IOP Con e ence Se ies: Ma e ials Science and Enginee ing
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Op imiza ion-Based In e se Iden i ica ion o he
Pa ame e s o a Conc e e Cap Ma e ial Model
To ci e his a icle: Pe K ál e al 2017 IOP Con . Se .: Ma e . Sci. Eng. 245 032078
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IOP Con . Se ies: Ma e ials Science and Enginee ing 245 (2017) 032078 doi:10.1088/1757-899X/245/3/032078
Op imiza ion-Based In e se Iden i ica ion o he Pa ame e s
o a Conc e e Cap Ma e ial Model
Pe K ál 1, Filip Hokeš 1, Ma in Hušek 1, Jiří Kala 1, Pe H adil 1
1 Facul y o Ci il Enginee ing, Ins i u e o S uc u al Mechanics, B no Uni e si y o
Technology, Ve eří 331/95, 602 00 B no, Czech Republic
[email p o ec ed]
Abs ac . Issues conce ning he ad anced nume ical analysis o conc e e building s uc u es in
sophis ica ed compu ing sys ems cu en ly equi e he in ol emen o nonlinea mechanics
ools. The e o s o design sa e , mo e du able and mainly mo e economically e icien
conc e e s uc u es a e suppo ed ia he use o ad anced nonlinea conc e e ma e ial models
and he geome ically nonlinea app oach. The applica ion o nonlinea mechanics ools
undoub edly p esen s ano he s ep owa ds he app oxima ion o he eal beha iou o conc e e
building s uc u es wi hin he amewo k o compu e nume ical simula ions. Howe e , he
success a e o his applica ion depends on ha ing a pe ec unde s anding o he beha iou o
he conc e e ma e ial models used and ha ing a pe ec unde s anding o he used ma e ial
model pa ame e s meaning. The e ec i e applica ion o nonlinea conc e e ma e ial models
wi hin compu e simula ions o en becomes e y p oblema ic because hese ma e ial models
e y o en con ain pa ame e s (ma e ial cons an s) whose alues a e di icul o ob ain.
Howe e , ge ing o he co ec alues o ma e ial pa ame e s is e y impo an o ensu e
p ope unc ion o a conc e e ma e ial model used. Today, one possibili y, which pe mi s
success ul solu ion o he men ioned p oblem, is he use o op imiza ion algo i hms o he
pu pose o he op imiza ion-based in e se ma e ial pa ame e iden i ica ion. Pa ame e
iden i ica ion goes hand in hand wi h expe imen al in es iga ion while i ying o ind
pa ame e alues o he used ma e ial model so ha he esul ing da a ob ained om he
compu e simula ion will bes app oxima e he expe imen al da a. This pape is ocused on he
op imiza ion-based in e se iden i ica ion o he pa ame e s o a conc e e cap ma e ial model
which is known unde he name he Con inuous Su ace Cap Model. Wi hin his pape ,
ma e ial pa ame e s o he model a e iden i ied on he basis o in e ac ion be ween nonlinea
compu e simula ions, g adien based and na u e inspi ed op imiza ion algo i hms and
expe imen al da a, he la e o which ake he o m o a load-ex ension cu e ob ained om he
e alua ion o uniaxial ensile es esul s. The aim o his esea ch was o ob ain ma e ial model
pa ame e s co esponding o he quasi-s a ic ensile loading which may be u he used o he
esea ch in ol ing dynamic and high-speed ensile loading. Based on he ob ained esul s i can
be concluded ha he se goal has been eached.
1. In oduc ion
Con inuous and ex ensi e use o conc e e o he pu pose o building he new s uc u es cu en ly
leads o he e o s o e ine he design o conc e e s uc u es h ough he compu e nume ical
simula ions based on he ini e elemen me hod [1-3]. These e o s ela ed o he design o sa e , mo e
du able and mo e economically e icien conc e e s uc u es, howe e , equi e he in ol emen o
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IOP Con . Se ies: Ma e ials Science and Enginee ing 245 (2017) 032078 doi:10.1088/1757-899X/245/3/032078
ools o he ad anced nume ical analysis. The in ol emen o ools o he ad anced nume ical analysis
