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STUDY OF THE EFFICIENCY AND ACCURACY OF OPTIMISATION
ALGORITHMS WITHIN INVERSE IDENTIFICATION OF THE
PARAMETER VALUES OF A NONLINEAR CONCRETE MATERIAL
MODEL
Pe KRÁL1, Jiří KALA1, Pe HRADIL1
1Ins i u e o S uc u al Mechanics, Facul y o Ci il Enginee ing, B no Uni e si y o Technology,
Ve eří 331/95, 602 00 B no, Czech Republic
[email p o ec ed], [email p o ec ed], [email p o ec ed]
DOI: 10.35181/ ces-2019-0013
Abs ac . The in e se iden i ica ion o he pa ame e
alues o nonlinea ma e ial models, which ha e been
de eloped o , in e alia, conc e e modelling, is cu en ly a
p ocess ha is widely used and in es iga ed in he ield o
esea ch and de elopmen . Today he e a e se e al
app oaches ha can be employed o he in e se
iden i ica ion p ocess. One o he mos signi ican o hese
app oaches in ol es he use o op imisa ion algo i hms
which, howe e , o en demons a e a ying le els o
p ecision and e iciency wi hin speci ic asks. These
aspec s a e he subjec o he esea ch p esen ed in his
con ibu ion.
Keywo ds
Compu a ional model, conc e e, expe imen al da a, ou -
poin bend, global op imisa ion, ma e ial model, objec i e
unc ion, op imisa ion algo i hms.
1. In oduc ion
The modelling o con inuum mechanics asks using
nonlinea mechanics ools is cu en ly he main a ea o
ocus a many scien i ic ins i u ions ([1], [2], [3] and [4]).
The e m “use o nonlinea mechanics ools” e e s in
pa icula o he use o geome ic and ma e ial (physical)
nonlinea i ies wi hin con inuum mechanics asks. The
necessi y o using geome ic o ma e ial nonlinea i ies
wi hin nume ical calcula ions is dependen on he ype o
s uc u e in ques ion and pa icula ly on he selec ed
ma e ial om which i will be made. Mode n
compu a ional sys ems based on he ini e elemen me hod
([5], [6], [7] and [8]), which a e cu en ly seeing
widesp ead use in he in es iga ion o con inuum
mechanics asks, con ain a se ies o me hods o he
conside a ion o he nonlinea beha iou o s uc u es,
along wi h many nonlinea ma e ial models ha can be
employed o desc ibe he beha iou o p ac ically any
ma e ial in he con ex o nume ical simula ions. Howe e ,
he applica ion o nonlinea ma e ial models wi hin
calcula ions o a nume ical na u e gi es ise o a
undamen al di icul y in e ms o he necessi y o de ine
he pa ame e s o hese models co ec ly in o de o hem
o unc ion p ope ly. This ask o en is no e y easy as
nonlinea ma e ial models (and nonlinea ma e ial models
o conc e e in pa icula ) e y o en include pa ame e s o
a pu ely ma hema ical na u e o pa ame e s which can only
be de i ed using special expe imen al da a. I ele an da a
which would enable he alues o he ma e ial model’s
pa ame e s o be de i ed di ec ly a e no a ailable, he
p ocess o he in e se iden i ica ion o ma e ial pa ame e
alues can cu en ly be used o deal wi h his p oblem ([9],
[10], and [11]).
Du ing in e se iden i ica ion, ou pu da a usually
consis ing o expe imen al da a a e gene ally used o
ob ain alues o ma e ial and o he pa ame e s. The
in e se iden i ica ion p ocess is he e o e based on he
combina ion o expe imen al da a wi h nume ical
calcula ions and iden i ica ion app oaches. The goal is o
ob ain he bes possible app oxima ion o expe imen al
da a om nume ically simula ed da a. The mos widely
used iden i ica ion app oaches oday a e me hods based on
he exe cise o a i icial neu al ne wo ks [12] and
op imisa ion algo i hms [13]. The capabili y o using
op imisa ion algo i hms o pe o m he in e se
iden i ica ion o pa ame e alues is o e ed by, e.g.
op iSLang so wa e [14], which con ains a o al o i e
op imisa ion algo i hms whose e iciency and accu acy
can a y o di e en asks.
