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J. Eliáš, Frattura ed Integrità Strutturale, 39 (2017) 1-6; DOI: 10.3221/IGF-ESIS.39.01

1 

Focussed on Modelling in Mechanics 

On adaptive refinements in discrete probabilistic fracture models 

J. Eliáš
Brno University of Technology, Faculty of Civil Engineering, Veveří 331/95, Brno, 60200, Czech Republic
elias.j@fce.vutbr.cz

ABSTRACT. The possibility to adaptively change discretization density is a 
well acknowledged and used feature of many continuum models. It is 
employed to save computational time and increase solution accuracy. 
Recently, adaptivity has been introduced also for discrete particle models. 
This contribution applies adaptive technique in probabilistic discrete 
modelling where material properties are varying in space according to a 
random field. The random field discretization is adaptively refined hand in 
hand with the model geometry. 

KEYWORDS. Adaptivity; Discrete model; Probability; Random field. 

Citation: Eliáš, J., On adaptive refinements in 
discrete probabilistic fracture models, Frattura 
ed Integrità Strutturale, 39 (2017) 1-6. 

Received: 11.07.2016  
Accepted: 12.09.2016  
Published: 01.01.2017  

Copyright: © 2017 This is an open access 
article under the terms of the CC-BY 4.0, 
which permits unrestricted use, distribution, 
and reproduction in any medium, provided 
the original author and source are credited. 

INTRODUCTION 
 

he adaptivity of model geometry has been originally developed for elastic problems [1,2] and later applied also in 
inelastic problems with localization [3,4]. The classical rigorous approach involves an error estimation, remeshing 
criterion, mesh re-generation and transfer of variables onto the new mesh. Recently, the adaptive concept was 

applied also in discrete modelling [5]. The goal of this work is to extend it for probabilistic discrete models. 
Discrete models represent the material via collection of interconnected rigid bodies organized into a net structure. There 
are several versions of discrete models developed and used for many purposes. In case of simulating fracture in concrete, 
the lattice models are often employed [6-8]. These models represent the concrete meso-structure by projecting it onto the 
independently generated lattice. They are excellent in describing fracture phenomena, but applicable only for small 
laboratory specimens due to their extreme computational demands. Another group of discrete meso-level modelling 
approaches, sometimes called particle models, generates the network geometry directly according to the meso-structure of 
concrete [9,10]; typically one node for each mineral aggregate. We focus here on the latter group with geometry generated 
via Voronoi tessellation [11-14]. 
Though some reduction of computational cost in particle models is achieved when compared to the lattice models, further 
reduction would be desirable. It can be done by adaptive construction of the discrete geometry as described in [5]. 
Availability of adaptive refinement allows starting simulation with coarse discretization and refining it adaptively during 
the simulation run only in areas where needed.  

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J. Eliáš, Frattura ed Integrità Strutturale, 39 (2017) 1-6; DOI: 10.3221/IGF-ESIS.39.01                                                                                      
 

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In some applications of fracture simulations, it might be important to consider additional material randomness (besides 
the one covered by the random location of nodes in the discrete model) usually represented by a random field [15-18]. An 
extension of the discrete model by fluctuation of material parameters according to a random field was developed in 
[18,19]. In this contribution, the adaptive concept is extended for such probabilistic discrete fracture models. 
 
 
PROBABILISTIC DISCRETE MODEL 
 

he model uses random geometry to avoid directional bias that occurs in any regular structure. Domain of the 
modeled body is filled with nuclei with randomly generated positions. These nuclei are added sequentially with 
restricted minimal distance lmin. The parameter lmin controls size of the discrete bodies and therefore it should 

correspond to the size of heterogeneities in the material. In concrete, this is typically a size of the mineral aggregates. Each 
of the nuclei will serve as one model node with associated six degrees of freedom, three translational and three rotational. 
The connectivity of the nodes is given by Delaunay triangulation. Dual diagram of Delaunay triangulation called Voronoi 
tessellation then creates geometry of the rigid bodies. Rigid bodies have common contact facets, which are perpendicular 
to their connections because of the Voronoi tessellation properties. There is a complex damage-mechanics based 
constitutive law used at the facets. Its deterministic version has been adapted from [9], where it is also described in detail. 
The main material parameters for fracture behavior are tensile strength, ft, and tensile fracture energy, GF.  
The probabilistic extension of the model is elucidated in [19,20]. Here, only brief description of the probabilistic part is 
given. Both the tensile strength and fracture energy in tension are assumed to be governed by single random field H with 
mean value 1 and probabilistic distribution with Gaussian core and Weibull left tail. The correlation structure of the 
random field is given by square exponential function with single parameter, lρ, called the correlation length.   
The strength and fracture energy of every model contact with centroid c  are given by  
 

   f f Ht t c c                (1) 
  