especially means ha i is necessa y o conside he nonlinea beha iou o conc e e wi hin he con ex
o compu e simula ions in ended o he analysis and design o conc e e s uc u es. Cu en
compu a ional sys ems based on he ini e elemen me hod, which include o example he p og ams
ANSYS [4], LS-Dyna [5] o A ena [6], o e a ela i ely la ge amoun o ma e ial models which a e
sui able o desc ibing he nonlinea beha iou o conc e e. These nonlinea ma e ial models o
conc e e may ind he use in he s a ic nume ical simula ions bu also in he dynamic nume ical
simula ions and a numbe o publica ions is de o ed o hei desc ip ion and p ac ical applica ion [7-
10]. Howe e , he bigges p oblem ela ed o he use o he nonlinea ma e ial models o conc e e is
o en he inabili y o p ope ly de ine he alues o he ma e ial pa ame e s (cons an s) because hey can
o en be ob ained only on he basis o he special es s o conc e e. Some pa ame e s ha e e en only
pu ely ma hema ical meaning and de ining o hei alues is no , he e o e, so easy. Howe e , he
co ec de ini ion o pa ame e alues o he used ma e ial model is ex emely impo an o he co ec
desc ip ion o he beha iou o conc e e wi hin he compu e simula ion. Solu ion o he men ioned
p oblem is cu en ly possible wi h he use o so-called he in e se analysis [11, 12].
In e se analysis, o he wise known as in e se iden i ica ion, allows o ind such pa ame e alues o
he used nonlinea conc e e ma e ial model whe ein he esul ing esponse o he s uc u e ob ained
om he compu e simula ion is e y simila o he expe imen ally measu ed esponse o he conc e e
s uc u e. The p inciple o he in e se analysis is based on a combina ion o nume ical and
expe imen al analysis wi h op imiza ion algo i hms, me hods o p ocedu es. Cu en ly, he mos used
me hods o he in e se pa ame e iden i ica ion o nonlinea ma e ial models o conc e e a e me hods
based on he exe cise o a i icial neu al ne wo ks [13]. A e y powe ul ool in he ield o he in e se
analysis is also he op iSLang p og am [14] which includes a a ie y o op imiza ion algo i hms
sui able o he op imiza ion-based in e se iden i ica ion o he ma e ial pa ame e s [15, 16].
This pape is ocused on he op imiza ion-based in e se iden i ica ion o he pa ame e s o a
conc e e cap ma e ial model which is known unde he name he Con inuous Su ace Cap Model and
which is implemen ed in an explici ini e elemen sol e LS-Dyna [17]. Wi hin his pape , pa ame e s
o he ma e ial model a e iden i ied on he basis o in e ac ion be ween nonlinea compu e
simula ions, g adien based and na u e inspi ed op imiza ion algo i hms implemen ed in he op iSLang
p og am and expe imen al da a, he la e o which ake he o m o a load-ex ension cu e ob ained
om he e alua ion o uncon ined uniaxial ensile es esul s.
2. Uncon ined uniaxial ensile es s
The p ocess o he op imiza ion-based in e se pa ame e iden i ica ion pe o med in his pape
equi ed he expe imen al da a. Fo his pu pose, he expe imen al da a ob ained om he e alua ion o
uncon ined uniaxial ensile es esul s ha we e made wi hin [18] we e used. Used expe imen al da a
ook he o m o a load-ex ension cu e which desc ibed he nonlinea beha iou o conc e e es
specimens du ing he uncon ined uniaxial ensile loading. The load-ex ension cu e is shown in igu e
1.
Conc e e es specimens used wi hin he o iginal uncon ined uniaxial ensile es s pe o med in he
con ex o [18] had dimensions 305 x 60 x 19 mm3 (leng h x wid h x dep h o he c i ical c oss-
sec ional a ea). The es specimens we e manu ac u ed and hen hey we e subjec ed o he conc e e
ha dening p ocess which las ed 28 days. The uniaxial comp ession s eng h o he 28 days old and
ha dened conc e e was 44 MPa. The maximum agg ega e size used was 10 mm. The expe imen al da a
