A s udy on ini e elemen s o cap u ing s ong
discon inui ies
J. Oli e ∗A. E. Huespe †
E. Samaniego ‡
E.T.S. Enginye s de Camins, Canals i Po s,
Technical Uni e si y o Ca alonia
Campus No d UPC, M`odul C1, G an Capi ´an s/n
08034 Ba celona, Spain
Ma ch 2, 2002
Abs ac
The wo k ocuses on he p esen ly exis ing amilies o ini e elemen s wi h embedded
discon inui ies and explo es he possibili ies o ob ain symme ic s a ically consis en i-
ni e elemen s ha alle ia e he s ess locking p oblem. Fo his pu pose, mixed ( educed
in eg a ion) and assumed enhanced s ain echniques a e applied o he basic symme ic
ou -noded elemen . Nume ical simula ions show he e ec i eness o he p oposed mea-
su es.
keywo ds: ini e elemen s , embedded discon inui ies, s ong discon inui ies,
mixed elemen s, assumed enhanced s ains
1 Mo i a ion
In ecen yea s, he so called ini e elemen s wi h embedded discon inui ies ha e been objec
o inc easing s udy and de elopmen [5], [8], [10], [24], [21], [13], [1], [9], [3] ,[2], [27], [23]. I s
∗P o esso . e-mail: oli [email p o ec ed]c.es
†Resea che om: CIMEC/ CONICET-UNL, A gen ina. e-mail: ahuespe@in ec.unl.edu.a
‡Resea ch assis an
1
ising popula i y comes om he ac ha , by using hese elemen s, a displacemen discon inui y
model can be in oduced in he bulk o he elemen (and he e o e ega dless he o ien a ion
o i s sides) in combina ion wi h app op ia e p opaga ion mechanisms. Mo eo e , i has been
shown [13] ha in s ain localiza ion scena ios, some o hose elemen s comple ely o e come he
well known p oblem o he spu ious mesh size and mesh o ien a ion dependence o he esul s
[4].
Al hough he e a e qui e di e en amilies o such elemen s, appa en ly, no all hem beha e
he same way. A ai ly comp ehensi e s udy o he di e en amilies can be ound in [6]. They
can be classi ied in h ee di e en g oups:
I) Symme ic s a ically consis en elemen s. T ac ion con inui y ac oss he discon inui y in e -
ace is in oduced in a ully consis en a ia ional en i onmen ha esul s in a symme ic
o mula ion. Howe e he way o in oducing he discon inuous kinema ics does no gua -
an ee ee igid body ela i e mo ions o he wo po ions o he elemen spli up in o by
he discon inui y1. A desc ip ion o such o mula ion can be ound in [9].
II) Symme ic kinema ically consis en elemen s. The kinema ics is in oduced in a way ha
does no es ic he igid body ela i e mo ions o he wo po ions o he elemen . How-
e e , ac ion con inui y ac oss he discon inuous in e ace is no gua an eed a elemen al
le el. A ypical iangula elemen based on his o mula ion can be ound in [10].
III) Non-symme ical s a ically and kinema ically consis en elemen s. Bo h he igid body
ela i e mo ions and he ac ion con inui y a e in oduced a he elemen al le el, bu
he second is in oduced in a s ong o m ha makes he esul ing o mula ion non-
symme ical. Typically, hese a e he o mula ions used in [13] and [1].
F om he au ho s expe iences he las amily o elemen s iii) is he one ha p o ides mo e
obus an eliable esul s and hey ha e used i success ully o nume ical simula ions in many
di e en se ings [12], [16], [17], [14], [18] ,[19], [20]. Family ii) exhibi s also a obus beha iou
al hough a much slowe con e gence wi h mesh e inemen whe eas amily i), as i will be shown
below, exhibi s in many cases he well known s ess locking phenomenon.
Howe e , al hough he ac ha o mula ion iii) is no symme ic is nei he a undamen al
d awback no a sou ce o unbea able compu a ional cos s, he a ia ional consis ency o o mu-
la ion i) ag ees wi h he adi ional ini e elemen echnology in Compu a ional Solid Mechanics
1As i will be shown in subsequen sec ions his can esul in s ess locking beha iou .
2
and con e s addi ional appeal upon i . Mo e impo an , he symme ic elemen s o amily i)
can be heo e ically de i ed wi hou necessi y o he acking algo i hm concep which appea s
o be a undamen al issue on he o he wo amilies and se s p ac ical di icul ies o gene alize
i s use o cap u ing mul iple c acking and b anching phenomena.
This is why his pape is de o ed o explo e and de elop se e al new ypes o ini e elemen s
wi h embedded discon inui ies belonging o his amily. Howe e , and since he a o emen ioned
s ess locking phenomena makes he basic o mula ion unsui able, speci ic ea men s a e ex-
plo ed o ace ha p oblem. Mixed and assumed enhanced s ain echniques a e en isaged as
app op ia ed emedies, so ha hey a e in oduced in he basic elemen and hei e ec s on he
s ess locking a e analyzed.
1.1 The Bounda y Value p oblem
Le us conside he body Ω o igu e 1-a, unde going a ( a e o ) displacemen discon inui y
[[ ˙u]](x, ) ac oss he ma e ial ( ixed) su ace S ha spli s he body in o Ω+(poin ed by he uni
no mal n o S) and Ω−such ha Ω+∪Ω−= Ω S2. The esul ing eloci y, ˙u(x, ), and s ain
a e, ˙ε(x, ) , ields3can be exp essed as in [19]:
˙u(x, ) = .
¯u(x, ) + MS[[ ˙u]](x, );
[[ ˙u]] = ˙u|x∈∂Ω+∩S −˙u|x∈∂Ω−∩S ;.
¯u =˙u∗in ∂uΩ
˙ε(x, ) = ∇S˙u(x, ) = .
¯ε
|{z}
egula
(bounded)
+δS([[ ˙u]] ⊗n)S
| {z }
singula
(unbounded)
(1)
MS(x) = HS(x)−ϕ(x) ; ϕ(x)∈H1(Ω) = 