The aim o his pape is o in es iga e he e iciency
and accu acy o op iSLang op imisa ion algo i hms du ing
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he in e se iden i ica ion o alues o a small quan i y o
pa ame e s o he modi ied e sion o a nonlinea ma e ial
model o conc e e known as he Con inuous Su ace Cap
Model, which is implemen ed in he explici ini e elemen
compu a ional sys em LS-Dyna [15]. Fo his pu pose, a
ask in he o m o a ou -poin bending es is ca ied ou
on a high, s eel- ein o ced conc e e beam. To achie e he
se goal, he applica ion o his conc e e model and o he
ma e ial models wi hin he compu a ional model o a es
c ea ed in LS-Dyna is equi ed as well as he use o
expe imen al da a ob ained om a eal ou -poin bending
es . These aspec s a e desc ibed in he ollowing chap e s.
2. Ma e ial Models
A o al o h ee ma e ial models implemen ed in he LS-
Dyna p og amme we e used in he compu a ional model o
he ou -poin bending es . A modi ied e sion o he
Con inuous Su ace Cap Model was used o model he
beha iou o he high conc e e beam. The beha iou o he
conc e e ein o cemen was modelled using he Plas ic
Kinema ic Model and he beha iou o he washe s was
modelled using he Linea Elas ic Model.
2.1. The Con inuous Su ace Cap Model
The Con inuous Su ace Cap Model ([16] and [17]) is a
nonlinea ma e ial model o conc e e based on elas o-
plas ic cons i u i e heo y. The occu ence o plas ic
de o ma ions is, wi hin he model, con olled by he
achie emen o a yield su ace [18] whose unc ional
ela ionship can be exp essed as:
22
123 2 3 1 1
(, , ) ( ) () (,),
c
YI J J J J F I FI
κ
=−ℜ (1)
whe e he second membe on he igh side o he equa ion
is a combina ion o he shea ailu e unc ion F (I1) and he
ha dening model Fc (I1,
κ
) ia a mul iplica i e o mula ion.
The shea ailu e unc ion and he ha dening model can be
exp essed ma hema ically as:
1
11
() exp ,
I
FI I
β
αλ θ
−
=− + (2)
2
1
12
(())
(,) 1(() ())
c
IL
FI XL
κ
κκκ
−
=− −
o 1(),IL
κ
> (3)
1
(,)1
c
FI
κ
= o 1(),IL
κ
≤ (4)
whe e:
()L
κκ
= o 0,
κκ
> (5)
0
()L
κκ
= o 0,
κκ
≤ (6)
1
() () ().
XLRFI
κκ
=+ (7)
The desc ip ion o he pa ame e s con ained in Eqs.
(1)-(7) is as ollows: I1 is he i s in a ian o he s ess
enso , J2 and J3 a e he second and hi d in a ian s o he
de ia o ic pa o he s ess enso , ℜ(J3) is he educ ion
ac o acco ding o Rubin,
κ
is he ha dening pa ame e ,
α
,
β
, λ, and
θ
a e ma e ial cons an s which a e de i ed om
he expe imen al es ing o conc e e in iaxial
comp ession, and R is he a io pa ame e o he ha dening
model.
As he Con inuous Su ace Cap Model is pa o an
explici ini e elemen sol e , i enables he e ec o he
s ain a e on he s ess s a e o be conside ed wi hin i s
o mula ion. Howe e , his abili y o he model can be
neglec ed du ing he calcula ions ia he pe inen se ing
o he model pa ame e IRATE (IRATE = 0: calcula ion
wi h he e ec o he s ain a e on he s ess s a e → he
iscous componen o he model is swi ched o ; IRATE =
1: calcula ion wi h he e ec o he s ain a e on he s ess
s a e → he iscous componen o he model is swi ched
on). I can be concluded om he abo e ac s ha i he
pa ame e IRATE equals ze o, he nume ically simula ed
esponse o he model co esponds o s a ic (slow) loading,
which means ha i is independen o he eloci y o
loading used. Fo his eason, he Con inuous Su ace Cap
Model can be used no only o he nume ical modelling o
he dynamic loading o conc e e s uc u es bu also o
model he quasi-s a ic o s a ic loading o conc e e
s uc u es. Wi hin he s udy desc ibed in his pape , he
pa ame e IRATE equals ze o was used because he s a ic
esponse o he s uc u e was modelled. In o de o p e en
he dependence o nume ical simula ion esul s on he
ini e elemen mesh, an algo i hm is implemen ed in he
ma e ial model which is based on he p inciple o he c ack
band model and hus ul ils he unc ion o a localisa ion
limi e .
