Recei ed: 17 Augus 2022 Re ised: 4 No embe 2022 Accep ed: 5 No embe 2022
DOI: 10.1002/nme.7167
RESEARCH ARTICLE
No el me hodologies o sol ing he in e se uns eady hea
ans e p oblem o es ima ing he bounda y hea lux in
con inuous cas ing molds
Umbe o Emil Mo elli1,2,3 Pa icia Ba al1,2 Pe eg ina Quin ela1,2
Gianluigi Rozza3Gio anni S abile3
1Depa amen o de Ma emá ica Aplicada,
Uni e sidade de San iago de Compos ela,
San iago de Compos ela, Spain
2Cen o de In es igación e Tecnoloxía
Ma emá ica de Galicia (CITMAga),
San iago de Compos ela, Spain
3Scuola In e nazionale Supe io e di S udi
A anza i (SISSA), T ies e, I aly
Co espondence
Umbe o Emil Mo elli, Cen o de
In es igación e Tecnoloxía Ma emá ica de
Galicia (CITMAga), Ins i u o de
In es igacións Tecnolóxicas, plan a -1,
Rúa de Cons an ino Candei a, s/n, 15782
Campus Vida, San iago de Compos ela,
Spain.
Email: [email p o ec ed];
[email p o ec ed]
Funding in o ma ion
Agencia Es a al de In es igación,
G an /Awa d Numbe :
PID2019-105615RBI00/AEI; Eu opean
Resea ch Council, G an /Awa d Numbe :
765374; H2020 Ma ie Skłodowska-Cu ie
Ac ions, G an /Awa d Numbe : 681447;
Minis e io de Economía, Indus ia y
Compe i i idad, Gobie no de España,
G an /Awa d Numbe :
MTM2015-68275-R
Abs ac
In his a icle, we in es iga e he es ima ion o he ansien mold-slab hea lux
in con inuous cas ing molds gi en some he mocouples measu emen s in he
mold pla es. Ma hema ically, we can see his p oblem as he es ima ion o a
Neumann bounda y condi ion gi en poin wise s a e obse a ions in he in e-
io o he domain. We o mula e i in a de e minis ic in e se p oblem se ing.
A e in oducing he indus ial p oblem, we p esen he mold he mal model
and ela edassump ions.Then,we o mula e hebounda yhea luxes ima ion
p oblem in a de e minis ic in e se p oblem se ing using a sequen ial app oach
acco ding o he sequen iali y o he empe a u e measu emen s. We conside
di e en o mula ions o he in e se p oblem. Fo each one, we de elop no el
di ec me hodologies exploi ing a space pa ame e iza ion o he hea lux and
he linea i y o he mold model. We cons uc hese me hods o be di ided in o
a compu a ionally expensi e o line phase ha can be compu ed be o e he
p ocess s a s, and a cheape online phase o be pe o med du ing he cas ing
p ocess. To conclude, we es he pe o mance o he p oposed me hods in wo
benchma k cases.
KEYWORDS
bounda y condi ion es ima ion, con inuous cas ing, da a assimila ion, hea ans e , in e se
p oblem, op imal con ol
1INTRODUCTION
Mos o he s eel p oduced e e yday wo ldwide is made by con inuous cas ing (CC).1Con inuous cas e s ha e been
a ound o many decades now and a long sequence o imp o emen s ha e inc eased h ough he yea s hei p oduc i i y
(i.e., he cas ing speed) and he quali y o he cas ed p oduc s.
This is an open access a icle unde he e ms o he C ea i e Commons A ibu ion License, which pe mi s use, dis ibu ion and ep oduc ion in any medium, p o ided he
o iginal wo k is p ope ly ci ed.
© 2022 The Au ho s. In e na ional Jou nal o Nume ical Me hods in Enginee ing published by John Wiley & Sons L d.
1344 wileyonlinelib a y.com/jou nal/nme In J Nume Me hods Eng. 2023;124:1344–1380.
MORELLI e al. 1345
The CC p ocess s a s by apping he liquid me al om he ladle in o he undish. In he undish, he me al low is
egula ed and smoo hed. Th ough he subme ged en y nozzle (SEN), he me al is d ained in o a mold. The ole o he
mold is o cool down he s eel un il i has a solid skin which is hick and cool enough o be suppo ed by olle s in he
seconda y cooling egion.
A he ou le o he mold, he me al is s ill mol en in i s inne egion. Suppo ed by olle s, i is cooled un il comple e
solidi ica ionbydi ec lysp ayingwa e o e i .A heendo his seconda y cooling egion, he cas ingiscomple ed.This
is jus a b ie o e iew on he CC p ocess. We e e he in e es ed eade o I ing’s monog aph on he subjec .2
In his wo k, we ocus on CC o hin slabs, ha is, slabs wi h ec angula c oss sec ion wi h hickness smalle han
70 mm and wid h be ween 1 and 1.5 m. Thanks o he small hickness, he solidi ica ion in he slab is ela i ely as ,
consequen ly he cas ing speed is gene ally high, be ween 7 and 14 m pe minu e.
Thin slab molds a e made o ou di e en pla es: wo wide pla es and wo la e al pla es, all made o coppe (see
Figu e1).Ingene al,la e alpla escanbemo edo changed omodi y heslabsec iondimensions.Thegeome yo hese
pla esismo ecomplex hanonecanexpec : heyha ed illedchannelswhe e hecoolingwa e lows,slo sin heou side
ace o he malexpansion, he mocouples, and as eningbol s. Tocompensa e hesh inkageo he slab wi h hecooling
and minimize he gap, he molds a e ape ed. Mo eo e , he uppe po ion o he mold o ms a unnel o accommoda e
he SEN.
Due o he high cas ing speed and he ela ed s ong he mal g adien , se e al complex and coupled phenomena
ela ed o s eel low, solidi ica ion, mechanics, and hea ans e appea in he mold egion. This complexi y makes he
mold he mos c i ical pa o he CC p ocess. He e, sa e y and p oduc i i y issues mus be add essed.
Fo example, a common issue is he s icking o he s eel o he mold. In his case, i is essen ial o quickly de ec
he p oblem and educe he cas ing speed, o he wise i can lead o dange ous e en s ha could o ce he shu down o
he cas e . Less equen bu mo e ca as ophic e en s a e he liquid b eak-ou and he excessi e inc ease o he mold
empe a u e. The o me is due o a non-uni o m cooling o he me al wi h he skin being so hin o b eak. The la e
is gene ally conside ed he mos dange ous e en in a cas ing plan . In ac , i he mold empe a u e is high enough o
cause he boiling o he cooling wa e , we ha e a d ama ic dec ease in he hea ex ac ion. Then, he empe a u e in he
mold quickly ises, ha could cause he mel ing o he mold i sel . Bo h hese inciden s a e e y dange ous and cos ly.
In ac , hey gene ally equi e he shu down o he cas e , he subs i u ion o expensi e componen s and an ex ended
u na ound.
Fo all hese easons, he ea ly de ec ion o p oblems in he mold is c ucial o a sa e and p oduc i e ope a ion o
con inuous cas e s. Thei de ec ion becoming mo e di icul as cas ing speed ( hus p oduc i i y) o he cas e s inc eases.
Un il now, ope a o s aced all hese p oblems by equipping he molds wi h senso s. Among o he pa ame e s, hey
measu e he poin wise empe a u e o he mold by he mocouples (see Figu e 1) and he cooling wa e empe a u e as
well as i s low a he inle and ou le o he cooling sys em. On one hand, he mocouples empe a u es a e used o ha e
insigh o he mold empe a u e ield. On he o he , he wa e empe a u e ise is used o app oxima e he hea ex ac ed
om he s eel.
This app oach allowed o un con inuous cas e s o decades. Ne e heless, i has se e al d awbacks: i elies on he
expe ience o ope a o s, gi es e y limi ed in o ma ion abou he hea lux a he mold-slab in e ace, and is cus omized
o each geome y so i equi es new e o o be applied o new designs. So, wi h he always inc easing cas ing speed o
mode n cas e s, a new and mo e eliable ool o analyzing he mold beha io is necessa y.
S eel
Cooling wa e
The mocouple
FIGURE 1 Schema ic o a ho izon al sec ion o he mold ( he cas ing di ec ion is pe pendicula o he image)
10970207, 2023, 6, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.1002/nme.7167 by Uni e sidade de San iago de Compos ela, Wiley Online Lib a y on [15/06/2023]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
1346 MORELLI e al.
Acco ding o CC ope a o s and designe s, knowing he local hea lux be ween mold and slab is he mos impo an
in o ma ion in moni o ing he mold. Mo eo e , we should es ima e i in eal- ime o he ea ly de ec ion o issues and a
p ope moni o ing.Byconside ing hemoldpla es obeou domainand ocusingou in e es oni s he malbeha io , he
mold-slab hea lux can be seen as a Neumann Bounda y Condi ion (BC) in he model. To compu e i s alue, we pose he
ollowing in e se p oblem: gi en he empe a u e measu emen s p o ided by he he mocouples, es ima e he bounda y
hea lux a hemold-slabin e ace.Ina p e iouspublica ion,3we de eloped a no elme hodology o hesolu ion o his
p oblem using a s eady-s a e mold model. The p esen wo k is an ex ension o he p e ious one conside ing he mo e
challenging uns eady-s a e case.
A e de i ing he mold hea ans e model in Sec ion 2, we discuss in Sec ion 3 he s eel-mold hea lux es ima ion
p oblem and p opose no el me hodologies o i s solu ion. Finally, we design in Sec ion 4some nume ical benchma k
es cases ha we use o s udy he pe o mance o he p oposed in e se sol e s.
2MATHEMATICAL MODEL
A de ailed desc ip ion o he physical phenomena ha occu in he mold egion o a cas e can be ound in ou p e ious
wo k.3He e, we only men ion ha hese phenomena a e ex emely complex and igh ly coupled ( he modynamic eac-
ions,mul iphase low, eeliquidsu acesandin e aces,solidi ica ion,e c.).Then,moni o ing hecas ingbysimula ing
all o hem om he SEN o he seconda y cooling egion would be ex emely complex and compu a ionally expensi e o
deal wi h, especially o eal- ime applica ions.
Acco ding o CC ope a o s, o moni o he mold beha io i is su icien o know he mold-slab hea lux. Then, gi en
he mold pla es physical p ope ies, i s geome y and he cooling wa e empe a u e, ou app oach is o sol e an in e se
p oblemha ingas inpu da a he empe a u emeasu emen smadeby he he mocouples ha a ebu iedinside he mold
pla es.
Asmen ioned, hemoldsolidpla esa e hecompu a ionaldomainwhile hemold-slabhea luxisaNeumannBCon
a po ion o i s bounda y o be de e mined as solu ion o an in e se p oblem. The di ec p oblem co esponds o a model
o he hea ans e in he mold pla es. In he es o his sec ion, we desc ibe his model and he ela ed assump ions.
In modeling he he mal beha io o he mold, we conside he ollowing well es ablished assump ions:3
•Thecoppe moldisassumedahomogeneousandiso opicsolidma e ial.
•The he mal expansion o he mold and i s mechanical dis o ion a e negligible.
•The ma e ial p ope ies a e assumed cons an .
•The bounda ies in con ac wi h ai a e assumed adiaba ic.
•The hea ansmi ed by adia ion is neglec ed.
•The cooling wa e empe a u e is known a he inle and ou le o he cooling sys em. Mo eo e , i is assumed o be
cons an in ime and linea wi h espec o he zcoo dina e (see Figu e 2).
•No boiling in he wa e is assumed.
We e e o ou p e ious wo k3 o he mo i a ions ela ed o hese assump ions.
Acco ding o hese assump ions, we conside an uns eady-s a e h ee-dimensional hea conduc ion model posed on
he(solid)coppe mold,wi hacon ec i eBCin hepo iono hebounda yincon ac wi h hecoolingwa e ,aNeumann
BC in he po ion o he bounda y in con ac wi h he s eel, and adiaba ic BC in he po ion o he bounda y in con ac
wi h ai .
