Recei ed: 12 Ap il 2018 Re ised: 15 Oc obe 2018 Accep ed: 28 No embe 2018
DOI: 10.1002/nme.6004
RESEARCH ARTICLE
Modeling polyc ys alline ma e ials wi h elonga ed g ains
I in Pé ez1Má cio Muniz de Fa ias2,3 Manuel Cas o1Robe o Roselló1
Ca los Reca ey Mo a1,3 Liosbe Medina4E. Oña e5
1Cen e o Compu a ional Mechanics and
Nume ical Me hods in Enginee ing,
CIMNE-UCLV Class oom, Cen al
Uni e si y o Las Villas, San a Cla a, Cuba
2In aLab, Uni e si y o B asilia, B asilia,
B azil
3Facul y o Technology, Uni e si y o
B asilia, B asilia, B azil
4Depa men o T anspo Enginee ing,
Fede al Uni e si y o Goiás, Goiânia,
B azil
5In e na ional Cen e o Nume ical
Me hods in Enginee ing, Poly echnic
Uni e si y o Ca alonia, Ba celona, Spain
Co espondence
I in Pé ez, Cen e o Compu a ional
Mechanics and Nume ical Me hods in
Enginee ing, CIMNE-UCLV Class oom,
Cen al Uni e si y o Las Villas,
San a Cla a, Cuba.
Email: [email p o ec ed]
Funding in o ma ion
Coo dina ion o he Imp o emen o
Highe Educa ion Pe sonnel (CAPES),
B azilian Minis y o Educa ion,
G an /Awa d Numbe : CAPES-MES
P ojec No. 208/13; In aLab, Facul y o
Technology, Uni e si y o B asilia;
In e na ional Cen e o Nume ical
Me hods in Enginee ing (CIMNE)
Summa y
A no el algo i hm o ep oduce he a angemen o g ains in polyc ys alline
ma e ials was ecen ly published by he au ho s. In his o iginal app oach,
a dense package o ci cles (o sphe es) wi h he same dis ibu ion as he
g ains is gene a ed o p oduce a se o Vo onoi cells ha a e la e modi ied o
Lague e cells ep esen ing he o iginal s uc u e. This algo i hm was success-
ully applied o ma e ials wi h somewha equidimensional g ains; howe e , i
ails o long-shaped g ains. In his pape , modi ica ions a e p o ided in o de o
o e come hese d awbacks. This is accomplished by mo ing each e ex o he
Vo onoi cells in such a way ha he e ex should be equidis an om he pa i-
cles wi h espec o he Euclidean dis ance. The algo i hm is applied o packages
o ellipses and sphe ocylinde s in 2D. An example o a package o sphe es is
also p o ided o illus a e he applica ion o a simple 3D case. The adhe ence
be ween he gene a ed packages and he co esponding essella ions is e i ied
by means o he Jacca d coe icien (J). Se e al packages a e gene a ed andomly
and he dis ibu ion o Jcoe icien s is in es iga ed. The ob ained alues sa is y
he heo e ical es ain s and he quali y o he p oposed algo i hm is s a is ically
alida ed.
KEYWORDS
Lague e, RVE, simila i y, essella ion, Vo onoi
1INTRODUCTION
The beha io o he mic os uc u e o some polyc ys alline ma e ials can be success ully simula ed wi h he ini e elemen
me hod,1-3by modeling a ep esen a i e olume elemen (RVE).4Fo his pu pose, he medium should be ep esen ed
by pa i ioning he domain in o polyhed a. One way o ob ain such pa i ion can be by ob aining polyhed a ha a e
ci cumsc ibed o bodies o a packing ha ills he domain wi h a high olume ac ion (see Figu e 2 bo om o an example
in 2D). The sizes and shapes o he polyhed a mus ollow some gi en s a is ical dis ibu ion, and hey a e de e mined by
he elemen s o he packing con ained inside he polyhed a.
Sphe es a e one o he simples shapes o gene a ing a dense packing5and ob aining a polyhed al pa i ion om i .
