GENERALIZED LINEAR MODELS FOR SIMULATED HYDROFORMING PROCESSES

Transcription

1 Kolloquum - Forschergrue FreFormFlächen GENERALIZED LINEAR MODELS FOR SIMULATED HYDROFORMING PROCESSES U. Gather, S. Kuhnt Char of Mathematcal Statstcs and Industral Alcatons, Deartment of Statstcs, Unversty of Dortmund, 44 Dortmund, Germany Summary In hgh-ressure sheet metal hydroformng (HBU) smulatons rovde a useful tool for off-lne qualty mrovement. We analyse such fnte element smulatons by usng extended generalzed lnear models. Esecally, we determne settngs of the desgn arameters whch mnmze the estmated varance of the qualty measures n these models. At the same tme the arameter settng wll fulfl the estmated mean model equaton at the desred target. The otmal rocess s carred out smultaneously for the two qualty characterstcs blank thnnng and geometrcal accuracy. Ths s done usng a multvarate loss functon aroach. Introducton Off-lne lannng methods for hydroformng rocesses can nowadays rely on modern fnte element smulatons. Thereby they save exensve test seres. Nevertheless t remans unfeasble to smulate the hydroformng rocess for all ossble arameter settngs. In order to fnd an otmal settng wth resect to qualty measures for the formed workece we can use smulatons accordng to a sutable desgn of exerments. The stochastc nature of the real rocess should be taken nto account here by ncludng so-called nose arameters. These arameters can be set n the fnte element smulaton but are uncontrollable n the actual hydroformng rocess. The task of fndng values of the controllable arameters for whch the mean of the qualty measures hts the desred target wth mnmal varance defnes a tycal robust arameter desgn roblem [6]. The classcal soluton gven by Taguch [3] s based on exermental desgns whch are a roduct of a desgn for the controllable arameters and a desgn for the uncontrollable or nose arameters. The varance s estmated by means of reeatng exerments wth fxed desgn onts for the controllable art. As an alternatve we may emloy generalzed lnear models for modellng the mean and the varance. Ths aroach can be based on exermental desgns wthout reeated settngs of the controllable arameters. Hence, less exerments or smulatons are needed. Generalzed lnear models can also be used to redct the mean and varance for all ossble arameter settngs, where the redcton s otmal wthn the exermental regon. The model also rovdes estmates of the exected loss functon. Ths loss functon must be mnmzed n the Taguch aroach. In ths aer, we revew the dea of modellng the mean and varance of a qualty measure smultaneously. We wll aly these methods to the hydroformng rocess. As the hydroformng rocess cannot be sensbly measured by only one qualty characterstc, we

2 Kolloquum - Forschergrue FreFormFlächen also nvestgate the use of the estmated models n a multvarate robust arameter desgn strategy. Mult-Resonse Parameter Desgn usng Loss Functons We consder the stuaton of an ndustral rocess, the outcome of whch s gven n terms of qualty characterstcs Y, K,Y l. We further assume a vector of fnte target values τ= ( τ, K, τ l )' to be gven. The random vector Y = (Y, K,Y)' l s assumed to deend on a vector v = (v, K,v)' k of desgn factors (covarates) by a functon f, that s we have Y = f(v, ε ), where e descrbes the nose. The condtonal dstrbuton of Y gven v has exectaton E(Y v) =µ (v) and covarance matrx Cov(Y v) =Σ (v). Hence, the resonse mean and the covarance matrx deend on the desgn factors. The sngle resonse case s treated by the classcal Taguch aroach [3]. There, the overall qualty of the roduct s modelled n terms of a loss resultng from the devaton of a qualty characterstc Y from ts target τ. Taguch takes ths loss to be loss(y) = c(y τ ), the quadratc loss, where c s some constant. As a straghtforward extenson to multle qualty characterstcs the quadratc form loss(y) = (y τ)'c(y τ ) has been roosed n [0]. Here C denotes a l l symmetrc cost matrx. For the exected loss, the so-called rsk functon, one can wrte R(v) = E(loss(Y v)) = E(((Y v) τ)'c((yv) τ)) = trace(c' Σ (v)) + ( µ (v) τ )'C( µ (v) τ) () Hence, mnmzng the rsk functon wll, roughly seakng, brng the mean on target and smultaneously mnmze the varances. On the bass of desgned exerments, dfferent strateges have been roosed to fnd combnatons of the desgn factor (arameter) values whch mnmze the rsk functon (see e.g. [0,. 8-3]). For the hydroformng rocess analysed n Secton 4, we wll follow the aroach of [3] to ft generalzed lnear models for the mean and varance of each resonse. Thus, we can mnmze the exected loss resultng from these models for a sequence of cost matrces. 3 Generalzed Lnear Models wth Varyng Dserson Based on an exermental desgn wth n trals we assume that the corresondng resonses Y, =, K,n, for a sngle qualty characterstc are random varables wth means E(Y ) = µ and varances Var(Y ) = s. An arorate functon x = (x, K,x)' of

