line search ex
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CREPORT NO. UCD/CGM-01/04
LEC
B B
DCU S
ENTER FOR GEOTECHNICAL MODELING
INE SEARCH TECHNIQUES FOR LASTO-PLASTIC FINITE ELEMENT OMPUTATIONS IN GEOMECHANICS
Y
. JEREMIC
EPARTMENT OF CIVIL & ENVIRONMENTAL ENGINEERING OLLEGE OF ENGINEERING NIVERSITY OF CALIFORNIA AT DAVIS
EPTEMBER 2001
UCD-CGM report
University
ofCalifo
rinaatDavis,
Center
forGeotech
nica
lModelin
gReport
UCD-CGM-00
Publish
edin
theCommunica
tionsin
Numerica
lMeth
odsin
Engineerin
g,Vol.
17,issu
e2,
pages
115-125,2001.
LineSearchTechniques
forElasto
{Plastic
Finite
ElementComputatio
ns
inGeomechanics
BorisJeremi�c
Departm
entof
CivilandEnviron
mental
Engin
eering,
University
ofCaliforn
ia,Davis,
CA95616,
phone530.754.9248,
fax530.752.7872,
Email:
Abstract
Inthispaper
wepresen
taglobally
converg
entmodi�catio
nofNew
ton'smeth
od
forinteg
ratin
gconstitu
tiveequatio
nsin
elasto
{plasticity
ofgeomateria
ls.New
ton's
meth
odis
knownto
beq-quadratica
llyconverg
entwhen
thecurren
tsolutio
nap-
proxim
atio
nis
adequate.
Unfortu
nately,
itis
notunusualto
expendsig
ni�cant
computatio
naltim
ein
order
toachiev
esatisfa
ctory
results.
Wewill
presen
ta
techniquewhich
canbeused
when
theNew
tonstep
isunsatisfa
ctory.
Thissch
eme
canbeconsid
eredasamodi�ed
versio
nofthetra
ditio
nalconcep
tofbacktra
cking
alongtheNew
tondirectio
nifafullNew
tonstep
provides
unsatisfa
ctory
results.
Themeth
odisalso
knownaslin
esea
rchtech
nique.
Thetech
niqueis
applied
tothefully
implicit
New
tonalgorith
mforaharden-
ingorsoften
inggenera
liso
tropic
geomateria
lsattheconstitu
tivelev
el.Vario
us
solutio
ndeta
ilsandvisu
aliza
tionsare
presen
ted,which
emerg
efro
mtherea
lis-
ticmodelin
gofhighly
nonlin
earconstitu
tivebehaviorobserv
edin
theanalysis
of
cohesio
nless
granularmateria
ls.
KeyWords:
Elasto
{plastic
ity,Geomechanics,Constitu
tiveintegratio
ns,
Newtoniterativ
emethod,Linesearchtechnique
1
UCD-CGM report
1Introductio
n
Prob
lemof
accurately
followingtheequilib
rium
path
innumerical
modelin
gof
geoma-
terialselasto{p
lasticityhas
been
researched
forsom
etim
enow
.It
has
been
show
n
(Runesson
andSam
uelson
[12],Sim
oandTaylor
[14])that
consisten
tuse
ofNew
toniter-
ativealgorith
mon
theglob
al,�nite
elementandlocal
constitu
tivelevels
prov
ides
forfast
convergen
ce.How
ever,com
plex
itiesof
elasto{plastic
constitu
tivemodels
forgeom
ateri-
alsputhigh
dem
andon
thecon
stitutive
driver.
Inparticu
lar,low
con�nem
entregion
,
usually
foundnear
thesoil
surface
ordurin
gliquefaction
behavior
ofsan
ds,with
high
ly
nonlin
earhard
eningandsoften
ingbehavior,
high
dilatan
cyangles
(non{asso
ciativebe-
havior)
andhigh
lycurved
yield
surface,
canlead
tothenumerical
failure
(divergen
ce)
ofthecon
stitutive
driver.
Thislead
sto
theinterru
ption
of�nite
elementcom
putation
s
andcon
sequently
requires
rerunof
theprob
lemwith
smaller
loadingstep
s.
While
theuse
offullNew
tonsch
emeim
proves
rateof
convergen
ce,in
somecases
theiterative
algorithm
does
not
converge
atall.