Once a sphe e packing exis s, he e a e wo main app oaches o calcula ing he polyhed al pa i ion om i . The i s
app oach1consis s in calcula ing a powe diag am (also known as Lague e diag am), which can be ob ained by eloca ing
In J Nume Me hods Eng. 2018;1–11. wileyonlinelib a y.com/jou nal/nme © 2018 John Wiley & Sons, L d. 1
2PÉREZ ET AL.
he cell e ices o he Vo onoi essella ion o he sphe es' cen e s. I has he disad an age ha , in 3D, i is only sui able
o sphe es. The second app oach6is mo e gene al and consis s in i e a i ely calcula ing he planes de ining he aces o
each cell. This me hod is he one implemen ed in he popula lib a y Vo o++7 o sphe es.
Two o he publicly a ailable compu a ional lib a ies o calcula ing Vo onoi essella ions mus be men ioned. The i s
one is CGAL,8which allows he calcula ion o bo h weigh ed diag ams and diag ams wi h espec o segmen s. The
o he one is Qhull,9which allows calcula ing Vo onoi diag ams and also con ex hulls. Howe e , none o hem p oduce
essella ions in 3D om se s o shapes o he han sphe es.
When he cen e s o a Vo onoi diag am a e se s o he han single poin s, he essella ion may ha e he disad an age o
con aining cells wi h cu ed aces. Howe e , he in e es ed eade can check he wo k o Emi is e al10 o ob aining such
diag ams o pseudo-ci cles and he wo k o Dong e al11 o a bi a y shapes as well. The o me me hod is based on a
polynomial ep esen a ion o he diag am, while he la e is based on a pixel app oxima ion.
Polyc ys alline ma e ials do no always ha e g ains wi h aspec a ios close o 1. Howe e , as a as we can ell, no
p e ious esul s ha e been p esen ed o model polyc ys alline ma e ials wi h g ains ha ing aspec a ios e y di e en
om 1, by means o essella ions buil om dense packings o nonci cula shapes. In his pape , an ini ial a emp in ha
di ec ion is p esen ed, by using dense packings o nonci cula shapes such as ellipses and sphe ocylinde s in o de o
ob ain he essella ions.
2MODELING POLYCRYSTALLINE MATERIALS
The whole p ocess o modeling g anula solids using mic oscopic images, acco ding o he wo ks o Benabbou e al,1
Hi i,2and Mo a e al,3can be summa ized in he ollowing Pseudocode 1.
The i s wo s eps o he p e ious pseudocode will no be explained in de ail since hey a e ou o he scope o he
p esen pape and consis in he acquisi ion and segmen a ion o images o polyc ys alline ma e ials (Figu e 1). This seg-
men a ion allows o es ima e he g ain size dis ibu ion (s ep 3) ha can be used as inpu o a sphe e packing algo i hm,
in o de o ob ain a dense sphe e packing whose sizes ha e he same dis ibu ion as he g ains o he mic oscopical o ig-
inal image (s ep 4 and Figu e 2 op). The packing o sphe es allows he calcula ion o a Lague e diag am o polyhed al
cells con aining he sphe es (s ep 6 and Figu e 2), as explained in he ollowing.
FIGURE 1 Polyc ys alline ma e ial: closed cell oam.2Le : o iginal 2D image. Righ : segmen ed image
PÉREZ ET AL.3
FIGURE 2 De ails o Lague e diag am ob ained om a dense se o disks. (A) Packing wi h Vo onoi essella ion supe imposed.
(B) Packing wi h Lague e essella ion supe imposed
The algo i hm chosen o gene a e he packings in his esea ch is an ad ancing on algo i hm ha allows o gene a e
high densi y packings o bodies ha ing an a bi a y dis ibu ion o shape and size, in easonable imes.5,12,13 Such algo i hm
is a good cos -e ec i e al e na i e agains o he me hods like g a i y deposi ion, la ice me hods, o me hods o sol e he
bin-packing p oblem.
