Metamodels in design of GFRP composite stiffened deck structure

Transcription

1 Metamodels n desgn of GFRP composte stffened deck structure Kaspars Kalnns * and Olgerts Ozolns Rga Techncal Unversty, Insttute of Materals and Structures, Azenes 6/0, Rga, LV-048, Latva Gnts Jekabsons Rga Techncal Unversty, Insttute of Appled Computer Systems, Meza /3, Rga, LV-048, Latva Ths paper deals wth the comparson of parametrc and non-parametrc metamodels used n desgn procedure of pultruded glass fbre renforced plastc (GFRP) deck panels. Stffened deck structure loaded under the three-pont bendng and unform dstrbuted load has been examned. A set of fve hundred sample ponts has been evaluated by means of fnte element analyss and stffness and strength responses extracted for buldng of metamodels. Moreover, set of complmentary physcal experments has been carred out n order to verfy the numercal model. Both advantages and dsadvantages of parametrc and non-parametrc metamodels have been drawn for a specfc engneerng applcaton. I. Introducton OMMON optmum desgn practce of stffened structures under any loadng combnatons should nvolve C extensve fnte element analyss wth complmentary expermental valdaton. However, such a procedure s only partly effcent as besdes trade-off desgn the desgner s seekng for alternatves n the overall perspectve. Despte the advances n rapdly growng computatonal capacty, as n Hgh Performng Computng or n GRID technologes, the enormous computatonal cost of complex engneerng smulatons makes t mpractcal to rely exclusvely on smulaton for the purpose of desgn optmzaton. As a good practce, one could use mathematcal approxmatons nstead of full scale analyses, thus reducng the level of numercal optmzaton complexty. Metamodels, also called surrogate models, are constructed from response approxmatons extracted from actual smulaton models. In partcular, for determnaton of the most sutable metamodelng technque fttng to deck under the bendng load desgn procedure a dfferent parametrc and non-parametrc approxmatons have been compared low order global polynomals, locally weghted polynomals, partal polynomals, and Multvarate Adaptve Regresson Splnes. Important research ssue assocated wth metamodelng s how to acheve good accuracy of a metamodel wth reasonable number of sample ponts. The samplng technques, often referred to as desgn of experments, should be mplemented to reduce the number of smulaton runs wthout decreasng the accuracy of the metamodel. The dfferences between samplng strateges for physcal experments and for computer experments should be noted. Whlst physcal experments have statstcal expermental errors, numercal analyses are determnstc and results are obtaned wth 00% repetton and no statstcal varance of model parameters,3,4. Currently, there s a wde range of lterature concernng dfferent methods for DoCE 5, whch nclude many approaches for space-fllng desgns. It should be noted that the frst space-fllng desgn crteron,3 for numercal experments was proposed at Rga Techncal Unversty by Audze and Eglajs. Whle the accuracy of a metamodel s drectly related to the approxmaton technque used and to propertes of the problem tself, the type of samplng approaches 6 have a drect nfluence on the approxmaton performance. It s generally accepted that space-fllng desgns, for example the Latn Hypercube desgn, are preferable for buldng of metamodels. In the current study, to reduce the number of computatons, expermental desgn optmzed accordng to Mean Square Error (MSE) crteron 7,6 has been selected. * Leadng researcher, Insttute of Materals and Structures, kaspars.kalnns@sgmanet.lv Ph.D. student, Insttute of Materals and Structures, olgerts@bf.rtu.lv Ph.D. student, Insttute of Appled Computer Systems, gntsj@cs.rtu.lv

