Ad ances in Enginee ing So wa e 190 (2024) 103598
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Resea ch pape
Robus imp o emen o he ini e-elemen -model upda ing o his o ical
cons uc ions ia a new combina i e compu a ional algo i hm
Ja ie Na anjo-Pé ez a, Rubén Rod íguez-Rome o b, Pablo Pachón c,∗, Víc o Compán c,
And és Sáez b, Aleksanda Pa ic d, Ja ie Fe nando Jiménez-Alonsob
aDepa men o Con inuum Mechanics and S uc u es, E.T.S. Ingenie os de Caminos, Canales y Pue os. Uni e sidad Poli écnica de Mad id, Mad id, Spain
bDepa men o Con inuum Mechanics and S uc u al Analysis, Uni e sidad de Se illa, Se ille, Spain
cDepa men o Building S uc u es and G ound Enginee ing, Uni e sidad de Se illa, Se ille, Spain
dVib a ion Enginee ing Sec ion, College o Enginee ing, Ma hema ics and Physical Sciences, Uni e si y o Exe e , Exe e , UK
ARTICLE INFO
Keywo ds:
Fini e elemen model upda ing
Maximum likelihood me hod
Bi-objec i e op imiza ion
Decision-making p oblems
His o ical building s one
ABSTRACT
Fini e-elemen -models a e usually employed o simula e he beha iou o his o ical cons uc ions. Howe e ,
despi e he high complexi y o hese nume ical models, he e a e always disc epancies be ween he ac ual
beha iou o he s uc u e and he nume ical p edic ions ob ained. In o de o imp o e hei pe o mance,
an upda ing p ocess can be implemen ed. Acco ding o his p ocess, he alue o he mos ele an physical
pa ame e s o he model is adjus ed o be e mimic he ac ual beha iou o he s uc u e. Fo his pu pose, he
ac ual s uc u al beha iou is usually cha ac e ized ia i s expe imen al modal p ope ies (na u al equencies
and associa ed ib a ion modes). Fo p ac ical enginee ing applica ions, he maximum likelihood me hod is
no mally conside ed o cope wi h his p oblem, due o i s easy implemen a ion oge he wi h an unde s andable
in e p e a ion o he upda ing esul s. Howe e , he complexi y o hese nume ical models makes un easible
he p ac ical implemen a ion o he p ocess due o he simula ion ime equi ed o i s compu a ion. In o de o
shed some ligh o his p oblem, a new combina i e compu a ional algo i hm is p oposed he ein. Addi ionally,
he pe o mance o he p oposal has been assessed success ully ia wo applica ions: (i) a alida ion example,
he model upda ing o a labo a o y oo b idge, in which he p ac ical implemen a ion o he algo i hm has been
desc ibed in de ail; and (ii) a case-s udy, he model upda ing o a complex his o ical cons uc ion, in which
he main ad an age o he p oposal has been highligh ed, a clea educ ion o he simula ion ime equi ed o
sol e he upda ing p oblem wi hou comp omising he accu acy o he solu ion ob ained.
1. In oduc ion
Fini e elemen (FE) me hod is usually employed o simula e nume -
ically he s uc u al beha iou o his o ical cons uc ions [1]. Howe e ,
despi e he g ea complexi y o hese models [2], he e a e always
di e ences be ween he p edic ions p o ided by hese nume ical ools
and he ac ual beha iou o he s uc u e [3]. These di e ences a e
no mally o igina ed by some o he ollowing sou ces o e o s and un-
ce ain y [4]: (i) he app oxima e solu ion o di e en ial equa ions ha
go e ns he dynamic beha iou o a s uc u e (nume ical unce ain y);
(ii) he imp ecise alue o he model inpu pa ame e s (pa ame e
unce ain y); (iii) he incomple e de ini ions o unde lying physics due
o assump ions and idealiza ions (bias e o ); and (i ) he a iabili y in
measu emen s (expe imen al unce ain y).
In o de o educe he e ec o hese sou ces o unce ain y, he
FE model upda ing me hod is no mally conside ed [5]. Acco ding o
∗Co esponding au ho .
E-mail add ess: [email p o ec ed] (P. Pachón).
his me hod, some physical pa ame e s o he FE model a e modi ied
in o de o be e mimic he eal beha iou o he s uc u e. In he case
o complex his o ical cons uc ion, he upda ed FE model is usually
employed bo h o be e assess he s a ic o dynamic esponse o he
s uc u e and o es ablish a damage de ec ion applica ion based on a
s uc u al heal h moni o ing s a egy.
Di e en c i e ia can be conside ed o he classi ica ion o he FE
model upda ing me hods. Rega ding he ime a ailable o i s compu a-
ion, wo ypes o me hods can be conside ed: (i) o -line me hods; and
(ii) on-line me hods. Thus, unde he o -line app oach, he FE model
o he s uc u e is di ec ly upda ed ia he modi ica ion o he alue
o i s mos ele an physical pa ame e s. A high compu a ional ime
is no mally equi ed o sol e he upda ing p oblem acco ding o his
app oach. This me hod is usually used o imp o e he accu acy o FE
model when hey a e employed o assess he s uc u al beha iou o
h ps://doi.o g/10.1016/j.ad engso .2024.103598
Recei ed 29 Augus 2023; Recei ed in e ised o m 1 Feb ua y 2024; Accep ed 2 Feb ua y 2024
Ad ances in Enginee ing So wa e 190 (2024) 103598
2
J. Na anjo-Pé ez e al.
ci il enginee ing s uc u es and his o ical cons uc ions [2]. In addi-
ion, unde he on-line app oach, he FE model is app oxima ed ia a
sub oga e model and subsequen ly his model is calib a ed acco ding
o some upda ing me hod. The use o his sub oga e model allows
educing he simula ion ime equi ed o sol e he upda ing p oblem.
This me hod is commonly used o damage de ec ion unde s uc u al
heal h moni o ing s a egies [6]. In his manne , he main di e ence
be ween his wo ype o upda ing me hods is ela ed o he conside ed
nume ical model. The e o e, he same p ocess is commonly employed
o sol e he upda ing p oblem o his wo me hods. This s udy is
ocused on he o -line upda ing app oach, bu i s esul s can be easily
ex apola ed o he on-line app oach.
Addi ionally, FE model upda ing p oblems can be classi ied, ega d-
ing he pa ame e selec ion, in wo main g oups [7]: (i) di ec me hods
(non-i e a i e) and (ii) indi ec me hods (i e a i e).
