mapping the input–output relationship in hsla steels through expert neural network

Materials Science and Engineering A 420 (2006) 254–264

Mapping the input–output relationship in HSLA steelsthrough expert neural network

S. Datta ∗, M.K. BanerjeeDepartment of Metallurgy and Materials Engineering, Bengal Engineering and Science University, Shibpur, Howrah 711 103, India

Received 16 September 2004; received in revised form 4 January 2006; accepted 10 January 2006

Abstract

Modification of the architecture of the artificial neural network is done to accommodate the information available from the knowledge base in thefield of materials science for thermomechanically processed HSLA steel. The complicated architectures of these networks are made to satisfy thewell-understood physical metallurgy principles, which administer the property response to the combined actions of the compositional and processparameters. The networks developed have been found to give very good convergence during training. The number of epochs required to reach thetargeted error was found less for these networks than the conventional networks.© 2006 Elsevier B.V. All rights reserved.

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eywords: HSLA steel; Thermomechanical-controlled processing; Artificial neural network; Composition; Process parameters; Yield strength

. Introduction

In order to achieve desired set of mechanical properties theigh strength low alloy (HSLA) grade of steel are subjectedo thermomechanical-controlled processing (TMCP). The influ-nce of composition and process parameters on the microstruc-ure and mechanical properties of a thermomechanically pro-essed HSLA steels is well documented in literatures [1–3].owever, in majority of those works the roles of compositional

nd process variables in TMCP steels are assessed qualitativelyn the basis of available knowledge in physical metallurgy. Thereas only been a limited effort in the formulation of a suitableodel, which can determine the response variables quantita-

ively from a given set of input variables. Although regressionnalysis is sometimes carried out to best fit a set of data to apecified relationship, its main drawback lies in the fact that theorrelation between the input (composition and process param-ters) and the output (mechanical properties) is to be pre-chosenithout much reasons. Artificial neural network (ANN) is aind of learning system, which maps the existing input–outputelationship in a more precise way [4]. It is capable to accom-

particular technique has been found to be very useful in pre-dicting the mechanical properties of TMCP steels, which areknown to depend upon a large number of independent inputvariables [5–9]. Attempts have been made earlier to model themechanical properties of HSLA steel by means of neural net-works [10–14]. It has been noticed that the performance ofANN models, in terms of the achievable error level in leastcomputational time, is essentially determined by a successfuloptimisation of the number of neurons in the network, effectivealgorithms for accurate prediction and suitable transfer function.But the main criticism faced by the concept of artificial neuralnetwork is that the relation it develops between the inputs andoutputs are mostly unknown to its user. The learning process ofthe network is inside a ‘black box’. Though suitably designednetworks are capable of making accurate predictions after beingappropriately trained, it is not at all clear if the learning envis-aged in the network has any similarity with that of a materialsscientist. It is also not known whether the process of predictionused by the trained neural network makes use of the elementalknowledge acquired by a scientist in respect of the effects ofcomposition and process variables on the strength properties of

odate the non-linearity of the relationships existing amonghe variables. When used with backpropagation algorithm this

∗

thermomechanically processed HSLA steel. Since mapping ofinput–output relationships for a given problem is mathematicallyfeasible even without the understanding of scientific reasoningsbehind the phenomenological relationships between the inputand the output, it is possible for such models to predict outputv

Corresponding author.E-mail address: [email protected] (S. Datta).

921-5093/$ – see front matter © 2006 Elsevier B.V. All rights reserved.oi:10.1016/j.msea.2006.01.037

alues from a given set of fresh inputs.

mailto:[email protected]

dx.doi.org/10.1016/j.msea.2006.01.037

S. Datta, M.K. Banerjee / Materials Science and Engineering A 420 (2006) 254–264 255

Considering this aspect, an attempt is made here to modify thearchitecture of a classical neural network in such a manner thatit is able to give due consideration to the expert knowledge baseavailable in the field of materials science. The complicated archi-tectures of these custom networks, as they are called, are madeto satisfy the well-understood physical metallurgy principles,which govern the property response to the combined actions ofthe input variables like compositional and process parameters.