1∀x∈Ω+ Ωh
0∀x∈Ω− Ωh
(2)
whe e [[ ˙u]] s ands o he eloci y jump, ∂uΩ is he pa o he ex e nal bounda y o Ω (wi h
ou wa d no mal ν) whe e displacemen s a e p esc ibed o u∗(x, ), HSis he Hea iside’s jump
unc ion placed on S(HS(x) = 1 ∀x∈Ω+and HS(x)=0 ∀x∈Ω−), MSis a uni jump
unc ion, whose suppo is a ce ain domain Ωhcon aining S(see igu e 1-(b)) and cons uc ed
2No a ion A Bs ands o he esul o subs ac ion o domain B om domain A.
3The ma hema ical exp essions o he esul ing con inuum o ma kinema ics is e e ed o as s ong discon-
inui y kinema ics [19].
3
as i is indica ed in equa ion (2), and δSis he Di ac’s del a unc ion placed on S ha a ises
om he de i a ion o he discon inuous unc ion MS(x).
The co esponding quasis a ic Bounda y Value P oblem (B.V.P.) can be desc ibed, in a e
o m, as he ollowing h ee ield, u−ε−σ, p oblem:
FIND :








˙u(x, )
˙ε(x, )
˙σ(x, )
sa is ying:
∇ · ˙σ−˙
b=0in Ω S (in e nal equilib ium) (a)
˙ε− ∇S˙u =0in Ω (kinema ical compa ibili y) (b)
˙σ−˙
Σ(ε) = 0in Ω (cons i u i e compa ibili y) (c)
˙σ·ν=˙
ex on ∂σΩ (ex e nal equilib im) (d)
˙σΩ+·n−˙σΩ−·n
| {z }
de
= [[ ˙σ]]Ω S ·n
=0on S(ou e ac ion con inui y (e)
˙σΩ+·n−˙σS·n
| {z }
de
= [[ ˙σ]]S·n
=0on S(inne ac ion con inui y) ( )
(3)
whe e σ(x, ) s ands o he s esses, b(x, ) a e he speci ic body o ces, Σ(ε) s ands o he
cons i u i e unc ion e u ning he s esses in e ms o he s ains εand ∂σΩ is he pa o
he bounda y o Ω whe e ac ions ex a e p esc ibed. Equa ions (3)-(e) and (3)-( ) s a e he
con inui y o he ac ion ec o T=σ·n h ough he discon inui y in e ace S. A e explici
imposi ion o condi ion (3)-(c) B.V.P. (3) can be ew i en as a wo ield, u−ε, p oblem: :
FIND :


˙u(x, )
˙ε(x, )
sa is ying:
∇ · ˙
Σ−˙
b=0in Ω S (in e nal equilib ium) (a)
˙ε−∇S˙u =0in Ω (kinema ical compa ibili y) (b)
˙
Σ·ν=˙
ex on ∂σΩ (ex e nal equilib im) (c)
˙
ΣΩ+·n−˙
ΣΩ−·n
| {z }
de
= [[ ˙
Σ]]Ω S ·n
=0on S(ou e ac ion con inui y (d)
˙
ΣΩ+·n−˙
ΣS·n
| {z }
de
= [[ ˙
Σ]]S·n
=0on S(inne ac ion con inui y) (e)
(4)
4
2 Non-symme ic (Pe o -Gale kin) app oach
A weak o m o he B.V.P. (4) can be de ised as ollows. In iew o he eloci y ield (1) le
us conside he unc ional spaces o he eloci ies, Vu, and i ual (kinema ically admissible)
eloci ies, ¯
V0
u:
Vu
de
≡©η(x) = ¯η+MSα;η∈[H1(Ω)]ndim ;α∈L2(S)ª
¯
V0
u
de
≡©¯η0(x)∈[H1(Ω)]ndim ;¯η0|∂uΩ=0ª(5)
whe e ndim s ands o he numbe o dimensions o he p oblem, H1(Ω) is he unc ional space
o unc ions de ined in Ω wi h squa e in eg able i s de i a i es and L2(S) he unc ional space
o squa e in eg able unc ions de ined in S4.
Rema k 1 No ice ha spaces Vuand ¯
V0
udi e no only o he homogeneous bounda y condi-
ion imposed o he elemen s o ¯
V0
u, bu also o he in insic na u e o hei elemen s η(x) =
¯η+MSαand ¯η0(x) espec i ely. This ac en ails he nonsymme ic cha ac e o he esul ing
o mula ion.
In iew o he p eceding no a ion he weak o m o he p oblem can be w i en:
PROBLEM 1 CONTINUOUS NON-SYMMETRIC PROBLEM
FIND:
˙u =.
¯u +MS[[ ˙u]]
.
;˙u ∈ Vu
˙ε=∇S˙u =.
¯ε+δs([[ ˙u]]⊗n)S(6)
SUCH THAT:
δΠu(˙
Σ;η) = ZΩ S
˙
Σ(∇S˙u) : ∇SηdΩ−ZΩ S
˙
b·ηdΩ + Z∂σΩ
˙
·ηdΓ
| {z }
Gex
= 0 ∀η∈¯
V0
u(7)
Some s anda d calcula ions show ha he s ong o m o P oblem 1 is:
δΠu(u;η) = 0 ⇒
∇ · ˙
Σ−˙
b=0in Ω S
˙
Σ·ν=˙
ex on ∂σΩ
[[ ˙
Σ]]Ω S ·n= 0 on S
(8)
4Roughly H1(·) con ains con inuous unc ion de ined in (·) wi h discon inuous i s de i a i es and L2(·)
con ains piecewise discon inuous bounded unc ions de ined in (·).
5