On he one hand, acco ding o he di ec me hods, he upda ing
pa ame e s a e di ec ly he componen s o he ma ices ha cons i u e
he sys em equa ions conside ed o nume ically mimic he beha iou
o he s uc u es. The main ad an age o hese me hods is i s di ec
and easy implemen a ion bu hey p esen a clea d awback, he non-
physical meaning o he upda ed alue o he pa ame e s. On he o he
hand, acco ding o he indi ec me hods, he upda ing pa ame e s a e
some o he mos ele an physical pa ame e s o he s uc u e (as o
ins ance, Young’s modulus o he ma e ials, s i ness o he suppo ...)
ha a e i e a i ely modi ied in o de o adjus he p edic ions o he
nume ical model o he ac ual esponse o he s uc u e. In con as o
he p e ious me hods, he adjus ing p ocess is mo e complica ed, being
necessa y he use o complex compu a ional ools, bu hey p esen s as
ad an age an easy physical in e p e a ion o he upda ed alue o he
conside ed pa ame e s. This ad an age allows ha he upda ing p ocess
can be conside ed a sys em iden i ica ion me hod and i has o igina ed
ha i e a i e me hods ha e been widely used o he FE model upda ing
o building and ci il enginee ing s uc u es.
Among he indi ec me hods, a new classi ica ion can be es ablished
in e ms o he quan i ica ion o he le el o unce ain y. Two ypes o
me hods can be conside ed [7]: (i) de e minis ic and (ii) s ochas ics
me hods. Al hough, s ochas ics me hods, as p obabilis ic [8] and uzzy
app oaches, allow he de e mina ion o he le el o unce ain y o he
es ima ion o he upda ed alues o he physical pa ame e s [9], he
high simula ion ime equi ed o sol e he upda ing p oblem, acco ding
o hese app oaches, ha e educed, o da e, hei ex ensi e p ac ical im-
plemen a ion. Fo obus FE models o complex his o ical cons uc ion,
de e minis ic me hods a e no mally used.
Among he de e minis ic me hods, he maximum likelihood me hod
has been widely employed due o i s easy implemen a ion. Acco ding
o his me hod, he upda ing p oblem can be o mula ed as ei he a
sensi i i y-based weigh ed single-objec i e op imiza ion p oblem o he
combina ion o a bi-objec i e op imiza ion sub-p oblem and a decision-
making sub-p oblem. Some ecen esul s, [10] abou he pe o mance
o bo h me hods when hey a e implemen ed o he FE model upda ing
o ci il enginee ing s uc u es; ha e concluded ha he second al e -
na i e allows ob aining a be e adjus men since his second me hod
explo es in de ail all he domain space o he possible alue o he
conside ed physical pa ame e s.
The e o e, he upda ing p oblem, acco ding o his second me hod,
is o mula ed conside ing a bi-objec i e unc ion whose componen s
a e de ined in e m o he ela i e di e ences ( esiduals) be ween he
expe imen al (ac ual) and nume ical beha iou o he s uc u e [11]. In
o de o cha ac e ize he ac ual beha iou o he s uc u e, i s expe i-
men al modal p ope ies, iden i ied conside ing ei he an expe imen al
(EMA) [12] o ope a ional modal analysis (OMA), a e usually consid-
e ed [13]. In his manne , hese esiduals a e usually de ined in e ms o
he ela i e di e ences be ween he expe imen al and nume ical modal
p ope ies (na u al equencies and associa ed ib a ion modes) o he
s uc u e.
Addi ionally, as he ela ion be ween he upda ing pa ame e s and
his bi-objec i e unc ion is clea ly nonlinea and, as a consequence,
mul iple local minimums can be eached; global op imiza ion algo-
i hms mus be used o sol e his p oblem [14]. As esul o his
op imiza ion p ocess, a se o possible solu ions is ob ained, he so-
called Pa e o on [12]. Finally, a subsequen decision-making p oblem
mus be sol ed, he selec ion o he bes balanced solu ion ( he ‘‘knee’’
poin ) among he di e en elemen s o he Pa e o on . Fo his pu -
pose, di e en me hods ha e been conside ed [15] wi hou exis ing o
da e a uni ied c i e ion o cope wi h his p oblem.
In spi e o all he ad an ages o he abo emen ioned me hod, when
i is implemen ed o sol e he FE model upda ing o complex his o ical
cons uc ion s uc u es, i p esen s a clea d awback, he high simula-
ion ime equi ed o sol e he p oblem. In o de o imp o e he pe o -
mance o his me hod, se e al nume ical echniques can be adop ed: (i)
o inc ease he sea ch speed o he op imiza ion algo i hm combining
local and global compu a ional op imiza ion algo i hms [16]; (ii) o
educe he complexi y o he FE model app oxima ing i s beha iou ia
su oga e models [17]; (iii) o ind he op imum selec ion o he phys-
ical pa ame e s o he FE model; (i ) o use al e na i e ma hema ical
ools o sol e he upda ing p oblem mo e e icien ly [7]; and ( ) o use
some clus e ing echnique o he pa allel compu a ion o he upda ing
p oblem [18].
This esea ch s udy ocuses in he i s al e na i e, he p oposal o
a new combina i e compu a ional algo i hm o educe he simula ion
ime equi ed o pe o m he FE model upda ing o his o ical con-
s uc ion wi hou comp omising he accu acy o he solu ion ob ained.
Fo his pu pose, his p oposal akes ad an age o he i ues o wo
p e ious algo i hms [19,20], p oposed by some o he au ho s o he
manusc ip , and i combines hem o imp o e he e iciency o he
upda ing p ocess.
Fo his pu pose, he p oposed algo i hm makes use o a local–
global op imiza ion algo i hm, he UKF-MHS algo i hm [19], bo h o
inc ease he con e gence speed o he sea ching algo i hm and o e-
duce he unce ain y associa ed wi h he a iabili y o he expe imen al
measu emen s, oge he wi h a collabo a i e algo i hm [20], which
allows a obus es ima ion o he ‘‘knee’’ poin ia he linking o se e al
s a is ical lea ning echniques. The combina ion be ween hese wo
algo i hms imp o es clea ly he pe o mance o he FE model upda ing
o complex his o ical cons uc ions. Thus, i is necessa y o highligh
ha he main con ibu ion o his pape is no only he combina ion o
wo p e iously epo ed algo i hms o ob ain an e icien compu a ional
ool ha assis s a chi ec s and s uc u al enginee s in he FE model
upda ing o his o ical cons uc ions bu also i s p ac ical implemen a-
ion in wo examples. These applica ion examples y o highligh he
i ues o he p oposal. Fi s , his new algo i hm has been implemen ed
o he o -line FE model upda ing o a labo a o y oo b idge. In his
manne , his i s applica ion example has been used as u o ial o
explain in de ail he di e en s eps o his compu a ional algo i hm.
