Robus op imiza ion s a egies o shee me al sp ingback compensa ion
XIII In e na ional Con e ence on Compu a ional Plas ici y. Fundamen als and Applica ions
COMPLAS XIII
E. O˜na e, D.R.J. Owen, D. Pe ic & M. Chiumen i (Eds)
ROBUST OPTIMIZATION STRATEGIES FOR SHEET
METAL SPRINGBACK COMPENSATION
A. Maia∗, E. Fe ei a†, M.C. Oli ei a†, L.F. Menezes†and A.
And ade-Campos∗
∗Depa men o Mechanical Enginee ing, Cen e o Mechanical Technology & Au oma ion,
GRIDS Resea ch G oup, Uni e si y o A ei o
Campus Uni e si ´a io de San iago, 3810-193 A ei o, Po ugal
e-mail: [email p o ec ed], web page: h p://www.ua.p /
†Cen e o Mechanical Enginee ing o he Uni e si y o Coimb a (CEMUC)
Depa men o Mechanical Enginee ing, Uni e si y o Coimb a
P´olo II, Rua Lu´ıs Reis San os, Pinhal de Ma ocos, 3030-788 Coimb a, Po ugal
e-mail: cem[email p o ec ed], web page: h p://www.uc.p /en/iii/ esea ch cen e s/CEMUC
Key wo ds: Shee me al o ming, Sp ingback Compensa ion, Fini e Elemen Model
Upda ing S a egy, Robus Op imiza ion s a egies, NURBS pa ame iza ion
Abs ac . Shee me al o ming is a majo indus ial p ocess, mainly due o i s cos
efficiency a e he es ablishmen o he p ocess design. Howe e , he p ocess design
om ools geome y o load condi ions is no s aigh o wa d, as a consequence o he
side effec s associa ed wi h shee me al o ming. The emphasis in his a ea goes o he
sp ingback effec o elas ic eco e y, which is one o he main causes o pa ’s inaccu acy,
demanding ool compensa ion. This wo k p oposes o compa e diffe en obus op i-
miza ion s a egies o shee me al o ming sp ingback compensa ion. The me hodology
adop ed eso s o Response Su ace Me hod (RSM), as well as o Fini e Elemen Model
Upda ing (FEMU) s a egies, o adjus he design a iables. These include he ools’
su aces, which a e pa ame ised wi h NURBS. These s a egies a e hen compa ed using
he U-Rail benchma k. The esul s achie ed e eal a educ ion o 99% on he geome ical
e o o he final piece o he bes me hodology.
1 INTRODUCTION
The e olu ion obse ed in he indus y, whe e new pieces a e cons an ly needed, c ea es
he necessi y o accele a e he design s age o new o ming p ocesses while main aining
i s accu acy. Conce ning sp ingback compensa ion, his means ha he necessa y ools
adjus men s and o he a iables need o be swi and igo ous. This challenge is no effi-
cien ly answe ed by he adi ional ” ial-and-e o ” p ocess. The e o e, he employmen
o op imiza ion s a egies, among which s a is ical me hods, such as he Response Su -
ace Me hod (RSM), o Fini e Elemen Model Upda ing (FEMU) s a egies associa ed
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o op imiza ion algo i hms, ha e been s udied. Though he e a e comme cial so wa e
[1] ha can p edic he final geome y o a o med shee gi en he ini ial a iables and
adjus hem o achie e he desi ed piece (di ec and in e se p oblem espec i ely), hei
efficiency is s ill no op imal.
The wo k he eby p esen ed p oposes o implemen an in eg a ed me hodology ha
sea ches o he se o design a iables ha be e compensa es he sp ingback effec . The
esul s ob ained a e hen checked o easibili y and a compa ison be ween he p oposed
op imiza ion s a egies is made. The design a iables conside ed a e he con ol poin s o
he NURBS ha define he o ming ools geome y and he Blank Holde Fo ce (BHF).
2 SPRINGBACK COMPENSATION STRATEGIES
A single-s ep shee me al o ming p ocess is di ided in h ee phases: (i) he ini ial
posi ioning o he ools; (ii) he o ming by he ools and (iii) he emo al o he ools
wi h subsequen sp ingback.
