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Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Cor
nell
Uni
vers
ity
A C
ON
TIN
UU
M S
EN
SIT
IVIT
Y A
NA
LY
SIS
OF
LA
RG
E D
EF
OR
MA
TIO
NS
WIT
H A
PP
LIC
AT
ION
S T
O
ME
TA
L F
OR
MIN
G D
ESI
GN
Spec
ial C
omm
ittee
:Pr
of. N
icho
las
Zab
aras
(M
&A
E)
Prof
. Sub
rata
Muk
herj
ee (
T&
AM
)
Supp
ort :
WPA
FB, A
FOSR
, CT
C
Prof
. Tho
mas
Col
eman
(C
S)
B-e
xam
pre
sent
atio
n
Srikanth Akkaram
Nov
embe
r 20
th, 2
000
Fin
ite th
erm
o-in
elas
tic d
efor
mat
ion
anal
ysis
Mul
tista
ge s
ensi
tivity
ana
lysi
s
Met
al fo
rmin
g op
timiz
atio
n ex
ampl
es
The
sen
sitiv
ity d
efor
mat
ion
prob
lem
Def
initi
on a
nd c
ompu
tatio
n of
sen
sitiv
ity fi
elds
OU
TL
INE
OF
TH
E P
RE
SEN
TA
TIO
N
Sug
gest
ions
for
futu
re w
ork
Cor
nell
Uni
vers
ity
An
exam
ple
to m
otiv
ate
the
need
for
met
al fo
rmin
g de
sign
Obj
ectiv
es o
f thi
s pr
ojec
t
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup1
Effi
cien
t man
ufac
ture
of d
esire
d sh
ape
with
desi
red
mat
eria
l pro
pert
ies
in th
e fin
al p
rodu
ct
Met
al fo
rmin
g :
MO
TIV
AT
ION
AN
D O
BJE
CT
IVE
S Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Cor
nell
Uni
vers
ity2
Pla
ne s
trai
n fo
rgin
g
Initi
al p
rodu
ct
Fin
al p
rodu
ct
MO
TIV
AT
ION
AN
D O
BJE
CT
IVE
S
Fea
ture
s of
form
ing
proc
esse
s
Mec
hani
sms
coup
led
in a
hig
hly
non-
linea
r fa
shio
n
Pre
dict
the
resp
onse
of e
ach
of th
ese
mec
hani
sms
tova
riatio
ns in
con
trol
var
iabl
es -
Sen
sitiv
ity fi
elds
Larg
e de
form
atio
n pl
astic
ity
Def
orm
atio
n in
duce
d m
icro
stru
ctur
e ev
olut
ion
Tim
e va
ryin
g co
ntac
t and
fric
tion
cond
ition
s
The
rmal
effe
cts
: res
ult o
f mec
hani
cal d
issi
patio
n
Dam
age
accu
mul
atio
n le
adin
g to
mat
eria
l ru
ptur
e
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Cor
nell
Uni
vers
ity3
Inte
rmed
iate
sta
ge
Fini
shin
g st
age
Initi
al w
orkp
iece
Fina
l pro
duct
Pref
orm
ing
stag
e
MO
TIV
AT
ION
AN
D O
BJE
CT
IVE
S
For
min
g de
sign
obj
ectiv
es :
Min
imiz
e en
ergy
req
uire
d to
def
orm
wor
kpie
ce
Des
ired
final
sha
pe o
f the
pro
duct
Des
ign
clas
sific
atio
n
Des
ired
mic
rost
ruct
ure
in th
e fin
al p
rodu
ct
Des
ign
of s
eque
nces
Mul
tista
ge d
esig
n
Sin
gle
stag
e de
sign
Die
& P
roce
ss p
aram
eter
s
Pref
orm
* *
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Cor
nell
Uni
vers
ity4
OB
JEC
TIV
E
Pre
form
des
ign
- S
hape
sen
sitiv
ity
Mul
tista
ge d
esig
n -
Sha
pe &
Par
amet
er s
ensi
tivity
Die
and
pro
cess
par
amet
er d
esig
n -
Par
amet
er s
ensi
tivity
Sen
sitiv
ity a
naly
sis
prov
ides
the
basi
s fo
r gr
adie
nt b
ased
form
ing
desi
gn o
ptim
izat
ion
Dev
elop
a d
efor
mat
ion
proc
ess
desi
gn m
etho
dolo
gy
* A
ccur
ate
desc
ript
ion
of t
he m
echa
nics
of
def
orm
atio
n
* E
ffic
ient
and
acc
urat
e co
mpu
tati
on o
f de
sign
der
ivat
ives
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Cor
nell
Uni
vers
ity5
HO
T F
OR
MIN
G A
NA
LY
SIS
Nat
ure
of c
oupl
ing
betw
een
defo
rmat
ion
and
ther
mal
fiel
ds
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Cor
nell
Uni
vers
ity
Ver
satil
e la
rge
obje
ct o
rient
ed p
rogr
am d
evel
oped
Var
ious
con
stitu
tive
mod
els,
con
stitu
tive
inte
grat
ion
s
chem
es, c
onta
ct a
nd fr
ictio
n m
odel
s, fi
nite
ele
men
t
typ
es, r
emes
hing
sch
emes
, for
min
g ap
plic
atio
ns
Fea
ture
s of
the
dire
ct th
erm
omec
hani
cal s
imul
ator
6
Diff
pack
C+
+ li
brar
y pr
ovid
ed th
e ba
sic
FE
M e
nviro