we e measu ed in 85 mm leng h o each es specimen. Du ing he uncon ined uniaxial ensile loading,
conc e e es specimens we e s e ched a a quasi-s a ic cons an loading eloci y.
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IOP Con . Se ies: Ma e ials Science and Enginee ing 245 (2017) 032078 doi:10.1088/1757-899X/245/3/032078
I is clea om igu e 1 ha du ing he uncon ined uniaxial ensile loading, he conc e e es
specimens i s exhibi ed linea ly elas ic beha iou be o e eaching he maximum ensile o ce (i.e. he
maximum ensile load capaci y o he conc e e es specimens). A e eaching he maximum ensile
o ce, he conc e e es specimens began o show signs o ensile s ain so ening. The ensile s ain
so ening o he conc e e began o asse i sel as a esul o he damage o he conc e e es specimens
as he ul ima e s eng h o he conc e e in uniaxial ension was eached.
Figu e 1. Expe imen ally-measu ed load-ex ension cu e.
3. Compu e simula ions
The p ocess o he op imiza ion-based in e se pa ame e iden i ica ion pe o med in his pape u he
demanded pe o ming o nonlinea compu e simula ions. Fo his pu pose, he simpli ied
compu a ional model o he uncon ined uniaxial ensile es was c ea ed in LS-Dyna so wa e which is
based on an explici ini e elemen me hod and in which he nonlinea compu e simula ions we e
pe o med.
3.1. The compu a ional model
Agains he eal uncon ined uniaxial ensile es , he compu a ional model o his es c ea ed wi hin
his pape was e y simpli ied. Only he c i ical pa o he es specimen was modelled. This means
ha he ini e elemen model o he es specimen ook he o m only o he measu ed pa o he es
specimen wi h he leng h o 85 mm. Explici 3-D s uc u al ini e elemen s we e used o c ea ion o
he ini e elemen model. In e ms o bounda y condi ions, suppo s we e no applied wi hin he
con ex o he ini e elemen model. Howe e , linea ly inc eased e ical displacemen s o e ime we e
p esc ibed o nodes o bo h bases o he ini e elemen model, see igu e 2. These displacemen s
simula ed he axial s e ching o he model a a cons an loading eloci y. The nonlinea ma e ial
beha iou o he model was modelled h ough he Con inuous Su ace Cap ma e ial model [19, 20].
The men ioned simpli ica ions in oduced in o he compu a ional model o he uncon ined uniaxial
ensile es we e accep able because leng h and c oss-sec ional a ea o he ini e elemen model
co esponded o leng hs and c i ical c oss-sec ional a eas o he conc e e es specimens o which he
expe imen al da a we e measu ed. The simpli ica ions we e also accep able om he an age poin o
damage. In eal es s he damage occu ed always jus on he measu ed leng h o he es specimen in
he place o he c i ical c oss-sec ional a ea. I ollows ha in e ms o he damage, i was necessa y o
cons uc a ini e elemen model a leas as he measu ed pa o he es specimen wi h he c i ical
c oss-sec ional a ea. This necessi y was me wi hin he con ex o his pape .
The compu a ional model used wi hin his pape is shown in igu e 2.
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IOP Con . Se ies: Ma e ials Science and Enginee ing 245 (2017) 032078 doi:10.1088/1757-899X/245/3/032078
Figu e 2. The compu a ional model.
3.2. Nonlinea ma e ial model o conc e e
In his pape , nonlinea ma e ial beha iou was included in o he compu a ional model h ough he
ma e ial model which is known unde he name he Con inuous Su ace Cap Model [19, 20]. This
ma e ial model is a pa o he ma e ial model lib a y which is implemen ed in LS-Dyna so wa e.
Theo e ical backg ound o he Con inuous Su ace Cap Model is based on a yield su ace which is
de ined as a unc ion o h ee s ess in a ian s acco ding o he equa ion [21, 22]:
22
123 2 3 1 1
(, , ) ( ) () (,)
c
YI J J J J F I F I