Sp ingback o elas ic eco e is an undesi ed side-effec o shee me al o ming which is
p opo ional o he a io be ween esidual s esses and Young Modulus [2]. This ela ion
leads o a fi s app oach o he minimiza ion o his side-effec : an inc ease on he plas ic
de o ma ion, done by inc emen ing he BHF and/o o he es aining o ces [3]. Howe e ,
his app oach may induce se ious quali y p oblems in he final pa as his leads o a
educ ion on he ma e ial flow du ing he o ming phase. This may gene a e necking o
he blank and consequen ac u e.
Ano he possible app oach a e Sp ingback Compensa ion me hodologies. These me hod-
ologies acknowledge he sp ingback as una oidable. Thus ins ead o ying o educe i ,
hey use he sp ingback so ha he final piece has he desi ed shape. This ype o app oach
comp ises se e al s a egies. The Displacemen Adjus men me hod (DA) [4] consis s on
he displacemen o he ool’s su ace on he opposi e di ec ion o he sp ingback, which
can be done in jus one s ep [3, 5, 6] o i e a i ely [7]. The la e concep has e ealed
i sel mo e effec i e in a eas s ongly affec ed by plas ic de o ma ion. In spi e o i s good
p ac ical esul s, his me hodology has some d awbacks, such as he difficul y in aligning
he CAD model and he o ming piece o he endency o he compensa ed su ace o
become oughe [3]. The o me p oblem may be sol ed by he Smoo h Displacemen
Adjus men (SDA). This me hodology app oxima es he geome ic e o by smoo h unc-
ions, using bounda y cons ain s i es ic ions a e needed [5]. O he imp o emen s on
he DA s a egy is he Comp ehensi e Compensa ion (CC), whe e ins ead o adop ing he
opposi e di ec ion o he sp ingback, he bes di ec ion and magni ude o he adjus men
is compu ed. This app oach combined wi h he i e a i e DA has achie ed p omising e-
sul s [6]. The Di ec Cu a u e Me hod (DCA) is ano he possible app oach. Ins ead o
ea ing he ool as an whole, i changes diffe en pa s o he ool independen ly, h ough
a dynamic compensa ion ac o [3]. The compu a ion o he co ec compensa ion ac o s
/ adjus men s can be done h ough a ” ial-and-e o ” me hodology using nume ical ials
o achie e he op imal adjus men , o using op imiza ion algo i hms.
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Op imiza ion algo i hms a e me hods o compu e he se o inpu a iable ha min-
imize o maximize a cos - unc ion. These algo i hms may be classified in (i) Na u e-
inspi ed algo i hms, (ii) G adien -based me hods and (iii) A ificial in elligence (AI).
While he G adien -based algo i hms p esen a as e con e gence and a e ai ly efficien ,
hey may be apped in local minima. The Na u e-inspi ed me hods s a is ically con e ge
o he global minimum equi ing howe e many e alua ions o he cos unc ion, which
leads o high compu a ional cos s. The AI algo i hms also ha e he disad an age o e-
qui ing massi e quan i ies o da a o hei aining. Howe e , when well calib a ed, hey
p o ide e y accu a e esul s. In sum, all me hods ha e ad an ages and disad an ages,
being he choice o hei employmen s ongly dependen on he specific applica ion de-
si ed and on he amoun o da a a ailable. The e is also he possibili y o combine se e al
s a egies and app oaches in o de o ake ad an age o hei bes ea u es [8].
In his wo k, a simila app oach o he DCA is adop ed. Howe e , he ools modelling
and adjus men s is done h ough NURBS, being he design a iables he NURBS con ol
poin s. The geome ic e alua ion howe e emains using geome ical pa ame e s, as hese
be e exp ess he desi ed shape o he componen .
3 METHODOLOGY AND IMPLEMENTATION
In o de o find he design a iables ha op imize (minimize) he cos unc ion, he
p ocess p esen ed in Fig. 1 is ollowed. The cycle s a s by p e-p ocessing he o iginal
inpu files which s o e he ini ial design a iables, including he desi ed pa design and
he ini ial ool design, defined by IGES and Mesh files (one pe ool). Following, he
simula ion so wa e ou pu s he nume ical esul s co esponding o he o med me al
shee . An e alua ion o hese esul s is hen compu ed, based on he geome ical e o
be ween he ob ained piece and he e e ence one.
