,„/,«« J. Phy^- 79 (5), 473-483 (2005)
§ U P ^
N euro-fuzzy m odelling o f the strength o f thermomechanically processed
HSLA steels
S D ana* and M I« ^ a n c rjc e
D epartm ent o f M etallurgy & M aterials n ig in c c iin g , B E S. U niversity, Shibpur, Uowrah-711 10^, '^ vsi Bengal, India
'f
E-m ail sd a n a @ m a ts|.b c c s.a c in
R e c e iv e d 21 Septem ber, 2004, a t^ e p te d 10 M a tch , 2 0 0 5I i'
\bsiract : T h e m e c h a n ic a l p ro p e r tie s o f H S L A s te e ls are m o d e lle d by 'the ap p lica tio n o f ncuro-fii-tzy sy ste m s in resp ect o f the e ffe c t o f u inposinon and process param eters. N eu ro -fu zz y sy stem generated through data clustering show ed a good perform ance from the p icd iction point o f Also die increase in the number o f rules im proved the predictabiliiy o f the system A new design «>f n c u r o -fu //y system through division in sub- j.i,ses h:»s enabled the sy ste m to m odel a co m p lic a ted sy stem o f non-lincar inpur-output relation.ship.
k m io r d s . H SLA ste e l, m ech a n ica l property, n e u io -fu z z y system , prediction I'\( S Nos. K1 0 5 B x . 8 4 35 -H, 0 7 0 5 Mh
1. Introduction
The mechanical properties o f steels are know n to depend on independent variables like com position and process param eters.
1 tioits lo develop a m odel relatin g the independent variiibles wiih the dependent ones are still co ntinuing. T he m ain reason ioi ihc lack o f progress in p erfect pred ictio n o f m echanical ptopciiies as a function o f co m p o sitio n and process param eters
•sthui a particular property, say, yield strength (Y S) is dependent in i\ very com plex w ay on a n u m b er o f variables. N evertheless, ihere ought to exist som e specific hidden patterns, w hich relate the mpuis with the outputs. So far, these have been recognised t>nly qualitatively by the experts in m aterials science. In order to make a quantitative assessm en t o f th e effect o f com position/
ptocess variables in steel o n to its u ltim a te p ro p erties, it is necessary to en v isag e a b lack b ox, cap ab le o f developing a i^tlationship betw een th e variables.
Artificial neural n e tw o rk (A N N ), d ifferen tial equations, if^nltidunensional analyses can act as such black boxes, which [Enable the accurate m ap p in g o f inputs to an appropriate, output i^pacc. Several efforts have been m ade to m odel the m echanical properties o f high strength low alloy (H S L A ) steel using neural l^twork 11-5 j. Even fuzizy logic can m ake up a black box. This is
^ ojresponding A u t h ^
p articularly useful as fuzzy logic is con cep tu ally e a sier to understand. Fuzzy logic i.s capable o f m odelling non -lin ear arbitrary relationships and is highly tolerant o f im precise data.
Since fuzzy logic can be built with the help o f the experience o f expert, the system rem ains highly com patible with the real life situation [6-9], H ow ever the prediction process can be m ade m uch m ore p re c ise by u sin g n e u ro -fu z z y sy ste m , w h ich incorporates adaptive technique to develop the final relationship betw een the inputs and the output variable. Fuzzy inference systems (FIS) use the experts’ km^wledge for prediction, w hereas adaptive neuro-fuzzy inference system s (A N FIS) possess the capacity to utilise the prior know ledge o f the expert and to further refine the results o f F iS through the artificial learning process.
Thus, the lack o f transparency in A N N m odelling, w hich does not take into account the expert know ledge, can be avoided.
A pplication o f neuro-fuzzy system s in case o f fatigue o f Ni- based superalloys 110] and structure-property c o n elalio n in Al- Zn-M g alloys [ 11J are docum ented in recent literatures. E fforts have been m ade by the present authors to apply fuzzy system in assessing the effects o f co m p o sitio n al variables and the therm om echanical control processing (T M C P) param eters on the m echanical propeities o f steel. It has been dem onstrated that the phenom ena can be described in fuzzy inference system s through som e if-then rules 112]. O bservations have suggested
0 2 0 0 5 lA C S
4 7 4 S D o lt a a n d M K B a n e r je e
that a high degree o f precision in p redicting the strength o f steels may not be possible w ithout further im provem ent in PIS.
A N FIS is capable o f taking care o f this gap in FIS m odelling through adaptive learning. A s it is very useful to assess the u n c e rta in ty in th e q u a n tita tiv e d e te rm in a tio n o f stre n g th p ro p e rtie s o f H S L A s te e ls as a fu n c tio n o f its c h e m ic a l com position and T M C P param eters, attem pts have been m ade to study the effect o f com position and process param eters on the m echanical properties tifH S F A steels by the application o f neuro-fuzzy system s.