and, he e o e, only condi ion (4)-(e) emains o be ul illed in he B.V.P. de ined by equa ions
(4). This condi ion is going o be imposed ia a di e en p ocedu e in he ini e elemen
o mula ion, essen ially in s ong o m. In summa y he B.V.P. (4) is app oached as:
δΠu(˙
Σ(.
¯u,[[ ˙u]]); η) = 0 ∀η∈¯
V0
u( a ia ional/weak o m)
[[ ˙
ΣS]] ·n=0on S(s ong o m)
(9)
2.1 Fini e elemen disc e iza ion (s anda d non-symme ic elemen :
U4n)
Le us conside he ma e ial domain Ω disc e ized in a ou -noded5 ini e elemen mesh wi h
nelem elemen s and nnode nodes c ossed by he discon inui y in e ace S(see igu e 2-(a)). Le
us assume ha an a ailable discon inui y acking algo i hm [13] de e mines he subse Jo
he nJelemen s ha a e c ossed by Sa he conside ed ime :
J:= {e|Ωe∩ S 6=∅} ={ei,...., em,....ep, ...}(10)
Fo e e y elemen o J, he acking algo i hm6also p o ides he posi ion o he elemen al
discon inui y in e ace Se(see igu e 2-(b) o leng h lewhich de ines he domains Ω+
eand Ω−
e
and he nodes i+∈ {i+
1, .., i+
n+
e}and i−∈ {i−
1, .., i−
n−
e}. Now, he ollowing in e pola ion o he
eloci y ield ˙u(e)inside a gi en elemen eis conside ed[13]:
˙u(e)(x, ) = Σi=4
i=1 N(e)
i(x)˙
di( )
| {z }
.
¯u(e)
+M(e)
S(x) [[ ˙u]]e( )
| {z }
.
˜u(e)
(11)
whe e .
¯u(e)is he s anda d C0 eloci y ield, in e pola ed by he shape unc ions {N(e)
1, N(e)
2,
N(e)
3, N(e)
4}o he linea isopa ame ic quad ila e al elemen [28], in e ms o he nodal eloci ies
di( ) a node i. The e m
.
˜u(e), in equa ion (11), cap u es he singula (discon inuous) pa o
he eloci y ield (1) in e ms o he elemen al eloci y jump [[ ˙u]]eand Me
S(x) is he disc e e
coun e pa o he uni jump unc ion in equa ion (2) de ined as ollows:
M(e)
S(x) = 








0∀e6∈ J
H(e)
S(x)−ϕ(e)
(ϕ(e)= Σn+
e
i+=1Ni+)


∀e∈ J (12)
5Fo he sake o simplici y, om now on only wo-dimensional p oblems will be conside ed.
6This acking algo i hm is a c ucial ing edien o he non-symme ic o mula ions and cons i u e one o hei
mos ypical se i udes.
6
whe e H(e)
Sis he s ep unc ion. Figu e 2-(c) shows he M(e)
S unc ion and emphasizes i s
elemen al suppo .
F om equa ions (11) and (12), he disc e e ( a e o ) s ain ield eads:
˙ε(e)=∇S˙u(e)= Σi=4
i=1 (∇N(e)
i⊗˙
di)S−(∇ϕ(e)⊗[[ ˙u]]e)S+δS([[ ˙u]]e⊗n)S(13)
No ice ha equa ion (13) ma ches he s ong discon inui y kinema ics (1). In o de o o e -
come he nume ical di icul ies o ea ing wi h he Di ac’s del a unc ion δSis eplaced by a
egula ized7 unc ion δe
Sde ined wi hin he elemen eas:
δ(e)
S=µ(e)
S
1
k(14)
whe e µ(e)
Sis a colloca ion unc ion whose suppo is he domain Sk
ein igu e 2-(d) de ined in
e ms o he egula iza ion pa ame e k:
µ(e)
S(x) = 1 ∀x∈ Sk
e
µ(e)
S(x) = 0 ∀x/∈ Sk
e
(15)
By conside ing equa ions (14) and (15) he egula ized o m o he s ain a e ield eads:
˙ε(e)=∇S˙u(e)= Σi=4
i=1 (∇N(e)
i⊗˙
di)S−(∇ϕ(e)⊗[[ ˙u]]e)S+µ(e)
S
1
k(n⊗[[ ˙u]]e)S(16)
In o de o in eg a e he discon inuous e ms eme ging om he second e m o he igh -
hand-side o equa ion (16), in addi ion o he s anda d sampling poin s o he linea quad ila -
e al (PG1 o PG4 in igu e 2-(e)), he elemen is equipped wi h ano he in eg a ion poin (SSP
in igu e 2-(d)) placed a he cen e o he elemen and whose associa ed a ea is (see igu e
2-(d)) :
measu e(Sk
e) = kle(17)
The egula iza ion pa ame e khas an a bi a y small alue (as small as pe mi ed by he
machine p ecision).
In his con ex he inne ac ion con inui y condi ion in equa ions (4)-(e) and (9) can be
imposed on an elemen basis in e ms o a e ages:
˙
ΣΩ+·n= ( ˙
ΣΩ−·n) = ˙
ΣS·n→1
ΩeZΩe
˙
Σ·ndΩ
| {z }
mean alue on Ω S
=1
leZSe
˙
Σ·ndΩ
| {z }
mean alue on S
⇒(18)
7This p ocedu e and he esul ing o mula ion ha e been some imes e med in he li e a u e he egula ized
s ong discon inui y app oach.
7
⇒ZΩe
(µ(e)
S
1
k−le
Ωe
)n·˙
ΣdΩ = 0∀e∈ J (19)
In iew o he p e ious ini e elemen disc e iza ion, and by eso ing o he classical ini e
elemen p ocedu es, he disc e e coun e pa o he B.V.P. in equa ions (9) can be w i en as
ollows:
PROBLEM 2 DISCRETE NON-SYMMETRIC PROBLEM
GIVEN:
Vh
u
de
≡nηh(x) = Σi=nnode
i=1 Ni(x)ηi+ Σe∈J M(e)
S(x)αeo
¯
Vh0
u
de
≡nηh0(x) = Σi=nnode
i=1 Ni(x)η0
i;η0
i|∂uΩ=0o(20)
FIND:
˙uh= Σi=nnode
i=1 Ni˙
di+ Σe∈J M(e)
S(x) [[ ˙u]]e;˙u h∈ Vh
u
˙εh=∇S˙uh= Σi=nnode
i=1 (∇Ni⊗˙
di)S+ Σe∈J ³[µ(e)
S1
kn− ∇ϕ(e)]⊗[[ ˙u]]e´S(21)
SUCH THAT:
δΠu(˙
Σ;ηh) = Pe=nelem
e=1 RΩe∇Sηh:˙
Σ(εh)dΩ−Gex = 0 ∀ηh∈¯
Vh0
u
[[ ˙
Σ]]S·n=0→RΩe(µ(e)
S1
k−le
Ωe)n·˙
ΣdΩ = 0∀e∈ J (22)
The s uc u e o equa ions (22) co esponds o a ypical Pe o -Gale kin esidual weigh ing
p ocedu e [28] o he o iginal B.V.P. (4).
2.1.1 2D implemen a ion
Fo he wo-dimensional case, in a ca esian coo dina e sys em (x, y), using he ec o o ma
o he s ains {ε}= [εxx, εyy,2εxy]Tand he s esses {σ}= [σxx, σyy,σxy]T(whe e (·)Ts ands
o he anspose o (·)), conside ing he ou noded quad ila e al as unde laying elemen and
using he s anda d ini e elemen B- o ma [28] equa ions (22) yield:
δΠu(˙
Σ;ηh) = 0
[[ ˙
Σ]]S·n=0