Subsequen ly, he algo i hm has been implemen ed o he o -line FE
model upda ing o a complex his o ical cons uc i e. In his second
applica ion example, he highe pe o mance o he p oposal in com-
pa ison wi h he abo emen ioned con en ional bi-objec i e app oach
has been ema ked.
In addi ion o his in oduc o y sec ion, he emaining pape is
o ganized as ollows. Sec ion 2desc ibes bo h he basics o he FE
model upda ing, acco ding o he maximum likelihood me hod, and
he o mula ion o he p oposed new combina i e algo i hm. Sec ion 3
ocuses on he de ailed desc ip ion o he p oposal ia a alida ion
example, he p ac ical implemen a ion o he algo i hm o he FE
model upda ing o a labo a o y oo b idge. Sec ion 4p esen s he
pe o mance assessmen o his p oposal when i is implemen ed o
he FE model upda ing o a his o ical cons uc ion. Fo his pu pose, he
model upda ing o he chu ch o he Royal Monas e y o San Je ónimo
(G anada, Spain) is de eloped in de ail. Finally, Sec ion 5summa izes
he main conclusions ha can be d awn om he wo k.
Ad ances in Enginee ing So wa e 190 (2024) 103598
3
J. Na anjo-Pé ez e al.
2. Sol ing he maximum likelihood FE model upda ing p oblem
ia a new combina i e compu a ional algo i hm
2.1. Basics o FE model upda ing acco ding o he maximum likelihood
me hod
The goal o FE model upda ing is o acqui e a p ecise nume ical FE
model ha co esponds o he expe imen al dynamic p ope ies o he
analysed ci il o building enginee ing s uc u e. This objec i e can be
accomplished by app oxima ing he mos signi ican physical pa ame-
e s o he model ha educe he di e ences be ween he nume ical and
expe imen al da a ( esiduals). Consequen ly, he FE model upda ing
issue can be in e p e ed as a pa ame e iden i ica ion p oblem.
In his ega d, he maximum likelihood me hod is he mos com-
monly u ilized es ima o o ci il o building enginee ing applica ions.
Assuming ha he esiduals ollow a no mal dis ibu ion, he maximum
likelihood me hod is analogous o he o dina y leas squa es me hod.
Hence, he model upda ing can be con e ed in o an op imiza ion p ob-
lem, whose p ima y goal is o educe he o al o he ela i e de ia ions
be ween he expe imen al and nume ical modal p ope ies. Thus, he
upda ing p oblem can be o mula ed al e na i ely as wo di e en op i-
miza ion p oblems [7]: (i) a sensi i i y-based weigh ed single-objec i e
op imiza ion p oblem; o (ii) a combina ion o a bi-objec i e op i-
miza ion sub-p oblem oge he wi h a decision-making sub-p oblem.
Due o he highe e iciency o he second o mula ion [10] o cope
wi h his p oblem, he mul i-objec i e app oach has been conside ed
he ein. The ele an physical pa ame e s o he model a e aken as
he design a iables o his pu pose. The e o e, he o mula ion o he
FE model upda ing issue conside ing he bi-objec i e app oach can be
ep esen ed as:
min [𝑓1(𝜽)𝑓2(𝜽)] = min [ 1
2
𝑚𝑓
∑
𝑗=1
𝑟𝑓
𝑗(𝜽)21
2
𝑚𝑚
∑
𝑗=1
𝑟𝑚
𝑗(𝜽)2]
𝜽𝑙≤𝜽≤𝜽𝑢
(1)
whe e 𝑟𝑓
𝑗(𝜽) and mj 𝑟𝑚
𝑗(𝜽) e e o he disc epancies be ween he
𝑗 h na u al equency and mode shapes o he s uc u e and hei he-
o e ical coun e pa s; 𝑚𝑓 ep esen s he quan i y o conside ed na u al
equencies; 𝑚𝑚 ep esen s he quan i y o conside ed mode shape; 𝜽is
a ec o comp ising he upda ing pa ame e s o he FE model; and 𝜽𝑙
and 𝜽𝑢deno e he lowe and uppe limi s o he explo a ion ange o
hese physical pa ame e s, espec i ely.
The esiduals may be de ined as:
𝑟𝑓
𝑗(𝜽) = 𝑓𝑛𝑢𝑚,𝑗 (𝜽) − 𝑓𝑒𝑥𝑝,𝑗
𝑓𝑒𝑥𝑝,𝑗
𝑗= 1,2,…, 𝑚𝑓(2)
𝑟𝑚
𝑗(𝜽) = √
√
√
√
√⎛
⎜
⎜
⎝
(1 − √𝑀𝐴𝐶,𝑗 (𝜽))2
𝑀𝐴𝐶,𝑗 (𝜽)⎞
⎟
⎟
⎠
𝑗= 1,2,…, 𝑚𝑚(3)
conside ing he nume ical na u al equency 𝑗as 𝑓𝑛𝑢𝑚,𝑗 (𝜽), he ex-
pe imen al na u al equency 𝑗as 𝑓𝑒𝑥𝑝,𝑗 (𝜽), and he Modal Assu ance
C i e ion alue 𝑀𝐴𝐶,𝑗 (𝜽)[21] as a measu e used o assess he simila i y
be ween he mode shapes o he nume ical ib a ion mode 𝑗,𝜑𝑛𝑢𝑚,𝑗 (𝜽)
and he expe imen al ib a ion mode 𝑗,𝜑𝑒𝑥𝑝,𝑗 (𝜽). As he accu acy
o he 𝑀𝐴𝐶,𝑗 (𝜽) a io depends on he geome ical de ini ion o he
mode shapes, a su icien e ined g idline is needed o cha ac e ize he
alue o his magni ude [10]. Fo his pu pose, a sensi i i y analysis
can be pe o med. In his s udy, he a ia ion o he 𝑀𝐴𝐶,𝑗 (𝜽) a io
in e m o he numbe o coo dina es ha de ined he g idline (o
he expe imen al iden i ica ion es ) is analysed in de ail. A balanced
alue be ween he complexi y o he g idline and he accu acy o he
𝑀𝐴𝐶,𝑗 (𝜽) a io is aken in o accoun .
The physical pa ame e s o he upda ing p ocess mus be selec ed
ca e ully since hey condi ion he accu acy and eliabili y o he so-
lu ion. Based on a FE model, a di e en upda ing p oblem can be
o mula ed depending on he ela ion be ween he esiduals and he
conside ed upda ing pa ame e s. Fo he pa ame e selec ion pu pose,
di e en c i e ia can be conside ed [7]. Among he di e en c i e ia,
a sensi i i y analysis is commonly employed. Acco ding o his s udy,
he a ia ion o he na u al equencies and mode shapes in e m o
he conside ed physical pa ame e s is compu ed. As his a ia ion is
p opo ional o he modal s ain ene gy his magni ude is conside ed o
pe o m he sensi i i y analysis. The physical pa ame e s, which e lec
a highe a ia ion, a e selec ed as upda ing pa ame e s [10].
