REPORT NO.
UCB/SEMM-2000/02
STRUCTURAL ENGINEERING,
MECHANICS AND MATERIALS
BY
DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING
UNIVERSITY OF CALIFORNIA
BERKELEY, CALIFORNIA
JANUARY 2000
AND
ON THE FORMULATION OF CLOSEST-POINT
PROJECTION ALGORITHMS IN
ELASTOPLASTICITY.
PART II: GLOBALLY CONVERGENT SCHEMES.
Wi h Appplica ions o De ia o ic and P essu e-Dependen
Plas ic Models.
F. ARMERO
A. PEREZ
-
FOGUET
'
On he Fo mula ion o Closes -Poin P ojec ion
Algo i hms in Elas oplas ici y.
Pa II: Globally Con e gen Schemes.
Wi h Applica ions o De ia o ic and P essu e-Dependen Plas ic Models
by
A. P´
e ez-Fogue †& F. A me o∗
S uc u al Enginee ing, Mechanics and Ma e ials
Depa men o Ci il and En i onmen al Enginee ing
Uni e si y o Cali o nia, Be keley CA 94720, USA
Abs ac
This pape p esen s he o mula ion o nume ical algo i hms o he solu ion o he closes -poin
p ojec ion equa ions ha appea in ypical implemen a ions o e u n mapping algo i hms in
elas oplas ici y. The main mo i a ion behind his wo k is o a oid he poo global con e gence
p ope ies o a s aigh applica ion o a New on scheme in he solu ion o hese equa ions, he so-
called New on–CPPM. The ma hema ical s uc u e behind he closes -poin p ojec ion equa ions
iden ified in Pa I o his wo k delinea es clea ly diffe en s a egies o he success ul solu ion o
hese equa ions. In pa icula , p imal and dual closes -poin p ojec ion algo i hms a e p oposed,
in non-augmen ed and augmen ed Lag angian e sions o he imposi ion o he consis ency con-
di ion. The p imal algo i hms in ol e a di ec solu ion o he o iginal closes -poin p ojec ion
equa ions, whe eas he dual schemes in ol e a wo le el s uc u e by which he o iginal sys em o
equa ions is s agge ed, wi h he imposi ion o he consis ency condi ion d i ing alone he i e a i e
p ocess. New on schemes in combina ion wi h app op ia e line sea ch s a egies a e conside ed,
esul ing in he desi ed asymp o ically quad a ic local a e o con e gence and he sough global
con e gence cha ac e o he i e a i e schemes. These p ope ies, oge he wi h he compu a ional
pe o mance o he diffe en schemes, a e e alua ed h ough ep esen a i e nume ical examples
in ol ing diffe en models o fini e s ain plas ici y. In pa icula , he a oidance o he la ge
egions o no con e gence in he ial s a e obse ed in he s anda d New on–CPPM is clea ly
illus a ed.
KEY WORDS: elas oplas ici y; e u n mapping algo i hms; closes -
poin p ojec ion; globally con e gen schemes; aug-
men ed Lag angian.
Submi ed o In e na ional Jou nal o Nume ical Me hods in Enginee ing.
†On lea e om he Depa men o Ma em`a ica Aplicada III, ETSECCPB, UPC,
Ba celona, Spain, du ing he Fall semes e 1999.
∗Co esponding au ho ([email p o ec ed]).
A. P´e ez-Fogue & F. A me o 2
1. In oduc ion
The de elopmen s p esen ed in A me o & P´
e ez–Fogue [2000] ( e e ed simply
as Pa I he ea e ) ha e iden ified a ich ma hema ical s uc u e behind he closes -poin
p ojec ion equa ions cha ac e is ic o he local in eg a ion o elas oplas ic models, including
hei iscoplas ic ex ensions, in bo h he infini esimal and fini e de o ma ion anges. Box
1.1 summa izes hese equa ions ollowing he same no a ion employed in his p e ious
pa o his wo k. Mo e specifically, he upda ed alues (·)n+1 o he s esses and in e nal
a iables a he end o a ypical ime inc emen [ n,
n+1] in he solu ion o he global
mechanical bounda y- alue p oblem a e ob ained om hei known coun e pa s (·)n o
a gi en s ain inc emen ∆ε=εn+1 −εn. Associa ed o his inc emen , we ha e he
ial alues (·) ial
n+1 o he diffe en a iables ob ained unde he assump ion o inc emen al
elas ic esponse. The case o a e-independen elas oplas ici y in a ypical plas ic co ec o
s ep, cha ac e ized by he en o cemen o he consis ency condi ion n+1 = 0 om an
ini ial ial solu ion wi h ial
n+1 >0, has been conside ed in Box 1.1. The disc e e o ms
o he flow ule and ha dening laws ha e been w i en wi hou conside ing explici ly he
usual assump ions o con exi y and associa i i y equi ed o a igo ous cha ac e iza ion o
he a o emen ioned a ia ional s uc u e. I is in his gene al con ex whe e we p esen he
nume ical algo i hms p oposed in his wo k, e en hough he s a emen s on he con e gence
p ope ies o he diffe en schemes apply igo ously o he associa ed con ex case only.
Box 1.1 summa izes also he applica ion o a New on scheme in he solu ion o he
esul ing nonlinea algeb aic equa ions. To his pu pose, he equa ions ha e been w i en
in esidual o m and he Jacobian ma ix u ilized in he i e a i e p ocess iden ified. The
final scheme, e e ed o as he New on–CPPM he ein, can be ound in Simo & Hughes
[1998] and has become a e y popula s a egy in he nume ical solu ion o elas oplas ic
p oblems. As no ed in Sec ion 1 o Pa I o his wo k, he asymp o ic quad a ic a e
o con e gence o his i e a i e scheme makes his app oach e y a ac i e. Howe e ,
i s limi ed global con e gence p ope ies leads o en o a lack o con e gence o la ge
anges o he ini ial ial s a e. This si ua ion is ela ed o he s ong nonlinea i y o he
equa ions which may a ise, o example, om he high cu a u e o he yield unc ion, as
i can be ound in ypical p ac ical applica ions. The nume ical esul s epo ed in de
Souza Ne o e al. [1994], Bi´
cani´
c&Pea ce[1996], P´
e ez–Fogue e al. [2000a]
and he ein illus a e hese difficul ies. These obse a ions iden i y clea ly he goal o
his wo k: he de elopmen o efficien globally con e gen algo i hms o he solu ion o
he closes -poin p ojec ion equa ions in elas oplas ici y. We ocus only on schemes ha
p ese e he local asymp o ic a e o con e gence cha ac e is ic o he o iginal New on–
CPPM. We no e also ha all he algo i hms p oposed in his wo k lead o he exac
solu ion o he o iginal closes -poin p ojec ion equa ions. This implies, in pa icula , ha
he consis en algo i hmic angen employed in a ypical New on-Raphson solu ion o he
global mechanical bounda y- alue p oblem coincides in all cases; see e.g. Simo & Hughes
[1998].
Closes -Poin P ojec ion Algo i hms in Elas oplas ici y 3
BOX 1.1. The New on closes -poin p ojec ion algo i hm o a e-
independen elas oplas ici y.
•Fo he plas ic co ec o s ep (i.e. ial
n+1 >0), define he esiduals
ε:= εe
n+1 −εe, ial
n+1 +∆γmσ(σn+1,qn+1),
α:= −αn+1 +α ial
n+1 +∆γmq(σn+1,qn+1),
:= (σn+1,qn+1),
o he elas ic s ain εe
n+1 and s ain-like ha dening a iables αn+1,
and whe e
σn+1 =∂eψ(εe
n+1,αn+1)andqn+1 =−∂αψ(εe
n+1,αn+1)
he s esses σn+1 and s ess-like ha dening a iables qn+1, o he
s o ed ene gy unc ion ψ(εe,α) and yield unc ion (σ,q).
•Le
(x):=
α
o x:=
εe
n+1
−αn+1
∆γ
.
Conside he New on i e a i e scheme
x(k+1) =x(k)+d(k) o d(k)=−J−1(x(k)) (x(k)),
wi h x(0) =εe, ial
n+1 ,−α ial
n+1 ,0Tand he Jacobian
J=∇ =
I+∆γ∇m˜
Gm
∇ T˜
G0
,
whe e
∇ =∂σ
∂q ,m=mσ
mq,∇m=∂σmσ∂qmσ
∂σmq∂qmq,
and
˜
G=∂2
εeεeψ−∂2
εeαψ
−∂2
αεeψ∂
2
ααψ.
Rema k: The condi ion ∇ ·˜
Gm >0 s ic ly is assumed o he
unde lying con inuum model.
A. P´e ez-Fogue & F. A me o 4
An algo i hm is globally con e gen i all he sequences ha i gene a es con e ge o
a solu ion o he p oblem. We e e o he classical global con e gence heo em p esen ed
in Luenbe ge [1989] (page 187) o de ails. In he case o in e es he ein, his p ope y
co esponds o con e gence o any ini ial ial s a e. As s udied in de ail in his las
e e ence and o he classical e e ences on nonlinea op imiza ion, his desi ed p ope y
can be ob ained h ough he conside a ion o he app op ia e line sea ch scheme unde
ce ain echnical condi ions (usually in ol ing he con exi y o he a ia ional p oblem in
op imiza ion and he smoo hness o he unc ions in ol ed). We explo e his al e na i e
fi s , by combining he o iginal New on scheme wi h a line sea ch designed specifically
o he equa ions o in e es he e. In pa icula , he cons ained cha ac e o he o iginal
closes -poin p ojec ion equa ions implied by he non-nega i e cha ac e o he plas ic
mul iplie (∆γ≥0 in Box 1.1) is aken in o accoun explici ly in he i e a i e p ocess.
Appendix I p esen s he de ails o he line sea ch schemes de eloped in his wo k. We
e e o he esul ing algo i hm as he p imal closes -poin p ojec ion me hod ( he p imal–
CPPM in sho ), gi en i s clea ela ion o he p imal a ia ional o mula ion o he closes -
poin p ojec ion equa ions unde he usual assump ion o con exi y and associa i i y. This
fi s p oposed algo i hm consis s hen simply o he New on scheme wi h line sea ch.
The special conside a ions equi ed by he cons ained cha ac e o he go e ning
equa ions, wi h he consequen need o special a ia ions in he compu e code, mo i a e
he de elopmen o al e na i e echniques. The conside a ion o he new augmen ed p imal
o mula ions iden ified in Pa I appea s hen as a clea op ion, as i is de eloped nex in
his pape . We e e o he final scheme as he augmen ed–p imal–CPPM. In addi ion o
in ol ing a egula ized uncons ained p oblem, he augmen a ion adds also a simple mech-
anism o imp o ing he compu a ional efficiency o he o iginal p imal–CPPM. Nume ical
expe imen s iden i y in his way a ange o he egula izing penal y pa ame e leading o
his imp o emen .
I is in e es ing o obse e ha he o iginal New on–CPPM has been o en in e p e ed
in he li e a u e as imposing he closes -poin p ojec ion o each i e a e, (·)(k)in he
no a ion o Box 1.1; see Simo & Hughes [1998], page 143 (Figu e 3-10 in his e e ence,
in pa icula ). Howe e , a look a he esidual o m o he equa ions clea ly shows ha
his in e p e a ion is no alid since, simply, he esiduals (k)co esponding o he flow
ule and ha dening law do no anish, in gene al, du ing he i e a ion p ocess, excep a
he con e ged alue. This obse a ion iden ifies clea ly he ole played by he consis ency
condi ion and led o he dual a ia ional o mula ions p esen ed in Pa I. In his con ex ,
he solu ion can be ob ained simply as he oo s o he yield unc ion unde s ood as a
scala unc ion o he plas ic mul iplie . The e alua ion o his “dual consis ency unc ion”
equi es he solu ion o he disc e e o ms o he flow ule and ha dening law o a fixed
plas ic mul iplie and, hence, uncons ained in na u e. These conside a ions lead na u ally
o a wo-le el algo i hm ha we e e as he dual–CPPM. The conside a ion o a New on
scheme in combina ion wi h a line sea ch in bo h le els assu es he asymp o ic quad a ic
Closes -Poin P ojec ion Algo i hms in Elas oplas ici y 5
local a e o con e gence o he algo i hm and i s globally con e gen cha ac e , his la e
p ope y holding again unde he usual echnical condi ions (i.e. he associa ed con ex
case wi h smoo h ma e ial ela ions). The augmen ed ex ension o hese ideas, e e ed o
as he augmen ed–dual–CPPM he ein, p o es also o be a use ul ool o add compu a ional
efficiency o he final nume ical scheme.