→
e=nelem
[
e=1 ·ZΩe
B(e)T·˙
Σ({ε}(e))dΩ−˙
Fex (e)¸=0(23)
whe e ˙
Fex (e)s ands o he classical elemen al ex e nal o ces ec o and:
{˙ε}(e)=B∗(e)·˙
d(e)
˙
d(e)=h˙
d1,˙
d2,˙
d3,˙
d4,[[ ˙u]]eiT(24)
8
B(e)=hB(e)
1,B(e)
2,B(e)
3,B(e)
4,G(e)i
B∗(e)=hB(e)
1,B(e)
2,B(e)
3,B(e)
4,G∗(e)i
B(e)
i=




∂xN(e)
i0
0∂yN(e)
i
∂yN(e)
i∂xN(e)
i





G(e)= (µ(e)
S1
k−le
Ωe)




nx0
0ny
nynx





G∗(e)=µ(e)
S1
k





nx0
0ny
nynx





−




∂xϕ(e)0
0∂yϕ(e)
∂yϕ(e)∂xϕ(e)





(25)
whe e Ss ands o he classical assembling ope a o and n= [nx, ny]T.
Rema k 2 No ice ha ma ices B(e)and B∗(e)in equa ion (25) di e in he e ms G(e)6=
G∗(e).This ac makes he esul ing angen s i ness ma ix non-symme ical as could be ex-
pec ed om he o iginal non-symme ic cha ac e o he app oach s a ed in Rema k 1.
Rema k 3 The s uc u e o equa ions (23) o (25) sugges s he in oduc ion o an in e nal
addi ional i h node o each elemen e, ha is ac i a ed only o he elemen s c ossed by he
discon inui y in e ace (e∈ J ) and whose co esponding deg ees o eedom and associa ed
shape unc ion a e, espec i ely, he displacemen jumps [[u]]eand M(e)
Sin equa ions (11) and
(12). Since he suppo o M(e)
Sis only Ωe, hose in e nal deg ees o eedom can be e en ually
condensed a he elemen al le el and emo ed om he global sys em o equa ions.
3 Symme ic assumed enhanced s ain app oach
The assumed enhanced s ain me hods, which can be conside ed a pa icula case o he mo e
gene al assumed s ain me hods o mixed me hods [25], p o ide a di e en se ing o app oach
displacemen discon inui ies. In nex sec ions he co esponding o mula ion is p esen ed ol-
lowing he guidelines in ([26]).
9
a) Dec ease he polynomial deg ee o he e m ∇S˙u(e) o ze o (cons an ).
b) Inc ease he polynomial deg ee o he enhancemen e m o one (linea polynomial).
S a egy a) is ollowed in he mixed app oach p esen ed in Sec ion 5 whe eas s a egy b)
leads o he s ain e-enhancemen me hodology p esen ed in Sec ion 612.
5 Mixed app oach
5.1 Assumed s ain and s ess ields
Le us conside he ollowing unc ional spaces:
Vu
de
≡©η(x)∈[H1(Ω)]2ª(a)
V0
u
de
≡©η0(x)∈[H1(Ω)]2;η0|∂uΩ=0ª(b)
Vε
de
≡ {ξ(x) = ¯
ξ+δS(α⊗n)S;¯
ξij ∈L2(Ω) ; αi∈L2(S)}(c)
Vσ
de
≡nτ(x) ; τij ∈L2(Ω ) ; [[τ]]Ω S ·n= [[τ]]S·n=0o(d)
(52)
The a ia ional h ee ield (u−ε−σ) mixed p oblem can we w i en as:
PROBLEM 5 CONTINUOUS MIXED PROBLEM
FIND:
˙u(x, )˙u ∈ Vu
˙ε(x, ) = .
¯ε+δs(˙
β⊗n)S.
ε∈ Vε
˙σ(x, )˙σ∈ Vσ
(53)
SUCH THAT
δΠu(˙σ;η) = RΩ S ˙σ:∇SηdΩ−Gex = 0 ∀η∈ V0
u
δΠε(˙u,˙ε;τ) = RΩ¡˙ε− ∇S˙u¢:τdΩ = 0 ∀τ∈ Vσ
δΠσ(˙σ,˙
Σ;ξ) = RΩ³˙σ−˙
Σ´:ξdΩ = 0 ∀ξ∈ Vε
(54)
S anda d calcula ions on equa ions (54) lead o he co esponding s ong o ms:
i)
δΠu(˙σ;η) = 0 ⇒
∇ · ˙σ−˙
b=0in Ω S
˙σ·ν=˙
on ∂σΩ
[[ ˙σ]]Ω S ·n=0on S
(55)
12Indeed, hese s a egies o alle ia ing he s ess-locking p oblem can only be applied o ini e elemen s
sensi i e o mixed /assumed s ain o assumed enhanced s ain echnologies, like he ou -noded quad ila e al.
Fo ins ance, s a egies a) and b) canno be applied, a elemen al le el, o linea iangles.
16