The esolu ion o his bi-objec i e op imiza ion p oblem gene a es a
se o pa ame e ec o s, each ep esen ing a po en ial solu ion. These
pa ame e s can be isually ep esen ed by o ming a cu e known
as he Pa e o on . Each poin along he Pa e o on ep esen s an
op imal upda ed model whe e imp o ing one objec i e would equi e
sac i icing ano he . Consequen ly, selec ing he bes solu ion om he
Pa e o on equi es a decision-making p ocess. Se e al me hods ha e
been p oposed in li e a u e o ackle his challenge by balancing he
di e en e ms o he Pa e o on . Gene ally, hese me hods iden i y
he bes solu ion on he Pa e o on as he poin whe e a sligh
imp o emen in one objec i e would signi ican ly de e io a e he o he
objec i e. This pa icula poin on he Pa e o on is o en e e ed o as
he ‘‘knee’’ poin , and each me hod sugges s a sligh ly di e en c i e ion
o i s de e mina ion.
Following he abo e app oach and unde he maximum likelihood
me hod, he FE model upda ing is usually add essed by implemen ing
compu a ional in elligence algo i hms [22]. Fo his pu pose, a FE
analysis package is commonly linked o an op imiza ion package. As
illus a ion o his connec i i y, Fig. 1 shows he lowcha o he
upda ing p ocess gi ing a special emphasis o he linking be ween
he wo abo emen ioned packages oge he wi h he low o da a.
Du ing he upda ing p ocess, a pai ing p oblem can be sol ed. The
nume ical and expe imen al mode shapes mus be ma ched. Fo his
pu pose, he 𝑀𝐴𝐶𝑗(𝜽) a io is commonly conside ed. Fo his pu pose,
he 𝑀𝐴𝐶𝑗(𝜽) a io among all he nume ical and expe imen al mode
shape is compu ed and o ganized in a ma ix. The ows o his ma ix
a e associa ed wi h he nume ical mode shapes and he columns wi h
he expe imen al mode shapes. The componen o he ma ix wi h
a highe alue allows de e mining he linking among nume ical and
expe imen al ib a ion modes.
Al hough compu a ional algo i hms ha e been used ex ensi ely o
cope wi h his upda ing p oblem, hey p esen wo clea limi a ions.
Fi s , hey elapse a high simula ion ime when he complexi y o he
FE model inc eases. As i is he case o obus FE models. Second, hey
a e no able o deal wi h unce ain ies associa ed wi h he expe imen al
modal p ope ies o he sys em.
To ackle hese wo issues, a hyb id local–global algo i hm com-
bined wi h a p edic i e model is used in his pape . In pa icula , he
hyb id Unscen ed Kalman Fil e (UKF)-Ha mony Sea ch (HS) algo i hm
is used o ob ain he Pa e o on o he bi-op imiza ion p oblem [19,
23]. This algo i hm allows conside ing he unce ain ies o bo h he
measu emen s and he es ima ion p ocess and akes ad an age o he
accele a ion scheme p o ided by he UKF and he global cha ac e o
he HS. Once he Pa e o on is ob ained, a decision-making p oblem
mus be sol ed o selec he bes solu ion. To cope wi h his, he Pa e o
on is p ocessed h ough a PC analysis and an a i icial neu al ne -
wo k (ANN) is designed o map he ela ionship be ween he physical
pa ame e s o he model and he bi-objec i e unc ion. Finally, he bes
solu ion is de e mined om a local minimiza ion p oblem. In his sense,
he ANN de ines a con inuous p edic i e model ha allows educing
he numbe o i e a ions, hus, he simula ion ime o he op imiza ion
algo i hm [20].
In his sec ion, he mul i-objec i e HS, he UKF algo i hm and he
pos -p ocess o he Pa e o on a e de ailed.
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Fig. 1. Flowcha ep esen ing he linking be ween he FE analysis package and he ma hema ical so wa e o he FE model upda ing implemen a ion.
2.2. MHS algo i hm
The global me aheu is ic HS algo i hm [24] is inspi ed in he men al
p ocess ca ied ou by musicians when hey imp o ise sea ching o
ha mony acco ding o aes he ic equi emen s. The goal o his algo-
i hm is o ind he global minimum o an objec i e unc ion ia he
i e a i e modi ica ion o a se o physical pa ame e s o he model.
When compa ed o Gene ic Algo i hm, he complexi y o he ma he-
ma ical ope a ions implemen ed in he HS algo i hm o moni o he
e olu ion o he popula ion a e designed o imp o e i s pe o mance.
The HS algo i hm has been implemen ed success ully o se e al
p ac ical enginee ing applica ions [25]. This algo i hm has shown a
g ea e ec i eness when compa ed wi h o he con en ional me a-
heu is ic algo i hms o sol e he FE model upda ing p oblem o com-
plex ci il enginee ing s uc u es [10].
The HS algo i hm consis s o he ollowing s eps [10]. Fi s , he
ha mony ma ix, H, ha s o es he ini ial candida e solu ions is c ea ed.
A e ha , o each candida e solu ion he objec i e unc ion is assessed.
The new ha monies a e gene a ed using h ee mechanisms (memo y
conside a ion, pi ch adjus men and andomiza ion) and he objec i e
unc ion is e alua ed again. Finally, he ha mony ma ix, H, is upda ed
wi h he bes ha monies and he s eps a e epea ed un il a con e gence
c i e ion is eached.
The MHS is an ex ension o he abo e desc ibed algo i hm which
allows minimizing mul i-objec i e unc ions. When a new ha mony is
gene a ed, each elemen o a new candida e ec o can be de ined in
e ms o ei he a p e ious alue s o ed in he ha mony ma ix, H, o
adop ing a new andom alue. This ac is con olled by he ha mony
memo y conside a ion a e, HMCR, Thus, o each pa ame e ,a andom
numbe , be ween 0 and 1, is gene a ed and i his numbe is lowe han
o equal o HMCR, he pa ame e is chosen andomly om he ma ix
H. O he wise, a andom alue among he possible alues o he sea ch
domain is assigned o he pa ame e . Addi ionally, when some elemen s
adop s he alue o a p e ious one, i can be mu a ed acco ding o he
pi ch adjus men a e, PAR. This pa ame e wo ks in he same way
as he p e ious one. In case he pa ame e mus be adjus ed, his is
based on an addi ional pa ame e , he so-called bandwid h, b𝑤, which
is added o sub ac ed o mu a e he conside ed candida e ec o . The
classi ica ion o he non-domina ed solu ions is pe o med using bo h
he non-domina ed so ing me hod and he c owding dis ance [26].
