Forward and reverse modeling of electron beam ... - Semantic Scholar

Forward and reverse modeling of electron beam welding process using radial basis function neural networks Vidyut Dey, Dilip Kumar Pntihar* and Ci .L. Dattn D c p ~ v t m e ncij f iJVccl~cr)rir~l E n ~ i l ; ~ r r i r ihdiutr g, br.rtitritr, ~ f ' T c c h n n l o ~Ki:llc~r-ngptcr.??l v, 302, l~dirl

.4bstract. An atrznlpt has k c n madr In ihr preqent i t u d ~10rntdcl 111pur-oulpurrelalir?nshi]~sot'.ji~clcclrt~r)k a m wclding prtkchs in both tunvard as well ss rcvcr.;u Cii~~ecti,)rl~ using radial haxi+ t'ur~cr~on nei~rslnctwt>rks. The performance ot this ntt work is dcpcndenr on irs aruhiiecrurc wgr~iticat~rly, which. in lum. depzr~ds,MI the number nf hiddcn neurons, as the n u m b u! input nodes and that of output neurl)n\ be decidcd hcir3rch~ndit rr modeling a par~ivi~lar process. l11pi11-ot~tput data can hr c(i15rcredbascd on t h e ~ similarity r among thet11. The number of hidden neurons of this nrluvrh i l generally kcpt alui~l10 that t7i clusters made hy ihe &la-szl. Two popular f u ~ z yuluslerir~galgcrr~rhrns,namely U U L L ~ C'-tncal~\and m~ropy-hascdfurz! clusrering h a w k e n u s d for grouping t l ~ cdata into some clu.c~rn.4s hnth these algorithms hate rr~hel-e~lr limilations, a m;)dificd clustering slguritbm ha' bcm propcasd by eliminating rhrir demen 1s iitld combining their ;~dvnntages.Radial hnsis funsttam neural network devtlopd using the prop~xedclustering algorithm is found to perfr)rm M t e r th:in [hat dcsipntd h:~$ctlon the ahorc lwa we!l-kr~nu.~~ clvrrering algorithms. Kcywordh: Electron keain wclding. Radial 0 s > 1Fi11:stion ~ Nzul-al Nt.l..cork5. F o r w ~ r dhIappinp, Rcvcrse Mappir~g.Clustcrin~

Nunlher of input notles kept equal to rht. dimtnsiol~of thc , data Nurrtbcr of hidden neurons ma rang on^ nurrlber Ni~rnbcri)l'null>utneurons N u m b e r of lrainrng scenaricw ur data pninn Population si7.c Calculated o u t p u t ul'the output Ihr ! , I h

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Cun~ranlIn wmilarity-d~c~;lni.r:

rel;l~ron.;hip Cuordinares of polnt P 1 ckn head profilc Coordinates of pulnt Pi un head profile Number ol'clusrrr~ Eucl~dcandir1ani.e herween I", data point and ,if'' clusicr ccnlcr Drk131:onin p r e d i c t i ~ na1 kt/: output neurtms F,flrrr)py c l f a data point X: Fitness l a l u e o f a GA->elution 1-eve1of zluq~crf'lrzzincss Maximurn nunlher of g e n c r ~ l i o n c

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t: Td : *ComsprbnJin,r aurhor Tel : +91 3222 ?Y?Ocil: F a x , +Y 1 3212 182278: E-r~lail,[email protected]~.ernaIn.

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P r u b a h ~ l ~i)f i j rnutation Peclet nurnbcr Randtl n u ~ n h r r S ~ m ~ l a r ibctwccn ty dara point, 1 and , j Iteration numbcr Txrpet ourput of kt';ourput rorpth train~ng case

GA: GMA:

Input data s c ~ Membership rnatrix A matrix denoting thc tra~nlng scenarios Momeniurn \ern1 Threshold value of si~nilarity Learning rate Constant of log-sigmoid transfer function Artificial Neural Nerworks Auslen~tii:SLainlcss Sleel Bead Hcighl Bead-on-platc Bead Penetration Bead Width Central Composite Design Canuolled Random Search Elec'lrnn Bearn Electron Beam Welding Enrropy-based Fuzzy Clustering Fuzzy C-Means Gaussian Radial Basis Fu~lc~ion Neural Ne~wilrls G c n c ~ i cA lgor~~lim Gas M m l Arc

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CCD: CRS: EB :

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Rad~alBasis Funz~ionNeural Ne~wilrki

RSM : SAON:

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R z ~ p o n wSurt'kt: Methodology Self Adaptive Ollfsrt Nerworkb Vic hers Hardness

1. Introduction bltectrun Beam l'telding (EBU') is J ~ Iautagrnous welding process, in which the momentum (!I' highly accelerated electrons is converted into heat on striking the work-piece. This heal is so immense that it fuscs ~ h pxts c lo bc joined without u s ~ n gany filler material. Thc montllithic rc-sol~dificdstruclurc fcrmed al the junction bctwccn thc two parts to be joincd is popularly known as jksio11 zone. The material surrounding this re-solidified tnonolithic structurt: cannot remain aloof from the influence of the in~enseheat and also gets metallurgicatly ilffccted. This porrion of rnatenal is called heat ofiected : o w pclpularly known as HAZ. To successfully join two parts together, weld~nghas to be carried out at aproper comh~nationof different welding parameters, Mechanical properties of a welded joint depend o n various metallurgical features and geome-

try nf the weld-had, which, in lurn. are dependent on inpul process paramctcrs. Thus. i l rnigh~he required to eswblish input-output rela~~onships of lhis process in order to have a proper con~rol~ r l 'thc sarne In fvrwqrd mapping. outputs of h proccss are expressed 3s

the functicln of input paramelers, whereas the ]alter ih rcpresentd i n terms of the fonncr in rcvcrse mapping. To automate any procelc. ilc inpul-output relal~onships arc tu be known in bolh forward as well as rcvcrsc Jirections. It 1s a challenging ask. which haq attracted a large numbcr of rcscarchers, recen~ly.

2. Literature review A prclpcr undrri~andingt ~the i de\rl~prnrtitof wcld pool shape in EBU' had dlrvays hzen A difticult tilsk. Dud to inhercnt i.c~mplcrilyof rhr process and a huge amounr of running cclst of thc EB W sct-up, liirec.~cxprrirnental investigations m only a fcw. Petrov et d l . [ I I iilrncci \hc formarion of the shape and size of the w d d poul along w i h h e keyhole with a charged cc~uplcdcvice camera, during an exper~~nental investigatlon. They noticed that with the increait: in welding speed. ihc pool width decreased. The shapes of the weldjng pool and keyholc wcrc apparenrly asymmetrical. There had buen many theoretical approaches lo sludy he effect of various physical phcnomcna on wcld-pot)] shape. It had always has heen a challcngin2 task mudel the irlrmaricln and solidification of a weld-pool, as there are loo many physical phenomcne such as turbulence, multi-phase Aow, electm-magnetic snd surface rension flaws. transittit heat lransfers and evapi~ration,plasma formation. X-ray radiation, and others. 4 s a lirst stcp towards ~nathcmatical mtldeling, Roscnthal I21 establ~sheda relationship hctween different welding machine sertings and the geometric shapc c7f the weld pool in 1941. He utilired thu concept of point-source heating tu establish that relationship. In 1965. Hashimoto and Malsuda 131 postulated a rela~ionihiphcrwccn rhc dcprh of penetration, beam parnlnrlrrs and lrlarerial characteristics. They a5sumt.d 1ha1 the hram wn, c,f srluiuc cross-scction and had a conftant panel- dens~ly. No intcrac~iclnbetween the ulcclr~mhealn and ~netalvapor in rhe cap11Ia1-ywas considered in their slud!. Thr fused yeriphcral zilnc M . ~ S assumed 10 he at the mr.lt~ngtrmperaruru. Hcnl ~ I S S I patton 10 thc a~rnl,q?hcrewas considered to he isutrop~c and nu hcu~was assumed lo he transferred by canvection w i ~ h ~~ nh cI'USC~ pcr~pheral7nnz. .4 innre realis-