ii)
δΠε(˙u,˙ε;τ) =0 ⇒RΩ¡˙ε− ∇S˙u¢:τdΩ = 0 ⇒˙ε=∇S˙u in Ω(56)
iii)
δΠσ(˙σ,˙
Σ;ξ) = 0 ⇒RΩ³˙σ−˙
Σ´:ξdΩ⇒˙σ=˙
Σin Ω(57)
The e o e, equa ions (3)-(a) o (3)-(e) o he o iginal B.V.P. a e ul illed in weak o m
by he a ia ional equa ions (54) whe eas equa ion (3)-( ) is ul illed om he pa icula
choice made o he space Vσin equa ion (52)-(d).
5.2 Fini e elemen disc e iza ion (cons an s ess/s ain mixed ele-
men : M4n)
Fo he 4-noded elemen disc e iza ion o igu e 2-(a) le us conside he ollowing disc e e
coun e pa o he spaces in equa ion (52):
Vh
u
de
≡©ηh(x) = Σi=nnode
i=1 Ni(x)ηiª(a)
V0
u
de
≡©ηh(x) =Σi=nnode
i=1 Ni(x)ηi;ηi|∂uΩ=0ª(b)
Vh
ε
de
≡ {ξh(x) =Σe=nelem
e=1 χe(x)(¯
ξe+µ(e)
S1
k(αe⊗n)S)};χe(x) = 


1 o x∈Ωe
0 o he wise
(c)
Vh
σ
de
≡ {τh(x) =Σe=nelem
e=1 χe(x)τe}(d)
(58)
and he co esponding disc e e p oblem:
PROBLEM 6 DISCRETE MIXED PROBLEM
FIND:
˙uh= Σi=nnode
i=1 Ni˙
di;˙uh∈Vh
u(a)
˙εh= Σe=nelem
e=1 (χe(x)˙
¯εe+µ(e)
S1
kχe(x)(˙
βe⊗n)S).
εh∈Vε(b)
˙σh=Σe=nelem
e=1 χe(x).
σe˙σh∈Vσ(c)
(59)
SUCH THAT:
δΠu(˙σ;ηh=RΩ˙σh:∇SηhdΩ−Gex = 0 ∀ηh∈Vh0
u(a)
δΠε(˙uh,˙εh;τh=RΩ³˙εh− ∇S˙uh´:τhdΩ = 0 ∀τh∈Vh
σ(b)
δΠσ(˙σh,˙
Σh;ξh) = RΩ³˙σh−˙
Σh´:ξhdΩ = 0 ∀ξh∈Vh
ε(c)
(60)
17
By inse ing he s ain a e ield ˙εho equa ion (59)-(b) in o equa ion (60)-(b) one ge s:
δΠε(˙uh,˙εh) = 0 ⇒
e=nelem
X
e=1 ZΩe³˙εh− ∇S˙uh´:τedΩ = (61)
=
e=nelem
X
e=1 µΩe
.
¯εe+le(˙
βe⊗n)S−ZΩe
∇S˙u hdΩ¶:τe= 0 ∀τe⇒
⇒µΩe
.
¯εe+le(˙
βe⊗n)S−ZΩe
∇S˙u hdΩ¶= 0 e∈ {1...nelem} ⇒
and sol ing o .
¯εeand, hen, o .
¯εhin equa ion (59)-(b):
.
¯εe=1
ΩeZΩe
∇S˙uhdΩ−le
Ωe
(˙
βe⊗n)Se∈ {1...nelem} ⇒ (62)
.
¯εh=
e=nelem
X
e=1
χe(x)³1
ΩeZΩe
∇S˙uhdΩ
| {z }
de
=∇S˙u(e)
−(le
Ωe
−µ(e)
S
1
k)( ˙
βe⊗n)S´
whe e ∇S˙u(e)s ands o he mean alue o ∇S˙uh(x) inside he elemen e. In summa y:
δΠε(˙uh,˙εh:τh) =0 →˙εh=Pe=nelem
e=1 χe(x)[∇S˙u(e)+ (µ(e)
S1
k−le
Ωe)( ˙
βe⊗n)S](63)
Now, om equa ion (60)-(c) and he expansion o ξh(x) in equa ion (58)-(c):
δΠσ(˙σh,˙
Σ;ξh) = 0 ⇒
e=nelem
X
e=1 ZΩe³˙σe−˙
Σ´:ξhdΩ = (64)
=
e=nelem
X
e=1 ZΩe³˙σe−˙
Σ´: (¯
ξe+µ(e)
S
1
k(αe⊗n)S)dΩ =
=
e=nelem
X
e=1 µ·Ωe˙σe−ZΩe
˙
ΣdΩ¸:¯
ξe+·le˙σe·n−ZSe
˙
Σ·ndS¸·αe¶= 0 ∀¯
ξe∀αe
⇒hΩe˙σe−RΩe
˙
ΣdΩi= 0
hle˙σe·n−RSe
˙
ΣdS · ni