2. Database

The database used for training and testing the differentlydesigned custom networks comprise of the chemical compo-sition (viz. weight percentage of carbon (C), manganese (Mn),silicon (Si), nickel (Ni), copper (Cu), molybdenum (Mo), nio-bium (Nb), chromium (Cr), titanium (Ti), and boron (B)) and theTMCP parameters (viz. slab reheating temperature, deformationgiven in three different temperature zones, finish-rolling temper-ature and cooling rate) are used as the input variables and theyield strength is taken as the output variable. The data used forthe present exercise have been mostly generated in the labora-tories. The chemical analyses are done in atomic spectrometer.Controlled rolling has been carried in a laboratory scale twohigh rolling mill. The mechanical testing has been carried out inINSTRON 4204 machine. Some data from the published liter-ao

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3. Different models

3.1. Basic concept

The type of ANN used in the present case is a supervisedmultilayered feed forward network trained with backpropaga-tion algorithms. The inputs and outputs are connected throughhidden units. The inputs pi are multiplied by weights Wji for ahidden node aj; summation of all the Wjipi is then added to abias value bji and finally operated by a suitable transfer function(f). The operation may therefore be written as:

aj = f(∑

Wjipi + bji

)(2)

Similar operations may be repeated for varying number ofhidden layers in order to find out the most suitable networkarchitecture.

Hidden layers contribute to the output nodes through a linearoperation. The output Y can be written as:

Y =(∑

wjaj + b′)

(3)

where wj and b′ are new sets of weights and bias values. Inthe process of learning, the error of the calculated or predictedoutput in relation to the actual output is backpropagated to adjustall the weight and bias values.

3

miioarrIcbrceamnoa

noatsc

a

tures are also taken into consideration in getting a wide rangef variables.

The ranges of variables used in the present work are listed inable 1. Each variable is normalized within the range of 0–1 forNN modelling by the operation given below:

N = x − xmin

xmax − xmin(1)

here xN is the normalized value of a variable x, xmax and xminre the maximum and minimum values of x, respectively.

able 1he minimum and maximum limits of the parameters

arameters Minimum Maximum

0 0.1n 0 2

i 0 0.5i 0 4u 0 2o 0 1b 0 0.1r 0 1i 0 0.05

0 0.003RT 1000 12501 0 302 10 403 10 50RT 650 850R 0 35TS 600 1200S 300 1100el 10 25

.2. Modelling the synergism phenomenon

It is known that boron increases the bainitic fraction in theicrostructure; but at the same time the � grain size is also

ncreased [15]. The singular effect of boron on the mechan-cal properties of HSLA steel is therefore marginal. On thether hand, if boron is added in combination with niobium,remarkable improvement in strength can be achieved. It is

eported that the combined addition of niobium and boronetards recrystallization of austenite grains during TMCP [16].n this regard niobium and boron are known to act synergisti-ally to increase the recrystallization temperature of austenitey about 25 ◦C. This is ascribed to the non-equilibrium seg-egation of boron at dislocations and to the formation Nb–Bomplexes [17]. Addition of Ti with B has almost identicalffect, though the grain refinement is relatively less significants compared to Nb–B steel [18]. On the basis of this infor-ation a network architecture is so designed (Fig. 1) that the

eural network is able to take into consideration the phenomenonf the synergism between two elements in the predictivectivity.

In the customised network 1 (Fig. 1) niobium, boron and tita-ium are used as inputs whereas the yield strength is used as theutput. The inputs after they are multiplied by input weights, aressigned bias values, and are operated with hyperbolic tangentransformation function, which has been found to be the mostuitable in previous work [12]. Following Fig. 1, the operationan be written as:

1(k) = tanh{IW1,1p1(k) + b1} (4)

256 S. Datta, M.K. Banerjee / Materials Science and Engineering A 420 (2006) 254–264

Fig. 1. The diagram of custom network 1.

where a1(k) is operational output place used as input in the hid-den layer, generated from the kth element of the input vector p1,IW1,1 the input weight and b1 is the bias value.