A e in oducing some no a ion and he compu a ional domain, we de o e he p esen sec ion o he o mula ion o
he mold model. As common when dealing wi h in e se p oblems, we e e o i as he di ec p oblem. We conclude his
sec ion by discussing i s nume ical disc e iza ion.
2.1 Compu a ional domain and no a ion
Conside a solid domain, Ω, which is assumed o be an open Lipschi z bounded subse o R3, wi h smoo h bounda y Γ
(see Figu e 2). Le Γ=Γ
sin ∪Γ
sex ∪Γ
s whe e
Γsin,
Γsex ,and
Γs a e disjoin se s. Mo eo e , gi en ∈R+, we conside he
10970207, 2023, 6, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.1002/nme.7167 by Uni e sidade de San iago de Compos ela, Wiley Online Lib a y on [15/06/2023]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
MORELLI e al. 1347
FIGURE 2 Schema ic o he mold domain, Ω, and i s bounda ies (images aken om Mo elli e al.3)
ime domain (0, ]. The Eule ian Ca esian coo dina e ec o is deno ed by x∈Ωand n(x) ep esen s he uni no mal
ec o ha is di ec ed ou wa ds om Ωa poin x∈Γ.
In his se ing, Ωco esponds o he egion o he space occupied by he mold. The in e ace be ween he mold and
he cooling sys em is deno ed by Γs .WhileΓsin is he po ion o he mold bounda y in con ac wi h he solidi ying s eel.
Finally, we deno e he emaining pa o he mold bounda y wi h Γsex .
2.2 Di ec p oblem
We shall assume all along he ollowing assump ions on he da a:
(H1) The he mal conduc i i y is cons an and s ic ly posi i e: ks∈R+.
(H2) The e is no hea sou ce inside he mold domain.
(H3) The densi y and speci ic hea a e cons an and s ic ly posi i e: 𝜌∈R+,Cp∈R+.
(H4) The hea ans e coe icien on Γs is cons an and s ic ly posi i e: h∈R+.
(H5) The cooling wa e empe a u e, T , is known, cons an in ime, and belongs o Lq(Γs ).
(H6) The ini ial empe a u e, T0,isknownandbelongs oL2(Ω).
(H7) The s eel-mold hea lux, g,belongs oL (0, ;Lq(Γsin )).
In (H5), (H7) we assume ha ,q∈(2,+∞) and
1
+1
q<1
2.(1)
No ice ha i implies ,q>2.
Unde he assump ions (H1)–(H7), we p opose he ollowing h ee-dimensional, uns eady-s a e, hea conduc ion
model
P oblem 1. Find Tsuch ha
𝜌Cp𝜕T
𝜕 −ksΔT=0,in Ω×(0, ],(2)
10970207, 2023, 6, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.1002/nme.7167 by Uni e sidade de San iago de Compos ela, Wiley Online Lib a y on [15/06/2023]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
1348 MORELLI e al.
wi h BCs and Ini ial Condi ion (IC)
⎧
⎪
⎪
⎨
⎪
⎪
⎩
−ks∇T⋅n=gon Γsin ×(0, ],
−ks∇T⋅n=0onΓsex ×(0, ],
−ks∇T⋅n=h(T−T )on Γs ×(0, ],
T(⋅,0)=T0in Ω.
(3)
A weak solu ion is now de ined by es ing agains a smoo h unc ion and o mally in eg a ing by pa s.
De ini ion 1. We say ha a unc ion T∈C([0, ];L2(Ω)) ∩ L2(0, ;H1(Ω)) is a weak solu ion o P oblem 1on [0, ] o
some >0i
−𝜌Cp∫
0∫ΩT(x, )𝜕𝜓(x, )
𝜕 dxd +ks∫
0∫Ω∇T(x, )∇𝜓(x, )dxd +∫
0∫Γs
hT(x, )𝜓(x, )dΓd
=𝜌Cp∫ΩT0(x)𝜓(x,0)dx−∫
0∫Γsin
g(x, )𝜓(x, )dΓd +∫
0∫Γs
hT (x)𝜓(x, )dΓd ,(4)
o all 𝜓∈H1(0, ;H1(Ω)) ha sa is y 𝜓(⋅, )=0inΩ.
Thanks o Ni ka4(i s Theo em 2.11 and Co olla y 2.13), we ha e
Theo em 1. Le assump ions (H1)–(H7) and (1) hold. Then, he e exis s a unique weak solu ion o P oblem 1on [0, ].
Finally, we ecall Theo em 3.3 in Ni ka4
Theo em 2. Le assump ions (H1)–(H7) and (1) hold. Then, he weak solu ion T o P oblem 1is in C([0, ];C(Ω)).So,in
pa icula ,T(x, )→T0(x)uni o mly on Ωas →0.
Rega ding he nume ical solu ion o P oblem 1, we use he ini e olume me hod o i s disc e iza ion. Gi en a es-
sella ion o he domain, Ω, we w i e he disc e e unknown (TC( ))C∈as he eal ec o T( ),belonging oRNhwi h
Nh=size(). Then, we w i e he spa ially disc e ized p oblem as
MdT( )
d +AT( )=b( ), ∈(0, ],(5)
whe e M∈MNh×Nhis he mass ma ix, A∈MNh×Nhis he conduc i i y ma ix and b∈RNh hesou ce e m.The alueo
each elemen o M,A,andbdepends on he pa icula ini e olume scheme o he disc e iza ion and he mesh used.
Since ou p oblem is a classic di usion p oblem, we e e o u he de ails ega ding he ini e olume disc e iza ion o
he Eyma d’s monog aph.5
To disc e ize (5) in ime, we di ide he ime in e al o in e es in o NT egula s eps
0=0, n+1= n+Δ ,n=0,…,NT−1,Δ =
NT.(6)
F om now on, we deno e by jan app oxima ion o a gi en unc ion ( )a ime j.
Fo he ime disc e iza ion, we conside he implici Eule scheme. I is a i s -o de implici scheme. Wi h his
disc e iza ion, (5) becomes
(M+Δ A)Tn+1=MTn+Δ bn+1,n=0,…,NT−1.(7)
No ice ha , hanks o hypo heses (H1)–(H7), ma ices Aand Ma e ime independen .
3INVERSE PROBLEM
In his sec ion, we discuss he o mula ion and solu ion o he bounda y hea lux es ima ion p oblem. We s a e i in
an in e se p oblem se ing using da a assimila ion. We begin his sec ion wi h a li e a u e su ey, hen we discuss he
ma hema ical o mula ion o he p oblem and, inally, he me hodology ha we de eloped o i s solu ion.
10970207, 2023, 6, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.1002/nme.7167 by Uni e sidade de San iago de Compos ela, Wiley Online Lib a y on [15/06/2023]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
MORELLI e al. 1349
3.1 S a e o he a
The li e a u e on in e se hea ans e p oblems is as .6-9 We e e o Ali ano ’s,10 O lande’s,11 Beck and Clai ’s,12 and
Chang’s13 wo ks o ade ailed e iew.In heli e a u e,o he esea che salso in es iga ed he pa icula p oblemo com-
pu ing he mold-slab hea lux om empe a u e measu emen s in he mold.14-17 F om a ma hema ical poin o iew, he
p esen p oblem i s in he amewo k o es ima ing a Neumann BC ( he hea lux) ha ing as da a poin wise measu e-
men s o he s a e ( he empe a u e) inside he domain. Such p oblems we e also add essed in in es iga ions no ela ed
o hea ans e .18-20
The s o y o in e se hea ans e p oblems s a ed in he 50 s when ae ospace enginee s we e in e es ed in know-
ing he he mal p ope ies o hea shields and hea luxes on he su ace o space ehicles du ing e-en y. The i s
app oach was pu ely heu is ic, hen in he 60 and 70 s, esea che s mo ed o a mo e ma hema ically o mal app oach.
In ac , mos o he egula iza ion heo y ha we use nowadays o ea ill-posed p oblems was de eloped du ing hese
yea s.10,21-24
The i s app oach o es ima ing hebounda yhea luxinCCmoldswas oselec ahea luxp o ile,and henby ial
and e o adap i o ma ch he measu ed empe a u es.17 Pinhe o e al.25 we e he i s o use an op imal con ol ame-
wo k and egula iza ion me hods. They used a s eady-s a e e sion o he 2D mold model p oposed by Sama aseke a and
B imacombe26 and pa ame e ized he hea lux wi h a piecewise cons an unc ion. Finally, hey used Tikhono ’s egu-
la iza ion o sol ing he in e sep oblemand alida ed he esul swi h expe imen almeasu emen s.Asimila app oach
was used mo e ecen ly by Rau e e al.15,27,28 They es ima ed he hea lux ans e ed om he solidi ying s eel o he
mold wall bo h in a 2D and 3D domain. They used a s eady-s a e hea conduc ion model o he mold and pa ame e -
ized he hea lux wi h a piecewise linea p o ile in 2Dand symme ic cosine p o ile in 3D. Fo he solu ion o he in e se
p oblem, hey used he Conjuga e G adien Me hod (CGM) and a mixed GA-SIMPLEX algo i hm29 in 2D while in 3D
hey only used he GA-SIMPLEX algo i hm. Thei esul s we e also es ed wi h expe imen al da a.
Usinga3Duns eady-s a ehea conduc ionmodelin hes andand hemoldwi haRobincondi iona hemold-s and
in e ace, Hebi e al.30,31 a emp ed o es ima e he solidi ica ion in CC ound bille s. Simila ly o he p esen wo k, hey
looked o he hea ans e coe icien ha minimizes a dis ance be ween measu ed and compu ed empe a u es a he
he mocouples’ poin s. Assuming he hea ans e coe icien o be piecewise cons an , hey i e a i ely adap ed each
piece oma ch hemeasu ed empe a u e.Howe e ,in he alida ionwi hplan measu emen s, heydidno ob aingood
ag eemen . A simila app oach was used by Gonzalez e al.32 and Wang e al.,33-36 he la e using a Neumann condi ion
a he mold-s and in e ace.
Uday aj e al.16 applied he conjuga e g adien me hod wi h adjoin p oblem o he solu ion o he s eady-s a e 2D
mold-slab hea lux es ima ion p oblem. This me hodology was i s p oposed by Ali ano 10 o he egula iza ion o
bounda y in e se hea ans e p oblems wi hou he need o pa ame e izing he hea lux. Howe e , as we also p o ed
in ou p e ious wo k,3 his me hod unde es ima es he hea lux away om he measu emen s. To o e come his issue,
Uday aj e al. p oposed o a e age he compu ed hea lux a each s ep and use he uni o m a e aged alue as ini ial
es ima ion o he ollowing s ep. Howe e , he ob ained esul s we e no sa is ying.
Since he eal- ime equi emen is common in indus ial applica ions, eal- ime me hodologies o he solu ion o
hese p oblems ha e al eady been in es iga ed in he li e a u e. In pa icula , Videcoq e al.37 used a b anch eigenmodes
educed model38 o he eal- ime iden i ica ion o he hea sou ce s eng h a ia ions in a 3D nonlinea in e se hea
conduc ionp oblem.La e , o sol ing he samep oblem,Gi aul e al.39 used heModalIden i ica ionMe hod40 o gen-
e a ing he educedmodel.Finally,Aguadoe al.41 coupledclassicalha monicanalysiswi h ecen modelo de educ ion
echniques(p ope gene alized decomposi ion) osol ein eal- ime he ansien hea equa ion a moni o edpoin s, also
showing he applicabili y o hei me hod o in e se p oblems.
To conclude, also deep lea ning echniques we e in es iga ed. Wang and Yao42 used he in e se p oblem solu ion
echnique de eloped by Hebi e al.31 and a se o expe imen al empe a u e measu emen s o ain a Neu al Ne wo k
(NN) o on-line compu a ion.Simila ly, Chen e al.43 used he uzzy in e enceme hod o es ima ing he mold hea lux.
In bo h wo ks, hey modeled he mold wi h a 2D s eady-s a e hea conduc ion model in he solid and pa ame e ized he
bounda y hea lux.