e∈ {1...nelem}(65)
and sol ing o ˙σein equa ion (65):
˙σe=1
ΩeRΩe
˙
ΣdΩ = ˙
ΣΩe(a)
˙σe·n=1
leRSe
˙
ΣdS · n=˙
ΣS·n(b)
(66)
18
whe e ˙
ΣΩe=1
ΩeRΩe
˙
ΣdΩ and ˙
ΣS=1
leRSe˙
ΣdSa e, espec i ely, he mean alues o ˙
Σ(ε(x))
in Ωeand Se.F om equa ion (66) i i ially ollows:
˙
ΣΩe·n=˙
ΣS·n(67)
which s a es he inne ac ion con inui y condi ion (4)-(e) in e ms o he mean alues o ˙
Σ.
Equa ion (66)-(b) can be ew i en in a mo e con enien o ma :
1
leZSe
˙
ΣdS · n−1
ΩeZΩe
˙
ΣdΩ·n=ZΩe
(µ(e)
S
1
k−le
Ωe
)˙
Σ·ndΩ = 0(68)
In summa y, om equa ions (64) and (68):
δΠσ(˙σh,˙
Σ;ξh) = 0 →RΩe(µ(e)
S1
k−le
Ωe)˙
Σ·ndΩ = 0∀e∈ {1...nelem}(69)
Finally, s anda d algeb aic ope a ions on equa ion (60)-(a), aking in o accoun equa ions (66)
lead o:
δΠu(˙σh;ηh) = 0 →
e=nelem
[
e=1 ZΩe
.
σe:∇SηhdΩ−Gex =
=
e=nelem
[
e=1 ZΩe
˙
ΣΩe:∇SηhdΩ−Gex =
=
e=nelem
[
e=1
˙
ΣΩe:ZΩe
∇SηhdΩ
| {z }
= Ωe∇Sη(e)
−Gex = (70)
=
e=nelem
[
e=1
Ωe˙
ΣΩe
| {z }
RΩe
˙
ΣdΩ
:∇Sη(e)−Gex = 0 ⇒
⇒
e=nelem
S
e=1 RΩe∇Sηh:˙
Σ(εh)dΩ−Gex = 0 ∀ηh∈ V0
u(71)
Rema k 6 No ice om equa ion (63) ha he mixed app oach eco e s he same enhanced
s ain coun e pa
.
˜ε= (µ(e)
S1
k−le
Ωe)( ˙
βe⊗n)Spos ula ed in equa ion (37) o he assumed
enhanced s ain app oach. Howe e in he mixed app oach he e m ∇S˙u(e)in equa ion (63) is
cons an and, he e o e, so is he componen ∇S˙u(e)≈ ∇S˙u(e)in equa ion (51). This ac is
expec ed o acili a e he cancella ion o he elemen al bulk s ain ˙ε(e)
Ω S =∇S˙u(e)−le
Ωe(˙
βe⊗n)S
and con ibu e o alle ia ing he s ess locking phenomenon.
19
5.2.1 2D implemen a ion
Again, o he wo-dimensional case, and he ou noded quad ila e al elemen , he B- o ma
o mula ion o he ini e elemen me hod, om equa ions (63), (69) and (71) eads:
δΠu(˙
Σ;ηh) = 0
δΠσ(˙σh,˙
Σ;ξh)=0
⇒
e=nelem
[
e=1 ·ZΩe
¯
B(e)T·n˙
ΣodΩ−˙
Fex (e)¸=0(72)
{˙ε}(e)=¯
B(e)·˙
d(e)
˙
d(e)=h˙
d1,˙
d2,˙
d3,˙
d4,˙
βeiT(73)
¯
B(e)=h¯
B(e)
1,¯
B(e)
2,¯
B(e)
3,¯
B(e)
4,G(e)i
¯
B(e)
i=




∂xN(e)
i0
0∂yN(e)
i
∂yN(e)
i∂xN(e)
i





; (·)(e)de
=1
ΩeZΩe
(·)dΩ
| {z }
mean alue o (·) on Ωe
G(e)= (µ(e)
S1
k−le
Ωe)




nx0
0ny
nynx





(74)
Rema k 7 Fo p ac ical pu poses he compu a ion o he mean elemen al alues (·)(e) e e ed
o in equa ion (74) can be app oxima ely (o e en exac ly, o undis o ed elemen s) compu ed,
o he quad ila e al elemen , h ough sampling a he cen e o he elemen . Consequen ly
by compa ison o equa ions (47) and (74) he mixed M4n elemen can be e ie ed om he
symme ic S4n elemen by educing he ou egula sampling poin s (PG1 o PG4 in igu e
2-(e)) o one placed a he cen e o he elemen ( (RSP in ha igu e). The e o e, he classical
link be ween mixed elemen s and educed in eg a ion ([11]) is also eco e ed o ini e elemen s
wi h embedded discon inui ies.
6 S ain e-enhancemen : elemen E4n
Nex s a egy is based on p o iding new e ms o he enhanced s ain componen (
.
˜εin equa ion
(28)) o he symme ic elemen S4n. Two condi ions a e equi ed on his enhancemen :
i) Ful ill he o hogonali y condi ion (27) :
20
ZΩ
.
˜ε:τdΩ=0 ∀.
˜ε∈ V˜ε∀τ∈Vσ(75)
ii) Include linea polynomial e ms o con ibu e o alle ia e he s ess locking phenomenon.
Wi h hese condi ions in mind, he ollowing s ain enhancemen is p oposed:
.
˜εh(e)
= (µ(e)
S
1
k−le
Ωe
)(˙
βe⊗n)S
| {z }
.
˜ε(e)
1
+1
Js˙
Se+1
J ˙
Te
| {z }
.
˜ε(e)
2
(76)
whe e sand s and o he isopa ame ic coo dina es o he s anda d 4-noded quad ila e al and
Jis he jacobian o he isopa ame ic ans o ma ion ela ing di e en ial a eas in he egula
and isopa ame ic spaces h ough:
dΩ = J ds d (77)
In equa ion (76)
.
˜ε(e)
1is he basic s ain enhancemen , al eady p esen in he basic S4n ele-
men , and
.
˜ε(e)
2is a e-enhancemen o he s ain ield ha supplies he equi ed linea poly-
nomial componen s o he elemen al bulk s ain. The alues n˙
Seo= [ ˙
Sxx,˙
Syy,˙
Sxy]T
eand
n˙
Teo= [ ˙
Txx,˙
Tyy,˙
Txy]T
ea e (cons an ) in ensi y ac o s ha cons i u e six ( o he 2D p ob-
lem) addi ional deg ees o eedom o he elemen . F om exp ession (76) i is clea ha he
o hogonali y condi ion (75) is ul illed o he elemen -wise cons an assumed s ess ield in
equa ion (35) since:
ZΩe
.
˜ε(e)
1:τedΩ = ZΩe
(µ(e)
S
1
k−le
Ωe
)(˙
βe⊗n)S:τedΩ =
=ZΩe
(µ(e)
S
1
k−le
Ωe
)dΩ
| {z }
le−le= 0
(˙
βe⊗n)S:τe= 0 (78)
ZΩe
.
˜ε(e)
2:τedΩ = ZΩe
1
J(s˙
Se+ ˙
Te) : τedΩ = (79)
=Z+1
−1Z+1
−1
s dsd
| {z }
= 0
˙
Se:τe+Z+1
−1Z+1
−1
dsd
| {z }
= 0
˙
Te:τe= 0
21