nmen
t
X
xn
xn
+1
Bo
Bn
Bn
+1
xn
=~ x(X;
tn
)
xn
+1
=~ x(X;
tn
+1)
xn
+1
=^ x(xn
;
tn
+1)
Fn
Fn
+1
Fr
r
n
�Pr
+
fr
=
0
ZBn
Pr�r
n
~ udVn
=
Z �
��~ udAn
+
ZBn
fr�~ udVn
DE
FO
RM
AT
ION
P
RO
BL
EM
Prin
cipl
e of
virt
ual w
ork
: UL
form
ulat
ion
New
ton-
Rap
hson
Met
hod
with
Lin
e S
earc
h
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Cor
nell
Uni
vers
ity7
Alg
orith
mic
div
isio
n of
the
defo
rmat
ion
prob
lem
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Cor
nell
Uni
vers
ity
Kin
emat
ic s
ubpr
oble
m-
Con
stitu
tive
subp
robl
em-
Con
tact
& fr
ictio
n su
bpro
blem
-
Rem
eshi
ng &
dat
a tr
ansf
er s
ubpr
oble
m-
8
Giv
en d
ispl
acem
ents
Com
pute
def
orm
atio
n gr
adie
nt a
nd s
trai
ns
Giv
en d
efor
mat
ion
grad
ient
Giv
en o
ld m
esh
Upd
ate
stre
sses
, sta
te v
aria
ble
& d
amag
e p
aram
eter
s
Upd
ate
regi
ons
of c
onta
ct, s
tick,
slip
& tr
actio
ns
Com
pute
new
mes
h &
tran
sfer
def
orm
atio
n fie
lds
DE
FO
RM
AT
ION
PR
OB
LE
M
Giv
en lo
catio
n of
die
Initialcon�guration
Intermediatethermalcon�guration
Stressfree(relaxed)con�guration
Deformedcon�guration
Temperature:�o
Temperature:�
Temperature:�
Temperature:�
Voidfraction:fo
Voidfraction:fo
Voidfraction:f
Voidfraction:f
B
o
B
F
e
F
F
p
F
�
CO
NST
ITU
TIV
E S
UB
PR
OB
LE
M
ep
F =
F F
Fθ
The
follo
win
g m
odel
is p
ropo
sed: M
ater
ials
Pro
cess
Des
ign
and
Con
trol
Gro
upC
orne
ll U
nive
rsity9
CO
NST
ITU
TIV
E S
UB
PR
OB
LE
M
F
Fθ
θ-1
= β
θI
The
rmal
Exp
ansi
on :
Mec
hani
cal d
issi
patio
n
Hyp
erel
astic
con
stitu
tive
law
Inel
astic
res
pons
e :
Flo
w r
ule
:
Voi
d fr
actio
n (
dam
age
) ev
olut
ion
ΦD =
sym
( L
) =
F F
=
Φd T
-1
γ
pp
pp
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Cor
nell
Uni
vers
ity
is th
e vi
scop
last
ic p
oten
tial
( G
urso
n et
al.
)
Inte
rnal
var
iabl
e ev
olut
ion
( A
nand
et a
l. )
10
Initial
Con�guration In
termediatethermalcon�guration
Stressfreecon�guration
Deformedcon�guration
B
o
B
n
B
n+1
�o
�n
�n
�n+1
�n+1
Fe n
Fe �
Fe n+1
Fp n
Fp n+1
F� n
F� n+1
FT
F
Fn
Fr
FC M
ater
ials
Pro
cess
Des
ign
and
Con
trol
Gro
upC
orne
ll U
nive
rsity11
CO
NST
ITU
TIV
E T
IME
IN
TE
GR
AT
ION
g(xn
+
1)�
0
�N
=
�
��
�N
g(xn
+
1)=
0
�
T
=
�
�
+
�N
�
�
:=
jj�
T
jj
�
��N
�
0
v
T
=
�
�
T
jj�
T
jj
�
�
0
��
=
0
�=
0
�=
1
e1
e2
e3
AdmissibleregionK
(g<
0)
Inadmissibleregion(g>
0)
@K
(g=
0)
Reference
con�guration
�
Currentcon�guration
B
n
xn
+1
xn
B
n
+1
xn
+1
=
^ x(xn
;t n+1)
g(x)
�
r
n
� y
CO
NT
AC
T A
ND
FR
ICT
ION
SU
BP
RO
BL
EM
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Cor
nell
Uni
vers
ity
Sim
o &
Lau
rsen
(1
992)
12
CO
NT
AC
T A
ND
FR
ICT
ION
SU
BP
RO
BL
EM
mul
tiplie
r es
timat
es
en
forc
e no
rmal
con
tact
and
fric
tiona
l (st
ick)
con
ditio
ns
Nor
mal
con
tact
- p
enet
ratio
n fu
nctio
n
Tan
gent
ial c
onta
ct -
Cou
lom
b fr
ictio
n m
odel
Aug
men
ted
Lagr
angi
an fo
rmul
atio
n us
ed to
Acc
urat
e en
forc
emen
t of c
onst
rain
ts w
ith m
odes
t pen
altie
s
Con
tact
trac
tions
giv
en b
y co
nver
ged
Lagr
ange
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Cor
nell
Uni
vers
ity
13
1.Computetheequilibratedbodycon�gurationB
(old)
n+1correspond-
ingtothemeshdiscretizationM
(old)
u(old)
n+1
=
x(old)
n+1�
x(old)
n
2.Performaremeshingoperationonthespatialcon�gurationB
n+1
toyieldameshdiscretizationM
(new) .
u(new)
n+1
=
T1
" u(old)
n+1
#
x(new)
n
=
x(new)
n+1
�
u(new)
n+1
3.Qn
representsthenecessarysetof�eldvariablesthatcharacter-
izesthehistoryofthematerialdeformationattimet n.