Ou con ibu ion o he li e a u e is he de elopmen o no el me hods o sol ing he uns eady-s a e 3D in e se
hea ans e p oblem in CC molds ha exploi s he pa ame e iza ion o he hea lux. We p opose di e en no el di ec
me hodologies ha exploi ano line-onlinedecomposi ion.In ac ,wedi ide heminacompu a ionallyexpensi eo line
phase and an online phase whose compu a ional cos is much smalle . The ad an age is ha we compu e o line phase
once and o all be o e s a ing he cas ing p ocess.Then, while he machine is unning, we only need o sol e he cheap
10970207, 2023, 6, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.1002/nme.7167 by Uni e sidade de San iago de Compos ela, Wiley Online Lib a y on [15/06/2023]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
1350 MORELLI e al.
online phase. Mo eo e , in his wo k, we design some benchma k cases o his applica ion, and we use hem o es he
pe o mances o he p oposed me hodologies.
3.2 In e se p oblem o mula ion
Be o e p oceeding wi h he ma hema ical o mula ion o he in e se p oblem, we do some echnical conside a ions
ha will guide us in he p ocess. Fi s , he he mocouples measu e he empe a u e a he sampling equency
samp. This sampling equency is ypically o 1Hz and we will assume his alue all along his in es iga ion (no ice
ha di e en alues o samp a e compa ible wi h he ollowing discussion). Second, e e y sampling pe iod Tsamp =
1∕ samp =1 s, he he mocouples p o ide a new se o measu emen s, so we ha e a egula sequence o measu emen s
in ime.
Tha said, we conside he p oblem o es ima ing he hea lux, g,onΓsin , in be ween he las acqui ed measu emen
ins an and he p e ious one. In his way, we ollow he sequen iali y o he measu ed da a in ou solu ion p ocedu e
acco ding o he moni o ing pu pose o his esea ch.
Wein oduce he ollowingno a ion.Le Ψ∶={x1,x2,…,xP}beacollec iono poin sinΩandΥ∶={𝜏0,𝜏1,…,𝜏P }
a collec ion o poin s in [0, ]such ha 𝜏k= kN (see Figu e 3). Acco ding o he in oduced sequen ial app oach, we
conside he ollowing es ic ion o P oblem 1 o (𝜏k−1,𝜏k],1≤k≤P , as di ec p oblem
P oblem 2. Le 1 ≤k≤P and gk(x, )be a gi en hea lux on Γsin ×(𝜏k−1,𝜏k].FindTksuch ha
𝜌Cp𝜕Tk
𝜕 −ksΔTk=0,in Ω×(𝜏k−1,𝜏k],(8)
wi h BCs and IC
⎧
⎪
⎪
⎨
⎪
⎪
⎩
−ks∇Tk⋅n=gkon Γsin ×(𝜏k−1,𝜏k],
−ks∇Tk⋅n=0onΓsex ×(𝜏k−1,𝜏k],
−ks∇Tk⋅n=h(Tk−T )on Γs ×(𝜏k−1,𝜏k],
Tk(⋅,𝜏k−1)=Tk−1(⋅,𝜏k−1)in Ω,
(9)
whe e T0(⋅,𝜏0)=T0,beingT0 he ini ial empe a u e.
So basically, we a e di iding he ime domain in o chunks going om one measu emen ime o he nex one in
a way ha acili a es he de ini ion o he in e se p oblems below. Be o e o mula ing i , we in oduce some u he
no a ion. We de ine he applica ion (xi,𝜏k)∈Ψ×Υ→
T(xi,𝜏k)∈R+,1≤i≤P,1≤k≤P ,
T(xi,𝜏k)being he expe i-
men ally measu ed empe a u e a (xi,𝜏k)∈Ψ×Υ. Mo eo e , o simpli y he no a ion, and i he e is no oom o e o ,
we deno e
Tk(xi)∶=
T(xi,𝜏k),1≤i≤P,1≤k≤P ,(10)
and we le Tk[g] ep esen he solu ion o P oblem 2co esponding o hea lux gon Γsin ×(𝜏k−1,𝜏k].
A eachmeasu emen in e alk,1≤k≤P ,wep oposeani e a i e p ocedu e,assuming ha , o k≥1, gland Tl[gl],
0≤l≤k−1, ha e been compu ed. Using a leas squa e, de e minis ic app oach, we s a e wo di e en in e se p oblems
o P oblem2.In he i s one,weconside as unc ional obeminimizedadis ancebe ween hemeasu edandcompu ed
empe a u es a he he mocouples. Then, we s a e i as
FIGURE 3 Time line o he in e se p oblem
10970207, 2023, 6, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.1002/nme.7167 by Uni e sidade de San iago de Compos ela, Wiley Online Lib a y on [15/06/2023]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
MORELLI e al. 1351
P oblem 3 (In e se). Being gland Tl[gl],1≤l≤k−1, known, and gi en he empe a u e measu emen s
Tk(xi),1≤
i≤P, indgk∈C(𝜏k−1,𝜏k;Lq(Γsin)) which minimizes he unc ional
Sk
1[gk]=1
2
P
∑
i=1
[Tk[gk](xi,𝜏k)−
Tk(xi)]2.(11)
He e, we deno e T0[g0]=T0.
The second in e se p oblem ha we conside includes in he cos unc ional he L2-no m o he hea lux. Thus, we
w i e i as
P oblem4 (In e se).Being gland Tl[gl],1≤l≤k−1, known, and gi en he empe a u emeasu emen s
Tk(xi),1≤i≤
P, indgk∈C(𝜏k−1,𝜏k;Lq(Γsin)) which minimizes he unc ional
Sk
2[gk]=1
2
P
∑
i=1
[Tk[gk](xi,𝜏k)−
Tk(xi)]2+pg⟨gk(𝜏k),gk(𝜏k)⟩L2(Γsin),(12)
whe e pg[K2∕W2]is a weigh applied o he hea lux no m.
3.3 In e se sol e o Sk
1
In hissec ion,wediscussano elme hodology o sol ingP oblem3.Inpa icula ,wemimic heme hodologyde eloped
by he au ho s o a s eady-s a e mold model,3expanding i o he uns eady case.
Weexploi asui ablepa ame e iza iono hehea lux,gk.Top ope lypa ame e izei ,wes a byconside ing ha we
wan o pa ame e ize an unknown unc ion gkin L (𝜏k−1,𝜏k;Lq(Ω)),1≤k≤P . Then, we no ice ha in hin slab cas ing
molds, he he mocouples a e all loca ed ew millime e s inwa d om Γsin . All oge he hey o m a uni o m 2D g id on
a su ace pa allel o he Γsin bounda y. Thus, a possible choice o he space pa ame e iza ion o gkis o use Radial Basis
Func ions(RBFs)cen e eda hep ojec ionso he he mocouplespoin sonΓsin.44 Byusing hispa ame e iza ion,weend
up ha ing as many basis unc ions as he mocouples. No e ha he me hodology is e y well adap ed o he applica ion
in use.
Inpa icula , wepa ame e izegkbyGaussian RBFs which allow us osepa a e he imeand space dependence. These
a e con inuous unc ions wi h global suppo in Γsin. Howe e , he ollowing discussion can be applied o o he basis
unc ions.
The pa ame e iza ion o he bounda y hea lux eads (see P ando’s appendix45)
gk(x, )≈ ℊk(x, )= P
∑
i=1
gk
i( )𝜙i(x), o ∈(𝜏k−1,𝜏k],(13)
whe e he 𝜙i(x)a e Pknown basis unc ions, and he gk
i( )a e he espec i e ime dependen unknown weigh s.
To de ine he RBFs, le 𝝃i,1≤i≤P, be he p ojec ion o he poin xi∈Ψon Γsin , ha is, such ha
𝝃i=a gmin
𝝃∈Γsin ‖xi−𝝃‖2,xi∈Ψ.(14)
By cen e ing he RBFs in hese poin s, hei exp ession is
𝜙j(x)=e−(𝜂‖x−𝝃j‖2)2, o j=1,2,…,P,(15)
whe e 𝜂is he shape pa ame e o he Gaussian basis. By inc easing (dec easing) i s alues, he adial decay o he basis
slows down (speeds up).
In his wo k, we explo e wo di e en app oaches o he ime pa ame e iza ion. In he i s one, we conside gk
i
independen o ime
gk
i( )=wk
i, o ∈(𝜏k−1,𝜏k],1≤i≤P,(16)
10970207, 2023, 6, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.1002/nme.7167 by Uni e sidade de San iago de Compos ela, Wiley Online Lib a y on [15/06/2023]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
1352 MORELLI e al.
beingwk
i ealnumbe s.In hisway, hehea luxisassumed obepiecewisecons an , ha is,cons an be weenconsecu i e
measu emen ins an s.
The second app oach is o conside he hea lux o be con inuous piecewise linea in (0, ], being a polynomial o
deg ee 1 be ween he sampling imes. Then, we assume he weigh s gk
i( ) o be linea in ime in he in e al (𝜏k−1,𝜏k].
Mo eo e , in his second case, he ollowing con inui y is assumed
gk
i( )| ↓𝜏k−1=gk−1
i( )| ↑𝜏k−1.(17)
In u n, we cha ac e ize gk
i( )as
gk
i( )=wk−1
i+( −𝜏k−1)wk
i−wk−1
i
𝜏k−𝜏k−1, o ∈(𝜏k−1,𝜏k].(18)
No ice ha bydoingpa ame e iza ion(13),wechange he p oblem omes ima ing a unc ioninanin ini edimensional
space a each ime in e al ( (k−1)N , kN ]=(𝜏k−1,𝜏k], o es ima ing he ec o wk=(wk
1,wk
2,…,wk
P)Tin RP, o each 1 ≤
k≤P .
Now, a each ime in e al (𝜏k−1,𝜏k], he objec i e o he in e se p oblem is o de e mine wkwhich iden i ies ℊkonce
he elemen s o he basis 𝜙i,i=1,2,…,Pa e ixed.Wes a e hein e sep oblemas
P oblem 5 (In e se). Gi en he empe a u e measu emen s
T(Ψ,𝜏k), ind
wk∈RP,1≤k≤P , which minimizes he
unc ional
Sk
1[wk]=1
2
P
∑
i=1
[Tk[wk](xi,𝜏k)−
Tk(xi)]2,(19)
whe e i he e is no oom o con usion Tk[wk]deno es he empe a u e Tk[ℊk],wi hℊkde ined as in (13)andgk
i( )
gi en by (16)o (18).
Fo a la e use, we de ine he gene al ec o ak∈RPas he ec o o he alues o a gene al ield a(x, )a he
measu emen poin s and a he measu emen ime 𝜏k,suchas
(ak)i=a(xi,𝜏k).(20)
Mo eo e , gi en wk, we de ine he esidual ec o Rk[wk]∈RPas
(Rk[wk])i∶= (Tk[wk])i−(
Tk)i,i=1,2,…,P.(21)
Thanks o (21), we ew i e he cos unc ional (19)as
Sk
1[wk]=1
2Rk[wk]TRk[wk].(22)
To minimize i , we w i e he c i ical poin equa ion
𝜕Sk
1[
wk]
𝜕wk
j
=P
∑
i=1
(Rk[
wk])i𝜕(Tk[
wk])i
𝜕wk
j
=0, o j=1,2,…,P.(23)
Thus, o each k,1≤k≤P , he solu ion o his equa ion will p o ide he weigh s ec o
wkco esponding o a c i ical
poin o Sk
1.
To explici ly ob ain om (23) an equa ion o he weigh s ha minimize ou unc ional Sk
1,weexploi helinea i yo
P oblem 2. To de i e i , we conside sepa a ely he piecewise cons an (16) and he piecewise linea (18)cases.
