J. Eliáš, F a u a ed In eg i à S u u ale, 39 (2017) 1-6; DOI: 10.3221/IGF-ESIS.39.01
1
Focussed on Modelling in Mechanics
On adap i e e inemen s in disc e e p obabilis ic ac u e models
J. Eliáš
B no Uni e si y o Technology, Facul y o Ci il Enginee ing, Ve eří 331/95, B no, 60200, Czech Republic
elias.[email p o ec ed] b .cz
ABSTRACT. The possibili y o adap i ely change disc e iza ion densi y is a
well acknowledged and used ea u e o many con inuum models. I is
employed o sa e compu a ional ime and inc ease solu ion accu acy.
Recen ly, adap i i y has been in oduced also o disc e e pa icle models.
This con ibu ion applies adap i e echnique in p obabilis ic disc e e
modelling whe e ma e ial p ope ies a e a ying in space acco ding o a
andom ield. The andom ield disc e iza ion is adap i ely e ined hand in
hand wi h he model geome y.
KEYWORDS. Adap i i y; Disc e e model; P obabili y; Random ield.
Ci a ion: Eliáš, J., On adap i e e inemen s in
disc e e p obabilis ic ac u e models, F a u a
ed In eg i à S u u ale, 39 (2017) 1-6.
Recei ed: 11.07.2016
Accep ed: 12.09.2016
Published: 01.01.2017
Copy igh : © 2017 This is an open access
a icle unde he e ms o he CC-BY 4.0,
which pe mi s un es ic ed use, dis ibu ion,
and ep oduc ion in any medium, p o ided
he o iginal au ho and sou ce a e c edi ed.
INTRODUCTION
he adap i i y o model geome y has been o iginally de eloped o elas ic p oblems [1,2] and la e applied also in
inelas ic p oblems wi h localiza ion [3,4]. The classical igo ous app oach in ol es an e o es ima ion, emeshing
c i e ion, mesh e-gene a ion and ans e o a iables on o he new mesh. Recen ly, he adap i e concep was
applied also in disc e e modelling [5]. The goal o his wo k is o ex end i o p obabilis ic disc e e models.
Disc e e models ep esen he ma e ial ia collec ion o in e connec ed igid bodies o ganized in o a ne s uc u e. The e
a e se e al e sions o disc e e models de eloped and used o many pu poses. In case o simula ing ac u e in conc e e,
he la ice models a e o en employed [6-8]. These models ep esen he conc e e meso-s uc u e by p ojec ing i on o he
independen ly gene a ed la ice. They a e excellen in desc ibing ac u e phenomena, bu applicable only o small
labo a o y specimens due o hei ex eme compu a ional demands. Ano he g oup o disc e e meso-le el modelling
app oaches, some imes called pa icle models, gene a es he ne wo k geome y di ec ly acco ding o he meso-s uc u e o
conc e e [9,10]; ypically one node o each mine al agg ega e. We ocus he e on he la e g oup wi h geome y gene a ed
ia Vo onoi essella ion [11-14].
Though some educ ion o compu a ional cos in pa icle models is achie ed when compa ed o he la ice models, u he
educ ion would be desi able. I can be done by adap i e cons uc ion o he disc e e geome y as desc ibed in [5].
A ailabili y o adap i e e inemen allows s a ing simula ion wi h coa se disc e iza ion and e ining i adap i ely du ing
he simula ion un only in a eas whe e needed.
T
J. Eliáš, F a u a ed In eg i à S u u ale, 39 (2017) 1-6; DOI: 10.3221/IGF-ESIS.39.01
2
In some applica ions o ac u e simula ions, i migh be impo an o conside addi ional ma e ial andomness (besides
he one co e ed by he andom loca ion o nodes in he disc e e model) usually ep esen ed by a andom ield [15-18]. An
ex ension o he disc e e model by luc ua ion o ma e ial pa ame e s acco ding o a andom ield was de eloped in
[18,19]. In his con ibu ion, he adap i e concep is ex ended o such p obabilis ic disc e e ac u e models.
PROBABILISTIC DISCRETE MODEL
he model uses andom geome y o a oid di ec ional bias ha occu s in any egula s uc u e. Domain o he
modeled body is illed wi h nuclei wi h andomly gene a ed posi ions. These nuclei a e added sequen ially wi h
es ic ed minimal dis ance lmin. The pa ame e lmin con ols size o he disc e e bodies and he e o e i should
co espond o he size o he e ogenei ies in he ma e ial. In conc e e, his is ypically a size o he mine al agg ega es. Each
o he nuclei will se e as one model node wi h associa ed six deg ees o eedom, h ee ansla ional and h ee o a ional.
The connec i i y o he nodes is gi en by Delaunay iangula ion. Dual diag am o Delaunay iangula ion called Vo onoi
essella ion hen c ea es geome y o he igid bodies. Rigid bodies ha e common con ac ace s, which a e pe pendicula
o hei connec ions because o he Vo onoi essella ion p ope ies. The e is a complex damage-mechanics based
cons i u i e law used a he ace s. I s de e minis ic e sion has been adap ed om [9], whe e i is also desc ibed in de ail.
The main ma e ial pa ame e s o ac u e beha io a e ensile s eng h, , and ensile ac u e ene gy, GF.
The p obabilis ic ex ension o he model is elucida ed in [19,20]. He e, only b ie desc ip ion o he p obabilis ic pa is
gi en. Bo h he ensile s eng h and ac u e ene gy in ension a e assumed o be go e ned by single andom ield H wi h
mean alue 1 and p obabilis ic dis ibu ion wi h Gaussian co e and Weibull le ail. The co ela ion s uc u e o he
andom ield is gi en by squa e exponen ial unc ion wi h single pa ame e , lρ, called he co ela ion leng h.
The s eng h and ac u e ene gy o e e y model con ac wi h cen oid c a e gi en by
 