Similarly, for boron and titanium, we get:

a3(k) = tanh{IW3,2p2(k) + b2} (5)

and

a5(k) = tanh{IW5,3p3(k) + b3} (6)

To incorporate the synergistic effect of Nb–B and Ti–B, twoseparate neurons are added in the network. The neuron combin-ing the niobium and boron gives the equation:

a2(k) = tanh{IW2,1p1(k) + IW2,2p2(k)} (7)

And the operation describing the synergism of Ti–B can bewritten as:

a4(k) = tanh{IW4,2p2(k) + IW4,3p3(k)} (8)

where p1(k), p2(k) and p3(k) are the kth element of input vectorsp1 (for Nb), p2 (for B) and p3 (for Ti).

The places after the input layers (a1, a2, . . . a5) are all mul-tiplied with the layer weights (LW) and are added with a biasvector to get the final output (yield strength) value. This opera-tion can be written as:

Y

Watg

0.55 wt.% molybdenum, 0.78 wt.% chromium and being sub-jected to TMCP of fixed process parameters like slab reheatingtemperature of 1150 ◦C, deformation of 30, 20 and 25% at therecrystallization temperature range, unrecrystallized tempera-ture range and in the (α + γ) two phase regions, respectively.Out of the three input variables, niobium and titanium arevaried within the range of 0–0.04 wt.%, while the other vari-able boron is varied within 0–0.002 wt.%. The correspondingyield strength values are found to lie within a range of 800–1000 MPa.

The above-designed network is trained with both theLevenberg-Marquardt (LM) backpropagation and scaled con-jugate gradient algorithm, which were found to be quite suitablein previous work [12]. The training patterns in both the caseshave shown a good convergence (Figs. 2 and 3). LM backprop-

Fb

S =5∑

x=1

LW6,xax + b4 (9)

Each layer’s weights and biases are initialised with Nguyen-idrow layer initialisation method. The training was done fromdata set obtained from experimental results of a steel with

he basic composition of 0.06 wt.% carbon, 1.38 wt.% man-anese, 0.3 wt.% silicon, 1.12 wt.% nickel, 1.05 wt.% copper,

ig. 2. Training pattern of custom network 1 trained with Levenberg-Marquardtackpropagation algorithm.


Fig. 3. Training pattern of custom network 1 trained with scaled conjugate gra-dient algorithm.

agation algorithm is found to have reached the targeted meansquare error (MSE) level with fewer numbers of epochs. How-ever, consistent with the observation in a previous work [12],this algorithm has taken higher training time than the scaledconjugate gradient algorithm.

The variation of layer weights during training of the networkis shown in Fig. 4. Since layer weights are connected to the nodesrepresenting individual element and also those representing syn-ergism, Fig. 4 depicts the effects of individual elements as wellas their combined effects on the yield stress of the experimentalsteel. Unfortunately the results in Fig. 4 are not able to describethe well-known synergism between Nb and B in a convincingmanner.

Fig. 4. Layer weights of the trained custom network 1 signifying the effects ofthe alloying elements individually and in combination on the yield strength ofthe steel.

Hence another network is designed with the same alloyingelements as inputs, the synergism effects among elements beingdescribed in a different manner (Custom network 2). The net-work architecture (Fig. 5) shows that the combined effect ofNb–B and Ti–B are used as separate input nodes.

In the network (Fig. 5), niobium, boron and titanium are usedas three inputs, combination of Nb–B and Ti–B as other twoinputs and yield strength is used as output. All the inputs havebias values, and are operated with hyperbolic tangent as transferfunction. The operation can be written as:

an(k) = tanh{IWn,npn(k) + bn} (10)

where an(k) is operational output place used as the input in thehidden layer, generated from the nth element of the input vectorp, IWn,n the input weight and bn is the bias value.

of c
Fig. 5. The diagram ustom network 2.


Fig. 6. Training pattern of custom network 2 trained with Levenberg-Marquardtbackpropagation algorithm. The combined effects are designated as summationof the wt.% of the elements.

