1222
EXPLICIT INTEGRATION SCHEME FOR GENERALIZED
PLASTICITY CONSTITUTIVE MODELS WITH AUTOMATIC ERROR
CONTROL
MIGUEL M. STICKLE
1
*, PABLO DE LA FUENTE
2
AND CARLOS OTEO
3
1*: Applied Ma hema ics and Compu e Science Depa men
ETSI Caminos, Canales y Pue os
Uni e sidad Poli écnica de Mad id
A d. P o eso A angu en s/n, 28040 Mad id, Spain
e-mail: miguels ick[email p o ec ed]pm.es
2: Con inuum Mechanics and S uc u es Depa men
ETSI Caminos, Canales y Pue os
Uni e sidad Poli écnica de Mad id
A d. P o eso A angu en s/n, 28040 Mad id, Spain
e-mail: pdela @caminos.upm.es
3: P o esso on G ound Eng.
C / To pede o Tucumán 26, 28016 Mad id, Spain
e-mail:
ca loso eo@ ele onica.ne
Key wo ds:
Gene alized Plas ici y, explici in eg a ion.
Abs ac . An explici algo i hm o in eg a ing Gene alized Plas ici y cons i u i e models is
p esen ed. This au oma ically di ides he applied s ain inc emen in o subinc emen s using
an es ima e o he local e o con olling he global in eg a ion e o in he s ess. The
algo i hm modi ies he well known S. W. Sloan subs epping scheme o accoun o
Gene alized Plas ici y cons i u i e models, in which, unlike Classical Elas oplas ici y, he
yield su ace is no explici ly de ined. The in eg a ion scheme is desc ibed and esul s a e
p esen ed o a igid oo ing es ing on a laye o speci ic Gene alized Plas ici y model o
sands, in which a hype elas ic o mula ion is in oduced o desc ibe he e e sible componen
o he soil esponse ins ead o he hypoelas ic app oach o iginally p oposed. The explici
algo i hm wi h au oma ic subs epping and e o con ol is shown o be eliable and e icien
o hese complex cons i u i e laws.
1 INTRODUCTION
Nowadays i is well ecognized ha he selec ion o an adequa e cons i u i e model,
oge he wi h he use o accu a e, e icien and obus in eg a ion algo i hms o he
elas oplas ic equa ions, is a key poin in ini e elemen analysis o geo echnical p oblems.
As obse ed by Hughes [1], he in eg a ion o he cons i u i e equa ions a he local le el
plays a c ucial pa in compu a ional plas ici y, since i s ongly a ec s he pe o mance o he
cons i u i e equa ion in ac ual compu a ions.
Implemen a ion o an ad anced elas oplas ic cons i u i e model in o a ini e elemen
XI In e na ional Con e ence on Compu a ional Plas ici y. Fundamen als and Applica ions
COMPLAS XI
E. Oña e, D.R.J. Owen, D. Pe ic and B. Suá ez (Eds)
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Miguel M. S ickle, Pablo De la Fuen e, Ca los O eo.
2
p og am equi es he de elopmen o a obus and e icien nume ical p ocedu e in o de o
pe o m he in eg a ion o he cons i u i e equa ions along a gi en loading pa h.
In he con ex o classical plas ici y o mula ions, in which a yield unc ion is de ined in an
explici manne and he en o cemen o he consis ency condi ion is a key ea u e o he
in eg a ion algo i hm, a a ie y o implici and explici in eg a ion schemes migh be used [2,
3].
On he o he hand, based on he assump ion ha explici in eg a ion schemes o highly
non-linea models may po en ially lead o inaccu acy and uns able beha io [4], implici
in eg a ion algo i hms ha e been conside ed mos ly in he con ex o non-s anda d
elas oplas ic models. The a o emen ioned assump ion canno be u he suppo ed, in he
con ex o classical plas ici y o mula ions, i explici schemes a e endowed wi h e o con ol
echniques [5, 6]. The same si ua ion is obse ed o Gene alized Plas ici y based models [7].
The ou line o he pape is as ollows. We i s p esen he undamen als o Gene alized
Plas ici y, wi h pa icula a en ion paid o he SandPZ cons i u i e equa ions including he
modi ica ions in he elas ic componen in oduced by Mi a and cowo ke s in 2009 pape . A
no el explici algo i hm o in eg a ing Gene alized Plas ici y cons i u i e models is p esen ed
in he ollowing sec ion. Finally, Resul s and conclusions a e p esen ed o a igid oo ing
es ing on a sand laye modeled by a SandPZ cons i u i e ela ion.
2 GENERALIZED PLASTICITY FRAME WORK. MODIFIED SAND PZ MODEL.
The Gene alized Plas ici y basic idea, in oduced by Zienkiewicz and M oz [8] la e
ex ended by Pas o and cowo ke s [9, 10] and also by Mi a and cowo ke s [4], is ha no yield
nei he plas ic po en ial su ace a e explici ly de ined, bu he g adien s o he unc ions
hemsel es. The elas oplas ic beha io o he ma e ial wi hin he Gene alized Plas ici y heo y
is desc ibed by he gene al inc emen al ela ionship,
(
)
:
ep ep
ij ijkl kl
d Dd d d
σε
′′
=⋅ =
σDε
(1)
In which he angen elas oplas ic s i ness ou o de enso
ep
D
depends no only on he
in e nal s a e a iables bu also on he cu en e ec i e s ess s a e
′
σ
, on he s ain-s ess
his o y and he di ec ion o he e ec i e s ess inc emen s
d
′
σ
.
The dependence o
ep
D
on he di ec ion o
d
′
σ
is exp essed by simply dis inguishing
be ween wo di e en loading classes, namely Loading (L) and Unloading (U). The e o e a
no malized di ec ion
n
is de ined in he e ec i e s ess space o any gi en
′
σ
, de e mining
loading/unloading/neu al loading condi ion.
The e a e wo possibili ies o he angen elas oplas ic s i ness enso in (1) depending on
whe he loading,
ep
L
D
o unloading
ep
U
D
is occu ing. To gua an ee con inui y be ween loading
and unloading,
ep
L
D
and
ep
U
D
a e de ined as
( ) ()
[]
( ) ()
[]
11
11
1
1
ep e
L L
L
ep e
U U
U
H
H
−−
−−
= +⋅⊗
= +⋅⊗
D D mn
D D mn
(2)
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Miguel M. S ickle, Pablo De la Fuen e, Ca los O eo.
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In exp ession (2),
L
m
and
U
m
a e di ec ions o uni no m ep esen ing he plas ic low
di ec ion in loading (L) and unloading (U) condi ions espec i ely,
L
H
and
U
H
a e wo scala
unc ions de ined as plas ic moduli while
e
D
is he angen elas ic s i ness enso . By sui able
manipula ion o (2),
ep
L
D
and
ep
U
D
can be ob ained gi ing:
::
::
::
::
ee
ep e L
Le
LL
ee
ep e U
Ue
UU
H
H
⊗
=− +
⊗
=− +
D m nD
DD
nD m
D m nD
DD
nD m
(3)
The s ain inc emen
d
ε
can be decomposed in o elas ic and plas ic pa s as
ep
dd d
=+
εε ε
whe e
(
)
1
:
1
: o loading
1
: o unloading
ee
p
L
L
p
U
U
dd
dd
H
dd
H
−
′
=

