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Combination of an adaptive remeshing technique

with a coupled FEM-DEM approach for analysis of

crack propagation problems

Alejandro Cornejo ·Vicente Mataix ·

Francisco Zárate ·Eugenio Oñate

Received: date / Accepted: date

Abstract This paper presents an enhanced coupled approach between the

Finite Element Method (FEM) and the Discrete Element Method (DEM) in

which an adaptative remeshing technique has been implemented. The remesh-

ing technique is based on the computation of the Hessian of a selected nodal

variable, i.e. the mesh is reﬁned where the curvature of the variable ﬁeld is

greater. Once the Hessian is known, a metric tensor is deﬁned node-wise that

serves as input data for the remesher (MmgTools) that creates a new mesh.

After remeshing, the mapping of the internal variables and the nodal values

is performed and a regeneration of the discrete elements on the crack faces

of the new mesh is carried out. Several examples of fracturing problems us-

ing the enhanced FEM-DEM formulation are presented. Accurate results in

comparison with analytical and experimental solutions are obtained.

Keywords Remeshing technique ·Coupled formulation ·Fracture mechanics ·

FEM-DEM ·Finite element method ·Discrete element method

1 Introduction

The modelling and simulation of the mechanical process of fracturing of ma-

terials and structures is one of the most challenging topics in computational

mechanics. The laboratory predictions of the ultimate strength of materials

and the detection/propagation of fractures is also a complex research topic

due to their typical prompt or abrupt behaviour of crack, which diﬃculties

the experimental analysis.

In this paper a novel coupled FEM-DEM formulation is presented. The

method is based on previous works of the authors in coupling the FEM and

International Centre for Numerical Methods in Engineering

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E-mail: acornejo@cimne.upc.edu

2 Alejandro Cornejo et al.

DEM procedures [2][3]. The existing coupled FEM-DEM procedure is en-

hanced with an adaptive remeshing technique based on the Hessian of the

distribution of a nodal variable of interest. Unlike in other fracture mechanics

formulations [4][5][6][7][8][42][43], this methodology only uses the remeshing

technique to improve the quality of the crack path. As shown in [2][3], the

FEM-DEM formulation is capable of obtain accurate, consistent and mesh-

independent results [2][3] due to the use of the super-convergent patch recov-

ery technique [9], this avoids the need of stabilization of the stress ﬁeld, as it

is required in other alternatives [12][13][14].

The selection of a proper nodal variable whose Hessian is going to be

computed is crucial in order to reﬁne the mesh near the crack openings and

not close to the Dirichlet or Newman boundaries.

The enhanced coupled FEM-DEM formulation here presented has been

applied to a collection of benchmark problems. The numerical results presented

in this paper prove the accuracy and correctness of the adaptive remeshing

technique implemented.

2 State of the Art

2.1 State of the art in Fracture Mechanics

In the ﬁeld of continuum mechanics, a fracture can be expressed mathemat-

ically as a discontinuity of the displacement ﬁeld in a certain zone of the

domain. In this way one must separate the body in two domains Ω+and Ω−

and deﬁne the boundaries between them. This approach is called continuum

strong discontinuity. Another option is to regularize the strong discontinuity

by imposing a ﬁnite zone in which the displacement ﬁeld is continuous and the

strains are discontinuous but not inﬁnite [14]. This technique is usually called

continuum smeared approach (Fig. 1).

If one moves on from the continuum to the discretized problem, i.e. to the

FEM, the discrete strong discontinuity technique was introduced by Clough

[15], Ngo and Scordelis [16], Nilson [17]. It consists in separating the elements

that have achieved a certain stress threshold by duplicating the aﬀected nodes

(Fig. 2). This methodology has the drawback that is inherently mesh depen-

dent due to the fact that the propagation directions are the element bound-

aries.

In order to solve the previous limitation, a remeshed strong discontinuity

procedure was introduced by Shephard[4] and Wawrzynek and Ingraﬀea [15]

and improved by Bittencourt et al. [16]. In this case the mesh is reﬁned in

zones near the crack tip so the aﬀected elements are split into smaller ones.

This methodology overcomes part of the problems of the discrete strong dis-

continuity method but it has an additional computational cost regarding the

remeshing procedure. Moreover, the remeshing tool needs to know the prop-

agation direction of the crack, which is diﬃcult to predict using conventional

displacement-based formulations.