The modi ied e sion o he Con inuous Su ace Cap
Model includes, abo e and beyond he basic e sion (25
ma e ial pa ame e s), a o al o 3 ma e ial pa ame e s
whose nume ical alues mus be de ined. The alues o he
o he pa ame e s a e gene a ed au oma ically based on he
alues o hese h ee pa ame e s. On he basis o he
in e se iden i ica ion o he alues o hese h ee
pa ame e s, he e iciency and accu acy o selec ed
op imisa ion algo i hms we e es ed in his pape .
Desc ip ions and used uni s a e lis ed o he pa ame e s o
he modi ied e sion o he Con inuous Su ace Cap Model
in Tab. 1 [15].
Tab.1: Ma e ial pa ame e s o he modi ied e sion o he Con inuous
Su ace Cap Model.
Pa ame e Desc ip ion o he pa ame e Uni
RO Mass densi y. Mg/mm3
FPC Uncon ined uniaxial comp essi e s eng h. MPa
DAGG Maximum agg ega e size. mm
2.2. The Plas ic Kinema ic Model
The Plas ic Kinema ic Model is a bilinea model based on
elas o-plas ic cons i u i e heo y. I is sui able o
modelling he beha iou o cons uc ion s eel o s eel
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eba , o o model he beha iou o plas ic ma e ial ha
beha es in a simila manne o s eel. The model enables
he modelling o a ma e ial wi h ha dening, which can be
iso opic o kinema ic, o wi hou ha dening. As he model
is pa o an explici ini e elemen sol e , i includes he
iscous beha iou al componen . I hus enables, wi hin i s
o mula ion, he conside a ion o he e ec o he s ain
a e on he s ess s a e. Howe e , his abili y o he model
can be neglec ed by using ze o alues o he ele an
pa ame e s, as is he case wi h he Con inuous Su ace Cap
Model.
In calcula ions ca ied ou o his pape , he iscous
componen o he model was neglec ed because he s a ic
esponse o he s uc u e was modelled, and he
o mula ion o he model wi hou ha dening was used. The
desc ip ions and uni s used o he ma e ial pa ame e s o
he Plas ic Kinema ic Model, whose alues needed o be
de ined, a e lis ed in Tab. 2 [15].
Tab.2: Ma e ial pa ame e s o he Plas ic Kinema ic Model.
Pa ame e Desc ip ion o he pa ame e Uni
RO Mass densi y. Mg/mm3
E Young’s modulus o elas ici y. MPa
PR Poisson’s a io. -
SIGY Yield s eng h. MPa
ETAN Tangen modulus (ETAN = 0: ma e ial wi hou
ha dening). MPa
2.3. The Linea Elas ic Model
As a mo e de ailed ma e ial model was no needed o he
modelling o he beha iou o he washe s, he Linea
Elas ic Model, o in o he wo ds a cons i u i e model
espec ing gene alised Hooke’s Law, was used o his
pu pose. The desc ip ions and uni s used o he ma e ial
pa ame e s o he Linea Elas ic Model, whose alues
needed o be de ined, a e lis ed in Tab. 3 [15].
Tab.3: Ma e ial pa ame e s o he Linea Elas ic Model.
Pa ame e Desc ip ion o he pa ame e Uni
RO Mass densi y. Mg/mm3
E Young’s modulus o elas ici y. MPa
PR Poisson’s a io. -
3. Expe imen al Da a and he
Compu a ional Model
3.1. Expe imen al Da a
The expe imen al da a used in his pape a e he esul s o
a ou -poin bending es ca ied ou on a high, s eel-
ein o ced conc e e beam. This es was ca ied ou and
desc ibed wi hin [19]. A schema ic ep esen a ion o he
es can be seen in Fig. 1. I shows he geome y o he
conc e e beam oge he wi h he geome y and loca ion o
he ein o cing ba and he washe s a he poin s whe e he
loading and suppo o he beam ook place. The ma e ial
pa ame e s o he ha dened (28 days old) conc e e and s eel
eba we e he ollowing, as s a ed in [19]:
Pa ame e s o he conc e e:
• Modulus o elas ici y: 20.68 GPa,
• Poisson’s a io: 0.15,
• Uniaxial comp essi e s eng h: 24.13 MPa,
• Uniaxial ensile s eng h: 3.10 MPa.