2. Database
T he composiMonal variables viz. carlx»n (C'), m anganese (M n), silicon (vSr), nickel (Ni), copper (Cu). m olybdenum (M o), niobium (N b), chrom ium (Cr), titanium ( I i) and boron (B ), the process variables like slab reheating tem p eratu re (SR T), percen tag e deform ation in d ifferent tem p eratu re zones (designated as D l . D2 and D 3), finish rolling tem perature (FRT) and cockling rate (CR) are u.sed as input variables and yield strength (0.2% p ro o f stress) is u.sed as the output variable. T h e alloys used for the present w ork have been p repared m the laboratory and then control rolled in a laboratory scale tw o high rolling m ills with 10 H P motor. T he m echanical properties have been mea.sured in INSTRON 4204. Sim ilar data from published literatures have also been included to develop a datab ase w ith w ide variations. The range o f variables used in the H SLA steel data is described in Table I .
T a b le 1. T he m inim um and m axim u m lim its o f the param elers
P aram eters M in im u m M a x im u m
C 0 0 .1
M n 0 2
SI 0 0 5
Ni 0 4
C'u 0 2
M o 0
Nb 0 0 . 1
Cr 0 1
T i 0 0 0 5
B 0 0 0 0 3
SRT 1 0 0 0 1 2 5 0
D l 0 3 0
D 2 10 4 0
D 3 1 0 5 0
FR T 6 5 0 8 5 0
CR 0 3 5
U T S 6 0 0 1 2 0 0
YS 3 0 0 1 1 0 0
% cl 1 0 2 5
3. IViodelling technique
Fuzzy inference system enables m apping from a set of inputs to an output space by m eans o f fuzzy logic. The FIS invt^lvcs ( m em bership function, (b) fuzzy logic operator and (c) it rule. A m em bership function (M F ) is a curve that descnbvrs the m apping o f each point in the input space to a membership value betw een O and 1, called the degree o f m em bership ( /v ). 7’hcrc aie q u ite a few types o f m e m b ersh ip functions, viz., triann-ijiai trapezoidal, G aussian, sigm oidal, asym m etrical poJyntmna) o f w hich the G aussian function is the m ost commonly function and has been used in the present work. There are se\ ur.,, types o f fuzzy logic operator, o f w hich the Sugeno-type | , used here. If-then rule statem en ts are uhcd to formulate ih * co n d itio n al sta te m e n ts b etw een the inputs and the nuipius I'h e if-then rule assum es the form .
If V is A then y is 7?,
w here A and B are linguistic values defined by the fuzz\ .seis ofi the specific arrays. In o u r system , w e can .say if carbon is )nu then strength is low, (say).
On the o th er hand, the con cep t o f artificial neural neuv Mk used in the present case for adaptive learning o f the FIS happen.
to be a supervised feed forw ard netw ork trained with stand.,!,:
gradient descent biickpropagation algi>nthms along w ith i!i„
least squares type o f m ethod. In the process o f lcarmn<i (fu erro r o f the calcu lated or predicted output in relation to (hi actual output is b ackpropagated to adjust all the w e ig h t an ! bias values. A n euro-fuzzy inference system m aps the inputs! • an output space in a sim ilar way. It com prises o f niemhei-'hip fu n ctio n , fuzzy logic o p e ra to r and if-then rules The i\i concept is to provide a m ethod for the fuzzy modelling pioeedui to learn inform ation about a data set in order to compute i!k
m em bership function p aram eters that best allow the nssoeiainJ fuzzy inference system to track the given input/outpui data U ltim ately, it c o n stru c ts a FLS w hose m em bership funetK'n param eters are adjusted using certain learning algorithm rhi>
allo w s th e fu z z y s y ste m s to learn fro m th e data (hey an- m odelling. A netw ork-type structure resem bling that ot a nciii ii n e tw o rk th e n m a p s th e in p u ts th ro u g h th e ir membership fu n c tie m s a n d a s s o c i a t e d p a r a m e t e r s , a n d fin ally . m em bership fu n ctio n s and associated param eters o f the ourpm is used to interpret the in p u t/o u tp u t relations. The parameters associated with the m em b ersh ip fu n ctio n s will change thiough the learning process. T h e m o d ellin g approach used by ANH^
is sim ilar to any o f the sy stem id entification techniques prim ary jo b is to hy p o th esise a param eterised model .siruciun.
(relating inputs to m em b ersh ip functions to rules to outputs m em bership fu nctions), then to train the FIS model to emulate the training data p resen ted to it by m o difying the mciTibcrshiP function p aram eters acco rd in g to a ch o sen error criterion.
type o f m odelling yields best results if the training data preseniei to the A N H S for train in g m em b ersh ip function parameters
N e u r o - fu z z y m o d e llin g o f th e stre n g th o f th c r m o m e c h a n ic a lly p r o c e s s e d H S L A s te e ls 4 7 5 toll) representative o f the featu res o f the data that the trained
IS intended to model. So, if a large am ount o f data is collected, ii will contain all the n ecessary rep resen tativ e features. There aic certain other constraints in using A N FIS, e,g. it only supports Simcno-type system s, and that is also with a single output,
which is obtained by using the w eighted average defuzzification -linear or constant o u tput m em b ersh ip functions). M oreover, it caimot accept all the cu sto m isatio n o p tio n s that basic fuzzy inlcrcnce allo w s. T h a t is, o w n m e m b e rsh ip fu n ctio n s and dciLi/zification functions can n o t be used in this system .
