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Real g ain shape analysis: cha ac e iza ion and gene a ion o ep esen a i e i ual g ains. Applica ion o ailway
ballas
IV In e na ional Con e ence on Pa icle-based Me hods – Fundamen als and Applica ions
PARTICLES 2015
E. Oña e, M. Bischo , D.R.J. Owen, P. W igge s & T. Zohdi (Eds)
REAL GRAIN SHAPE ANALYSIS: CHARACTERIZATION AND
GENERATION OF REPRESENTATIVE VIRTUAL GRAINS.
APPLICATION TO RAILWAY BALLAST.
N. OUHBI¹2, C. VOIVRET2, G. PERRIN3 AND J-N. ROUX1
¹ Labo a oi e Na ie (UMR CNRS 8205)
CNRS, Ecole des Pon s e Chaussées, IFSTTAR
Champs-su -Ma ne, F ance
e-mail: {jean-noel. oux}@i s a .
²Inno a ion and Resea ch Depa men , SNCF, Pa is, F ance
Immeuble Lumiè e
40 A enue des Te oi s de F ance, F-75611 PARIS CEDEX 12
e-mail: {nou a.ouhbi, cha les. oi e }@snc .
³CEA/DAM/DIF
F-91297, A pajon, F ance
Keywo ds: Ballas , pa icle shape, Cha ac e iza ion, DEM, polyhed a, POD, gene a ion.
Abs ac . G ain shape signi ican ly in luences he mechanical p ope ies o g anula media.
In o de o explo e his e ec and o simula e ealis ic ma e ial mo phology, we designed a
me hod which well cha ac e izes eal g ains shape. S a ing om a ep esen a ion o he
pa icle su aces as a poin s cloud, his pape p esen s a me hod o gene a e a se o i ual
g ains ha a e mo phologically ep esen a i e o eal ballas g ains. The model elies on a
s a is ical modelling o he ballas g ain mo phology based on a dimensionali y educ ion
app oach (P ope O hogonal Decomposi ion) leading o an op imal and nea ly exhaus i e
shape cha ac e iza ion by ex ac ing a hie a chy o shape unc ions ha ully desc ibe he
g ain sample. We will show he e iciency o he bo h cha ac e izing and gene a ing me hods
and desc ibe hei ad an ages, as well as a u u e ou look
1 INTRODUCTION
G anula ma e ials a e widely used in di e en applica ions anging om ood indus y o
ci il enginee ing. The e o e, a be e unde s anding o he o e all beha iou o hese ma e ials
is pi o al o imp o e and con ol hei pe o mance. Nume ous s udies ha e been ca ied ou
and en iched o e he las yea s, abou he impac o pa icle size, shape and mine alogy on
he mechanical beha iou o he g anula media. As o shape p ope ies, se e al expe imen al
analyses as well as nume ical s udies using Disc e e Elemen Me hods (DEM) [1] ha e shown
he signi ican in luence o pa icle shape on he e olu ion o g anula assemblies [2-6, 11-13,
16, 17, 24, 26]. Unde s anding his in luence is hen a opic o in e es in his s udy.
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N. Ouhbi, C. Voi e , G.Pe in and J-N. Roux
2
Fo DEM simula ions, ep esen ing pa icle shape is a eal challenge. Mos disc e e elemen
codes ep esen pa icles as discs (2D) o sphe es (3D), because o implemen a ion simplici y
and compu a ional e iciency. Howe e , i has been p o ed ha hese simple geome ies ail
o ep oduce ealis ic ma e ial beha iou o an indi idual g ain, as well as o a g anula
assembly [7, 14, 20]. O he a emp s ha e hen been made o ake in o accoun complexi y
ha alls oughly in wo classes: 1) pa ame ic model based on gi en geome ic cons uc ion
ules and 2) i ed model based on eal g ain shapes. Bo h app oaches can be ul illed wi h
simple p imi i e (sphe e, ellipsoid, plane, clumps...) o wi h mo e e sa ile geome ic shape as
polyhed on [8, 10]. O e he las decades, SNCF (F ench ailway company) has chosen he
second app oach o simula e he beha iou o he ballas wi h DEM based on polyhed on
shaped pa icles. Railway ballas is a g anula laye o med by i egula ock g ains o a
cen ime ic size ex ac ed om ha d s one qua ies by c ushing ((BS EN 13450, 2003) [1]).
Cu en ly, i ual g ains used in simula ions a e se s o nea ly 1000 sampled g ains ha ha e
been 3D digi alized and meshed.
