Applica ions in Enginee ing Science 15 (2023) 100141
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Applica ions in Enginee ing Science
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P edic ion o mic oscopic esidual s esses using gene ic p og amming
Lau a Millán a, Gab iel K onbe ge b,∗, Rica do Fe nández a, Gizo Bokucha a c,
Pa ice Halodo a d, Albe o Sáez-Made uelo e, Gaspa González-Doncel a, J. Ignacio Hidalgo
aCen o Nacional de In es igaciones Me alú gicas, CSIC, Mad id, Spain
bJose Ressel Cen e o Symbolic Reg ession, Heu is ic and E olu iona y Algo i hms Labo a o y, Uni e si y o Applied Sciences Uppe
Aus ia, Hagenbe g, 4232, So wa epa k 11, Aus ia
cF ank Labo a o y o Neu on Physics, Join Ins i u e o Nuclea Resea ch, Dubna, Russia
dCen um Výzkumu Řež, Řež, Czech Republic
eCen o de In es igaciones Ene gé icas, Medioambien ales y Tecnológicas, CIEMAT, Mad id, Spain
Uni e sidad Complu ense de Mad id, Mad id, Spain
ARTICLE INFO
Keywo ds:
Ma e ial science
Machine lea ning
Symbolic eg ession
Residual s ess
Neu on di ac ion
Mic os uc u e
ABSTRACT
Me allu gical manu ac u ing p ocesses commonly used in he indus y ( olling, ex usion, shaping, machining,
e c.) usually cause esidual s ess de elopmen which can emain a e he mal hea ea men s. These s esses
can be de imen al o he in-se ice pe o mance o s uc u al componen s, which makes hei s udy and
unde s anding impo an . Residual s ess a ia ions a e usually de e mined a a mac oscopic scale (commonly,
using di ac ion me hods). Howe e , s ess a ia ions a he mic oscopic scale o he indi idual c ys alli es
(g ains), a e also ele an . Con a y o he mac oscopic esidual s esses, mic oscopic esidual s esses a e
di icul o quan i y using con en ional p ocedu es. We p opose o use machine lea ning o ind equa ions ha
desc ibe mic oscopic esidual s esses. Conc e ely, we show ha we a e able o lea n equa ions o ep oduce he
di ac ion p o iles om mic os uc u al cha ac e is ics using gene ic p og amming. We e alua e he lea ned
equa ions using eal neu on di ac ion peaks as a e e ence, ob aining accu a e esul s o he mos equen
g ain o ien a ions wi h un imes o a ew minu es.
1. In oduc ion
Residual s esses (RS) a e commonly de eloped du ing manu ac-
u ing p ocesses o me allic componen s. Me al o ming ope a ions
such as machining and shaping, applied o me allic pa s, induce non-
homogeneous plas ic de o ma ions wi h he esul o RS ields gene a-
ion (Hauk,1997). These s ess ields may emain a e hea ea men s
and can a ec he pe o mance o componen s. Mos impo an ly, RS
can educe he du abili y and lead o sudden unexpec ed ailu es. Many
acciden s occu ing in ac o ies, buildings, ehicles, o ci il s uc u es
can be di ec ly associa ed wi h hese RS. An impo an s a egy o
p e en hese acciden s is o ensu e a de ailed unde s anding o hese
RS ields o hei con ol in manu ac u ing o componen s. Howe e ,
one o he p incipal p oblems s ems om he lack o knowledge o RS
and om he di icul ies when i comes o de e mining he ex en o
he damage hey can p oduce.
RS a e di ided in o wo ypes depending on he scale on which
hey mani es hemsel es: mac oscopic and mic oscopic RS. The mac o-
scopic s esses (M-RS) ex end o e he sample o componen size,
and he mic oscopic ones (m-RS) a y a he scale o he mic os uc-
u e (Cha e jee e al.,2016;Ganguly e al.,2011;Gloaguen e al.,
∗Co esponding au ho .
E-mail add ess: [email p o ec ed] (G. K onbe ge ).
2015). Wi hin hese m-RS, in e g anula ( ype II) and in ag anula
( ype III) esidual s ess can be dis inguished ( he e m ype I is usually
ese ed o he M-RS). Da a o calcula e he M-RS can be ob ained
wi h neu on o synch o on adia ion di ac ion me hods. Howe e ,
he measu emen o quan i ica ion o he m-RS is s ill an open p oblem.
To unde s and he di icul y o add essing hese kind o s udies, i is
necessa y o ocus on he complex na u e o ma e ial mic os uc u e as
well as he limi ed capabili y o di ac ion me hods.
The mic os uc u e o c ys alline ma e ials is subdi ided in o small
c ys alli es (g ains) wi h a size o ypically a ew mic ons. The o ien a-
ions o g ains wi hin he polyc ys al di e among neighbo ing g ains.
Due o he aniso opic cha ac e o he la ice s uc u e, a s ess ield
associa ed wi h he indi idual g ains, he m-RS o ype II, is de eloped
du ing me al manu ac u ing p ocesses. Fu he mo e, c ys al de ec s,
such as disloca ions, which a e esponsible o plas ic de o ma ion,
dis o he o de ed c ys alline s uc u e and lead o m-RS.
Se e al s udies ha e employed di e en echniques such as Elec-
on Back-Sca e Di ac ion (EBSD) / Kikuchi lines, mic o-Raman,
X- ay di ac ion, mic omagne ic and ul asonic me hods o calcula e
h ps://doi.o g/10.1016/j.apples.2023.100141
Recei ed 14 Ap il 2023; Accep ed 9 Augus 2023
Applica ions in Enginee ing Science 15 (2023) 100141
2
L. Millán e al.
his mic oscopic s ess. Di ac ion me hods a e non-des uc i e and
sui able o in es iga e m-RS ields. Di ac ion peaks a e p oduced by
many g ains, all o hem wi h a common c ys allog aphic di ec ion.