   G G H
2

F F    c c  
 
with X  being the mean value of the material parameter X . The square in the equation for fracture energy is added to 
preserve constant material characteristic length [20]. In the adaptive model, new contacts are created after every 
refinement. Therefore, the random field values at the new contact centers must be generated after every refinement. This 

is effectively done using kriging. Initially, standard Gaussian random field realizations ( Ĥ ) are generated on points 
arranged in a regular grid with spacing lρ/4. Random field value at point  c is then estimated using the optimal linear 
estimation method [21]  
 

 
K

Tk
k cg

kk

H
1

ˆ 


c ψ C           (2) 

            

and finally standard Gaussian field is transformed onto the Weibull-Gauss field ( H Hˆ  ) using isoprobabilistic 
transformation. Vector ξ  collects realizations of K independent standard Gaussian variables, λ and ψ are K eigenvalues 
and eigenvectors of the grid covariance matrix and cgC  is the covariance vector between the grid points and point c .  
 
 
ADAPTIVITY 
 

nly brief description of the adaptive concept in deterministic model is given. Deep elucidation is provided in [5]. 
The refinement criterion is intuitive. It is based on an average stress in the rigid bodies calculated using the 
fabric stress tensor. For rigid body associated with node i, the average stress components st   are   

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                                                                                   J. Eliáš, Frattura ed Integrità Strutturale, 39 (2017) 1-6; DOI: 10.3221/IGF-ESIS.39.01 
 

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   j j

st s t
j

F c
V

1             (3) 

 
where j runs over all nodes in contact with node i, F is a vector of contact force,  c is the centroid of the contact facet and 
V is a volume of the i-th rigid body. The Mazar's equivalent stress serves as measure of the stress level, σeq.  

 
 

 
 

Figure 1: Adaptive refinement of discretization in steps; a) schematic explanation; b)-g) application to a 2D model. 
 
 

In probabilistic model, the contacts have random strength. Assuming that the random field does not change too much 
within one discrete body of the model ( lρ<lmin), it is reasonable to estimate strength of hypothetical newly created contacts 
within the i-th coarse discrete body by strength at node i at coordinates ix .  
After every solution step, average stress tensors in all rigid bodies belonging to coarse discretization are evaluated and the 
Mazar‘s equivalent stresses are calculated. The refinement takes place whenever the equivalent stress exceeds chosen 
strength level γ 
 

 if
eq

t




x
           (4) 

 
The node associated with the rigid body satisfying Eq. (4) serves as a center of the refinement sphere. The safe value of 
parameter γ was determined as 0.7, i.e. whenever equivalent stress reaches 70% of the tensile strength, the refinement 
takes place.  
The refinement is sketched in Fig. 1. All the nuclei inside the refinement sphere that does not belong to the fine 
discretization are removed. New nuclei are added into the refinement sphere according to the sequential algorithm 
described in the previous section. The parameter lmin controlling the discretization density changes based on two 
additional length parameters, rf and rc. The linear transition from coarse (lmin=lc) to fine (lmin=lf) discretization is included 
within the circular ring of outer (inner) radius rc (rf) in order to minimize the shape distortion of the bodies. If the 
transitional regime is omitted, the sharp change in discretization density would produce significantly elongated body 
shapes inducing directional bias and anisotropy.  
 



 

J. Eliáš, Frattura ed Integrità Strutturale, 39 (2017) 1-6; DOI: 10.3221/IGF-ESIS.39.01                                                                                      
 

4 
 

 
NUMERICAL EXAMPLE 
 

erformance of the proposed adaptive algorithm is demonstrated on simulation of four-point bending test with 
incorporated material randomness. The computer code used for calculation is an in-house software. The beam 
geometry is shown in Fig 2. The deterministic model parameters were taken from simulation of experimental 

series in three-point bending [22]. The average tensile strength is ft=2.2 MPa, fracture energy in tension is GF=35 J/m2 and 
elastic modulus is 60 GPa. All these parameters are applied on the meso-level, they are not equal to the corresponding 
macroscopic properties of the model. The adaptive algorithm uses the following parameters: rf=60 mm, rc=120 mm, lf=10 
mm and lc=30 mm. The parameters of the probabilistic extension are arbitrarily chosen according to [19]. The correlation 
length and the coefficient of variation of the random field is 80 mm and 0.25, respectively.  
Three model types are used: (i) the fine model, that uses fine discretization everywhere from the beginning; (ii) the coarse 
model, that uses coarse discretization all the time; and (iii) the adaptive model, that starts with coarse discretization and 
refines it adaptively.  
Fig. 3 shows on the left-hand side identical responses of one simulation using the fine model and one simulation using the 
adaptive model with the same refined meso-structure and also the same random field realization. The resulting crack 
patterns as well as the random field applied are shown in Fig. 4.  
 