6.1 2D implemen a ion
In iew o he p eceding o mula ion, he 2D implemen a ion o elemen E4n becomes simila
o ha o he elemen S4n in equa ions (45) and (47) bu including he addi ional enhanced
s ain e ms in equa ion (76). Tha is:
δΠu(˙
Σ;ηh) = 0
δΠσ(˙σh,˙
Σ;ξh) = 0
⇒
e=nelem
[
e=1 ·ZΩe
B(e)T·n˙
ΣodΩ−˙
Fex (e)¸=0(80)
{˙ε}(e)=B(e)·˙
d(e)
˙
d(e)=h˙
d1,˙
d2,˙
d3,˙
d4,[˙
βe,n˙
Seo,n˙
Teo]iT(81)
B(e)=hB(e)
1,B(e)
2,B(e)
3,B(e)
4,G(e)i
B(e)
i=




∂xN(e)
i0
0∂yN(e)
i
∂yN(e)
i∂xN(e)
i





G(e)=




γnx0
0γny
γnyγnx
| {z }
.
˜ε1
s0 0
0s0
00s
0 0
0 0
00





| {z }
.
˜ε2
γ= (µ(e)
S1
k−le
Ωe)
(82)
whe e he eigh in e nal deg ees o eedom [ ˙
βe,n˙
Seo,n˙
Teo] can be condensed a elemen al
le el.
7 Nume ical es s.
7.1 Homogeneous pla e
The basic es o Sec ion 4 and igu e 3 is now epea ed using he modi ied elemen s M4n and
E4n. The esul s, again in e ms o σxx −δcu es, a e p esen ed in igu e 4 oge he wi h he
ones ob ained wi h he o iginal elemen S4n and he exac esul (U4n).
I can be obse ed he d ama ic educ ion in he s ess-locking e ec ob ained wi h he
new elemen s which p o e he e ec i eness o he adop ed s a egies. Besides, one can obse e
ha he esul s ob ained wi h elemen M4n (mixed s a egy) and elemen E4n ( e-enhancemen
22
s a egy) a e e y simila o each o he , which shows ha he imp o emen easoning based
on he cancella ion o he bulk s ain, s a ed in Sec ion 4 and implemen ed in wo di e en
s a egies, was essen ially co ec .
7.2 No ched specimen
In o de o assess he pe o mance o he p oposed s a egies in mo e complex p oblems, he
es o igu e 5-(a) is also conside ed. I is a no ched specimen whose expe imen al es ing is
epo ed in [7]. The couples o o ces F1and F2a e p og essi ely applied, as i is indica ed, in
igu e 5-(b) in o de o induce a mixed mode c ack which de elops om he no ch ip wi h an
inclina ion angle o 71o(see igu e 5-(c). Fo he nume ical simula ion he same ma e ial model
(iso opic con inuum damage wi h linea so ening) as in he p e ious case has been adop ed,
wi h he ollowing pa ame e s : he elas ic modulus and Poisson’s a io a e E= 30580.[Mpa]
and ν= 0.2 espec i ely, he ac u e ene gy G = 100.[N/m] and he ensile s eng h is
assumed o be σu= 3.[MPa].The wid h o he specimen is = 50.8[mm].
The esul s, in e ms o he applied o ce e sus he CMOD, ob ained wi h he U4n el-
emen , he o iginal S4n elemen and he modi ied symme ic M4n and E4n elemen s, as well
as he expe imen al alues om he abo e e e ence, a e p esen ed in igu e 5-(d). Again he
d ama ic imp o emen , in e ms o he nume ical s ess locking e ec , ob ained wi h he de-
ised s a egies in compa ison wi h he o iginal symme ic elemen S4n can be obse ed. Also
he iso-displacemen cu es shown in igu e 5-(e) display a modelled discon inui y ha ma ches
e y well he expe imen al c ack.
7.3 Th ee poin no ched beam
We conside now he h ee poin no ched conc e e beam, in plane s ain, shown in igu e 6-(a).
Expe imen al da a o his es ha e been p esen ed by Pe e son [22].
The conc e e pos -c i ical beha io is modeled by he same iso opic con inuum damage
model han in ([15]). The ma e ial pa ame e s a e aken: E= 30580.[Mpa], ν= 0.2 and he
ac u e ene gy G = 126.[N/m]. The ensile s eng h is aken σu= 3.4[MPa].The wid h o
he specimen is = 100.[mm].The mesh consis s o 615 quad ila e als e ined nea he no ch
ip.
A c ack opening in mode I is expec ed o de elop om he no ch ip, p opaga ing in he
e ical di ec ion h ough he beam, as shown in igu e 6-(b). The plo s in igu e 6-(c)(d)
23
co espond, espec i ely, o he load F s. he e ical displacemen δcu es using linea and
exponen ial so ening laws.
Once again i can be no iced he s iking beha io al eady obse ed in he p e ious simula-
ions: he M4n and E4n elemen s d ama ically educe he s ess locking phenomenon exhibi ed
by he o iginal S4n elemen , and p o ide a esponse ha is e y close o he one ob ained wi h
he non-symme ic (U4n) elemen .
8 Concluding ema ks
Th oughou his wo k we ha e explo ed he possibili y o using s a ically consis en symme ic
elemen s wi h embedded discon inui ies o cap u e s ong discon inui ies. As a ma e o ac i
has been shown ha he use o mixed (M4n elemen ) o assumed enhanced s ain (E4n elemen )
echniques con ibu e o subs an ially alle ia e he s ain locking phenomenon appea ing in he