An ou line o he es o he pape is as ollows. Sec ions 2 and 3 desc ibe he de ails
behind he p imal and dual algo i hms, espec i ely, in hei non-augmen ed and aug-
men ed o ms. Appendix I includes again he de ails o he line sea ch schemes de eloped
in his wo k o cons ained and uncons ained p oblems. We ocus he de elopmen s on
he a e-independen p oblem, wi h ex ensions o he simple iscoplas ic p oblem consid-
e ed b iefly in ema ks in he end o each sec ion. The p ope ies and pe o mance o he
p imal and dual classes o algo i hms a e e alua ed nume ically in Sec ions 4 and 5, e-
spec i ely. Se e al ep esen a i e examples a e conside ed in ol ing pu ely de ia o ic and
p essu e dependen plas ic models (bo h in an associa ed amewo k, wi h he second one
in ol ing non-smoo h flow ec o s), s ain ha dening and so ening, as well as cons an and
non-cons an elas ici ies. Fini e de o ma ion mul iplica i e plas ici y models a e consid-
e ed in all cases. Appendix II includes a summa y o he ma e ial models conside ed. The
solu ion o a fini e elemen p oblem using he p imal–CPPM is shown in Sec ion 6, as an
example o p ac ical applica ion. Finally, Sec ion 7 p esen s se e al concluding ema ks,
compa ing in pa icula he diffe en p oposed algo i hms.
2. P imal Algo i hms
We de elop in his sec ion nume ical algo i hms based on he p imal a ia ional o -
mula ions de eloped in Sec ions 3 and 4 o Pa I o his wo k o he a e-independen
and iscoplas ic p oblems, espec i ely. Mo e p ecisely, we add ess he solu ion o he co -
esponding Eule -Lag ange equa ions o , mo e gene ally, hei coun e pa s in he con ex
o non-associa ed o mula ions wi hou his a ia ional s uc u e. Ex ensions in ol ing he
co esponding augmen ed Lag angian o mula ions, as de eloped in Sec ion 5 o Pa I,
a e p esen ed in Sec ion 2.2. We conside he a e-independen elas oplas ic p oblem, wi h
he iscoplas ic p oblem discussed b iefly in addi ional ema ks in he end o each sec ion.
2.1. The p imal closes -poin p ojec ion me hod
A simple and efficien choice when ying o imp o e he con e gence p ope ies o he
o iginal New on closes -poin p ojec ion me hod o he a e-independen p oblem sum-
ma ized in Box 1.1 is he addi ional conside a ion o a line sea ch echnique, as desc ibed
in Appendix I. We conside again he esidual o m o he equa ions p esen ed in his box,
A. P´e ez-Fogue & F. A me o 6
namely,
(x):=
εe
n+1 −εe, ial
n+1 +∆γmσn+1
−αn+1 +α ial
n+1 +∆γmqn+1
n+1
, o x:=
εe
n+1
−αn+1
∆γ
,(2.1)
he elas ic s ains, s ain-like in e nal ha dening a iables, and he disc e e plas ic mul i-
plie , espec i ely, all o hem a he end n+1 o a ypical ime s ep [ n,
n+1]. The flow
ec o s mσn+1 and mqn+1 , and yield unc ion n+1 in (2.1) a e gi en unc ions o he conju-
ga e s ess a iables σn+1 =∂εeψn+1 and qn+1 =−∂αψn+1 o he s o ed ene gy unc ion
ψn+1 =ψ(εe
n+1,αn+1), as conside ed in Box 1.1. The in e es he ein is he conside a ion
o a plas ic co ec o s ep, de ec ed by he condi ion ial
n+1 := (σ ial
n+1 ,q ial
n+1 )>0 o he
ial elas ic alues {εe, ial
n+1 ,−α ial
n+1 ,0}, and he co esponding conjuga e s ess a iables
σ ial
n+1 and q ial
n+1 . The cons ained cha ac e o he p oblem a ises om he es ic ion on
he las componen (say, componen nx)o x, ha is,
[x]nx:= ∆γ≥0,(2.2)
o he plas ic mul iplie ∆γ. The p esence o his cons ain equi es hen he use o
he line sea ch scheme desc ibed in Sec ion I.2 o Appendix I o unila e ally cons ained
p oblems. We deno e he final algo i hm by he p imal closes -poin p ojec ion me hod
(o , in sho , he p imal–CPPM). Box 2.1 includes a summa y o his scheme o a ypical
plas ic co ec o s ep.
The esul ing algo i hm is hen e y simila o he o iginal New on–CPPM, bu wi h
he added conside a ions associa ed o he line sea ch and cons ain de ec ion. In pa ic-
ula , he condi ion (I.15) in Appendix I o he ac i a ion o he cons ain (2.2) eads
m(k)
σn+1 ·(εe,(k)
n+1 −εe, ial
n+1 )−m(k)
qn+1 ·(α(k)
n+1 −α ial
n+1 )>0,wi h ∆γ(k)=0,(2.3)
in a gene ic i e a ion (k). I he condi ion (2.3) is sa isfied, he modified Jacobian (I.17)
p esen ed in Box 2.1 is o be used. We obse e ha bo h he ial s ep and he final
solu ion o he plas ic co ec o s ep o in e es he ein ( ha is, wi h ∆γ>0 s ic ly) do
no sa is y his condi ion, and hence he ac i a ion o he cons ain by he algo i hm nea
hen. To gain a be e unde s anding o he ela ion (2.3), Figu e 2.1 depic s a g aphical
in e p e a ion o his condi ion o on Mises and T esca yield condi ions wi h associa ed
pe ec plas ic flow and cons an elas ici ies. We no e, as an aside, ha he cons ain
(2.2) has no become ac i a ed in any o he examples p esen ed in Sec ion 4, in ol ing
mo e complex yield condi ions. We conclude ha , e en hough he condi ion (2.3) needs
o be conside ed om a heo e ical poin o iew, i does no lead o added compu a ional
cos in p ac ice. We e e o his sec ion o u he de ails and discussion.
As discussed in Appendix I, he global con e gence o he algo i hm equi es he Ja-
cobian o be non-singula and con inuously diffe en iable, wi h he special conside a ions
Closes -Poin P ojec ion Algo i hms in Elas oplas ici y 7
BOX 2.1. Implemen a ion o he p imal closes -poin p ojec ion
me hod (p imal–CPPM).
1. Inpu da a: {εe, ial
n+1 ,α ial
n+1 }, wi h he co esponding {σ ial
n+1 ,q ial
n+1 }
and ial
n+1 >0.
2. Ini ializa ion: se k=0,
x(0) :=
εe, ial
n+1
−α ial
n+1
0
,and (0) :=
0
0
ial
n+1
.
3. Compu e he Jacobian:
J(k):= I+∆γ(k)∇m(k)˜
G(k)m(k)
∇ (k)T˜
G(k)0
whe e all he quan i ies in his exp ession a e e alua ed a x(k)=
{εe(k)
n+1,−α(k)
n+1,∆γ(k)}Tacco ding o hei defini ions in Box 1.1.
4. Compu e he di ec ion o ad ance:
aux := 0
IF ∆γ(k)=0THEN aux := m(k)
σn+1 ·(εe,(k)
n+1 −εe, ial
n+1 )−
m(k)
qn+1 ·(α(k)
n+1 −α ial
n+1 )
IF aux ≤0THEN
lag := . ue.
d(k):= −(J(k))−1 (k)
ELSE
lag := . alse.
[D(k)]ij :=
(J(k))−1(J(k))−Tij i, j =nxo i=j=nx
0i, j =nxand i=j
d(k):= −D(k)(J(k))T (k)
ENDIF
5. Apply he line sea ch scheme o Box I.2 o he unc ion (x) by (2.1).
INPUT: x(k), (k),d(k)and lag.
OUTPUT: x(k+1) and (k+1),aswellas(σ,q)(k+1)
n+1 om he
compu a ion o (k+1), and o he auxilia y quan i-
ies used in I em 3. o he Jacobian J(k+1).
6. Check con e gence →se (εe,α,σ,q)n+1 =(εe,α,σ,q)(k+1)
n+1
and EXIT.
7. Se k←k+1andGO TO 3.
A. P´e ez-Fogue & F. A me o 8
sn+1
on Mises
n+1
s ial
n+1
s ial
( s ) 0
n+1
s (k)
n+1
s (k)
sn+1
T esca
( s ) 0
(k) = 0
n+1
m (k)
n+1
m (k)
egion whe e he cons ain
is no ac i a ed
n+1n+1
m (k)· ( s (k) - s ial ) < 0
n+1n+1n+1
m (k)· ( s (k) - s ial ) < 0
n+1
FIGURE 2.1. G aphical in e p e a ion o he condi ion (2.3) o
he a) on Mises and b) T esca yield condi ions, bo h wi h associa ed
pe ec plas ic flow (m=∇ ) and cons an elas ici ies. The egion o
he de ia o ic s ess plane (s:= σ−3
i=1[σ]i) whe e he cons ain
∆γ≥0 is no ac i a ed, leading o he modified Jacobian in he line
sea ch scheme, is shown in g ay.
on he esidually-based descen unc ion p esen ed in Rema k I.1 due o he cons ained
cha ac e o he p oblem. A non-singula Jacobian is implied by he usual assump ions
o an associa i e con ex p oblem (i.e., m=∇ wi h ∇2 posi i e semi-defini e) and
s ic con exi y o he elas ic ela ions (i.e. ˜
Gposi i e defini e), as conside ed in Pa I
o his wo k. Unde hese assump ions, he equa ions (2.1) co espond o he fi s o de
necessa y and sufficien condi ions o he con ex p imal a ia ional p oblem conside ed
he ein (P oposi ion 3.1, o be specific). The equi ed smoo hness ollows om he egu-
la i y o he ma e ial model, ha is, he yield unc ion, and elas ic and ha dening laws.
Howe e , he p e ious ul ima e equi emen s on he Jacobian can be sa isfied in mo e gen-
e al si ua ions. As a ma e o ac , we conside in Sec ion 4 an example in ol ing s ain
so ening. The pe o mance o he algo i hm is also assessed in ha sec ion o a case
in ol ing a non-diffe en iable esidual due o he conside a ion o a yield unc ion leading
o a non-diffe en iable flow ec o . In bo h cases, he p imal–CPPM shows a d ama ic
imp o emen in i s global con e gence p ope ies o e he o iginal New on–CPPM, while
exhibi ing locally he desi ed asymp o ic quad a ic a e o con e gence.
Rema ks 2.1.
1. Globally con e gen p imal closes -poin p ojec ion algo i hms can be easily de ised
o he iscoplas ic p oblem gi en i s uncons ained cha ac e ; see Sec ion 4 o Pa I.
Closes -Poin P ojec ion Algo i hms in Elas oplas ici y 15
/ c
( )
(1) (k)
=
( )
(1) (k)
ial
n+1
c ial
n+1
ial
n+1
c
c
c = 0c > 0
FIGURE 3.1. G aphical in e p e a ion o he uppe le el scheme o
he dual–CPPM o he non-augmen ed (c= 0) and augmen ed (c>0)
o mula ions.
o an ypical i e a ion (k) in e ms again o he Maucalay b acke s ·. The de i a i e ¯
in (3.4) is ob ained in closed- o m using he equa ions (3.1) and (3.2) as
¯
=−∇ ·˜
G−1+∆γ∇m−1
m.(3.5)
A he bounda y ∆γ= 0 o he admissible se , his de i a i e educes o he alue
¯
∆γ=0 =−∇ ·˜
Gm<0 s ic ly ,(3.6)
gi en he undamen al assump ion o he unde lying con inuum model (see Box 1.1), e en
in he gene al non-con ex (e.g. wi h s ain so ening) and non-associa ed plas ic flow
(m=∇ ). This impo an esul implies ha e en hough he i e a ion p ocess (3.4) may
lead o he alue ∆γ(k)= 0 a some i e a ion (k), he cons ain i sel is no ac i a ed, ha
is, he i e a ion di ec ion (3.4)2leads na u ally o ∆γ(k+1) >0 and no modified ad ancing
ules like (I.17) in Appendix I a e needed. This si ua ion, oge he wi h he a o emen ioned
uncons ained cha ac e o he p oblem in he lowe le el, a oids al oge he he need o
he use o complex line sea ch ules as equi ed in he cons ained p imal o mula ion
p esen ed in Sec ion 2.1.