2 II. Case study of pultruded GFRP deck structure Among many other composte materals, the glass fbre renforced polymer (GFRP) compostes has been demonstrated to have great success n brdge engneerng applcatons. It can be generalzed that characterstcs of a composte elements or structures can be talored and desgned to meet any desred specfcatons. The favorable characterstc of composte brdge decks are a hgh strength to weght rato effcency and relatvely low materal and manufacturng costs. There have been already developed and nstalled dfferent deck systems wth comparatvely small spans up to 0 m 8, however span lengths of GFRP deck applcatons are foreseen to expend over the next years. Most of the GFRP brdges constructed up to now 9 used mult-cellular pultruded deck systems. In prncple, two constructon forms are used: mult-cellular deck panels from adhesvely bonded pultruded shapes and sandwch panels wth dfferent core structures 0,. In present work, desgn of GFRP composte stffened panel structures has been consdered smlar to ones currently manufactured by Rshon-Inter.Ltd ( Eleven I-type stffener deck desgn, as shown n Fgure, has been elaborated n parallel wth expermental tests performed at Rga Techncal Unversty, Insttute of Materals and Structures usng dedcated test equpment. Three-pont bendng test case has been consdered for numercal analyses and physcal tests along wth unformly dstrbuted load test case that has been elaborated only numercally. Fgure. GFRP Deck geometry wth parametrcal varables The flexural stffness response of GFRP pultruded deck structure has been evaluated numercally by fnte element method commercal software ANSYS employng SHELL 8 4-node shell element. The mechancal propertes for glass fbre composte used for full-scale analyss have been extracted by small coupon tests n tenson and bendng. Numercal values of load-deformaton curve, stress, and stran dstrbuton over the tested deck structure were extracted and ncorporated n desgn of stran gauge locatons for expermental valdaton. The numercal deflecton and stress graphs of three-pont bendng test are shown n Fgure. A comparatve study between numercal and expermental results wll be gven n the chapter III. Fgure. Three pont bendng numercal test deflecton (left) and stress-dstrbuton (rght) graphs The choce of desgn varables should represent all geometrcal parameters n optmum desgn procedure. However most of those varables are ratonally nterconnected. Exploraton of non-ratonal varable combnaton mostly leads to relatvely hgh approxmaton errors. In partcular, when non-proportonal ratos of span and heght are used for bendng problems ths usually causes sngularty n approxmatons. Therefore geometrcal desgn varables wth correspondng rato varables have been proposed for metamodelng procedure. As geometrcal

3 varables the panel length parameter L and the panel heght parameter h along wth two plate thcknesses have been taken: the cover plate thckness t and the stffener thckness t. Moreover, ratonal desgn varables as kb (rato between the panel length and wdth) and kh (stffener spacng parameter rato between I-stffener foot wdth and panel heght) have been proposed. Such a procedure s requred to restran the combnaton where stffener spacng s narrower than the deck heght h. The numercal bounds of desgn varables are gven n Table. Table. Desgn space for deck structure Name and notaton Lower bound Upper bound Unts Deck length L m Deck stffener heght h m Plate thckness t m Stffener thckness t m Deck length to wdth rato kb.5 3 Stffener spacng rato kh 3 In order to acheve the best performance (mnmal predcton error of metamodels), a space-fllng desgn of fve hundred sample ponts optmzed accordng to the Mean Squared Error 6,7 unformty crteron has been selected. A numercal samplng procedure nvolves a large amount of analyzed numercal data that are unacceptably tme demandng. Therefore, all numercal data sets have been analyzed n parallel, explorng the LatvanGrd ( computng capabltes, thus reducng the necessary tme and outsourcng the computatonal capactes. III. Physcal experments Three three-pont bendng tests untl collapse of the structure for real deck panels have been performed. Addng the valdaton test that has been outftted wth stran-gauges and loaded untl 60% of the total collapse load. The expermental setup of a one-meter span length and correspondng deflected collapse mode shape s shown n Fgure 3. Durng the tests, the load-versus-deflecton curves (outlned n Fgure 4) and strans have been recorded by means of load cell and the stran gauge readngs. Thus the data from physcal experments has been mplemented n valdaton procedure of ANSYS fnte element model as summarzed n Table. Fgure 3. The expermental three-pont bendng test setup and collapse mode of the stffened deck structure 3