Pa ame e s o he s eel eba :
• Young’s modulus o elas ici y: 210.00 GPa,
• Yield s eng h: 344.75 MPa,
• C oss-sec ional a ea o he ein o cing ba :
0.71x10-4 m2.
Fig. 1: Schema ic ep esen a ion o he ou -poin bending es .
Fig. 2: Measu ed expe imen al da a.
As shown in Fig. 1, he beam was loaded wi h o ces P
du ing he es . The in ensi y o he o ces inc eased
linea ly o e ime un il 80 kN was achie ed. The eloci y
o loading was e y slow, hus he loading was s a ic.
Du ing he es , he e ical displacemen (de lec ion) o
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beam U was measu ed a midspan. The exac loca ion
whe e he de lec ion o he beam was measu ed is ma ked
wi h poin B in Fig. 1. The measu ed expe imen al da a a e
shown in Fig. 2.
3.2. Compu a ional Model
The compu a ional model o he ou -poin bending es
was c ea ed in LS-Dyna p og amme, du ing which he es
scheme om Fig. 1 was espec ed. The compu a ional
model equi ed he c ea ion o ini e elemen models o he
high beam, ein o cing ba and washe s, and he bounda y
condi ions (suppo s and loading) had o be de ined.
The ini e elemen model o he beam was c ea ed
using 3D eigh -node explici s uc u al ini e elemen s. The
ini e elemen mesh o ming he model o he beam was
egula (see Fig. 3). As he in e se iden i ica ion p ocess
equi ed he epea ed execu ion o nume ical simula ions
o he ask, he symme y o he ask was exploi ed (only
hal o he beam was modelled) and he consis ence o he
ini e elemen mesh o he beam model was chosen in such
a way ha he ime equi ed o he calcula ion was no oo
high, which i migh o he wise ha e been due o he use o
an explici ini e elemen algo i hm.
The ein o cing ba was modelled using 3D wo-node
explici beam ini e elemen s. The consis ence o he beam
ini e elemen mesh was adap ed o he ini e elemen
model o he beam in such a way ha con inui y be ween
he beam and he ein o cing ba model was ensu ed. The
dimensions o he ec angula c oss sec ion o he beam
ini e elemen s we e en e ed in such a way ha he
esul an c oss-sec ional a ea co esponds o he alue
om he expe imen (0.71x10-4 m2) which is a pa o his
pape .
The washe s we e modelled using 3D eigh -node
explici s uc u al ini e elemen s. The size o he ini e
elemen s o ming he models o he washe s co esponded
o he size o he ini e elemen s o ming he model o he
beam (see Fig. 3).
Fig. 3: Fini e elemen model o he in es iga ed ask.
The bounda y condi ions we e, wi hin he
compu a ional model o he in es iga ed ask, de ined in
he places whe e washe s we e loca ed and on he axis o
symme y o he high beam. The model o he bo om
washe had i s bounda y condi ions de ined in such a way
ha i s displacemen was p e en ed in he ho izon al and
e ical di ec ion. The model o he op washe had i s
bounda y condi ions de ined in such a way ha i s
displacemen was p e en ed only in he ho izon al
di ec ion because o he applica ion o loading. A he
loca ion o he axis o symme y, he symme ic bounda y
condi ions we e de ined o he model o he beam.
Loading was applied o he model o he op washe in he
o m o p essu e linea ly inc easing o e ime wi h a inal
o ce o 80 kN. Loading was also applied by conside ing
he s uc u e’s sel -weigh .
4. The In e se Iden i ica ion P ocess
4.1. Global Op imisa ion and he
Op imisa ion Algo i hms Used
Fo his pape , “global op imisa ion” was used in o de o
execu e he in e se iden i ica ion o he alues o he
pa ame e s o he conc e e ma e ial model. The aim o
global op imisa ion was o ind alues o hose pa ame e s
whose iden i ica ion we e equi ed and whose applica ion
would esul in nume ically simula ed da a displaying he
smalles possible de ia ion om he expe imen al da a
used. This would ep esen he “ e e ence esponse” wi hin
he in e se iden i ica ion p ocess. In o he wo ds, he
global minimum o he selec ed objec i e unc ion was
sough , along wi h op imal alues o he iden i ied
pa ame e s [13]. Global op imisa ion was ca ied ou using
op imisa ion algo i hms in he op iSLang p og amme,
du ing which hei accu acy and e iciency we e e alua ed.