The global con e gence o he uppe le el algo i hm defined by he i e a i e p ocess
(3.4) is hen assu ed i he de i a i e in (3.5) is con inuous and does no anish. The
smoo hness condi ion is again implied by he p ope smoo hness o he ma e ial model,
ha is, he yield su ace and elas ic and ha dening laws. The non-ze o condi ion on he
de i a i e is sa isfied unde he usual assump ions o con exi y and associa ed plas ic
A. P´e ez-Fogue & F. A me o 16
n+1 , qn+1
(1) > 0
n
+1
= 0
ial> 0
n
+1
ial, q ial
n+1 n+1
(1), q (1)
n+1 n+1
Elas ic
domain
< 0
n+1
(k), q (k)
n+1 n+1
(k) > 0
n+1
n+1
o hogonal p ojec ion
in he -1 me ic
FIGURE 3.2. G aphical in e p e a ion o bo h he dual–CPPM and
he augmen ed–dual–CPPM. The s ess a iables {σ(k)
n+1,q(k)
n+1}a
each i e a ion (k) a e he closes -poin p ojec ion on o he cu en i -
e a e o he yield unc ion ¯
(∆γ(k))(o ¯
c(∆λ(k)) o heaugmen ed
algo i hm) o he plas ic mul iplie ∆γ(k)(o ∆λ(k)) ob ained by he
uppe le el scheme. A simila figu e can be ound in he li e a u e bu
e oneously associa ed wi h he o iginal New on–CPPM o Box 1.1.
flow. Unde hese assump ions, equa ion (3.5) e eals ha he unc ion ¯
(∆γ) is s ic ly
mono onically dec easing ( ¯
<0) as ob ained in Pa I o his wo k. Figu e 3.1 illus a es
hese esul s g aphically. Unde hese condi ions, he line sea ch scheme conside ed in his
uppe le el scheme did no need o be ac i a ed in he nume ical examples epo ed in
Sec ion 5, bu i s conside a ion is needed o assu e he global con e gence o he i e a i e
p ocess in gene al si ua ions.
The e o e, we conclude ha he p oposed wo le el scheme based on he dual o m o
he closes -poin p ojec ion equa ions is globally con e gen unde hese assump ions. The
nume ical examples p esen ed in Sec ion 5 ha e confi med his p ope y. Figu e 3.2 depic s
g aphically he idea behind he p oposed dual–CPPM o he associa ed case gi en by he
no mali y ule m=∇ , and cons an elas ici ies and linea ha dening (i.e. ˜
G= cons an ).
Equa ions (3.2) imply in his case ha in a gi en i e a ion o he uppe le el scheme, ha
is, o each alue ∆γ(k), he ec o Σ(k)
n+1 −Σ ial
n+1 in s ess space ( o Σ={σ,q}) is indeed
no mal in he me ic ˜
G−1 o he cu en alue o he yield su ace (k). This p ope y is
illus a ed in Figu e 3.2, showing each i e a ion alue o he s ess a iables as he closes -
poin p ojec ion on he elas ic domain (k)
n+1 =¯
(∆γ(k))≤0 o he cu en i e a e ∆γ(k)
in he uppe le el algo i hm depic ed in Figu e 3.1. We no e ha a simila figu e can
Closes -Poin P ojec ion Algo i hms in Elas oplas ici y 17
be ound in Simo & Hughes [1998], among o he s, and i is e oneously associa ed wi h
he o iginal New on–CPPM. The de elopmen s p esen ed in his sec ion clea ly iden i y
he app oach depic ed in his figu e wi h he newly p oposed algo i hm based on he dual
o mula ion o he go e ning equa ions.
Rema k 3.1. The dual–CPPM desc ibed in his sec ion ex ends o he iscoplas ic
p oblem, ollowing he de elopmen s o he dual o mula ion o his p oblem p esen ed in
P oposi ion 4.3 o Pa I. Tha is, simply eplace he scala equa ion (3.1) by he ela ion
g(¯
(∆γ)) −η∆γ=0,(3.7)
o a gene al iscoplas ic unc ion g(·) and iscoplas ic pa ame e η≥0. We obse e ha
his is he only needed modifica ion, since he p oblem (3.2) is no changed, wi h he lowe
le el algo i hm o Box 3.2 applying en i ely o he iscoplas ic p oblem as well. Simila ly,
he same conside a ions ega ding he non-ac i a ion o he cons ain ∆γ= 0 and, indeed,
he same global con e gence p ope ies apply unde he usual mono onici y assump ion o
he iscoplas ic unc ion g(·). Addi ional de ails a e omi ed.
3.2. The augmen ed dual closes -poin p ojec ion me hod
E en hough he cons ained cha ac e o he uppe le el p oblem in he dual scheme
conside ed in he p e ious sec ion was shown no o affec he algo i hm i sel , he con-
side a ion o augmen ed Lag angian ex ensions a oiding al oge he his cons ain leads
o algo i hms imp o ing on hei nume ical pe o mance, as illus a ed in he nume i-
cal examples p esen ed in Sec ion 5. Following he de elopmen s o he dual augmen ed
Lag angian o mula ion summa ized in P oposi ion 5.2 o Pa I, he plas ic consis ency
unc ion ¯
(·) in (3.1) is eplaced by
¯
c(∆λ):= (σc(∆λ),qc(∆λ)) = 0 ,(3.8)
wi h {σc(∆λ),qc(∆λ)}defined by he augmen ed lowe le el p oblem ˜ c(˜x)=0 o he
augmen ed esidual
˜ c(˜x)=εe
n+1 −εe, ial
n+1 +∆λ+c (σn+1,qn+1)mσn+1
−αn+1 +α ial
n+1 +∆λ+c (σn+1,qn+1)mqn+1 ,(3.9)
o a penal y pa ame e c≥0 and in e ms, again, o he p ima y unknowns ˜x:=
{εe
n+1,−αn+1}T. Following he same no a ion employed in he p imal augmen ed La-
g angian o mula ions o Sec ion 2.2, we deno e he scala field appea ing in hese equa-
ions by ∆λemphasizing i s en i ely uncons ained cha ac e in on o ∆γ≥0. The final
plas ic mul iplie ∆γis eco e ed by he simple ela ion
∆γ=∆λ,(3.10)
A. P´e ez-Fogue & F. A me o 18
BOX 3.3. Implemen a ion o he uppe le el o he augmen ed–dual–
CPPM, in ol ing a New on i e a i e p ocess in ∆λ.
1. Inpu da a: {εe, ial
n+1 ,α ial
n+1 }, wi h he co esponding {σ ial
n+1 ,q ial
n+1 }
and ial
n+1 >0.
2. Ini ialize: se k=0,(εe,α,σ,q)(0)
n+1 =(εe,α,σ,q) ial
n+1 ,∆λ(0) =0
and ¯
(0)
c= ial
n+1 .
3. Compu e upda e di ec ion:
δ(∆λ)(k):= ¯
(k)/∇ (k)·˜
G(k)(I+H(k)˜
G(k))−1m(k),
whe e
s(k):= 0, o ∆λ(k)+c (k)<0,
1, o ∆λ(k)+c (k)>0,
H(k):= ∆λ(k)+c (k)∇m(k)+cs
(k)m(k)⊗∇ (k).
5. Apply he line sea ch scheme o Box I.1 wi h he scala esidual and
descen unc ion
c∆λ(∆λ):= ¯
c(∆λ),and Mc∆λ(∆λ)=1
2¯
c(∆λ)2
o x(k)←∆λ(k), and whe e ¯
c(∆λ)= (σc(∆λ),qc(∆λ)) o he
alues {σc(∆λ),qc(∆λ)}compu ed by he lowe le el algo i hm in
Box 3.4 o a fixed ∆λ.
INPUT: ∆λ(k),¯
(k)
cand δ(∆λ)(k).
OUTPUT: ∆λ(k+1),¯
(k+1)
c,aswellas(εe,α,σ,q)(k+1)
n+1
and o he auxilia y quan i ies used in he upda e
o mula in 3., om he e alua ion o ¯
(k+1)
c h ough
he algo i hm in Box 3.4 (lowe le el).
5. Check con e gence →se (εe,α,σ,q)n+1 =(εe,α,σ,q)(k+1)
n+1
and EXIT.
6. Se k←k+1andGO TO 3.
Closes -Poin P ojec ion Algo i hms in Elas oplas ici y 19
BOX 3.4. Implemen a ion o he lowe le el o he augmen ed–dual–
CPPM.
1. Inpu da a: he fixed alues εe, ial
n+1 ,α ial
n+1 and ∆λ(k+1), and he
ini ial alues (εe,α)(k)
n+1,wi h(σ,q)(k)
n+1.
2. Ini ialize: se i=0,˜x(0) := εe,(k)
n+1 ,−α(k)
n+1Tand
˜ c(0)
:= ˜ c(˜x(0)) by (3.9).
3. Compu e he Jacobian:
˜
Jc(i):= I+∆λ(k+1) +c
(i)∇m(i)+cs
(i)m(i)⊗∇ (i)˜
G(i),
whe e
s(i):= 0,∆λ(k+1) +c (i)<0,
1,∆λ(k+1) +c (i)>0.
4. Compu e he di ec ion o ad ance:
˜
dc(i):= −(˜
Jc(i))−1˜ c(i).
5. Apply he line sea ch scheme o Box I.1 based on he unc ion ˜ c(x).
INPUT: ˜x(i),˜ c(i)and ˜
dc(i).
OUTPUT: ˜x(i+1),and˜ c(i+1),aswellas(σ,q)(i+1) om he
compu a ion o ˜ c(i+1), and o he auxilia y quan i-
ies used in I em 3. o he Jacobian ˜
Jc(i).
6. Check con e gence →se (εe,α,σ,q)(k+1)
n+1 =(εe,α,σ,q)(i+1)
and EXIT.
7. Se i←i+1andGO TO 3.
co esponding ac ually o ∆λ>0 in he plas ic co ec o s ep o in e es . We obse e ha
he o iginal dual p oblem (3.1)–(3.2) is eco e ed o c= 0. The solu ion o he uppe and
lowe p oblems defined by (3.8) and (3.9), espec i ely, is app oached again wi h New on
schemes in combina ion wi h he uncons ained line sea ch echnique p esen ed in Sec ion
I.1. The final scheme is summa ized in Boxes 3.3 and 3.4 o he uppe and lowe le el
algo i hms, espec i ely.
The ully uncons ained cha ac e o he p oblem in he uppe le el o he algo i hm
A. P´e ez-Fogue & F. A me o 20
leads o he di ec conside a ion o he line sea ch scheme o Box I.1 wi hou he need
o he added imposi ion o he non-nega i e cons ain on ∆γby (3.4)1 o he p e ious
cons ained dual o mula ion. The de i a i e o he augmen ed plas ic consis ency unc ion
(3.8) can also be ob ained in closed- o m a e using equa ions (3.9) o he lowe le el
augmen ed p oblem as
¯
c=−∇ ·˜
G−1+H−1
m.(3.11)
whe e
H:= ∆λ+c ∇m+csm⊗∇ , (3.12)
o
s:= 0, o ∆λ+c < 0,
1, o ∆λ+c > 0.(3.13)
E en hough no p oblems ha e been obse ed in he ac ual nume ical simula ions due o
he discon inui y o sa ∆λ+c = 0 (in ac , alues ∆λ+c < 0 a e ne e eached),
we assign he alue s= 0 a his poin . We can conclude om hese esul s he he same
p ope ies o he augmen ed plas ic consis ency unc ion ¯
c(∆λ) as i s coun e pa ¯
(∆γ)
in he o iginal cons ained dual o mula ion; addi ional de ails a e omi ed. In pa icula ,
i s s ic ly mono onically dec easing cha ac e (i.e., ¯
c<0) is concluded unde he usual
con exi y and associa i i y assump ions. The same globally con e gen cha ac e o he
p oposed scheme is concluded in hese cases.