4 Bendng load P, [N] st panel deflecton curve nd panel deflecton curve 3 rd panel deflecton curve 4 th panel deflecton curve Deck deflecton u, [mm] Fgure 4. Load-versus-deflecton curves obtaned expermentally The test results obtaned expermentally for four tested deck structures have been compared wth values obtaned numercally by ANSYS. One could observe from the Fgure 4 and the Table that all panels have practcally the same loadng stffness. However, there s a certan dvergence between obtaned crtcal load levels. Nevertheless, n valdaton procedure the numercal deflecton results have about 0% dscrepancy wth physcal test results what should be consdered as a good agreement between actual (manufactured) and numercal model. Moreover, numercal stress threshold value practcally corresponds to the value obtaned by small specmen tenson tests. Valdaton procedure outlned that the load level correspondng to deflecton lmt legslated by buldng codes [/50 to /50 of the deck span] are practcally one ffth of the ultmate stress value. Table. Valdaton between expermental and numercal results st panel test, nd panel test, 3 rd panel test, 4 th panel test, Load(N) deflecton (mm) deflecton (mm) deflecton (mm) deflecton (mm) ANSYS, deflecton (mm) ANSYS stresses (MPa) N.A N.A N.A N.A N.A

5 IV. Employed metamodelng technques Ths secton brefly overvews the employed metamodelng technques: full global polynomals and locally weghted polynomals of nd, 3 rd, and 4 th order, Multvarate Adaptve Regresson Splnes (MARS), and partal polynomals constructed usng Adaptve Bass Functon Constructon (ABFC) approach. As descrbed by Smpson et al. 3, t s assumed that the nputs to the actual computer analyss are suppled n matrx x, and the outputs (or responses) from the analyss n vector y. Then the true computer analyss code evaluates y = g(x) () where g(x) s a complex engneerng analyss functon. The computatonally effcent metamodel approxmaton s such that where ε ncludes both approxmaton and random errors. y ˆ = f ( x) () y = yˆ + ε (3) A. Full global polynomals Low-order polynomals are the most wdely used metamodels,3,4. For example, second-order polynomal can be defned as follows: d d d yˆ = β + β x + β x x (4) 0 j= where d s the number of nput varables; β 0, β, β j are coeffcents usually determned by the ordnary least squares method mnmzng j n ( yˆ ( ) y( ) ) j β = arg mn (5) β where β are the calculated coeffcents; n s the number of sample ponts; y s the value of the metamodel s ( ) response for the -th sample pont; y s the actual value of the response of the computer analyss code n vector y. () A more complete dscusson on the polynomal metamodels and least squares method can be found n Myers & Montgomery 4. In the present study full polynomals of nd, 3 rd, and 4 th order have been employed. B. Locally weghted polynomals Locally weghted polynomal approxmaton was orgnally proposed by Cleveland 5. It was desgned to address stuatons n whch the global polynomals do not perform well or cannot be effectvely appled wthout undue effort. The approxmaton s carred out by pontwse fttng of low-order polynomals to localzed subsets of the data. The advantage of ths method s that the analyst s not requred to specfy a global functon of the data. However, the method requres consderably hgher computatonal resources. The assumpton of the local polynomal approxmaton s that near the query pont the value of the actual response changes smoothly and can be approxmated usng a low-order polynomal. The coeffcents of the polynomal are then calculated usng the weghted least squares method gvng the largest weghts to the nearest (usually accordng to the Eucldan dstance) sample ponts and the lowest or zero weghts to the farthest sample ponts. The coeffcents β are calculated by the weghted least squares mnmzng ˆ 5