The addi i ely weigh ed powe dis ance (AWPD) dP(p,c) be ween wo poin s pand cis de ined by he exp ession
dP(p,c)=d(p,c)2−w2,whe edis he Euclidean dis ance and wis any posi i e numbe .14 I cis he cen e o a ci cle Co
adius and w= , hendP(p,c) is equal o he squa ed leng h o he angen line ha goes om p o C, as a consequence
o he Py hago as heo em, and dP(p,c) can be conside ed as he AWPD be ween pand C.
Le Vbe a Vo onoi diag am whose cen e s a e he cen e s o a se o ci cles, like he one o Figu e 2 op. Each Vo onoi
cell has o con ain exac ly one o he ci cle's cen e s, and each e ex o each Vo onoi cell mus be equidis an om h ee
ci cle's cen e s, bu he ci cles do no ha e o necessa ily be con ained in he cells, as can be seen in ha igu e. Howe e ,
he cells' e ices can be modi ied o his o happen. Le ibe a e ex o Vequidis an wi h espec o he Euclidean
dis ance d om h ee ci cle cen e s ci𝟏,ci𝟐,andci𝟑,being 1, 2,and 3 he espec i e adii (Figu e 2 op). I each e ex i
o Vis ans o med in o a e ex ′
iequidis an om ci𝟏,ci𝟐,andci𝟑wi h espec o he AWPD dPwi h weigh s equal o 1,
2,and 3, espec i ely, hen he cells o he esul ing essella ion (known as Lague e diag am) will con ain he ci cles, as
can be seen in Figu e 2 bo om. The 3D case o sphe es is o ally analogous.
Wi h he p e ious s eps, a i ual ep oduc ion o polyc ys alline ma e ials can be ob ained. Howe e , his app oach
has he disad an age ha i will no be ealis ic o polyc ys alline ma e ials ha ing elonga ed g ains, such as he ones
in Figu e 3.15 The igu e shows he mic os uc u e o an ul a ine-g ained Ni mic os uc u e a e s ains o 1600% and
3200%. One solu ion o his p oblem is o modi y s ep 4 o Pseudocode 1 by gene a ing packings o nonsphe ical pa icles
ha ing he same aspec a io dis ibu ion as he polyhed al g ains and o modi y s ep 6 o Pseudocode 1 by using a dis ance
o he han he AWPD.
3MODIFICATION OF THE VORONOI DIAGRAM IN 2D
Suppose ha a Vo onoi diag am has been calcula ed wi h espec o he cen e s o a se o pa icles and ha i is necessa y
o modi y he diag am in such a way ha i s cells ully con ain he pa icles. A possibili y o accomplish his could be
o modi y each Vo onoi e ex so as o make i equidis an om he bodies ha con ain he cen e s om which i was
o iginally equidis an . Figu e 2 shows an example o his modi ica ion, whe e he pa icles a e ci cles and each eloca ed
Vo onoi e ex is equidis an om h ee ci cles wi h espec o he AWPD.1In he op essella ion, each e ex (which is no
in he bounda y box) is equidis an om h ee disk cen e s, and in he bo om essella ion, each e ex is equidis an om
h ee disks wi h espec o he AWPD and he pa icles a e ully con ained in he cells. In his case o ci cula pa icles, he
AWPD was used o calcula e he dis ance om a poin o a body, bu o o he pa icle shapes, o he dis ances can be used.
4PÉREZ ET AL.