The above c o n cep t is used for d ev elo p in g tw o system s, whcic com position and p ro cess p aram eters are used as inpuls ,ind yield strength is used as t)utput. T he database is used to iinin the Sugeno type F IS dev elo p ed on the basis o f som e if- iht n rules relating the co m position and process param eters with the Yield strength o f th crm o m ech an ically controlled processed HSLA steel. B esides train in g the F IS w ith a dataset through a
learning algorithm , all the d a ta w ere subjected to an operation Liillcd Ihe clustering o f num erical data. The purpose o f clustering IS to identify natural gro u p in g s o f data from a large data set to piuduce a con cise rep resen tatio n o f the sy stem ’s behaviour.
The cluster inform ation can be used to generate a S ugeno-type (u//y inference system to m odel the data behaviour using a
m in im u m n u m b er o f rules. T h e ru le s p a rtitio n th em selv es
according to the fuzzy q u alities associated with each o f the data duslcis I 14]. A co m p arativ e study betw een the predictions o f the FIS itself, prediction o f the FIS after neurevadaptive learning and the learning o f the generated FIS through data clustering is
done
4, The N euro-fuzzy m o d els
4 I R elation o f s o m e a llo y a d d itio n s w ith y ie ld stren g th [\N F I S -l):
Here the role o f som e alloy ad d itio n s viz. niobium , titanium , copper and borem, in the enhancem ent o f yield strength o f HSI >A
steel have been m odelled w ith the help o f prior know ledge o f physical metallurgy.
It has been found by earlie r w orkers that boron, as a single itddition, has no ap p reciab le e ffe c t on the strength o f H SL A steels. Although it in creases th e bainitic fraction, it increases the a grain size at the sam e tim e [15]. If boron is added in
^combination with niobium , a rem arkable im provem ent in strength can be achieved. It is also rep o rted that the presence o f niobium and boron in co m b in atio n re ta rd s th e recry stallizatio n o f grains during TM CP [16]. N iobium and boron exerts a synergistic effect
^’n the non- re c ry s ta lliz a tio n te m p e ra tu re o f au ste n ite and increases it by about 25®C. T h is is ascribed to the non-equilibrium 'vegrcgation o f boron at d islo catio n s and to the form ation Nb-B complexes [17]. A ddition o f Ti w ith B has alm ost identical effect, although the grain re fin e m e n t is relativ ely less significant than
Nb-B steel [18]. Though the strengthening m echanism o f copper is different from that o f niobium or titanium , it also exhibits .synergism with boron in a therm om echanically proccs.sed H SLA steel, as the precipitation o f copper is delayed in boron treated Steels. So m icroalloying o f niobium andA>r titanium in boron treated copj^er bearing H SL A steel has m anifold effects on the strength property o f steel 119-22). This understanding has been
; used to develop a model o f the relationship o f these m icroalloying , elem ents with the yield strength o f the steel.
Regarding the data, niobium , titanium , copper and boron is {taken w ith in the ra n g e as sla te d in T able J. R e st o f th e
\ com position o f the steel is 0.06 wt% C\ J .38 wt% Mn, 0 .30 wt^X I Si, 1.12 wi% Ni, 0.55 wt9f M o and 0.78 Cr. The fixed process i param eters in all cases are slab reheating tem perature (SRT)
1150”C, deform ation in the recrystallisation tem perature range (D J) 3 0 ^ , in the non recrystalliscd tem perature region (D 2), , 20% and in the (o: + y ) tw o pha.se region (D 3), 25% . T he finish
rolling tem perature (F"RT) is taken as 750‘*C and cooling rate is 30‘^C/s. The resultant yield strength (YS) o f the steel has varied betw een 800 to I 100 M Pa. P rim arily the pn>cess o f a u to generation o f FLS through data clustering wa.s done to achieve m inim um error level. The output values were clustered to five groups, VIZ. low, low-medium, medium-medium, high-medium and high. Sim ilarly all the inpuis are also clustered into five groups (F igure 1). W hen the A N FIS is trained w ith lin ear o u tp u t membership function (a feature o f Sugeno type FIS) the inference
Copper (5)
F ig u re 1. Stfuctiirc ol ihc A NPIS I generated by data clustering technique With five ruic.s.
system is found to result into m crgerance o f a few input m em bers together thereby leading to a less num ber o f m em berships. Thus, it may be noted that the m em bership functions o f niobium , titanium and boron can be virtually treated as three m em bership functions, viz. low, m edium and high after the training o f the A NFIS is over whereas in ca.se o f copper it is four, viz. low, low- m edium , high-m edium and high. T he five rules developed by the FIS a r e :
(i) if N b is low and Ti is low and B is low and C u is low then YS is low.