6 n ( y y ) β = arg mn w( xquery, x( ) ) ˆ (6) ( ) ( ) β where w s a weght functon; x query s the query pont nearest neghbors of whch wll get the hghest weghts; x s () the -th pont n vector x. The weght functon w depends on the Eucldean dstance (n scaled [-,] d space) between the pont of nterest x query and the ponts of observatons x. One of the most wdely used weght functons s the Gaussan weght functon : where α s a coeffcent and the μ can be calculated as ( ) ) = exp ( ) w ( x query, x αμ (7) μ = x x ) ( x x ) (8) ( query ( ) query farthest where s the Eucldan norm; x farthest s the farthest pont n the neghborhood of the pont x query. In general, the Gaussan weght functon wth constant value α = s used n local approxmatons varyng only the value of the consdered nearest neghbors (unlke n equaton (6) where all the sample ponts are used). However, n the present study all the sample ponts have been used and the localty of the approxmaton has been controlled by varyng the value of the coeffcent α. If α s equal to zero then local approxmaton transforms nto global approxmaton. The best value of α s found usng the leave-one-out cross-valdaton technque 6. In the present study locally weghted polynomals of nd, 3 rd, and 4 th order have been employed. C. Multvarate Adaptve Regresson Splnes Multvarate Adaptve Regresson Splnes 7,8 was proposed as a method for flexble regresson modelng of hgh dmensonal data (.e., a large number of nput varables). The model takes the form of an expanson n product splne bass functons, where the number of bass functons as well as the parameters assocated wth each one (product degree and knot locatons) are automatcally determned by the data through a forward/backward teratve approach. Compared to polynomal approxmatons, the use of MARS for engneerng desgn s relatvely new. However, ts applcaton s drawng an ncreasng attenton of the researchers (e.g., Jn et al. ). MARS model can be defned as a sum of bass functons 7,8 : k 0 + β f ( x) yˆ = β (9) where f (x) s a bass functon; k s the number of bass functons n the model except for the constant bass functon f ( x) coeffcent of whch s the β 0. The bass functons are of the form 0 = d = [ s j ( xv( t j ] j, ) ) f ( x) (0) j= where d s the number of varables (nteracton order) n the -th bass functon; s j = ± ; x s the v-th varable, v( j, ) v( j, ) d ; t j s knot locaton on each of the correspondng varables. The subscrpt + means that the functon s a truncated power functon 7,8. The coeffcents β are agan determned by the ordnary least squares method (equaton (5)). In the present study pecewse-cubc MARS verson 3.6 wthout a specfc restrcton of the number of bass functons or nteracton orders has been employed. D. Partal polynomals Low-order global polynomal approxmatons have been well accepted n engneerng practce, as they requre low number of sample ponts and are computatonally very effcent. On other hand they can not approxmate hghly 6 +

7 nonlnear behavor. Instead, hgher-order polynomals can be employed. However, f no specal care s taken they tend to overft the data and produce hgh errors n regons where the sample ponts are relatvely sparse. One possble remedy for the overfttng problem s employment of the subset selecton (also called model buldng) technques 3,4. The technques are amed to dentfy the best subset of polynomal terms (or bass functons) to nclude n the model and to remove the unnecessary ones, n ths manner creatng a partal polynomal model (n leu of full model) of ncreased predctve performance. However, the approach of subset selecton assumes that the chosen fxed full set of predefned bass functons (usually just by choosng a fxed maxmal order of the polynomal) contans a subset that s suffcent to descrbe the target relaton suffcently well. Hence the effectveness of subset selecton largely depends on whether or not the predefned set of bass functons contans such a subset. There exsts a dfferent approach for polynomal model buldng whch does not assume a predefned set of bass functons Adaptve Bass Functon Constructon 9,0. The approach allows generatng polynomals