Title Suppressed Due to Excessive Length 3

Fig. 1: Discontinuities in a continuum (a) strong discontinuity (b) continuum

smeared approach. Image from [18]

Fig. 2: Discontinuities in a discrete medium (a) strong discontinuity (b) dis-

crete smeared approach. Image from [18]

After analysing those technical issues, the previous formulation lead to the

Extended Finite Element Method (XFEM) [19][20][21][22][23], which uses an

enriched set of shape functions to interpolate the displacement ﬁeld in the

elements crossed by the crack.

As for the remeshing techniques, the embedding of the crack path requires

to know the directions of the evolution of the crack and this is not accurately

predicted in displacement-based formulations. In the XFEM an additional set

of integration quadratures are required to properly integrate the split elements

that, in normal conditions, have a singular stiﬀness. Additionally, these tech-

niques usually require tracking techniques in order to preselect the elements

that are going to be enriched [26].

Anther option is the so-called smeared crack approach. This model was ini-

tially proposed by Rashid [27] and it describes the crack path by a band of

elements whose displacements ﬁeld is continuous and its strain ﬁeld discon-

tinuous but bounded. Fig. 2b shows the ﬁnite elements band aﬀected by the

4 Alejandro Cornejo et al.

weak discontinuities. The smeared crack approach was very popular but after

some years of its adoption Pietruszczak and Mroz [28] and Bazant and Oh

[29] noticed that the fracture process is not only dependant on the fracture

energy, but also on the characteristic length of the mesh analysed. This issue

provokes that the ﬁner the mesh is, the more brittle the behaviour is, which

is an unacceptable inconsistency.

Despite the many diﬀerent discrete weak discontinuities approaches, it was

seen that the standard displacement-based formulations appeared to be inac-

curate when dealing with the onset and propagation of fractures. This is due

to the fact that on the crack tip the strains and stresses are not well predicted

and, in general, the direction of the crack path is strongly dependant on the

mesh orientation.

Regarding the enhanced or enriched ﬁnite element formulations, the mul-

tidimensional generalization of the displacement jump is not straight forward

and it is usually accompanied by ill-conditioning problems. Additionally, since

the mentioned enhancements are applied depending on the stress ﬁeld of the

previous time step and, taking into account that the standard displacement-

based formulations do not guarantee the local convergence of the stress ﬁeld,

these methods often require the use of auxiliary crack tracking techniques.

Subsequently, other FEM formulations were developed in order to provide

computational enhancements, such as mixed formulations. Mixed formulations

are computationally more expensive than the standard ones since they require

to solve multiple unknown ﬁelds. In addition, the ﬁnite elements have to satisfy

the Inf-Sup condition [30][31] to ensure the stability of the solution. This

condition is not easy to fullﬁl and several developments were introduced to

mitigate this problem [32][33].

If one moves on from the methods based on the FEM to the DEM [36][37],

the kind of problematic is intrinsically diﬀerent. There are several approaches

that discretize the continuum as a set of discrete elements (DE) (hereafter

termed particles) attached by a bonding between them ruled by local or non-

local constitutive laws [38][39]. However, the calibration of the local material

parameters of the bonds between the particles is complex [40] and only under

certain conditions behaves as a continuum. In addition, the large number of

DE needed to solve practical problems discourages its use.

Bearing all this information in mind and trying to combine the best features

of the FEM and the DEM, the FEM-DEM methodology was developed [2][3].

The continuum is initially represented with FE whose material behaviour is

represented by an isotropic damage model. A smoothing procedure is used

by computing the stresses at the element edges. When damage in a certain

element achieves a maximum threshold, the element is removed from the mesh

DE are placed at its nodes. The new DE avoid the indentation between the

crack faces by the contact forces among them and these forces are transferred

to the FE nodes as equivalent nodal forces.

Title Suppressed Due to Excessive Length 5

2.2 State of the art in mesh reﬁnement

2.2.1 Mesh generation

The early works of mesh generation by Zienkiewicz and Phillips[57] on the

1970s were based on the geometry boundary of the domain size and the re-

quired distribution of the element size. Since then many diﬀerent technolo-

gies have been proposed, including mapping techniques and semi-automatic

remeshing methods, where the domain has to be subdivided manually in a ini-

tial stage into simpler subdomains[49]. These methodologies depend on know-

how from the user and are generally limited to structured meshes (quadrilat-

erals and hexahedra).