FIGURE 3 Example o an ul a ine-g ained Ni mic os uc u e a e s ains o (A) 1600% and (B) 3200% (see he wo k o Hohenwa e and
Pippan15) [Colou igu e can be iewed a wileyonlinelib a y.com]
FIGURE 4 Le : Vo onoi diag am wi h espec o he cen e s o a se o ellipses. Righ : essella ion wi h cells con aining he ellipses
A mo e na u al way o de ine o calcula e he dis ance om a poin o a body is he in imum dis ance om he poin o
he body. I pand Ba e a poin and a se o Rn, espec i ely, hen he in imum dis ance din (p,B) omp o Bis de ined by
he exp ession
din (p,B)=in
q∈Bd(p,q),(1)
whe e qis any poin (o elemen ) o Band dis he Euclidean dis ance in Rn. The equali y din (p,B)=d(p,q) holds o
some poin qin ( he bo de o ) Bi Bis a closed se . Using his de ini ion, Vo onoi diag ams ha e been eadjus ed o
packings o ellipses and sphe ocylinde s, which a e elonga ed shapes. Examples o hese shapes can be seen in Figu es 4
and 5, which ha e he Vo onoi diag am o he le and he co esponding modi ied essella ion o he igh .
Ellipses a e a e y common conic sec ion, while a sphe ocylinde is a capsule-like body de e mined by a line segmen
and a posi i e eal numbe called adius. I is de ined as he se o all poin s ha lie a a dis ance om he segmen equal
o o smalle han he adius. Exp ession (1) o ellipses can be ound by minimiza ion.16 On he o he hand, he dis ance
om a poin o a sphe ocylinde is he dis ance om he poin o he line segmen ha de e mines he sphe ocylinde ,
minus i s adius.
Despi e all he 2D examples in his pape a e based on he da a o Figu es 4 and 5, whe e equal pa icles ha e o ien a-
ions dis ibu ed acco ding o he con inuous uni o m dis ibu ion on in e al [0,2𝜋], i is no p oblem a all o base he
analysis on di e en sized pa icles and o he ypes o o ien a ions, such as in Figu e 6. High densi y packings wi h a bi-
a y sizes and o ien a ions can be ela i ely easily ob ained wi h ad ancing on packing algo i hms such as hose in he
wo ks o Mo ales e al12 and Hohenwa e and Pippan.16
PÉREZ ET AL.5
FIGURE 5 Le : Vo onoi diag am wi h espec o he cen e s o a se o sphe ocylinde s. Righ : essella ion wi h cells con aining he
sphe ocylinde s
FIGURE 6 Tessella ion con aining di e en sized sphe ocylinde s wi h a ai ly common o ien a ion
4QUANTIFICATION OF THE RESEMBLANCE BETWEEN THE SHAPES OF
A PACKING IN 2D AND THE CELLS OF A TESSELLATION
I may be use ul o e alua e he esemblance be ween he cells o a essella ion and he pa icles hey con ain. This helps
choosing he mos app op ia e body shapes o gene a e he packing and con ibu es o make he subsequen physical
simula ions mo e ealis ic. One way o measu e he esemblance be ween wo se s Aand Bcan be he Jacca d simila i y
coe icien ,17 which is de ined by he exp ession
J(A,B)=M(A∩B)
M(A∪B),(2)
whe e M(X) deno es he a ea o olume o se Xin case X∈R2o X∈R3, espec i ely. The componen s o he Jcoe icien
a e illus a ed in Figu e 7, o an ellip ical g ain Ao e lapping a hexagonal cell B. O cou se, simila i y Jshould be equal
o o smalle han one and g ea e han ze o.
6PÉREZ ET AL.
FIGURE 7 Componen s o he Jacca d coe icien . Two bodies Aand B(le ); hei in e sec ion A∩B(cen e ) and hei union A∪B( igh )
FIGURE 8 Powe diag am buil om an in ini e hexagonal a angemen o disks
A use ul benchma k o assessing a essella ion in 2D can be he powe diag am buil om an in ini e honeycomb
hexagonal a angemen o equal disks (Figu e 8), which has, acco ding o o mula (2), a simila i y o 𝜋
2√3=0.91 be ween
each disk and he cell ha con ains i . Such packing has he highes possible simila i y o equal disks in he plane.18 In he
analogous 3D case, he maximum possible simila i y is 0.74, acco ding o Keple 's conjec u e,19 being he cells i egula
dodecahed a. This maximum simila i y alue can be achie ed wi h bo h he cubic close packing20 and he hexagonal close
packing21 (Figu e 9 le and igh , espec i ely).