of arbtrary complexty and order wthout the requrement to predefne any bass functons or to set the maxmal order of the polynomal (or any other hyperparameters) all the requred bass functons are constructed adaptvely. Generally a polynomal model can be defned as a lnear summaton of bass functons: k yˆ = β f ( x) () where the coeffcents β are stll calculated by the ordnary least squares method (equaton (5)); f (x) s a bass functon whch generally can be defned as a product of the nput varables each rased to some order: d j= rj f ( x) = x () where r j s the order of the j-th varable n the -th bass functon (a non-negatve nteger). It should be noted that when all r j s of a bass functon become equal to 0, the bass functon becomes equal to, ergo t s the ntercept term. The matrx r completely defnes all the bass functons n the model each row corresponds to one bass functon wth all of ts orders. Constructon of the model s carred out n an teratve manner drectly wth the matrx r usng fve smple so-called model refnement operators whch allow addng, copyng, modfyng, and deletng the rows of r,.e., addng, copyng, modfyng, and deletng the bass functons of the model 9,0. As a search procedure a modfcaton of the Sequental Floatng Forward Selecton algorthm s employed whle models are evaluated usng the Corrected Akake s Informaton Crteron. Addtonally, n order to lower the model buldng ssues of selecton bas and selecton nstablty a technque of model averagng (also called ensemblng or combnng) s carred out 0. E. Metamodel evaluaton To evaluate the metamodels, 5-fold cross-valdaton technque 6 has been used where the full data set s dvded n fve equally-szed subsets. In each of the fve cross-valdaton teratons four of the subsets are used for metamodel buldng and one subset s used as an ndependent test set for evaluaton of the metamodel. As the metamodel accuracy measure the Relatve Root Mean Square Error has been used: j n t ( y yˆ ( ) ( ) ) nt RRMSE = 00 % (3) STD where n t s the number of test ponts; STD s the standard devaton n test sample: STD = n n t t ( y( ) y) (4) 7

8 where y s the mean value of all y values n the test sample. It should be noted that RRMSE and STD are calculated usng strctly only the test sample and averaged over all the cross-valdaton runs. V. Metamodelng results As structural response parameters the followng enttes have been taken: global deflecton of the deck panel u, the relatve deflecton rato between the deck length and the plate deflecton Δu, the maxmum equvalent stress at the upper plate and the stffeners σ top, σs tft and the maxmum shear stresses n stffeners τ. A total of 500 samplng ponts have been generated and a cross-valdaton procedure together wth the RRMSE measure has been carred out comparng the dfferent metamodelng technques. Two loadng scenaros have been selected for metamodelng: the concentrated three-pont bendng load case and unformly dstrbuted load case wth smple support boundary condtons. The obtaned results are summarzed n Tables 3 and 4. Table 3. Cross-valdaton RRMSE errors of dfferent metamodels bult usng 500 sample ponts n the case of concentrated three-pont bendng load Locally weghted Parametrc polynomal approxmatons Metamodels polynomals MARS nd 3 rd 4 th ABFC nd 3 rd 4 th Panel deflecton, u Comparatve deflecton, Δu Max_stresses, σ top Max_stresses, σ stf Max_shear_stresses, τ Table 4. Cross-valdaton RRMSE errors of dfferent metamodels bult usng 500 sample ponts n the case of unformly dstrbuted load Locally weghted Parametrc polynomal approxmatons Metamodels polynomals MARS nd 3 rd 4 th ABFC nd 3 rd 4 th Panel deflecton, u Comparatve deflecton, Δu Max_stresses, σ top Max_stresses, σ stf Max_shear_stresses, τ The conventonal nd order polynomals, whch are mostly assocated wth engneerng problems of the response surface methodology, gave the worst approxmaton results for almost all the response values. It has been noted that locally weghted polynomals of the nd order consderably ncreased the predctve performance. As overall observaton could be stated that, by ncreasng the order of polynomals, the approxmaton performance