Thenecessary
condition
forgoodcon
-
vergence
behavior
ofNew
tonmeth
odsisthat
curren
tsolu
tionapprox
imation
isgood
enough,i.e.
itiswith
inthecon
vergence
regionof
New
tonmeth
od.Ifthecurren
tso-
lution
approx
imation
isnot
goodenough,New
tonmeth
odwill
not
converge
butrath
er
bounce
throu
ghsolu
tionspace
until
erroneou
siteration
sare
interru
pted
.Oneof
the
strategiesused
topreven
tfailu
reof
New
tonmeth
odsis
thelin
esearch
technique.
It
should
benoted
that
linesearch
will
not
cure
allof
theprob
lemsasso
ciatedwith
lo-
cal,con
stitutive
levelNew
toniteration
s.How
ever,as
longas
thefunction
(andtheir
derivatives)
used
initeration
s(yield
function
,poten
tialfunction
,hard
ening/soften
ing
function
s)are
analy
tic,lin
esearch
will
prov
idefor
global
convergen
cein
thesen
sede-
�ned
byDennisandSchnabel[3].
Equilib
rium
iterationsfor
material
nonlin
ear�nite
elementcom
putation
scan
bein
general
separated
intwolevels.
First
iterationlevel
istied
tothecon
stitutive,
elasto{
plastic
computation
s.Onthislevel,
constitu
tivedriver,
foragiven
strainincrem
ent1
iteratesin
stressandintern
alvariab
lespace
until
convergen
cecriteria
ismet.
Onthe
secondlevel,
nonlin
ear�nite
elementsystem
ofequation
ssolver
isiteratin
guntil
balan
ce
ofintern
alandextern
alforces
isach
ieved.Glob
allevel
iterationto
achieve
balan
ceof
intern
alandextern
alforces
have
seenuse
oflin
esearch
techniques
(eg.Cris�
eld[2],
Larsson
etal.
[10],Sim
oandMesch
ke[13]).
Intheir
recentpaper,
Dutko
etal.
[4],
1Assu
mingdisp
lacem
entbased
implem
entatio
nofFEM.
2
UCD-CGM report
used
avarian
tof
linesearch
algorithm
forcon
stitutive
leveliteration
s.They
applied
thistech
niqueto
thebiax
ialanisotrop
icyield
criterion(Barlat
etal.
[1])andpresen
ted
interestin
gandusefu
lresu
lts.Ofparticu
larinterest
arestatistics
onnumber
oflin
e
searchiteration
sfor
asharp
lycurved
regionof
theyield
line.
Unfortu
nately,
theactu
al
linesearch
techniqueused
was
just
brie
ydescrib
ed.
Thepaper
isorgan
izedas
follows.
Section
2brie
ydescrib
eselastic{p
lasticalgo-
rithmicform
ulation
based
onthefully
implicit,
New
tonproced
ure
andtheBMaterial
Model
used
incom
putation
s.Section
3describ
estheory
behindthelin
esearch
tech-
niqueandsection
4describ
esapplication
ofthelin
esearch
techniques
tothecon
stitutive
integration
prob
lems.
Section
5presen
tsnumerical
exam
ples
that
illustrate
describ
ed
develop
ments.
Itisworth
while
notin
gthat
while
thepresen
tedtech
niques
isapplied
tosm
alldefor-
mation
elasto{plastic
prob
lemsithas
been
used
inthelarge
deform
ationelasto{p
lastic
algorithmsas
well.
How
ever,inheren
tcom
plex
itiesof
LDEPalgorith
msdevelop
ments
migh
thidethebasic
ideas
ofusin
glin
esearch
techniques
sowerestrict
ourpresen
ta-
tionto
small
deform
ationelasto{p
lasticity.Alth
ough
develop
ments
presen
tedhere
are
simpli�
edto
small
deform
ationform
at,thegen
eralityof
theapproach
isnot
lost.
2Elastic
{Plastic
Geomechanics
Inthissection
webrie
ypresen
tBack
ward
Euler
algorithmsfor
thesolu
tionsof
elastic{
plastic
prob
lemsin
geomech
anics.