4 7 6 S D a tta a n d M K B a n e rje e
(ii) (iii) (iv)
if N b is low and Ti is m edium and B is medium and Cu is low-m edium then YS is low -m edium ,
it'N b is low and Ti is medium and B is m edium and Cu is high-m edium then YS is m edium -m edium .
(V )
if N b is m edium and Ti is m edium and B is high and Cu is high then YS is high-m edium ,
if N b is high and Ti is high and B is high and Cu is high then YS is high.
The average error value o f the A N FIS after training is found to be i 2.5 M Pa (Figure 2), w hich is m ore o r less acceptable.
F ig u r e 2. A ctual predicted y ie ld strength o f A N F IS J generated through data clustering with fiv e rules (after training).
In another exercise, an inference system is auto-generated with all the inputs and the o u tp u t values clu stered to seven groups, viz. very-low, low, low -m edium , m edium -m edium , high- m edium , high and very-high. H ere also, FIS is found to have m erged som e o f the m em bership functions o f the inputs after training and each o f niobium , copp>er and boron has been divided into four m em bership functions (against the seven taken initially) viz. low, low -m edium , h ig h -m ed iu m and high, m em bership functions; w hereas in the case o f titanium the total num ber o f membership functions is five (low, low-medium, medium-medium, high-m edium and high). T he seven rules is seen to have reduced to five after the training o f the A N FIS and these are:
(i) if N b is low and Ti is low and B is low and C u is low then YS is very-low,
(ii) if Nb is low and Ti is low-medium and B is kiw-medium and Cu is low -m edium then YS is low,
(iii) if N b is low and Ti is m edium -m edium and B is high- m edium and Cu is h igh-m edium then YS is low- medium,
(i v) if N b is low -m edium and Ti is m edium -m edium and B is high and C u is high-m edium then Y S is m edium - medium,
(v) if Nb is low-medium and Ti is medium-medium and B is high and Cu is high then Y S is high-medium.
(vi) if N b is high-m edium and Ti is high-m edium and B is high and C u is high then YS is high,
(vii) if N b is high and T i is high and B is high and Cu is
high then YS is very-high.
The average error value is further reduced to 10.3 MPa (Fjguie 3). It is noted from the above tw o cases that increasing the
num ber o f clusters and rules leads to a reduction in the error level. B ut further increase in clusters or rules will reduce the
8 5 0 9 0 0 9 5 0 1000
Actual YS
1050
F ig u r e 3. A ciu a l v e rstis prcdiclecl y ield strength o f A N F IS I genet,iicfl through data clustering with .seven rules after training
advantage o f fuzzy system s, w hose beauty lies in describing j system through sim ple linguistic expressions. So, taking clue
from the above auto -g en erated F IS , it is decided to stick to the
system o f seven m em bership functions for the output and tnui for the inputs. I'h e com plicated rules generated by FIS through
data clustering are m ade to look a bit sim plified so that, with the
understanding o f the physical m etallurgy o f such steels, one
can easily prepare the rules. T he rules are as follows:
(i) if N b is low and Ti is low and B is low and Cu is 1o\n
then YS is very-low ,
(ii) if Nb is low and Ti is low and B is low-medium and C u is low -m edium then YS is low,
(iii) if N b is low -m edium and Ti is low -m edium and B is low -m edium and C u is low -m edium then YS is low medium,
(iv) if N b is low -m edium and T i is low -m edium and B is h igh-m edium and C u is high-m edium then Y S is
medium-medium,
(v) if N b is high-m edium and Ti is high-m edium and B is high-m edium and C u is high-m edium then YS is high- medium,
(vi) if N b is high-m edium and Ti is high-m edium and B li' high and C u is high then Y S is high,
(vii) if N b is high and Ti is high and
B
is high and Cu is high then Y S is very-high.N e u r o - fu z z y m o d e llin g o f th e stren g th o f th e n n o m e c h a n ic a lly p r o c e s s e d H SL A ste e ls 4 7 7 Figure 4 show s th e su rface view s d epicting the relation
je d u c e c l by the fuzzy system . W hen the system is used to plot
ihe predicted v ersu s actu al o u tp u t v alu es b efore learning,
d e p e n d in g only on the rule base, it w as found tf) show a huge
prediction error (average testing error 54.2 M Pa) (Figure 5). After
learning through hybrid algorithm the error value reduces to 1 ^ 9 MPa (Figure 6), which is reasonable for all practical purposes K)i an output range o f 800 to 1100 M Pa.
0 04
(a)
0 04
(b )
I
I
J'iRUre 4. Surface 'll. (b) Nb-B and
0.04
( c )
view o f the relations betw een yield strength and (a) Nb>
(c) T i-B .