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uc l o ~ ~ lwas c l developed by E a y r ~ n c Tsal l [4], where they assun~cda Gaussian energy d ~ s t r ~ b ~ tn ~ tpredict ~nn I ~ Lgcc~metric ' shape clf thc wclds. Tlus rrltidel cumplcmented Rosenthal'\ rntdrl as a special casc. when the width of the Ciauqilaa d~btributionwas considtrcd 110 be equal to zero 1fl horh the above mtldcls, it waq assumed t h a ~the hidtlng svurcc was vn ~ h surface z rial). This assumption, howevcr, could not he true fcr :i high energy ilvnsily welding like the EHW. Elmer et al. [5] investigalwj lhc ~nnuenceof vari0115 wclding parameter5 on lhc ~cld-pucll shape for 3 wide range ofenergy densitlrs. In their study. [he weld aspecr-ratio was ut~nsiderrdro LK a iunction of average energ) dcnsity. It was also show11 t h a ~the average cncrpy drnsi~y~vouldhe used to predict the iransitlon of the c h ; l r ~ l : ~ r r ~ sof ~ iacdistributed s heal svurcc. those of a ]in< h c a ~source. '[he critical energy dc.nsi~y which separ~redthe differenl healing mode< was tound to hc mucrial dependent. In case of ASS 304, rtl~swah found 10 be equal to 1 0 . J . m IT?-'. Finall!, rhr ;luthal.s could dcvelop empirical rr l ~ u o n s h ~ pbetween s penerraticn depth and El3 w r l d ~ nparameters ~ for thc distrihutcd, point and line qnurct heating modcs. In their tntdel, thc weld depth r.oulcl he predicted. only if the weld width wah known. Sirlcr [he weld width could ntil be know11 17 priori. they substituted that with rhe focal spot disme~er. The greatest problem associated with t h i ~mudcl w3.s that any error in thu mcasurcmenl or the focal spot diiin~eterof he electroll hcum. would have resulted in an erroneous relationship bciwc.cn welding parameters and weld dcpth. Before conducting nc~ualwelding c)f massive sircd plalcs. generally, the work~npwclding paramelers arc tried out on smallur plate$ ro check the full pcnelratlon. The width of wcldmcn~on the tnal-plates migh~widen due to ovcr-hrat~np. it' the overall width of the trial plates is key1 IUa low value. To overcome this, CouCJrl el a]. 16.71 dcvclopcd an analytical hrat rransfcr mctdel, which madc it possible to estima~ccrillual condi~ i u n sol' wcld-bead widening. Thc mrdrl was equally capable of predicting hlth [he full as well as partla1 penetration welds. The 2D analytiz~ls~ilutinnsuscd vibrated and non-vibrated Gaussian cyl~ndricalor line models of heal sourzc. Howuver. h i s model did nut consider latent heat. cunvection hcat transfer and n l h r ~ properties of lht. ivcldcd rneral that were indcpondenr of lcmpcrature. Rru el a[. [8] developed a 3D numer~calrnoticl at' hc':rt rranster and Huid flow I n a keyhole m d e of ihc EBW. T h c modcl ~tlokinti) account thc variat~oni l i wall tenlpcrature with depth and effect ut hfarangnni