′
= ⋅⊗



′
= ⋅⊗


εDσ
εmnσ
εmnσ
(4)
The e o e, in a Gene alized Plas ici y app oach, he non-linea i e e sible beha io o
soils can be ully desc ibed by simply speci ying h ee di ec ions,
,
L
nm
and
U
m
, wo
scala s,
L
H
and
U
H
and a ou h o de enso
e
D
.
Since he ha dening moduli
L
H
and
U
H
as well as he plas ic low di ec ions
L
m
and
U
m
a e ully de e mined wi hou e e ence o any yield su ace no plas ic po en ial, di e en
exp essions can be selec ed o hem whe he he s ess inc emen implies loading o
unloading. Mo eo e consis ency canno be en o ced and he consis ency pa ame e
d
λ
is
simply de ined as:
::
::
e
e
LU LU
d
dH
λ
=+
nD ε
nD m
(5)
Al hough no explici ly de ined, plas ic po en ial and yield su ace can be es ablished a
pos e io i, by in eg a ing
LU
m
and
n
, espec i ely.
SandPZ model was de eloped by Pas o and cowo ke s [9] as a pa icula ype o
Gene alized Plas ici y o mula ion wi h he aim o p edic ing g anula soil beha io unde
bo h mono onic and cyclic loading.
The model assumes an iso opic ma e ial esponse. As a esul , he plas ic low di ec ion
m
, as well as he loading di ec ion
n
, is exp essed in he in a ian space de ined by
,,
pq
θ
′
as
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Miguel M. S ickle, Pablo De la Fuen e, Ca los O eo.
4
mmm
s
ij ij ij
pq
θ
θ
σσσ
′
∂∂∂
=++
′′′
∂∂∂
m
(6)
The alue o he coe icien s
m ,m ,m
s
θ
a e loading class (loading o unloading) dependen .
In o de o ake in o accoun he main ea u es o sand esponse, i.e. he exis ence o a c i ical
s a e condi ion, dila i e esponse a e peak, lique ac ion in loose sands, memo y o p e ious
s ess pa h, Pas o and cowo ke s [9] p oposed o he plas ic modulus
L
H
he ollowing
ela ionship:
(
)
0
L s DM
H H pH H H H
′
= ⋅⋅ ⋅ + ⋅
(7)
Toge he wi h
()
4
01 0
1
max
1 ; 1 ; exp
1
; ; 1 1
s
g
p
Dm s
H HH
MM
H dd p M
γ
α
α
ηη
ββ βξ
α
α
ζη
ξ ε ξζ
ζα
−