Alternatively to structured meshing algorithms, unstructured meshing tech-

niques have been extensively developed for simplex element geometries: trian-

gles in 2D and tetrahedra 3D. These techniques are based mainly in three

algorithms[1]:

–Delaunay triangulation methods[50].

–The advancing front method[51].

–Tree methods: quadtrees for 2D and octrees for 3D cases[52].

2.2.2 Adaptive ﬁnite element reﬁnement techniques

Mesh adaption is widely used in numerical simulations to improve the accuracy

of the solutions, as well as to capture the behaviour of physical phenomena[60].

This technique allows to reduce considerably the computational cost, associ-

ated with a reduction of degrees of freedom (DoF from now on), while yielding

an accurate solution[67].

Adaptive mesh reﬁnement allows to compute complex problems with good

results in 3D without requiring the initial remeshing during pre-processing

step, which can be a time consuming and error-prone[1] task. Additionally,

adapting the computation during the simulation, avoids the creation of an

initial mesh that ﬁts all the problem evolution, which can be a priori not

known.

The mesh reﬁnement process depends on the previous numerical results[1].

These methodologies were introduced originally by Babuka and Rheinboldt[53][54]

in the late 1970s.

One of the most popular mesh remeshing strategies is based on the recov-

ery by equilibrium of patches (REP) techniques. A widely used approach based

on REP is the Superconvergent Patch Recovery (SPR) technique proposed by

Zienkiewicz and Zhu [9]. The methodologies based on REP are not the only

ones available to measure the error. Zienkiewicz and Zhu proposed other tech-

niques by using diﬀerent recovery methods[10]. Recently, techniques based on

the Hessian of a solution ﬁeld have been developed[66]. This requires that the

variable to be used as error estimation has to be twice continuously diﬀer-

entiable. This methodology has the advantage of giving a proper measure in

6 Alejandro Cornejo et al.

Fig. 3: DE generation after removing a FE

order to create anisotropic meshes, which reduces the number of new elements

necessary.

3 Coupled FEM-DEM Formulation

The coupled FEM-DEM formulation was developed by Zárate and Oñate[2]

as an eﬀective procedure for predicting the onset and propagation of cracks in

concrete and rocks. Zárate, Cornejo and Oñate[3] extended the formulation to

3D problems.

Initially the continuum is modelled with simplex FE (3-noded triangles in

2D and 4-noded tetrahedra in 3D). The FE solution is obtained by reaching

the dynamic equilibrium via an implicit transient dynamic solution scheme.

An isotropic damage constitutive law is chosen in order to verify failure at the

edges of the FE (using the SPR technique [9]). Once one of the failure modes

of the FE is achieved, this FE is removed from the mesh and DE are placed

at the nodes of the removed FE (see Fig. 3 and [2][3]).

Some important aspects inherent to the FEM-DEM formulation guarantee

the good results obtained, such as a smoothed stress ﬁeld, mass conservation

and the use of a simple algorithm to ensure the post-fracture contact between

the fractured edge and the adjacent FE and DE in the mesh[2][3].

3.1 FEM formulation

The predictive stress tensor ¯

σon all the elements of the mesh is initially

computed, as:

¯

σ=C0:ε(1)

where C0is the elastic constitutive tensor and εthe strain tensor.

Once the predictive stresses at the integration points of all the elements of

the mesh are computed, the smoothed stress ﬁeld is evaluated at the edges of

Title Suppressed Due to Excessive Length 7

the FE. The smoothing procedure is based on the average stress between the

current element and the neighbour one sharing an edge, i.e.

¯

σedge =1

2(¯

σcurrent +¯

σneighbour )(2)

Next, the constitutive equation is integrated at the edges. An isotropic

damage model is used if the stress state is outside the yield surface Φ, i.e.

Φ := f(¯

σ)−κ > 0(3)

where f(¯

σ)is the uniaxial stress that is computed according to diﬀerent yield

surfaces and κis the mechanical threshold that is related to the yield strength.

In the examples performed in this work we have used the Rankine and Modiﬁed

Mohr-Coulomb[41] yield surfaces. Once the initial threshold κ0is achieved it

has to be updated according to the maximum historical stress state.