Tothisend,weuse
theadditive
decom
position
of
thestrain
increm
entinto
elasticandplastic
parts
together
with
ow
theory
ofplasticity
andKaru
sh{K
uhn{T
ucker
condition
sto
formulate
theelastic{p
lasticprob
lem.Detailed
descrip
tionof
theform
ulation
scan
befou
ndelsew
here
(eg.Jerem
i�candSture
[8]).The
fully
implicit
Back
ward
Euler
algorithm
isdevelop
edin
general
stressten
sor{intern
al
variableten
sorsettin
gandisbased
onthefollow
ingequation
s
n+1�
ij=E
ijkl �
n+1�kl �
n+1�pkl �
;n+1�pij=
n�pij+�n+1m
ij
n+1q�=
nq�+�n+1h�
;Fn+1=0
(1)
where,
�ij ;
�eijand�pijare
thetotal,
elasticandplastic
strainten
sorsresp
ectively,�ij
istheCauchystress
tensor,
q�rep
resents
suitab
leset
ofintern
alvariab
les,h�is
the
plastic
moduliandFn+1istheyield
surface
function
atthe�nal
position
.Theasterisk
3
UCD-CGM report
intheplace
ofindices
inq�rep
lacesnindices,
sothat
forexam
plein
thecase
ofisotrop
ic
hard
eningq�isascalar
while
forkinem
atichard
eningq�issecon
dord
erten
sor.Equation
s
(1),are
thenonlin
earalgeb
raicequation
sto
besolved
fortheunknow
ns
n+1�
ij ,n+1�pij ,
n+1q�and�.New
toniterative
schem
eisused
tosolve
forthesin
glevector
return
instress
tensor
{intern
alvariab
lespace.
Wecan
circumven
ttheprob
lemof
�ndingthesolu
tion
inthesoften
ingregion
,which
,in
general
stressten
sor{intern
alvariab
lespace
belon
gs
tonon{con
vexspace,
byde�ningtheten
sorof
stressresid
uals.
Then,bywork
ingon
theprob
lemof
minim
izingthestress
residualwecon
vertthenon{con
vexprob
lemto
the
convex
one.
Ofcou
rse,as
usual,
byusin
gNew
toniterative
meth
od,wehave
toprov
ide
forcon
tinuation
ofiterative
proced
ure
eveniftheNew
tonstep
isnot
satisfactory.That
isprecisely
thepoin
twhere
thelin
esearch
techniqueshow
situsefu
lness.
TheBack
ward
Euler
algorithm
isbased
ontheelastic
pred
ictor{plastic
corrector
strategy:
n+1�
ij=
pred�
ij��E
ijkln+1m
kl
(2)
where
pred�
ij=E
ijkl�klistheelastic
trialstress
stateand
n+1m
kl=
(@Q=@�kl )jn
+1isthe
gradien
tto
theplastic
poten
tialfunction
instress
space
atthe�nal
position
.Wede�ne
aten
sorof
residuals
rij=�ij� �
pred�
ij��E
ijkln+1m
kl �
(3)
that
represen
tsthedi�eren
cebetw
eenthecurren
tstress
state�ijandtheBack
ward
Euler
stressstate
pred�
ij��E
ijkln+1m
kl .Byusin
g�rst
order
Taylor
seriesexpansion
of
theten
sorof
residuals
rijandthe�rst
order
Taylor
expansion
oftheyield
function
F,
andafter
somealgeb
raicmanipulation
sweobtain
thechange
incon
sistency
param
eter
�
d�=
n+1F
old�
n+1n
mn
oldr
ijn+1T�1
ijmn
n+1n
mnE
ijkln+1H
kl n+1T�1
ijmn�
n+1��h�
(4)
Wehave
alsointro
duced
thefou
rthord
erten
sorsTijmnandH
ijmn :
n+1T
ijmn=ÆimÆnj+�E
ijkl
@m
kl
@�mn �����n
+1
;n+1H
kl=
n+1m
kl +
�@m
kl
@q� �����n
+1 h�
(5)
where
nmn=
@F=@�mn ,
��=
@F=@q�anddq�=
d�h� (�
ij ;q� ).
With
thesolu
tionsfor
d�wecan
write
theiterative
solution
ford�mnanddq� ,in
theexten
ded
stress{intern
al
variablespace
as:
d�mn=�
oldr
ij+
n+1F
old�
n+1n
mn
oldr
ijn+1T�1
ijmn
n+1n
mnE
ijkln+1H
kl n+1T�1
ijmn�
n+1��h�
Eijkln+1H
kl !
n+1T�1
ijmn
(6)
4
UCD-CGM report
dq�=
n+1F
old�
n+1n
mn
oldr
ijn+1T�1
ijmn
n+1n
mnE
ijkln+1H
kl n+1T�1
ijmn�
n+1��h� !
h�
(7)
Iterativeproced
ure
iscon
tinued
until
theob
jectivefunction
krij k
=0issatis�
edgiven
acertain
tolerance.