I
U’F ig u r e 5. Actual predicted yield slicn g ih o f A N P IS 1 with se v en I lilie s , g e iu 'ia le d on the b a sis ol iiie ta lliiig ic a l u iid crsia n d in g , (b e fo r e I n a m in g )
Target Y S
F ig u r e 6. Actual piedietod yield sticn glh o f A N PIS I with se v en rules (after training)
4.2. Relationship o f com position a n d p rocess param eters with yield strength (A N F /S-II) :
T he re la tio n s h ip s b e tw e e n th e c o m p o s itio n and p ro c e s s parameters with the strength are quite complicated. The available data have ten compt^sitional and six process param eters to model the strength o f Iherm om echanically processed H SLA steel. B ut It is difficult and in som e way im practical to develop a single ncuro-fuzzy system relating all these variables. T he num ber o f rules that will be required to defin e such a system w ill be enorm ously high io m ake the A N FIS predict reasonable results.
So, all the inputs and th eir relations with the strength w ere divided into four subsystem s, the outputs o f those subsystem s were then com pounded into a final A N FIS to give the final strength o f the steel (Figure 7).
4.2 J . C ontribution o f carbon, m anganese, silicon, nickel a n d chrom ium (A N F IS IIA) :
C arbon, m anganese, silicon, nickel and chrom ium are the fiv e elem ents m ost com m only presen t in steels. C arbon increases the tensile strength considerably by increasing the am o u n t o f
4 7 8 S D a n a a n d M K B a n e rje e
pcarlite in air-cooled steels, but it does so at the expense o f w eldability, cold form ability and toughness. In hot rolled air ccK)led steel carbon exerts a negligible effect on the yield strength
ANFIS
HE f(u)
7 rules E
YS(7)
F ig u r e 7. The structure o f the A N FIS II.
123]. T he prim ary beneficial effect o f m anganese is its affinity for sulphur, which prevents the form ation o f the detrim ental intergranular iron sulphides. In ad d itio n to this, m anganese c a u se s so lid so lu tio n h ard e n in g . A s a c o n se q u e n c e o f its austenite stabilizing effect, m anganese depresses the y or tra n sfo rm a tio n te m p e ra tu re, and th u s re fin es the cir-grain, especially on rolling with large am ounts o f deform ation in the low er tem perature range. H igh addition o f m anganese increases yield strength m arkedly d ue to tran sfo rm atio n hardening [24].
O n the other hand, nickel and chrom ium enhances the strength o f H SLA steel through solid solution hardening but also aids in precipitation o f niobium carbides.
So from the available d ata, the contribution o f each o f these elem ents to the strength property has been calculated and is
norm alised w ithin a range o f 0 to l. T he yield strength of the steel due to the m inim um and m axim um addition o f each ol the above elem ents, with o ther elem ents and process parameters rem aining constant, are designated as and x^,^^ for that elem ent. T hen any value o f yield strength, say jc, for specific value o f any the above elem ents is represented as the normalised contribution o f that elem ent to the yield strength, by the relation.
(I) As in the preceding case, the inputs are again divided imo four m em bership functions and the output is divided into seven m em bers. T he seven rules designed to explain the system is stated below :
(i) if C is low and M n is low and Si is low and Ni is |t>w and C r is low then YS is very-low,
(ii) if C is low and M n is low and Si is low-medium and Nt IS low -m edium and C r is low-m edium then YS is low, (in) if C is low -m edium and M n is low -m edium and Si is
low -m edium and Ni is low -m edium and C r is lou m edium then YS is low-medium,
(iv) if C is low -m edium and M n is low -m edium and S i is
high-m edium and Ni is high-m edium and Cr is high medium then YS is m edium -m edium ,
(v) if C is high-m edium and M n is high-m edium and Si is high-m edium and Ni is high-m edium and Cr is high m edium then YS is high-m edium ,
(vi) if C is high-m edium and M n is high-m edium and Si is high and Ni is high and C r is high then YS is high.
(vii) if C is high and M n is high and Si is high and Ni is high and C r is high then YS is very-high
F ig u r e 8 . S u r fa c e v ie w o f th e r e la t io n s b e tw e e n y ie ld strength an(J (a) C -M n, (b) C -S i, (c) C -N i and (d) C-Cr.
N e u r o - fu z z y m o d e llin g o f th e stre n g th o f tlie n n o m e c h a n ic a lly p r o c e s s e d H S L A ste e ls 4 7 9 The relations betw een the strength and the concentration
ol ihc five elem ents are show n through surface plots (Figure 8).
Belore training, the data show an average error equal to 0.142 was obtained. A fter training the A N F IS , the error is finally
to 0.005 (Figure 9).
r y’-*
Target Y S
Fi^urt* 9. Actual versus predicted contribution to yield strength o f AN HIS IIA altci training.