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cunvectinn ijn bcyhole walls. Ci.rnvcctitin was the dominanl mrch;ini\m of heat transfer in he weld yclol and gradicnt ot' surtace tension played an irnp~!rlun~role in the fluid AOH. The effcct of Lorcntz force ivaa t'ound to h e insignificant ct~rnparediu that of Marangoni h r c c I n their mudel. The welding parameters, such as hc:im rildius. input powcr and weld~nsspccd were seen ro havc signiiicant contrihu~ionscxn ~ h cweld pool geometry. In 2005, Ho [9]s~udirdthe electrun beam focusing char;icrcristics o n the ~ h ~ pofr :fusion Lone. The author de\:c)opud an analyt~calrnc~rlclfor a focuscd clectrun h ~ a l nirradii~ti~~g i n ~ ua paraboloid (d'rcvoluticn-shapd cat, ily. Tile influences of focal s p o ~ .ci/c. focal lvcatian relai~vc.to ~ o r k p i e c csurt'acc and bt~n~-ct)nvcrgence angle ntl thr: shapc o f fusion zone wcrc studled. It was 'found r h a ~ he fusion zone uf thc elec.rron-beam was deep and narrow, whcn [he focal spot SIZC and angle uT convergence u cre small. Thc deepest netr ration uccurrcd, when thc 1-oc31SPCII was kept midway inlo thc t l ~ i u k n ~of s s the sheets tir hc wclded. Koleva ct al. 1 101 uunsidcred a Inc!rillg I~nearthermal source w ~ t hthe added effects of turhulcnoe and Marangoni t'orct', w h ~ k the. d z p ~ hand wid& of the wclds were predic~ctl~ i r h Ihe help of nomograms for glren proccss paramelers. There had hccn a few attempis 1 1 1 . I 21. where a cnrrclii~iun for )oinlng ma~crialrundrr dirtcrcn~operatlng ~onditrons.in case ul'lacer ubc.lcllng,was ~riedr!ut wirh some dimensionless nunliurs. 1'0cs~ablishthis. Peulct { P r ) and Mnr;ingclni jlllul rlutltbers wcrc urilized. It wnk slinwn {hat for the materials with high Prandtl (PI) numhcr, hoth Pe and Jfu were h ~ g lheal ~. was transported ~rr~rnarlly by cunvectinn. i d thc resulting weld pcwls were shallotv and wide. Hr!ucver. fora low Prandtl nu~nhcr,{heresulting pooli wcrr drcp and narrow. ~ numzrThe ahovc study gives an ~ n s i p h~~>cotnplex lsal models invtllving heal tramfer and fluid flow fur Eleclron Reatn and Lascr weldrng proccsscs. These ~ncldclscould perform accurijtr quanti tativt: calculations 110 set thc dcsirrd rr.iults hilt [heir applications wcrc 111ni1rddue to scveral facton. Though these models uscd uh~qujlouscquaticws of consertution of mass, momentum and cncrgy, the results dii'icrcd from tltust: obta~nrJerper~rnentallydue lo a lack ot proper inpur vari,ihlzs. Kun~arand Debroy [13] dc~urminedsuch uncertain inpuis by optimizing thcm within crrtn~nIimits. Nexl, they couplrd a Gtncric Algor~thni(CiA) Lo lhc developed mudcl grnerale a number oizombinaiitnns of thosc pararnrterq ro achicvc the target weld g t ometry. The proposed spproauh showed an incrcascll ~rliabiliryof prcdic~ingrhd held featurcs in case of G35 Melul Arc (GMA) welding.