 

 
 

Figure 2: Dimensions of the simulated beam loaded in four-point bending. 
 
 

 
 

Figure 3: Left: Load-displacement response of one four-point-bending test simulation using the fine model and the adaptive model 
with the same refined meso-structure. Right: Average response of 30 simulation of four-point-bending test. 
 
 
All three model types were then compared statistically. The same 30 realizations of the random field were used for every 
model type. The average responses together with standard deviations are shown in Fig. 3 on the right hand side. The fine 
and the adaptive model exhibit the same behavior while the coarse model deviates from them. In average, the 
computational time consumed by the adaptive model was only 47% of the time consumed by the fine model. 
 

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                                                                                   J. Eliáš, Frattura ed Integrità Strutturale, 39 (2017) 1-6; DOI: 10.3221/IGF-ESIS.39.01 
 

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Figure 4: Damage patterns and random field discretizations developed during the simulation of four-point bending test for the 
adaptive and fine model.  
 
 
CONCLUSIONS 
 

he probabilistic discrete model has been extended by an adaptive technique that allows significant reduction of the 
computational time with no effect on the obtained results. The probabilistic model had both fracture energy and 
tensile strength assigned according to the random field. The random field discretization was adaptively refined on 

the run hand to hand with the discretization of the model. The adaptive algorithm was verified by simulating four-point 
bending test. 
Usage of the adaptive concept is limited to the specific types of the discrete models that have elastic behavior independent 
on discretization density. Moreover, the presented concept is available only for static models. In dynamics, the translations 
and rotations and their first and second order derivatives cannot be computed from scratch and needs to be somehow 
estimated from replaced coarse mesh via some transfer algorithm.  

 
 
ACKNOWLEDGEMENT 
 

he financial support provided by the Ministry of Education, Youth and Sports of the Czech Republic under the 
project LO1408 ‘‘AdMaS UP - advanced Materials, Structures and Technologies’’ under ‘‘National Sustainability 
Programme I’’ is gratefully acknowledged. 

 
 

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J. Eliáš, Frattura ed Integrità Strutturale, 39 (2017) 1-6; DOI: 10.3221/IGF-ESIS.39.01                                                                                      
 

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    /HRV (Za stvaranje Adobe PDF dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke.  Stvoreni PDF dokumenti mogu se otvoriti Acrobat i Adobe Reader 5.0 i kasnijim verzijama.)
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    /NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
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    /UKR <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>
    /ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing.  Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
  >>
  /Namespace [
    (Adobe)
    (Common)
    (1.0)
  ]
  /OtherNamespaces [
    <<
      /AsReaderSpreads false
      /CropImagesToFrames true
      /ErrorControl /WarnAndContinue
      /FlattenerIgnoreSpreadOverrides false
      /IncludeGuidesGrids false
      /IncludeNonPrinting false
      /IncludeSlug false
      /Namespace [
        (Adobe)
        (InDesign)
        (4.0)
      ]
      /OmitPlacedBitmaps false
      /OmitPlacedEPS false
      /OmitPlacedPDF false
      /SimulateOverprint /Legacy
    >>
    <<
      /AddBleedMarks false
      /AddColorBars false
      /AddCropMarks false
      /AddPageInfo false
      /AddRegMarks false
      /ConvertColors /ConvertToCMYK
      /DestinationProfileName ()
      /DestinationProfileSelector /DocumentCMYK
      /Downsample16BitImages true
      /FlattenerPreset <<
        /PresetSelector /MediumResolution
      >>
      /FormElements false
      /GenerateStructure false
      /IncludeBookmarks false
      /IncludeHyperlinks false
      /IncludeInteractive false
      /IncludeLayers false
      /IncludeProfiles false
      /MultimediaHandling /UseObjectSettings
      /Namespace [
        (Adobe)
        (CreativeSuite)
        (2.0)
      ]
      /PDFXOutputIntentProfileSelector /DocumentCMYK
      /PreserveEditing true
      /UntaggedCMYKHandling /LeaveUntagged
      /UntaggedRGBHandling /UseDocumentProfile
      /UseDocumentBleed false
    >>
  ]
>> setdistillerparams
<<
  /HWResolution [2400 2400]
  /PageSize [612.000 792.000]
>> setpagedevice