In o de o p ope ly s udy he impac o he ballas shape on he mechanical beha iou wi h
DEM, a gene a o o i ual g ains is needed, ap o p oduce la ge se s o i ual g ains ha
a e ep esen a i e o a limi ed se o eal g ain, and ha also allows o each an accu a e
cha ac e iza ion o he g ain shape. In his pape , we p opose a me hod o achie e hese goals.
In Sec ion 2, he p oposed app oach o eal ballas g ain shape modelling as well as i s
alida ion a e p esen ed and analysed. Some illus a ions a e p esen ed in Sec ion 3.
2 REAL BALLAST GRAINS MODELLING
2.1 Global iew
In he li e a u e, gene a ion me hods we e p esen ed, such as Fou ie -Shape-Desc ip o s
[19, 22], and sphe ical-based andom ields [9, 18, 29]. These app oaches ha e gi en good
esul s in e ms o simila i y o gene a ed g ains and eal ones, bu in oduce shape unc ions
ha a e imposed.
We p esen in his pape an inno a i e app oach based on a dimensionali y educ ion
me hod leading o an op imal and nea ly exhaus i e shape cha ac e iza ion o eal g ains.
By means o P ope O hogonal Decomposi ion (POD) [15], we iden i y he op imal
hie a chy o shape unc ions ha desc ibe he g ain se . The main ad an age o his app oach
is o educe he numbe o needed shape unc ions o ep esen he g ain shape wi h a
quan i a i e con olled app oxima ion (e o based), such ha we educe he pa ame ic space
o he op imal one.
2.2 P e-p ocessing: Sample p epa a ion
A da abase o 121 di e en ballas g ains is p o ided by SNCF. Ballas ma e ials in F ance
a e selec ed based on he Eu opean ailway ballas speci ica ion. The g ains a e ep esen ed
by poin clouds ob ained expe imen ally by 3D digi iza ion (3D scan) o he pa icle su aces,
and o m i egula polyhed ons o 4000 aces, and abou 2000 e ices. Fig.1 shows some o
hese g ains.
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Figu e 2: A g ain o ballas : (a) Raw da a (b) A e p ojec ion and o a ion
The p e-p ocessing is based on he ollowing s eps:
 P ojec ion o he expe imen al poin s cloud on an imposed basis (B1) wi h cons an
angula s ep and N di ec ions. Since he expe imen al poin cloud a e composed o
many poin s (up o millions o ine digi aliza ion) wi h andom di ec ion in a
sphe ical ame, his s ep aims a educing he numbe o poin and ha ing he same
di ec ion o each g ain.
 Compu a ion o all g ains o olume, su ace, ine ia enso , and mean adius.
 Cen e ing g ains and o a ing e ices un il he p incipal di ec ions o he ine ia enso
a e pa allel o he global coo dina e axes, and in e pola ing (Fig.2)
 P ojec ion on a new basis (B2) wi h uni o m densi y o numbe o di ec ion by solid
angle. Indeed, he p e ious basis, wi h cons an angula s ep, lead o a high densi y o
di ec ion nea he pole ha can bias he s a is ic. This new basis wi h 500 di ec ions is
gene a ed wi h a epulsion poin i e a i e p ocedu e. (Fig. 3)
 By conca ena ing all g ains oge he , ou se o expe imen al g ain is hen ep esen ed
by a 500 x 121 ma ix. Each column ep esen s g ain e ices dis ances along he 500
o he basis B2 (Fig.4). This ma ix will be he inpu da a o POD p ocedu e.
Figu e 1 : Example o eal ballas g ains ep esen ed by dense poin clouds - SNCF
(a) (b)
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Figu e 4: In e pola ion on he gene a ed basis wi h 500 di ec ions and eal g ain
2.3 G ain cha ac e iza ion
A model educ ion me hod, namely he P ope O hogonal Decomposi ion o P incipal
Componen Analysis (PCA) [27] is used. S a ing om he da a ma ix o he Sec ion 2.2, his
mul i a ia e s a is ical me hod aims a ob aining a compac ep esen a ion o he da a. I
iden i ies an op imal and use ul se o basic unc ions ha allows o achie e a sa is ac o y
app oxima ion o he sys em.
This me hod se es wo pu poses, namely o de educ ion by p ojec ing high-dimensional
da a in o a lowe -dimensional space and ea u e ex ac ion by e ealing ele an , bu
unexpec ed, s uc u e hidden in he da a. The i s pu pose allows o quan i y he con olled
app oxima ion (e o based), while choosing he dimension o he p ojec ion educed space.