We also obse e ha he augmen ed–dual–CPPM de eloped in his sec ion has he
same g aphical in e p e a ion depic ed in Figu e 3.2 o he non-augmen ed dual algo i hms
(i.e c= 0). The uppe le el algo i hm in ol es, howe e , a egula ized consis ency unc ion,
a si ua ion ha has been depic ed in Figu e 3.1 o c>0. As a gued in he end o
Rema k 3.2 below and e ified in he nume ical examples p esen ed in Sec ion 5, his
simple modifica ion leads o some imp o emen in he compu a ional pe o mance o he
dual algo i hms conside ed in his sec ion.
Rema k 3.2. We obse e ha by conside ing he ini ial es ima e o he uppe le el
algo i hm by ∆λ(0) = 0, he fi s solu ion o he lowe le el p oblem (3.9) ob ained by
he scheme in Box 3.4 co esponds o he solu ion o a iscoplas ic p oblem defined by he
iscoplas ic pa ame e c=1/ηand linea iscoplas ic model g( )=< >. In ac , his
p oblem co esponds o he p imal o mula ion o he iscoplas ic p oblem conside ed in
Rema k 2.1 abo e o his case. The conside a ion o mo e gene al iscoplas ic models is
easily accomplished h ough he use o augmen ed o mula ions based on he co esponding
egula iza ion unc ion g(·) as p esen ed in Rema k 5.1.2 ( ha is, eplace <·>by g(·)in
he ela ions o Box 3.4). In his iscoplas ic case, he equa ion (3.10) gi ing he plas ic
mul iplie ∆γis o be eplaced by he iscous ela ion
∆γ=1
ηg(¯
c(0)) .(3.14)
Closes -Poin P ojec ion Algo i hms in Elas oplas ici y 21
The e o e, he choice c=1/ηleads he final solu ion o he iscoplas ic p oblem in a
single i e a ion o he uppe le el algo i hm. I he iscoplas ic p oblem is unde s ood as
he penal y egula iza ion o he a e-independen p oblem as η→0, he imp o emen
in he compu a ional pe o mance gained by he conside a ion o a la ge egula iza ion
pa ame e cis o be expec ed when sol ing he limi a e-independen p oblem. The
usual conside a ions on he well-condi ioning o he esul ing p oblem is o be weighed in
he a gumen , as i is in es iga ed h ough he nume ical examples p esen ed in Sec ion
5below.
4. Nume ical Assessmen : P imal Algo i hms
We assess in his sec ion he nume ical pe o mance o he new p imal algo i hms
p oposed in Sec ion 2. Mo e specifically, i is ou in e es o e alua e he global con e gence
p ope ies o hese schemes. To his pu pose, we s udy in Sec ion 4.1 he con e gence
p ope ies o he p oposed schemes o a single inc emen ( om, say, n o n+1) o diffe en
imposed s ain inc emen s. Sec ion 4.2 e alua es he o e all pe o mance o he schemes
in a gi en imposed pa h o he de o ma ion g adien , in ol ing diffe en numbe s o ime
inc emen s.
Rema ks 4.1.
1. The esul s p esen ed below a e exp essed in e ms o he in a ian s
p∗=−I1(∗)
3,q
∗=3J2(∗)andθ∗=1
3a ccos 3√3J3(∗)
2J2(∗)3/2,(4.1)
o a gene ic second o de enso ∗,whe eI1(∗), J2(∗)andJ3(∗) deno e he fi s
in a ian o ∗and he second and hi d in a ian s o he de ia o ic pa o ∗, [de (∗)]i=
[∗]i+p∗, espec i ely,
I1(∗)=
3
i=1
[∗]i,J
2(∗)=1
2
3
i=1
([de (∗)]i)2and J3(∗)=
3
i=1
[de (∗)]i,(4.2)
in e ms o he h ee p incipal poin s [∗]i o i=1,2,3. The unc ion θ∗co esponds
o he so-called Lode’s angle and a ies om 0◦ o 60◦. These exp essions a e used
o he Ki chhoff p incipal s esses ∗=σand he loga i hmic p incipal elas ic s ains
∗=εe; see Sec ion 2.3 o Pa I o his wo k.
2. All he con ou plo s p esen ed in his and subsequen sec ions (see e.g. Figu e 4.1)
ha e been ob ained wi h a uni o m g id o 80 ×80 sampling alues, independen ly o
he anges o he a iables depic ed in bo h axis.
A. P´e ez-Fogue & F. A me o 22
3. In all he examples p esen ed in his pape , he con e gence o he algo i hm is de-
ec ed h ough he exp ession
e(k)
ˆx=||ˆx(k+1) −ˆx(k)||∞
||ˆx(k+1)||∞≤TOLˆx(4.3)
measu ing he ela i e e o in i e a ion (k+ 1) in he maximum no m || · ||∞=
maxi|[·]i|, o each componen [ ·]idespi e hey may ha e diffe en dimensions, in
gene al. He e ˆx e e s he d i ing a iable employed in he pa icula algo i hm unde
conside a ion ( ha is, xgi en by (2.1) in he p imal algo i hms, ˜xgi en by (3.2) in
he lowe le el o he dual algo i hms, and ∆γo ∆λin he uppe le el o he dual
algo i hms). The ole ance alue o TOLˆx=10
−12 has been employed in all cases.
4.1. Single inc emen es s
We e alua e in his sec ion he con e gence p ope ies o he i e a i e schemes unde
in es iga ion du ing a plas ic co ec o s ep o an imposed ial elas ic s a e. The ini ial
s a e is assumed elas ic (i.e., anishing ini ial alues o he plas ic in e nal a iables) wi h
anishing s ain. De ia o ic and p essu e dependen models a e conside ed in Sec ions
4.1.1 and 4.1.2, espec i ely, bo h in combina ion o pe ec plas ici y and Hencky’s hy-
pe elas ic law, esul ing in a se o cons an elas ici ies in he loga i hmic p incipal elas ic
s ains εe; see Appendix II. The imposed ial s a e is hen cha ac e ized in he de ia o ic
Π-plane by he co esponding Lode angle θεe, ial (= θσ ial ) and he adial measu e qεe, ial
(= qσ ial /2µ o he shea modulus µ). The p essu e pa ame e pεe, ial (= pσ ial /3κ o
he bulk modulus κ) defines comple ely he imposed ial s a e in he p essu e-dependen
models conside ed in Sec ion 4.1.2.
4.1.1. De ia o ic plas ic models
We conside fi s he on Mises–T esca ype yield su aces desc ibed in Sec ion II.2 o
Appendix II, defining in e ms o he ma e ial pa ame e ma amily o de ia o ic yield
su aces exhibi ing diffe en le els o cu a u e, as i is o he in e es in his wo k. In
pa icula , he choice m= 0 co esponds o he smoo h on Mises ci cula cylinde , wi h
highe alues o mleading o inc easing o he cu a u e o his yield su ace a θσ=0,30
( he T esca non-smoo h yield su ace is eco e ed o m→∞). Only he wo de ia o ic
pa ame e s qεe, ial and θεe, ial a e needed in his case o define he ial s a e. Because o
he symme y o he esul ing yield unc ions (see Figu e II.1 in Appendix II), Lode angles
be ween 0◦and 30◦need only o be conside ed. The alues o κ= 164.206 kN/mm2and
µ=80.1938 kN/mm2a e aken o he bulk and shea modulus, espec i ely, in Hencky’s
law (II.2)-(II.3), wi h he cons an yield s ess σyo=0.45kN/mm
2.
Figu e 4.1 compa es he pe o mance o he New on–CPPM and he p imal–CPPM in
his case. The numbe o i e a ions needed o con e gence o bo h schemes o se e al
Closes -Poin P ojec ion Algo i hms in Elas oplas ici y 23
m
=5
m
=10
m
=20
p imal{CPPM p imal{CPPM p imal{CPPM
New on{CPPM New on{CPPM New on{CPPM
"
e; ial
0
Æ
30
Æ
"
e; ial
0
Æ
30
Æ
"
e; ial
0
Æ
30
Æ
2
q
"
e; ial
=
y
o
40
2
q
"
e; ial
=
y
o
4
0
1 3 5 7 9 11 13 15 17 19 NC
FIGURE 4.1. Single inc emen es s: de ia o ic plas ic mod-
els. Numbe o i e a ions needed o con e gence by he New on–
CPPM and he p imal–CPPM o h ee diffe en on Mises–T esca ype
yield su aces (m= 5, 10 and 20). “NC” deno es no con e gence a e
mo e han 100 i e a ions; ex ended es s in hese egions lead o no
con e gence a e se e al hund eds o i e a ions.
elas ic ial s a es {qεe, ial ,θ
εe, ial }and diffe en alues o he ma e ial pa ame e ma e
depic ed in his figu e. The egions whe e no con e gence is de ec ed a e mo e han 100
i e a ions a e deno ed by “NC”; ex ended simula ions show no con e gence a e se e al
hund eds o i e a ions in hese egions. The alue o qεe, ial a ies in he ange [0,4·
σyo/(2µ)], ha is, wi h co esponding ial s ess s a es up o ou imes he yield limi
σyo.
The esul s depic ed in Figu e 4.1 e eal ha no con e gence is de ec ed wi h he
A. P´e ez-Fogue & F. A me o 24
m
=20
New on{CPPM p imal{CPPM (
j
max
=1) p imal{CPPM
"
e; ial
0
Æ
30
Æ
"
e; ial
0
Æ
30
Æ
"
e; ial
0
Æ
30
Æ
2
q
"
e; ial
=
y
o
10
0
1 4 8 12 16 20 24 28 32 NC
FIGURE 4.2. Single inc emen es s: de ia o ic plas ic models (m=
20). Numbe o i e a ions needed o con e gence by he New on–
CPPM and he p imal–CPPM (wi h a maximum o one cu e fi ing
(jmax = 1), and wi hou a limi on he numbe o cu e fi ings) o
la ge excu sions ou side he elas ic domain.
s anda d New on–CPPM when he solu ion is close o θεe, ial =0
◦(and 60◦). As no ed
abo e, hese alues co espond o poin s whe e he cu a u e o he yield su ace is highe .
Mo eo e , as minc eases he size o he egion o no con e gence inc eases. In ac ,
he New on–CPPM becomes basically useless e en o mode a e alues o m. Wi h he
p imal–CPPM hese egions o no con e gence a e a oided. Con e gence is a ained o any
ial s a e, wi h less han 17 i e a ions e e ywhe e. This imp o emen is obse ed mainly
in he egions o no con e gence o he o iginal New on–CPPM whe e e he New on–
CPPM con e ges, he p imal–CPPM does no educe he numbe o i e a ions sensibly.
No e also ha o θεe, ial exac ly equal o 0◦(and 60◦) he con e gence is achie ed always
in a educed numbe o i e a ions. This e y special si ua ion is due o he ac ha he
g adien o he yield su ace has always he same di ec ion a hese Lode angles.
Bo h schemes a e compa ed again in Figu e 4.2, bu wi h he added conside a ion o
he maximum numbe o cu e fi ings in he line sea ch scheme employed in he p imal–
CPPM, ha is, he jmax pa ame e in Boxes I.1 and I.2 o Appendix I. La ge excu sions
ou side he elas ic domain, in he ange [0,10 ·σyo/(2µ)] o he adial measu e qεe, ial ,
a e also conside ed. We obse e ha he p imal–CPPM wi hou a limi in he numbe o
cu e fi ings pe i e a ion a oids comple ely he egions o no con e gence o he o iginal
New on–CPPM. The maximum numbe o cu e fi ings needed is only wo, hough. The
small egion o no con e gence emaining a om he yield su ace wi h a limi o jus one
Closes -Poin P ojec ion Algo i hms in Elas oplas ici y 31
which in ol e ensile p essu es wi h his yield su ace.