This is because he egion illumina ed by he beam is signi ican ly
la ge han he g ain size. As a consequence, he esul ing di ac ion
peak in he di ac og am is he sum o he di ac ed signal p oduced
by he con ibu ing g ains. I is clea ha di ac ion peaks ob ained
om hese expe imen s con ain he s ess in o ma ion o he indi idual
g ains (Fe nández e al.,2018). In addi ion, some da a om indi idual
g ains, such as hei size, aspec a io, and slope o he long g ain axis
wi h espec o a gi en sample di ec ion can be eadily ob ained om
isual inspec ion o EBSD maps besides he c ys allog aphic o ien a ion.
In his wo k we p opose o use gene ic p og amming (GP) o ind
equa ions ha link indi idual g ain cha ac e is ics ob ained om EBSD
maps, o peak shi s in he di ac og ams, and hus allow o p edic he
m-RS con ibu ion o each g ain. Fo ou s udy, we use di ac og ams
and EBSD maps om mul iple samples om an aluminum alloy cylinde
ha was apidly cooled (quenching) o de elop ype I and ype II RS
ields.
Machine lea ning is undamen al o he de elopmen o a i icial
in elligence. One impo an a ea o machine lea ning is supe ised
lea ning om obse a ional da a. In his app oach we lea n a model
ha maps inpu alues o a ge alues. The model can la e be used
o p edic alues o he a ge a iable. In enginee ing applica ions
o a i icial in elligence, explainabili y and in e p e abili y o models
is an impo an conce n because we o en equi e o ully unde s and
he in e nal wo kings o he model, o ga he new insigh s and knowl-
edge, o o inc ease eliabili y and sa e y. Equa ion lea ning, equen ly
called symbolic eg ession, p o ides he po en ial o lea n in e p e able
models in he o m o sho equa ions (Koza,1992;A enzelle e al.,
2009;Sahoo e al.,2018). Wi h hese equa ions we can calcula e
p edic ed di ac og ams om mic os uc u al cha ac e is ics o he
g ains p o ided by expe imen al EBSD maps and analyze he e ec s
o mic os uc u al cha ac e is ics on peak shi s.
The es o he pape is o ganized as ollows: Sec ion 2, desc ibes
p e ious wo k applying machine lea ning o sol e p oblems in he ield
o ma e ials science. In Sec ion 3, he da a collec ion echniques and he
p oposed me hod a e explained. In Sec ion 4we alida e he equa ions
iden i ied by GP using in e sec ion plo s o isualize he dependencies
be ween a iables ha a e cap u ed by he models. This wo k inishes
wi h he mos ele an conclusions in Sec ion 5.
2. Rela ed wo k
Gene ic p og amming is an e olu iona y machine lea ning me hod
de eloped o gene a e exp essions and po en ially e en ull compu e
p og ams ha sol e a gi en p og amming p oblem (Koza,1992). I
is he oldes and mos es ablished me hod o symbolic eg ession
whe e he goal is o lea n equa ions om obse a ional da a. Recen ly,
he e has been an inc eased in e es in equa ion lea ning me hods
and symbolic eg ession o scien i ic machine lea ning (Sahoo e al.,
2018;Ud escu and Tegma k,2020;Guime à e al.,2020;Mundhenk
e al.,2021). Since i s beginning, symbolic eg ession has been used
success ully in many applica ions o science and enginee ing.
O ele ance o he p esen s udy is i s applicabili y in indus ial
p ocesses, pa icula ly in p ocesses ha in ol e he shaping o me allic
ma e ials. B ezonick’s wo k measu ing cold de o ma ion (B ezocnik
and Gusel,2004), su ace oughness (B ezocnik and Ko acic,2003) and
impac s eng h (Gusel and B ezocnik,2006) o coppe and aluminum
alloys, o example, shows he po en ial use o GP in his sec o .
Ano he impo an example is i s use in he momechanical p ocesses
such as ic ion s i welding (Yunus and Alsou i,2018;Hidalgo e al.,
2016). Symbolic eg ession using GP has been used o ex end a physics-
based model in o a semi-analy ical hyb id model o a me al shee
bending p ocess (Asadzadeh e al.,2021). We ha e p e iously used
symbolic eg ession and GP in he a ea o ma e ial science o lea ning
ic ion models om la ge-scale indus ial ic ion es s (K onbe ge
e al.,2018), o lea n semi-analy ical models o s ess–s ain cu es
o aluminum alloys (Kabliman e al.,2021;K onbe ge e al.,2022),
and o p edic he phase kine ics o s eel (Pi inge e al.,2022).
We we e also able o use GP o calcula e he elaxed la ice pa am-
e e ac oss he weld bead o an age-ha dening aluminum alloy using
neu on di ac ion da a (Cio i e al.,2014). In his way, i was possible
o de e mine he M-RS p o ile ac oss he weld. This s udy, in which he
p oblem o he m-RS could no be add essed due o i s complexi y, was
a p ecu so o he de elopmen s p esen ed in his wo k.
3. Me hodology
Fo he p esen in es iga ion we ha e selec ed a single-phase alu-
minum alloy AA5083, ha has a ela i ely simple mic os uc u e, and
in which he la ice pa ame e does no a y wi h annealing ea men s.