Q(new)
n
=
T2
� Q(old)
n
�
Q1=(F
e;s)orQ
2=(F;F
p;s).
4.�n
=
(�N
n
;�Tn
)representsthenormalandtangentialcontact
tractionsattimencorrespondingtomeshM
(old) .
�(new)
n
=
T3
� �(old)
n
�
5.Solvethedirectdeformationproblem
forthetimeincrement
[tn;tn+1]onthemeshM
(new) .
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Cor
nell
Uni
vers
ity
RE
ME
SHIN
G A
ND
DA
TA
TR
AN
SFE
R P
RO
CE
DU
RE
With
out r
emes
hing
With
rem
eshi
ng
Initi
alD
efor
med
14
34
6
6
7
7
7
88
36
7
Stat
e va
riabl
e
841
.000
0
739
.714
3
638
.428
6
537
.142
9
435
.857
1
334
.571
4
233
.285
7
132
.000
0
3
45
5
5
6
66
7
77
88
86
Equi
vale
nt s
tress
819
.000
0
718
.142
9
617
.285
7
516
.428
6
415
.571
4
314
.714
3
213
.857
1
113
.000
0
1
1
2
2
3
3
44
4 5
55
5
5 66
78
5
Plas
tic s
train
82.
0000
71.
7500
61.
5000
51.
2500
41.
0000
30.
7500
20.
5000
10.
2500
46
6 6
7 7
7
Stat
e va
riabl
e
841
.000
0
739
.714
3
638
.428
6
537
.142
9
435
.857
1
334
.571
4
233
.285
7
132
.000
0
56
77
856
Equi
vale
nt s
tress
819
.000
0
718
.142
9
617
.285
7
516
.428
6
415
.571
4
314
.714
3
213
.857
1
113
.000
0
2
3
3 44
55
66
7
5
Plas
tic s
train
81.
8500
71.
6214
61.
3929
51.
1643
40.
9357
30.
7071
20.
4786
10.
2500
RE
ME
SHIN
G A
ND
DA
TA
TR
AN
SFE
R O
PE
RA
TO
RS
Issu
es in
volv
ed in
a r
obus
t rem
eshi
ng p
roce
dure
for
larg
e de
form
atio
ns
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Cor
nell
Uni
vers
ity
No
rem
eshi
ngW
ith r
emes
hing
15
Pre
form
B, w
ith d
amag
e
Pre
form
A, n
o da
mag
e
Pre
form
A, w
ith d
amag
e
IMP
OR
TA
NC
E O
F M
ET
AL
FO
RM
ING
DE
SIG
N
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Cor
nell
Uni
vers
ity16
1
2
2
3
3 4
4
5
55
6
Leve
lV
oid
frac
tion
60.
0390
50.
0319
40.
0248
30.
0177
20.
0107
10.
0036
1
2
2
3
3
4
4
5
6
Leve
lV
oid
frac
tion
60.
0514
50.
0487
40.
0428
30.
0309
20.
0128
10.
0023
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0
500
1000
1500
2000
2500
3000
Force (N)
Str
oke
(mm
)
Pre
form
B
Pre
form
A
For
ce v
s st
roke
cha
ract
eris
tics
Dam
age
dist
ribut
ion
in fi
nal p
rodu
ct
Pre
form
AP
refo
rm B
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
IMP
OR
TA
NC
E O
F M
ET
AL
FO
RM
ING
DE
SIG
N Cor
nell
Uni
vers
ity17
SCH
EM
AT
IC O
F T
HE
SE
NSI
TIV
ITY
AL
GO
RIT
HM
equi
libri
um e
quat
ion
Con
tact
and
fri
ctio
nco
nstr
aint
s
Sens
itivi
ty w
eak
form
mat
eria
l con
stitu
tive
law
s
law
sM
ater
ial c
onst
itutiv
e
Sens
itivi
ty c
onst
itutiv
e s
ub-p
robl
emSe
nsiti
vity
con
tact
sub-
prob
lem
inte
grat
ion
inte
grat
ion
Tim
e
Tim
e
Equ
ilibr
ium
equ
atio
n
Reg
ular
ized
des
ign
deri
vativ
e of
the
cont
act
and
fric
tion
cons
trai
nts
Des
ign
deri
vativ
e of
the
Des
ign
deri
vativ
e of
the
wea
k fo
rm
Inpu
tIn
put
Ass
umed
kin
emat
ics
assu
med
kin
emat
ics
mod
ifie
d w
eak
form
Tim
e an
d sp
ace
disc
retiz
ed
Tim
e an
d sp