In con as , he p imal–CPPM leads o con e gence o all he ial s a es. S ill, he
o e all beha io o he p imal–CPPM in he egion behind he apex is wo se han in he
examples o Figu e 4.6. The e is a high a iabili y in he dis ibu ion o he numbe o
i e a ions, which o some ial s a es a e a ela i ely la ge, al hough less han 50 i e a ions
a e needed anywhe e. The la ge numbe o i e a ions is again ela ed wi h o he non-
diffe en iabili y o he flow ec o , wi h he associa ed lack o con inui y o he Jacobian
a θσ=5
◦and θσ=55
◦. The high high sensi i i y in he ini ial ial s a e can be aced
back o he closeness o he i e a i e p ocess o hese poin s o low alues o qσ.The
esul s o Figu e 4.7 has been ob ained wi h a maximum numbe o cu e fi ings fixed o
h ee. Fo some ial s a es, his pa ame e influences in he con e gence esul s, al hough
he o e all beha iou is he same
To illus a e be e he effec o he non-diffe en iabili y o he esidual ec o and
he obse ed dependence on diffe en algo i hmic pa ame e s in he esul ing highly sen-
si i e cases, we conside he specific ial s a e defined by pεe, ial =−3·CMC co (φ)/(3κ),
qεe, ial =2.25 ·√3CMC /(2µ)andθεe, ial =30
◦in his egion o high a iabili y. The
p imal–CPPM akes 49 i e a ions wi h a maximum o h ee cu e fi ings (jmax =3),as
shown in Figu e 4.7. This is he maximum numbe o i e a ions obse ed in his figu e.
The e olu ion o he ela i e e o , he yield unc ion, he numbe o cu e fi ings, he
alue o he line sea ch pa ame e and he Lode angle o he s esses du ing he i e a i e
p ocess is depic ed in Figu e 4.8 o a maximum numbe o cu e fi ings jmax =5,and
10. In bo h cases, mo e cu e fi ings han his fixed maximum numbe a e needed in a
ew i e a ions. These i e a ions a e indica ed wi h a black ci cle in he plo s depic ing he
equi ed numbe o cu e fi ings. No e ha bo h i e a i e p ocesses ( o jmax = 5 and 10)
coincide un il his poin . Du ing hese ea ly s ages o he i e a i e p ocess, he i e a ions
a e “cap u ed” a he Lode angle θσ=55
◦, whe e he Jacobian o he esidual is no con-
inuous. The numbe o cu e fi ings inc eases and he i e a i e inc emen becomes e y
small (bounded om below by he back acking pa ame e η=0.1 and maximum numbe
o cu e fi ings; see Box I.1) while he ela i e alue o he yield unc ion emains basi-
cally cons an and he ela i e e o based on he no m o he inc emen o he unknowns
diminishes. This ac is di ec ly ela ed wi h he alue o he line sea ch pa ame e .
The es wi h jmax = 10 allows he addi ional numbe o cu e fi ings a e his poin .
A e he maximum numbe o cu e fi ings jmax is eached, he beha io o he i e a i e
scheme changes d ama ically. One cu e fi ing is needed du ing some o he subsequen
i e a ions and finally a quad a ic a e o con e gence is achie ed. The i e a i e p ocess has
“escaped” om he discon inui y o he Jacobian. These conside a ions, oge he wi h he
discussion p esen ed wi h Figu e 4.2 o smoo h cases, allow o conclude ha a mode a e
alue o he pa ame e jmax (≈3−5 seems app op ia e) should be used o he maximum
numbe o cu e fi ings.
A. P´e ez-Fogue & F. A me o 32
0
2
4
6
8
10
1 1121314151617181
0
15
30
45
60
1 1121314151617181
1,E-16
1,E-12
1,E-08
1,E-04
1,E+00
1 1121314151617181
1,E-11
1,E-08
1,E-05
1,E-02
1,E+01
1 1121314151617181
0
2
4
6
8
10
1 1121314151617181
0
15
30
45
60
1 1121314151617181
1,E-16
1,E-12
1,E-08
1,E-04
1,E+00
1 1121314151617181
1,E-11
1,E-08
1,E-05
1,E-02
1,E+01
1 1121314151617181
(k)
ex
| / ial|
n+1
ex
| / ial|
n+1
(k)
FIGURE 4.8. Single inc emen es s: p essu e-dependen plas ic
models. Influence o he non-diffe en iabili y o he flow ec o in he
RHMC model o he es gi en by pεe, ial =−3·CMC co (φ)/(3κ),
qεe, ial =2.25 ·√3CMC/(2µ)andθεe, ial =30
◦. E olu ion o he
he ela i e e o e(k)
x, he yield unc ion (k)/ ial
n+1 , henumbe o
cu e fi ings, he line sea ch pa ame e α(k)and he s ess Lode angle
θσ(k) o a maximum o cu e fi ings jmax = 5 and 10.
Closes -Poin P ojec ion Algo i hms in Elas oplas ici y 33
c
=0
:
001
=C
MC
c
=0
:
03
=C
MC
p
"
e; ial
3
C
MC
co (
)
3
0
p
"
e; ial
3
C
MC
co (
)
3
0
q
"
e; ial
2
p
3
C
MC
5
0
-25 -15 -7 -1 1 7 14 21
FIGURE 4.9. Single inc emen es s: p essu e-dependen plas-
ic models. Inc emen o he numbe o i e a ions o he p imal–
CPPM wi h diffe en alues o he penal y pa ame e c, wi h espec
o he non-augmen ed o mula ion in Figu e 4.7 o θεe, ial =30
◦..
To conclude his sec ion, we conside again he augmen ed e sions o he algo i hms.
The inc emen o he numbe o i e a ions o he augmen ed p imal–CPPM wi h espec
o he non–augmen ed e sion is depic ed in Figu e 4.9 (elas ic ial s eps in he me idian
plane θεe, ial =30
◦). The pe o mance o he scheme imp o es in a e age wi h small
alues o he penal y pa ame e cwhen compa ed wi h he non-augmen ed o mula ion.
Fo la ge alues o c, al hough a high educ ion o he numbe o i e a ions is ound in
some egions, a significan inc ease is ound in ano he ones (no e he esul s ob ained wi h
c=0.03/CMC in he egion close o he apex). This effec is accen ua ed as cinc eases,
and, as his effec occu s in he egion closes o he apex, he me hod becomes less
compe i i e wi h la ge alues o c. The effec s o he non-diffe en iabili y o he esidual
ollow he same pa e n in his augmen ed case as discussed in he p e ious pa ag aphs
o he dual–CPPM.
4.2. Imposed de o ma ion g adien pa h
We e alua e nex he pe o mance o he p imal algo i hms in mo e gene al si ua-
ions in ol ing Ogden hype elas ic models (i.e., non-cons an elas ici ies in he loga i hmic
s ains) as well as s ain ha dening and so ening ela ions. To his pu pose we conside
imposed pa hs o he de o ma ion g adien . Mo e specifically we conside he de o ma ion
g adien his o y
F( )=b( )1/3diag(a( ),a( )− ,a( ) −1) (4.4)
o he pseudo ime , unc ion a( ) (such as a(0) = 1 and a( )>0) ha guides he e olu ion
A. P´e ez-Fogue & F. A me o 34
TABLE 4.2. Elas ic pa ame e s used wi h he Ogden hype elas ic
model.
κ= 164.206 kN/mm2ˆµ1=1.4911 ˆα1=1.3000
µg=80.1938 kN/mm2ˆµ2=0.0028 ˆα2=5.0000
N=3 ˆµ3=−0.0237 ˆα3=−2.0000
o he axial s e ch in he fi s p incipal di ec ion, and he unc ion b( ) con olling he
olume ic esponse (de F=b( )). The exponen in (4.4) con ols he desi ed Lode
angle o he elas ic ial inc emen (∆εe, ial =εe, ial
n+1 −εn), since
=1
21−√3 an(θ∆εe, ial )(4.5)
In pa icula , he alue =0.5 co esponds o pu e ension (θ∆εe, ial =0
◦) and he alue
=−1 o pu e comp ession (θ∆εe, ial =60
◦).
4.2.1. De ia o ic plas ic models
We conside he de ia o ic plas ic models defined by he on Mises-T esca ype yield
su aces desc ibed in Sec ion II.2 o Appendix II. The Ogden hype elas ic model defined in
(II.4) o Appendix II, wi h ma e ial pa ame e s summa ized in Table 4.2 (Miehe [1998]),
is conside ed. Simila ly, he ha dening/so ening po en ial (II.9) is conside ed o he
sa u a ion exponen δ= 20 and wi h he sa u a ion s ess σy∞=0.6kN/mm
2 o he
examples wi h s ain ha dening and σy∞=0.3kN/mm
2in he examples wi h s ain
so ening. In all cases, including he examples wi h pe ec plas ici y, he ini ial yield
s ess is σyo=0.45kN/mm
2.
The imposed unc ion o
a( )=1+0.2 o ∈[0,1] ,(4.6)
is conside ed, wi h b( )≡1 leading o a pu ely isocho ic de o ma ion. We conside solu ions
in ol ing 5, 50 and 500 equal ime inc emen s. The esul ing Ki chhoff s ess pa h in
qσis depic ed in Figu es 4.10, including de ails in he small ange o he imposed axial
s e ch. We conside he alues o θ∆εe, ial =0
◦,15
◦and 30◦, leading o he co esponding
exponen s by (4.5). No e he smoo h ansi ion om elas ic o elas oplas ic egime in
he cu es co esponding o θ∆εe, ial =15
◦. This is p oduced by he change o he Lode’s
angle o he s esses du ing he e olu ion o he simula ion. The e olu ion o qσ o a la ge
ange o he axial s e ch and θ∆εe, ial =15
◦is shown in Figu e 4.10.b. The e olu ion o
he Lode angle θσo he Ki chhoff s ess is depic ed in Figu e 4.10.c o he so ening case.
The solid cu es o co espond o he p oblems sol ed wi h 500 equal inc emen s and he
Closes -Poin P ojec ion Algo i hms in Elas oplas ici y 35
a)
0,3
0,35
0,4
0,45
0,5
1 1,005 1,01 1,015 1,02
0,3
0,35
0,4
0,45
0,5
1 1,005 1,01 1,015 1,02
0,3
0,35
0,4
0,45
0,5
1 1,005 1,01 1,015 1,02
0o
15o
30o
0o
15o
30o
0o
15o
30o
q
a( )
q
q
a( )
a( )
b) c)
0,2
0,3
0,4
0,5
0,6
0,7
1 1,05 1,1 1,15 1,2
0
4
8
12
16
1 1,05 1,1 1,15 1,2
q
a( )a( )
FIGURE 4.10. Imposed de o ma ion g adien pa hs: de ia o ic plas-
ic models (m= 20). a) De ails o he e olu ion o he equi alen Ki ch-
hoff s esses qσin he ange a( )∈[1,1.02], o h ee p esc ibed pa hs
(θ∆εe, ial =0
◦,15
◦and 30◦) and pe ec plas ici y, ha dening and
so ening. b) E olu ion o qσin he ull ange a( )∈[1,1.2] o pe ec
plas ici y, ha dening and so ening models, and θ∆εe, ial =15
◦. (solid
lines o ob ained wi h 500 ime inc emen s, wi h he s a s co espond-
ing o he solu ion ob ained wi h 5 ime inc emen s, bo h ob ained wi h
he p imal–CPPM). c) E olu ion o he Lode angle θσ o he so ening
case and θ∆εe, ial =15
◦.
s a s o he p oblems sol ed wi h jus 5 equal inc emen s. The line sea ch scheme has no
been ac i a ed in he p oblem wi h 500 inc emen s eco e ing hen he New on–CPPM. I
has been ac i a ed wi h he simula ion in ol ing 5 ime inc emen s only. S ill, we obse e
in Figu e 4.10.b a good ag eemen be ween bo h solu ions, hence implying ha he use o
e y la ge elas ic ial s eps does no imply a loss o accu acy.