The isotropic damage constitutive model is written as:

σ= (1 −d)¯

σ= (1 −d)C0:ε(4)

where dis the damage parameter that takes into account material degra-

dation as well as the irreversibility of the constitutive model. As far as the

computation of damage is concerned, we have used the exponential softening

law[69]:

d(¯

σ) = 1 −κ0

f(¯

σ)exp A1−f(¯

σ)

κ0 (5)

where the Aparameter is determined from the energy dissipated in an

uniaxial tension test as[69]

A= GfE

ˆ

lf 2

t−1

2!−1

(6)

where ftis the tensile strength, Gfis the speciﬁc fracture energy per unit area

(taken as a material property) and ˆ

lis the characteristic length of the element.

In this way one can compute the damage at the midpoint of the element edges.

Next, the damage of the whole element is be evaluated. By analysing all the

fracture modes that can occur, the damage of the element corresponding to

the mode with less energy is computed (Fig. 4). In 2D problem one can use

average the two maximum values at the element edges, as:

dF E =1

2(dedge,max +dedge,max−1)(7)

8 Alejandro Cornejo et al.

Fig. 4: Diﬀerent fracture modes in 2D and 3D element geometries

3.2 Tangent constitutive tensor approximations

The FEM solution is obtained via an implicit transient dynamic solution

scheme. Thus, the tangent constitutive matrix is required at each iteration

of the loading step. For this purpose, several numerical techniques have been

developed and adapted to the FEM-DEM formulation.

The most robust but slower option is to use the secant constitutive ten-

sor Cs, computed as a function of the initial constitutive tensor C0and the

damage d:

Cs= (1 −d)C0(8)

Another alternative is based on the derivatives approximation via ﬁnite

diﬀerences, i.e. the tangent constitutive tensor relationship can be expressed

as ˙

σ=CT:˙

ε. A column of the tangent constitutive tensor CTis deﬁned as

[34]:

CT,j =δjσ

δεj

(9)

An approximation of the tangent constitutive tensor can be obtained by

deﬁning nsmall perturbations of the strain tensor δεjin order to obtain n

stress tensor increments δjσ. This can be done in several ways, as stated below

(depending on the ﬁnite diﬀerence scheme):

CT,j 'σ(ε+δ εj)−σ(ε)

δεj

;CT,j 'σ(ε+δ εj)−σ(ε−δεj)

2δεj

(10)

where δεjis a zero vector except for the jth component whose value is the

strain perturbation δεj.

The most general option consists in perturbing the displacement ﬁeld of

the FEM solution [35]. This method is appropriate for small and large strain

computations (the strain perturbation method is limited to small strains) and

Title Suppressed Due to Excessive Length 9

for any kind of constitutive model. In this way, the approximation of the

tangent stiﬀness matrix can be computed as:

KT,j 'Fint (un,i +∆un)−Fint(un,i)

∆un(11)

Where KT,j is the jth column of the tangent stiﬀness matrix, ∆unis the

displacement increment of that node in the previous time step, Fint is the

internal force vector that depends of the displacement ﬁeld and is a small

constant computed as:

=√κ 1 +

un,i−1

k∆un,i−1k!(12)

being κthe computer precision. Note that the components of the vector ∆un

are null except for the jth component whose value is ∆un.

3.3 DEM formulation

The DEM methodology used in the FEM-DEM formulation implemented in

this work is based on the work of Casas et al [44], Oñate et al [38] and Thornton

et al. [45].

The motion of the DE is computed by solving the dynamic equilibrium of

forces at the center of each particle using an explicit dynamic solution scheme.

A sub-stepping procedure has been implemented in order to combine the DEM

explicit calculations with the implicit solution scheme[2][3] for the FEM.

Aspring-dashpot type soft-sphere approach for the contact between spheres

has been selected. Considering two contacting spheres, whose centres are r1

and r2, the normal vector that connects the centers of the spheres can be

computed as follows:

n21 =r2−r1

kr2−r1k,n21 =−n12 (13)

The normal indentation δnbetween the discrete particles is computed as:

δn=R1+R2− kr21k(14)

where Riare the radii of the particles. The total contact force between two

particles is deﬁned as the sum of a normal and a tangential force:

F=Fnn+Ftt(15)

The normal contact force Fnis obtained as a combination of an elastic and

a viscous contribution:

Fn=Fn,el +Fn,damp (16)

Where the elastic part can be computed as:

Fn,el =4

3˜

R1

2˜

Eδ

3

2

n(17)

10 Alejandro Cornejo et al.

where ˜

R:= (1/R1+ 1/R2)−1,˜

Ei:= Ei/(1 −ν2),˜

E=1/˜

E1+ 1/˜

E2−1

.