Inord
erto
starttheNew
toniterative
proced
ure,
weuse
forward
Euler
solution
asa
startingpoin
t
start�mn=E
mnpqd�pq�E
mnpq
crossn
rsE
rstud�tu
crossn
abE
abcdcrossm
cd��� h�
crossm
pq
(8)
startq�=
previousq�+
crossn
mnE
mnpqd�pq
crosn
mnE
mnpqcrosm
pq��� h� !
h�
(9)
where
cross()
denotes
thepoin
twhere
trialstate
crossesyield
surface.
Consisten
t,New
toniteration
son
thecon
stitutive
levelare
re ected
ontheglob
al,
�nite
elementlevel
throu
ghuse
ofalgorith
mictan
gentsti�
ness
(ATS)ten
sorATSE
ep
pqmn
de�ned
as
d�pq=
ATSE
ep
pqmnd�mn
with
ATSE
ep
pqmn=R
pqmn�
Rpqkl n+1H
kl n+1n
ij Rijmn
n+1n
ot R
otpqn+1H
pq+
n+1��h�
(10)
where
wehave
used
reduced
sti�ness
tensor
Rmnkl=(n+1T
ijmn )�1E
ijkl .Itisim
portan
t
tonote
that
thelin
esearch
techniquedescrib
edlater
doesnotalter
theATSten
sor
ATSE
ep
pqmn .
Adetailed
derivation
isgiven
inJerem
i�candSture
[6].
Weuse
small
deform
ationversion
oftheB
material
model
(Jeremi�c
etal.
[5])for
ourcom
putation
s.Themodelrelies
onthedevelop
mentbehindtheso
calledMRS{L
ade
model(Sture
etal.
[16]).TheB{M
odelisasin
glesurface
model,
with
uncou
pled
cone
portion
andcap
portion
hard
ening.
Very
lowcon
�nem
entregion
was
carefully
modeled
andtheyield
surface
was
shaped
insuch
away
tomim
icrecen
t�ndings
obtain
eddurin
g
Micro
Grav
ityMech
anics
testsaboard
Space
Shuttle
(Sture
etal.
[15]).Figu
re(1)
depicts
merid
iananddeviatoric
tracesandafullyield
surface.
Italso
depicts
coneand
caphard
ening/soften
ingfunction
s.Detailed
descrip
tionofthemodelisgiven
byJerem
i�c
etal.
[5].
3LineSearchTechnique
Inthissection
wedevelop
theory
behindthelin
esearch
techniques.
Webase
ourde-
velopments
oncon
ceptof
minim
izationtheory
forascalar
function
f(x� ).
Weshow
5
UC
D-C
GM
rep
ort
a)
qη
pcp
0.71(pc- pt)ηinit
pt
b1
b)
σ3σ2
σ1
low confinment trace
high confinment trace c)
-0.5
0
0.5
0 0.25 0.5 0.75 1
-0.5
0
0.5-0.5
0
0.5
d)1 2 3 4
0.5
1
1.5
2
2.5
3
cone
peak
start
residual
cone
peak
η
η
ηη
κ
κ
e)
1040
1020
1060
0.2 0.4 0.6 0.8 1.0
’
k
pc
cap
Figure 1: (a) Meridian trace of yield, ultimate or potential surface. (b) Change of the
deviatoric trace of yield surface changes along the mean stress axes. (c) Yield and/or po-
tential surface in principal stress space. (d) Cone hardening function. (e) Cap hardening
function.
later that in our case, function to be minimized will be the energy norm of the vector of
residuals rij (Eq. 3).
The minimization problem is de�ned as (eg. Dennis and Schnabel [3].):
minx�2Rn
f(x�) : Rn �! R (11)
The basic idea of a global method for minimization is to take the step that lead downhill
for the function f(x�). One chooses a direction p� from the current point xc� in which
f(x�) decreases initially and a new point x+� in this direction from xc� is such that f(x+
� ) <
f(xc�). Such a direction p� is called a descent direction. From the mathematical point of
view, p� is a descent direction from xc� if the directional derivative of f(x�) at xc� in the
direction p� is negative:@f (xc�)
@x�p� < 0 (12)
6
UC
D-C
GM
rep
ort
If (12) holds, then it is guaranteed that for a small positive �, f(xc� + �p�) < f(xc�). The
idea of line search algorithm can be described as follows:
� At iteration k do:
{ calculate a descent direction pk�,
{ set xk+1� � (xk� + �kpk�) for some �k that makes xk+1� an acceptable next
iterate.
Figure (2) shows the basic concept: select xk+1� by considering the half of a one{
dimensional cross section of f(x�) in which f(x�) decreases initially from xk� .