The con e gence esul s a se e al pseudo imes a e p esen ed in Figu e 4.11 o he
ha dening and he so ening examples, including he e olu ion o he ela i e e o s e(k)
x
(see equa ion (4.3)) in x={εe
n+1,−αn+1,∆γ}. The esul s co espond o he pa hs sol ed
A. P´e ez-Fogue & F. A me o 36
Ha dening (5 inc emen s)
1,E-16
1,E-12
1,E-08
1,E-04
1,E+00
1 3 5 7 9 1113151719
= 0.2
= 0.4
= 0.6
= 0.8
= 1.0
So ening (5 inc emen s)
1,E-16
1,E-12
1,E-08
1,E-04
1,E+00
1357911131517
= 0.2
= 0.4
= 0.6
= 0.8
= 1.0
ex
ex
Ha dening (50 inc emen s)
1,E-16
1,E-12
1,E-08
1,E-04
1,E+00
1357911131517
= 0.2
= 0.4
= 0.6
= 0.8
= 1.0
So ening (50 inc emen s)
1,E-16
1,E-12
1,E-08
1,E-04
1,E+00
1357911131517
= 0.2
= 0.4
= 0.6
= 0.8
= 1.0
ex
ex
Ha dening (500 inc emen s)
1,E-16
1,E-12
1,E-08
1,E-04
1,E+00
1357911131517
= 0.2
= 0.4
= 0.6
= 0.8
= 1.0
So ening (500 inc emen s)
1,E-16
1,E-12
1,E-08
1,E-04
1,E+00
1 3 5 7 9 11 13 15 17
= 0.2
= 0.4
= 0.6
= 0.8
= 1.0
ex
ex
FIGURE 4.11. Imposed de o ma ion g adien pa hs: de ia o ic plas-
ic models (m= 20). Con e gence esul s wi h he p imal–CPPM o
se e al pseudo imes (ha dening and so ening, and 5, 50 and 500 in-
c emen s).
Closes -Poin P ojec ion Algo i hms in Elas oplas ici y 37
So ening, 5 inc emen s ( = 0.2)
0
1
2
13579111315
Line-sea ch pa am.
Num. cu e i ings
Ha dening, 5 inc emen s ( = 0.2)
0
1
2
13579111315
Line-sea ch pa am.
Num. cu e i ings
So ening, 50 inc emen s ( = 0.2)
0
1
2
1 3 5 7 9 11 13 15
Line-sea ch pa am.
Num. cu e i ings
Ha dening, 50 inc emen s ( = 0.2)
0
1
2
13579111315
Line-sea ch pa am.
Num. cu e i ings
FIGURE 4.12. Imposed de o ma ion g adien pa hs: de ia o ic plas-
ic models (m= 20). E olu ion o he line sea ch a =0.2 o he
examples wi h ha dening and so ening, and 5 and 50 ime inc emen s.
wi h 5, 50 and 500 ime inc emen s. In all cases he p imal–CPPM con e ges wi h he a e
o con e gence being asymp o ically quad a ic o each ime inc emen . The e olu ion o
he line sea ch pa ame e and he numbe o cu e fi ings is shown in Figu e 4.12, o
he uns wi h 5 and 50 inc emen s, pseudo ime =0.2, and bo h he ha dening and
so ening p oblem. As occu ed in he examples o Sec ion 4.1, he maximum numbe o
cu e fi ings is wo. The e o e, he compu a ional cos added in each i e a ion by he
conside ed line sea ch scheme is no significan .
We can obse e ha , al hough he numbe o i e a ions pe inc emen inc eases as he
numbe o inc emen s educes, he a io is clea ly a o able o he p oblems wi h less ime
inc emen s. A educ ion o he numbe o inc emen s by a ac o 10 implies a educ ion
o he numbe o accumula ed i e a ions by a ac o o 5, a he leas . This educ ion is
di ec ly ela ed wi h a educ ion o he o al compu a ional cos . The e o e, he p imal–
CPPM is compu a ionally e y efficien , including cases wi h non-cons an elas ici ies.
A. P´e ez-Fogue & F. A me o 38
0
1
2
3
-2 -1 0 1 2 3
s = -1 2 /(3 )
0
15
30
45
60
-2 -1 0 1 2 3
s = 0 2 /(3 )
s = 1 2 /(3 )
s = 2 2 /(3 )
s = -1 2 /(3 )
s = 0 2 /(3 )
s = 1 2 /(3 )
s = 2 2 /(3 )
FIGURE 4.13. Imposed de o ma ion g adien pa hs: p essu e-
dependen plas ic models. E olu ion o he Ki chhoff s esses in he
me idian and he de ia o ic planes o he pa hs defined by =0and
s=−1, 0, 1, 2 ·2µ/(3κ)(wi h =1.58,2.24, 1.58 and 1, espec i ely),
ob ained by he p imal–CPPM wi h 10 ime inc emen s
4.2.2. P essu e-dependen models
We conclude he e alua ion o he p imal algo i hms wi h he conside a ion o he
ounded hype bolic Moh -Coulomb model o imposed pa h o he de o ma ion g adien .
We conside again he pa h defined by he ela ion (4.4), wi h he s e ch unc ion
a( )=1+6·10−5 ∈[0,
],(4.7)
o a final ime (see below) and wi h he olume ic unc ion
b( )=(a( ))3s,(4.8)
allowing o p esc ibe he pa ame e
p∆εe, ial =−s
√ 2− +1
q∆εe, ial
√3,(4.9)
o he elas ic ial s ain inc emen ∆εe, ial in he me idian plane. Hencky’s law is
conside ed in his case which esul s in a linea ela ion be ween he s esses and loga i hmic
s ains. Equa ion (4.9) is equi alen in his case o
s=− 2− +1 2µ
3κ
p∆σ ial
n+1
q∆σ ial
n+1 /√3.(4.10)
whe e ∆σ ial
n+1 =σ ial
n+1 −σn. The e olu ion o he Ki chhoff s esses is depic ed in Figu e
4.13 o he pa hs defined by =0ands= o−1, 0, 1, 2 ·2µ/(3κ)(wi h =1.58,2.24,
Closes -Poin P ojec ion Algo i hms in Elas oplas ici y 39
2 inc emen s
1,E-16
1,E-12
1,E-08
1,E-04
1,E+00
1357911
= 0.5
= 1.0
1 inc emen
1,E-16
1,E-12
1,E-08
1,E-04
1,E+00
1357911
= 1.0
ex
ex
10 inc emen s
1,E-16
1,E-12
1,E-08
1,E-04
1,E+00
1357911
= 0.6
= 0.7
= 0.8
= 0.9
= 1.0
50 inc emen s
1,E-16
1,E-12
1,E-08
1,E-04
1,E+00
1357911
= 0.6
= 0.7
= 0.8
= 0.9
= 1.0
ex
ex
FIGURE 4.14. Imposed de o ma ion g adien pa hs: p essu e-
dependen plas ic models. Con e gence esul s o se e al pseudo imes
o he pa h s=2·2µ/(3κ)( = 1) ob ained wi h he p imal–
CPPM and 1, 2, 10 and 50 inc emen s.
2 inc emen s ( = 1)
0
1
2
1357911
Line-sea ch pa am.
Num. cu e i ings
1 inc emen ( = 1)
0
1
2
1357911
Line-sea ch pa am.
Num. cu e i ings
10 inc emen s ( = 1)
0
1
2
1357911
Line-sea ch pa am.
Num. cu e i ings
FIGURE 4.15. Imposed de o ma ion g adien pa hs: p essu e-
dependen plas ic models. E olu ion o he line sea ch pa ame e a
each las inc emen o he pa h s=2·2µ/(3κ)( = = 1) ob ained
wi h he p imal–CPPM and 1, 2 and 10 inc emen s.
A. P´e ez-Fogue & F. A me o 40
TABLE 4.3. Imposed de o ma ion g adien pa hs: p essu e-
dependen plas ic models. Accu acy o he final esul s ob ained wi h
he p imal–CPPM o he ou pa hs p esen ed in Figu e 4.14 wi h
espec o he e e ence solu ion compu ed wi h 1000 inc emen s.
Numbe inc . Me hod App ox. θεe
n+1 θεe
n+1 e o qεe
n+1 e o pεe
n+1 e o
50 New on–CPPM 57.7◦0◦0.04% 0.01%
10 p imal–CPPM 57.7◦0.005◦0.23% 0.06%
2p imal–CPPM 57.0◦0.7◦0.87% 0.40%
1p imal–CPPM 56.5◦1.2◦1.14% 0.61%
1.58 and 1, espec i ely). The s ess pa hs in he me idian and he de ia o ic planes a e
shown.
The pa h defined by =0ands=2·2µ/(3κ) has been ob ained wi h he p imal–
CPPM and wi h 1, 2, 10 and 50 equal ime inc emen s. The con e gence esul s o se e al
pseudo imes a e shown in Figu e 4.14. Excep o he solu ion ob ained wi h 50 ime
inc emen s, he line sea ch scheme has been ac i a ed in hese solu ions. The cha ac e is ic
asymp o ic quad a ic a e o con e gence is obse ed in all he cases. The alues o he
line sea ch pa ame e , αk, and he numbe o quad a ic cu e fi ings has been depic ed
in Figu e 4.15 o he las ime inc emen ( = 1) o he pa hs sol ed wi h 1, 2 and
10 inc emen s. In all he cases he maximum numbe o cu e fi ings is wo, and he
ac i e cons ain condi ion, equa ion (2.3), has no been ac i a ed. The e o e, he cos
pe i e a ion o he p imal–CPPM is o he same o de han ha o New on–CPPM. In
ac , he numbe o accumula ed i e a ions dec eases wi h he numbe o inc emen s, hence
esul ing in a educed compu a ional cos wi h he use o he p imal–CPPM.
The accu acy o he compu ed final solu ion is assessed in Table 4.3. The same final
de o ma ion g adien has been imposed wi h 1000 inc emen s, wi h he final esul aken
as a e e ence. Al hough he esul s a e less accu a e as less inc emen s a e done, he e o s
a e low (o he same o de as he esul s o Table 4.1) e en o only one inc emen , hus
jus i ying he use o e y la ge ial inc emen s in combina ion o he p imal–CPPM.
5. Nume ical Assessmen : Dual Algo i hms
We e alua e in his sec ion he nume ical p ope ies o he new dual closes -poin
p ojec ion algo i hms p esen ed in Sec ion 3. Mo e specifically, i is ou goal o e i y hei
globally con e gen cha ac e , showing locally an asymp o ic quad a ic a e o con e gence,
and o e alua e he compu a ional cos gi en hei wo-le el s uc u e. Fo b e i y in he
Closes -Poin P ojec ion Algo i hms in Elas oplas ici y 47
c
=0
:
1
=C
MC
c
=0
:
5
=C
MC
c
=0
c
=0
:
01
=C
MC
p
"
e; ial
3
C
MC
co (
)
3
0
p
"
e; ial
3
C
MC
co (
)
3
0
q
"
e; ial
2
p
3
C
MC
5
0
q
"
e; ial
2
p
3
C
MC
50
110 20 30 40 50 60 NC
FIGURE 5.6. Single inc emen es s: p essu e-dependen plas ic
models. Numbe o i e a ions needed o con e gence by he dual–
CPPM (c= 0) and he augmen ed–dual–CPPM o diffe en alues o
he penal y pa ame e c,andθεe, ial =30
◦.
i e a ions is lowe : wi h he p imal–CPPM i is 49, and only 40 wi h he dual–CPPM.
These maximum alues a e eached wi h diffe en ial s a es. The maximum numbe o
cu efi ingsisse ojmax = 3 as in he example in Sec ion 4.1.2. The same o e all
beha io discussed in his sec ion is obse ed on he dependence o his pa ame e in he
nume ical pe o mance o he algo i hms in hese non-smoo h cases.
6. Fini e elemen applica ion
In his sec ion, he p imal–CPPM is applied o sol e he necking o a cylind ical ba ,
as an example o p ac ical applica ion o he algo i hms p esen ed in p e ious sec ions.
This p oblem is a well–known benchma k es in la ge–s ain solid mechanics, see Simo
A. P´e ez-Fogue & F. A me o 48
A
B
C
D
E
F
AC = DF = 26.667 mm
CD = BE = 6.413 mm
FA = 0.99 CD
.
FIGURE 6.1. Necking o a cylind ical ba . P oblem defini ion and
compu a ional mesh.
0
10
20
30
40
50
60
70
80
01234567
Edge displacemen [mm]
Edge eac ion [kN]
on Mises
T esca
FIGURE 6.2. Necking o a cylind ical ba . Load–displacemen cu es
ob ained wi h he on Mises model and he T esca model.
& Hughes [1998] o he analysis o his es wi h he on Mises model, and Miehe
[1998] and Pe i´
c & de Souza Ne o [1999] o he solu ion wi h T esca–like models.