3.3.1 Piecewise cons an app oxima ion o he hea lux
Suppose o ha e he solu ions o he ollowing auxilia y p oblems
10970207, 2023, 6, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.1002/nme.7167 by Uni e sidade de San iago de Compos ela, Wiley Online Lib a y on [15/06/2023]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
MORELLI e al. 1359
and
ak
w=[wk
12wk
1wk
2··· wk
P2]T.(63)
We can now ew i e (61)as
P
∑
l=1
P
∑
q=1wk
lwk
q∫Γsin
𝜙l(x)𝜙q(x)dΓ=𝝓T
𝜙ak
w.(64)
Fu he mo e, de i ing (64) wi h espec o he weigh s, we ob ain
𝜕[∫Γsin (∑P
l=1wk
l𝜙l(x))(∑P
q=1wk
q𝜙q(x))dΓ]
𝜕wk
j
=𝜕𝝓T
𝜙ak
w
𝜕wk
j
=𝝓T
𝜙𝜕ak
w
𝜕wk
j
, o j=1,2,…,P.(65)
Conside ing he case j=1, we ha e
𝝓T
𝜙𝜕ak
w
𝜕wk
1
=𝝓T
𝜙[2wk
1wk
2··· wk
Pwk
20··· 0wk
30··· 0wk
P0···
]T,(66)
ha we can ew i e as
𝝓T
𝜙𝜕ak
w
𝜕wk
1
=2𝝓𝜙1wk
1+𝝓𝜙2wk
2+𝝓𝜙3wk
3+···+𝝓𝜙Pwk
P+𝝓𝜙P+1wk
2+𝝓𝜙2P+1wk
3+···+𝝓𝜙(P−1)P+1wk
P.(67)
Now, by no icing ha
𝝓𝜙( −1)P+s=∫Γsin
𝜙 (x)𝜙s(x)dΓ=𝝓𝜙(s−1)P+ ,1≤ ,s≤P,(68)
we ob ain
𝝓T
𝜙𝜕ak
w
𝜕wk
1
=2𝝓𝜙1wk
1+2𝝓𝜙2wk
2+2𝝓𝜙3wk
3+···+2𝝓𝜙Pwk
P=2𝝓T
𝜙1∶Pwk.(69)
Simila ly, i we conside he gene al case, we ha e
𝝓T
𝜙
𝜕ak
w
𝜕wk
j=2𝝓𝜙(j−1)P+1wk
1+2𝝓𝜙(j−1)P+2wk
2+2𝝓𝜙(j−1)P+3wk
3+···+2𝝓𝜙(j−1)P+Pwk
P=2𝝓T
𝜙(j−1)P+1∶(j−1)P+Pwk, o j=1,2,…,P.(70)
The e o e, hanks o (65) and (70), we can w i e
𝜕[∫Γsin (∑P
l=1wk
l𝜙l(x))(∑P
q=1wk
q𝜙q(x))dΓ]
𝜕wk
j
=𝝓T
𝜙𝜕ak
w
𝜕wk
j
=2𝝓T
𝜙(j−1)P+1∶(j−1)P+Pwk, o j=1,2,…,P.(71)
Le us de ine he ma ix Φ∈MP×Psuch ha
Φ ,s∶= ∫Γsin
𝜙 (x)𝜙s(x)dΓ.(72)
I we now conside he minimiza ion o Sk
2wi h espec o he weigh s, wk,asin(23), we ha e
𝜕Sk
2[
wk]
𝜕wk
j
=P
∑
i=1
(Rk[
wk])i𝜕(Tk[
wk])i
𝜕wk
j
+2pg𝝓T
𝜙(j−1)P+1∶(j−1)P+Pwk=0, o j=1,2,…,P.(73)
10970207, 2023, 6, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.1002/nme.7167 by Uni e sidade de San iago de Compos ela, Wiley Online Lib a y on [15/06/2023]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
1360 MORELLI e al.
Algo i hm 3. In e se sol e o he solu ion o P oblem 4 wi h piecewise cons an pa ame e iza ion in ime o he hea
lux, g
OFFLINE
Inpu RBF shape pa ame e , 𝜂; he mocouples measu emen poin s and imes, Ψ,Υ; cos unc ional pa ame e , pg
1: Se up RBF pa ame e iza ion by (15)
2: Compu e T𝜙i o i=1,2,…,Pby sol ing P oblem 6
3: Assemble ma ix Θby (38)
4: Assemble ma ix Φby (72)
ONLINE
Inpu Ini ial condi ion, T0
1: Se k=1
2: while k≤P do
3: Read he he mocouples measu emen s,
Tk
4: Compu e Tk
IC by sol ing P oblem 7
5: Assemble Tk
IC
6: Compu e
wkby sol ing (75)
7: Compu e gk
i( )by (16)
8: Compu e he hea lux ℊ(x, ) o ∈(𝜏k−1,𝜏k]by (13) and (16)
9: Use (30) o compu e Tk[ℊk]
10: k=k+1
11: end while
Conside ing he piecewise cons an case, hanks o (40) and (65), we ew i e (73)as
ΘT(Θ
wk+Tk
IC −
Tk)+2pgΦ
wk=0,(74)
being Θ he ma ix de ined in (38). The e o e, o each k,1≤k≤P , he solu ion o he in e se p oblem,
wk, is ob ained
by sol ing he linea sys em
(ΘTΘ+2pgΦ)
wk=Θ
T(
Tk−Tk
IC).(75)
Simila ly, o he piecewise linea case, hanks o (56) and (65), we ew i e (73)as
(
Θ)T(
Θ
wk−Θ
d
wk−1+Tk
IC −
Tk)+2pgΦ
wk=0,(76)
being
Θand Θd he ma ices de ined in (55). The e o e, o each k,1≤k≤P , a solu ion o he in e se p oblem,
wk,is
ob ained by sol ing he linea sys em
(
ΘT
Θ+2pgΦ)
wk=
ΘT(
Tk+Θ
d
wk−1−Tk
IC).(77)
The esul ing in e se sol e s a e s aigh o wa d modi ica ions o Algo i hms 1and 2. Then, we show hem in he
ollowing Algo i hms 3and 4.
As a inal ema k, no ice ha o pg=0K2
W2, we end up wi h he same solu ion as o Sk
1.
3.5 Regula iza ion
A e he de elopmen o no el in e se sol e s, we p o ide a b ie discussion abou egula iza ion. I is well known ha
in e se p oblems as he ones he e conside ed a e ill-posed. This means ha o ou p oblem a leas one o he ollowing
10970207, 2023, 6, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.1002/nme.7167 by Uni e sidade de San iago de Compos ela, Wiley Online Lib a y on [15/06/2023]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
MORELLI e al. 1361
Algo i hm 4. In e se sol e o he solu ion o P oblem 4 wi h piecewise linea pa ame e iza ion in ime o he hea
lux, g
OFFLINE
Inpu RBF shape pa ame e , 𝜂; he mocouples measu emen poin s and imes, Ψ,Υ; cos unc ional pa ame e , pg
1: Se up RBF pa ame e iza ion by (15)
2: Compu e T𝜙i o i=1,2,…,Pby sol ing P oblem 6
3: Compu e Tdi o i=1,2,…,Pby sol ing P oblem 8
4: Assemble ma ices
Θand Θd
5: Assemble ma ix Φby (72)
ONLINE
Inpu Ini ial condi ion, T0
1: Se k=1
2: while k≤P do
3: Read he he mocouples measu emen s,
Tk
4: Compu e Tk
IC by sol ing P oblem 7
5: Assemble Tk
IC
6: Compu e
wkby sol ing (77)
7: Compu e gk
i( )by (18)
8: Compu e he hea lux ℊ(x, ) o ∈(𝜏k−1,𝜏k]by (13) and (18)
9: Use (46) o compu e Tk[ℊk]
10: k=k+1
11: end while
p ope ies does no hold: o all admissible da a, a solu ion exis s; o all admissible da a, he solu ion is unique; he
solu iondependscon inuouslyon heda a.46 Inou discussion,we u ned hein ini edimensionalin e seP oblem3in o
hesolu iono hedisc e elinea sys ems(42)and(58)bymakingsomeassump ionson hehea lux(i.e.,pa ame e izing
i ). In his new se ing, i he ma ices ΘTΘand
ΘT
Θa e in e ible, we ha e he exis ence o a unique solu ion o ou
in e se p oblem.
As we will see in he nume ical es s sec ion, i u ns ou ha hese ma ices a e e y ill-condi ioned. This can cause
hema ix obenume ically ankde icien ,losing heuniquenesso asolu ion.Howe e , hisisno heonlyconce n.We
s ill ha e he p oblem o a con inuous dependence o he solu ion on he da a. The ill-condi ioning o he linea sys em
causes ha , i we ha e some noise in he da a ec o (as usual in an indus ial measu emen equipmen ), he solu ion o
he linea sys em di e ges om he co ec alue.
To add ess bo h hese p oblems, we equi e egula iza ion. The e a e se e al echniques a ailable o egula izing a
disc e eill-posedp oblemas hep esen one.Ingene al, heya edi idedin odi ec me hodslikeT unca edSingula Val-
ues Decomposi ion (TSVD) and Tikhono egula iza ion, and i e a i e me hods such as he conjuga e g adien me hod.
Fo adeepdiscussiono all egula iza ionme hods,we e e hein e es ed eade oHansen’smonog aphon hesubjec .47
In he p esen in es iga ion, we use TSVD. To b ie ly desc ibe his egula iza ion echnique, we deno e he Singula
Values Decomposi ion (SVD) o a ma ix Kby
K=UΣVT=
∑
i=1ui𝜎i T
i,(78)
whe e 𝜎ideno es he i h singula alue o K(numbe ed acco ding o hei dec easing alue), deno es he las no null
singula alue (i.e., he ank o K), uiand ia e he i h column o he semi-uni a y ma ices Uand V espec i ely (bo h
belonging o MP× ), and Σis he squa e ma ix o M × such ha Σii =𝜎iand Σij =0i i≠j. Then, gi en 𝛼TSVD ≤ , he
TSVD egula ized solu ion o he gene al linea sys em Kz=cis
z=𝛼TSVD
∑
i=1(uT
ic
𝜎i) i.(79)
10970207, 2023, 6, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.1002/nme.7167 by Uni e sidade de San iago de Compos ela, Wiley Online Lib a y on [15/06/2023]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
1362 MORELLI e al.
This solu ion di e s om he leas squa e solu ion only in ha he sum is unca ed a i=𝛼TSVD ins ead o i= .In his
way, we cu o he smalles singula alues ha a e esponsible o he e o s p opaga ion. Fo a de ailed discussion on
he solu ion o disc e e ill-posed in e se p oblems, we e e he eade o Hansen’s monog aph on he subjec .47
Toge he wi h heclassicala o emen ioned egula iza ionme hod,wein es iga ealso he egula iza ionbydisc e iza-
ion.48,49 Using his me hod, we exploi he egula izing p ope ies o coa sening he ime and/o space disc e iza ion o
imp o e he hea lux es ima ion. In he nex sec ion, we will es he pe o mance o hese egula iza ion me hods also
by adding noise o he he mocouples measu emen s.
3.6 Disc e iza ion selec ion algo i hm
To conclude his sec ion, we p opose an algo i hm o he au oma ed selec ion o some o he pa ame e s equi ed by
Algo i hm 4. As will be shown in Sec ion 4, he nume ical es s highligh ha his algo i hm is e y sensi i e o he mesh
and imedisc e iza ion e inemen as wellas o he pa ame e pg. We an icipa ehe e ha his in e sesol e showsse e e
ins abili ies o ine disc e iza ions. Howe e , hese ins abili ies a e e ec i ely elimina ed o alues o pg ha a e abo e
a h eshold ha depends on he disc e iza ion e inemen . In o hese alues o pg,weno icead as icdec easeo he
dependency o he algo i hm om he disc e iza ion.
Howe e , he uncon olled inc ease in pgdoes no lead o a mono onic imp o emen o he in e se sol e pe o -
mances. As can be obse ed in he nume ical esul s o Sec ion 4(see Figu es 10,11,12,18,19), he dependency o he
algo i hm om pgis such ha i is uns able o low alues o pg hen, inc easing u he pg, i sha ply achie es an op i-
mum o pe o mance be o e eaching a pla eau a which he algo i hm is s able bu he e m ⟨gk(𝜏k),gk(𝜏k)⟩L2(Γsin)in (12)
o e comes he measu emen s dis ance one, Sk
1de ined in (11). Thus, o oo high alues o pg, we ha e a s able algo i hm
ha is almos independen om he disc e iza ion e inemen bu ha p o ides poo hea lux es ima ions.