Quadraticapproximation
slope
=0
p* < 0
xf(*k )
x*dd
xf(*k+ p
*k )ζ
ζζ
Figure 2: A cross section of f(x�) from xk� in the direction pk�.
The term \line search" refers to the procedure of choosing the acceptable �k. The so
called \exact line search" accounts for �nding the exact solution of the one{dimensional
minimization problem, i.e. �nding the exact �k so that f(xk� + �kpk�) attains minimum.
This was the preferred approach to the problem until mid 1960s. More careful compu-
tational testing has led to the use of \slack line search" which has a weak acceptance
criteria for �k as a more computationally eÆcient procedure. The common procedure
now is to try the full Newton step �rst (with �k = 1) and then, if �k = 1 fails to satisfy
criterion, to reduce �k in a systematic way, along the direction de�ned by that step.
Systematic reduction of �k along the descent direction can be achieved by applying
the line search techniques through backtracking algorithm:
Given � 2 (0; 12):
�k = 1;
7
UCD-CGM report
while
f(x
k�+�kp
k� )>f(x
k� )+��k@f(
xk� )
@x�
pk�do:
�k ���kwhere,
usually
�=
12 ;
xk+1
� �xk�+�kp
k� ;
Dennis
andSchnabel
[3]prov
iderath
erpow
erfulcon
vergence
results
forprop
erly
chosen
steps.
4Applicatio
nto
Elasto
{Plastic
Computatio
nsinGe-
omechanics
Inthissection
wedescrib
etheapplication
oflin
esearch
techniqueto
theelasto{p
lastic
constitu
tivelevel
equilib
rium
iterations.
Theob
jectivefunction
that
isfollow
edisthe
Euclid
eannorm
often
sorof
residuals
rij .
Tothisendwerew
ritetheelastic
pred
ictor
plastic
correctorEq.(2)
as:n+1�
ij=
pred�
ij���E
ijkln+1m
kl
(13)
Itisim
portan
tto
note
that
theaddition
ofscalar
linesearch
param
eter�does
not
change
theinitial
equation
s.Moreover,
as�is
only
used
inlin
esearch
improvem
ent
andisnot
afunction
ofanystate
variable(stress,
intern
alvariab
lesor
disp
lacements)
�rst
order
Taylor
expansion
sused
tocom
eupwith
theiterative
steps(Eq.(4)
{(7)
arenot
changed
.Theonly
di�eren
ceisthat
now
solution
forthechange
incon
sistency
param
eterd�read
s:�d�=
n+1F
old�
n+1n
mn
oldr
ijn+1T�1
ijmn
n+1n
mnE
ijkln+1H
kl n+1T�1
ijmn�
n+1��h�
(14)
with
changed
tensors
TijmnandH
kl
n+1T
ijmn=ÆimÆnj+��E
ijkl
@m
kl
@�mn �����n
+1
;n+1H
kl=
n+1m
kl +
��@m
kl
@q� �����n
+1 h�
(15)
With
thischanges
thelin
esearch
algorithm
cannow
bespecialized
toelasto{p
lastic
implicit
computation
s.In
eachiterative
stepkwedo:
�Given
�2(0;12 ):
��k=1;
(FullNew
tonstep
)
8
UCD-CGM report
�while
krij (
k�ij+�kd
k�ij ;
kq�+�kd
kq� )k
>krij (
k�ij ;
kq� )k
+��k
@krij k
@�mn
k�mn+@krij k
@q�
kq� !
do:{
�k ���kwhere,
usually
�=
12 ;
{k+1�
ij �
k�ij+�kd
k�ij
and
k+1q� �
kq�+�kd
kq�
Wealw
aysstart
with
�k=
1(th
atis,
afullNew
tonstep
)andonly
ifthat
iteration
fail,weapply
theback
trackalgorith
m.Asweare
followingvalu
eof
theEuclid
eannorm
often
sorof
residuals
rij ,
failure
ofNew
tonstep
isde�ned
asdivergen
ceof
that
scalar
variable.
Inoth
erword
s,increase
thevalu
eof
Euclid
eannorm
often
sorof
residuals
rij
mean
sthat
ourNew
tonstep
intheexten
ded
stress{intern
alvariab
lespace
isnot
valid.
Italso
mean
sthat
convergen
ceof
theNew
toniterative
proced
ure
isquestion
able
and
that
itwill
most
prob
ably
failin
thisiterative
step.