= − ⋅ =− = −


+



 ′
= ===−⋅


 
+




∫∫
In hese exp essions
001
,,,
H
ββγ
a e cons i u i e pa ame e s,
ξ
is he accumula ed
de ia o ic plas ic s ain and
max
ζ
s ands as he maximum alue o he mobilized s ess unc ion
ζ
accoun ing o he soil s ess his o y.
In he case o unloading he plas ic modulus
U
H
is gi en by:
0
0
o 1
o 1
u
gg
Uu
uu
g
Uu
u
MM
HH
M
HH
γ
ηη
η

=>


=<
(8)
Fo whe e
0
u
H
is a cons i u i e pa ame e and
u
η
, e e ed as unloading s ess a io, is he
s ess a io
qp
′
om which unloading akes place.
Finally, he PZ model assumes a non-linea elas ic esponse o he soils. As in a la ge
numbe o cons i u i e models, he non-linea e e sible beha io is desc ibed h ough a
hypoelas ic app oach, in which he angen bulk modulus
K
and shea modulus
G
only
depend on he hyd os a ic pa o he e ec i e s ess enso , acco ding o he ollowing
ela ionships
00
00
,
pp
KK GG
pp
′′
=⋅ =⋅
′′
(9)
Al hough widely used, one o he majo sho comings o such hypoelas ic o mula ion is
ha i esul s in a non-conse a i e elas ic esponse and ene gy dissipa ion o e closed s ess
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Miguel M. S ickle, Pablo De la Fuen e, Ca los O eo.
5
pa hs [11].
An al e na i e is o desc ibe he elas ic esponse o soils wi hin a conse a i e amewo k
adop ing he hype elas ic app oach based on he exis ence o an ene gy po en ial om which
he e e sible esponse can be de i ed. This na u ally leads o a conse a i e elas ic esponse,
gua an eed o obey he Fi s Law o The modynamics, and hus a oiding he p oblems on
cycling desc ibed abo e [12, 13]
Among he di e en o mula ions ecen ly p oposed in he geo echnical li e a u e, in his
wo k he hype elas ic app oach desc ibed by Houslby and cowo ke s in 2005 ( om now on
e e ed as HAR acco ding o au ho s’ ini ials) has been adop ed o desc ibe he e e sible
componen o he soil esponse. This Gene alized Plas ici y PZ model o g anula soils has
been p oposed i s ly by Mi a and cowo ke s [4] unde i s iaxial o mula ion, ex ending he
model in he p esen wo k o deal wi h a gene al s ess o mula ion.
The s o ed ene gy unc ion
ϒ
o he HAR model in gene al s ess o mula ion has wo
di e en exp essions depending on he alue assigned o he dimensionless p essu e exponen
HAR
n
, which go e ns he amoun o nonlinea i y in ol ed in he o mula ion. Fo
1
HAR
n
≠
,
ϒ
akes he ollowing o m:
()
()()
( )( )
21
0
1
2
HAR HAR
nn
ea
ij HAR HAR
HAR HAR
pkn
kn
ευ
−−
ϒ = ⋅ ⋅− ⋅