The places after the input layers (a1, a2, . . . a5) were allmultiplied with the layer weights (LW) and were added with abias vector to get the final output (yield strength) value. Thisoperation can be written as:

YS =5∑

n=1

LW6,nan (11)

For the input places of Nb–B and Ti–B, the data used for train-ing the network are (i) summation, (ii) product and (iii) ratio ofthe wt.% of the respective elements. The three alternatives aretried to find out how best the input data can be made respon-sive to the predictive process in order to yield output results ofsynergism compatible to real life situation. On training with theLevenberg-Marquardt algorithm, a good convergence of trainingpattern is noted in all the three cases (Figs. 6–8). The variationof layer weights due to training with this architecture is shownin Fig. 9. In the instant case, summation of input values of nio-

Fbt

Fig. 8. Training pattern of custom network 2 trained with Levenberg-Marquardtbackpropagation algorithm. The combined effects are designated as ratio of thewt.% of the elements.

bium and boron or titanium and boron is used to describe theinput values at the nodes describing synergistic effects. From thefigure it appears that combined weightage values due to Nb–Bor Ti–B are higher than the values obtained in custom network1. Weightages due to individual elements have also increased.But comparative increase in weightage is far more for the nodesdescribing synergism than those depicting the effect of individ-ual elements.

The variation of layer weights of the network where productvalues have been computed in the combined input nodes is plot-ted in Fig. 10. The synergistic effects of the elements should behigher than that of the individual elements. But here the weigh-tages due to Nb–B or Ti–B are found to be lower than those dueto individual elements. However, when the input nodes denot-ing combined effects are trained with the ratio of the individualinput values, the layer weight plot is found to be highly accept-able from the above point of view (Fig. 11). The steady statelayer weights values for different cases are furnished in Table 2.

Fto

ig. 7. Training pattern of custom network 2 trained with Levenberg-Marquardtackpropagation algorithm. The combined effects are designated as product ofhe wt.% of the elements.

ig. 9. Layer weights of the trained custom network 2 signifying the effects ofhe alloying elements individually and in combination (summation of the wt.%f the elements) on the yield strength of the steel.


Fig. 10. Layer weights of the trained custom network 2 signifying the effects ofthe alloying elements individually and in combination (product of the wt.% ofthe elements) on the yield strength of the steel.

Fig. 11. Layer weights of the trained custom network 2 signifying the effects ofthe alloying elements individually and in combination (ratio of the wt.% of theelements) on the yield strength of the steel.

From the table it is quite clear that computation with the ratioof input values at nodes describing synergism, is most appropri-ate as it makes the relative weightage of Nb–B or Ti–B appeardistinctly appreciable in Fig. 11. Thus the synergistic effects arevery well predicted.

3.3. Model incorporating all the variables

After determining the means to arrive at a reasonably accept-able output reflecting synergism between two elements in HSLAsteels, the extension of the model is made to incorporate allthe variables operative in TMCP steels. This takes account ofthe combined and other relationships between Nb and B, B

and Ti, Cu and B, Cu and Ni, Cu–Ni–Nb, Nb–FRT, Ti–SRT,Nb–Ti–SRT, different stages of deformation with SRT and FRT.

All pairs of input variables for which synergistic effect onstrength property of TMCP steel is expected are placed as theratio of weight percents of the concerned elements. The inputvariables are connected to six nodes representing the six knownstrengthening mechanisms operative in HSLA steels. Thesenodes represent precipitation hardening, solid solution hard-ening, grain refinement, microstructural modifications, disloca-tion hardening and texture hardening (Fig. 12). A connectivitybetween an input node and a hidden layer insures a definite biasin accordance with the role of the concerned input variable in therepresentative strengthening mechanism of the connected nodein the hidden layer. For example the presence of elemental cop-per in HSLA steel is known to have appreciable influence onsolid solution strengthening, precipitation hardening and grainrefinement. Hence input node representing copper singularlyis connected only to those three hidden layer nodes, whichdescribes the three above-mentioned strengthening mechanism.