3 Kolloquum - Forschergrue FreFormFlächen the orgnal covarates v = (v, K,v)' k s used n the modellng rocess. The values of x for the ndvdual exerments are denoted by x = (x, K,x )', =, K,n. Consder now classcal lnear regresson models Y = x β+ε, =,...,n, j j j= where β IR denotes an unknown arameter vector. The resonses Y, =, K,n, are suosed to be normally dstrbuted wth constant varances (σ =σ, =, K,n). The systematc effects are suosed to be lnear (xβ+ K + x β). The errors e, =, K,n, are ndeendent and normally dstrbuted wth zero mean, such that E(Y) = x β =,...,n. j j j = Sometmes a transformaton of the resonse hels to fulfl these assumtons, at least aroxmately. However, often ths cannot be acheved by a sngle transformaton. In ths case generalzed lnear models [4,8] can be aled, whch extend the assumtons n two ways. Frstly, for each Y, =, K,n, the condtonal dstrbuton of Y gven x s assumed to belong to an exonental famly wth mean µ and, n some cases, on a common scale arameter φ not deendng on. These dstrbutonal famles ncludes for examle famles of gamma, nverse Gaussan, Posson and bnomal dstrbutons. Secondly, for each, =, K,n, the exectaton µ s related to the lnear redctor by a so-called lnk functon g, whch s a known one-to-one, suffcently smooth functon, ( ) g E(Y) 3 = x β. j j j= Further, the varance of the resonse can be wrtten as Var(Y) =φ V( µ ), =,...,n, where V s a known varance functon. The usual lnear regresson model s art of ths model class; then the resonses are normally dstrbuted, g s the dentty functon, φ=σ and V( µ ) =. The am n qualty-mrovement exerments s the selecton of covarate values, whch brng the mean on target whle keeng the varablty of the roduct at a mnmum. We must allow the varances to be nfluenced by covarates as well. For ths urose the model can be extended to a jont model of mean and dserson, a so-called double generalzed j= x β j

4 Kolloquum - Forschergrue FreFormFlächen lnear model [,7,,]. In ths sense we now assume that the mean model gven above s extended to ( ) g E(Y) = x β wth Var(Y) =φ V( µ ), =,...,n. () j j j= As a second generalzed lnear model a dserson model s formulated for a sutable statstc d as a measure of dserson for each =,,n. Here φ= E(d ), m h( φ ) = z γ, Var(d ) = tv( D φ ), =,...,n,, (3) j j j = where z = (z, K,z m)' s a further functon of the orgnal covarates v, h a dserson lnk functon, γ IR m an unknown arameter vector, V( D φ ) the dserson varance functon, and t IR a known scalar. In [8] we fnd two ossble choces for the dserson statstc d, namely the generalzed Pearson contrbuton (Y µ ) rp ( ) = V( µ ) and the contrbuton to the devance of unt, resectvely, Y Y D ( ) µ To ft such a model, we must choose a sutable dserson varance and sutable lnk r have a φχ functons. In the case of normally dstrbuted Y the statstcs r ( ) and ( ) dstrbuton, such that a gamma model wth V() D φ = φ s arorate. The most natural lnk functons are the dentty, gvng addtve varance comonents, and the log functon, gvng multlcatve effects of the covarates. The mean model () and the dserson model (3) are related n the sense that the mean model requres /φ to be used as a ror weght, whle the dserson model requres µ to be known n order to form the dserson resonse varable d. The estmaton of the unknown arameter vectors β and γ s usually acheved by the obvous algorthm, whch alternates between fttng the model for the means for gven weghts /φ ˆ and fttng the model for the dserson usng the resonse varable d(y, ˆ µ ). The ndvdual estmates are based on extensons of the maxmum lkelhood rncle. Hence they am at maxmsng the jont robablty functon of the gven samle [,,8,]. t r = dt =,...,n. V(t) D P 4 Alcaton to a hydroformng rocess In the develoment stage of a sheet metal formng rocess, often both rocess arameters and the geometry of the workece tself can be vared wthn a gven range [5]. To study the effect of geometrcal features on the qualty of the hydroformng result a de wth a square ocket s consdered, see Fgure. 4