To allow an indus ial use o he p oposed in e se sol e , he objec i e o his sec ion is o de elop a me hod o
au oma icallyselec ing heΔ , hemeshand he alueo pgsuch ha he algo i hmiss able and accu a ely es ima es he
mold-s eel hea lux.
In de eloping such me hod, we assume o ha e a ailable a eliable da ase o he mocouples measu emen s,
T ain(Ψ,Υ ain), ha we can use o pe o m his uning o line. Mo eo e , we assume ha , independen ly om he mold
physical pa ame e s and he hea lux alues, his in e se sol e always shows he p e iously desc ibed beha io wi h
espec o pg. In pa icula , we assume ha , o alues o pghighe han a p oblem speci ic h eshold, he algo i hm is
s able o all he disc e iza ions and independen om hem (i.e. we ob ain simila solu ions o any gi en mesh and Δ ).
We ecall, ha in he eal indus ial case, we do no ha e any in o ma ion abou he ue hea lux ha we wan o
es ima e. Thus, his selec ion me hodology canno be based on he hea lux es ima ion e o . Howe e , Figu es 13 and
20 show ha he measu emen disc epancy unc ional Sk
1and he hea lux es ima ion e o ha e a simila beha io as
unc ions o pgand we will use his quan i y o de e mine he quali y o he hea lux es ima ion.
All ha said, webeginby selec ingan o de edse o meshes (Δx1,Δx2,…,ΔxnM)andan o de edse o imes epsizes
(Δ 1,Δ 2,…,Δ n ). We o de hem om he ines o he coa ses disc e iza ion (i.e. Δx1<Δx2<···<ΔxnMand Δ 1<
Δ 2<···<Δ n ). Then, ou i s objec i e is o iden i y a p0
gwi hin he a o emen ioned s abili y egion.
Todoi ,wes a wi ha en a i ep0
g.Fo his alueo hepa ame e ,wesol e hein e sep oblemon he ainingmea-
su emen da ase o all Δxand Δ . Le us deno e by T[Δx,Δ ] he co esponding solu ion. Ha ing done so, we compu e
ΔT∶= max
i,j,q,p‖‖T[Δxi,Δ j]−T[Δxq,Δ p]‖‖L∞((0, ];L2(Ωs)).(80)
I we ha e
ΔT>ΔxnM+Δ n ,(81)
we conside ha he solu ion is oo dependen on he disc e iza ion e inemen . Then, we inc ease he alue o p0
gand
edo he calcula ions un il
ΔT≤ΔxnM+Δ n ,(82)
is sa is ied.
10970207, 2023, 6, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.1002/nme.7167 by Uni e sidade de San iago de Compos ela, Wiley Online Lib a y on [15/06/2023]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
MORELLI e al. 1363
Algo i hm 5. O line selec ion o he mesh, he imes ep size and pg o he in e se sol e in Algo i hm 4
Inpu O de ed se o meshes, (Δx1,Δx2,…,ΔxnM); o de ed se o imes ep sizes, (Δ 1,Δ 2,…,Δ n );p0
g; aining se ,
T ain(Ψ,Υ ain)
1: while ΔT>ΔxnM+Δ n do ⊳Iden i y s abili y egion
2: o i=1 onMdo
3: o j=1 on do
4: Sol e he in e se p oblem on he aining se ,
T ain, by using Algo i hm 4 wi h Δxi,Δ jand pg=p0
g
5: Compu e mS[Δxi,Δ j,p0
g]by (83)
6: end o
7: end o
8: Compu e ΔTby (82)
9: i ΔT>ΔxnM+Δ n hen
10: p0
g=10p0
g
11: end i
12: end while
13: Choose Δx0and Δ 0co esponding o mini,jmS[Δxi,Δ j,p0
g] o 1 ≤i≤nM,1≤j≤n
14: l=1, =0
15: while =0do
16: Find pl
g=a gminpg(mS[Δxl−1,Δ l−1,pg])
17: o i=1 onMdo
18: o j=1 on do
19: Sol e he in e se p oblem on he aining se ,
T ain, by using Algo i hm 4 wi h Δxi,Δ jand pg=pl
g
20: Compu e mS[Δxi,Δ j,pl
g]by (83)
21: end o
22: end o
23: Choose Δxland Δ lco esponding o mini,jmS[Δxi,Δ j,pl
g] o 1 ≤i≤nM,1≤j≤n
24: i Δxl=Δxl−1ςΔ l=Δ l−1 hen
25: =1
26: end i
27: l=l+1
28: end while
Once we ind a alue o p0
gwi hin he s abili y egion, we choose he disc e iza ion se up (Δx1,Δ 1) ha co esponds
o he minimum o p0
go
mS[Δx,Δ ,pg]∶=meank(Sk
1[Δx,Δ ,pg]).(83)
Once we selec Δx1and Δ 1, we choose p1
gas he alue o pg ha minimizes mS[Δx1,Δ 1,pg]. Then, we ix pg=p1
g
and we sol e again he in e se p oblem o all he conside ed disc e iza ion se ups. I he p e iously selec ed dis-
c e iza ion is he one ha co esponds o he lowes alue o mS[Δx,Δ ,p1
g], we choose Δx1,Δ 1,andpg=p1
g,andwe
s op he p ocess. O he wise, we con inue i e a ing by selec ing Δx2and Δ 2as he ones co esponding o he small-
es mS[Δx,Δ ,p1
g]and looking o he p2
g ha minimizes mS[Δx2,Δ 2,pg], and so on. We summa ize all his p ocess in
Algo i hm 5.
This me hod allows a da a-d i en, au oma ed selec ion o he disc e iza ion e inemen and he pgpa ame e . This
esul comes o he cos o compu ing nM⋅n solu ions o he in e se p oblem a each i e a ion. I he a ailable memo y
allows i , we can keep in he memo y he esul s o he o line compu a ions ela ed o each disc e iza ion. O he wise,
we ha e he ecompu e e e y ime hese o line phases. Howe e , we designed his algo i hm o be used o line. Then,
e en i i is compu a ionally expensi e, we can un i be o e he cas e s a s o wo k and i only equi es he da ase o
he mocouples measu emen s
T ain.
10970207, 2023, 6, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.1002/nme.7167 by Uni e sidade de San iago de Compos ela, Wiley Online Lib a y on [15/06/2023]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
1364 MORELLI e al.
4NUMERICAL TESTS
To es he p e iously de eloped me hodologies, we design di e en benchma k cases. Th ough hese es s, we alida e
andanalyze hepe o manceso hein e sesol e s ha wep oposedin hep e ioussec ions.Wedesign wobenchma ks
o pe o m di e en es s o he in e se p oblem sol e s p oposed in Sec ion 3.
No ice ha all he compu a ions a e pe o med in ITHACA-FV50,51 which is a C++ lib a y based on OpenFOAM52
de eloped a SISSA Ma hlab.
4.1 Benchma k 1
In his sec ion, we es he pe o mances o he in e se sol e s p oposed in Sec ion 3in he econs uc ion o a linea in
ime hea lux, which is nonlinea in space.
4.1.1 Se up o he es case
To design a nume ical es case o he in e se p oblems, we p oceed as ollows: we a bi a ily de ine a bounda y hea
lux, g (x, ), and he he mocouples posi ions, Ψ, and sampling equency, samp. Then, we sol e he di ec P oblem 1
associa ed wi h g (x, )in he ime domain (0, ], ob aining he ela ed empe a u e ield. Finally, we use i s al-
ues a he he mocouples poin s and sampling imes as inpu measu emen s o he in e se p oblem,
T. Using his
app oach, we a e able o analyze he in e se p oblem pe o mance in he econs uc ion o he bounda y hea lux,
g (x, ).
Table 1shows he geome ical and physical pa ame e s selec ed o he p esen benchma k case. In he a emp o
mimicking he eal indus ial si ua ion o es ima ing he bounda y hea lux in a pla e o a CC mold, hese pa ame e s
a e close o eal indus ial alues. We use he compu a ional domain in Figu e 4A whe e L,Wand Ha e se as in a eal
mold pla e. Finally, Figu e 4B shows he he mocouple loca ions.
To es he e ec o he egula iza ion by disc e iza ion, we use di e en space and ime disc e iza ions. Fo he ime
disc e iza ion, we use homogeneous ime disc e iza ion wi h Δ =0.1, 0.2, 0.25, and 0.5 s. Fo he space disc e iza ion,
we use he uni o m, s uc u ed, o hogonal, hexahed al meshes p esen ed in Table 2.
TABLE 1 Geome ical and physical pa ame e s used o he benchma k es cases
Pa ame e Value
The mal conduc i i y, ks383 W/(m K)
Densi y, 𝜌s8940 kg/m3
Speci ic hea capaci y, Cps390 J/(kg K)
Hea ans e coe icien , h5.66e4W/(m
2K)
Wa e empe a u e, T 350 K
Ini ial condi ion, T0350 K
L2m
W0.1 m
H1.2 m
Sampling equency, samp 1Hz
a1100 W/(m2s)
b1200 W/(m2s)
c3000 W/(m2s)
Final ime, 50 s
10970207, 2023, 6, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.1002/nme.7167 by Uni e sidade de San iago de Compos ela, Wiley Online Lib a y on [15/06/2023]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
MORELLI e al. 1365
(A)
(B)
FIGURE 4 Schema ic o he domain used in he benchma k es cases (A) and posi ion o he 100 he mocouples a he plane
y=0.02 m (B) used o he in e se sol e es s (images aken om Mo elli e al3)
TABLE 2 Summa y o he di e en meshes used in he nume ical es s
Mesh 1 Mesh 2 Mesh 3 Mesh 4 Mesh 5
Numbe o elemen s 1.7e54.5e42.1e47.5e31.5e3
Fo his es case, we selec he hea lux g o be linea in ime and quad a ic in space. In pa icula , gi en g1(x)=
bz2+cwi h band cas in Table 1, we selec he hea lux
g (x, )=−ks(0.5 g1(x)+g1(x)).(84)
Mo eo e , o analyze he pe o mance o he in e se sol e s, we in oduce he ela i e e o
e el(x, )∶=g (x, )−gc(x, )
g (x, ),(85)
whe e gcis he hea lux compu ed wi h he di e en me hodologies desc ibed in Sec ion 3 ha will be es ed in he
ollowing.
4.1.2 E ec o Time and space disc e iza ion e inemen
Now, p o ided all he de ails o he i s benchma k se up, we can p oceed p esen ing he esul s. Fi s ly, we show he
e ec s o mesh and ime disc e iza ion e inemen . To do i , we do no add any noise o he empe a u e measu emen s
and do no apply any egula iza ion in he solu ion o he linea sys ems sol ing hem by a LU ac o iza ion wi h ull
pi o ing.
Wes a byanalyzing hecaseinwhichweminimize he unc ionalSk
1(i.e.,pg=0).WeshowinFigu e5 hemaximum
andmean alueo heL2-andL∞-no mo he ela i ee o , e el,in hein e al(0, ],as he imeandspacedisc e iza ion
changes o Algo i hm 1(i.e., piecewise cons an app oxima ion in ime o he hea lux).
F om he igu es,weapp ecia eononeside ha he imedisc e iza ioncoa seninghas e yli lee ec sonAlgo i hm1
wi h a small dec ease o he e o as Δ inc eases. On he o he , he space disc e iza ion does no ha e any e ec on his
in e se sol e .