Le Vi,i=1,n, be he Vo onoi cells con aining he cen e s o a se o disjoin bodies Bi, espec i ely, and le Libe he
cell in a essella ion ob ained om Vi o each i∈{1, …,n}, espec i ely, as explained in he p e ious sec ion. Then, i
makes sense o expec in p ac ice he ollowing inequali ies o hold:
0<J(Vi,Bi)<J(Li,Bi)<Jmax ={0.91,i he space dimension is 2
0.74,i he space dimension is 3 <1,(3)
which means ha he simila i y be ween a cell in he modi ied Vo onoi essella ion and i s associa ed body should usually
be g ea e han he simila i y be ween ha same body and he associa ed Vo onoi cell bu should usually be smalle han
he simila i y be ween he cells and he espec i e bodies in a hexagonal packing. An illus a ion o his idea in 2D can be
seen in Figu e 10, which shows, o he le , a Vo onoi cell Vicon aining he cen e o an ellipse Bi, in he cen e , a cell Li
o he essella ion ob ained om he Vo onoi diag am, con aining Bi, and o he igh , a hexagonal cell con aining a disk.
Howe e , he hi d inequali y om le o igh in (3) migh no hold in some cases, gi en ha some shapes could be mo e
adjus ed o he con aining cell han a ci cle o a sphe e in a la ice packing is, and ha is why a ew spikes in Figu es 12,
13, and 15 a e abo e he h esholds o 0.91 (2D cases) o 0.74 (3D case).
PÉREZ ET AL.7
FIGURE 9 Samples o he in ini e cubic close packing (le ) and hexagonal close packing ( igh ) o sphe es, wi h hei espec i e Vo onoi
essella ion o i egula dodecahed a cells
<<
BiVi
Li
V0
BiB0
0 < J(Vi, Bi) < J(Li, Bi) < 0.91
FIGURE 10 Simila i y be ween cells and bodies in Vo onoi and powe diag ams
Cell numbe
025
0.6
0.7
0.8
0.91
50 75 100
Simila i y o ellipses
Vo onoi
Mod. Vo .
FIGURE 11 Simila i y alues o he packing o ellipses, wi h he cases o he o iginal Vo onoi diag am (dashed line) and he essella ion
ob ained by modi ying i (con inuous line)
Fo each cell in Figu es 4 and 5, he simila i y coe icien be ween he cell and he co esponding ellipse o sphe ocylin-
de was calcula ed. Such calcula ions we e ca ied ou wi h Mon e Ca lo me hods22 o simplici y, and he esul s can be
seen in Figu es 11 and 12 o ellipses and sphe ocylinde s, espec i ely. In hese wo igu es, a simila i y coe icien in
he modi ied Vo onoi diag am, which is g ea e han he simila i y coe icien o he co esponding Vo onoi cell, can be
obse ed in mos cases, as expec ed. The alidi y o inequali ies (3) can also be obse ed o almos all cells.
The simila i y alues o he modi ied Vo onoi diag ams a e appa en ly g ea e han he analogous alues in he unmodi-
ied Vo onoi diag ams, as can be seen in Figu es 11, 12, and 15, and he same hing happens be ween he simila i y alues
o sphe ocylinde s and ellipses in Figu e 13. We ha e chosen he Wilcoxon's signed ank es o con i m hese di e ences,
because he dis ibu ions o simila i y alues a e skewed.
8PÉREZ ET AL.
Cell numbe
025
0.6
0.7
0.8
0.91
50 75 100
Simila i y o sphe ocylinde s
Vo onoi
Mod. Vo .
FIGURE 12 Simila i y alues o he packing o sphe ocylinde s, wi h he cases o he o iginal Vo onoi diag am (dashed line) and he
essella ion ob ained by modi ying i (con inuous line)
Cell numbe
025
0.6
0.7
0.8
0.91
50 75 100
Simila i y wi h mod. Vo onoi
Ellipses
Sphe oc.