Fu he mo e, a cylind ical sample, which was annealed a ele a ed
empe a u e ollowed by a quenching, is selec ed because he cylin-
d ical symme y simpli ies he s udy. E en in his case, add essing he
p oblem o calcula ing he s ess s a e o indi idual g ains is s ill a e y
complex p oblem. I equi es no only he expe imen al da a supplied
by di e en , ep esen a i e, di ac ion peaks bu also he in o ma ion
o many indi idual g ains. We use he EBSD echnique o de e mine
mic os uc u al da a o indi idual g ains.
3.1. Elec on back-sca e ing di ac ion
The EBSD echnique, used in a scanning elec on mic oscope (SEM),
allows measu ing he indi idual o ien a ions o g ains and o gene a e
o ien a ion imaging mic oscopy (OIM) maps. I is a su ace echnique,
as consequence, a special p epa a ion o he samples is necessa y be o e
measu emen . A 4 mm hick disc was cu om he cen al a ea o
he cylinde . Cubes o 4 ×4×4mm3we e cu ou a posi ions 0, 6
and 12 mm om he cen e . The aces o hese cubes pe pendicula
o he ex usion axis di ec ion we e sanded and polished o ensu e a
good su ace inish. Fo each o he cubes, EBSD da a we e collec ed
on a Ly a GMU SEM equipped wi h a No dlys II de ec o (Ox o d
Ins umen s) wi h AZTec acquisi ion so wa e. The di ac ion maps
we e pos -p ocessed using Channel5 so wa e.
Fig. 1 shows he maps ob ained using his echnique. These maps
a e ex ac ed om he axial su ace o each cube. The colo pa e n
ep esen s he di e en o ien a ions o he g ains, blue o 111, ed
o 200, g een o 220, and pink o 311. Despi e s emming om he
same sample, i is possible o obse e a clea a ia ion among he h ee
cubes. The mic os uc u al g adien obse ed in Fig. 1 a ises om he
ic ional o ces be ween he ex usion die and he me allic alloy. This
ic ion p o okes a di e en plas ic low beha io in he inne egion
(away om he die) o he ex uded ba wi h espec he ou e one
(close o he die) du ing he ex usion p ocess. The e o e, his gene a es
a di e en mic os uc u e and, consequen ly, an indi idual p edic ion
model o each posi ion and o ien a ion should be sough ha depends
on he p ope ies and neighbo hood o he g ains. F om he EBSD maps,
we collec ed da a o each o he isible g ains including he o ien a ion
(ℎ𝑘𝑙), he a ea, he diame e , he numbe o neighbo s, he posi ion
(x,y), he aspec a io and he slope (which is he angle be ween he
long axis g ain di ec ion and he adial axis o he cylinde s which
in e cep s he cen e o mass o he g ain). These da a a e collec ed
in a able wi h one ow o each g ain as shown in Table 1. Fo each
one o hese h ee posi ions on he adial di ec ion o he cylinde we
p epa ed a sepa a e able.
Each g ain in he lis was included in one o he ou main ℎ𝑘𝑙-
amilies based on he dis ance be ween i s di ec ion ec o and he
ec o o he nea es p incipal o ien a ion. The g ain coun s ob ained
pe o ien a ion a e ep esen ed in Table 2. The las column includes he
alues o he elas ic modulus 𝐸ℎ𝑘𝑙 o each o ien a ion. This p ope y
Applica ions in Enginee ing Science 15 (2023) 100141
3
L. Millán e al.
Fig. 1. EBSD maps ob ained a h ee di e en posi ions, le o igh ; a =0 mm (cen e ), a =6 mm, and a =12 mm, ac oss he adial di ec ion o he cylinde . Each colo
ep esen s a di e en c ys al o ien a ion wi h he axial di ec ion, acco ding wi h he e e ence iangle on he igh . The black band in he lowe le co ne ep esen s he scale
o he images, i s alue being 200 μm. The ho izon al di ec ion ( o maps a 6 and 12 mm) is coinciden wi h he adial di ec ion.
Table 1
An ex ac o he da a ob ained using he EBSD echnique. This able includes in o ma ion o g ains loca ed in he cen al posi ion (𝑟= 0).
belonging o all ou ex u e componen s.
G ain ID A ea μm2Diame e μm Neighbo s x y Aspec a io Slope ℎ𝑘𝑙
0 717 30.214 9 15.67 12.9 1.621 152.5 111
1 201 15.998 3 38.13 3.86 2.0907 0.49718 237
2 295 19.381 4 57.21 6.12 2.7134 19.073 111
3 117 12.205 3 83.32 3.34 1.8991 165.59 013
4 242 17.553 6 94.92 9.48 1.4032 51.102 015
5 31 6.2825 2 101.71 2.52 1.3311 108.08 001
6 124 12.565 4 114.77 2.4 3.9348 7.0842 32 17
7 771 31.332 6 151.98 9.39 1.9345 0.56682 112
8 47 7.7358 2 173.55 2.43 1.4229 150.11 111
9 368 21.646 4 186.9 7.79 1.4882 30.399 111
10 633 28.389 6 202.38 16.81 1.3586 25.153 102
11 41 7.2252 2 212.02 1.71 2.1311 163.3 111
12 1066 36.841 11 233.9 19.91 1.7427 130.14 158
13 37 6.8637 1 234.11 5.84 2.6223 63.921 011
14 155 14.048 3 252.44 2.75 3.3816 177.12 111
15 389 22.255 3 277.08 7.51 1.4817 165.74 223
16 1920 49.443 10 306.66 19.66 1.7247 12.195 213
17 87 10.525 4 334.02 3.67 1.5737 143.37 111
18 21 5.1709 2 341.38 1.24 1.7178 19.178 328
19 1407 42.326 10 360.94 14.87 1.8775 26.352 111
20 944 34.669 7 409.84 20.62 2.0654 90.54 001
... ... ... ... . . . . . . . . . ... ...