o iginal S4n elemen .
In essence hese echniques a e in ended o eco e ing he capaci y o he non-symme ic
elemen U4n o ep oduce igid body mo ions o he po ions o he elemen s spli up in o by he
discon inui y, while keeping he symme ic cha ac e and a ia ional consis ency o he o iginal
symme ic S4n elemen .
An impo an issue, ha has no been add essed so a , is he s abili y p ope ies o he
de i ed elemen s. I is well known in he li e a u e ha he use o mixed elemen echniques
[28] and assumed enhanced s ain echniques [26] may in oduce spu ious singula modes whose
p opaga ion h ough he ini e elemen mesh can des oy bo h he eliabili y o he solu ion
and he con e gence o he non-linea p oblem. Fo ins ance: i is known ha he ou -noded
elemen wi h educed in eg a ion (one sampling poin ) conside ed he e in elemen M4n, leads
o he de elopmen and p opaga ion o he so called hou -glass spu ious modes [28]. Also he
linea e-enhancemen modes,
.
˜ε(e)
2in equa ion (76), conside ed o he E4n elemen do no ul ill
he s abili y condi ion (V˜ε∩ Vε={0}) [26].
Howe e , he e is a c ucial aspec in he way ha he elemen s de i ed in his wo k a e
implemen ed, ha makes hei s abili y beha iou e y di e en om he classical ones.
Th oughou his wo k, hose ini e elemen o mula ions ha e been de i ed and p esen ed
o he B.V.P. in a e (inc emen al) o m (see equa ions (4),(45),(72) o (80)). This allows o
conside he p oblem as a sequence, along ime, o inc emen al p oblems, each one ha ing i s
own ini e elemen o mula ion. As a ma e o ac he implemen a ion is done in such a way
24
ha he modi ica ions on he basic elemen S4n, ha lead o he de i ed elemen s M4n and
E4n, a e e ec i e only o he band o elemen s ha cap u es he discon inui y and only beyond
he ime ha his discon inui y appea s . The elemen s ou side his band beha e as he o iginal
S4n since hey do no exhibi he s ess locking p oblem ha he modi ied elemen s y o
o e come. This dec eases he numbe o elemen s a ec ed by he educed in eg a ion o e-
enhancemen echniques o a single band o essen ially one elemen bandwid h. Consequen ly
he de elopmen and p opaga ion o spu ious ins abili y modes a e ex emely es ained. In
addi ion, he compu a ional cos s associa ed o he p esence o new deg ees o eedom ( o he
e-enhancemen s a egy) a e hen e y small.
Al hough speci ic s abili y analysis ha e no been conduc ed in his pape , and hey a e le
o subsequen wo ks, du ing he nume ical simula ions p esen ed abo e, and o he s ca ied ou
du ing his s udy, he possible ins abili y modes ha e no been obse ed and ha e no placed
special di icul ies on he nume ical p ocedu e. Howe e , he au ho s a e awa e ha his can
no be gene alized o any ype and size o he p oblems, and ha speci ic s udies on he s abili y
issue should be ca ied ou in he u u e.
Also he possibili y o using some o he de eloped elemen s o dispense wi h he discon i-
nui y acking algo i hm should be explo ed in subsequen wo ks.
Re e ences
[1] F. A me o and K. Ga ikipa i. An analysis o s ong discon inui ies in mul iplica i e ini e
s ain plas ici y and hei ela ion wi h he nume ical simula ion o s ain localiza ion in
solids. In .J. Solids and S uc u es, 33(20–22):2863–2885, 1996.
[2] T. Bely schko, N. Moes, S. Usui, and C. Pa imi. A bi a y discon inui ies in ini e elemen s.
In . J. Nume . Me h. Engng., (50):993–1013, 2001.
[3] C.C.Celigoj. On s ong discon inui ies in anelas ic solids. a ini e elemen app oach aking
a ame indi e en g adien o he discon inuous displacemen s. In . J. Nume . Me h.
Engng., (49):769–796, 2000.
[4] R. de Bo s , L. J. Sluys, H. B. Muhlhaus, and J. Pamin. Fundamen al issues in ini e
elemen analyses o localiza ion o de o ma ion. Enginee ing Compu a ions, 10:99–121,
1993.
25
0
0.4
0.8
1.2
U4n(non-symme ic) U4n(non-symme ic)
M4n(mixed
symme ic)
M4n(mixed
symme ic)
E4n
(enhanced
symme ic)
E4n
(enhanced
symme ic)
S4n(symme ic) S4n(symme ic)
E= 30580.MPa
= 0.20
=0.1mThickness
n
F[kN] F[kN]
0.1m
1. m1. m
0.2 m
F
d
(a) (b)
(d)
Expe imen Expe imen
0
0.4
0.8
1.2
00.2 0.4 0.6 0.8 1.
00.2 0.4 0.6 0.8 1.
(c) d[x10 m]
-3
d[x10 m]
-3
G = 126.Nw/m
= 3.4 MPas
u
Figu e 6: Th ee poin s no ched beam: a) geome ical model; b) discon inui y pa h; c) esul s
o linea so ening; d) esul s o exponen ial so ening.
32