The second pu pose allows o cha ac e ize he shape ea u es by associa ing hem o POD
ou pu s, i.e. basis unc ions (eigenmodes) and coe icien s. Eigenmodes, o p incipal
componen s, a e he eigen ec o s o he co a iance ma ix co esponding o o iginal da a,
whe eas coe icien s a e p ojec ion coo dina es on he educed space o p ojec ion. These wo
elemen s, in addi ion o eigen alues o he co a iance ma ix a e he h ee key elemen s o he
me hod. Fo u he explica ions o he me hod, see [15].
While applying POD on he da a, a co esponding e o ε is exp essed in e ms o he
p ojec ion e o s ha a e con olled in he cons uc ion o POD bases. In o he wo ds, i is
de ined as he de ia ion o he ans o med da a o he new space om he o iginal da a,
no malized by he aw da a and induced by he unca ion o he POD basis.
While POD e o gi es an indica ion o he magni ude o he “missing” in o ma ion, he
Figu e 3: Basis wi h 500 di ec ion: (a) basis B1 (b) basis B2
(a) (b)
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ene gy o he sys em 𝑒𝑒(𝑘𝑘) = 1 − 𝜀𝜀(𝑘𝑘) ep esen s he quan i y o in o ma ion cap u ed by he
k i s POD basis ec o s. (Fig. 5) shows he quan i ies o e o /ene gy as a unc ion o he
dimension o he educed space.
Fo ze o e o we ha e o keep all 121 modes. We see ha wi h only 12 modes we ep esen
90% o he in o ma ion and 99% o he in o ma ion is ep esen ed wi h oughly 50 modes.
Wi h his p ecision, we ha e hal - educe ou da a.
As he accu acy o he g ain app oxima ion depends on he numbe o modes, i is
impo an o check i he p ope ies o g ains (a e age adius, su ace, olume) a e sensi i e o
he numbe o modes. In (Fig.6), we compa e hose p ope ies dis ibu ions o di e en e o
h esholds.
In o de o quan i y mo e p ecisely he compa isons, we pe o m Kolmogo o -Smi no (KS)
es s be ween o iginal da a su aces/ olumes/mean adius and econs uc ed g ains
cha ac e is ics. The esul s a e p esen ed in Table 1.
Table 1: Values o Kolmogo o -Smi no es s be ween o iginal and econs uc ed da a S, V, Rm dis ibu ions
E o on shape
KS Su aces
KS Volumes
KS Rayons moyens
0.1%
0.999
0.997
0.947
1%
0.999
0.997
0.879
5%
0.999
0.997
0.785
10%
0.997
0.984
0.465
15%
0.984
0.879
0.223
Figu e 5: E o and ene gy quan i ica ion

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We see ha e en wi h signi ican e o (mo e han 10%) we ha e a e y good shape
app oxima ion bu wi h a much mo e educed space.
Fu he mo e, by s a is ically iden i ying POD coe icien s dis ibu ions, we can gene a e as
many equi alen da a as wan ed. Fo a pe ec ly exhaus i e shape cha ac e iza ion, we decide
o keep all o he eigenmodes and he e o e an e o o 0% is made. The s a is ical analysis
will conce n he coe icien s ha a e unco ela ed. (Fig. 7) shows s a is ical dis ibu ions o
he i s 8 coe icien s as an example.
Figu e 6: Recons uc ion o he eal g ains o di e en e o h esholds – Compa ison o (a) Su aces
dis ibu ions (b) Volumes dis ibu ions (c) Mean adius dis bu ions
Figu e 7: S a is ic ep esen a ion o he i s 8 coe icien s
Expe imen al da a
ε = 0.1%
ε = 1%
ε = 5%
ε = 10%
ε = 15%
Expe imen al da a
ε = 0.1%
ε = 1%
ε = 5%
ε = 10%
ε = 15%
Expe imen al da a
ε = 0.1%
ε = 1%
ε = 5%
ε = 10%
ε = 15%
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The challenge is hen o build a model o each dis ibu ion. While i is possible o sea ch
he bes ma ching dis ibu ion o o i he CDF, we p esen ly p e e o keep he exac CDFs
co esponding o disc e e da a, wi hou ans o ming i o con inuous i ed unc ions (Fig. 8).
Figu e 8: Fi s coe icien CDF
2.4 Gene a ion
E en hough he coe icien s a e unco ela ed (no shown), hese coe icien s can be
dependen in a mo e complex way. The e o e, he gene a ion p ocess has o ake in o accoun
he dependence. Since he coe icien s a e o di e en ma ginal dis ibu ions, his canno be
done by only using a mul i a ia e dis ibu ion, which su e s om he es ic ion ha he
ma ginal should be o he same ype. One e icien way o do his is copula unc ions [28].