As the geometry of the weld-head was a good indicator o f the quatity o f he welding, researches did not stop a1 thc outuomcs from analytical formulac. Thia could b r duc to improper undcrsiandinp of the complex physical phcnomcna involvcd in rhc wclding proccss or application of inaccurate boundary conditions. Even with a proper understanding nf the physical phenomena and selection of !he boundary conditions often the mathematical models could nor prcdict with a high level of accuracy due to inherent cotnputational complexity. Thus, the need of a non-deterministic approach in modeling of welding process is felt. Amwgst various non-dctcrministic approaches, Arrificial Ncural Nciwarks (AWNS) wcrc widely used to predict the bead parameters 114- 161. Li et al. [ 171 developed a network, where the single output was modified to accornmodatc an offset layer which could offset the inputs. This type of network was called as SelfAdaptive Offset Nerwork (SAON). For the non-deterministic models, the experiments should he pcrformcd in a holis~icmanncr in accordance with some statislical models. If, so done, the modPIS wou Id be capable of predicting, hy interpolation. within the entire search space. The outcome of an experimenr depends upon different experimental faztors having either direct or indirect effecr. Thest nondeterministic methods were found to predict and optimize the weld-bead geometries and fused metal volumes successfully [ 1 a]. Response Surface Methodology (RSM) was successfully ut~lixciIn develop mode1.c fur predicling he eifecl $11' welding paramelcrs on ~ht: heat input and weld-head proiile [19]. Kim CI al. 1201 proposed a method for determining near-optimal setting of welding precess parameters using a Controlled Random Search (CRS) technique. There had heen a few tcchniqucs also, whcrc mult~plcregression analysis and neural networks were used simultaneously. The results of the studies carried ow by Kim el al. 1211, Lee and Urn 1221 and Dutta and Pratihar 1151 showed that neural networks cvuld predict even better than the lincar and curvilinex cquatiilns of empirically developed models. Kim et at. [23], developed an intelligen~ system using SIMULINK of Matlab. based on multiple rcgrcssions and neural networks. The developed system enabled the user to select e ~ t h e regression r model or neural network-based niodel to predict the outpuls. The model could generate graphical ourputs of the weId head geometry predicted by the said algorithms. Olabi et al. 1241 used both back-propagation ncuml network and Taguchi method to design h e i r experiments. Their model was applied to find nut thc opti-

mum lcvds of diflerent welding parameters. The optimal solution was found ro be valid lying in the ranges 01' he welding parameters used for training the neural networks. Taguchi method was medifred by Tarny et al. (25, 261 as Grey-based Taguchi method and could sucucssfully optimizc wclding paramctcrs. Gaussian Radial Basis Function Ncural Nctworks (GaRBFNNs) were also ut~lizedand found to b~:cfficicnt for input-ourpul njodelitjg of weldinp processes [16.27]. Not much work has becn carricd out on f~rwarding ma.pping of EBW process and consequently, there is a scopc of'further study. Moreover, the problems related to rcvcrsc mapping in EBW process have not been reported to the best of thc aulhors' knowledge. To automate this prwcss, thc problcms rclateli lo both forward and revcrsc mappings are ta be solved. In the present study, both the problcms u i forward and reverse mappings have been tackled using a GaRBFNN. whose architecture has bccn dccidcd ~hroughclustering of the data by utilizing some fuzzy clusrering algorithms. The remaining part uf the paper is organized as fol11,~s:Tnols and techniques used in the present study are explained in Section 3. The merhod of data collcction has been discussed in Section 3. Section 5 deals wirh the developed approaches. Results are slated and discussed in Scztion 6. Somc concluding remarks are made in Section 7.

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3. Tools and techniques used Thc tor!ls and tcchniqucs utilizcd in the present work are illustrated in this scction.

Radial Basis Function Neural Nelwork (RBFNN) proposcd by Broomhcad and Lowe [ 2 8 ] , is a special type uf ANN capablc of fitling a surface in multidimensional s p x c aftcr cnsuring thc bcst match to the training data. A schematic view of the same ih shown in Fig. 1. This network consists of an input layer containing a number of nodes equal to rhal of independent

inputs. The hidden layer of the RBFNN contalns a few neurons having the radial basis functions as their transfer functions. Similarly, the autput layer consists of a luw ncurons. whose numkr is kept equal to that of the outputs. The performance of the RBFNN i s dependent nn the number of hidden neurnns, means and standard

dcvia~ionst.f the Gauhhidn distribution< u\cJ a> their transfcr luncrlnns. The nurnhcr of hiddrri neurons of the RBFNN is assumed Lr) hc rqusl ro that 01- clllsters formed hy ttlc iraining data poinls. The Inearl and I;tdndarddeviatiori o f thc Gaussiari dil;tributions may he calculated from the req-wctrvc cluster center and standsrd deviat~onuf the d a points ~ ~ contained in i ~ . Figure 1 shou.5 thc schematic view nf nn RBFNN. in which there are L i n p u ~nodes (equal ((1 ~ h t dimcnsiun ! of the dara), m hidden and n clutput ncunlna. The neuronh lying on thc h~ddcnand uurput laycrs arc assumed ti) havc Gaussian and Iog-; denales thc dcv~rrt~on it? prcd~ct~nn at k t h output neuroni.