Table 2
Numbe o g ains, a each loca ion ac oss he adial di ec ion 1belonging o each one
o he ou main ℎ𝑘𝑙 o ien a ions. The alue o he elas ic modulus 𝐸ℎ𝑘𝑙 o each ℎ𝑘𝑙
is also indica ed.
hkl 0 mm 6 mm 12 mm 𝐸ℎ𝑘𝑙 GPa
111 754 661 626 75.5
200 871 279 159 63.2
220 103 399 96 72.0
311 976 1765 633 69.0
To al 2704 3104 1514 69.9 (a e age)
in o ms us abou he s i ness o he g ains (111 g ains a e he s i es
and he 200 g ains he mos complian ).
In his way, ou da ase s we e c ea ed om each lis , acco ding
o he ou p e e ed ℎ𝑘𝑙-o ien a ions (as shown in Table 2). Fo each
o ien a ion a di e en mean la ice spacing 𝑑ℎ𝑘𝑙
𝑥𝑖is ele an in he
di ac og ams (see Fig. 2). The la ice spacing indica es he dis ance be-
ween c ys al la ice planes wi h he same o ien a ion. In o he wo ds,
he o ien a ion mainly de e mines he posi ion and con ibu ion o he
di ac ion peak o each g ain, howe e o he g ain cha ac e is ics such
as he size o aspec a io may also ha e an e ec . Thus, each g ain has
a di e en alue o 𝑑ℎ𝑘𝑙
𝑥𝑖 ha will con ibu e o a g ea e o lesse ex en
o i s co esponding peak. The sum o he con ibu ions o all g ains
𝑑ℎ𝑘𝑙
𝑥𝑖will gi e ise o he p o ile o he di ac ion peak co esponding o
he o ien a ion o hose g ains. These alues o 𝑑ℎ𝑘𝑙
𝑥𝑖ha e an associa ed
alue o m-RS. The e o e, i we can p edic he 𝑑ℎ𝑘𝑙
𝑥𝑖 om he g ain
cha ac e is ics we can also calcula e he m-RS associa ed wi h his g ain
using Hooke’s law and he s ess equilib ium exp essions desc ibed in
p e ious wo ks (Millán e al.,2020,2021a,b). We selec ed he a ea,
aspec a io, numbe o neighbo s, and slope as po en ial pa ame e s
ha a ec he 𝑑ℎ𝑘𝑙
𝑥𝑖and used hose a iables as inpu s o equa ion
lea ning.
3.2. Neu on di ac ion
The neu on di ac ion measu emen s p o ide he la ice spacing,
𝑑ℎ𝑘𝑙
𝑥𝑖, o di e en (ℎ𝑘𝑙) planes in g ains inside he sample. Peak shi s in
he di ac og ams a e used o analyze he s ains, and he M-RS using
he gene alized Hooke’s law o elas ici y. The neu on di ac ion da a
was ob ained a nea ly simila posi ions a which he EBSD maps we e
ob ained. The measu emen s we e conduc ed be o e sec ioning he
quenched cylinde o ex ac he samples o he EBSD measu emen s.
In his way, i is gua an eed ha he RS ields gene a ed wi h he hea
ea men s we e no al e ed.
The expe imen was ca ied ou on he neu on Fou ie s ess
di ac ome e (FSD) ope a ing a he IBR-2 pulsed eac o in he
F ank Labo a o y o Neu on Physics (FLNP) in Dubna (Russia) (see
Bokucha a e al.,2019 o u he de ails). Fo he measu emen s, he
sample was posi ioned in wo con igu a ions, e ical and ho izon al,
o ob ain di ac ion da a o he axial, adial, and hoop di ec ions.
The gauge olume used o each con igu a ion was 2 ×2×25 mm3
and 2 ×2×2mm3, espec i ely. The measu emen s we e aken a
ou di e en posi ions, 𝑟={0,3.5,7, and 10}mm along he adial
Applica ions in Enginee ing Science 15 (2023) 100141
4
L. Millán e al.
Fig. 2. Neu on di ac ion pa e n ob ained o he axial di ec ion a he cen al posi ion, =0, o he cylinde . A magni ica ion o he 311 and 111 peaks is shown.
di ec ion, a he mid-heigh o he cylinde . The measu emen s gene a e
di ac ion pa e ns which include he con ibu ion o he di e en se s
o (ℎ𝑘𝑙) planes. Fig. 2 shows he di ac ion pa e n (axial di ec ion)
ob ained a dis ance =0 mm in he sample. He e, he in ensi y is
shown on he 𝑦-axis and he la ice spacing 𝑑ℎ𝑘𝑙
𝑥𝑖on he 𝑥-axis. The ℎ𝑘𝑙
co esponding o he ou p incipal di ac ion peaks a e indica ed. A
magni ied iew o he 311 and 111 peaks is also shown. The echnique
is e y sensi i e and is capable o de ec ing elas ic de o ma ions o he
la ice wi h g ea p ecision.
Fo he pu pose o he p esen s udy, i will be aken in o accoun
ha he peak in ensi y dis ibu ion o a gi en ℎ𝑘𝑙 is p opo ional o he
o al numbe o g ains wi h he co esponding o ien a ion, as ob ained
om he EBSD measu emen s, using p opo ionali y ules.