⋅−
(10)
Whe e
() () ()
2
0
2
11
1 11
ee
ij ij
HAR
ee
ii jj
HAR HAR HAR HAR HAR HAR
g ee
kn knkn
υε ε
  
  
  
  
=+ ⋅+ +
⋅− ⋅− −
, while he asymp o ic
exp ession o
1
HAR
n
=
is
(
)
()
a HAR
e ee
ii ij ij
HAR HAR HAR
k g k ee
e
ij
pk
e
ε
ε



⋅+ ⋅ ⋅
ϒ= ⋅ . ,
HAR HAR
kg
a e dimensionless
cons an s ep esen ing he shea and bulk s i ness ac o s, espec i ely, while
a
p
is he
a mosphe ic p essu e, adop ed as e e ence s ess.
The e ec i e s ess enso
′
σ
and he angen elas ic enso
e
D
can be unambiguously
de e mined by aking he i s and second o de de i a i es o (10), ob aining he ollowing
exp ession
()
0
2
0
1
12
3
HAR
n
ij kl
e
ijkl a HAR HAR HAR HAR ij kl HAR ik jl kl ij
a
p
D p nk k n g
pp
σσ δδ δδ δδ
′′
  

=⋅ ⋅ ⋅ + − + −
  


  
(11)
Whe e
()
(
)
2
0
1
92
HAR HAR mn mn
mm nn
HAR
k n ss
pg
σσ
⋅− ⋅
′′
=+
Fo he p esen model he e a e 12 ma e ial pa ame e s equi ing de ini ion. Gene ally, all
pa ame e s a e iden i ied om mono onic and cyclic iaxial es s, hough in ce ain cases
some pa ame e s a e adop ed om p e ious expe iences i ull es eco ds a e una ailable.
3 EXPLICIT INTEGRATION OF GENERALIZED PLASTICITY MODELS.
Du ing a ypical s ep o i e a ion o an elas oplas ic ini e elemen analysis, he o ces a e
applied in inc emen s and he co esponding displacemen inc emen s a e ound om he

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Miguel M. S ickle, Pablo De la Fuen e, Ca los O eo.
6
global s i ness equa ions. Once he nodal displacemen inc emen s
d
u
a e known, he s ain
inc emen s a a disc e e numbe o in eg a ion poin s wi hin each elemen a e de e mined
using he s ain-displacemen ela ion
dd
=
ε
Bu
. I he s esses associa ed wi h an imposed
s ain inc emen cause plas ic yielding, i is necessa y o sol e he sys em o i s o de
o dina y di e en ial equa ions (12)-(13):
::
ep ep
∆

′==

∆

ε
σDDε
ɺɺ
(12)
p
LU
d
λ
=⋅εm
ɺ
(13)
Whe e
::
::
ee
LU
ep e
e
LU LU
H
⊗
=− +
D m nD
DD nD m
(14)
::
::
e
e
LU LU
d
dH
λ
=+
nD ε
nD m
(15)
In hese exp essions,
′
σ
deno es he e ec i e s ess enso ,
ε
he small s ain enso and
p
ε
he plas ic s ain enso . The supe io do ep esen s a de i a i e wi h espec o ime while
∆
is he ime in e al o e which he ex e nal o ces ha e been applied. Since he e ec i e
s ess and he plas ic s ains a e kwon a he beginning o he ime in e al, and he known
s ain a es may be assumed o be cons an h ough he ime in e al wi h alue
∆∆
ε
, he
equa ions (12) and (13) de ine an ini ial alue p oblem.
In o de o in eg a e hese equa ions nume ically, i is con enien [14] o in oduce a
pseudo ime,
T
, de ined by
(
)
0
T
=− ∆
, whe e
0
is he ime a he s a o he load
inc emen , while
0
+∆
is he ime a he end o he load inc emen , wi h
01
T
≤≤
. Since
1
dT d
=∆
applica ion o he chain ule o
′
ɺ
σ
and
p
ɺ
ε
in (12) and (13) gi es
::
: ::
::
ee
LU
ep e ee
LU
e
LU LU
d
dT H
λ