 (1)
whe e I1 is he i s in a ian o he s ess enso , J2 and J3 a e in a ian s o he de ia o y s ess enso
(second and hi d), (J3) is he Rubin s eng h educ ion ac o and

is he cap ha dening pa ame e .
The yield su ace is composed o wo pa s. These being he shea ailu e su ace F (I1) and he
ha dening compac ion su ace Fc (I1,

). The exp ession o he shea ailu e su ace is gi en by
equa ion:
1
11
() expI
FI I




  (2)
whe e

,

, λ, and

a e he ma e ial cons an s which a e usually de e mined on he basis o he iaxial
comp ession es s. The ha dening compac ion su ace is de ined by equa ions:
2
1
12
(())
(,) 1 (() ())
c
IL
FI XL



  o 1()IL

 (3)
1
(,) 1
c
FI


o 1()IL


(4)

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IOP Con . Se ies: Ma e ials Science and Enginee ing 245 (2017) 032078 doi:10.1088/1757-899X/245/3/032078
in a con ex wi h equa ions:
()L



o 0


 (5)
0
()L



o 0



(6)
1
() () ()
XLRFI



 (7)
whe e R is he cap aspec a io. Wi hin he yield su ace, he shea ailu e su ace and ha dening
compac ion su ace a e combined using a mul iplica i e o mula ion which allows hei combina ion
o be con inuous and smoo h a hei in e sec ion.
The Con inuous Su ace Cap Model allows wi hin i s o mula ion o ake in o accoun he e ec o
s ain a e on he esul ing s ess s a e. Howe e , his model capabili y can be neglec ed du ing he
calcula ions. The esponse o he model is hen always quasi-s a ic and independen o compu a ional
ime. Due o he men ioned ac s, i is clea ha he ma e ial model can be used in dynamic, bu also in
quasi-s a ic o s a ic, compu e simula ions. The ac , ha he ma e ial model allows o calcula e he
quasi-s a ic esponse, was used wi hin his pape .
Table 1. The iden i ied ma e ial pa ame e s o he Con inuous Su ace Cap Model.
Ma e ial
pa ame e Pa ame e desc ip ion Uni
RO Mass densi y, ρ. Mg/mm3
E Young’s modulus, E. MPa
PR Poisson’s a io, ν. -
ALPHA T iaxial comp ession su ace cons an e m, α. MPa
THETA T iaxial comp ession su ace linea e m, θ. -
LAMDA T iaxial comp ession su ace nonlinea e m, λ. MPa
BETA T iaxial comp ession su ace exponen , β. MPa-1
ALPHA1 To sion su ace cons an e m, α1. -
THETA1 To sion su ace linea e m, θ1. MPa-1
LAMDA1 To sion su ace nonlinea e m, λ1. -
BETA1 To sion su ace exponen , β1. MPa-1
ALPHA2 T iaxial ex ension su ace cons an e m, α2. -
THETA2 T iaxial ex ension su ace linea e m, θ2. MPa-1
LAMDA2 T iaxial ex ension su ace nonlinea e m, λ2. -
BETA2 T iaxial ex ension su ace exponen , β2. MPa-1
R Cap aspec a io, R. -
X0 Cap ini ial loca ion, X0. MPa
W Maximum plas ic olume compac ion, W. -
D1 Linea shape pa ame e , D1. MPa
D2 Quad a ic shape pa ame e , D2. MPa2
B Duc ile shape so ening pa ame e , B. -
GFC F ac u e ene gy in uniaxial s ess, G c. N/mm
D B i le shape so ening pa ame e , D. -
GFT F ac u e ene gy in uniaxial ension, G . N/mm
GFS F ac u e ene gy in pu e shea s ess, G s. N/mm
The Con inuous Su ace Cap ma e ial model is implemen ed in LS-Dyna so wa e in wo
modi ica ions, speci ically as he gene al e sion *MAT_CSCM and he modi ied e sion
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IOP Con . Se ies: Ma e ials Science and Enginee ing 245 (2017) 032078 doi:10.1088/1757-899X/245/3/032078
*MAT_CSCM_CONCRETE [17]. The gene al e sion *MAT_CSCM o he ma e ial model was used
in calcula ions pe o med wi hin his pape . In o de o ob ain he mos ealis ic esponse o he model,
a o al o 25 pa ame e s (cons an s) o he ma e ial model e sion *MAT_CSCM we e iden i ied.
Howe e , he ma e ial pa ame e s G (shea modulus) and K (bulk modulus), which o iginally
belonged be ween iden i ied pa ame e s, we e eplaced by pa ame e s E (Young’s modulus) and PR
(Poisson’s a io, ν) due o hei dependency on hese pa ame e s acco ding o he equa ions:
2(1 )
E
G