Finally, in o de o es o e he ini ial size o he ha mony ma ix, H,
he wo s solu ions, acco ding o a c owding dis ance c i e ion, a e
emo ed. When he con e gence c i e ion is me , he so-called Pa e o
on is ob ained in e ms o he non-domina ed solu ions.
2.3. Unscen ed Kalman il e
The Unscen ed Kalman Fil e (UKF) is amed wi hin he amily o
sigma-poin s Kalman il e s [27]. This local algo i hm is a de i a i e
ee es ima o , i.e., no Jacobians o Hessians need o be calcula ed,
b oadly employed o nonlinea dynamic sys ems. The o mula ion o
a pa ame e es ima ion p oblem is ep esen ed as [28]:
𝜽𝑘=𝜽𝑘−1 +𝐰𝑘−1 (4)
𝐳𝑘=ℎ(𝜽𝑘) + 𝐯𝑘(5)
being 𝜽 he pa ame e ec o ; ℎ he nonlinea modelling unc ion,
𝐳 he ou pu s o he dynamic sys em; 𝐰 he s a is ical noise o he
iden i ica ion p ocess; and 𝐯 he s a is ical noise o obse a ion p ocess.
Bo h ype o noise a e assumed o be unco ela ed and whi e Gaussian
noise wi h ze o-mean and co a iance ma ices 𝑸and 𝑹 espec i ely.
The ma ix 𝑹may be compu ed by means o wo e ms [29]: (i) he
measu emen noise; and (ii) he modelling noise. I he same FE model
is conside ed o each s ep, he i s componen o he ma ix 𝑹can be
neglec ed [16].
This algo i hm pe o ms he nonlinea es ima ion h ough he de -
ini ion o 2𝑛+ 1 (being 𝑛 he numbe o pa ame e s) de e minis ic
sampling poin s (sigma poin s). Thei p opaga ion h ough he non-
linea unc ion, ℎ, leads o he ue pos e io mean and co a iance o
he es ima ed pa ame e s up o he second o de o he Taylo se ies
expansion o he nonlinea unc ion ( hi d o de o he Taylo se ies
expansion o a Gaussian inpu s). Hence, i is an ex ension o he
unscen ed ans o ma ion [30]. The main compu a ional e o o his
algo i hm is he compu a ion o he new sigma poin s which is based
on he squa e- oo decomposi ion o he pos e io co a iance ma ix,
𝐏. Thus, his ma ix mus be posi i e semide ini e a each s ep. To
employ he Cholesky ac o iza ion (𝐀=√𝐏=𝑐ℎ𝑜𝑙(𝐏)being 𝐏=𝐀𝐀𝑇)
may p o ide an e icien decomposi ion bu he ma ix 𝐏needs o
be upda ed a each i e a ion and nume ical e o s can gi e a non-
posi i e semide ini e ma ix. To o e come his issue, he squa e- oo
UKF [31] p opaga es di ec ly he ma ix 𝐴a oiding he ac o iza ion
and ensu ing ha he ma ix is posi i e semi-de ini e. These sigma
poin s and hei associa ed weigh s a e calcula ed as:
(𝝌𝑘−1)0=
𝜽𝑘−1|𝑘−1 (6)
(𝝌𝑘−1)𝑖=
𝜽𝑘−1|𝑘−1 +√(𝑛+𝜆)(𝐀𝜃
𝑘−1|𝑘−1)𝑖𝑖= 1,2,…, 𝑛 (7)
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(𝝌𝑘−1)𝑖+𝑛=
𝜽𝑘−1|𝑘−1 −√(𝑛+𝜆)(𝐀𝜃
𝑘−1|𝑘−1)𝑖𝑖= 1,2,…, 𝑛 (8)
𝑊0=𝜆
𝑛+𝜆(9)
𝑊𝑖=𝑊𝑖+𝑛=1
2(𝑛𝑑+𝜆)𝑖= 1,2,…, 𝑛 (10)
whe e 
𝜃𝑘−1|𝑘−1 a e he pos e io pa ame e s es ima ed a he p e ious
s ep and 𝜆is a scaling pa ame e .
As an algo i hm belonging o he Kalman Fil e amily, he squa e-
oo UKF has also wo s eps: p edic ion and co ec ion. The o me
conside s he p io model o assess he sigma poin s, p edic he es-
ima ion e o co a iance, 𝐀𝜃, he model ou pu s e o co a iance, 𝐒𝑧,
and he c oss co a iance be ween he es ima ion e o and he model
ou pu s e o co a iances, 𝐏𝜃𝑧.
The es ima ion e o co a iance is gi en by 𝐀𝜃
𝑘=𝛾−1∕2𝐀𝜃
𝑘−1, whe e
𝛾is a scala ac o sligh ly less han he uni [31]. Since 𝑊𝑖>0 o all
𝑖≥1, he model ou pu e o co a iance, 𝐒z, can be exp essed as [32]:
𝐒z
𝑘=
2𝑛𝑑
∑
0
𝑊𝑖[[(𝑧𝑖
𝑘|𝑘−1 −
z𝑘|𝑘−1)] ⋅(𝑧𝑖
𝑘|𝑘−1 −
z𝑘|𝑘−1)𝑇] + 𝐑
= [√𝑊𝑖(𝑧𝑖
𝑘|𝑘−1 −
z𝑘|𝑘−1),√𝐑]
⋅[√𝑊𝑖(𝑧𝑖
𝑘|𝑘−1 −
z𝑘|𝑘−1)𝑇,√𝐑
𝑇
]𝑇
+𝑊0[(𝑧0
𝑘|𝑘−1 −
z𝑘|𝑘−1)⋅(𝑧0
𝑘|𝑘−1 −
z𝑘|𝑘−1)𝑇]𝑓𝑜𝑟 𝑖 =1∶𝑛𝑑
(11)
The i s e m can be exp essed by means o a QR ac o iza ion. The
las e m, can be aken in o accoun pe o ming a ank 1 upda e o
Cholesky ac o iza ion. The c oss co a iance is hen calcula ed as:
𝐏𝜃z
𝑘|𝑘−1 =
2𝑛𝑑
∑
0(𝑊𝑖[(𝝌𝑘−1)𝑖−
𝜽𝑘|𝑘−1]⋅[(𝑧𝑘|𝑘−1)𝑖−
z𝑘|𝑘−1]𝑇)(12)
Finally, he co ec ion s ep uses bo h he measu emen s, z𝑜𝑏𝑠, and
he model ou pu s o de e mine he pos e io mean and co a iance o
he pa ame e es ima ion by conside ing he Kalman’’s gain ma ix, 𝐊.