This cos unc ion is subsequen ly in oduced in o he op imiza ion algo i hm, which
compu es he new design a iables. A his s age, an e alua ion o he whole p ocess
akes place: i he con e gence c i e ia o he algo i hm is me , hen hese a iables a e
ou pu ed as op imised alues; o he wise he inpu files a e upda ed wi h he new a iables
p oposed, p e-p ocessed and inpu on he simula ion so wa e, s a ing a new cycle.
Figu e 1: P ocess Schema ics
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A. Maia, E. Fe ei a, M.C. Oli ei a, L.F. Menezes and A. And ade-Campos
I should be no iced ha his cycle only applies o he Di ec Sea ch and G adien -
based algo i hms. Fo he RSM, fi s all he FEM-simula ions a e un o a p ede e mined
se o a iables combina ions, and only a he end he op imiza ion is applied.
Each ac ion lis ed is pe o med by an indi idual block o code. This op ion o modu-
la i y lends flexibili y o he code and eases he e o de ec ion and he code e olu ion.
3.1 Fini e Elemen Analysis (FEA)
The simula ions a e ca ied ou using he fini e elemen code DD3IMP [9], de eloped
specifically o simula e shee me al o ming p ocesses, using an upda ed Lag angian o -
mula ion and a p edic -co ec ion scheme o de e mina e he equilib ium s a e [10].
A p oblem inhe en o he use o he FEM is he nume ical noise due o ound off
e o s, mesh disc e isa ion and ins abili y o con ac condi ions [11]. This noise may
lead o diffe en meshes, a ia ions on he numbe o inc emen s o he simula ions and,
consequen ly, o diffe en esul s and e olu ions o he op imiza ion algo i hm. In ex eme
cases, he noise can c ea e local minima apping he op imisa ion p ocess, and lead o
inaccu a e op imiza ion esul s. As he noise c ea es mo e local minima o he objec i e
unc ion, i is necessa y o accoun o i s impac in he op imiza ion e olu ion. To his
end, a mul i-s a s a egy in conjunc ion wi h he chosen op imiza ion algo i hms is
employed.
3.2 Sp ingback compensa ion h ough ool design
This implemen a ion elies on Fini e Elemen Model Upda ing (FEMU) s a egy [12].
Once s a ed, he op imiza ion algo i hm does i s i e a ions eso ing o a simula ion e e y
ime a new alue o a se o design a iables is necessa y. To do so, he algo i hm ou pu s
a se o design a iables and au oma ically ini ia es a cycle o ew i ing and p e-p ocessing
new inpu files, pe o ming a simula ion and e alua ing he ob ained piece, using his alue
o he nex i e a ion. Al hough his app oach is compu a ionally expensi e and subjec
o nume ical noise, i is eliable in he sense ha i is no based on in e pola ions o he
cos unc ion beha iou . Ins ead, i compu es he ac ual alues o he measu e a iables
o each se o design a iables. In he scope o his a icle a leas -squa es G adien -based
algo i hm, a di ec sea ch algo i hm and a linea Response Su ace Me hod a e used. The
esul s achie ed by all he me hods a e compu ed and discussed. I should be no iced
ha all he design a iables a e conside ed as ele an inpu s o bo h he in e pola ion
and he op imiza ion algo i hms.
3.2.1 Linea Response Su ace Me hod (RSM)
This app oach eso s o an in e pola ion o a se o p e-exis en esidues in o de
o es ima e equa ions ha desc ibes he beha iou o he cos unc ion. P e iously o
he op imiza ion p ocess, a Sensi i i y Analysis was conduc ed o asce ain he gene al
beha iou o he blank upon pe u ba ions on he p ocess design a iables and o help o
choose he op imiza ion s a egies s a ing poin s. Ha ing all he da a, a simple Linea
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Fi ing is applied, esul ing on a se o linea equa ions ha desc ibe a plana Response
Su ace o he geome ic e alua ion pa ame e s.