⊗
′′
= ∆= − ∆=∆ −∆


+

D m nD
σD
εDεσ Dm
nD m
(16)
p
LU
d
dT
λ
=∆ ⋅
ε
m
(17)
Whe e
::
::
e
e
LU LU
H
λ
∆
∆= +
nD ε
nD m
(18)
Equa ions (16) and (17) de ine a classical ini ial alue p oblem which needs o be
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Miguel M. S ickle, Pablo De la Fuen e, Ca los O eo.
7
in eg a ed o e he pseudo ime in e al om
0
T
=
o
1
T
=
, whe e he known alues known
alues in hese ela ions a e he imposed s ain inc emen s,
∆
ε
, oge he wi h he e ec i e
s esses and plas ic s ain a he s a o he pseudo ime inc emen . The quan i ies
/
and
LU
mn
a e e ec i e s ess unc ions, while pa ame e
/
LU
H
is a unc ion o bo h he e ec i e s ess
and he plas ic s ain.
In o de o sol e he sys em o i s o de o dina y di e en ial equa ions (16)-(17) Sloan
de eloped a subs epping algo i hm [5] whe e he cons i u i e law is in eg a ed by
au oma ically di iding he s ain inc emen in o a numbe o subs eps. An app op ia e size o
each subs ep is ound h ough he use o modi ied Eule o Runge-Ku a-Do mand-P ince
o mulae, which a e specially cons uc ed o p o ide an es ima e o he local e o . La e ,
Sloan and cowo ke s [6] gene alized he 1987 scheme, inco po a ing new algo i hms o
handling elas oplas ic unloading, compu ing he yield in e sec ion poin , and es o ing he
s ess o he yield su ace.
Sloan and cowo ke s schemes we e de eloped exclusi ely o classical plas ici y based
models, including classical and gene alized c i ical s a e models, whe e non-linea elas ic
beha io inside he yield su ace is exhibi ed. In all hese models he admissible s a es in he
s ess space a e cons ained o lie wi hin he in e io o he bounda y o he domain explici ly
de ined by he yield su ace. As his is no he case o gene alized plas ici y based models,
Sloan subs epping algo i hm should be adjus ed in o de o be able o in eg a e his kind o
models.
The p oposed in eg a ion scheme s a s wi h he known s ain inc emen ,
∆
ε
, he ini ial
s ess
0
σ
and ini ial plas ic s ain
0
p
ε
a he s a o he inc emen whe e
0
T
=
and
0
=
. A
he end o he in eg a ion p ocess he s esses and plas ic s ains a e ob ained a he end o he
inc emen whe e
0
T
=
and
0
=
.
Conside a pseudo ime subinc emen in he ange
01
n
T
≤∆ ≤
and le he subsc ip s
1
n
−
and
n
, deno e quan i ies e alua ed a he pseudo imes
1
n
T
−
and
1
nn n
TT T
−
= +∆
, espec i ely.
Plas ic modulus and plas ic low di ec ion in exp essions (16)-(17) a e dependen on he
di ec ion o he e ec i e s ess inc emen s he e o e di e en ia ion be ween he wo loading
classes should be pe o med be o e he p ope in eg a ion p ocess s a s. By means o he
s ain subinc emen
nn
T∆ =∆ ∆
εε
he loading class is i s ly es ablished h ough he ollowing
exp ession
(
)
(
)
::
e
n nn
′′
∆
n
σDσε
(19)
I exp ession (19) is posi i e an elas oplas ic loading p ocess is pe o med, i nega i e an
elas oplasc ic unloading p ocess is implied, while an elas ic p ocess comes om a ze o alue.
In he explici Eule me hod, he solu ion o
,
p
′
σε
a he end o he pseudo ime s ep
n
T
∆
is ound om
11
11
nn
pp p
nn
−
−
′′ ′
= +∆
= +∆
σσ σ
εε ε
(20)
Whe e
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Miguel M. S ickle, Pablo De la Fuen e, Ca los O eo.
8
(
)
()
()
1 11
1 11 1
,:
,,
ep p
nn n
pp
n n n LU n
λ
−−
−− −
′′
∆= ∆
′
∆ =∆ ∆ ⋅
σDσε ε
ε σε εmσ
(21)
A mo e accu a e es ima e o he s ess and plas ic s ains a he end o he in e al
n
T
∆
can
be ound using he modi ied Eule p ocedu e, which is gi en by
()
()
1 12
1 12
1
2
1
2
nn
pp p p
nn
σσ
εε
−
−
′′ ′ ′
= + ∆ +∆
= + ∆ +∆
σσ
εε
⌢
⌢
(22)
Whe e
11
and
p
′
∆∆
σε
a e ob ained om Eule scheme and
(
)
()
()
2 1 11 1
2 1111 11
,:
,,
ep p p
nn n
ppp
n n n LU n
λ
−−
−− −
′ ′′
∆ = +∆ +∆ ∆
′′ ′′
∆ =∆ +∆ +∆ ∆ ⋅ +∆
σ
D
σ σε ε ε
ε σ σε ε ε
m
σσ
(23)
Since he local unca ion e o [15] in he Eule and modi ied Eule solu ions is
(
)
(
)
23
and
OT OT
∆∆
, espec i ely, he e o in
n
σ
and
p
n
ε
can be es ima ed om
()
()
21
21
1
2
1
2
nn
pp
pp
nn