5NumericalExample
Prev
ioustheoretical
develop
ments
has
been
implem
ented
inFEI
(Finite
Elem
entInter-
preter)
�nite
elementprogram
.FEI
isourexperim
ental
software
platform
,written
in
C++
with
someFortran
modules.
Itisbuild
ontop
ofnDarrayandFEMtools
class
libraries
(Jeremi�c
andSture
[9]).Thelin
esearch
algorithm
isim
plem
ented
onboth
global,
�nite
elementlevel
aswell
ason
thelocal,
constitu
tivelevel,
describ
edin
this
paper.
Wefollow
nonlin
ear�nite
elementiteration
sin
analy
zingdirection
alshear
cell
(DSC)experim
entsdoneat
theUniversity
ofColorad
oat
Boulder.
For
exam
pleMcFad-
den
([11])used
DSCapparatu
sin
order
toinvestigate
shear
behavior
ofvariou
ssan
ds.
Here
wepresen
tmodelin
gof
particu
larplan
estrain
experim
entK
0 14:5�30
in�gure
(3).Load
ingisdivided
intwostages.
The�rst
stagecom
prises
isotropiccom
pression
to
targetcon
�nem
entstate
(inthiscase
p=180
:0kPa).
Thesecon
dstage
isashear
loading
until
instab
ilityoccu
rs(in
stability
intheexperim
ental
test,numerically
wecan
follow
thespecim
enbeyon
dlim
itpoin
t).Itisnot
cleariftheinstab
ilitiesin
DSCexperim
ents
(lossof
control
overload
ingprocess)
were
dueto
thebifu
rcationphenom
enainsid
ethe
specim
enor
totheglob
alrotationofthespecim
en.Since
DSCisaload
controlled
device,
only
shear
deform
ationof
xy'3:5
%was
reached
inthelab
oratoryexperim
ent.
9
UCD-CGM report
a)S
hear deformation
γ [%]
Shear stress τ [kN/m^2]
0.01.0
2.03.0
4.05.0
0.0
50.0
100.0
150.0
200.0
b)
Shear deform
ation γ [%]
0.01.0
2.03.0
4.05.0
-2.0
-1.5
-1.0
-0.5 0.0
0.5
1.0
volumetric strain [%]
Figu
re3:
Numerical
modelin
gof
ashear
testK
0 14:5�30
a)Shear
stress{axial
strain
curves.
b)Volu
metric
strain{axial
straincurves.
Asthenumerical
experim
entisproceed
ing,
thelen
gthof
theload
ingstep
siscon
-
trolledbyusin
gthevariab
lehypersp
herical
arc{length
constrain
t([7]).
Inthisparticu
lar
case,theNew
toniterative
algorithmfailed
atcon
stitutive
levelafterstress
stateadvan
ced
into
elastic{plastic
regime.
Thispart
oftheelastic{p
lasticcom
putation
sissom
etimes
prob
lematic,
since
there
isasudden
activationof
thehard
eningmech
anism
.In
this
particu
larcase,
wehave
stopped
thecom
putation
sat
thebegin
ningof
theprob
lematic
increm
ental
step.Then,weresu
med
iterationsbymanually
decreasin
gthestep
sizefrom
0.5%to
0.1%.Figu
re4(a)
show
svalu
esof
thenorm
oftheten
sorof
residualkrij k
in
such
asuccessfu
literation
.Itcan
beseen
fromFig.
4(a)that
convergen
ceisquite
fast,
leadingto
thecon
clusion
that
theinitialestim
ateofthestress
andintern
alvariab
lestate
was
with
intheNew
toncon
vergence
region.
Sim
ilarto
theprev
iousnumerical
experim
ent,weresu
med
computation
sat
thebe-
ginningof
theprob
lematic
step,butthistim
ewith
largerincrem
ental
deform
ationof
0.5%andwith
thelin
esearch
algorithm
turned
o�.Figu
re4(b
)show
svalu
esofthenorm
oftheten
sorof
residualkrij k
forthe�rst
increm
ental
step.Wecan
observe
that
the
initial,
uncorrected
iterationproduces
non{con
vergingset
ofvalu
esfor
thenorm
ofthe
tensor
ofresid
ualkrij k.
Itisinterestin
gto
note
that
thevery
�rst
iterativestep
takesthe
valueofkrij k
tohigh
erthan
initial
values.
Moreover,
Figu
re5(a)
show
sthat
although
valueof
evaluated
yield
surface
Finitially
poin
tdow
nward
fromtheinitial
pred
ictor,it
iseven
tually
divergin
gtoo.