The peak p o iles a e ex ac ed o he ou p incipal ℎ𝑘𝑙 and used
as he a ge o he equa ion lea ning p ocess.
To esol e he s ess o indi idual g ains ( he m-RS) om he
neu on di ac ion da a, i mus be aken in o accoun ha hese da a
include also he M-RS, in he way calcula ed in a p e ious wo k (Millán
e al.,2020). This M-RS mus be sub ac ed om he o al s ess o
ob ain he co ec ed m-RS using he equa ion 𝜎𝑚,ℎ𝑘𝑙
𝑥𝑖=𝜎𝑇 ,ℎ𝑘𝑙
𝑥𝑖−𝜎𝑀
𝑥𝑖,
whe e 𝜎𝑚,ℎ𝑘𝑙
𝑥𝑖is he m-RS o 𝑖posi ion ac oss he adial di ec ion and
ℎ𝑘𝑙 o ien a ion and 𝜎𝑀
𝑥𝑖is he M-RS o 𝑖posi ion.
3.3. Equa ion lea ning o p edic ing he g ain la ice spacing
Fig. 3 gi es a isual o e iew o he equa ion lea ning p ocess.
The aim is o ind a unc ion (equa ion) o each ℎ𝑘𝑙 o ien a ion ha
depends on he mic os uc u al pa ame e s men ioned in Sec ion 3.1.
This unc ion p edic s he la ice spacing alue o each g ain, and
mus be such ha i ep oduces he speci ic expe imen al di ac ion
peak p o ile. To calcula e he p edic ed peak p o ile, we sum up he
con ibu ions o each o he g ains, whe eby he p edic ed la ice
spacing is used o loca e he peak o each g ain. The mean o absolu e
e o s be ween he p edic ed and he a ge peak p o ile is used as
he objec i e unc ion o be minimized. I is assumed ha all g ains
con ibu e equally o he peak in ensi y, and he exp ession mus be
common o all g ains wi h he same o ien a ion.
Howe e , he exp ession iden i ied should no be equal o all peaks
since he mechanical esponse o g ains will depend on he speci ic
ℎ𝑘𝑙. The e o e, we ea he ou mos equen g ain amilies ℎ𝑘𝑙 (111,
220, 200, and 311) independen ly o each o he . This is conduc ed
by spli ing he EBSD da a-se o each g ain amily and selec ing he
co esponding peak om he di ac ion spec um, as shown in he
igu e.
Some app oxima ions a e assumed:
•Only he di ac ion pa e ns (EBSD and neu on di ac ion) co -
esponding o he axial di ec ion will be analyzed he e, as i
coincides wi h he axi-symme ic di ec ion and co esponds o
he uncons ained di ec ion o he ex usion p ocess. This is a
simpli ica ion o he p oblem since a comple e analysis o he m-
RS would equi e also he s udy o he spec a ob ained o he
adial and hoop componen s. This much mo e complex p oblem
is being he subjec o u he analysis.
•Each un will s udy only one indi idual peak and i s co espond-
ing g ains, o a gi en posi ion in he sample.
•The in luence o he ins umen al sca e and he s esses as-
socia ed wi h la ice de ec s (i.e., ype III m-RS) is conside ed
negligible. Peak b oadening o he measu ed peaks is, he e o e,
only a ibu ed o ype II m-RS de elopmen .
3.4. Gene ic p og amming
The pseudo-code o he GP algo i hm adap ed o his p oblem is
shown in Fig. 4. I akes as inpu s he able o g ain cha ac e is ics
𝑋 o one ℎ𝑘𝑙-di ec ion as o example shown in Table 1 as well as
he ex ac ed peak 𝑦 o he same di ec ion om he di ac og am.
The esul is a o mula o calcula e he la ice spacing o each g ain
𝑑ℎ𝑘𝑙
𝑥𝑖. The algo i hm is a a ia ion om A enzelle e al. (2009) and
uses o sp ing selec ion o inc ease di e si y o solu ion candida es.
The main adap a ion necessa y o his p oblem is he i ness unc ion
desc ibed in Fig. 5.
Fo ou expe imen s, we ha e used he algo i hm pa ame e alues
shown in Table 3. The e minal se o GP con ains nume ic pa ame-
e s as well as he inpu a iables: g ain a ea, numbe o neighbo s,
Applica ions in Enginee ing Science 15 (2023) 100141
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L. Millán e al.
Fig. 3. O e iew o he equa ion lea ning p ocess. We use GP (Fig. 4) o ind ou models ep esen ed as sho equa ions, namely; 𝑓311, 𝑓220, 𝑓200, 𝑓111, o p edic he la ice spacing
𝑑ℎ𝑘𝑙
𝑥𝑖 o each g ain in each ℎ𝑘𝑙 g oup. The p edic ed peak p o ile is calcula ed ia ke nel densi y es ima ion (Fig. 6) wi h a Gaussian ke nel. GP sea ches o he equa ions ha
bes i he a ge p o ile (Fig. 5), which a e ex ac ed o all ou ℎ𝑘𝑙 om he ull di ac og am.
Table 3
GP pa ame e se ings.