ace
disc
retiz
ed
Mod
ify
Des
ign
deri
vativ
e of
the
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Cor
nell
Uni
vers
ity18
B
Bo
BR
B0
B0 o
x
X
Y
x+
Æ x
X
+
Æ X
F F+
Æ F
FR F
R
+
Æ FRI
+LR
X
=
� X(Y;�s
)
X
+
Æ X
=
� X(Y;�s
+��s
)
x=
~ x(X;
t
;�s
)
x+
Æ x=
~ x(X+
Æ X
;
t
;�s
+��s
)
Æ �=
~Æ �(X
;
t
;�s
;
��s
)=
�Æ �(Y;
t
;�s
;
��s
)=
dd
�
� �(Y;
t
;�s
+�
��s
)� � � � � � �=
0
Æ �=
� �(Y;
t
;�s
+��s
)�
� �(Y;
t
;�s
)+O
� jj��s
jj2
�
SHA
PE
SE
NSI
TIV
ITY
OF
TH
E D
EF
OR
MA
TIO
N
Gat
eaux
diff
eren
tial
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Cor
nell
Uni
vers
ity19
Æ
r
o
�P
+
Æ f=
0
ZB
o
Æ P
�r
o
~ �dVo
�
ZB
o
P
� ro
�LT R
� �~ �dVo
�
ZB
o
� PLT R
�r
o
~ �� dVo
=
Z@B
o
(Æ �
�
[LR
�
(N
N
)]�
)�
~ �
dA
o
Æ F
=
Æ
r
o
x
=
r
o
Æ x
�
F
LR
Æ P
=
A
" Æ F# +
B
Der
ive
a w
eak
form
for
the
shap
e se
nsiti
vity
of t
he
equ
ilibr
ium
equ
atio
n
SEN
SIT
IVIT
Y D
EF
OR
MA
TIO
N P
RO
BL
EM
x -
sen
sitiv
ity o
f the
def
orm
ed c
onfig
urat
ion
oPrim
ary
unkn
own
of th
e w
eak
form
TL
wea
k fo
rm
Cor
nell
Uni
vers
ityM
ater
ials
Pro
cess
Des
ign
and
Con
trol
Gro
up20
SE
NSI
TIV
ITY
DE
FO
RM
AT
ION
PR
OB
LE
M
Alg
orith
mic
div
isio
n of
the
sens
itivi
ty d
efor
mat
ion
prob
lem
Kin
emat
ic s
ubpr
oble
m
Con
stitu
tive
subp
robl
em
Con
tact
& fr
ictio
n su
bpro
blem
Rem
eshi
ng &
dat
a tr
ansf
er s
ubpr
oble
m
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Cor
nell
Uni
vers
ity
Rel
atio
n be
twee
n th
e se
nsiti
vity
of
( de
form
atio
n gr
adie
nt &
dis
plac
emen
t ) e
tc.
Giv
en th
e de
form
atio
n hi
stor
y,
Rel
atio
n be
twee
n th
e se
nsiti
vity
of
( st
ress
, sta
te &
def
orm
atio
n gr
adie
nt )
Giv
en th
e co
ntac
t reg
ions
& tr
actio
ns,
Rel
atio
n be
twee
n th
e se
nsiti
vity
of
( tr
actio
ns &
dis
plac
emen
t )
Giv
en th
e ne
w m
esh,
Tra
nsfe
r se
nsiti
vitie
s be
twee
n th
e ol
d m
esh
and
new
mes
h
21
Giv
en th
e de
form
ed c
onfi
gura
tion,
Parent
con�guration
Initial
con�guration
Unstressed
con�guration
Deformed
con�guration
B
Bo
BR
B0
B0 o
x
X
Y
x+
Æ x
X
+
Æ X
FF+
Æ F
Fe
Fp
Fe+
Æ Fe
Fp+
ÆF
p
FR
Æ Fi
FR+
Æ FR
SEN
SIT
IVIT
Y C
ON
STIT
UT
IVE
SU
BP
RO
BL
EM
Bad
rinar
ayan
an a
nd Z
abar
as (
1996
)
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Cor
nell
Uni
vers
ity22
SEN
SIT
IVIT
Y C
ON
STIT
UT
IVE
SU
BP
RO
BL
EM
Obt
ain
the
linea
r re
latio
nshi
p be
twee
n st
ress
, sta
te
and
defo
rmat
ion
sens
itivi
ty fi
elds
Obt
ain
rate
law
s go
vern
ing
the
sens
itivi
ty o
f ine
last
ic
Sen
sitiv
ity o
f the
hyp
erel
astic
con
stitu
tive
law
sens
itivi
ty fi
elds
at t
he e
nd o
f the
load
ing
incr
emen
tT
ime
inte
grat
ion
of th
e ra
te c
onst
itutiv
e la
ws
to c
ompu
te
Pro
blem
iden
tical
for
shap
e an
d pa
ram
eter
sen
sitiv
ity
varia
bles
( p
last
ic d
efor
mat
ion
rate
, sta
te v
aria
bles
)
To
achi
eve
this
( Z
abar
as e
t al.