𝐊𝑘= (𝐏𝜃z
𝑘|𝑘−1∕𝐒z
𝑘|𝑘−1
𝑇)∕𝐒z
𝑘|𝑘−1 (13)

𝜽𝑘|𝑘=
𝜽𝑘|𝑘−1 +𝐊𝑘(z𝑜𝑏𝑠 −
z𝑘|𝑘−1)(14)
2.4. Pos -p ocess o he pa e o on
The Pa e o on is p ojec ed om he unc ional space o he
p incipal componen (PC) space by pe o ming a p incipal componen
analysis (PCA) o he cons i uen poin s. PCA is a s a is ical ool o
ex ac pa e ns om co ela ed a iables ollowing he nex s eps [33]:
i s , he da a a e no malized and hei co a iance ma ix is compu ed;
subsequen ly, he eigen alues and eigen ec o s o his ma ix a e cal-
cula ed, and inally, he PC sco es a e de e mined. I is wo h poin ing
ou ha he eigen ec o associa ed o he highes eigen alue is he
i s PC and so on and ha he sco es ep esen he p ojec ion o he
da a in he PC space. The esul ing Pa e o on exhibi s a clea con ex
cha ac e which eases he de e mina ion o he bes solu ion among
all he possible ones as his is add essed sol ing a local minimiza ion
p oblem.
The app oxima ion o he p e ious Pa e o on o a con inuous
p edic i e model is achie ed employing an a i icial neu al ne wo k
(ANN). This is a supe ised machine lea ning echnique ha maps
nonlinea and coupled ela ionships be ween a se o inpu s and ou -
pu s. The mul ilaye pe cep on (MLP) is he mos widely used ne wo k
opology and consis s o an inpu laye , an ou pu laye and one
o se e al hidden laye s, which a e in e connec ed in a eed- o wa d
manne : one neu on sends in o ma ion o he subsequen laye bu does
no ecei e om hem. The numbe o hidden laye can be assumed
o be equal o one as i has been demons a ed ha only a single
hidden laye may app oxima e any con inuous unc ion [34,35]. A
ial and e o p ocedu e o empi ical ela ionships may me employed
o de e mine he numbe o neu ons. The applica ion o he ANN o
he Pa e o on allows mapping he ela ionship be ween he physical
pa ame e s o he model and he mul i-objec i e unc ion playing he
ole o a con ex unc ion. The ela ionship needs o be ained and
es ed on he basis o a backp opaga ion aining algo i hm ha adjus s
he weigh s among he neu ons’ connec ion by minimizing he e o
be ween he es ima ed ou pu and he desi ed ou pu p opaga ed in a
backwa d di ec ion [36].
Taking ad an age o he con ex cha ac e o he con inuous unc-
ion, he decision-making p oblem o ob ain he bes solu ion is ans-
o med in o a local minimiza ion p oblem due o i s con e gence speed
and good accu acy. The solu ion ob ained is he knee-poin o he
o iginal Pa e o on . As local op imiza ion algo i hm, he Ac i e-Se
(A-S) is he eby adop ed [37]. Unde his app oach, he decision-making
p oblem may be o mula ed as:
minimize𝑔(𝜃) = 𝑦(𝜃)(15)
Subjec ed o ∶ ⎧
⎪
⎨
⎪
⎩
𝜃𝑙≤𝜃≤𝜃𝑢
𝑥(𝜃)≥𝑥𝑚𝑖𝑛
𝑦(𝜃)≥𝑦𝑚𝑖𝑛
(16)
being 𝑔(𝜃) he objec i e unc ion, 𝜃𝑙and 𝜃𝑢a e espec i ely he mini-
mum and maximum alues o he physical pa ame e s which o m he
Pa e o on , (𝑥(𝜃), 𝑦(𝜃)) a e he ANN ou pu s in he PC space, and 𝑥𝑚𝑖𝑛
and 𝑦𝑚𝑖𝑛 a e he minimum alues o he ANN ou pu s in he PC space.
2.5. P oposed algo i hm
The implemen ed algo i hm combines he main i ues o he wo
desc ibed algo i hms (MHS and UKF) and he echniques employed o
pos -p ocess he Pa e o on . Fi s , he unce ain ies a e conside ed
and, second, he compu a ional cos is educed due wo easons. The
i s one is he accele a ion scheme o he hyb id algo i hm i sel . The
second one lies in he ANN which allows educing he i e a ions and
he popula ion o ob ain a non-c owded Pa e o on as his is hen
simula ed by he ANN. Finally, he PCA imp o es he e iciency o he
decision-making p oblem since i p o ides a con ex ep esen a ion o
he Pa e o on which is u he employed by he A-S algo i hm o
ob ain he bes solu ion in a s aigh o wa d manne (Fig. 2).
3. Valida ion example: FE model upda ing o a labo a o y oo -
b idge
A eal labo a o y oo b idge is conside ed as benchma k s uc u e
o alida e he p oposed hyb id-collabo a i e algo i hm. Sec ion 3.1
desc ibes he geome y and main cons i u i e elemen s o he s uc u e.
Sec ion 3.2 in oduces he ini ial FE model o he oo b idge. Then,
Sec ion 3.3 p esen s he dynamic cha ac e iza ion o his s uc u e
ia a o ced ib a ion es (FVT) oge he wi h an EMA algo i hm.
Subsequen ly, Sec ion 3.4 de ails he pa ame e selec ion o o he
upda ing p ocess. Finally, Sec ion 3.5 desc ibes in de ail he upda ing
p ocess unde he maximum likelihood me hod.
3.1. Geome ical con igu a ion
This benchma k s uc u e is a econ igu able s eel oo b idge om
he Vib a ion Enginee ing Sec ion o he Uni e si y o Exe e (Fig. 3).
The oo b idge is a ame s uc u e wi h a leng h o 15 m. The s uc u e
is comp ised o wo la e al s eel beams sepa a ed ans e sally 2.5 m.
Rec angula pla es o 200 ×12 mm spaced 1.25 m b ace hese wo
beams. S eel bol s connec he deck, made o composi e SPS pan-
els [38], wi h bo h he longi udinal beams and ans e sal pla es. Fou
columns di ec ly pinned o he g ound a e used as suppo o he ou
ends o he s eel beams. Fu he in o ma ion abou his oo b idge can
be ound in Re . [39].