3.2.2 Di ec Sea ch Algo i hm
The Nelde -Mead Simplex Algo i hm [13], a di ec sea ch me hod, elies on he con-
s uc ion o a simplex o N+1 e ices, o a N-dimensional p oblem. Then, i i e a i ely
eplaces i s e ices o new ones wi h lowe alues o he cos unc ion. I s main ad an age
is independence o he g adien o he cos unc ion o any app oxima ion.
To s udy he influence o he uncons ained na u e o he Nelde -Mead algo i hm a
simula ion whe e he e is no cons ain s o he alues o he inpu a iables, and ano he
whe e a a iable ans o ma ion is applied a e pe o med. The ans o ma ion is applied
be ween he uncons ained a iable xi, o be op imized, and he cons ained a iable Xi,
subjec ed o a maximum Xmax
iand a minimum Xmin
i, such as
Xi=Xmin
i+(Xmax
i−Xmin
i)exp(xi)
exp(xi)+ exp(−xi).(1)
This ans o ma ion is applied be o e each simula ion, so ha i is pe o med wi h
easible a iables. I is also necessa y o apply he in e se ans o ma ion a he beginning
o he op imiza ion, meaning ha he cons ained ini ial design a iables Xioshould be
ans o med in he uncons ained ones xio.
The ole ance o 10−5is employed, along wi h he s anda d alues o he algo i hm
coefficien s: α=1., γ=2. and β=δ=0.5.
3.2.3 Leas -Squa es G adien Based Algo i hm
The Le enbe g-Ma qua d is a g adien -based me hod, simila o he New on-Raphson,
ha ing howe e a s abiliza ion pa ame e , µk, in o de o imp o e he algo i hm’s be-
ha iou a ound minima. As his pa icula p oblem doesn’ ha e an a-p io i g adien o -
mula, he Jacobian ma ix is compu ed a each i e a ion h ough o wa d fini e-diffe ence
calcula ions.
A majo difficul y on his me hodology o he sensi i i y compensa ion is i s sensi i i y
o noise. This noise no only affec s he e alua ions o he cos unc ions (possibly e en
in oducing ex a local minima), bu also he Jacobian cons uc ion, which is c ucial
o g adien -based me hods. In o de o o e come his obs acle, a mul i s a s a egy
is implemen ed, equi alen o he one desc ibed on he Nelde -Mead subsec ion. The
influence o he noise on he Jacobian, howe e , needed ano he app oach. An inc ease on
he fini e-diffe ences pe u ba ion is made so ha , e en affec ed by noise, he diffe ence
ansla es he eal end o he cos unc ion, which o e comes he noise. The chosen s ep
o his pa icula applica ion is 2% o he a iable alue. This algo i hm implemen a ion
has a ole ance o 10−5, iden ical o he one adop ed by he Nelde -Mead.
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3.3 P ocess Pa ame iza ion
Classical ypes o ep esen a ion, such as B´ezie cu es o geome ical pa ame iza ions
(leng hs, adius and angles), hough being mo e in ui i e o he human use , do no ha e
he necessa y flexibili y o pa ame ize e y complex geome ies o became e y hea y
o compu e. Mo e ad anced pa ame iza ions such as he T-splines, hough p esen ing
impo an ad an ages, a e s ill no sp ead in indus y and p esen ew in-use so wa e.
The use o NURBS [14] as pa ame iza ion p esen s a good balance. Thei flexibili y
and accu acy on ep esen ing complex shapes ha e lead o hei adop ion and s anda d-
iza ion a an indus ial le el, being used on a wide ange o applica ions and so wa e.
In his wo k, NURBS a e used as pa ame iza ion me hodologies o define he design
a iables: hei con ol polygon coo dina es and weigh s. All he da a conce ning NURBS
con ol poin s is ully codified in s anda d IGES files, widely applied in indus y. The
IGES files p e-p ocessing occu s in wo phases (i) diffe en IGES files wi h he new design
a iables a e w i en; (ii) he cons ain s imposed by he p oblem unde analysis a e
e ified and he necessa y meshes gene a ed, using GiD so wa e h ough a ba ch file.