1996
) :
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
22a
Cor
nell
Uni
vers
ity
y
=y(�)Die
y
=y(�)Die
� y=y(� �)
� y+y;�
Æ � �
B
R
B
B
o
B
0
B
0 o
x
XX
+
Æ X
Y
x+
Æ x
�
�
r
r
x
=~ x(X;t;�s
)
X
=
� X(Y;�s
)
X
=
� X(Y;�s
+��s
)
x
=~ x(X+
Æ X
;t;�s
+��s
)
CO
NT
AC
T S
HA
PE
SE
NSI
TIV
ITY
Ess
entia
l for
pre
form
des
ign
Cha
nges
in th
e in
itial
pre
form
str
ongl
y in
fluen
ce
cont
act h
isto
ry
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Cor
nell
Uni
vers
ity23
Diey
=
y(�)
PerturbedDie
y
=
y+
Æ y
� y=
y(� �)
� y+
Æ [� y]
B
B
o
B
0
x
X
x+
Æ x
�
r �+
Æ � r+
Æ r
x
=
~ x(X;t;�p)
x
=
~ x(X;t;�p
+��p)
CO
NT
AC
T P
AR
AM
ET
ER
SE
NSI
TIV
ITY
Ess
entia
l for
die
des
ign
Cha
nges
in d
ie s
hape
are
the
driv
ing
forc
e
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Cor
nell
Uni
vers
ity24
Reg
ular
izin
g as
sum
ptio
ns fo
r no
n-di
ffere
ntia
bilit
yco
ntac
t sen
sitiv
ity a
ssum
ptio
nfr
ictio
n se
nsiti
vity
ass
umpt
ion
Diff
eren
tiate
the
stro
ng fo
rm o
f con
tact
con
stra
ints
as o
ppos
ed to
the
time
disc
rete
trac
tion
upda
te
Use
hig
her
(diff
eren
t) p
enal
ties
than
thos
e us
ed in
the
co
ntac
t pro
blem
to e
nfor
ce s
ensi
tivity
con
stra
ints
Sen
sitiv
ity s
tiffn
ess
cont
ribut
ion
- im
plic
it na
ture
of th
e co
ntac
t alg
orith
m
A [
x ]
+ b
λ =
o
SEN
SIT
IVIT
Y C
ON
TA
CT
SU
BP
RO
BL
EM
o
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Cor
nell
Uni
vers
ity25
Æ �=
Æ �N
�(� y)+�N
Æ
[�(� y)]�
Æ �T
�1(� y)�
�T
Æ
[�1(� y)]
Æ �N
=
Æ �Nn
+� N
Æ g(xn+1)
Normalcontact
_Æ � �
=
1 � T_Æ �
T
Stick
Æ �T
=
Æ
0 B @ ��N
�T
jj�Tjj
1 C A
Slip
Æ � �
=
a
�
Æ x
+
b
a
=
�
1(� y)
jj�
1(� y)jj2
f
1
+
g�(� y)g
b
=
�
( g�
(� y)�
Æ � y;
�
+
�
1(� y)�
Æ � y)
jj�
1(� y)jj2
f
1
+
g�(� y)g
Æ g
=
�
(� y)�
(
Æ � y
�
Æ x
)
SEN
SIT
IVIT
Y C
ON
TA
CT
SU
BP
RO
BL
EM
26
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Cor
nell
Uni
vers
ity
Sen
sitiv
ity o
f con
tact
trac
tions
Sen
sitiv
ity o
f the
gap
func
tion
and
inel
astic
slip
Solve
the i
ncrem
ental
defo
rmati
on pr
oblem
Chec
k if c
ontac
t con
strain
tsare
satis
fied.
Augm
ent to
obtai
nmo
re ac
curat
e tra
ction
estim
ates.
Solve
sens
itivit
y defo
rmati
onpr
oblem
( us
e ove
rsize
dpe
nalti
es fo
r con
tact )
YES
NO
direc
t defo
rmati
onSo
lve th
e inc
remen
tal
prob
lem
Solve
the s
ensit
ivity
defo
rmati
on pr
oblem
Chec
k if c
ontac
t co
nstra
ints a
re sa
tisfie
d.
Post-
proc
ess
YES
NO
Solve
direc
t and
sens
itivit
y pro
blem
estim
ates o
fus
ing m
ore a
ccur
ate
tracti
ons a
nd tr
actio
nse
nsiti
vities
SEN
SIT
IVIT
Y C
ON
TA
CT
SU
BP
RO
BL
EM
Ove
rvie
w o
f the
sen
sitiv
ity c
onta
ct a
lgor
ithm
A d
iscr
ete
itera
tive
sche
me
A c
ontin
uum
line
ar s
chem
e
Pre
ferr
edap
proa
ch
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Cor
nell
Uni
vers
ity27
B
B
n
B
o
B
0
B
0 n
x
xn
X
x+
Æ x
xn
+
Æ xn
F
r
F
r+
Æ F
r
F
nF
n
+
Æ F
n
I
+
Ln
xn
=
~ x(X
;
t
n
;�p)
Q
n
=
~Q
(X
;
t
n
;�p)
xn
+
Æ xn
=
~ x(X
;
t
n
;�p
+
��p)
Q
n
+