The es consis s in he uniaxial ex ension o a cylind ical ba wi h ci cula c oss–
sec ion, wi h a adius o 6.413 mm and 53.334 mm leng h. A sligh geome ic impe ec ion
(1% educ ion in adius), see Figu e 6.1, induces necking in he cen al pa o he ba .
An axisymme ic analysis is ca ied ou wi h he mesh shown in Figu e 6.1. Eigh –noded
quad ila e al elemen s wi h 2 ×2 Gauss–poin s a e used.
The Hencky’s hype elas ic law and a T esca–like plas ic model ( he on Mises–T esca
Closes -Poin P ojec ion Algo i hms in Elas oplas ici y 49
2
4
6
8
10
12
14
16
18
0
ZOOM
ZOOM
d = 7 mmd = 5.6 mmd = 2.8 mm d = 4.2 mm
FIGURE 6.3. Necking o a cylind ical ba wi h he T esca model and
he p imal–CPPM. Dis ibu ion o numbe o i e a ions a Gauss-poin
le el o ou load inc emen s, op displacemen s equal o 2.8, 4., 5.6
and 7 mm.
model wi h a shape pa ame e m= 20, see Sec ion II.2), a e used he e. The load–
displacemen cu e is depic ed in Figu e 6.2. The esul s ob ained wi h he on Mises model
a e included in he same figu e o compa a i e pu poses. Bo h esul s ag ee wi h hose
p esen ed by Miehe [1998] and Pe i´
c & de Souza Ne o [1999]. The load inc emen s
used in he simula ions a e ma ked on he cu es. Load inc emen s o educed size ha e
been needed in he middle and he end o he es o ensu e con e gence o he p oblem
wi h he T esca model. Wi h he p imal–CPPM he con e gence a Gauss–poin le el
A. P´e ez-Fogue & F. A me o 50
does no limi he size o he load inc emen s, see Figu e 6.3 wi h he dis ibu ion o
numbe o i e a ions a Gauss-poin le el o ou load inc emen s. This is in con as
wi h he New on–CPPM, whe e he size o he load inc emen s is es ic ed by he poo
con e gence p ope ies o he algo i hm a Gauss–poin le el, and, he e o e, he mo e
demanding closes –poin p ojec ion p oblem guides he inc emen al–i e a i e solu ion o
he o e all fini e elemen p oblem.
7. Conclusions
We ha e p esen ed in his pape wo new amilies o algo i hms o he solu ion o
he closes-poin p ojec ion equa ions in elas oplas ici y, e e ed o as p imal and dual
algo i hms. Augmen ed Lag angian ex ensions ha e been conside ed in bo h cases, as well
as ex ensions o he iscoplas ic p oblem. The p oposed algo i hms ha e been e alua ed
in he con ex o mul iplica i e models o fini e s ain iso opic plas ici y, based on gene al
models o fini e hype elas ici y (Hencky’s law and Ogden models), including examples
wi h pe ec plas ici y, s ain ha dening and s ain so ening. This gene al se ing has
been conside ed wi hou equi ing he con exi y assump ions necessa y o he igo ous
heo e ical cha ac e iza ion o he a ia ional s uc u e de eloped in Pa I o his wo k,
and which mo i a ed all hese de elopmen s. Plas ic models based on de ia o ic and
p essu e-dependen yield su aces ha e been conside ed, he la e leading in addi ion o
non-diffe en iable flow ec o s and hus allowing he e alua ion o he nume ical schemes
unde hese condi ions.
The p imal–CPPM consis s o he applica ion o a New on scheme wi h he app op i-
a e line sea ch scheme applied di ec ly o he p imal equa ions o he p oblem in he elas ic
s ains, s ain-like in e nal a iables and plas ic mul iplie ( ha is, he Eule -Lag ange
equa ions in he a ia ional con ex p o ided by he assump ions o con exi y and asso-
cia i i y). The p oposed line sea ch scheme accoun s o he cons ain o posi i e plas ic
mul iplie and, up o he exis ence o spu ious accumula ion poin s shown no o appea in
common ma e ials models (see Rema k I.1.1 in Appendix I o al e na i e modifica ions),
lead in p ac ice o he globally con e gen cha ac e o he p imal–CPPM. The co espond-
ing augmen ed ex ension, he augmen ed–p imal–CPPM, has been p oposed a oiding his
difficul y.
In con as , he dual algo i hms spli he closes -poin p ojec ion equa ions in wo,
leading o wo-le el algo i hms: he uppe le el consis ing o he en o cemen o he plas ic
consis ency condi ion h ough an i e a ion in he plas ic mul iplie , wi h he lowe le el
consis ing o he solu ion o he closes -poin p ojec ion equa ions o a fixed plas ic mul i-
plie . Bo h le els a e sol ed wi h a New on scheme in combina ion o a line sea ch scheme,
assu ing he asymp o ic quad a ic a e o con e gence o bo h i e a ion p ocesses. In pa -
icula , he dual–CPPM, in ol ing a cons ained uppe le el p oblem because again o he
Closes -Poin P ojec ion Algo i hms in Elas oplas ici y 51
non-nega i e cha ac e o he plas ic mul iplie , has been shown no o equi e he ac i a-
ion o special line sea ch schemes, since his cons ain canno be ac i a ed. The esul ing
algo i hms ha e been igo ously p o en o be globally con e gen unde he usual assump-
ions o con exi y. Augmen ed ex ensions, e e ed o as he augmen ed–dual–CPPM, has
s ill been p esen ed o imp o e in hei nume ical pe o mance as summa ized below.
Based on he nume ical esul s p esen ed in Sec ions 4, 5 and 6, we conclude ha all
he newly p oposed algo i hms show a d ama ic imp o emen o e he s anda d New on–
CPPM. The la ge egions o no con e gence obse ed wi h his scheme a e a oided en i ely,
illus a ing he imp o ed global con e gence p ope ies o he p oposed schemes while
main aining locally he desi ed asymp o ic quad a ic a e o con e gence. Focusing fi s
on he p imal–CPPM, we obse e ha his imp o emen comes a no ex a compu a ional
cos o all p ac ical pu poses. In ac , he scheme educes o he New on–CPPM when he
la e does no exhibi difficul ies in he con e gence. The dual–CPPM has shown o be
compu a ionally compe i i e when he p imal–CPPM equi es a la ge numbe o i e a ions
o con e ge, namely, he egions whe e he New on–CPPM exhibi s no con e gence.
The augmen ed e sions o he algo i hms ha e imp o ed p ope ies wi h espec o
he non–augmen ed ones. They do no need o en o ce explici ly he cons ain o non-
nega i e plas ic mul iplie s, leading o uncons ained p oblems easily o ea analy ically
and a oiding al oge he complex line sea ch schemes in p ac ice. Mo eo e , hey ha e
been shown o educe he compu a ional cos o he o iginal algo i hms ( ha is, a educed
numbe o i e a ions) when he egula iza ion pa ame e is chosen in a gi en ange. A
dimensionless exp ession o his ange has been iden ified in he ep esen a i e nume ical
simula ions p esen ed in his pape , and i co e s app oxima ely one o de o magni ude.
This imp o emen in compu a ional cos is mo e significan o he dual–CPPM, esul ing
in he so-called augmen ed–dual–CPPM, e en hough i does no lead o a ully compe -
i i e scheme in egions whe e he p imal–CPPM, o e en he New on–CPPM, show no
difficul ies, as no ed abo e.
In conclusion, he local in eg a ion o he elas oplas ic models, in ol ing commonly
used yield su aces wi h high cu a u e and/o complex elas ic and ha dening/so ening
laws, lead o highly nonlinea equa ions whe e cu en ly a ailable echniques show clea
difficul ies, e en no con e gence unless small load inc emen s a e conside ed. Since his
limi a ion, e en a a single quad a u e poin o a ypical fini e elemen implemen a ion,
hinde s di ec ly he solu ion o he global mechanical bounda y- alue p oblem, i is o
he main in e es o conside obus globally con e gen schemes o he solu ion o he
esul ing closes -poin p ojec ion equa ions. We belie e ha he new algo i hms p esen ed
in his wo k p o ide efficien and compu a ionally compe i i e al e na i es in his espec .
Acknowledgmen s: Financial suppo o his esea ch has been p o ided by he ONR
unde con ac no. N00014-96-1-0818 and he NSF unde con ac no. CMS-9703000 wi h
UC Be keley. A. P´e ez-Fogue was suppo ed by he Gene ali a de Ca alunya (g an
A. P´e ez-Fogue & F. A me o 52
numbe 1998 BEAI200042) and he Commission o Cul u al, Educa ional and Scien ific
Exchange be ween he US and Spain (p og am o 1999, numbe 99258). All his suppo
is g a e ully acknowledged.
Re e ences
A me o, F. & P´
e ez–Fogue , A. [2000], “On he Fo mula ion o Closes -Poin P o-
jec ion Algo i hms o Elas oplas ici y. Pa I: The Va ia ional S uc u e, ” submi ed
o publica ion on In . J. Num. Me h. Eng .
Abbo, A.J. & Sloan, S.W. [1995], “A smoo h hype bolic app oxima ion o he Moh -
Coulomb yield c i e ion”, Compu e s and S uc u es, 54, 427-441.
Be sekas, D.P. [1982] Cons ained Op imiza ion and Lag ange Mul iplie Me hods,
A hena Scien ific, Belmon ( ep in o 1996).
Bi´
cani´
c, N. & Pea ce, C.J. [1996], “Compu a ional aspec s o a so ening plas ici y
model o plain conc e e”, Mechanics o Cohesi e and F ic ional Ma e ials 1, 75-94.
Dennis & Schnabel [1983] Nume ical me hods o uncons ained op imiza ion and non-
linea equa ions, P en ice-Hall, New Je sey ( ep in o 1996).
Luenbe ge , D.G. [1989] Linea and Nonlinea P og amming, Addison-Wesley, Read-
ing.
Miehe, C. [1998], “A o mula ion o fini e elas oplas ici y based on dual co- and con a-
a ian eigen ec o iads no malized wi h espec o a plas ic me ic”, Comp. Me h.
Appl. Mech. Eng ., 159, 223-260.
P´
e ez–Fogue , A., Rod ´
ıguez–Fe an, A. & Hue a, A. [2000a], “Consis en Tan-
gen Ma ices o Subs epping Schemes”, Comp. Me h. Appl. Mech. Eng . in p ess.
P´
e ez–Fogue , A., Rod ´
ıguez–Fe an, A. & Hue a, A. [2000b], “Nume ical di -
e en ia ion o local and global angen ope a o s in compu a ional plas ici y”, Comp.
Me h. Appl. Mech. Eng . 189, 277-296.
Pe i´
c, D. & de Souza Ne o, E. A. [1999], “A new compu a ional model o T esca
plas ici y a fini e s ains wi h an op imal pa ame iza ion in he p incipal space”,
Comp. Me h. Appl. Mech. Eng . 171, 463-489.
Shul z, G.A., Schnabel, R.B. & By d, R.H. [1985], “A amily o us – egion–based
algo i hms o uncons ained minimiza ion wi h s ong global con e gence p ope ies”,
SIAM J. Nume ical Analysis, 22, 47-67.
Simo, J.C. & Hughes, T.J.R. [1998] Compu a ional Inelas ici y, Sp inge , New Yo k.
de Souza Ne o, E.A., Pe i´
c, D. & Owen, D.R.J. [1994], “A Model o Elas oplas ic
Damage a Fini e S ains: Algo i hmic Issues and Applica ions”, Eng . Compu a ion
Closes -Poin P ojec ion Algo i hms in Elas oplas ici y 53
11, 257-281.
A. P´e ez-Fogue & F. A me o 54
Appendix I. Line Sea ch Schemes
We summa ize in his appendix some basic esul s on he o mula ion o solu ion
algo i hms o nonlinea algeb aic sys ems o equa ions as hey a e used in his pape . The
in e es he ein is he de elopmen o globally con e gen algo i hms o he closes -poin
p ojec ion equa ions o elas oplas ici y. To his pu pose, New on algo i hms combined
wi h he p ope line sea ch schemes a e p esen ed o gene al uncons ained and unila e ally
cons ained p oblems in Sec ions I.1 and I.2, espec i ely. Comple e de ails on he classical
esul s p esen ed in his appendix sec ion can be ound in Be sekas [1982], Dennis &
Schnabel [1983] and Luenbe ge [1989].