Finally,
once
againweresu
mecom
putation
sat
thebegin
ningof
theprob
lematic
increm
ental
step,now
with
thelin
esearch
algorithm
turned
on.It
canbeseen
from
10
UCD-CGM report
a)1
23
45
1e-06
1e-05
1e-04
1e-03
norm of residual r
nu
mb
er o
f itera
tion
sb)
05
1015
201e-06
1e-05
1e-04
1e-03
1e-02
1e-01
norm of residual r
nu
mb
er o
f itera
tion
s
erra
tic itera
tion
s(lin
e se
arch
off)
(line
sea
rch o
n)
corre
cted
itera
tion
s
first itera
tive ste
p
Figu
re4:
(a)Resid
ualnormkrij k
values
forasuccessfu
literation
with
small
increm
ental
steps(on
estep
of0.1%
shear
deform
ation).
(b)Resid
ual
norm
valueskrij k
values
for
erraticandcorrected
iterations(on
estep
of0.5%
shear
deform
ation).
Fig.
4(b)that
not
only
thealgorith
mhelp
sadvan
cetheiteration
swith
outanysign
sof
divergen
ce,italso
produces
avery
goodcon
vergence
rate.This,
ofcou
rse,istheresu
lt
ofouralgorith
malw
ays�rst
tryingthefullNew
tonstep
(forwhich
weset
�k=1).
Figu
re5(b
)depicts
theprob
lematic,
�rst
iterativestep
fromFigu
re4(b
)in
some
more
details.
Wefollow
thevalu
eof
theob
jectivefunction
krij k
throu
gh10
subincre-
ments
with
intheprob
lematic
iterativestep
.Atthispoin
titisim
portan
tto
remem
ber
that
New
toniterative
algorithm
makes
aquadratic
approx
imation
(inmultid
imension
al
space)
oftheob
jectivefunction
,here
theEuclid
eannorm
krij k.
Inthisparticu
larstep
,
theapprox
imation
forkrij k
does
apoor
job,andalth
ough
theob
jectivefunction
ini-
tiallydecreases,
eventually
its�nal
valueishigh
erthan
that
ithad
attheinitial
state.
Itisinterestin
gto
note
that
at5th
subincrem
ent,ob
jectivefunction
krij k
actually
does
attainaminimum
value,butfullNew
tonstep
(at10th
subincrem
ent)
leadsto
ahigh
er
than
initial
values
forkrij k.
Figu
re(6)
depicts
failedandcorrected
iterationsin
thespace
ofstress
invarian
tsp,
qand�
p=�13�kk
q= s
312sij s
ijcos
3�=
3 p3
2
13 sij s
jk s
ki
q(12 s
ij sij )
3
sij=�ij�
13�kk Æ
ij(16)
Inthisparticu
larview
,weare
lookingat
theBmodelyield
surface
fromtheten
sile
region(negative
p).Divergen
titeration
isoscillatin
gthrou
ghthestress
space
andthere
isnosign
ofrecovery.
Ontheoth
erhand,corrected
oscillation,after
modify
ingone
11
UC
D-C
GM
rep
ort
a)0 5 10 15 20
1e-06
1e-04
1e-02
1e+00
1e+02
1e+04
yiel
d fu
nctio
nF
number of iterations
erratic iterations
correct iterations
b)0.00 0.25 0.50 0.75 1.00
1e-04
1e-03
1e-02
1e-01
approximationquadratic
subincremented iterative step
norm
of r
esid
ual r
function
Figure 5: (a) Yield surface values F for erratic and corrected iterations (one step of 0.5%
shear deformation). (b) Dissection of failed iteration.
iterative step by the line search algorithm, converges successfully.
6 Summary
In this paper we have presented a globally convergent modi�cation of Newton's method
for integrating constitutive equations in elasto{plasticity of geomaterials. The method
has been developed in rigorous mathematical framework and implemented in experi-
mental �nite element program FEI. The practical use of method was illustrated in details.
The application of line search techniques in numerical modeling of elasto{plasticity of
geomaterials should provide for robust following of equilibrium path on both, global,
�nite element and local, constitutive levels.
Acknowledgment
The author gratefully acknowledge partial support by National Science Foundation's
Paci�c Earthquake Engineering Research Center Grant NSF-PEER 2132000.3.
12
UC
D-C
GM
rep
ort
q
p
φ
erratic iterations
yield surface
correct iterations
predictor point
Figure 6: Illustration of erratic and correct iterations in stress space.