ÆQ
n
=
~Q
(X
;
t
n
;�p
+
��p)
x
=
^ x(xn
;
t
;�p)
x+
Æ x=
^ x(xn
+
Æ xn
;
t
;�p
+
��p)
UL
SE
NSI
TIV
ITY
FO
RM
UL
AT
ION
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Cor
nell
Uni
vers
ity
Par
amet
er s
ensi
tivity
in th
e tim
e in
crem
ent [
n,n+
1]
28
B
Bn
Bo
BR
B0
B0 n
B0 o
x
xn
X
Y
x+
Æ x
xn
+
Æ xn
X
+
Æ X
Fr
Fr
+
Æ Fr
FR
FR
+
Æ FR
Fn
Fn
+
Æ Fn
I+Lo
I+Ln
X
=
� X(Y;�s
)
X
+
Æ X
=
� X(Y;�s
+��s
)
xn
=
~ x(X;
t
n
;�s
)
Q
n
=
~Q
(X;
t
n
;�s
)
xn
+
Æ xn
=
~ x(X+
Æ X
;
t
n
;�s
+��s
)
Q
n
+
ÆQ
n
=
~Q
(X+
Æ X
;
t
n
;�s
+��s
)
x=
^ x(xn
;
t
;�s
)
x+
Æ x=
^ x(xn
+
Æ xn
;
t
;�s
+��s
)
UL
SE
NSI
TIV
ITY
FO
RM
UL
AT
ION
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Cor
nell
Uni
vers
ity
Sha
pe s
ensi
tivity
in th
e tim
e in
crem
ent [
n,n+
1]
29
Æ
rn�Pr
+Æ f
r
=0
Z Bn
Æ Pr
�rn~ �dVn
�
Z Bn
Pr
� rn�L
T n� �
~ �dVn
�
Z Bn
� PrL
T n�rn~ �
� dVn
=Z �( Æ �
�[Ln�(nn)]�
) �~ �dAn
Æ
F
r
=
Æ
r
n
x
=
r
n
Æ x
�
F
r
L
n
Æ
F
=
Æ
F
r
F
n
+
F
r
Æ
F
n
UL
SE
NSI
TIV
ITY
FO
RM
UL
AT
ION
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Cor
nell
Uni
vers
ity
Wea
k fo
rm fo
r sh
ape
and
para
met
er s
ensi
tiviti
es
Kin
emat
ic r
elat
ions
hips
30
F-bar
metho
dB-b
ar me
thod
1. No s
tabiliz
ation (
mesh
a)
3. With
stabil
izatio
n (me
sh b)
2. With
stabil
izatio
n (me
sh a)
Reference
con�guration
Deformed(sti�/locked)
con�guration
Deformed(unstable)
con�guration
Bn
Fh
Fdev
h
� Fh
� Fvol
h
Fvol
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Cor
nell
Uni
vers
ity
Ext
ensi
on o
f the
F-b
arm
etho
d by
Ow
en (
96)
31
PE
RF
OR
MA
NC
E O
F A
SSU
ME
D S
TR
AIN
AN
AL
YSE
S
Æ Fave
h
=8 < :�
Æ Fh
+(1�
�)[
� J h J]1 3
Æ Fh
9 = ;+
1�
�3
8 < :2 4N
IN
T
X a=
1
Jha(� �a)tr[Æ Fh
(� �a)F
�
1
h
(� �a)]� Na
3 5� J�
1
h
� Fh
�
tr[Æ Fh
F�
1
h
]� Fh
9 = ;
Sint
h
=X e
[Z e
Æ Pr(Æ F
ave
h
)�rn~ � hdVn�
Z e
� Pr(F
ave
h
)h rn�
LT n
i� �
~ � hdVn
�
Z e
h Pr(F
ave
h
)LT n
i �
rn~ � hdVn]
TR
EA
TM
EN
T O
F I
NC
OM
PR
ESS
IBIL
ITY
Sen
sitiv
ity o
f the
ass
umed
def
orm
atio
n gr
adie
nt
Mod
ified
sen
sitiv
ity w
eak
form
( s
tabi
lized
F-b
ar m
etho
d )
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Cor
nell
Uni
vers
ity32
1.Solvethedirectdeformationproblemforthetimeincrement[tn;tn+1].
2.Useanappropriatetransferoperatortocomputethesensitivitiesofthenodesofmesh
M(new
)attimet n.ThistransferisusedinthecomputationofL
(new
)
n
intheULsensi-
tivityanalysis.
Æ x(new
)
n
=S1
� Æ x(old)
n
�
3.Let
Æ Qn
representthesensitivityofthenecessarysetofvariablesthatcharacterizesthe
historyofthematerialdeformationsensitivityattimet n.Thissetisknowncorre-
spondingtothemeshdiscretizationM
(old)andonemusttransferthesevariablesto
thenewmeshusingappropriatetransferoperators
Æ Q(new
)
n
=S2
" Æ Q(old)
n
#
4.Let
Æ �n=(Æ �N
n
;Æ �Tn
)representthenormalandtangentialcontacttractionssensitivities
attimencorrespondingtomeshM
(old) .
Æ �(new
)
n
=S3
" Æ �(old)
n
#
5.Solvethesensitivitydeformationproblemforthetimeincrement[tn;tn+1]onthemesh
M(new
) .