The additional elements added in this network have followedthe same logic for their connectivity with the six nodes. It isknown that, among the substitutional alloying elements molyb-denum exerts the largest strengthening effect on the HSLA steelsthrough solid solution hardening and microstructural modifica-tions, whereas copper and nickel result in weak strengtheningof steel on a single addition basis. According to Tamura et al.,atsaaA0gmtitcacbm

a

wh

Table 2Layer weights of different elements after attainment of steady state

Network name Combined effects as Niobium

Case 1 – 0.0690

Case 2 Summation 0.1297Product 0.1883Ratio 0.1213

simultaneous addition of two or three elements in combina-ions like copper–nickel or copper–nickel–molybdenum yieldsynergistic strengthening effects. It was found that combinedddition of 0.15 wt.% each of copper and nickel can strengthensteel with a basic composition of 0.25% Si–0.45% Nb–0.03%l to a larger extent, when compared with a single addition of.30% copper or 0.30% nickel. Molybdenum also bears syner-ism with niobium and titanium in grain refinement, structuralodifications and dislocation hardening [19]. If finish-rolling

emperature is lowered, particularly up to the two-phase region,t induces deformation bands, thereby resulting in a finer struc-ure. It further accentuates precipitation reactions. Acceleratedooling results in hardening through changes in microstructurend precipitation behaviour. Taking all these and other similaronsiderations into account the final network connectivity haseen operated with hyperbolic tangent transfer function. Thus itay be written as:

n(k) = tanh{∑

IWn,mpm(k)}

(12)

here an(k) is operational output place used as the input in theidden layer, generated from the summation of the multiplica-

Titanium Boron Nb and B Ti and B

0.0547 0.0645 0.0659 0.0455

0.1070 0.1515 0.1856 0.16700.1541 0.1902 0.1141 0.11260.1260 0.1307 0.2348 0.2118


Fig. 12. The diagram of custom network 4.

tion of different elements of the input vector p with IWn,m, theinput weight. Here n (1–6) corresponds to the nodes represent-ing the six strengthening mechanism and m can be of any valuewithin 1–23 corresponding to the inputs.

The places after the input layers were all multiplied with thelayer weights (LW) to get the final output (yield strength) value.This operation can be written as:

YS =6∑

n=1

LW6,nan

For training the network, the inputs are varied within therange as described in Table 1. Around 200 datasets were usedto train the network. The training patterns show good conver-gence with both the backpropagation training algorithms used(LM and SCG). As usual LM backpropagation algorithm hasreached the target MSE value in lesser epochs but has takenhigher simulation time (Figs. 13 and 14).

The custom network architecture has been modified by incor-porating additional hidden layers. Complete node-to-node con-nectivity has been made between the existing six hidden layers


Fig. 13. Training pattern of custom network 4 trained with Levenberg-Marquardt backpropagation algorithm. The combined effects are designated asratio of the wt.% of the elements.

depicting the strengthening mechanisms and the next additionalhidden layer, in case of network comprising two hidden lay-ers. In another modification another additional hidden layer isused. Here also the second and third hidden layer was com-pletely connected with each other. These were then trained withscaled conjugate gradient algorithm. The training patterns haveshown good convergence (Figs. 15 and 16). The error levelsachievable in the three cases are shown in Fig. 17. From the fig-ure, it is clear that the network with two hidden layers reachesminimum error level. However, the difference in the MSE israther insignificant for all practical purposes. From the com-putation of layer weights for this network, it is found that thecontributions from precipitation hardening, grain refinement,microstructural changes, solid solution hardening and disloca-tion hardening are quite high and are almost similar, whereas theeffect of texture hardening is rather low (Fig. 18). The steadystate input weights as connected to the node corresponding tosolid solution hardening are plotted in Fig. 19. The input weights

Fgt

Fig. 15. Training pattern of custom network 4 trained with scaled conjugategradient algorithm. The combined effects are designated as ratio of the wt.% ofthe elements (with one hidden layer).

Fig. 16. Training pattern of custom network 4 trained with scaled conjugategradient algorithm. The combined effects are designated as ratio of the wt.% ofthe elements (with two hidden layers).

Fig. 17. Achieved least mean square error in custom network 4 for differentnumber of hidden layers of the network.

ig. 14. Training pattern of custom network 4 trained with scaled conjugateradient algorithm. The combined effects are designated as ratio of the wt.% ofhe elements.


Fig. 18. Inputs weights of the trained custom network 4 signifying the contri-bution of different elements on solid solution hardening.