Pa ame e Value
Popula ion size 500
Func ion se {+,−,⋅,log,exp}
Te minal se {a ea A, num. neighbo s I, aspec a io AR,
slope S, andom nume ic pa ame e s }
Max. ee size 20 nodes
Max. ee heigh 12 le els
Fi ness unc ion (maximized) Nega i e sum o absolu e e o s
Ini ializa ion PTC2
Pa en selec ion Random & p opo ional
O sp ing selec ion S ic
C osso e a e 100%
Mu a ion a e 10%
Max. gene a ions 1000
Max. selec ion p essu e 10
aspec a io, and slope. The unc ion se con ains addi ion, sub ac ion,
mul iplica ion, as well as exponen ial and loga i hmic unc ions. The
size o exp ession ees is limi ed o 50 nodes and 12 le els o p e en
e olu ion o o e ly la ge and complex equa ions ha a e p one o
o e i ing.
The i ness unc ion is speci ic o his p oblem since he la ice
spacing o indi idual g ains canno be di ec ly measu ed. The i ness
unc ion in Fig. 5 compa es he p edic ed peak o he ac ual peak
ex ac ed om he di ac og am. Whe eby he p edic ed peak is cal-
cula ed om he p edic ed la ice spacing 𝑑ℎ𝑘𝑙
𝑥𝑖(𝑛) ia a smoo hing
ope a ion as shown in he pseudo-code o he unc ion CalcSpec um
in Fig. 6.
Fig. 6 shows he pseudo-code o calcula ing he p edic ed spec um
ia ke nel densi y smoo hing. Fi s , he exp ession is used o calcula e
he p edic ed la ice spacing alues o each g ain. Then we use a
Gaussian ke nel o calcula e he p edic ed peak p o ile. To ma ch he
a ge spec um, we i s b ing he p edic ed la ice spacing alues 𝑑p ed
o he same scale and o se as he la ice spacing alues in he a ge
spec um 𝑦. A he end, we scale he p edic ed peak o ma ch he a ge
peak.
4. Resul s
A o al o hi y uns o each con igu a ion o he algo i hm we e
made o each loca ion in he sample and c ys allog aphic o ien a ion
using an In el Co e i5 p ocesso wi h 16 GB RAM on Windows 10.
We a ied he mu a ion a e om 5% o 25% in ini ial expe imen s,
bu did no obse e a la ge di e ence in he esul s. So we used 10%
mu a ion a e, which is a common alue (A enzelle e al.,2009).

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L. Millán e al.
Fig. 4. Pseudo code o he GP algo i hm.
Fig. 5. Pseudo code o he e alua ion unc ion.
Applica ions in Enginee ing Science 15 (2023) 100141
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L. Millán e al.
Fig. 6. Pseudo code o calcula ing a p edic ed peak p o ile using a Gaussian ke nel.
The g ain da a iles a e spli in o wo hal es o model aining and
es ing. Rows om he g ain da a able a e assigned andomly o he
aining o es ing se s. Fo GP we used only he aining se s o i ing
and selec ing models. The es ing se s a e only used a e aining is
inished o he esul s epo ed in his sec ion.
4.1. P edic ed peak p o iles and exp essions
Figs 7,8and 9show he p edic ed dis ibu ions om he 30 uns
and he da a om he neu on di ac ion peaks. The 𝑥-axis ep esen s
he 𝑑ℎ𝑘𝑙
𝑥𝑖 ange o he peaks selec ed o each ℎ𝑘𝑙-o ien a ion. The 𝑦-
axis is de ined as he numbe o g ains wi h he same alue o 𝑑ℎ𝑘𝑙
𝑥𝑖,
i.e. belonging o he same sec ion. Each line co esponds o one model
p oduced by a GP un.
Fo he 111 and 311 g ains, he model is able o ep oduce he
di ac ion peak p o iles qui e well. I is impo an o conside he scale
o he 𝑦-axis alues. Since, o example, he p edic ed peaks o g ains
220 o he cen al posi ion, 𝑟= 0 mm, and a 6 mm ha e la ge
de ia ions om he a ge peaks. This mus be due o he ac ha he e
a e ewe g ains wi h his o ien a ion as shown in Table 2. Fu he mo e,
hese g ains ha e i egula i ies in he shape o hei peaks, i.e., hey
show dis o ions o he Gaussian p o ile. This is because hey a e
mo e complian han 111 g ains. The e o e hey a e subjec ed o mo e
de o ma ion.
Quali a i ely, he peaks o 111 and 311 a e p edic ed bes and mos
consis en ly o e all posi ions.
The a e ages o mean o absolu e e o s (MAE) o e all models
a e shown in Table 4. The uni o he e o is he g ain coun . When
in e p e ing he MAE numbe s we mus ake in o accoun he o al
numbe o g ains gi en in Table 2.
To demons a e he simplici y o he equa ions iden i ied by GP,
he equa ions wi h bes MAE on he es se a e shown below. Mos
o he exp essions a e polynomials o small deg ee e en hough we
Table 4
A e age o e 30 models o mean o absolu e e o s on he es se s.
0 mm 6 mm 12 mm
111 1.293 1.802 1.111
200 2.229 1.451 5.857
220 1.000 0.599 0.612
311 1.964 1.693 1.914
included he di ision ope a o s and nonlinea unc ions exp and log in
he unc ion se . GP au oma ically iden i ied, ha polynomials o low
deg ee a e su icien o p edic he di ac ion peaks.