I.1. Uncons ained p oblems
Conside he gene al uncons ained algeb aic sys em o equa ions
(x)=0,(I.1)
o he unknown ec o x∈Rnx, wi h nx≥1, and he unc ion :Rnx→Rnx.The
gene al p oblem (I.1) co esponds in some o he cases conside ed in his wo k o he fi s
o de necessa y condi ions o he uncons ained minimiza ion p oblem
min
x∈Rnxϕ(x)(I.2)
o a unc ion ϕ(x):Rnx→Rcon inuously diffe en iable wi h (x)=∇ϕ(x) (i s g adi-
en ). In his con ex , we conside a gene al algo i hm o he o m
x(k+1) =x(k)+α(k)d(k)k=0,1,... , (I.3)
o a gi en ini ial alue x(0), he upda e di ec ion d(k)∈Rnxand line sea ch pa ame e
α(k)∈R. We di ec ou a en ion o solu ion algo i hms exhibi ing an asymp o ically
quad a ic a e o con e gence h ough he conside a ion o he New on upda e di ec ion
d(k)=−J(x(k))−1
(x(k)) o J(x):=∇ (x)∈Rnx×nx,(I.4)
so J=∇2ϕin he case gi en by (I.2). Enough egula i y is aci ally assumed o he
defini ion (I.4) o make sense, as i is he in e ibili y o he Jacobian ma ix. The case o
a s ic ly con ex objec i e unc ion ϕ(x), as i appea s in some o he a ia ional p inciples
conside ed in Pa I o his wo k, assu es his las condi ion, gi en he posi i e defini e
cha ac e o J(x) in his case.
Ou main in e es is di ec ed o globally con e gen ex ensions o he pu e New on
scheme (α(k)≡1), assu ing he con e gence o he sequence {x(k)} om any ini ial ial
Closes -Poin P ojec ion Algo i hms in Elas oplas ici y 55
alue x(0) o a solu ion o (I.1). The classical Global Con e gence Theo em (see Luen-
be ge [1989], page 187) assu es his desi able p ope y o bounded sequences {x(k)}
i he algo i hm (I.3) defines a con inuous mapping om x(k) o x(k+1) (closed mapping
in he gene al con ex i mul iple alues x(k+1) exis ) and a con inuous descen unc ion
M:Rnx→Rexis s ( ha is, M(x(k+1))<M(x(k)), “≤”i x(k)is a solu ion). These wo
equi emen s a e sa isfied by he algo i hm (I.3) i he line sea ch pa ame e α((k)>0is
ob ained as he minimiza ion o he me i unc ion
ˆ
M(α):=M(x(k)+αd(k)),(I.5)
defined in e ms o he descen unc ion i sel , and he upda e di ec ion d(k)defines a
descen di ec ion in he sense ha
ˆ
M(0) = ∇M(x(k))·d(k)≤0.(I.6)
P ac ical line sea ch schemes a e ob ained by conside ing an app oxima e solu ion o he
minimiza ion p oblem o he me i unc ion (I.5). In his case, he global con e gence
o he final algo i hm is p ese ed i he so-called Golds ein’s condi ions (Luenbe ge
[1989], page 214)
M(x(k)+α(k)d(k))≤M(x(k))+βα
(k)∇M(x(k))·d(k)(I.7)
M(x(k)+α(k)d(k))≥M(x(k))+(1−β)α(k)∇M(x(k))·d(k),(I.8)
wi h β∈(0,1/2), a e sa isfied. Mo eo e , he alue α(k)= 1 is chosen whene e i e ifies
equa ions (I.7) and (I.8), assu ing a he same ime ha he asymp o ic quad a ic a e
o con e gence o he o iginal New on upda e (I.4) is main ained. The fi s Golds ein
condi ion (I.7) implies he educ ion o he me i unc ion, bounding also he line sea ch
pa ame e α(k)>0 om abo e. The second Golds ein condi ion (I.8) assu es ha his
pa ame e is bounded away om ze o. O he equi alen condi ions can be ound in he
li e a u e.
A descen unc ion o he p oblem (I.1) can be cons uc ed in e ms o he esidual
(x)i sel as
M(x):=1
2 (x)· (x),(I.9)
o he Euclidean inne p oduc “·”inRnx. The descen p ope y (I.6) ollows o his
unc ion and he New on upda e (I.4) a e no ing ha
∇M(x(k))·d(k)=− (x(k))· (x(k))=−2M(x(k))≤0.(I.10)
Fo he minimiza ion p oblem (I.2), an al e na i e descen unc ion is gi en by he simple
choice M(x)=ϕ(x) when his unc ion is s ic ly con ex. Gi en he gene ali y o (I.10),
A. P´e ez-Fogue & F. A me o 56
BOX I.1. A line sea ch scheme o uncons ained p oblems. The
pa ame e s η=0.1andβ=10
−4a e conside ed in he nume ical
simula ions p esen ed in his pape .
1. Inpu da a: x(k), (k)and d(k).
2. Ini ialize: se j=0,α(k)
(0) =1, ˆ
M(k):= (k)· (k)/2
and ˆ
M(k):= −2ˆ
M(k).
3. Compu e he new unknowns, esiduals and me i unc ion:
x(k+1)
(j):= x(k)+α(k)
(j)d(k)
(k+1)
(j):= (x(k+1)
(j))
ˆ
M(k+1)
(j):= (k+1)
(j)· (k+1)
(j)/2
4. Check Golds ein’s condi ion:
IF ˆ
M(k+1)
(j)≤1−2βα
(k)
(j)ˆ
M(k)THEN se x(k+1)
=x(k+1)
(j)and EXIT.
5. Check o maximum numbe o quad a ic cu e fi ings:
IF j=jmax THEN no i y, se x(k+1) =x(k+1)
(j)and EXIT.
6. Compu e new alue o line sea ch pa ame e :
α(k)
(j+1) := MAX
ηα
(k)
(j),−α(k)
(j)2ˆ
M(k)
2ˆ
M(k+1)
(j)−ˆ
M(k)−α(k)
(j)ˆ
M(k)
7. Se j←j+1andGO TO 3.
we ha e chosen he unc ion (I.9) in he de elopmen s ha ollow, e en i his con exi y
p ope y holds.
Se e al i e a i e echniques can be ound in he li e a u e o he cons uc ion o line
sea ch schemes sa is ying he condi ions (I.7)-(I.8). In his wo k we conside a cu e fi ing
echnique, consis ing o a quad a ic fi o he me i unc ion ˆ
M om he alues ˆ
M(k),ˆ
M(k)
and ˆ
M(k)
(j)a he i e a ion alue α(k)
(j), wi h he new alue α(k)
(j+1) gi en by he minimum o
he esul ing quad a ic cu e. This p ocess is epea ed o a fini e numbe o i e a ions
(j=0,1,...,j
max) un il he fi s Golds ein condi ion (I.7), which eads in his case
ˆ
M(α(k)
(j))=: ˆ
M(k)
(j)≤1−2βα
(k)
(j)ˆ
M(k),(I.11)
Closes -Poin P ojec ion Algo i hms in Elas oplas ici y 63
q y o
1
T esca
(m= )
m=
on Mises
(m =1)
m =1
m =5
m =10
m =20
on Mises - T esca
1
23
FIGURE II.1. T ace on he de ia o ic plane o he on Mises–T esca
ype yield su ace o diffe en alues o m.
pu pose, we conside wo yield unc ions, one in ol ing a de ia o ic plas ic model cha ac-
e is ic o me al plas ici y and a p essu e-dependen yield su ace ypical in geomechanics
applica ions. Associa ed plas ic e olu ions a e conside ed in bo h cases.
1. De ia o ic models. We conside he on Mises–T esca ype model defined by he
gene al o mula (see Miehe [1998])
(σ,q)=2m−1
2m
√33
i=1 [σ]i−[σ]mod(i+1,3)2m1
2m
−&2
3(σyo−q),(II.5)
o he ini ial yield limi σyoand he s ess-like in e nal a iable q=−∂αψh o he
conside ed iso opic s ain ha dening/so ening. The ma e ial pa ame e mde e -
mines he shape o he yield su ace in he s ess de ia o ic plane, eco e ing he on
Mises ci cle o m= 1 and he T esca hexagon o m→∞. The elas ic limi a
θσ=0
◦and 60◦is he yield s ess alue qσ=σyo o all alues o m. See Figu e
II.1 o an illus a ion. High alues o he cu a u e o he yield su ace a e ob ained
o high alues o he pa ame e m. Fo he alue m= 20, he diffe ence be ween he
yield s ess a θσ=30
◦be ween his yield su ace and he T esca hexagon is less han
2%. The combina ion o he yield su ace (II.5) wi h he egula ized elas ic models o
he p e ious sec ion leads o closes -poin p ojec ion equa ions in he de ia o ic s ess
plane only (i.e. in s:= σ−3
i=1[σ]i), wi h a pu ely elas ic olume ic esponse. The
final algeb aic equa ions conside ed in his pape ha e been implemen ed in his plane
o his case (i.e. he esidual associa ed wi h he flow ule is w i en in e ms o he
de ia o ic pa o he elas ic s ains de [εe]:=εe−3
i=1[εe]i).
A. P´e ez-Fogue & F. A me o 64
cos (
p an ( CM
C
q CMC
FIGURE II.2. T ace on he de ia o ic and me idian s ess planes o
he Rounded Hype bolic Moh –Coulomb yield su ace.
2. P essu e-dependen yield unc ions. We conside he Rounded Hype bolic Moh –
Coulomb (RMHC) yield unc ion (Abbo & Sloan [1995])
(σ)=J2(σ)K2(θσ)+(0.05 CMC cos(φ))2−(pσsin(φ)+CMC cos(φ)) ,(II.6)
o he ic ional angle φand cohesion CMC , and whe e he unc ion K(θσ) is defined
as
K(θσ)=
A1−A2sin(φ)+(B1sin(φ)−B2)cos(3θσ)θσ≤5◦
3 + sin(φ)
2√3cos(θσ)+1−sin(φ)
2sin(θσ)5
◦≤θσ≤55◦
A1+A2sin(φ)+(B1sin(φ)+B2)cos(3θσ)θσ≥55◦
(II.7)
wi h
A1=cos(25
◦)+2+√3
3sin(25◦)
A2=2+√3
3√3cos(25◦)−1
√3sin(25◦)
B1=2√2
3(3 −√3) cos(25◦)
B2=2√2
3(√3−1) sin(25◦).
(II.8)
The RMHC su ace (II.6) has been conside ed in Abbo & Sloan [1995,96] and
P´
e ez–Fogue e al. [2000a,b] in he con ex o infini esimal plas ici y. Figu e
II.2 depic s an illus a ion o he ace o his yield su ace wi h he me idian and
de ia o ic planes. The unc ion (II.6) is con inuously diffe en iable o all s ess s a es
Closes -Poin P ojec ion Algo i hms in Elas oplas ici y 65
(Abbo & Sloan [1995]), leading o a con inuous flow ec o e e ywhe e. Howe e ,
he second de i a i e o (II.6) in σis no con inuous a θσ=5
◦,θσ=55
◦,and
he e o e nei he a qσ= 0. This ac , and he high cu a u e a some he ounded
zones, causes he New on–CPPM no o con e ge in la ge egions o he ial s a e
space, in con as wi h he new schemes p oposed in his wo k.
II.3. Ha dening/so ening laws
The nume ical examples p esen ed in Sec ions 4and 5 conside pe ec plas ici y as
well as iso opic s ain ha dening and s ain so ening. These las wo cases a e conside ed
in combina ion wi h he de ia o ic yield unc ion (II.5), wi h he sa u a ion po en ial
ψh(α)=(σy∞−σyo)$α+1
δexp(−δα)%
=⇒q=−∂αψh=−(σy∞−σyo)(1−exp(−δα)) ,(II.9)
o he ini ial yield limi σyo, he sa u a ion yield s ess σy∞ he sa u a ion yield s ess
and he sa u a ion exponen δ. Pe ec plas ici y is conside ed only wi h he p essu e-
dependen yield su ace (II.6).
