43% over the FNN-1999 modeland about 65% over the WRC-1992 diagram and FunctionFit model.Table 3shows the RMS error values for all thefour methods. As the RMS error values represent the quan-

Fig. 5. Predicted FN vs concentration of the elements for 309 austenitic stainless steels weld. The plot shows the variation in the FN when one of theelement is varied and all other concentration are held constant at the 309 SS composition except Fe, which is adjusted to compensate for the varyingelement concentration.

titative measure of the degree of fit of the various modelsto the datasets on which they were trained, this compari-son clearly establishes that among the available methodsthe present BNN model is the most accurate model forprediction of FN in austenitic SS welds.


Fig. 5. (Continued).

Table 2Comparison of the errors (experimental FN–predicted FN) for the BNN model and the FNN-1999 model (training database)

BNN model FNN-1999 model

Absolute error Number of points Total no. of points (%) Number of points Total no. of points (%)

≤1.5 684 74.0 621 64.6≤2.5 820 88.7 764 79.5≤3.5 864 93.5 826 86≤4.5 888 96.1 – –≤5.5 900 97.4 – –≥5.5 20 2.16 – –≥9.5 4 0.4 32 3.3

4.3. Effect of compositional variations on the FN

The severe limitation of the WRC-1992 diagram is thatthe coefficients in the terms for Creg and Nieg formulas areconstant and hence the influence of an individual elementon FN is same irrespective of the change in the base com-position. As neural networks can take into account the in-teraction between the input variables on their influence overthe output variable, the interaction between the different el-ements on their influence over the FN is quantified for stain-less steel welds using the BNN analysis. The results for 308,308L, 316, 316LN have already been presented elsewhere[13,15]. In the present work, the effect of variations in theconcentration of the elements on FN have been quantifiedfor 309 stainless steel and duplex stainless steel welds. Thiswas done with two starting base compositions and then al-

Table 3Comparison of the RMS errors for complete training database for differentFN prediction methods

Prediction method RMS error

BNN model [12] 1.99FNN-1999 (back-propagation neural network) model[8] 3.5WRC-1992[6] 5.8Function Fit model[7] 5.6

lowing each element to vary over a limited range adjustingFe concentration accordingly but holding all other elementconcentrations constant.Table 4shows the base composi-tions of the 309 SS and duplex stainless steel welds used inthe present study.

4.3.1. The 309 stainless steel weldThe variation in the predicted FN as a function of the

variation in the concentration of the elements is found to benon-linear (Fig. 5). The FN is found to decrease with in-creasing concentration of the elements C, N and Ni. Theseelements acts as austenite stabilizers. The FN is found toincrease with increasing concentration of the elements Cr,Si and V and these elements are called ferrite stabilizers.The above observations are in agreement with the literature.The elements Mn, Mo, Nb, Ti, Cu and Co do not influence

Table 4Chemical composition of the 309 SS weld and duplex stainless steel weld

Material C Mn Si Cr Ni Mo

309 0.055 1.5 0.4 23 12.5 0.02DSS 2205 0.035 1.4 0.4 22.25 9.25 4.0N Nb Ti Cu V Co Fe0.09 0.02 0.02 0.03 0.08 0.03 62.2550.12 0.04 0.08 0.12 0.10 0.09 62.115


the FN value for 309 base composition used. However, inthe WRC-1992 diagram the composition of the elementCu is taken in to account for calculating the Nieq and thecomposition of Mo and Nb are included for calculating

Fig. 6. Predicted FN vs concentration of the elements for duplex stainless steel weld. The plot shows the variation in the FN when one of the element isvaried and all other concentration are held constant at the duplex stainless steel composition except Fe, which is adjusted to compensate for the varyingelement concentration.

the Creq. Thus, estimation of delta ferrite content by usingthe WRC-1992 diagram will always be less accurate. TheBNN model generated by us is more accurate compared tothe WRC-1992 diagram which was generated based on the


Fig. 6. (Continued).

linear regression analysis. Hence, the trends of the influenceof concentration of the elements on FN predicted by themodel is more useful in controlling FN through composi-tional modifications in this type of steel (Fig. 5).

4.3.2. Duplex stainless steel (alloy 2205) weldThe variation in the FN with variation in the concentra-

tion of the elements is found to be non-linear (Fig. 6). Theincrease in the concentration of the elements C, N, Mn andNi is found to decrease the FN. However, the effect of Mnis not as significant as the other austenite stabilizers. Theincrease in the concentration of the elements Cr, Si, Mo, Vand Co is found to increase the FN for the duplex stain-less steel welds. The effect of vanadium is not as significantas the other ferrite stabilizers. The elements Cu, Nb and Tiare found not to influence the FN for duplex stainless steelwelds. However, the elements Cu and Nb are included inthe WRC-1992 diagram in calculating the Nieq and Creq,respectively. The trends identified by this analysis of the in-fluence of concentration of the elements on the FN is veryuseful in controlling the FN by adjusting weld metal compo-sition in duplex stainless steel welds. Hence, depending onthe base composition, the influence of individual elementson the FN is different. However, the WRC-1992 diagramuses the same equation for all the stainless steel welds andis the severe limitation of the diagram (Fig. 6).

5. Conclusions

1. The generalized model for predicting the FN in stainlesssteel welds using BNN analysis has been developed. Theaccuracy of the BNN model in predicting FN is superiorcompared to the existing FN prediction methods.

2. Significance of the individual elements on FN has beenquantified. Neural network analysis has shown that ele-ments like manganese and niobium are insignificant ininfluencing the FN in stainless steel welds.

3. The effect of variation in the concentration of the ele-ments on the FN have been quantified for 309 and duplexstainless steel welds. Neural network analysis has shownthat there is a change in the role of elements when thebase composition is changed.

4. Cobalt is found to be ferrite stabilizer for the duplexstainless steel welds and is found not to influence the FNfor the austenitic stainless steel welds.

References

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Tesis Metodo de Analisis Por Soldadura