𝑑111,0
𝑥𝑖= 4.052 ⋅10−5 𝐴𝑅2
𝑖𝑆𝑖+ 3.042 ⋅10−4 𝐼𝑖+ 3.592 ⋅10−6 𝐴𝑅𝑖𝐴𝑖
− 7.828 ⋅10−6 𝐴𝑖− 0.0147 𝐴𝑅𝑖+ 2.372 (1)
𝑑111,6
𝑥𝑖= 2.542 ⋅10−5 𝑆𝑖− 0.00125 𝐴𝑅𝑖𝐼𝑖+ 2.549 ⋅10−4 𝐼𝑖
+ 0.00168 𝐴𝑅2
𝑖+ 0.00752 𝐴𝑅𝑖+ 2.3453 (2)
𝑑111,12
𝑥𝑖= 2.515 ⋅10−6 𝐼2
𝑖𝑆𝑖− 6.744 ⋅10−6 𝑆𝑖− 3.658 ⋅10−4 𝐴𝑅𝑖𝐼𝑖
− 3.856 ⋅10−5 𝐴𝑅𝑖𝐴𝑖+ 0.01291 𝐴𝑅𝑖+ 2.336 (3)
𝑑200,0
𝑥𝑖= −1.819 ⋅10−5 𝐼𝑖𝑆𝑖− 1.363 ⋅10−5 𝑆𝑖
− 4.270 ⋅10−5 𝐼2
𝑖− 1.493 ⋅10−4 𝐴𝑅𝑖𝐼𝑖
+ 0.0013 𝐼𝑖+ 4.893 ⋅10−4 𝐴𝑅𝑖+ 2.045
(4)
𝑑200,6
𝑥𝑖= 2.935 ⋅10−5 𝑆𝑖+ 1.407 ⋅10−4 𝐴𝑅𝑖𝐼𝑖− 0.00215 𝐼𝑖
+ 0.00307 𝐴𝑅2
𝑖− 0.02338 𝐴𝑅𝑖+ 2.071 (5)
𝑑200,12
𝑥𝑖= 4.7⋅10−6 𝐴𝑅𝑖𝐼𝑖𝑆𝑖− 3.416 ⋅10−5 𝐼𝑖𝑆𝑖
+ 1.048 ⋅10−6 𝐴𝑅𝑖𝐴𝑖𝐼𝑖− 5.371 ⋅10−5 𝐴𝑅𝑖𝐼𝑖
−9.453 ⋅10−4 𝐼𝑖−1.314 ⋅10−7 𝐴𝑖+ 2.0462
(6)
𝑑220,0
𝑥𝑖= 5.384 ⋅10−6 𝐴𝑅𝑖𝐼𝑖𝑆𝑖+ 4.453 ⋅10−6 𝐴𝑅𝑖𝑆𝑖+ 8.208 ⋅10−6 𝑆𝑖
+ 1.264 ⋅10−4 𝐼𝑖+ 1.612 ⋅10−6 𝐴𝑖+ 1.436
(7)
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L. Millán e al.
Fig. 7. De ia ions in hi y execu ions o GP o cen e posi ion.
𝑑220,6
𝑥𝑖= 7.815 ⋅10−5 𝑆𝑖− 9.574 ⋅10−5 (𝐴𝑅𝑖
− 8.026)(2.569 𝐼𝑖+𝐴𝑅𝑖+ 35.7)
− 9.11 ⋅10−6 𝐴𝑅𝑖+ 1.4024
(8)
𝑑220,12
𝑥𝑖= −1.869 ⋅10−6 𝐼𝑖𝑆𝑖+ 5.7151 ⋅10−5 𝑆𝑖
+ 5.4069 ⋅10−4 𝐴𝑅𝑖𝐼𝑖+ 5.3633 ⋅10−6 𝐼𝑖
+ 0.00118 𝐴𝑅2
𝑖− 0.00157 𝐴𝑅𝑖+ 1.426
(9)
𝑑311,0
𝑥𝑖= 1.227 ⋅10−5 𝐴𝑅𝑖𝑆𝑖− 7.621 ⋅10−6 𝑆𝑖
+ 0.00152 𝐴𝑅𝑖𝐼𝑖− 4.89 ⋅10−5 𝐼𝑖
− 1.626 ⋅10−5 𝐴𝑖+ 1.224
(10)
𝑑311,6
𝑥𝑖= 9.712 ⋅10−6 𝐼𝑖𝑆𝑖+ 3.093 ⋅10−6 𝑆𝑖− 9 ⋅10−4 𝐼𝑖
− 1.028 ⋅10−6 𝐴𝑅𝑖𝐴𝑖+ 1.23 (11)
𝑑311,12
𝑥𝑖= 1.22 − 3.064 ⋅10−6 (𝐼𝑖+ 2.629) (𝐴𝑅𝑖𝑆𝑖
− 5.413 𝑆𝑖+ 6.266 𝐴𝑅2
𝑖− 60.16 𝐴𝑅𝑖)(12)
Fig. 8. De ia ions in hi y execu ions o GP o posi ion 6 mm.
4.2. In e sec ion plo s
To u he explain and alida e he p edic ions made by he iden i-
ied equa ions, we use in e sec ion plo s, which show he esponse o
he models o e each o he inpu a iables. Wi h hese isualiza ions,
we can unde s and he dependency o p edic ed m-RS om mic os uc-
u al a iables as cap u ed by he equa ions. A de ailed desc ip ion o
his dependency would enla ge his pape unnecessa ily. This s udy is,
in ac , he subjec o a ollow-up publica ion.
To p oduce he in e sec ion plo s, he alue o h ee inpu s is kep
cons an a an a e age alue ep esen ed by he dashed g ay line, and
only he a iable o in e es is a ied. The 𝑦-axis ep esen s he ou pu
o he models ( he la ice spacing, 𝑑ℎ𝑘𝑙
𝑥𝑖). The dependence o 𝑑ℎ𝑘𝑙
𝑥𝑖o hese
pa ame e s is p opo ional o he m-RS. The plo s a e e iewed and
he highligh s obse ed o each o ien a ion o each posi ion in he
ollowing.