10970207, 2023, 6, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.1002/nme.7167 by Uni e sidade de San iago de Compos ela, Wiley Online Lib a y on [15/06/2023]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
1366 MORELLI e al.
(A) (B)
(C) (D)
FIGURE 5 Maximum (ci cles) and mean (squa es) alues o he L2-andL∞-no m o he ela i e e o , e el, in he in e al (0, ], o
Benchma k 1 as he ime and space disc e iza ion changes o Algo i hm 1(piecewise cons an ime app oxima ion o he hea lux and
pg=0K2
W2). (A) L2-no m o he ela i e e o as a unc ion o Δ ,(B)L∞-no m o he ela i e e o as a unc ion o Δ ,(C)L2-no m o he
ela i e e o as a unc ion o he mesh size, (D) L∞-no m o he ela i e e o as a unc ion o he mesh size
We now pe o m he same es o he piecewise linea ime app oxima ion o Algo i hm 2. Simila ly, Figu e 6shows
hemaximumandmean alueo heL2-andL∞-no mo he ela i ee o ,e el,in hein e al(0, ],as he imeandspace
disc e iza ion changes.
In his se ing, he ob ained esul s a e e y di e en om he p e ious case. Fi s o all, we no ice a massi e in lu-
ence o bo h he space and ime disc e iza ion e inemen on he pe o mances o he in e se sol e . As an icipa ed in
Sec ion 3.5, he egula iza ion by disc e iza ion plays an impo an ole as he algo i hm pe o mances a e imp o ed by
se e alo de so magni udeby hecoa seningo hedisc e iza ion.Mo eo e ,whencompa ing he esul s o Algo i hm1
and 2, we no ice ha he piecewise linea sol e is able o ou pe o m he cons an one by h ee o de s o magni ude bu
is also e y uns able depending on he disc e iza ion.
To be e unde s and he beha io o his in e se sol e , Figu e 7illus a es he L2-no m o he ela i e e o , e el,asa
unc iono ime o mesh3wi hdi e en Δ .F om hese esul s,wesee ha hehighe o sshowninFigu e6a ecaused
bydi e gingoscilla ionsin healgo i hm.Howe e ,wealsono ice om he igu e ha ,coa sening he imedisc e iza ion,
mono onically educes such ins abili y un il achie ing a s able solu ion, e en ually.
4.1.3 E ec o cos unc ional pa ame e , pg
In his sec ion, weanalyze he ole ha he cos unc ional pa ame e ,pg,in(12),has on he pe o manceo he p oposed
in e se sol e s. To do i , we sol e se e al imes his benchma k case using he di e en meshes o Table 2and di e en
imes ep sizes. Then, we plo he maximum and mean alue o he L2-no m o he ela i e e o , e el, o e he en i e
in e al =(0, ]as a unc ion o he cos unc ional pa ame e , pg.
10970207, 2023, 6, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.1002/nme.7167 by Uni e sidade de San iago de Compos ela, Wiley Online Lib a y on [15/06/2023]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
MORELLI e al. 1367
(A) (B)
(C) (D)
FIGURE 6 Maximum (ci cles) and mean (squa es) alues o he L2-andL∞-no m o he ela i e e o , e el, in he in e al (0, ], o
Benchma k 1 as he ime and space disc e iza ion changes o Algo i hm 2(piecewise linea ime app oxima ion o he hea lux and
pg=0K2
W2). (A) L2-no m o he ela i e e o as a unc ion o Δ ,(B)L∞-no m o he ela i e e o as a unc ion o Δ ,(C)L2-no m o he
ela i e e o as a unc ion o he mesh size, (D) L∞-no m o he ela i e e o as a unc ion o he mesh size
FIGURE 7 L2-no m o he ela i e e o , e el, in Benchma k 1 o Algo i hm 2(piecewise linea ime app oxima ion o he hea lux
and pg=0K2
W2). The p esen ed esul s a e ob ained wi h Mesh 3.
Wes a wi hAlgo i hm3(i.e.,piecewisecons an app oxima ionin imeo hehea lux).Figu e8shows heob ained
esul s o di e en imes ep sizes and a ixed space disc e iza ion.
F om he esul s, we no ice ha inc easing he alue o pgmono onically dec eases he quali y o he econs uc ion.
Mo eo e , i is ue o all conside ed Δ wi h a sligh imp o emen o he pe o mances as he ime disc e iza ion ge s
coa se .
Now,wepe o masimila es bu his imewekeepΔ =0.25 sand es hedi e en mesheso Table 2. We illus a e
in Figu e 9 he ob ained esul s.
10970207, 2023, 6, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.1002/nme.7167 by Uni e sidade de San iago de Compos ela, Wiley Online Lib a y on [15/06/2023]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
1368 MORELLI e al.
(A) (B)
FIGURE 8 Mean (a) and maximum (b) alues o he L2-no m o he ela i e e o , e el, in he in e al (0, ], o Benchma k 1 as he
alue o he cos unc ion pa ame e , pg, changes o Algo i hm 3(piecewise cons an ime app oxima ion o he hea lux). We show he
esul s o Mesh 3 and di e en Δ .
(A) (B)
FIGURE 9 Mean (a) and maximum (b) alues o he L2-no m o he ela i e e o , e el, in he in e al (0, ], o Benchma k 1 as he
alue o he cos unc ion pa ame e , pg, changes o Algo i hm 3(piecewise cons an ime app oxima ion o he hea lux). We show he
esul s o Δ =0.25 s and di e en meshes.
This igu e con i ms ha Algo i hm 3is badly a ec ed by he implemen a ion o he second e m o (12). In ac , i s
pe o mance d ama ically de e io a es as soon as his e m begins o play a ole (i.e., pg≳10−12K2∕W2). Mo eo e , he
esul sa ealmos independen om hedisc e iza ion e inemen .This u he con i ms heinsensibili yo hisalgo i hm
om he used disc e iza ion.
We con inue by pe o ming he same kind o es s on Algo i hm 3(i.e., piecewise linea app oxima ion o he hea
lux in ime). We s a by es ing di e en imes ep sizes while using Mesh 3 o he space disc e iza ion. We p esen he
esul s in Figu e 10.
A i s , we no ice ha his algo i hm has a e y di e en beha io wi h espec o he piecewise cons an case. In his
case, he imes ep size d ama ically a ec s he esul s. We can depic wo di e en beha io s as pgchanges o a chosen
Δ . In he i s one (i.e., Δ =0.1, 0.2, and 0.25 s), he in e se sol e is e y uns able and p o ides comple ely useless
solu ions o low alues o pg(i.e., pg≲10−12K2∕W2). As pginc eases, he quali y o he app oxima ion apidly ises up
un il he e o eaches a minimum. He e, we ha e s able solu ions and a good app oxima ion o he hea lux. Fo highe
alues o pg, he e o mono onically inc eases un il i eaches a pla eau a 100%.
On he o he hand, we ha e a di e en beha io o Δ =0.5 s. In his case, he in e se sol e pe o ms simila ly o
he piecewise cons an case, bu he quali y o he es ima ion is by almos wo o de so magni ude be e . Then, we ha e
s ableandaccu a esolu ions o low alueso pg.Fo pg≳10−12K2∕W2weha eamono onicdeg ada iono he hea lux
es ima ion un il we each he 100%pla eau.
I is in e es ing o no ice ha he second e m in he unc ional Sk
2can make he sol e insensible o he dis-
c e iza ion e inemen . In ac , a e a ce ain alue o pg, he ela i e e o no ms o he di e en Δ a e almos
coinciden .
10970207, 2023, 6, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.1002/nme.7167 by Uni e sidade de San iago de Compos ela, Wiley Online Lib a y on [15/06/2023]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
MORELLI e al. 1375
(A) (B)
FIGURE 20 Mean alues o Sk
1 o 1 ≤k≤P , o Benchma k 2 as he alue o he cos unc ion pa ame e , pg, changes. The esul s a e
ob ained using Algo i hm 4(piecewise linea ime app oxima ion o he hea lux). We show he esul s o Mesh 3 and di e en Δ in (A),
and o Δ =0.25 and di e en meshes in (B).
TABLE 5 Tes o Algo i hm 5 o Benchma k 2
I e a ion Mesh 𝚫 [s]pg[K2
W2]meank(Sk
1)[K2]
040.18e−78.46e2
1 5 0.5 3.6e−11 6.2e0
250.53.2e−13 5.5e−1
4.2.3 E ec s o measu emen s noise and egula iza ion
In hissec ion,we es hee ec ha addingnoise o hemeasu emen s ec o ,
T,hasin hepe o manceso Algo i hms1
and 2. F om he indus ial poin o iew, his analysis is o pa icula in e es o ou applica ion since in he eal case,
he mocouples measu emen s a e a ec ed by noise.
We pe o m his analysis by adding o he measu emen s ec o he Gaussian andom noise 𝜼=(𝝁,Σ),whe e𝝁∈
RMis he mean ec o and Σ∈MM×Mis he co a iance ma ix. Then, we ha e
Tk
𝜂=
Tk+𝜼.(86)
In pa icula , we choose 𝜼 o be an independen and iden ically dis ibu ed andom a iable wi h ze o mean, ha is,
𝜼=(0,𝜔2I),whe e𝜔deno es he noise s anda d de ia ion. To s udy he e ec o noise, we pe o m se e al solu ions
o he in e se p oblem using
Tk
𝜂as he mocouples measu emen s. Fo each es , we compu e 200 samples.
We show in Figu es 21 and 22 he ob ained esul s o he piecewise cons an and linea algo i hm, espec i ely. In
pa icula , we illus a e o each o hem he mean alues o e he samples o he mean and maximum o he ela i e
e o (85)in(0, ](wi h 90% quan ile ba s) o di e en alues o he noise s anda d de ia ion, 𝜔. The igu e com-
pa es he esul s ob ained using LU wi h ull pi o ing and TSVD wi h di e en alues o he egula iza ion pa ame e
𝛼TSVD.
The esul s show a e y di e en dependency om he measu emen noise in he wo algo i hm. The piecewise con-
s an Algo i hm 1shows in Figu e 21 o be qui e obus wi h espec o hese le els o noise. Mo eo e , he TSVD is
e ec i e in educing he noise p opaga ion and we a e able o keep a easonable le el o accu acy.
On he o he hand, he piecewise linea Algo i hm 2is much mo e a ec ed by he noise. By using he TSVD egula -
iza ion, we ha e an imp o emen o he noise obus ness. Howe e , he e o a e o inc ease is much highe han o he
piecewise cons an sol e .
10970207, 2023, 6, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.1002/nme.7167 by Uni e sidade de San iago de Compos ela, Wiley Online Lib a y on [15/06/2023]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
1376 MORELLI e al.
(A) (B)
FIGURE 21 E ec o he noise in he empe a u e measu emen s o Algo i hm 1in Benchma k 2. In he igu es, we show he mean
(A) and maximum (B) alues o he ela i e e o (85)in(0, ] o di e en alues o he noise s anda d de ia ion and using bo h LU wi h ull
pi o ing and TSVD o he solu ion o in e se p oblem linea sys em (42). Fo each case, we pe o med 200 uns. The ma ke s show he mean
alues while he ba s a e he 90% quan iles. In hese compu a ions, we conside ed pg=0K2
W2.
(A) (B)
FIGURE 22 E ec o he noise in he empe a u e measu emen s o Algo i hm 2in Benchma k 2. In he igu es, we show he mean
(A) and maximum (B) alues o he ela i e e o (85)in(0, ] o di e en alues o he noise s anda d de ia ion and using bo h LU wi h ull
pi o ing and TSVD o he solu ion o in e se p oblem linea sys em (58). Fo each case, we pe o med 200 uns. The ma ke s show he mean
alues while he ba s a e he 90% quan iles. In hese compu a ions, we conside ed pg=0K2
W2.
TABLE 6 A e age compu a ional cos o one i e a ion o Algo i hm 1in Benchma k 2
𝚫 ⧵Mesh Mesh 1 Mesh 2 Mesh 3 Mesh 4 Mesh 5
0.1 s 5184.0 ms 1402.6 ms 815.6 ms 482.8 ms 343.4 ms
0.2 s 2520.1 ms 741.2 ms 427.4 ms 260.2 ms 192.6 ms
0.25 s 2052.9 ms 600.6 ms 350.9 ms 215.8 ms 162.1 ms
0.5 s 1115.5 ms 333.4 ms 197.6 ms 128.2 ms 101.0 ms
4.2.4 Compu a ional cos
To conclude his analysis, we p esen in Tables 6and 7 he nume ical cos o pe o ming one i e a ion o he p oposed
in e se sol e s. No ice ha all he compu a ions we e pe o med in se ial on a In el®Co e™i7-8550U CPU p ocesso . As
expec ed he equi ed CPU ime inc eases wi h he e inemen o he disc e iza ion. Since we a e using ela i e coa se
meshes due o he simpli ied geome y, he compu a ional cos in many cases mee s he eal- ime equi emen o his
applica ion (i.e. 1 s). Howe e , he meshes equi ed o he disc e iza ion o he eal mold geome y a e such ha we
canno ensu e eal- ime pe o mances in hese cases.