The corresponding viscous damping contribution is modelled as:

Fn,damp =cnδ1/4

n˙

δn(18)

For particle-particle contact the constant cncan be expressed as:

cn=γq8˜

E˜

Mp˜

R(19)

being ˜

M:= (1/m1+ 1/m2)−1and γa viscous damping coeﬃcient.

On the other hand, the tangential force is computed as:

Ft=Ft,el td+Ft,damptν(20)

where the directions tdand tνare based on the kinematics during tangen-

tial deformation [40].

The elastic tangential contribution is obtained by:

Ft,el =δ1/2

nZa(t)dt (21)

and the tangential viscous contribution as

Ft,damp =ctδ1/4

n˙

δt(22)

with

ct= 2 γq8˜

G˜

Mp˜

R(23)

where ˜

G=G/(4 −2ν)and G=E/(2 + 2ν).

3.4 Coupling between the FEM and the DEM

Once the damage parameter for an individual element computed by Eq. (7)

reaches a maximum threshold, the damaged element is removed from the FE

mesh and a set of DE are placed at the nodes of the removed element (Fig.

3, [2][3]).Following this, the displacements and velocities of the element nodes

are transferred to the DE. The next step is the integration of the dynamic

equations of motion of the DE using an explicit scheme using a substepping

procedure.

After performing a contact search among all the DE, the contact forces at

each DE (as deﬁned in Section 3.3) are computed. Once these contact forces

are known, this information is transferred to the FEM mesh as an equivalent

nodal force (Fig. 3) whose objective is to prevent indentation between the crack

faces. More information about the time integration of the dynamic equations

for the DEM and the FEM is given in [2].

Title Suppressed Due to Excessive Length 11

4 Hessian Based Remeshing Technique

In this section we analyse on detail the techniques considered for remeshing.

We introduce ﬁrst the concepts of metrics (Section 4.1) and general Hessian

based error measures (Section 4.2). Then we present the transfer operators for

the damage parameter.

4.1 Metric based remeshing

In order to understand the concept involving the Hessian metric[58], we ﬁrst

introduce the concept of metric. Then, we will show the intersection operations

needed in case than more that one metric is taken into consideration.

(a) Metric analogy (b) Intersection

Fig. 5: Metric analogies. Images from [58]

4.1.1 Concept of metric

The notion of length in a metric space is related to the notion of metric[67] and

therefore to an adequate deﬁnition of the scalar product in the vector space

considered. We deﬁne a metric tensor at a point P, respect an element Kfrom

a mesh Th, represented by a matrix M(d×d) deﬁned symmetric positive and

not degenerated. In 3D, the following deﬁnition of M(24) is used, which can

be assimilated to the analogy of an ellipsoid (Fig. 5a).

M=

a b c

b d e

c e f

such that a > 0, d > 0, f > 0

and det(M)>0,considering a, b, c, d, e ∈R

(24)

Tensor Mcan be diagonalized because it is symmetrical. Then, Mcan be

written as M=RΛR−1, where Rand Λare the matrix of the eigenvectors

and eigenvalues of M, respectively.

12 Alejandro Cornejo et al.

Fig. 6 illustrates the eﬀect of the metric on the mesh. The tetrahedra

presented gets sketched accordingly to the metric computed at each node,

represented with ellipsoids (Figure 5a).

Fig. 6: Eﬀects of the metric on a tetrahedra

4.1.2 Metric intersection

In the case that several metrics are speciﬁed at the same point of the mesh

(for example if we want to use various nodal variables whose Hessians return

diﬀerent metrics) one have to deﬁne a procedure of intersection of all these

metrics into one.

To deﬁne the intersection of two metrics, we use the fact that a metric

tensor is represented geometrically by an ellipse (in 2D) or an ellipsoid (in

3D). The metric intersection consists then in keeping the most restrictive size

constraint in all the directions imposed by this set of metrics[58] (Fig. 5b).