The Blank Holde Fo ce (BHF) is an addi ional design a iable which has o be closely
moni o ed due o possible necking. This is ele an in o de o asce ain he easibili y o
he esul s ound a he blank s uc u al le el.
3.4 Cos Func ion Fo mula ion
The main goal is o minimize he geome ical de ia ion be ween a o med piece and
he e e ence one. This de ia ion may be measu ed by means o he euclidean dis ance
be ween co esponding pai s o nodes o he o med blank. Al e na i ely i can measu e
he dis ance be ween hose nodes along one well defined di ec ion (e.g. X-axis) [6] o
a he as he e o be ween ag eed se s o pa ame e s (such as he diffe ence be ween a
ce ain angle). The use o dis ances equi es a pai ing algo i hm be ween op imised piece
and e e ence nodes, which can be cos ly. Mo e o en, he op ion o compa ing a se o
deduced geome ic pa ame e s is enough. In his wo k, he cos unc ion is defined as
E(x)=1
n
n
∑
i=1
bi(xi−xi e )2,(2)
whe e xiand x e
ia e he obse ed and desi ed alues and biis he weigh associa ed o
he i h pa ame e diffe ence. The weigh is also used as a scale ac o in o de o le el he
diffe ences o magni ude o de s.
4 CASE STUDY: U-RAIL
The sp ingback compensa ion s a egies desc ibed on he p e ious sec ion a e compa ed
using he U-Rail benchma k. This pa icula case s udy is chosen due o la ge sp ingback
effec s which, i no dully compensa ed, may cause se ious quali y p oblems.
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4.1 Case S udy Fea u es
Fig. 2 p esen s he U-RAIL p oblem, as desc ibed in [15], which is aken as e e ence
o his wo k. The o iginal blank has 300x300x0.8 [mm] bu i can be simplified assuming
plane s ain condi ions. Thus only hal o he leng h and a s ip o 10 mm wid h is
simula ed. This is done o educe he compu a ional ime. The p ocesses is implemen ed
on a compu e equipped wi h an In el Co e I7 P ocesso wi h 8 co es and wi h 8 GB
RAM, whe e each simula ion akes an a e age o 711 seconds o be compu ed.
The blank ma e ial has he p ope ies lis ed on Table 1. This able also shows he
numbe o elemen s used in he FEM disc e iza ion. The ools a e assumed as igid and
handled by means o Naga a pa ches [16]. A ic ion coefficien o 0.15 is used.
(a) Tools and design
a iables
(b) Pe spec i e iew o he e e ence
piece (c) Desi ed piece geome y
Figu e 2: Schema ic o he expe imen ele an angles and poin s
P ope y Value
Young Modulus [MPa] 206629
Poisson’s Ra io 0.298
Swi Ha dening Law (K [MPa], n, ϵ0)488.35, 0.24, 0.015
Hill48 (F,G,H,N) 0.63974, 0.60976, 0.39024, 1.43693
Numbe o elemen s 990 (leng h) x 1 (wid h) x 3 ( hick.)
Table 1: Ma e ial P ope ies and model pa ame e s
In o de o e i y i he condi ions adop ed on DD3IMP accu a ely cap u e he eal
beha iou o he me al shee o ming, he esul s a e compa ed o he e e ence condi ion.
The flange angle compu ed by DD3IMP is o 13.66o, which has an e o o 0.07% o he
alues measu ed in [15].
The con ol poin s o he NURBS conside ed as design a iables a e showed in Fig.
2a: P1 o P4. Some cons ain s on hei mobili y a e assu ed in o de o gua an ee some
ools’ geome ic ea u es, such as o hogonali y be ween e ical and ho izon al su aces
o he ac ha he conco dance be ween hose su aces is desc ibed by a adius. They
can be w i en as
P3x= (30 −P1z); P4z= (28.85 −P2x),(3)
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whe e he subsc ip indica es he di ec ion. The e o e only h ee design a iables a e
s udied: he coo dina e zo poin P1 o de ine he die shape, he coo dina e xo P2 o
de ine he punch geome y, and he Blank Holde Fo ce. In cases whe e some limi a ion
o he pa ame e s is necessa y o he op imiza ion algo i hm, he inpu space conside ed
in [15] is adop ed. The same is alid o he alues adop ed in Eq. 1 o Xmax
iand Xmin
i.