Fig. 19. Inputs weights of the trained custom network 4 signifying the contri-bution of different input variables on dislocation hardening.

multiplied to the input values of the alloying elements and pro-cess parameters have shown that silicon and manganese havethe highest and nickel has the lowest contribution towards solidsolution hardening. When similar plots were developed for theeffects of input variables on the dislocation strengthening mech-anism (Fig. 20), it shows that the combined effect of copper andboron contributes maximum towards this mechanism, whereas

Fig. 20. Layer weights of the trained custom network 4 signifying the effects ofthe hardening mechanisms on the yield strength of the steel.

the combined effect of molybdenum and titanium does theleast.

4. Discussion

It is known that in the metallurgical system under investiga-tion there exists a complicated relationship between the inputvariables and the output responses. The relationships are notonly complex and non-linear but also very difficult to be prop-erly quantified. The design of neural network used for outputprediction in an earlier work, is seen to have learnt how to com-pute the output notwithstanding if the metallurgical reasoningare given due considerations in the concerned learning process.Node to node complex connectivity has enabled the learningprocess to attain a reasonable degree of accuracy in the predic-tion of outputs from a large number of input datasets. This act isaccomplished by more of mathematical manipulations to recog-nise the pattern of input–output relationship without regard tothe phenomenological relations between input and output avail-able in the qualitative knowledge base of materials science.

In the present work, the qualitative relationships have beenused in constructing the neural network. As a result, the presentnetwork behaves as a biased but intelligent network. Having hadthe expert knowledge inbuilt within its architecture, the devel-oped network is characteristically different from the classicalANN or even the Petri neural net. It is to be noted that thepf

whluwufiqbpaekd

vt[incnaaatsa

resently designed network is equally based on the concept of aeed forward network with backpropagation algorithm.

Minimisation of MSE has been carried out to train the net-ork. But the connectivity being biased, the present networkas acted intelligently and swiftly to come to a very low errorevel whence the weightages have attained their steady state val-es. This implies that at the steady state, the values of layereights signify the relative proportions to which an individ-al strengthening mechanism is responsible for influencing thenal properties. Similarly, a definite mechanism, being conse-uential to a specific set of input parameters, can be shown toe influenced by various input variables in different but definiteroportions. Therefore, this concept leads to the formulation ofn entirely new type of artificial neural network and is consid-red suitable for many problems in materials science where priornowledge exists in some qualitative form. Thus the network soesigned is called expert neural network (ENN).

The ENN so developed has used a very large number of inputariables and the entire dataset for training, as were used inhe classical ANN of the previous work by the same authors12,13]. It is observed that ENN could reach the targeted MSEn much less number of epochs than the previous work. It may beoted here that while a network with conventional connectivity,ould reach a training error up to a scale of 10−4 only [12], theewly designed ENN enables to achieve a training error as lows 10−12. Interestingly ENN takes much less number of epochsnd training time than the conventional ANN when the SCGlgorithm is used. Hence ENN is found to be more suitable thanhe conventional architecture of neural networks. This largelytems from the incorporation of the metallurgical understandingbout the relationships between the input variables and the out-


put, thereby making the system more intelligent and expert tolearn the non-linear relationship within shorter time and up to amuch lower error level.

The number of variables, which are used to develop the cus-tom networks 1 and 2 is small. Therefore, it has been difficult tosupply a large number of dataset while training the above net-works. However, the major intention behind the developmentof those networks has been to study the feasibility of training atailor made expert network with non-conventional architecture.The final custom network encompasses all the variables, whichare known to be the most common variations generally made inthermomechanically control processed HSLA steel to achievetarget strength. The concept of using input weights in the assess-ment of the sensitivity of the input variables and to the differenthardening mechanisms is represented in Figs. 18 and 19. Fig. 20gives us the idea of the relative proportions of contribution fromthe different strengthening mechanisms towards the final yieldstrength of TMCP steel.