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Cor
nell
Uni
vers
ity
DA
TA
TR
AN
SFE
R F
OR
SE
NSI
TIV
ITY
PR
OB
LE
M
33
Genericformingstage
Bo
Bi
B
B0
xx+
Æ x
X
Y
�X
�X
+��X
�Y
FX
FX
+
Æ FX
FY
x=~ x(X;
t;�X
;
�Y
)
x+
Æ x=~ x(X;
t;�X
+��X
;
�Y
)
X
=
� X(Y;
to
;�Y
)
Q
=
�Q
(Y;
to
;�Y
)
MU
LT
IST
AG
E S
EN
SIT
IVIT
Y A
NA
LY
SIS
Des
ign
sens
itivi
ty o
f the
cur
rent
form
ing
stag
e du
e to
varia
tions
in p
aram
eter
s of
the
curr
ent f
orm
ing
stag
e
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup34
Cor
nell
Uni
vers
ity
Genericformingstage
Bo
B0 o
Bi
B
B0
x x+
Æ x
XX
+
Æ X
Y
�X
�X
�Y
+��Y
�Y
FX
I+Lo
FX
+
Æ FX
FY
+
Æ FY
FY
x=
~ x(X;
t
;�X
;
�Y
)
x+
Æ x=
~ x(X+
Æ X
;
t
;�X
;
�Y
+��Y
)
X
=
� X(Y;
t
o
;�Y
)
Q
=
�Q
(Y;
t
o
;�Y
)
X
+
Æ X
=
� X(Y;
t
o
;�Y
+��Y
)
Q
+
ÆQ
=
�Q
(Y;
t
o
;�Y
+��Y
)
MU
LT
IST
AG
E S
EN
SIT
IVIT
Y A
NA
LY
SIS
Des
ign
sens
itivi
ty o
f the
cur
rent
form
ing
stag
e du
e to
varia
tions
in d
esig
n pa
ram
eter
s of
pre
viou
s fo
rmin
g st
ages
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup35
Cor
nell
Uni
vers
ity
� �=
@� �(Y;t;@Bo;Q)
@(@Bo)
2 6 4@(@Bo)
@�Y
[��Y
]3 7 5+
X i
@� �(Y;t;@Bo;Q)
@Qi
2 6 4@Qi
@�Y
[��Y
]3 7 5
2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4�
0
0
:
:
0
�
�
0
:
:
0
0
�
�
:
:
0
:
:
:
:
:
:
:
:
:
:
0
:
:
�
�
0
0
:
:
0
�
�
3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 58 > > > > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > > > > :
@�q
@�q
@�q+1
@�q
:
:
:
:
@�M
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MU
LT
IST
AG
E S
EN
SIT
IVIT
Y A
NA
LY
SIS
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Cor
nell
Uni
vers
ity
Effe
ct o
f pro
cess
ing
hist
ory
on th
e cu
rren
t for
min
g st
age
:
Gen
eral
izat
ion
to M
sta
ges
:q
= [
1 ..
M ]
Λq
and
prop
ertie
s af
ter
q s
tage
β q
repr
esen
ts p
refo
rm s
hape
is th
e de
sign
spa
ce o
f the
q s
tage
th
36
th
TL
or U
L se
nsiti
vity
form
ulat
ion
with
in e
ach
stag
e
4.0
mm
2.0
mm
Ela
stic
blo
ck
Rig
id o
bsta
cle
y
x
3.6
mm
p =
-20
0 M
Pa
p =
60
MP
a
01
23
4
-20
0
-10
00
10
0
Normal traction design derivative
Lo
catio
n in
th
e in
terf
ace
(m
m)
DD
M
FD
M
01
23
4-4
00
-30
0
-20
0
-10
00
Tangential traction design derivative
Lo
catio
n in
th
e in
terf
ace
(m
m)
DD
M
FD
M
NU
ME
RIC
AL
EX
AM
PL
ES
Sen
sitiv
ity v
alid
atio
n us
ing
FD
M a
s re
fere
nce
Nor
mal
con
tact
trac
tion
sens
itivi
tyT
ange
ntia
l con
tact
trac
tion
sens
itivi
ty
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Cor
nell
Uni
vers
ity37
11
11
12
22
22
33
33
44
4
5
5
66
7
77
8
8
85
7
Str
ess
sen
sitivi
ty (
DD
M)
80
.14
00
70
.12
86
60
.11
71
50
.10
57
40
.09
43
30
.08
29
20
.07
14
10
.06
00
1
11
1
2
22
23
3
4
44
5
55
6
66
77
88
2
35
8
Str
ess
sen
sitivi
ty (
FD
M)
80
.14
00
70
.12
86
60
.11
71
50
.10
57
40
.09
43
30
.08
29
20
.07
14
10
.06
00
1
1
1
2
2
3
3
34
4
45
56
66
77
78
8
Str
ess
sen
sitiv
ity
(DD
M)
80
.14
00
70
.12
86
60
.11
71
50
.10
57
40
.09
43
30
.08
29
20
.07
14
10
.06
00
11
1
1
12
22
2
2
3
3
3
333
4
44
4
5
5
55
5
66
6
6
7
7
77
8
12
3
Str
ess
sen
sitiv
ity
(FD
M)
80
.25
00
70
.20
71
60
.16
43
50
.12
14
40
.07
86
30
.03
57
2-0
.00
71
1-0
.05
00
DD
M
FD
M
No
rem
eshi
ngW
ith r
emes
hing
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
DE
SIG
N S
EN
SIT
IVIT
Y V
AL
IDA
TIO
N
38
Cor
nell
Uni
vers
ity
Objectives
Constraints
Variables
Materialusage
Pressforce
Identi�cationofstages
Plasticwork
Pressspeed
Numberofstages
UniformdeformationProductquality
Preformshape
Microstructure
Geometryrestrictions
Dieshape
Desiredshape
Cost
Mechanicalparameters
Residualstresses
ProcessingtemperatureThermalparameters
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
FO
RM
ING
DE
SIG
N P
RO
BL
EM
S
Fea
ture
s of
a ty
pica
l opt
imiz
atio
n pr
oble
m
Cor
nell
Uni
vers
ity39
510
1520
-0.0
10
0.00
0
0.01
0
0.02
0
0.03
0
0.04
0
0.05
0
Objective function (mm2)
Itera
tion
inde
x
No
rem
eshi
ng
3 re
mes
hing
ope
ratio
ns
5 re
mes
hing
ope
ratio
ns
7 re
mes
hing
ope
ratio
ns
Valid
ation
-with
out r
emes
hing
Inter
med
iate i
terati
onIn
itial
solu
tion
Optim
al so
lutio
n
P
refo
rm
Des
ired
sha
pe
H?