4.2 Case S udy E alua ion
(a) Poin s and angles (b) Zoom e ical su ace
Figu e 3: E o measu emen s
As he main objec i e is o ensu e he o hogonali y and plana i y o he su aces,
ins ead o compu ing he euclidean dis ance be ween ma ching pai s o poin s on he wo
pieces, he geome ical e o is compu ed eso ing o he e o s o he angles θ1and θ2
and he dis ance h o he espec i e e e ence alues (see Fig. 3). In o de o e alua e
hese a iables, auxilia y poin s a e de ined in he componen as shown in Fig. 3. Thei
coo dina es a e used o e alua e he de ia ions on he inal piece ough
α= an (Ez−Dz
Ex−Dx);θ1=180 − an (Bz−Az
Bx−Ax);h=√(Cx−Fx)2+ (Cz−Fz)2.(4)
whe e he le e in subsc ip desc ibe he coo dina e o he poin in case. The angle θ2is
measu ed conside ing he igonome ic ela ionships be ween θ1and α,i.e.,
θ2=θ1−α. (5)
The alues o he angles θ1,θ2and dis ance h, which a e e alua ion a iables, a e used
as inpu s on Eq. 2 esul ing in
E(θ1,θ
2,h)=1
3((θ1−90)2+ (θ2−90)2+(h∗10)2).(6)
As bo h angles a e desi ed o be o hogonal, he e e ence alues o bo h is 90o. The
cu a u e o he e ical su ace is supposed o be null, being he e e ence alue o h
0 mm. Addi ionally, as he scale o he cu a u e alues is a magni ude lowe han he
angle e o s, his a iable is mul iplied by 10, so ha i s in luence is no o e looked.
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5 RESULTS DISCUSSION
5.1 Sensi i i y and Noise Analysis
Fig. 4 p esen s he sensi i i y analysis pe o med o also e alua e he nume ical noise.
The leas sensi i e a iable is he P2x ha defines he punch adius, wha is in ag eemen
wi h li e a u e [15]. The o he a iables p esen a non negligible influence on he final
geome y o he piece, being he BHF he one who influences he angles he mos . Fig. 4c
shows ha an inc ease on he BHF alues lead o an e olu ion o he geome ic e alua ion
pa ame e s o he piece o he desi ed e e ences. The same is e ified, hough in a e y
smalle scale o low alues o he P1z(small die adius). On his analysis, only his
isibly affec ed by nume ical noise. Howe e , wi h hese noise le els, i is expec able ha
he basic s a egies employed in his wo k and p esen ed in Subsec ion 3.2, a e able o
o e came i . In ac , he p ocess p esen ed in Fig. 1 is classified as obus , as i is able
o o e come noise and ailu es on simula ions. Those ailu es only occu ed when he
design a iables ell in o alues ha lead o impossible ools (e.g. ools wi h null nega i e
adius). In hose cases, he p ocess ollows o he nex cycle wi h a new se o a iables.
Six s a ing poin s a e used in his wo k. Due o i s low sensi i i y, he P2xs a ing
poin alue emained cons an , being he poin s he combina ions o he alues −3, −5
and -7 o he P1zand 210 and 270 kN o he BHF.
(a) P2x= 23.85mm;
BHF = 210kN
(b) P1z= -5.0mm;
BHF = 210kN
(c) P1z= -5.0mm;
P2x= 23.85mm
Figu e 4: Sensi i i y Analysis T ials:
5.2 Op imiza ion Resul s
The ollowing exp essions a e ob ained using he Linea RSM:
θ1=−0.2014 ·P1z−0.0343 ·P2x−3.60 ×10−5·BHF + 102.65; (7a)
θ2=−0.2076 ·P1z+0.0117 ·P2x+2.21 ×10−5·BHF + 81.00; (7b)
h=−0.0129 ·P1z−0.0020 ·P2x−1.26 ×10−6·BHF +0.4502.(7c)
The minimum o equa ion 6 using he p e ious defini ions o θ1,θ2and h all in a
solu ion ha is no easible, conside ing ha he punch and he die would in e sec each
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