Du ing he op imisa ion, he selec ed objec i e unc ion
which cha ac e ised he calcula ion o he Roo -Mean-
Squa e De ia ion (RMSD) was hus minimised. I s
ma hema ical exp ession was de ined by he equa ion:
()
2
,
1
min,
n
calc i e ,i
i
UU
RMSD n
=
−
=→

(8)
whe e U
calc,i
was subs i u ed by nume ically simula ed
e ical displacemen (de lec ion) alues co esponding o
he pe inen loading o ce alues and U
e ,i
was subs i u ed
by e e ence (expe imen ally measu ed) e ical
displacemen alues co esponding o he same loading
o ce alues. Pa ame e n equalled he numbe o da a
de ining he esul an shape o he expe imen al loading
cu e in Fig. 2 (n = 32).
As sugges ed abo e, he iden i ied pa ame e s we e
only pa ame e s o he Con inuous Su ace Cap Model.
The pa ame e s o o he ma e ial models used we e no
iden i ied because o hei negligible in luence on he
esul an shape o he nume ically simula ed loading cu e,
which was e i ied by es calcula ions. The iden i ied
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pa ame e s o med, wi hin global op imisa ion, a design
ec o which can be exp essed as:
{}
T
,, .RO FPC DAGG=
d
X (9)
Fig. 4: Nume ically simula ed da a o ma e ial pa ame e s om he
expe imen .
The iden i ied pa ame e s en e ed he design ec o as
con inuous andom a iables wi h a dis ibu ion o
p obabili y a in e als gi en by de ined bounda y alues.
The limi alues o he iden i ied pa ame e s gi en by he
speci ica ion o he ma e ial model o conc e e we e used
as bounda y alues [15] (see Tab. 4). Ma e ial pa ame e
alues om he expe imen we e used as ini ial alues o
he iden i ied pa ame e s, which we e necessa y o he
i s gene a ion o i e a ion o he ele an op imisa ion
algo i hm. The nume ically simula ed loading cu e o
hese pa ame e alues is shown in Fig. 4. Ma e ial
pa ame e s which we e no subjec o iden i ica ion we e
en e ed de e minis ically using alues om he expe imen
(see Tab. 5).
Tab.4: Ini ial and bounda y alues o iden i ied pa ame e s.
Pa ame e Uni Ini ial alue
Minimum
bounda y
alue
Maximum
bounda y
alue
RO Mg/mm3 2.400x10-9 2.100x10-9 2.450x10-9
FPC MPa 24.13000 20.00000 58.00000
DAGG mm 16.00000 8.00000 32.00000
Tab.5: Pa ame e alues o he ma e ial models which we e no subjec
o iden i ica ion.
Pa ame e Uni Plas ic Kinema ic
Model Linea Elas ic Model
RO Mg/mm3 7.850x10-9 7.850x10-9
E MPa 210000 210000
PR - 0.3 0.3
SIGY MPa 344.750 -
ETAN MPa 0 -
The in e se iden i ica ion p ocess using op imisa ion
algo i hms equi ed he epea ed execu ion o nume ical
simula ions o he in es iga ed ask, du ing which he
objec i e unc ion was minimised. The necessa y numbe
o epe i ions (o in o he wo ds he necessa y numbe o
gene a ions o i e a ions) pe o med o ind he global
minimum o he objec i e unc ion o which he op imal
alues o he iden i ied ma e ial pa ame e s would
co espond di e ed g ea ly o he di e en op imisa ion
algo i hms (see Tab. 6). Fo he pu poses o his pape a
o al o i e op iSLang op imisa ion algo i hms (b ie ly
desc ibed below) we e used in o de o s udy hei
e iciency and accu acy.
Non-Linea P og amming by Quad a ic Lag angian
(NLPQL)
NLPQL ([14] and [20]) is a sequen ial algo i hm based
on nonlinea quad a ic p og amming. This op imisa ion
algo i hm is sui able o he solu ion o asks which u ilise
smoo h, con inuous as well as di e en iable objec i e
unc ions and cons ain s. The algo i hm u ilises quad a ic
app oxima ion o he Lag angian unc ion and he
linea iza ion o cons ain s.