13
UCD-CGM report
References
[1]Barlat,F.,Lege,D.J.,andBrem,J.C.Asix
componentyield
function
for
anisotrop
icmaterials.
Internatio
nalJournalforPlastic
ity7(1991),
693{712.
[2]Crisfield,M.A.Non-LinearFinite
ElementAnalysis
ofSolid
sandStru
ctures
Volume1:Esse
ntia
ls.Joh
nWiley
andSons,Inc.New
York
,1991.
[3]Dennis,Jr.,J.E.,andSchnabel,R.B.Numerica
lMethodsforUnconstra
ined
Optim
izatio
nandNonlin
earEquatio
ns.Pren
ticeHall,
Engelw
oodCli�
s,New
Jersey
07632.,1983.
[4]Dutko,M.,Peri �c,D.,andOwen,D.R.J.Universalan
isotropicyield
criterion
based
onsuperq
uadratic
function
alrep
resentation
:Part1.
algorithmic
issues
and
accuray
analy
sis.Computermethodsin
applied
mechanicsandengineerin
g109
(1993),73{93.
[5]Jeremi �c,B.,Runesson,K.,andSture,S.Amodelfor
elastic{plastic
pressu
re
sensitive
materials
subjected
tolarge
deform
ations.
Internatio
nalJournalofSolid
s
andStru
ctures36,
31/32(1999),
4901{4918.
[6]Jeremi �c,B.,andSture,S.
Implicit
integration
rules
inplasticity
:Theory
andim
plem
entation
.Report
toNASA
Marsh
allSpace
Fligh
tCenter,
Contract:
NAS8-38779,
University
ofColorad
oat
Boulder,
June1994.
[7]Jeremi �c,B.,andSture,S.Re�ned
�nite
elementanaly
sisof
geomaterials.
In
Proceed
ingsof11th
Engineerin
gMechanicsConference
(Fort
Lauderd
ale,Florid
a,
May
1996),Y.K.Lin
andT.C.Su,Eds.,
Engin
eeringMech
anics
Division
ofthe
American
Society
ofCivilEngin
eers,pp.555{558.
[8]Jeremi �c,B.,andSture,S.Im
plicit
integration
sin
elasto{plastic
geotechnics.
Internatio
nalJournalofMechanicsofCohesiv
e{Fric
tionalMateria
ls2(1997),
165{
183.
[9]Jeremi �c,B.,andSture,S.Tensor
data
objects
in�nite
elementprogram
ming.
Internatio
nalJournalforNumerica
lMethodsin
Engineerin
g41(1998),
113{126.
[10]Larsson,R.,Runesson,K.,andSture,S.Embedded
localization
bandin
undrain
edsoil
based
onregu
larizedstron
gdiscon
tinuity
{theory
andFE{an
alysis.
Internatio
nalJournalofSolid
sandStru
ctures33,
20-22(1996),
3081{3101.
[11]McFadden,J.J.Experim
ental
response
ofsan
ddurin
gpricip
alstress
rotations.
Master
ofScien
cethesis,
University
odColorad
oat
Boulder,
Decem
ber
61988.
14
UCD-CGM report
[12]Runesson,K.,andSamuelsson,A.Aspects
onnumerical
techniques
insm
all
deform
ationplasticity.
InNUMETA
85Numerica
lMethodsin
Engineerin
g,Theory
andApplica
tions(1985),
G.N.P.J.
Middleton
,Ed.,AA.Balkem
a.,pp.337{347.
[13]Simo,J.C.,andMeschke,G.Anew
classof
algorithmsfor
classicalplasticity
exten
ded
to�nite
strain.application
togeom
aterials.Computatio
nalMechanics11
(1993),253{278.
[14]Simo,J.C.,
andTaylor,R.L.
Consisten
ttan
gentoperators
forrate-
independentelastop
lasticity.ComputerMethodsin
Applied
MechanicsandEngi-
neerin
g48(1985),
101{118.
[15]Sture,S.,Costes,N.,Batiste,S.,Lankton,M.,AL-Shibli,K.,Jeremi �c,
B.,Swanson,R.,andFrank,M.Mech
anics
ofgran
ular
materials
atlow
e�ec-
tivestresses.
ASCEJournalofAerospace
Engineerin
g11,
3(Ju
ly1998),
67{72.
[16]Sture,S.,Runesson,K.,andMacari-P
asqualino,E.J.
Analy
sisand
calibration
ofathree
invarian
tplasticity
model
forgran
ular
materials.
Ingenieur
Archiv
59(1989),
253{266.
15