(8)
3(1 2 )
E
K

 (9)
Desc ip ions and used uni s o men ioned 25 iden i ied ma e ial pa ame e s a e gi en in able 1
[17].
4. Op imiza ion-based in e se pa ame e iden i ica ion
In his pape , he op imiza ion-based in e se pa ame e iden i ica ion was pe o med using he
op iSLang p og am and consis ed o h ee s eps:
1. Sensi i i y analysis
2. Global op imiza ion
3. Local op imiza ion
4.1. Sensi i i y analysis
The sensi i i y analysis [23, 24] o med he i s s ep o he whole ma e ial pa ame e iden i ica ion
p ocess. Wi hin he con ex o his i s s ep he sensi i i y o he inpu a iable da a o he de ined
e e ence esponse was analyzed. The inpu a iable da a we e ep esen ed by indi idual ma e ial
pa ame e s ha we e o be iden i ied and ha o med so-called design ec o . Re e ence esponse was
ep esen ed by indi idual poin s lying on he used expe imen ally-measu ed load-ex ension cu e. The
majo goal o he sensi i i y analysis was o educe he numbe o iden i ied ma e ial pa ame e s in he
design ec o o he necessa y minimum. Ano he goal was o modi y he ange o a iabili y o he
indi idual ma e ial pa ame e s con ained in he design ec o .
Sensi i i y analysis was ca ied ou ia he s a is ical me hod known as he La in Hype cube
Sampling (LHS) me hod [14]. Based on his me hod, a o al o 300 andom ealiza ions o he design
ec o we e gene a ed. The gene a ed amoun o andom ealiza ions su icien ly co e ed he design
space.
The esul s p oduced by he sensi i i y analysis indica ed ha only 11 iden i ied ma e ial
pa ame e s ou o he o al o 25 exe ed majo in luence on he esul an o m o he nume ically-
simula ed load-ex ension cu e. The e o e, o he subsequen global and local op imiza ion he
o iginal design ec o , which con ained all 25 iden i ied ma e ial pa ame e s gi en in able 1, was
educed o he design ec o which con ained only men ioned 11 pa ame e s exe ed majo in luence
on he esul an o m o he load-ex ension cu e. A e he educ ion he design ec o acqui ed he
ollowing o m:

T
,,, ,,2,2,2,,,
E
ALPHA THETA LAMDA BETA THETA BETA D D GFT GFS
educed
X (10)
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IOP Con . Se ies: Ma e ials Science and Enginee ing 245 (2017) 032078 doi:10.1088/1757-899X/245/3/032078
In he educed design ec o X educed you can see he men ioned 11 ma e ial pa ame e s.
The alues o he p e-op imized ma e ial pa ame e s and objec i e unc ion (EUCLID_NORM)
ob ained om he bes andom ealiza ion gene a ed by LHS me hod a e gi en in able 2.
Table 2. The esul ing iden i ied ma e ial pa ame e alues o he Con inuous Su ace Cap Model.
Ma e ial pa ame e Uni
Sensi i i y
analysis
(LHS me hod)
Global and local
op imiza ion
(NLPQL and PSO
me hods)
Global
op imiza ion
(NLPQL
me hod)
Local
op imiza ion
(PSO
me hod)
P e-op imized
alues De e minis ic alues Op imized
alues
Op imized
alues
RO Mg/mm3 2.32910-9 2.40010-9 - -
E MPa 31600 - 31600 30862
PR - 0.1855 0.1500 - -
ALPHA MPa 15.509 - 15.509 15.432
THETA - 0.3432 - 0.3216 0.3186
LAMDA MPa 10.388 - 10.388 10.257
BETA MPa-1 2.18510-2 - 2.18910-2 2.19710-2
ALPHA1 - 0.7441 0.6500 - -
THETA1 MPa-1 1.14810-3 0.70010-3 - -
LAMDA1 - 0.1908 0.1600 - -
BETA1 MPa-1 6.22410-2 4.50010-2 - -
ALPHA2 - 0.6184 0.5800 - -
THETA2 MPa-1 7.99510-4 - 7.99510-4 8.51410-4
LAMDA2 - 0.1645 0.1200 - -
BETA2 MPa-1 7.11210-2 - 7.11210-2 7.26410-2
R - 5.2871 4.7000 - -
X0 MPa 104.584 95.000 - -
W - 6.61410-2 4.00010-2 - -
D1 MPa 2.84910-4 2.00010-4 - -
D2 MPa2 3.44510-7 - 3.44510-7 3.42410-7
B - 81.988 100.000 - -
GFC N/mm 4.6363 2.6000 - -
D - 0.2678 - 0.1638 0.1000
GFT N/mm 4.97410-2 - 4.79810-2 4.82710-2
GFS N/mm 4.11610-2 - 6.34310-2 6.28410-2
EUCLID_NORM kN 0.344427 - 0.280801 0.252410
4.2. Global op imiza ion
The global op imiza ion o med he second s ep o he whole ma e ial pa ame e iden i ica ion p ocess.
Wi hin his second s ep he op imized ma e ial pa ame e alues we e sough so ha he esul o
compu e simula ion app oxima ed he expe imen al da a so well as possible. De ined objec i e
unc ion was used o he e alua ion o global op imiza ion esul s. The e o e, wi hin he global
op imiza ion he op imized ma e ial pa ame e alues we e sough so ha he alue o de ined
objec i e unc ion was minimized. Based on p e ious in o ma ion, i is clea ha he global
op imiza ion was based on minimizing he objec i e unc ion [25]. The objec i e unc ion de ined o
he pu poses o his pape ook he o m o Euclidean no m which was, o cou se, minimized:
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WMCAUS IOP Publishing
IOP Con . Se ies: Ma e ials Science and Enginee ing 245 (2017) 032078 doi:10.1088/1757-899X/245/3/032078