rose, however the hgher was the order the smaller was the mprovement of the locally weghted polynomals over the global ones. Addtonally, t should be expected that decreasng the number of the samplng ponts would lead the full polynomals of hgher orders to overfttng the data thus rapdly reducng ther predctve performance. The best results were obtaned usng the ABFC approach, leavng the MARS technque as the second best. One can conclude that, although usng hgher order global polynomals or locally weghted polynomals can mprove the predctve performance, an elaborated adaptve search for partal polynomals or regresson splnes has capabltes to provde an even further performance boost. Addtonally, t has been nvestgated how rapdly the performance of the metamodelng methods would deterorate whle decreasng the number of sample ponts. For the case of concentrated three-pont bendng load addtonal results for 00 sample ponts nstead of 500 were obtaned. The results are summarzed n Table 5. Wth the reduced number of sample ponts the 4 th order polynomals became mpossble to use because the number of ther bass functons was now hgher than the number of the sample ponts. For all metamodels a reducton n ther predctve performance has been observed, however the most rapd performance deteroraton was for MARS metamodels. The best results were stll obtaned usng the ABFC approach. Apparently, compared to the other technques, partal polynomals are relatvely more effcent than others when a small dataset of sample ponts s used whle reducng the number of bass functons n a polynomal model. 8

9 Table 5. Cross-valdaton RRMSE errors of dfferent metamodels bult usng 00 sample ponts n the case of concentrated three-pont bendng load Locally weghted Parametrc polynomal approxmatons Metamodels polynomals MARS nd 3 rd 4 th ABFC nd 3 rd 4 th Panel deflecton, u N/A N/A 7.5 Comparatve deflecton, Δu N/A N/A 4.69 Max_stresses, σ top N/A N/A 3.64 Max_stresses, σ stf N/A N/A 6.9 Max_shear_stresses, τ N/A N/A 8.39 Moreover, three-dmensonal graphcal valdatons of the developed metamodels for the deck panel deflecton u versus the panel length L and the panel heght h parameters n the case of concentrated three-pont bendng load wth 500 sample ponts were carred out as presented n Fgure 5. By graphcal valdaton one can easly dentfy that low order global polynomal approxmatons at the maxmum heght and the mnmum length behave dfferently than expected. In partcular the second order and to a lesser extent also the forth order polynomal surface plots show a decrease of stffness when ncreasng the panel heght. On the other hand, the thrd order polynomal functon shows non-negatve deflecton at the boundares, whch would ndcate bendng aganst gravty. The locally weghted polynomals reduce the unwanted behavor. However, by zoomng n at the regon of the maxmum heght and the mnmum length, the surface plots reveal that the hgher s the order of the locally weghted polynomals the more they behave lke global polynomals resultng agan n the unwanted behavor. a) b) c) d) e) f) g) h) Fgure 5. Graphcal valdaton of surrogate models for panel deflecton n the case of concentrated load. Full global polynomals of nd (a), 3 rd (b), and 4 th (c) order; locally weghted polynomals of nd (d), 3 rd (e), and 4 th (f) order; MARS (g); ABFC (h) 9

10 Overall, the low-order locally weghted polynomal approxmaton, MARS, and the ABFC gave the best overall perspectve of structural behavor by creatng a plateau-lke surface for the heght of stffened panel desgns. It seems that here for an optmzaton procedure all the three methods can be used wth some confdence however for what-f analyss the ABFC would be the most accurate. VI. Concluson The comparson study between parametrc and non-parametrc metamodels has been elaborated for desgn of pultruded GFRP deck structures under the bendng load. Two loadng scenaros have been selected to nvestgate the metamodelng effcency and have been valdated wth physcal experments. It has been concluded that the partal polynomals and MARS are capable to mprove the predcton accuracy compared to conventonal nd order