h
or
PR
EF
OR
M D
ESI
GN
EX
AM
PL
EO
bjec
tive
func
tion
in th
e fin
al p
rodu
ctM
inim
ize
barr
elin
g
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Cor
nell
Uni
vers
ity40
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0
50
10
0
15
0
20
0
Force (N)
Str
oke (
mm
)
Initia
l p
refo
rm
Op
tim
al p
refo
rm
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
For
ce
41
CL
OSE
D-D
IE P
RE
FO
RM
DE
SIG
N P
RO
BL
EM
Obj
ectiv
e is
the
desi
red
final
sha
pe
Cor
nell
Uni
vers
ity
Pre
form
ing
sta
ge
Fin
ish
ing
sta
ge
TW
O S
TA
GE
DE
SIG
N E
XA
MP
LE
Itera
tion
3
Itera
tion
6
Initi
al
Opt
imal
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Cor
nell
Uni
vers
ity42
0.19
1
0.37
9
0.56
8
0.75
7
0.94
5
1.13 37
.8
38.3
38.8
39.4
39.9
40.4
4141.5
TW
O S
TA
GE
DE
SIG
N E
XA
MP
LE
Var
iatio
n of
the
inte
rnal
sta
te v
aria
ble
in th
e fin
al p
rodu
ct
Var
iatio
n of
equ
ival
ent p
last
ic s
trai
n in
the
final
pro
duct
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Cor
nell
Uni
vers
ity43
1
1
22
222
2
33
3
33
3
4
4457
1
3
3
48
80.
0100
70.
0083
60.
0066
50.
0049
40.
0031
30.
0014
2-0
.000
3
1-0
.002
0
1
11
1
22
2
2
2
33
33
33
3
44
44
52
3
358
0.01
00
70.
0083
60.
0066
50.
0049
40.
0031
30.
0014
2-0
.000
3
1-0
.002
0
2
33
33
44
4
4
567
55
80.
0070
70.
0053
60.
0036
50.
0019
40.
0001
3-0
.001
6
2-0
.003
3
1-0
.005
0 2
3
33
3
33
44
44
5
55
6
6 455
67
80.
0070
70.
0053
60.
0036
50.
0019
40.
0001
3-0
.001
6
2-0
.003
3
1-0
.005
0
Stat
e se
nsiti
vity
(D
DM
)St
ress
sen
sitiv
ity (
DD
M)
Stat
e se
nsiti
vity
(FD
M)
Stre
ss s
ensi
tivity
(FD
M)
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Pre
form
ing
stag
e
DD
M
FD
M
Sta
teS
tres
s
Cor
nell
Uni
vers
ity
TW
O S
TA
GE
SE
NSI
TIV
ITY
VA
LID
AT
ION
44
55
6
66
66
66
6
77
77
7
77
7
8
8
88
1
5
67
78
80.
0100
70.
0043
6-0
.001
4
5-0
.007
1
4-0
.012
9
3-0
.018
6
2-0
.024
3
1-0
.030
0
55
6
66
66
66
77
77
7
77
8
82
56
7
78
80.
0100
70.
0043
6-0
.001
4
5-0
.007
1
4-0
.012
9
3-0
.018
6
2-0
.024
3
1-0
.030
0
4
55
55
55
6
666
6
6
6
7
77
77
88
8
8
4
45
6
678
8
80.
0050
70.
0029
60.
0007
5-0
.001
4
4-0
.003
6
3-0
.005
7
2-0
.007
9
1-0
.010
0
44
5
55
55
66
66
66
77
7
78
345
6
67
80.
0050
70.
0029
60.
0007
5-0
.001
4
4-0
.003
6
3-0
.005
7
2-0
.007
9
1-0
.010
0
Stat
e se
nsiti
vity
(DD
M)
Stre
ss se
nsiti
vity
(DD
M)
Stat
e se
nsiti
vity
(FD
M)
Stre
ss se
nsiti
vity
(FD
M)
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Fin
ishi
ng s
tage D
DM
FD
M
Sta
teS
tres
s
TW
O S
TA
GE
SE
NSI
TIV
ITY
VA
LID
AT
ION
Cor
nell
Uni
vers
ity45
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
IND
UST
RIA
L D
ESI
GN
PR
OB
LE
MS
Tw
o st
age
desi
gn fo
r en
gine
dis
k fo
rgin
g
Cor
nell
Uni
vers
ity46
FU
TU
RE
RE
SEA
RC
H A
ND
OP
EN
ISS
UE
S
Mat
eria
ls P
roce
ss D
esig
n an
d C
ontr
ol G
roup
Cor
nell
Uni
vers
ity
Mor
e co
mpl
ex fo
rgin
g ge
omet
ries
and
desi
gn fe
atur
es
Exp
licit
mic
rost
ruct
ure
optim
izat
ion
mod
els
Ada
ptiv
e an
alys
is d
riven
by
dire
ct &
sen
sitiv
ity e
rror
indi
cato
rs
Use
idea
l for
min
g m
etho
ds fo
r de
sign
of s
eque
nces
The
rmo-
mec
hani
cal d
esig
n
Tra
nsiti
on fr
om a
cade
mia
to in
dust
ry
47