Simplex Me hod (Simplex)
The Simplex Me hod ([14] and [21]) is an i e a i e
op imisa ion algo i hm based on linea p og amming
which is execu ed sys ema ically wi h he pu pose o
de e mining an op imal solu ion om a se o easible
solu ions. The me hod is sui able o he op imisa ion o a
low numbe o pa ame e s.
Adap i e Response Su ace Me hod (ARSM)
ARSM ([14] and [22]) is a di ec op imisa ion
algo i hm sui able o he op imisa ion o bo h low and
high numbe s o pa ame e s. The main ad an age o he
algo i hm is he ac ha i p o ides an o e iew o he
beha iou o an objec i e unc ion wi hin he whole design
space, i allows he simple inclusion o addi ional
equi emen s in o he objec i e unc ion, and i also
equi es a ela i ely small numbe o design poin s.
E olu iona y Algo i hm (EA)
The E olu iona y Algo i hm ([14] and [23]) is one o a
g oup o op imisa ion algo i hms ha u ilise p ocesses
inspi ed by biological e olu ion (i.e. occu ing in he
na u al wo ld), such as mu a ion, ep oduc ion and
ecombina ion. The E olu iona y Algo i hm included in
op iSLang p og amme is speci ically based on he
combina ion o a gene ic algo i hm wi h an e olu ion
s a egy.
Pa icle Swa m Op imisa ion (PSO)
PSO ([14] and [24]) is one o a g oup o op imisa ion
algo i hms inspi ed by na u al phenomena. To be speci ic,
PSO is a me hod whose algo i hm is inspi ed by and ies
o imi a e he beha iou o locks o bi ds looking o ood.
4.2. Resul s and Thei E alua ion
A able compa ison o he esul s ob ained om global
op imisa ion om i e di e en op imisa ion algo i hms is

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shown in Tab. 6. To be speci ic, Table 6 shows he esul an
op imal alues o iden i ied pa ame e s o he modi ied
e sion o he Con inuous Su ace Cap Model ob ained
om he bes gene a ions o i e a ions o he op imisa ion
algo i hms, including he ele an minimum alues o he
objec i e unc ion which de e mine he accu acy o he
indi idual op imisa ion algo i hms used. Fu he mo e,
Table 6 also shows he numbe o i e a ions o gene a ions
o he op imisa ion algo i hms necessa y o ind he global
minimum o he objec i e unc ion and hus also he
op imal alues o iden i ied ma e ial pa ame e s. These
da a de e mine he e iciency o he op imisa ion
algo i hms used. Figu e 5 shows a g aphic compa ison o
he esul s, oge he wi h he expe imen al da a.
Tab.6: Table compa ison o he esul s.
Pa ame e Uni Op imal alue
NLPQL (numbe o necessa y i e a ions o he algo i hm = 21)
RO Mg/mm3 2.400x10-9
FPC MPa 20.00000
DAGG mm 8.00000
RMSD mm 0.0074627
Simplex (numbe o necessa y i e a ions o he algo i hm = 75)
RO Mg/mm3 2.409x10-9
FPC MPa 20.00000
DAGG mm 8.00829
RMSD mm 0.0074597
ARSM (numbe o necessa y i e a ions o he algo i hm = 180)
RO Mg/mm3 2.450x10-9
FPC MPa 20.00000
DAGG mm 8.00000
RMSD mm 0.0074581
EA (numbe o necessa y gene a ions o he algo i hm = 400)
RO Mg/mm3 2.450x10-9
FPC MPa 20.00000
DAGG mm 28.10027
RMSD mm 0.0073995
PSO (numbe o necessa y gene a ions o he algo i hm = 400)
RO Mg/mm3 2.438x10-9
FPC MPa 20.00007
DAGG mm 8.00000
RMSD mm 0.0074575
The esul s shown in Tab. 6 and in Fig. 5 show ha he
accu acy o all o he i e op imisa ion algo i hms used
was e y sa is ac o y as a e y good app oxima ion o he
expe imen al da a was achie ed by he nume ical
simula ions, du ing which op imal alues we e gained o
he iden i ied pa ame e s o he ma e ial model o conc e e
used in he indi idual op imisa ion algo i hms. Mo eo e ,
i can be seen ha he di e ences in accu acy be ween he
indi idual algo i hms a e p ac ically negligible ( he cu es
p ac ically o e lap). Howe e , when looking a he RMSD
alues o he indi idual algo i hms in Tab. 6, i can be
concluded ha he mos accu a e op imisa ion algo i hm
o he gi en ask is unambiguously he E olu iona y
Algo i hm (EA).