′′
∆ −∆

′′
  
−=

  

   ∆ −∆


σσ
σσ
εε
εε
⌢
⌢
(24)
Using any con enien no m, his quan i y can be used o compu e he ela i e e o
measu e
21
21
1max ,
2
pp
np
nn
R

′′
∆ −∆
∆ −∆

=



εε
σσ
σε
(25)
Following 1987 Sloan wo k, he cu en s ain subinc emen is accep ed i
n
R
is no g ea e
han some p esc ibed ole ance,
STOL
, and ejec ed o he wise. Rega dless o whe he he
subinc emen is accep ed o ejec ed, he nex pseudo ime s ep is ound om he simple
ela ion
1
nn
T qT
+
∆ = ⋅∆
(26)
whe e
q
is chosen so ha
1
n
R
+
sa is ies he cons ain
0.8 , 0.1 1.1
n
q STOL R q
≤ ≤≤
(27)
1230
Miguel M. S ickle, Pablo De la Fuen e, Ca los O eo.
9
Two ypical con ols a e inally inco po a ed. A minimum absolu e s ep size,
min
T
∆
, and a
s ep size is no allowed o g ow immedia ely a e a ailed subinc emen .
4 RESULTS AND CONCLUSIONS.
The beha io o a smoo h igid s ip oo ing es ing on an elas oplas ic soil mass, go e ned
by he modi ied SandPZ model p esen ed abo e, is conside in o de o analyze he
pe o mance o he p oposed in eg a ion scheme. Due o he singula i y a he edge o he
oo ing and he s ong o a ion o he p incipal s esses, his example is a good es o
assessing he in eg a ion s a egy. As loading is p esc ibed in he o m o displacemen s, an
equi alen uni o m p essu e is ound by summing he app op ia e nodal eac ions.
To assess he accu acy o he scheme, an es ima e o he s ess in eg a ion e o is ound
di ec ly om
2
2
e
e o
e
σ
′′
−
=′
σσ
σ
(28)
Whe e
′
σ
a e he e ec i e s esses ob ained by he p oposed in eg a ion scheme,
e
′
σ
a e
he e e ence e ec i e s esses while
2
i
is he Euclidean no m. The e e ence e ec i e
s esses a e ob ained by he explici Do mand-P ince in eg a ion scheme wi h a s ess
ole ance o
9
10
STOL
−
=
. No e ha he e e ence s esses p o ide a e y accu a e se o
s esses o he gi en mesh and loading sequence and all alues a e compu ed a he end o he
las load inc emen .
The esul s o he analyses wi h 10 load inc emen s o equal size a e p esen ed in Table 1.
I can be obse ed how he uni o m p essu e o e he oo ing a e applying 4mm o e ical
displacemen is simila o all o he speci ied s ess ole ances wi h alues a ying by less
han 0.6% o he e e ence p essu e.
Table 1: Smoo h igid s ip oo ing on SandPZ laye . 10 load s eps.
S ess Tole ance
Equi alen uni o m
p essu e a e a e ical
displacemen o 4mm
[N/m
2
]
% o he equi alen
p essu e ob ained unde
Do mand-P ince
in eg a ion scheme
e o
σ
2
10
STOL
−
=
115320 0.14%
3
1.7 10
−
⋅
4
10
STOL
−
=
115450 0.02%
4
1.7 10
−
⋅
The e o in he compu ed s esses o he p oposed scheme, as de ined by equa ion (28), is
less han he in eg a ion ole ance o
2
10
STOL
−
=
and wi hin he o de o magni ude o
4
10
STOL
−
=
. The e o e he ole ance
STOL
hus gi es a equi ed e o con ol.
Figu e 1 shows he e o spa ial dis ibu ion induced by he p oposed local in eg a ion