Tha is, copula unc ions allow one o model he dependence s uc u e independen ly o he
ma ginal dis ibu ions. Any mul i a ia e dis ibu ion unc ion can se e as a copula. Tha
o e s a good modelling lexibili y.
Fo ou gene a ion p ocess, we use he Gaussian copula, since wo pa ame e s (Mean and
co ela ion ma ices) a e enough o desc ibe i . The algo i hm is explained in de ails in [21,
22, 25].
2.5 Valida ion
To alida e ou app oach, we gene a e di e en se s o 300, 500, 800, 1000, 2000 and
10000 i ual g ains. Su aces, olumes and mean adius dis ibu ions a e hen compu ed and
compa ed o hose o he o iginal da a. (Fig. 9) shows he esul s, and Kolmogo o -Smi no
es s esul s a e p esen ed in Table 2.
Expe imen al CDF
Fi ed CDF
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Table 2: Values o Kolmogo o -Smi no es s be ween o iginal gene a ed da a S, V, Rm dis ibu ions
By analysing Fig. 9, we see ha o su aces, olumes and a e age adii, he peaks a e he
poin s ha ma ch he leas . Tha can be explained by he ac ha he eal pa icle da ase (121
g ains) is ela i ely small, and insu icien o es ima ing accu a ely s a is ics o he g ains
shape, and hence cap u ing all shape ea u es. Tha was p o ed by Liu e al. [18] who
analysed he impac o da a se size on he accu acy o esul s using sphe ical ha monics
app oach, and by G igo iu e al. [9] who also p o ed he insu iciency o a da ase o 128
Numbe o
gene a ed
g ains
KS Su aces
KS Volumes
KS Rayons moyens
300
0.949
0.949
0.811
500
0.993
0.993
0.831
800
0.994
0.994
0.883
1000
0.998
0.939
0.705
2000
0.976
0.896
0.870
10000
0.939
0.968
0.613
Figu e 9: Sample alida ion – 10000 g ains gene a ed
Expe imen al da a
Gene a ed da a
Expe imen al da a
Gene a ed da a
Expe imen al da a
Gene a ed da a
Expe imen al da a
Gene a ed da a
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agg ega es o exac ly ep esen he conc e e g ains shape. They showed ha a “small da a se
was unable o accu a ely es ima e s a is ics o agg ega e geome y beyond second-momen
p ope ies and ma ginal dis ibu ion”.
Howe e , he high alues o KS es s ob ained and he ep esen a ions show ha we ge
sa is ac o y esul s, e en wi h a small da a se . Tha con i ms he accu acy o he app oach
and shows ha en iching ou da a se will ce ainly allow o ge an e en be e and exhaus i e
cha ac e iza ion o ballas shape, and o e an in e es ing ansi ion be ween eal and i ual
g ains in o de o inco po a e hem in DEM simula ions.
3 ILLUSTRATION
As shown p e iously, POD p ocedu e allows o e y well app oxima ing he g ains shape
h ough eigenmodes and coe icien s. One in e es ing ques ion would be how o link
eigenmodes and coe icien s o shapes ea u es.
One o he ad an ages o his me hod is o de ing he domina ing ea u es by dec easing
o de ( i s modes ha e he bigges con ibu ions o ene gy o he sys em, and he la ges
alues o a iance). We hen expec he i s mode o hold an impo an pa o shape ea u es
(wi h 45% o he ene gy o he sys em).
To obse e he e olu ion o g ains shape h ough modes, we ep esen he same g ain
econs uc ed wi h he i s 1 (ene gy = 45%) o 20 modes (95%) in Fig. 11. We can see he
p og essi e eme gence o shape de ails as we add new modes. As he quan i y o in o ma ion
inc eases (e o dec easing), we minimize he sum o he squa ed di e ences o he dis ances
be ween he poin on he eal g ain and he same one ep esen ed wi h a ini e numbe o basic
unc ions.
The i s ou h modes a e shown in Fig.12. We see ha he i s mode, con ibu ing mos o
he ene gy o he sys em, has a comple e shape o a g ain while he es o he modes
cons i u e he o he de ails o shape.
Finally, we ep esen econs uc ed g ains wi h only he i s mode in Fig. 13. Fo all 121
g ains, we ha e a “ ound” shape.
Figu e 11: G ain shape e olu ion h ough modes 1 o 20 and eal g ain