5 Kolloquum - Forschergrue FreFormFlächen z formed workece de heght of the feature (FH) heght of the de (DH) x radus of the de (DR) Fgure : De wth a square ocket as workece For every smulated hgh-ressure sheet metal hydroformng rocess the resultng vrtual workece s descrbed by two qualty characterstcs, whose values are to be otmsed smultaneously. They descrbe the form and geometrc accuracy (GA) as well as the maxmal thnnng of the blank sheet (MaT) caused by the formng oeraton. The target values are 0 and 0.5 resectvely. The de dameter and heght as well as the ocket dmensons and oston can be vared, such that screenng exerments wth 0 covarates have been carred out. Fve covarates have been dentfed as havng large effects on the qualty characterstc: PW: Pressure of the workng meda (600 Bar Bar), FH: Heght of the feature (.67 mm mm), DH: Heght of the de (0 mm - 40 mm), DR: Radus of the de (40 mm - 80 mm), TB: Orgnal blank sheet thckness (0.9 mm -. mm). Smulatons followng a so-called central-comoste desgn have been carred out, where and are the coded values corresondng to the settngs gven n blankets. The resultng data set s gven n Tab.. Nr. PW FH TB DH DR GA MaT Nr. PW FH TB DH DR GA MaT Tab : Data resultng from a central-comoste desgn 5

6 Kolloquum - Forschergrue FreFormFlächen We look for a model that exlans the data satsfactorly and fts wth the mechancal background, but s also arsmonous at the same tme. Ths desgn allows for the estmaton of man and nteracton effects, as well as quadratc effects. For each of the two qualty characterstcs we searately consder double generalzed lnear models as descrbed n Secton 3. The thckness of the orgnal blank sheet affords a nose factor whch cannot be controlled n the actual hydroformng rocess. Therefore ths factor s not ncluded actvely n the modellng rocess. We use the mean model wth lnk functon equal to the dentcal and wth normal dstrbuton. Further, we use the gamma dstrbuton and the log-lnk for the dserson model. Unknown model arameters are estmated by the restrcted maxmum lkelhood aroach descrbed n []. Startng wth the model ncludng all ossble effects we successvely remove the least sgnfcant covarates found by t-tests. Concernng the maxmal thnnng we fnd the followng reduced mean and dserson model equatons E(MaT) ˆ = PW+ 0.0FH+ 0.05DH 0.0DR FH 0.006DR PW DH FH DH 0.004FH DR DH DR +.553DH Var(MaT) ˆ = ex( PW+.36FH 0.59DH 0.67DR.45PW.83DH DR). The dagnostc lot n Fgure shows that the selected model s adequate. For both, the comlete model as well as the selected model the observed values are almost equal to the redcted values from the mean model. MaT MaT Ftted Comlete model Selected model Ftted Fgure : Dagnostc Plots Followng the above strategy we agan derve a model for the mean and varance of the geometrc accuracy: E(GA) ˆ = PW FH DH+ 0.53DR+.68DH +.99PW DH 0.89PW DR.09FH DH+.065DH DR 6