FIGURE 13 Simila i y alues o he modi ied Vo onoi diag ams in 2D (ellipses and sphe ocylinde s)
Acco ding o The Camb idge Dic iona y o S a is ics,23 “ he Wilcoxon's signed ank es is a dis ibu ion ee me hod o
es ing he di e ence be ween wo popula ions using ma ched samples. The es is based on he absolu e di e ences o
he pai s o obse a ions in he wo samples, anked acco ding o size, wi h each ank being gi en he sign o he o iginal
di e ence. The es s a is ic is he sum o he posi i e anks.” In he compu a ional implemen a ion o he Wilcoxon's
signed ank es used in his pape ,24 a small p- alue co esponding o he es s a is ic implies a high p obabili y o he
popula ions being di e en .
The obse ed di e ence be ween he simila i y alues in he non-Vo onoi essella ion and hei co esponding Vo onoi
simila i y alues is suppo ed by he (pai ed) Wilcoxon's signed ank es p- alues o 3.13 ×10−13 and 2.2 ×10−16 o
he ellipses and sphe ocylinde s, espec i ely, being he null hypo hesis ha he dis ibu ions o simila i y in Vo onoi
and Lague e diag ams ha e he same loca ion, wi h he al e na i e hypo hesis ha he loca ion o he dis ibu ion
co esponding o Vo onoi is smalle han he dis ibu ion co esponding o Lague e.
One mo e (unpai ed) Wilcoxon's signed ank es was also ca ied ou , in o de o know i he e is any signi ican
di e ence be ween he simila i y alues in he non-Vo onoi essella ions ( hose alues can be seen in Figu e 13). Such
di e ence can be conside ed signi ican , since 8.93 ×10−13 is he p- alue ob ained in he es .
5SIMPLE 3D EXAMPLE
The o mula ions in he p e ious sec ion canno be easily ex ended o 3D. Howe e , o he sake o comple eness, a simple
example wi h sphe es is p esen ed (Figu e 14). The sphe es a e con ained in a cube and ha e adii dis ibu ed acco ding
o he con inuous uni o m dis ibu ion in he in e al [1,2]. F om hem, a Vo onoi and a Lague e diag am we e ob ained
wi h he Vo o++ lib a y.7Analogously o he 2D examples, he simila i y alues (Figu e 15) a e also s a is ically di e en .
PÉREZ ET AL.9
FIGURE 14 Packing o sphe es wi h Lague e cells supe imposed [Colou igu e can be iewed a wileyonlinelib a y.com]
Cell numbe
0200
0.2
0.4
0.6
0.8
0.74
50 150
100
Simila i y o sphe es
Vo onoi
Mod. Vo .
FIGURE 15 Simila i y alues o he packing o sphe es, wi h he cases o he Vo onoi diag am (dashed line) and he powe diag am
(con inuous line)
Despi e he simila i y alues in he las Lague e diag am can be conside ed good (no ice ha he e inequali ies (3)
also hold in mos cases), an ex ension o 3D o he o mula ions o ellipses and sphe ocylinde s should be ca ied ou in
u u e wo ks. This is because, as p e iously men ioned, sphe es migh no be good enough o app oxima e g ain shapes
such as he ones in Figu e 3. The cu es co esponding o he Lague e and Vo onoi simila i y alues in Figu e 15 a e
no as dis inc as he cu es in he analogous Figu es 11 and 12. This ma ches he ac ha he p- alue ob ained wi h he
Wilcoxon's es , equal o 2.34 ×10−10, is in he o de o 1000 imes g ea e han in he o he es s.
6DISCUSSION
In his sec ion, we discuss some issues ega ding he alidi y and scope o ou app oach. I mus be said ha ou esul s
allow o ob ain essella ions in o de o ca y ou physical simula ions as in he wo ks o Hi i2and Benabbou e al.25 The
e o o quali y o he ob ained packings and essella ions can be es ima ed by measu ing p ope ies such as homogenei y,