Ad ances in Enginee ing So wa e 190 (2024) 103598
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J. Na anjo-Pé ez e al.
Fig. 2. Flowcha o he p oposed combina i e compu a ional algo i hm.
Fig. 3. Desc ip ion o he labo a o y oo b idge and layou o he FVT.
3.2. Ini ial FE model
An ini ial FE model o he oo b idge was buil using he Ansys
So wa e [40]. To model he s uc u e, h ee di e en ypes o ele-
men s we e employed: (i) shell elemen s (SHELL181) o he la e al
beams; he ans e sal beams and he SPS panel; (ii) 3D beam elemen s
(BEAM188) o he bol s o he connec ion be ween he SPS panel and
he s eel s uc u e; and (iii) equi alen sp ing elemen (COMBIN14)
wi h longi udinal and ans e sal s i ness o model each suppo whose
e ical displacemen was cons ained. The alue o he s i ness a each
di ec ion was es ima ed om a simpli ied FE model o jus he column,
esul ing in an equi alen s i ness o 5.5⋅107N∕m and 1.0⋅107N∕m
o he longi udinal and ans e sal di ec ion, espec i ely. The mesh
consis ed o 31 903 elemen s. The adop ed mechanical p ope ies o
Ad ances in Enginee ing So wa e 190 (2024) 103598
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Fig. 4. Sensi i i y s udy o selec he mos ele an physical pa ame e s o he oo b idge model.
Table 1
Expe imen al na u al equency, 𝑓𝑒𝑥𝑝 , ini ial nume ical na u al equency 𝑓𝑖𝑛𝑖, ela i e
di e ence, ▵𝑓, and he MAC a io.
Vib a ion mode 𝑓exp [Hz] 𝑓ini [Hz] ▵𝑓[%] MAC [-]
1 3.810 3.638 −4.51 0.998
2 5.144 5.433 5.62 0.994
3 8.485 9.106 7.32 0.988
4 12.366 11.310 −8.54 0.877
5 18.605 17.364 −6.67 0.985
6 20.459 19.519 −4.59 0.993
7 22.980 20.725 −9.81 0.634
he di e en ma e ials a e he ollowings: (i) o he s eel, a densi y
o 𝜌𝑠= 7850 kg∕m3, he Young’s modulus equals 𝐸𝑠= 210 ⋅109Pa and
a Poisson’s a io o 𝑣𝑠= 0.3; and (ii) o he polyu e hane he densi y
was assumed o be 𝜌𝑝= 1100 kg∕m3, he Young’s modulus 𝐸𝑝= 7.5⋅108
Pa and he Poisson’s a io 𝑣𝑝= 0.3.
The nume ical modal analysis o his model gi es as esul he se en
na u al equencies shown in Table 1 and he mode shapes depic ed in
Fig. 5.
3.3. FVT and EMA
The expe imen al modal iden i ica ion o he oo b idge was ca -
ied ou h ough he expe imen al modal analysis o he accele a ions
eco ded in a o ced ib a ion es . To do his, a se o p oo -mass
ac ua o s and accele ome e s we e employed (Fig. 3).
Random signals we e used as inpu o simul aneously d i e he
ac ua o s wi hin a Mul iple Inpu -Mul iple Ou pu (MIMO) con igu-
a ion [41]. F om he applied o ces and accele a ions esponse, he
F equency Response Func ion was calcula ed conside ing an o e lap
o 50%. A e his, a complex mode indica o unc ion was de ined o
iden i y p obable mode loca ion in he FRFs i ed cu es. To inish
he p ocess, a global polynomial cu e i ing me hod ex ac s he i s
se en expe imen al na u al equencies gi en in Table 1 and hei
mode shapes (shown in Fig. 5) om he abo emen ioned p obable
mode loca ions. The eade can e e o Re . [39] o u he explana-
ion ega ding he o ced ib a ion es and he expe imen al modal
analysis.
3.4. Pa ame e s selec ion and sea ch domain o he FE model upda ing
p ocess
Based on he di e ences be ween he nume ical and expe imen al
modal p ope ies, he FE model o he oo b idge is upda ed applying
he p oposed algo i hm. Fi s ly, he physical pa ame e s o he model
ha a e being upda ed mus be iden i ied. The a io be ween he
modal s ain ene gy associa ed wi h each pa ame e and he o e all
modal s ain ene gy is conside ed as c i e ion o e eal he in luence o
he pa ame e on he a ia ion o he na u al equencies [42]. Thus,
he pa ame e s wi h g ea e a ios a e selec ed. An ini ial se o 15
physical pa ame e s we e aken in o accoun in his analysis and, as
esul , en we e selec ed as design pa ame e s (designa ed as in Fig. 4):
Young’s modulus o he s eel o he longi udinal beams a 6 di e en
sec ions (𝜃𝑖𝑛,1 − 𝜃𝑖𝑛,6), Young’s modulus o he polyu e hane (𝜃𝑖𝑛,8),
Young’s modulus o he s eel o he bol s (𝜃𝑖𝑛,9), equi alen longi udinal
s i ness o he suppo s (𝜃𝑖𝑛,11) and he equi alen ans e sal s i ness
o he suppo s (𝜃𝑖𝑛,12). The selec ion o he Young’s modulus o he
di e en ma e ial as physical pa ame e has no go as objec i e he
iden i ica ion o his s uc u al p ope y ia he esolu ion o he in e se
p oblem. Howe e , his magni ude is used as es ima o o he s i ness
o he s uc u e. In his manne , he a ia ion o his quan i y wi h
espec o i s e e ence alue e lec s he educ ion o some unce ain-
ies (geome ical ole ances, cons i u i e laws ...) associa ed wi h he
modelling o he s uc u e.
In addi ion, a sea ch domain has been es ablished o each de-
sign pa ame e in o de o cons ain he op imiza ion p oblem and o
gua an ee an adequa e physical meaning o he upda ed alue. The
conside ed esea ch domain o each conside ed upda ing pa ame e
wi h i s lowe and uppe bound is illus a ed by Table 2. Once bo h he
design pa ame e s and hei sea ch ange ha e been se , he FE model
upda ing p ocess is pe o med.