Fig. 5: G aphic compa ison o he esul s.
When looking a Tab. 6 i can addi ionally be s a ed ha
he mos e icien algo i hm o he gi en ask was clea ly
he Non-Linea P og amming by Quad a ic Lag angian
(NLPQL) as his algo i hm needed only 21 i e a ions o
ind he global minimum o he objec i e unc ion. I can
also be s a ed ha , in con as , he leas e icien
op imisa ion algo i hms we e he E olu iona y Algo i hm
and Pa icle Swa m Op imisa ion (PSO), as hese
algo i hms needed up o a o al o 400 gene a ions o ind
he global minimum o he objec i e unc ion.
I can be concluded om he abo e-men ioned ac s
ha wi h ega d o he accu acy and he inding o he
global minimum o he objec i e unc ion du ing global
op imisa ion, he E olu iona y Algo i hm is he mos
ad an ageous, e en hough his algo i hm is no he mos
e icien wi h ega d o calcula ion ime consump ion.
Mo eo e , he op imal alue o DAGG pa ame e o he
E olu iona y Algo i hm di e ed signi ican ly om he
DAGG alues o o he algo i hms (see Tab. 6). This
di e ence was due o he exis ence o wo e y close
minimum peaks o he objec i e unc ion, o which jus
one can be classi ied as he global minimum. Gi en he
smalles RMSD alue o he E olu iona y Algo i hm, i
can be concluded ha he global minimum o he objec i e
unc ion was jus ound using his algo i hm. In he case o
o he algo i hms, a local minimum which is, howe e , e y
close o he global minimum was ound. Fo his eason,
he esul s o all algo i hms can be conside ed as
compa able.
Ano he conclusion is ha om he poin o iew o
e iciency, he use o he NLPQL algo i hm and possibly
also he Simplex Me hod can be e y ad an ageous o
global op imisa ion as hese op imisa ion algo i hms also
exhibi e y good accu acy, which is compa able wi h ha
o he E olu iona y Algo i hm.
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5. Conclusion
This pape ocused on he pe o ming o a s udy o he
e iciency and accu acy o i e op imisa ion algo i hms
included in op iSLang so wa e du ing he in e se
iden i ica ion o alues o a small numbe o pa ame e s
o he modi ied e sion o he Con inuous Su ace Cap
Model implemen ed in he LS-Dyna compu a ional
sys em. A ou -poin bending es ask was used o his
pu pose. I was execu ed on a high, s eel- ein o ced
conc e e beam o which a compu a ional model was
c ea ed, and expe imen al da a we e ob ained.
The esul s o he s udy showed ha he accu acy o all
i e o he op imisa ion algo i hms used was e y
sa is ac o y as a e y good app oxima ion o expe imen al
da a was achie ed by he nume ical simula ions in which
he op imal alues o iden i ied pa ame e s ob ained ia
he indi idual op imisa ion algo i hms we e used.
Mo eo e , he di e ences in accu acy be ween he
indi idual algo i hms we e p ac ically negligible.
Howe e , du ing global op imisa ion he mos accu a e
op imisa ion algo i hm was he E olu iona y Algo i hm,
hough i was no he mos e icien as a as calcula ion
ime consump ion is conce ned. Also, he esul s o he
s udy showed ha he use o he NLPQL algo i hm, and
possibly he Simplex Me hod, can be e y ad an ageous
o global op imisa ion wi h ega d o e iciency. The
accu acy o hese algo i hms was also compa able o ha
o he E olu iona y Algo i hm.
Acknowledgemen s
This pape was c ea ed wi h inancial suppo o p ojec
GACR 17-23578S p o ided by he Czech Science
Founda ion and wi h inancial suppo o he uni e si y
speci ic esea ch p ojec FAST-J-18-5604, which was
p o ided by B no Uni e si y o Technology.
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