4 2.2. C ontribution o f co p p er a n d m olybdenum (A N F IS
flB) :
A m o n g the s u b s titu tio n a l a llo y in g e le m e n ts , c o p p e r and molybdenum is m ost co m m o n and both have significant effects
on ihc yield strength o f H S L A steel. O ther than solid solution sticn g lh en in g , c o p p e r is k n o w n to p ro d u c e p re c ip ita tio n hardening w hen added in hig h er am ount. From the available
d a ta , the c o n trib u tio n s o f th e s e tw o e le m e n ts h av e been separated and are norm alized w ithin a range o f 0 to 1. The inputs
are divided into four m em b ersh ip functions and the output is divided into seven m em bers. T he seven rules designed to cx.plain
the system is stated below :
(i) if C u is low and M o is low then YS is very low, (ii) if Cu is low and M o is low -m edium then YS is low, (lii) if C u is low -m edium and M o is low -m edium then YS
is low -m edium ,
(iv) if C u is low -m edium and M o is high-m edium then Y S is m edium -m edium ,
(V) if Cu is high-m edium and M o is high-m edium then YS is high-m edium ,
(vi) if C u is high-m edium and M o is high then YS is high, (vii) if C u is high and M o is high then YS is very high.
The surface view in F ig u re 10 show s the relation betw een cop|>er, m olybdenum and Y ield strength. H ere, the data show an average erro r equal to 0 .1 1 9 b efo re training. The trained ANFIS has p re d ic te d o u tp u t w ith a v e ra g e e rro r o f 0.0 0 8 (b'igure 11).
F ig u r e 10 . Surface view of the lelatio ns between copper and yield strength
1.5
m olybdem irn
Target Y S
F ig u r e 11. A ctu a l v e r su s p red icted c o n trib u tio n to y ie ld stren g th by ANF-IS IIB (after training)
4 .2 .3 . C o n tr ib u tio n o f n io b iu m , tita n iu m a n d b o r o n (A N F IS I I C ) :
Niobium , titanium and boron are the mo.st im portant m icro- alloying elem ents in H SLA steel due to their ability to strengthen steel through grain refinem ent as well as precipitation hardening.
S im ilar to the earlier case, the inputs are divided into fo u r m em bership functions and the norm alised output is divided into seven m em bers. The seven rules designed to explain the system is staled below:
(i) if N b IS low and Ti is low and B is low then YS is very low,
(ii) if Nb is low and Ti is low-medium and B is low-medium then YS is low,
(iii) if N b is low -m edium and Ti is low -m edium and B is lo w -m ^ iu m then YS is low-m edium ,
(iv) if N b is low -m edium and Ti is high-m edium and B is high-m edium then Y S is m edium -m edium .
4 8 0 S Dcitta a n d M K B a n e r je e
(v) if N b is high-m edium and Ti is high-m edium and B js high-m edium then YS is high-m edium ,
(vi) if N b is high and Ti is high-m edium and B is high then YS is high,
(vii) if N b is high and Ti is high and B is high then YS is very high.
B efore training, the data show an average error equal to 0.142, w hen the A N FIS is trained, the eiTor is reduced to 0.022 (Figure 12).
Target Y S
F ig u r e 1 2 . A ctu a l v e r u ts p red icted c o n lr ib u tto n to y ie ld stren gth by A N FIS lie: (after training).
4 ,2 .4 . C o n t r ib u ti o n o f p r o c e s .s p a r a m e t e r s ( A N F I S H D ) : The process param eters viz, slab reheating tem perature (SRT), percentage deform ation (D), finish rolling tem perature (FRT) and cooling rate (C R ) are related to strength through the rules
(i) if SRT is high and D is low and FRT is high and CR is low then YS is very low,
(ii) if SRT is high-m edium and D is low -m edium and FRT is high and C R is low then YS is low,
(iii) if SRT is high-m edium and D is low -m edium and FRT is high-m edium and C R is low -m edium then YS is low-medium,
(iv) if SRT is low -m edium and D is high-m edium and FRT is high-m edium and C R is low -m edium then YS is medium-medium,
(v) if SRT is low -m edium and D is high-m edium and FRT is low -m edium and C R is high-m edium then YS is high-medium,
(vi) if SR T is low and D is high and FR T is low -m edium and C R is high-m edium then Y S is high,
(vii) if SRT
is
low and D is high and FR T is low and C R is high then YS is very high.R elation betw een the process param eters and yield strength is show n in F igure 13. B efore training the A N FIS predicted un average error equal to 0.089. W hen the A N FIS is trained, the error is reduced to 0.018 (Figure 14).
Deformation 1000
1150
1110 11
S R T
1200
( a )
250
Deformation
( b )
F ig u r e 13. Surface v iew t)f the relation s iKMwcen y ield ^iiengih .uiU (.d slab reheating temperature and deform ation pereent, and (b) finish mllmj tem perature and c o o lin g rate.
0 4 0 0 0 A
0 .4 0 0 0 0 .5 0 0 0 0 6 0 0 0 0 7 0 0 0 0 .8 0 0 0 0 9000 Target Y S
F ig u r e 1 4 . A c tu a l ver.su.s p r e d ic te d c o n tr ib u tio n to y ie ld strength h>
A N F IS I ID (after training).