7 Kolloquum - Forschergrue FreFormFlächen Var(GA) ˆ = ex( pw 0.6FH 0.744DH+.63DR 3.90PW PW DR.503FH DR). The estmated redcton models for the mean and varance of the qualty characterstcs are lugged nto the exected rsk functon (). The consdered rsk functon theoretcally becomes equal to zero f the rocess s always on target, ths s the means of all qualty characterstcs are equal to the target values and the varances are zero. The re-secfed weghtng C of the qualty characterstcs descrbes the relevance of reachng an otmal result for each characterstc. For dfferent ossble weghtngs values of the controllable desgn factors can be determned whch mnmze the estmated rsk functon. Accomlshng ths for a sequence of weght matrces C as descrbed n [3], we can summarze the result n the jont otmsaton lot gven n Fgure 3. Parameter settng PW FH DH DR Weght matrx Predcted maxmal thnnng Geometrc accuracy; target = target 0 = 0 Maxmal thnnng; target = target 0.5 = Weght matrx Predcted geometrc accuracy Fgure 3: Jont Otmsaton Plot On the left hand sde of Fgure 3, the derved desgn arameter settngs are lotted aganst the used weghts. More emhass s ut on reachng the target for maxmal thnnng to the left and on the geometrc accuracy to the rght. The lot on the rght hand sde shows the corresondng redcted means and varances of the qualty characterstc. The effects of the arameter factors on the qualty characterstcs are contradctory. A better geometrcal accuracy evokes a thnnng of the blank sheet, whch s then further away from the target. But what s more mortant: lower values of the feature heght lead to better geometrc accuracy but stretch the blank sheet below the desred target value. Therefore the otmum settngs have to be based on a comromse between geometrcal accuracy and thnnng of the blank sheet. 5 Outlook We have resented the sutablty of double generalzed lnear models n arameter desgn strateges for multle resonses. The suggested methods can be wdely aled to many engneerng and manufacturng rocesses, not only to hydroformng rocesses. Another advantage of the generalzed lnear model s ts alcablty n cases of contnuous 7

8 Kolloquum - Forschergrue FreFormFlächen resonses as well as dscrete resonses. Future research should nclude the develoment of jont generalzed lnear models for the mean vector and covarance matrx of multvarate resonses. Acknowledgements Fnancal suort for the research roject Process and Tool Otmsng Methods by Statstcal Desgn of Exerments and FEA for Sheet Metal Formng Oeratons by the central ublc fundng organsaton for academc research n Germany (DFG) s gratefully acknowledged. We also thank G. Smyth (htt:// for rovdng the S-lus functon dglm(). References [] Engel, J. and Huele, A.F.: A Generalzed Lnear Modelng Aroach to Robust Desgn. Technometrcs 38, S , 996. [] Engel, J. and Huele, A.F.: Taguch Parameter Desgn by Second-Order Resonse Surfaces. Qualty and Relablty Engneerng Internatonal (), S , 996. [3] Erdbrügge, M. and Kuhnt, S.: Strateges for Mult-Resonse Parameter Desgn Usng Loss Functons and Jont Otmsaton Plots. Techncal Reort, SFB 475, Unversty of Dortmund, Germany, 00. [4] Fahrmer, L.; Tutz, G.: Multvarate Statstcal Modellng Based on Generalzed Lnear Models. nd Edton, Srnger, New York, 00. [5] Gather, U.; Homberg, W.; Klener, M.; Klmmek, Ch. and Kuhnt, S.: Parameter Desgn for Sheet Metal Hydroformng Processes. In: Proceedngs of the 7 th Internatonal Conference on the Theory of Plastcty (ICTP), Yokohama, Jaan, October 7 November 4, 00. [6] Grze, Y.L.: Revew: A Revew of Robust Process Desgn Aroaches. Journal of Chemometrcs, 9, 39-6, 995. [7] Lee, Y. and Nelder, J.A.: Generalzed Lnear Models for the Analyss of Qualty- Imrovement Exerments. Canadan Journal of Statstcs 6(), S , 998. [8] McCullagh, P. and Nelder, J.A.: Generalzed Lnear Models, nd Edton, Chaman & Hall, London, 989. [9] Nelder, J.A. and Lee, Y.: Generalzed Lnear Models for the Analyss of Taguch- Tye Exerments. Aled Stochastc Models and Data Analyss 7, S. 07-0, 99. [0] Pgnatello, J.J.: Strateges for Robust Multresonse Qualty Engneerng. IIE Transactons, 5, S. 5-5, 993. [] Smyth, G.K.: Generalzed Lnear Models wth Varyng Dserson. Journal of the Royal Statstcal Socety B 5(), S , 989. [] Smyth, G.K. and Verbyla, A.P.: Adjusted Lkelhood Methods for Modellng Dserson n Generalzed Lnear Models. Envronmetrcs 0, S , 999. [3] Taguch, G. and Phadke, M.S.: Qualty Engneerng Through Desgn Otmzaton. Proceedngs of GLOBECOM 84 Meetng, Pscataway. NJ: IEEE Communcaton Socety, 06-3,