The Incan and ndariard deviation valucsclilhr Gausrian distribution> can he updated b~ ilollowlng rhe similar prl)uedurr rrp1aint.d above. ln~crustedreaders ma). rcfcr 10 [29] for a dctailed Jescr~ptiot~ or the rnethod r ~ i Inem and standard dl:[ laricrlr updating of thi: Gauss~an d~zrrtbur~ons. ,

3.2. Gr~rrhricnlgoiirhtti (GA) Genetic ,4lgorithrn is a populaiion-hased semch and uptimt7acion technique, which wvrbs I-lased on rhe 'mechanisms of natural gcnc~issdnd Darwin's principle ot natural selectiim. The GI\ iiarls with a po~ulsticln I~t'st~lurions created a1 random. Thi: ~olutn~ons are then mvtiified using vu'iou> crperators, n a m c l reproduction, crossrwC1-.3nd inutation 12'31, ah rliscuqsrd h c l ~ ~ . Repruduciion

In a barch Inode of irdining, all the tralnlng r.a>cs art. passed through the netuork onc aher anr)lllrr. and thc average vnluc ul'mean squared deviat iun In prrdictlons can b t delermincd as folluws 1291.

where iV dcnotzs tbr number oi wining cases: Tea, and O,,*,,icprcsenl the target and ci~lcula~ed outputs c d kl.I' output for p'h l r a r n ~ ncase, ~ respectively. A Back-Propagadnn IBP) algorithm may he used IL> update the connecting weighrs bctwccn the hidden and oukput layerf i o lutions. There arc screral sclirmcs LIE rzproduction. such as Roulr~tr-whcclszlzcticln, Ranking ielezrion, Tournament ct.lecrion. and ilt hers. Crossowr Hcrz, there is an cxchsngr: nf prupcrr~rrhetween two parents and as a result of which. L W U cilildru~isolutions are crrarcd. \'ariuus schcnlcb of cmssovcr arc available in thc literature. t~amclysirrgle-poinr c rorsirvcr, two-polnl rnmovet-. rnulfi-point cra.rsnwr, ~irr$ortn i.rnssoi8rr.ar.d others. Out of these. un~i'urmcrossoler i s the mc~stwidell; uscd one. ' hlulation In GA. Ihc concepl of' biological mutation has been ~nadeledartilioially to brlng a local chanpe I > \ - L - ~the

currznt solution t i ) avoid local minimum PI-oblcm,if any. In mutation. 1 1s converted to 0 and vice-versa. ,4 bit-wisc rnularion schcmc is generally uscd in a brnarycoded C A. In one ger~rrativnul'lhe GA. thc population of solutions is modified using the above 0pcr:itors. The pruc~sscontinuesunl~lrhc~,lg~rithin mects the pre-defined rerminativn critznon. 3.-3. C'lrr r r t , ~ - i tr.clrnryi~r.s r~~

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Cluitzrlng is a process of ]?ringing some ~ bL L-j I ~10gcthcr ba~tldon thcir similarity. Thc boundary defining ; I cluster may hc either fixed or ruzzy in nature. ?'hc present study dcals with fuzzy cluslrring. Two populilr fu7.7.y cluster~ngalgorithms are cxpla~~irci hrlnw.

Input Nodes

output layer Log+lgmdd T.F.

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Pig. 1 . Radial B a i a Funclion Neural Neluurk (RBI-NN I.