4,2,5, In te g ra te d m o d e l f o r y ie ld stren g th a g a in st all inpuf variables (A N F IS HE) :
In the above fo u r neuro-fuzzy m odels (A N FIS IIA to IlD).
outputs m ay b e con sid ered as the co ntribution o f elements (m
Neuro-fuzzy modelling of the strength of thermomechanically processed HSLA steels 481 the first three A N FISs) and process parameters (in the fourth
ANFIS) towards the overall yield strength o f the steel. It is seen that the outputs o f th ese four sy stem s have individually described the contributions o f carbon-manganese-silicoivnickel- chrom ium (C e t c ) , copper-m olybdenum (Cu-M o), niobium- iitanium-boron (N b -T i-B ) and the contribution o f process pai ameters (proc.param.) towards the final yield strength value.
T h e s e outputs ii . e , , the output o f su b sy stem s) are then integrated in ANFIS HE to find out the final yield strength o f the experimental steels. Therefore, these outputs o f the subsystems c o n s titu te th e inputs o f A N F IS H E , w hich a ssig n s fo u r memberships to each input. H owever, the output is divided into
s e v e n members. The rules formulated are as follow s :
(i)
(ii)
(iii)
(IV)
(V)
if C e t c is low and Cu-M o is low and Nb-Ti-B is low and process parameters is low then YS is very low, if C e t c is low-m edium and Cu-M o is low-medium and Nb-Ti-B is low and proc.param. is low then Y S is low,
if C e t c is low-medium and Cu-Mo is low-medium and Nb-Ti-B IS low-medium and prcx;.param. is low-medium then YS is low-medium,
if C e t c is high-medium and Cu-M o is high-medium and N b-Ti-B is low-m edium and proc.param. is low- medium then YS is medium-medium.
if C e t c is high-medium and Cu-M o is high-medium and Nb-Ti-B is high-medium and pn>c.param. is high- medium then YS is high-medium,
if C e t c is high and Cu-M o is high and Nb-Ti-B is high-medium and proc.param. is high-medium then YS is high,
if C e t c is high and Cu-M o is high and Nb-Ti-B is high and proc.param. is high then YS is very high.
fhe ANFIS HE predicted an average error equal to 58.51 MPa before training (Figure 15). The trained ANFIS prediction
(vi)
(VII)
has shown a reduced error level o f 12.5 MPa (Figure 16), which IS quite acceptable for all practical purposes.
I'lRure 15. A c tu a l versus p re d ic te d y ie ld .strength b y A N F IS HE (b e fo re blaming).
F ig u re 1 6 . A c tu a l »*crsi/s p rc L lictc il y ie ld s l i c i i g t h b y A N F I.S H E a f t e r i r a i m n g
5. Discussion
When the process o f learning is applied tc» the auto-generated neuro-fuzzy system, it has shown a good perfttrmance from the prediction point o f view (Figures 2 and 3). Both the figures have shown a good match belv/ecn the predicted and the target values.
Moreover the error level in the range o f 10 -12 MPa for steels o f above 800 MPa yield strength is considered to be acceptable for all practical purposes. This particular way o f designing a fuzzy system through data clustering can be u.seful to gather knowledge about the relationships between the inputs and the outputs from som e raw data w here prior k n o w le d g e is nonexistent. Generation o f neuro-fuzzy system through data clustering can even be u.sed to gel a piimary idea about the modelling o f a particular relationship, as is done in the present case. Here, in the case o f ANFIS 1, the data clustering technique has been used to optimise the number o f membership functions for the input and the output variables and to get an idea about the rules to be ftirmulaied. Then the membership functions as well as the rules are rationalised by follow ing the concept c^f physical metallurgy o f steel and the final result is found to be quite encouraging (Figure 6). The three-dimensional surface view s generated from the ANFIS I after training is also in connivance with the existing understanding o f the effects o f these three elem ents with the yield strength o f HSLA steel (Figure 4).
It is seen from Figures 2 and 4 that the prediction error level decreases with increase in the number o f rules. But the limitations o f using an ANFIS is that the number o f if-then rule.s has to be exactly equal to the number o f memberships o f the output and each o f these rules is assignable to only one o f the memberships.
It is disadvantageous for all practical purposes to divide an output range to a large number o f membership functions and thus the possibility o f de.scribing the com plicated relations
482 S Dana and M K Banerjee
betw een the inputs and the output shrinks. A s a result, predictability o f FIS has been found to be pH_^or when no training IS carried out (Figures 5 and 15).
As discussed earlier, the increase in the number o f rules leads to an improvement in the predictability o f the system.
This envisages the need to gather a thorough understanding of the relations betw een the inputs and outputs in order to su ccessfu lly design a neuro-fuzzy system . This is in total contradiction to the conventional concept o f artificial neural netwf^rk, where the network draws the relations between the inputs and the output without having a consideration o f the physical significance o f the system. Synthesisation o f neural network and fuzzy system has however, acquired the ability to overcome the so-called limitation o f neural network in modelling a metallurgical process with due em phasis on the existing knowledge.