polynomals, whch frequently are assocated wth engneerng problems of the response surface methodology. In partcular, the bendng deflecton responses could be mproved by an order of magntude compared to the nd order polynomals. In contrary, the mprovement n approxmaton predcton for equvalent stresses and shear stresses are less effcent. Elaborated metamodels have the capablty to be used n mplementaton of optmum desgn methodology for the bended deck structures. Acknowledgments Ths work was partly supported by the Latva Mnstry of Scences and Rga Techncal Unversty grant R7397 Optmsaton of rb-stffened composte structure load-carryng capacty wth expermental valdaton. The nformaton n ths paper s provded as s and no guarantee or warranty s gven that the nformaton s ft for any partcular purpose. The user thereof uses the nformaton at ts sole rsk and lablty. References Jn, R., Chen, W., and Smpson, T., Comparatve studes of metamodelng technques under multple modellng crtera, Journal of structural and Multdscplnary Optmzaton, Vol. 3, No., 00, pp. -3. Audze, P., and Eglas, V., New approach for plannng out of experments, Problems of Dynamcs and Strength, Vol. 35, Znatne Rga (n Russan), 977, pp Eglajs, V., Approxmaton of data by mult-dmensonal equaton of regresson, Problems of Dynamcs and Strength, Vol. 39, Znatne, Rga (n Russan), 98, pp Smpson, T. W., A Concept Exploraton Method for Product Famly Desgn, Ph.D. Dssertaton, Georga Insttute of Technology, Santner, T. J., Wllams, B. J., and Notz, W. I., The Desgn and Analyss of Computer Experments, Sprnger, Auzns, J., Drect optmzaton of expermental desgns, 0th AIAA/ISSMO Multdscplnary Analyss and Optmzaton Conf., AIAA paper, No , Fang, K. T., and Wang, Y., Number-Theoretc Methods n Statstcs, Chapman & Hall, London, Wllams, B., Shehata, E., and Rzkalla, S., Flament-Wound Glass Fber Renforced Polymer Brdge Deck Modules, ASCE Journal of Composte for Constructon, Vol. 7 No. 3, 003, pp Keller, T. Use of fber renforced polymers n brdge constructon, Structural Engneerng Documents, No. 7, Internatonal Assocaton for Brdge and Structural Engneerng IABSE, Kalnns, K., Eglts, E., Jekabsons, G., and Rkards, R., Metamodels for optmmum desgn of laser welded sandwch structures, Desgn, Fabrcaton and Economy of Welded Structures, edted by K. Jaram and J. Farkas, Horwood, Chchester, UK, 008, pp Kalnns, K., Skuks, E., and Auzns, J., Metamodels for I-core and V-core sandwch panel optmzaton, Proceedngs of 8th Intern. Conf. Shell Structures: Theory and Applcatons, edted by W. Petraszkewcz and C. Szymczak, Taylor & Francs, London, 005, pp ANSYS, software package, Ver., Canonsburg, USA, Smpson, T. W., Peplnsk, J., Koch, P. N., and Allen, J. K., Metamodels for Computer-Based Engneerng Desgn: Survey and Recommendatons, Engneerng wth Computers, Vol. 7, No., 00, pp Myers, R.H., and Montgomery, D.C., Response Surface Methodology: Process and Product Optmzaton Usng Desgned Experments, nd ed., John Wley & Sons, New York, 00 5 Cleveland, W. S., and Devln, S. J., Locally weghted regresson: An approach to regresson analyss by local fttng, Journal of the Amercan Statstcal Assocaton, 83, 988, pp Kohav, R., A study of cross-valdaton and bootstrap for accuracy estmaton and model selecton, Proceedngs of IJCAI- 95, edted by C. S. Mellsh, Morgan Kaufmann, 995, pp Fredman, J. H., 99, Multvarate Adaptve Regresson Splnes, The Annals of Statstcs, Vol. 9, No., pp Fredman, J. H., Fast MARS, Department of Statstcs, Stanford Unversty, Tech. Report LCS0,

11 9 Jekabsons, G., and Lavendels, J., Polynomal regresson modellng usng adaptve constructon of bass functons, IADIS Internatonal Conference, Appled Computng 008, Algarve, Portugal, 008, p. 8 0 Jekabsons, G., Ensemblng adaptvely constructed polynomal regresson models, Int. Journal of Intellgent Systems and Technologes (IJIST) [onlne journal], Vol. 3, No., Paper, 008, pp URL: [cted 4 May 008] Pudl, P., Novovcova, J., and Kttler, J., Floatng search methods n feature selecton, Pattern Recognton Letters 5, 994, pp Hurvch, C. M., and Tsa, C-L., Regresson and tme seres model selecton n small samples, Bometrka 76, 989, pp