H
cc
(1)
 
GGH
2
FF


cc
wi h X being he mean alue o he ma e ial pa ame e X. The squa e in he equa ion o ac u e ene gy is added o
p ese e cons an ma e ial cha ac e is ic leng h [20]. In he adap i e model, new con ac s a e c ea ed a e e e y
e inemen . The e o e, he andom ield alues a he new con ac cen e s mus be gene a ed a e e e y e inemen . This
is e ec i ely done using k iging. Ini ially, s anda d Gaussian andom ield ealiza ions ( H
ˆ) a e gene a ed on poin s
a anged in a egula g id wi h spacing lρ/4. Random ield alue a poin cis hen es ima ed using he op imal linea
es ima ion me hod [21]

KT
kkcg
k
k
H
1
ˆ




cψC (2)
and inally s anda d Gaussian ield is ans o med on o he Weibull-Gauss ield ( HH
ˆ) using isop obabilis ic
ans o ma ion. Vec o ξ collec s ealiza ions o K independen s anda d Gaussian a iables, λ and
ψ
a e K eigen alues
and eigen ec o s o he g id co a iance ma ix and c
g
C is he co a iance ec o be ween he g id poin s and poin c.
ADAPTIVITY
nly b ie desc ip ion o he adap i e concep in de e minis ic model is gi en. Deep elucida ion is p o ided in [5].
The e inemen c i e ion is in ui i e. I is based on an a e age s ess in he igid bodies calcula ed using he
ab ic s ess enso . Fo igid body associa ed wi h node i, he a e age s ess componen s s 

a e
T
O
J. Eliáš, F a u a ed In eg i à S u u ale, 39 (2017) 1-6;
DOI: 10.3221/IGF-ESIS.39.01
3

jj
s s
j
Fc
V
1



(3)
whe e j uns o e all nodes in con ac wi h node i,
F
is a ec o o con ac o ce,
c
is he cen oid o he con ac ace and
V is a olume o he i- h igid body. The Maza 's equi alen s ess se es as measu e o he s ess le el, σ
eq
.
Figu e 1: Adap i e e inemen o disc e iza ion in s eps; a) schema ic explana ion; b)-g) applica ion o a 2D model.
In p obabilis ic model, he con ac s ha e andom s eng h. Assuming ha he andom ield does no change oo much
wi hin one disc e e body o he model ( l
ρ
<l
min
), i is easonable o es ima e s eng h o hypo he ical newly c ea ed con ac s
wi hin he i- h coa se disc e e body by s eng h a node i a coo dina es
i
x
.
A e e e y solu ion s ep, a e age s ess enso s in all igid bodies belonging o coa se disc e iza ion a e e alua ed and he
Maza ‘s equi alen s esses a e calcula ed. The e inemen akes place whene e he equi alen s ess exceeds chosen
s eng h le el γ