On the other hand, the dependence o f a neuro-fuzzy system on the if-then rules, t.e, the prior understanding o f the relations, pose a definite limitation for m odelling a neuro-fuzzy system with large number o f input variables in the field o f materials science. As in the case o f the present exerci.se, the complex rclation.ships between the com position and process variables with the mechanical properties o f H SLA steel is difficult to be defined through a few if-then rules. It is necessary to formulate a large number o f rules to design an effective neuro-fuzzy system.
Unfortunately, this becom es a rigorous exercise and may not be considered suitable for practical use. With a view to overcom e this limitation, a process o f developing sub-clas.scs has been introduced here. All the input variables are divided into four sub-classes according to their resemblance in their manner o f stren g th en in g the ste e l. From th e a v a ila b le data, their con trib u tion s to the strength o f H SL A steel have been separately identified. The normalised values o f the contributions are used in the four separate neuro-fuzzy system s (ANFIS IIA to ANFIS IID). After training, the average prediction errors o f these neuro-fuzzy system s are seen to be quite low (Figures 9,11,12 and 14). This is so possible as each o f the individual system s are precisely dcscribable with a fewer number o f if- then rules.
A close look on the surface view s generated from ANFIS IIA (Figure 8) after training will reveal that the effects o f the elements on strength are plotted with carbon, the most common element present in the steel. The plots show that the individual strengthening effects o f silicon, nickel and chromium are quite less than that o f carbon (ANFIS IIA). In case o f relationship between copper and molybdenum with yield strength in ANFIS IIB (Figure 10). it is found that the contribution o f molybdenum towards increasing the strength o f the steel is appreciably higher than that o f copper. The copper addition also reaches saturation in its strengthening effect when added beyond 1.2 wt%. The surface views o f niobium, titanium and boron are almost similar
to that of ANFIS I (Figure 4), The surface views for the
processparameters show that higher slab reheating or finish
rollingtemperature has a negative effect on the strength (Figure I3j Due to increase in deformation percent, the strength is seen
U)rise sharply, whereas for increasing the cooling rate, the .strenuih value reaches a plateau after attainment of a rate more than jh.a due to oil quenching.
T h e o u tp u ts o f th e fo u r sy ste m s w ere fu rth e r used in ilic in te g rate d n e u ro -fu z z y sy ste m (A N F IS H E) to ob tain the 1‘inai p re d ic tio n ab o u t th e y ie ld stre n g th o f th e steel. H e re also a laojc im p ro v e m e n t in th e p re d ic tio n e rro r Ci)uld b e a c h ie v ed ihroicjli le a rn in g (F ig u re s 15 and 16). A N F IS IIE is a b le to dem onstrate th e fe a sib ility o f h a n d lin g a la rg e r p o p u la tio n o f input variables in th e s u c c e s s fu l d e s ig n o f n e u r o - fu z z y s y ste m by wa> o|
fo rm u la tin g sm all siib-clas.ses w ith fe w e r n u m b e rs o f variable^
in e a ch su b -c la ss.
W h e n the s u b c la ss e s fo rm an A N F IS in a c co rd an c e witi]
F ig u re 7, th e e rro r v a lu e can b e b ro u g h t dow n to a significam b low lev el, e .f * ., 12.5 M P a in steel o f yield stre n g th above SOd M P a. S im ila rly , F ig u re 16 sh o w s th at a very go o d agreemcfH b e tw e e n th e p re d ic te d an d th e ta rg e t v alu e o f yield slreivjth is a c h ie v a b le by th is ty p e o f n e u ro -fu z z y sy stem . It thus ap]-»c.n'.
that th e a p p n 'jp ria te d e s ig n o f a n e u ro -fu z z y sy stem enables u>
m o d e l a c o m p l i c a t e d s y s te m t)f n o n -lin c c u in p u t-o u tp u t re la tio n sh ip , e v e n if th e re e x is ts no p rio r k n o w le d g e about tiK*
sy ste m .
6. Conclusion
(i) N e u r o - f u z z y s y s t e m g e n e r a t e d t h r o u g h duta c lu s te rin g ca n .show^ a g o o d p e rfo rm a n c e from (l"‘
p re d ic tio n p o in t o f view . It can be u sed to gel a p rim a ry id e a a b o u t th e m o d e llin g o f a paitieiiiai re la tio n sh ip .
(ii) T h e increa.se in the n u m b e r o f ru les
improves
the p r e d ic ta b ility o f th e .system.(iii) A system with a large number of input
variables eanbe successfully designed in neuro-fuzzy system
byway of formulating small sub-classes with fewei numbers of variables in each sub-class.
(iv) A p p ro p ria te d e s ig n o f a n e u ro -fu z z y sy stem enahle^
to m o d e l a c o m p lic a te d sy ste m o f n o n -lin e ar inpui o u tp u t r e la tio n s h ip , e v e n i f th e re e x is ts no k n o w le d g e a b o u t th e sy ste m .
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