3.3.1. Frdzz? C - M ~ a r (FChf) ~s algorithm It i s onc of thc most popular fuzzy clustering techniques. in w h ~ o ha particular data of the set may be the member of several clusters wilh diffcrcnt V ~ U C of S memhcrship. In this approach. the number nf clusters into which the data set is tn be divided is pre-drcrded before the cclrnmenilemeat o f the ileraticms. Initially, a1I rhc point? arc assigncd mcmbcrship values with [he ciustcr ccntcrs Lying bc~wccn0.0 and 1.0 at random. Both thcsc mcmbcrship values as well as clusters are updated iteratively to obtain the final clusters. The following are the steps are used in lht: FCM algorithm 1291:

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- Step 1: Assume the number ofclusters to he made C , where2

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- Step 2: Choose an approximate level of clusler fuzziness y > 1.

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lnitializc thc rV A C sired membership matris [U]at random, such that T_,',>$ g [O.O. 1.01

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If ( I , , = 0. lhzn rhr dara pninr c n i n c ~ d rwith ~ the I": clus~cr ccntcr C; and ~t w ~ l havc l the tult membur\hip v;llut., t h a ~i urrcst csses during fonvard Inapprnl: uT.41-data !n terrrjs of 9'd e v ~ a ~ oInn prrd~cLnnof !a) BP, and (gl V H

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(b) bl. tu) u:. (d) b2, , aa obvained by the above three approaches. .4ppmaches 1. 2 and 3 habc yicldcd the value< nf averase ahct~lulepercent dev~nttonIn pred~ctions the outputs as I 1.2746. 1j.3 329 and 10.2044. respecrlvely. Thus. approdch 1 I~asuutperformcd thc othcr two approaches and the reason behind i t has hccn cxplaincd earlier.

Fig. h R e < u l ~n nt la) accelerilling wrliagr;. ( b l k i r l r md (c) weld spccrl

,. a;. and VB have been con.;idered, whercas that related ti) Stalnle55 Steel wcldlng data involves eight inpulr as me,,, inlIcd ~h~ uutpu,s. l,l,ucvcr, have bee,, kept 3s xcclcrating voltage. hram currcnt and welding speed for both the types of malerial.

6.2.1, BOP ~teld~rrg arr .4lumitlum Thc oplimirrci RBFNNs for carrying u u ~reverse mapping are obtairizd using Ihc ihrcr. de~clnpedapproache>. Their perf~rn~i~nces are the11 tested on eight tcsr cases. Figure 6 show~.b1hc rc\uI~so f reverse mapping on Aluminum wcldinp J a ~ ausing the above I ~ I - L ~ L . approaches. Thc valuc?; ot' aierage ahsolure pcrctn!

f\luncl 10 he equal to 5.1952. 5.0189 and 4.2152 for approaches 1 . 2 and 3, re.;pt.crively. All ~ h c a rthree apprudcbes arc ahlc 11> give reasonably good p r e d ~ c ttons I n reserse rnapplng and approach 3 1s seen tn pcrform btt1t.1 thdn the othcr two approaches. It ~ n a )hatc happrnrd so, due to [he reasons ~ l c n l l i ~ l eabove. ij

0.2.2. BOP clreldi~rg(:I: sttririless sfeel Figure 7 display( tllr results of reverse 111appingon .\rainless stccl wrldrng data. as ub~aineclby thr : ~ h v c : thrrz approac hcs. Approaches 1 , 2 and 3 hakc y yielded the talurs oi avcragc ahsulute percent devwt~onIII prediction ot' the process pwrlmclcrs 35 15.1 138. 2 1.4308 and I.;.O.;'i I . respectively. Onuu again, approach 3 1s seen to outperforn~the orhcr two approaches. It has happencd so, due 11) the reasons mrnrioned earlier.

0

10

Test cases

Fig. 7 . Results of lest cilszs dunng reversc ~uappinrnf ASS-&la in Ienns or 5 deviation in przdiclion of !a) iicceleixing rullilgr. current aid (cj weld swd

7. Conclusions

Input-output relattonships or EBW process have iu werse directions using radial haris function ncuritl networks, whose perfarrnancc dcpcnds on i t s uchi~ec.turr(decided by the nunher o f hidden neurons), connecttng wzighrs between I ~ L hidden . and output I3yzr.i. mean and standard dev~alionvd1uc.s ol the Gau