3.5. FE model upda ing p ocess
The p e ious desc ibed bi-objec i e algo i hm is applied o he
model upda ing o he oo b idge. The ollowings alues o he pa-
ame e s o he MHS algo i hm we e adop ed [22,43]: a HMCR a io
o 0.9, a PAR a io o 0.7 and a bandwid h, 𝑏𝑤, equals he 1% o he
sea ch domain o each pa ame e . Rega ding he pa ame e s o he
UKF algo i hm, he sensi i i y analysis ca ied ou by he au ho s in
Re . [19] was used, hus, he alues o he pa ame e s we e: numbe
o i e a ions o he UKF algo i hm, NUKF=3, ini ial es ima ion e o
co a iance, 𝑃𝜃
0=𝑑𝑖𝑎𝑔(((𝜃𝑢−𝜃𝑙)∕2000)2)and measu emen noise co-
a iance ma ix, 𝑅𝑖𝑖 = 0.001. The numbe o i e a ions o he MHS
algo i hm was se o 10, he numbe o ini ial ha monies was 20 and
he numbe o new ha monies, gene a ed a each i e a ion, equals 5. As
esul , he non-c owded Pa e o on con aining he possible solu ions
was ob ained (see Fig. 6a). The compu a ional cos o his s ep o he
Ad ances in Enginee ing So wa e 190 (2024) 103598
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J. Na anjo-Pé ez e al.
Fig. 5. Fi s se en mode shapes ob ained om: (a) he expe imen al iden i ica ion es ; and (b) he ini ial FE model.
p oposed algo i hm is he highes , as i ook 4678 s o ob ain he
non-c owded Pa e o on .
Nex , he PCA analysis o his Pa e o on is conduc ed acco ding o
he s eps o he p oposed algo i hm. The singula alue decomposi ion
me hod was employed o pe o m he decomposi ion o he co a iance
ma ix. Fig. 6b shows he p ojec ion o he Pa e o on in he PC
space. As men ioned in Sec ion 2, his analysis p esen s wo ad an ages.
Fi s ly, he accu acy o he ANN used in he nex s ep o app oxima e
he Pa e o on is imp o ed and, secondly, he selec ion o he bes
solu ion is mo e obus due o he con ex na u e o he Pa e o on
in he PC space. The ime o pe o m his analysis is negligible when
compa ed o he bi-objec i e op imiza ion as i akes less han one
second.
The ou h s ep o he algo i hm in ol ed he design o he ANN
o app oxima e he Pa e o on . The objec i e is o simula e he ela-
ionship be ween he en physical pa ame e s and he wo esiduals o
he bi-objec i e unc ion in a con inuous manne , yielding a con inuous
Pa e o on which leads o a simple decision-making p oblem. In his
s udy he MLP (Mul i-laye pe cep on) wi h one hidden laye has been
used as ANN opology. The ule o Ke manshahi [44] allows calcula ing
he numbe o neu ons o he hidden laye as neu ons = (𝑚+𝑛)∕2 + 𝛿,
being 𝑚and 𝑛 he numbe o neu ons o he inpu and ou pu laye s,
espec i ely, and 𝛿a no maliza ion ac o which can be equal o 1 o 2.
Thus, o his case 𝑚= 10,𝑛= 2 and 𝛿= 1, gi ing as esul 7 neu ons o
he hidden laye . The ANN was ained using he Le enbe g–Ma qua d
backp opaga ion algo i hm [36], de ining he e o unc ion in e ms
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J. Na anjo-Pé ez e al.
Fig. 6. Combina i e compu a ional algo i hm: (a) non-c owded Pa e o on , (b) PCA analysis, (c) solu ion using he ANN and (d) co esponding solu ion in he o iginal Pa e o
on .
Table 2
Upda ed alue o he physical pa ame e s o model, 𝜃𝑖𝑛, a e he upda ing p ocess.
Pa ame e Desc ip ion Ini ial alue Range o a ia ion Upda ed alue
Lowe Uppe
𝜃𝑖𝑛,1𝐸𝑠long. beam sec ion 1 [GPa] 210 190 240 230.08
𝜃𝑖𝑛,2𝐸𝑠long. beam sec ion 2 [GPa] 210 190 240 215.94
𝜃𝑖𝑛,3𝐸𝑠long. beam sec ion 3 [GPa] 210 190 240 202.43
𝜃𝑖𝑛,4𝐸𝑠long. beam sec ion 4 [GPa] 210 190 240 215.92
𝜃𝑖𝑛,5𝐸𝑠long. beam sec ion 5 [GPa] 210 190 240 193.52
𝜃𝑖𝑛,6𝐸𝑠long. beam sec ion 6 [GPa] 210 190 240 214.58
𝜃𝑖𝑛,8𝐸𝑝polyu e hane [MPa] 1000 500 1500 758.92
𝜃𝑖𝑛,9𝐸𝑠s eel bol s [GPa] 1000 210 2100 790.95
𝜃𝑖𝑛,11 KEqui alen long. s i ness [N/m2]6.0⋅1071.4⋅1071.1⋅1087.06 ⋅107
𝜃𝑖𝑛,12 KEqui alen ans . s i ness [N/m2]9.0⋅1064.8⋅1063.8⋅1077.79 ⋅106
o he mean squa e e o and conside ing 70% o he elemen s o he
Pa e o on o sampling. The es o he elemen s we e used o alida e
and es he ANN. Fig. 6b illus a es he accu acy o he ANN used o
con inuously app oxima e he Pa e o on in he PC space.
Finally, he decision-making p oblem is sol ed o ob ain he bes
solu ion (knee poin ). Fo his pu pose, he A-S algo i hm has been used
o conduc he minimiza ion p oblem. Once his poin is ob ained, i is
p ojec ed back in o he o iginal Pa e o on (Fig. 6c-d). The simula ion
ime o design he ANN and calcula e he knee poin was abou 5 s. The
physical pa ame e s associa ed o his op imum poin (upda ed alues)
a e collec ed in Table 2 and he co esponding modal p ope ies o he
upda ed FE model a e shown in Table 3.
I can be obse ed ha a e he upda ing p ocess, all he MAC
a ios a e abo e 0.89 and he ela i e di e ences ha e been educed.
In addi ion, he compu a ional cos o pe o m his p ocess has been
also less han adi ional bi-objec i e op imiza ion as i is no needed
Table 3
Expe imen al na u al equency, 𝑓𝑒𝑥𝑝 , upda ed nume ical na u al equency 𝑓𝑢𝑝𝑑 ,
ela i e di e ence, ▵𝑓, and he MAC a io.
Vib a ion mode 𝑓exp [Hz] 𝑓upd [Hz] 𝛥𝑓 [%] MAC [-]
1 3.810 3.844 0.892 0.999
2 5.144 5.458 6.104 0.994
3 8.485 8.388 1.143 0.988
4 12.366 11.858 4.107 0.890
5 18.605 18.148 2.456 0.986
6 20.459 20.185 1.339 0.993
7 22.980 21.607 5.975 0.963
a popula ed Pa e o on , and, he e o e, he numbe o i e a ions and
popula ion o he algo i hm can be se o low alues.
Ad ances in Enginee ing So wa e 190 (2024) 103598
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