n2
ii
i=1
_minEUCLID NORM y y 

 (11)
whe e, o yi, we subs i u ed he o ce alues ob ained om he app op ia e nume ically-simula ed
load-ex ension cu e a he ce ain de o ma ions, and yi* was subs i u ed wi h he o ce alues ob ained
om he expe imen al load-ex ension cu e a he same de o ma ions.
As poin ed ou abo e, he global op imiza ion in ol ed only hose ma e ial pa ame e s ha we e
pa o he educed design ec o . The emaining pa ame e s we e de ined by he cons an alues om
hei o iginal ange o a iabili y which was ob ained o each pa ame e on he basis o es
calcula ions. The global op imiza ion o he ma e ial pa ame e alues was pe o med using he
g adien based op imiza ion me hod known as he Non-Linea P og amming by Quad a ic Lag angian
(NLPQL) [14]. NLPQL is a sequen ial quad a ic p og amming me hod which sol es p oblems wi h
smoo h con inuously di e en iable objec i e unc ion and cons ain s. The algo i hm o his me hod
uses a quad a ic app oxima ion o he Lag angian unc ion and a linea iza ion o he cons ain s. Fo
he calcula ions pe o med ia he NLPQL me hod, he bes andom ealiza ion acqui ed om he LHS
me hod was used as he s a ing poin .
The op imized alues o he ma e ial pa ame e s p o ided by he bes gene a ion o he NLPQL
me hod a e, oge he wi h he ele an minimum alue o he objec i e unc ion, gi en in able 2.
4.3. Local op imiza ion
The local op imiza ion o med he hi d s ep o he whole ma e ial pa ame e iden i ica ion p ocess.
Wi hin he con ex o his hi d s ep, aim, pu pose and objec i e unc ion a e he same as in he case o
he global op imiza ion. The objec i e unc ion was, o cou se, minimized again. The local
op imiza ion was pe o med in an e o o sea ch he icini y o he global minimum wi h a goal o y
o e ine he men ioned global minimum.
As in he global op imiza ion case, he local op imiza ion in ol ed only ma e ial pa ame e s ha
we e pa o he educed design ec o . The emaining pa ame e s we e de ined by he same cons an
alues as in he case o he global op imiza ion. The local op imiza ion o he ma e ial pa ame e s was
ca ied ou using he na u e inspi ed op imiza ion me hod known as he Pa icle Swa m Op imiza ion
(PSO) me hod [14]. I is he me hod ha is inspi ed by he beha iou o bi d locks du ing he
sea ching o ood. Fo he calcula ions pe o med ia he PSO me hod, he bes gene a ion o he
NLPQL me hod was used as he s a poin .
The op imized alues o he ma e ial pa ame e s p o ided by he bes gene a ion o he PSO
me hod a e, oge he wi h he ele an minimum alue o he objec i e unc ion, gi en in able 2.
I is clea om able 2 ha he PSO me hod p o ided he mos op imized ma e ial pa ame e s
because he alue o he objec i e unc ion was he smalles o his me hod. Figu e 3 below compa es
he load-ex ension cu e ob ained ia he compu e simula ion, in which we applied he mos
op imized pa ame e alues o he Con inuous Su ace Cap Model om he PSO me hod, wi h he
expe imen ally-measu ed load-ex ension cu e. I is hen ob ious om he ep esen a ion ha he
pa ame e s o he Con inuous Su ace Cap Model we e iden i ied e y accu a ely ia he PSO me hod
because he esul o he compu e simula ion ensu es a e y good app oxima ion o he expe imen al
da a.