i
eq


x
(4)
The node associa ed wi h he igid body sa is ying Eq. (4) se es as a cen e o he e inemen sphe e. The sa e alue o
pa ame e γ was de e mined as 0.7, i.e. whene e equi alen s ess eaches 70% o he ensile s eng h, he e inemen
akes place.
The e inemen is ske ched in Fig. 1. All he nuclei inside he e inemen sphe e ha does no belong o he ine
disc e iza ion a e emo ed. New nuclei a e added in o he e inemen sphe e acco ding o he sequen ial algo i hm
desc ibed in he p e ious sec ion. The pa ame e l
min
con olling he disc e iza ion densi y changes based on wo
addi ional leng h pa ame e s,
and
c
. The linea ansi ion om coa se (l
min
=l
c
) o ine (l
min
=l
) disc e iza ion is included
wi hin he ci cula ing o ou e (inne ) adius
c
(
) in o de o minimize he shape dis o ion o he bodies. I he
ansi ional egime is omi ed, he sha p change in disc e iza ion densi y would p oduce signi ican ly elonga ed body
shapes inducing di ec ional bias and aniso opy.
J. Eliáš, F a u a ed In eg i à S u u ale, 39 (2017) 1-6;
DOI: 10.3221/IGF-ESIS.39.01
4
N
UMERICAL EXAMPLE
e o mance o he p oposed adap i e algo i hm is demons a ed on simula ion o ou -poin bending es wi h
inco po a ed ma e ial andomness. The compu e code used o calcula ion is an in-house so wa e. The beam
geome y is shown in Fig 2. The de e minis ic model pa ame e s we e aken om simula ion o expe imen al
se ies in h ee-poin bending [22]. The a e age ensile s eng h is
=2.2 MPa, ac u e ene gy in ension is G
F
=35 J/m
2
and
elas ic modulus is 60 GPa. All hese pa ame e s a e applied on he meso-le el, hey a e no equal o he co esponding
mac oscopic p ope ies o he model. The adap i e algo i hm uses he ollowing pa ame e s:
=60 mm,
c
=120 mm, l
=10
mm and l
c
=30 mm. The pa ame e s o he p obabilis ic ex ension a e a bi a ily chosen acco ding o [19]. The co ela ion
leng h and he coe icien o a ia ion o he andom ield is 80 mm and 0.25, espec i ely.
Th ee model ypes a e used: (i) he ine model, ha uses ine disc e iza ion e e ywhe e om he beginning; (ii) he coa se
model, ha uses coa se disc e iza ion all he ime; and (iii) he adap i e model, ha s a s wi h coa se disc e iza ion and
e ines i adap i ely.
Fig. 3 shows on he le -hand side iden ical esponses o one simula ion using he ine model and one simula ion using he
adap i e model wi h he same e ined meso-s uc u e and also he same andom ield ealiza ion. The esul ing c ack
pa e ns as well as he andom ield applied a e shown in Fig. 4.
Figu e 2: Dimensions o he simula ed beam loaded in ou -poin bending.
Figu e 3: Le : Load-displacemen esponse o one ou -poin -bending es simula ion using he ine model and he adap i e model
wi h he same e ined meso-s uc u e. Righ : A e age esponse o 30 simula ion o ou -poin -bending es .
All h ee model ypes we e hen compa ed s a is ically. The same 30 ealiza ions o he andom ield we e used o e e y
model ype. The a e age esponses oge he wi h s anda d de ia ions a e shown in Fig. 3 on he igh hand side. The ine
and he adap i e model exhibi he same beha io while he coa se model de ia es om hem. In a e age, he
compu a ional ime consumed by he adap i e model was only 47% o he ime consumed by he ine model.
P
J. Eliáš, F a u a ed In eg i à S u u ale, 39 (2017) 1-6;
DOI: 10.3221/IGF-ESIS.39.01
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Figu e 4: Damage pa e ns and andom ield disc e iza ions de eloped du ing he simula ion o ou -poin bending es o he
adap i e and ine model.
C
ONCLUSIONS
he p obabilis ic disc e e model has been ex ended by an adap i e echnique ha allows signi ican educ ion o he
compu a ional ime wi h no e ec on he ob ained esul s. The p obabilis ic model had bo h ac u e ene gy and
ensile s eng h assigned acco ding o he andom ield. The andom ield disc e iza ion was adap i ely e ined on
he un hand o hand wi h he disc e iza ion o he model. The adap i e algo i hm was e i ied by simula ing ou -poin
bending es .
Usage o he adap i e concep is limi ed o he speci ic ypes o he disc e e models ha ha e elas ic beha io independen
on disc e iza ion densi y. Mo eo e , he p esen ed concep is a ailable only o s a ic models. In dynamics, he ansla ions
and o a ions and hei i s and second o de de i a i es canno be compu ed om sc a ch and needs o be somehow
es ima ed om eplaced coa se mesh ia some ans e algo i hm.
A
CKNOWLEDGEMENT
he inancial suppo p o ided by he Minis y o Educa ion, You h and Spo s o he Czech Republic unde he
p ojec LO1408 ‘‘AdMaS UP - ad anced Ma e ials, S uc u es and Technologies’’ unde ‘‘Na ional Sus ainabili y
P og amme I’’ is g a e ully acknowledged.
T
T

J. Eliáš, F a u a ed In eg i à S u u ale, 39 (2017) 1-6; DOI: 10.3221/IGF-ESIS.39.01
6
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