The very architecture of the network is suggestive of thedependence of yield stress on the hardening mechanisms inaccordance with the corresponding layer weights in ENN.Fig. 21 clearly exhibits the superiority of grain refinement factorin determining the yield strength values. The observed responsesfrom individual mechanisms corroborate the common experi-ence of materials science. It therefore seems that Hall-Petchequation is now made resolvable by ENN into more numberoT

Y

wypooYc

Fef

value of yield stress obtainable solely by a particular hardeningmechanism, i, which again, is determined by a set of composi-tional and process variables.

The steel under investigation has been designed elsewherefrom several considerations [20]. The basic philosophy behindits design and development has been such as to maximize theeffect of all the strengthening factors other than texture harden-ing [21]. Texture hardening cannot be important in HSLA 100compatible steels owing to the very microstructural morphol-ogy normally attempted in this class of steel. From the resultsof Fig. 22, it is clear that all the strengthening parameters areenhanced significantly. This is the reason why the propertiesof this steel have been extremely good. Implicit in Fig. 22 isthe efficacy of the design of steel, which, beyond doubt, hasbeen able to exploit almost all the possible strengthening mech-anisms capable of imparting improved mechanical properties ofHSLA steel. This fact demonstrates that the presently designedENN not only learns superbly, but also can guide a metallurgistabout the strength and weakness in his design of steel. It is inthis sense, that the so-designed expert and intelligent network(ENN) is considered superior to others, at least in respect ofproblems related to physical metallurgy of steel.

Figs. 23 and 24 demonstrate the effects of different input vari-ables on strengthening mechanisms of HSLA steel. It is apparentfrom the figures that one makes use of such steels to developappropriate equation for yield stress of TMCP steels in termsoq

p

Y

we

Y

Fef

f mechanistic components in a much more deterministic way.hus yield stress may be given by

S = Y0 +n∑

i=1

WiYi (13)

here, Wi (i = 1, 2, . . . n) denotes the contributing factors onield stress of the alloy due to different mechanisms and areroportional to the layer weights at steady state; Y0 is the valuef yield stress at a standard state definable by pure single crystalf solvent metal in highly annealed/dislocation free situation.0 is determined by the characteristics of solvent atoms and therystal structure of the pure solvent. Yi, however, represents the

ig. 21. Bar chart showing a measure of the model-perceived significance ofach strengthening mechanism influencing the strength of the steel as deducedrom the network.

f its input variables. The following is proposed for the exactuantification of strength properties of experimental steels.

In consideration of the results on dislocation hardening it isossible to write:

DH = f (D2, D3, Cu/B, Mn/Nb, Mo/Ti)

here YDH is the obtainable yield stress due to dislocation hard-ning alone. This can be written as:

DH = tanhm∑

j=1

WDH,jpj

ig. 22. Bar chart showing a measure of the model-perceived significance ofach strengthening mechanism influencing the strength of the steel as deducedrom the network.


Fig. 23. Bar chart showing a measure of the model-perceived significance ofeach input variables influencing the solid solution hardening of the steel asdeduced from the network.

Fig. 24. Bar chart showing a measure of the model-perceived significance ofeach input variables influencing the dislocation hardening of the steel as deducedfrom the network.

where WDH,j is the input weight at the steady state for a variablej (say Cu/B) for dislocation hardening and pj is the normalizedvalue of the concerned variable.

On generalising:

Yi = tanhm∑

j=1

Wi,jpj (14)

where ‘i’ denotes the hardening mechanism and ‘j’ the inputvariables (compositional and process parameters).

Using Eq. (13), one may write:

YS = Y0 +n∑

i=1

Wi

⎛⎝tanh

m∑j=1

Wi,jpj

⎞⎠ (15)

Thus the yield stress of the thermomechanically processedHSLA steels is obtained in terms of the normalized values ofthe input variables and steady state values of the input and layerweights.

5. Conclusion

1 The custom networks (ENN) developed on the basis of gen-eral understanding of physical metallurgy principles of TMCPsteels are capable to yield very good convergence during train-ing.

2 The number of epochs required to reach the targetedMSE was found less for these networks than the classicalnetworks.

3 Efficacy of the design of an alloy can be elegantly studied byENN.

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[[[[[[

[

[

[

[

[

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mapping the input–output relationship in hsla steels through expert neural network

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