Applica ions in Enginee ing Science 15 (2023) 100141
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L. Millán e al.
Fig. 9. De ia ions in hi y execu ions o GP o posi ion 12 mm.
Figs. 10,11, and 12 show he in e sec ion plo s o he bes equa-
ions om all uns a posi ions 0, 6, and 12 mm.
4.2.1. Scena io 1: posi ion a 0 mm
Fi s ly, i can be obse ed ha peak 111 is symme ic (Fig. 7) and
he ends o he models o 111 p esen a lo o dispe sion. The plo
o he slope shows wo al e na i e ends. This implies ha 111 g ains
can be o de ed by inc easing o dec easing slope, and bo h cases would
lead o a Gaussian peak wi h small e o s. In summa y, o he Gaussian
peak o o ien a ion 111, GP iden i ied no clea unc ional dependency
o he a ge a iable and he inpu s.
Fo o ien a ion 200, he models show mo e de ined ends and he
peak is non-symme ic. Analyzing he a e ages, i is obse ed ha
he mos ele an pa ame e s in his o ien a ion a e he numbe o
neighbo ing g ains and he slope. I can be seen ha a highe alues
o hese pa ame e s, he la ice spacing dec eases. Fo his o ien a ion,
he e a e wo al e na i e explana ions o which he GP uns con e ged.
A ew models show a posi i e e ec o he aspec a io and slope, which
is con a y o he majo i y o he models. Howe e , hose ew models
a e also hose ha ma ch he a ge peak bes (see Fig. 7), while he
majo i y o models shows a mul i-modal peak ha signi ican ly de ia es
om he a ge peak. The h ee good models, consis en ly p edic lowe
la ice spacing o highe numbe o neighbo s, and highe la ices
spacing o highe aspec a io o slope.
The 220 g ains a e he la ges g ains and occu seldom and in
isola ion. Con a y o he 111 g ains, he da a o he a ge peak is no
symme ic and he in e sec ion plo s show ha he equa ions a e ela-
i ely consis en ega ding he e ec s o he p edic ion. Howe e , as
he p edic ions ha e la ge e o s we canno expec o ga he ele an
insigh s om he in e sec ion plo s o he 220 g ains.
Fo o ien a ion 311 he p edic ions a e be e . Howe e , he sensi i -
i y analysis plo s do no show clea co ela ions. The pic u e is simila
as o 111 because he 311 peak is symme ic as well. The e a e wo
al e na i e explana ions o which GP con e ged.
4.2.2. Scena io 2: posi ion a 6 mm
Fo he 6 mm posi ion, we obse e a simila si ua ion as o he
cen e posi ion. The peaks o 111 and 311 a e symme ic and i
well, bu he in e sec ion plo s show high a iance be ween models
indica ing ha GP was no able o iden i y e ec s consis en ly. The
p edic ed peaks o 220 ha e high e o as he e a e only ew 220 g ains.
The 200 g ains show a non-symme ic peak and a lowe a iance in
he in e sec ion plo . The p edic ions i he a ge peak ela i ely well,
and we see ha he equa ions consis en ly p edic lowe la ice spacing
o g ains wi h highe aspec a io o slope. The g ain a ea and numbe
o neighbo s ha e almos no e ec in he models.
A subs an ial a ia ion in he iden i ied ends is obse ed compa ed
o he cen al posi ion.
4.2.3. Scena io 3: posi ion a 12 mm
A 12 mm we obse e in Fig. 9 ha he peaks o 111 and 311 a e
p edic ed well, whe eby he p edic ions o 200 and 220 a e a he a
o . The peak o 111 is symme ic while he peak o 311 is asymme ic.
In he in e sec ion plo s in Fig. 12, we can igno e he plo s o
200 and 220 because hese equa ions do no i he a ge peak well.
In e es ingly, in his case he models o he 111 g ains show mo e
consis en ends, al hough he peak is symme ic. Fo example, i can
be seen ha a ea co ela es nega i ely wi h he la ice spacing. In he
co esponding EBSD map (see Fig. 1) i is possible o ind 111 g ains o
a ying sizes, some e y la ge and some e y small. Mo eo e , a his
posi ion he numbe o 111 g ains has dec eased and now hey a e no
a anged in clus e s bu a e loca ed in isola ion a he g ain bounda ies.
Many di e en equa ions allow o p edic he 311 peak well. The
main in luences appea o be he numbe o neighbo s and he slope.
Howe e , he e a e a leas wo dis inc ends. On end p edic s ha
he la ice spacing inc eases conside ably wi h he numbe o neighbo s
and he slope, compa ed o he second op ion whe e he numbe o
neighbo s has no e ec bu la ice spacing dec eases quickly wi h
inc easing slope.
5. Summa y
This wo k ex ends he s udy p esen ed a he EVOAPPS 2021 Con-
e ence (Millán e al.,2021b) in which m-RS in a quenched cylind ical
sample ex ac ed om aluminum alloy ex uded ba we e calcula ed
using da a om only one neu on di ac ion peak a he sample cen al
axis. Now, we comple e he s udy by inco po a ing he ou main peaks
o he spec a and da a om o he posi ions along he sample adial
di ec ion.
We use equa ion lea ning (symbolic eg ession) based on gene ic
p og amming, o de e mine he m-RS o each o ien a ion o ep oduce
he peaks’ p o iles. All 𝑑ℎ𝑘𝑙
𝑥𝑖 alues ha e been calcula ed o each g ain