10970207, 2023, 6, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.1002/nme.7167 by Uni e sidade de San iago de Compos ela, Wiley Online Lib a y on [15/06/2023]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
MORELLI e al. 1377
TABLE 7 A e age compu a ional cos o one i e a ion o Algo i hm 2in Benchma k 2
𝚫 ⧵Mesh Mesh 1 Mesh 2 Mesh 3 Mesh 4 Mesh 5
0.1 s 5966.4 ms 1621.2 ms 1442.5 ms 517.3 ms 400.1 ms
0.2 s 2836.2 ms 907.4 ms 518.1 ms 293.4 ms 206.2 ms
0.25 s 2258.8 ms 707.6 ms 382.5 ms 232.4 ms 182.9 ms
0.5 s 1195.8 ms 435.7 ms 216.8 ms 140.2 ms 108.0 ms
5CONCLUSIONS AND FUTURE WORKS
Thegoalo hep esen in es iga ionwas ode elopma hema ical ools omoni o hemoldbeha io inCCmachine ies.
In pa icula , we we e in e es ed in es ima ing he mold-s eel hea lux. We op ed o s a ing he p oblem in a da a assim-
ila ion, op imal con ol se ing in which we look o he hea lux ha minimizes a unc ional ha includes a measu e o
he dis ance be ween he compu ed and measu ed empe a u e a he measu emen poin s.
In de i ing he mold model, we conside edas compu a ional domain he mold pla es only and we o mula ed a h ee
dimensional uns eady-s a e hea conduc ion mold model. In his se ing, he mold-s eel hea lux is a Neumann BC on a
po ion o he bounda y o ou domain. Then, we can gene alize his ma hema ical in e se p oblem as he es ima ion o
a Neumann BC gi en poin wise s a e measu emen s in he in e io o he domain.
In his uns eady-s a e se ing, we used a sequen ial app oach o he in e se p oblem. In ac , o p o ide a eal- ime
solu ion in his se ing means o s ay always a he on o he ime line as i s e ches. Then, since ou measu emen s
come equally spaced in ime by one second, we conside ed he p oblem o es ima ing he hea lux only in be ween he
las measu emen and he p e ious one, assuming o ha e al eady he solu ion o olde imes.
In his amewo k, we s a ed wo no el o mula ions o he bounda y hea lux es ima ion p oblem. One looking o
he hea lux ha minimizes a measu e o he dis ance be ween compu ed and measu ed empe a u e only. While, in he
o he , we wan o minimize his dis ance plus a hea lux no m.
Fo bo h hese in e se p oblems, we de eloped no el me hodologies o hei solu ion ha exploi a RBFs pa ame-
e iza ion o he hea lux in space wi h ime dependen coe icien s. Wi h espec o hese coe icien s, we conside ed
bo h he piecewise cons an and he piecewise linea case. I means ha he es ima ed hea lux is cons an o linea in
be ween wo con iguous measu emen ins an s.
These no el me hodologies a e di ec me hods ha bene i om an o line-online decomposi ion. Thanks o his
decomposi ion, we ha e a i s compu a ionally expensi e o line phase, in which we sol e se e al di ec p oblems. This
o linephaseiscompu edonceand o allanddoesno equi eanymeasu emen .Then,when hecas e s a s owo k,we
onlyha e ocollec he he mocouplesmeasu emen sand un he onlinephasewhich iscompu a ionallymuch cheape .
To conclude, we es ed he p oposedin e sesol e s on somebenchma k cases. F om he ob ained esul s,we no iced
a g ea di e ence in he beha io o he piecewise cons an and linea in e se sol e s. The o me showed a e y s able
beha io and insensi i i y o he ime and space disc e iza ion used. The la e , a he , is e y much in luenced by he
disc e iza ion used. In pa icula , i can be e y uns able when using ine disc e iza ions bu his ins abili y is educed by
coa sening he ime and/o space disc e iza ion. In ac , o some disc e iza ions, we achie ed e y s able and accu a e
solu ions, e en ually.
We also es ed he e ec s ha adding he hea lux no m o he minimiza ion unc ional has on hese in e se sol e s.
We implemen ed his new e m mul iplying i by a pa ame e . Then, we es ed he e ec ha i s alue has on he sol e s
pe o mance.
Weno iced ha hepiecewisecons an algo i hmpe o mancemono onicallyde e io a esas hispa ame e inc eases.
The same goes o he piecewise linea sol e when using he coa ses , s able disc e iza ions. Howe e , he uns able
con igu a ions showed o be posi i ely a ec ed by he addi ion o his new e m and, o some alues o his pa ame e ,
we we e able o ob ain s able and accu a e solu ions o all he es ed disc e iza ions. While, o oo high alues o he
pa ame e , hesolu ioniss ablebu inaccu a e o all hemeshesand imes epsizes.Mo eo e ,weshowed ha , o alues
o he pa ame e abo e a h eshold, he in e se sol e pe o mance is almos independen om he disc e iza ion.
Due o his dependency om he disc e iza ion and he unc ional pa ame e , we de eloped an algo i hm o he
au oma ic selec ion o hese quan i ies. In he nume ical es s, i p o ed o be able o a selec ion ha co esponds o a
s able and accu a e in e se sol e .
10970207, 2023, 6, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.1002/nme.7167 by Uni e sidade de San iago de Compos ela, Wiley Online Lib a y on [15/06/2023]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
1378 MORELLI e al.
Tes ing he in e se sol e s o se e al noise le els showed again a di e en beha io be ween he piecewise cons an
and linea app oxima ions. The o me is much less sensi i e o he measu emen s noise han he la e . Fo bo h, he
TSVD egula iza ion p o ed o be able o mi iga e he noise p opaga ion and we we e able o ob ain accu a e and s able
solu ions also in he p esence o noise.
To conclude, we ecall ha he online phases o he p oposed algo i hms equi e he solu ion o a ull o de p oblem
whosecompu a ionalcos depends on hemesh and imes epsize.Asshown in henume ical es ssec ion, i means ha
wecanno ensu e eal- imepe o mance o hesealgo i hmsas heya e.Then,inou u u ewo k,wewillde elopmodel
o de educ ion echniques ha will allow us o educe he online phase compu a ional ime and make i independen
om he disc e iza ion.54,55
As a inal ema k, we discuss he applica ion o he new p oposed me hodology o o he p oblems. Recalling ha he
p esen ed con inuous cas ing p oblem is a Neumann BC es ima ion p oblem in a uns eady linea se ing wi h poin wise
s a e measu emen s in he in e io o he domain, we can apply he p oposed me hodologies o any p oblem sha ing
hese ea u es. An example can be a bounda y s ess es ima ion p oblem in linea elas ici y wi h poin wise de o ma ion
measu emen s.
O he possible u u e wo ks on he subjec could be ela ed o he s udy o heo e ical esul s ha can ensu e a p io i
he s abili y and accu acy o hese in e se sol e s wi h espec o he used disc e iza ion. I would inc ease he po en ial
o he p oposed me hodologies as well as hei eliabili y. In pa icula , i would be use ul o he inal use o know
a p io i he ime and space disc e iza ion o selec as well as he minimiza ion unc ional pa ame e . No ice ha i is
needed o hepiecewiselinea in e sesol e because hepiecewisecons an oneisalmos insensible o hedisc e iza ion
e inemen .
In he u u e,i wouldalsobein e es ing oin es iga e heuseo acomple elydi e en app oachin hesolu iono his
in e se p oblem. Thinking abou a mo e p ope handling o he measu emen noise, we could hink o using a Bayesian
app oach.56 Techniques such as ensemble Kalman il e could be sui able o his p oblem gi en he sequen iali y o he
measu emen s.Mo eo e ,conside ing he eal- ime equi emen o heapplica ion,i wouldp obably equi eane ec i e
use o model o de educ ion echniques o educe he demanding compu a ional cos o hese echniques.
ACKNOWLEDGMENTS
We would like o acknowledge he inancial suppo o he Eu opean Union unde he Ma ie Sklodowska-Cu ie G an
Ag eemen No. 765374. We also acknowledge he pa ial suppo by he Minis y o Economy, Indus y and Com-
pe i i eness h ough he Plan Nacional de I +D+i (MTM2015-68275-R), by he Agencia Es a al de In es igacion
h ough p ojec [PID2019-105615RB-I00/AEI/10.13039/501100011033], by he Eu opean Union Funding o Resea ch
and Inno a ion–Ho izon 2020 P og am–in he amewo k o Eu opean Resea ch Council Execu i e Agency: Consolida-
o G an H2020 ERC CoG 2015 AROMA-CFD p ojec 681447 “Ad anced Reduced O de Me hods wi h Applica ions in
Compu a ional Fluid Dynamics” and INDAM-GNCS p ojec “Ad anced in usi e and non-in usi e model o de educ-
ion echniquesandapplica ions”,2022.Wealsoacknowledge hesuppo om heMIURPRINp ojec NA-FROM-PDEs
and H2020 RISE ARIA p ojec . Mo eo e , we g a e ully hank Gian anco Ma coni, Fede ico Bianco and Ricca do Con e
om Danieli & C.O icine Meccaniche SpA o helping us in be e unde s anding he indus ial p oblem and o he
ui ul coope a ion.
DATA AVAILABILITY STATEMENT
All he da a used o he publica ion a e con ained in i .
ORCID
Umbe o Emil Mo elli h ps://o cid.o g/0000-0001-9929-5117
Gio anni S abile h ps://o cid.o g/0000-0003-3434-8446
REFERENCES
1. Wo ld S eel Associa ion. Wo ld S eel in Figu es 2018. Wo ld S eel Associa ion; 2018.
2. I ing WR. Con inuous Cas ing o S eel. The Ins i u e o Ma e ials; 1993.
3. Mo elliUE,Ba alP,Quin elaP,RozzaG,S abileG.Anume icalapp oach o hea luxes ima ionin hinslabscon inuouscas ingmolds
using da a assimila ion. In J Nume Me hods Eng. 2021;122(17):4541-4574. doi:10.1002/nme.6713
4. Ni ka R. Inhomogeneous pa abolic Neumann p oblems. Czechoslo Ma h J. 2014;64(3):703-742. doi:10.48550/a Xi .1108.6227
5. Eyma d R, Gallouë T, He bin R. Fini e olume me hods. Solu ion o Equa ion in Rn (Pa 3), Techniques o Scien i ic Compu ing (Pa 3).
Handbook o Nume ical Analysis. Vol 7. Else ie ; 2000:713-1018.
10970207, 2023, 6, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.1002/nme.7167 by Uni e sidade de San iago de Compos ela, Wiley Online Lib a y on [15/06/2023]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
MORELLI e al. 1379
6. Ling X, Keanini RG, Che uku i HP. A non-i e a i e ini e elemen me hod o in e se hea conduc ion p oblems. In J Nume Me hods
Eng. 2003;56(9):1315-1334. doi:10.1002/nme.614
7. Loulou T, Sco EP. An in e se hea conduc ion p oblem wi h hea lux measu emen s. In J Nume Me hods Eng. 2006;67(11):1587-1616.
doi:10.1002/nme.1674
8. Jin B. Conjuga e g adien me hod o he Robin in e se p oblem associa ed wi h he Laplace equa ion. In J Nume Me hods Eng.
2007;71(4):433-453. doi:10.1002/nme.1949
9. Huang CH, Yan JY. An in e se p oblem In p edic ing empe a u e dependen hea capaci y pe uni olume wi hou in e nal measu e-
men s. In J Nume Me hods Eng. 1996;39(4):605-618. doi:10.1002/(SICI)1097-0207(19960229)39:4<605::AID-NME872>3.0.CO;2-H
10. Ali ano OM. In e se Hea T ans e P oblems. 1s ed. Moscow Izda el Mashinos oenie; 1988.
11. O lande HRB. In e se p oblems in hea ans e : new ends on solu ion me hodologies and applica ions. J Hea T ans . 2012;134(3):1,
031011-13. doi:10.1115/1.4005131
12. Beck JV, Blackwell B, Clai J CRS. In e se Hea Conduc ion: Ill-Posed P oblems. James Beck; 1985.
13. ChangCW,LiuCH,WangCC.Re iewo compu a ionalschemesinin e sehea conduc ionp oblems.Sma Sci.2018;6(1):94-103.doi:10.
1080/23080477.2017.1408987
14. ¸Sahin HM, Koca epe K, Kay𝚤kc𝚤R, Aka N. De e mina ion o unidi ec ional hea ans e coe icien du ing uns eady-s a e solidi ica ion
a me al cas ing–chill in e ace. Ene gy Con e s Manag. 2006;47(1):19-34. doi:10.1016/j.enconman.2005.03.021
15. Ranu P. Op imiza ion and In e se P oblems in Hea T ans e . PhD hesis. Uni e si á degli S udi di Udine, Via delle Scienze, 206, 33100
Udine UD, I aly; 2012.
16. Uday aj CS, Ganguly S, Chacko E, Ajmani S, Talukda P. Es ima ion o su ace hea lux in con inuous cas ing mould wi h limi ed
measu emen o empe a u e. In J The m Sci. 2017;118:435-447. doi:10.1016/j.ij he malsci.2017.05.012
17. Mahapa a R, B imacombe J, Sama aseke a I. Mold beha io and i s in luence on quali y in he con inuous cas ing o s eel slabs: Pa II.
Mold hea ans e , mold lux beha io , o ma ion o oscilla ion ma ks, longi udinal o -co ne dep essions, and subsu ace c acks. Me all
Ma e T ans B. 1991;22(6):875-888. doi:10.1007/BF02651164
18. Raymond JP. Op imal Con ol o Pa ial Di e en ial Equa ions. Uni e si é Paul Saba ie , In e ne ; 2013.
19. Vi ale G, P eziosi L, Amb osi D. Fo ce ac ion mic oscopy: an in e se p oblem wi h poin wise obse a ions. JMa hAnalAppl.
2012;395(2):788-801. doi:10.1016/j.jmaa.2012.05.074
20. Huang CH, Chen CW. A bounda y elemen -based in e se-p oblem in es ima ing ansien bounda y condi ions wi h conjuga e g adien
me hod. In J Nume Me hods Eng. 1998;42(5):943-965. doi:10.1002/(SICI)1097-0207(19980715)42:5<943::AID-NME395>3.0.CO;2-V
21. Tikhono AN. Solu ion o inco ec ly o mula ed p oblems and he egula iza ion me hod. So ie Ma h Dokl. 1963;4:1035-1038.
22. Beck JV. Su ace hea lux de e mina ion using an in eg al me hod. Nucl Eng Des. 1968;7(2):170-178.
23. Beck JV. Nonlinea es ima ion applied o he nonlinea in e se hea conduc ion p oblem. In J Hea Mass T ans . 1970;13(4):703-716.
24. Chen CJ, Chiou JS. P edic ion o su ace empe a u e and hea lux om an in e io empe a u e esponse. Le Hea Mass T ans.
1976;3(6):539-548.
25. Pinhei o C, Sama aseke a I, B imacomb J, Walke B. Mould hea ans e and con inuously cas bille quali y wi h mould lux lub ica ion
pa 1 mould hea ans e . I onmak S eelmak. 2000;27(1):37-54. doi:10.1179/030192300677363
26. Sama aseke a I, B imacombe J. The in luence o mold beha io on he p oduc ion o con inuously cas s eel bille s. Me all T ans B.
1982;13(1):105-116.
27. Michelic S, Rau e W, E ke M, B andl W, Be nha d C. Hea T ans e in a Round CC Mould: Measu emen , Modelling and Valida ion.
S eelmaking & Plas ic De o ma ion S udy G oups. Associazione I aliana di Me allu gia; 2008:30.
28. Ranu P,Pe siC,NobileE,SpagnulS.Es ima iono Hea FluxDis ibu ioninaCon inuousCas ingMouldbyIn e seHea T ans e Algo-
i hms.P oceedingso heASME2011In e na ionalDesignEnginee ingTechnicalCon e ences&Compu e sandIn o ma ioninEnginee ing
Con e ence. Volume 2: 31s Compu e s and In o ma ion in Enginee ing Con e ence, Pa s A and B. The Ame ican Socie y o Mechanical
Enginee s. 2011: 389–398.
29. Nelde JA, Mead R. A simplex me hod o unc ion minimiza ion. Compu J. 1965;7(4):308-313.
30. Man Y, Hebi Y, Dacheng F. Real- ime analysis on non-uni o m hea ans e and solidi ica ion in mould o con inuous cas ing ound
bille s. ISIJ In . 2004;44:1696-1704. doi:10.2355/isijin e na ional.44.1696
31. Hebi Y, Man Y, Dacheng F. 3-D in e se p oblem con inuous model o he mal beha io o mould p ocess based on he empe a u e
measu emen s in plan ial. IISIJ In . 2006;46:539-545. doi:10.2355/isijin e na ional.46.539
32. GonzalezM,Goldschmi M,AssanelliA,D o kinE,Be dague E.Modelingo hesolidi ica ionp ocessinacon inuouscas ingins alla ion
o s eel slabs. Me all Ma e T ans B. 2003;34:455-473. doi:10.1007/s11663-003-0072-3
33. Wang X, Kong L, Du F, e al. Ma hema ical modeling o he mal esis ances o Mold lux and ai gap in con inuous cas ing Mold based on
an in e se p oblem. ISIJ In . 2016;56:803-811. doi:10.2355/isijin e na ional.ISIJINT-2015-601
34. ZhangH, Wang W.Moldsimula o s udyo hea ans e phenomenon du ing heini ial solidi ica ionin con inuouscas ingMold. Me all
Ma e T ans B. 2017;48:779-793. doi:10.1007/s11663-016-0901-9
35. HuP,WangX,WeiJ,YaoM,GuoQ.In es iga iono liquid/solidslagand ai gapbeha io inside he Molddu ingcon inuous slabcas ing.
ISIJ In . 2018;58(5):892-898. doi:10.2355/isijin e na ional.ISIJINT-2017-393
36. Tang L, Yao M, Wang X, Zhang X. Non-uni o m he mal beha io and shell g ow h wi hin mould o wide and hick slab con inuous
cas ing. S eel Res In . 2012;83(12):1203-1213. doi:10.1002/s in.201200075
37. VidecoqE,Quemene O,Laza dM,Ne euA.Hea sou ceiden i ica ionandon-line empe a u econ olbyab ancheigenmodes educed
model. In J Hea Mass T ans . 2008;51(19):4743-4752. doi:10.1016/j.ijhea mass ans e .2008.02.029
10970207, 2023, 6, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.1002/nme.7167 by Uni e sidade de San iago de Compos ela, Wiley Online Lib a y on [15/06/2023]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
1380 MORELLI e al.
38. Videcoq E, Ne eu A, Quemene O, Gi aul M, Pe i D. Compa ison o wo nonlinea model educ ion echniques: he modal iden-
i ica ion me hod and he b anch eigenmodes educ ion me hod. Nume ical Hea T ans B Fund. 2006;49(6):537-558. doi:10.1080/
10407790500344035
39. Gi aul M, Videcoq E, Pe i D. Es ima ion o ime- a ying hea sou ces h ough in e sion o a low o de model buil wi h he modal iden-
i ica ion me hod om in-si u empe a u e measu emen s. In J Hea Mass T ans . 2010;53(1):206-219. doi:10.1016/j.ijhea mass ans e .
2009.09.040
40. Gi aul M, Pe i D. Iden i ica ion me hods in nonlinea hea conduc ion. Pa II: in e se p oblem using a educed model. In J Hea Mass
T ans . 2005;48(1):119-133. doi:10.1016/j.ijhea mass ans e .2004.06.033
41. Aguado JV, Hue a A, Chines a F, Cue o E. Real- ime moni o ing o he mal p ocesses by educed-o de modeling. In J Nume Me hods
Eng. 2015;102(5):991-1017. doi:10.1002/nme.4784
42. WangX,YaoM.Neu alne wo ks o sol ing hein e sehea ans e p oblemo con inuouscas ingmould.P oceedingso he2011Se en h
In e na ional Con e ence on Na u al Compu a ion 2 IEEE Ci cui s and Sys ems Socie y. 2011;791-794.
43. ChenH, Su L, Wang G,ShibinW, Zhang L,LuoZ. Fuzzy es ima ion o hea lux dis ibu ion a he slab con inuous cas ingmoldsu ace.
In J The m Sci. 2014;83:80-88. doi:10.1016/j.ij he malsci.2014.04.012
44. Buhmann MD. Radial Basis Func ions: Theo y and Implemen a ions. Camb idge uni e si y p ess; 2003:12.
45. P ando G. Non-Pa ame ic Bayesian Me hods o Linea Sys em Iden i ica ion. PhD hesis. Uni e si á di Pado a, Via 8 Febb aio 1848, 2,
35122 Pado a PD, I aly; 2016.
46. Engl H. Regula iza ion o In e se P oblems. Kluwe Academic Publishe s; 1996.
47. Hansen P. Disc e e In e se P oblems: Insigh and Algo i hms. Socie y o Indus ial and Applied Ma hema ics; 2010.
48. As e R. Pa ame e Es ima ion and In e se P oblems. Else ie ; 2019.
49. Ki sch A. Regula iza ion by Disc e iza ion. Sp inge New Yo k; 2011:63-119.
50. S abile G, Hijazi S, Mola A, Lo enzi S, Rozza G. POD-Gale kin educed o de me hods o CFD using ini e olume disc e isa ion: o ex
shedding a ound a ci cula cylinde . Commun Appl Ind Ma h. 2017;8(1):210-236. doi:10.1515/caim-2017-0011
51. ITHACA-FV. h ps://ma hlab.sissa.i /i haca- . Accessed Oc obe 26, 2020.
52. Moukalled F, Mangani L, Da wish M. The Fini e Volume Me hod in Compu a ional Fluid Dynamics: an Ad anced In oduc ion wi h
OpenFOAM and Ma lab. 1s ed. Sp inge Publishing Company, Inco po a ed; 2015.
53. Olsson DM, Nelson LS. The Nelde -Mead simplex p ocedu e o unc ion minimiza ion. Den Tech. 1975;17(1):45-51. doi:10.1080/
00401706.1975.10489269
54. S azzulloM,Balla inF,RozzaG.POD-Gale kinmodelo de educ ion o pa ame izednonlinea ime-dependen op imal lowcon ol:
an applica ion o shallow wa e equa ions. JNume Ma h. 2022;30(1):63-84. doi:10.1515/jnma-2020-0098
55. Neg iF,RozzaG,ManzoniA,Qua e oniA.Reducedbasisme hod o pa ame izedellip icop imalcon olp oblems.SIAMJSciCompu .
2013;35(5):A2316-A2340. doi:10.1137/120894737
56. S abile G, Rosic B. Bayesian iden i ica ion o a p ojec ion-based educed o de model o compu a ional luid dynamics. Compu Fluids.
2020;201:104477. doi:10.1016/j.comp luid.2020.104477
How o ci e his a icle: Mo elli UE, Ba al P, Quin ela P, Rozza G, S abile G. No el me hodologies o sol ing
he in e se uns eady hea ans e p oblem o es ima ing he bounda y hea lux in con inuous cas ing molds. In J
Nume Me hods Eng. 2023;124(6):1344-1380. doi: 10.1002/nme.7167
10970207, 2023, 6, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.1002/nme.7167 by Uni e sidade de San iago de Compos ela, Wiley Online Lib a y on [15/06/2023]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