The simultaneous reduction enables us to ﬁnd a common basis (e1,e2,e3)

such that M1and M2are congruent to a diagonal matrix. In this basis we

can deﬁne a new tensor N, whose expresion is:

N=M−1

1M2(25a)

Ncan be diagonalized in Rbecause it is symmetrical in the metric M1. The

base in question is given by the normalized eigenvectors of Nthat we denote

e1,e2and e3(they form a base because Nis diagonalisable) . The eigenvalues

of M1and M2are found in this base using the Rayleigh quotient:

λi=et

iM1eiand µi=et

iM2ei(25b)

Considering P= (e1,e2,e3)be the matrix the columns of which are the

eigenvectors of N(common basis) one can obtain

M1=P−t

λ10 0

0λ20

0 0 λ3

P−1and M2=P−t

µ10 0

0µ20

0 0 µ3

P−1(25c)

Title Suppressed Due to Excessive Length 13

The metric intersection can be computed as:

M1∩2=M1∩M2=P−t

max(λ1, µ1) 0 0

0 max(λ2, µ2) 0

0 0 max(λ3, µ3)

P−1

(25d)

4.2 Hessian based error measure

Before introducing the theory involving the Hessian based metric, we summa-

rize the following properties[67]:

–The analysis and results obtained by this methodology are not asymptotic.

This means that the size of the mesh hdoes not tend to zero, avoiding

potential errors, like the collapse of the mesh at certain points.

–The metric is based in the Hessian of the solution.

–The metric is anisotropic.

–It is independent of the nature of the operator, so it can be used with any

type of equation.

4.2.1 Theory

We compute the Hessian[66] matrix Hof a scalar variable fas

H=

∂2f

∂x2

1··· ∂2f

∂x1∂xn

.

.

.....

.

.

∂2f

∂xn∂x1··· ∂2f

∂x2

n

or just: Hi,j =∂2f

∂xi∂xj

(26)

Once the Hessian matrix has been computed we compute the correspond-

ing anisotropic metric by [58].

M=Rt˜

ΛtRwhere ˜

Λ=diag(˜

λi)being

˜

λi= min max cd|λi|

,1

h2

max ,1

h2

min (27a)

Being λithe eigenvalues of Hand the error threshold and cda constant

ratio of a mesh constant. The interpolation ratio has been taken as 10−6. On

the other hand cdcan be taken as 2

9and 9

32 for 2D and 3D cases, respectively.

For an isotropic mesh the metric will be,

Miso =diag(max(˜

λi)) =

max(˜

λi) 0 0

0 max(˜

λi) 0

0 0 max(˜

λi)

(27b)

14 Alejandro Cornejo et al.

For an anisotropic mesh we have

Maniso =Rt˜

ΛanisoRwith

˜

Λaniso =diag(max(min(˜

λi,˜

λmax), Rλrel )) being

Rλrel =|˜

λmax −Rλ|where Rλ= (1 −ρ)|˜

λmax −˜

λmin|

(27c)

4.2.2 Example

Fig. 7: Initial mesh

The objective is to remesh the structured mesh of Fig. 7 according to the

Hessian of the nodal variable (objective function) deﬁned in Eq. (28). The

original mesh has 40000 structured elements. Our objective is to obtain an

unstructured mesh where the smaller elements will be in the vicinity of the

objective function.

The nodal variable values are computed according to:

f(x, y) = tanh(−100(y−0.5−0.25 sin(2πx)))

+ tanh(100(y−x)) (28)

The results obtained are depicted in Fig. 8b, using a mesh of 15000 ele-

ments. The smaller elements are placed around the χshape displayed in Fig.

8a showing also the nodal value of the funcion deﬁned in Eq. (28).

4.3 Hessian nodal indicator

In order to optimize the remeshing technique and reﬁne the elements close

to the crack opening we deﬁne a proper nodal variable Υwhose Hessian is

Title Suppressed Due to Excessive Length 15

(a) Nodal values of Eq. (28) (b) Remeshed mesh

Fig. 8: Nodal values of the remeshed mesh for the error function from Eq. (28)

computed. Initially, the nodal extrapolation of the predictive Cauchy’s stress

tensor was selected but the meshes generated with this indicator were subop-

timal, as it reﬁnes the zones near the boundary conditions where, in general,

there is no interest. In the end, a normalized energetic nodal variable indicator

was selected. The expression of the mesh reﬁnement indicator is:

Υ=1

2ρε:C0:ε(1 −d)r

gt

+1−r

gc (29)

where ρis the material density, dis the damage internal variable, gtand gc

are the regularized fracture energies in tension and compression, respectively

and ris a tension indicator computed as:

r=P3

i=1 hσii

P3

i=1 |σi|,hσii=1

2(σi+|σi|)(30)

being σithe principal components of the stress tensor. The mesh reﬁnement

indicator can be interpreted as the energy dissipated, normalized with the

total energy available.

4.4 Internal variables interpolation

The internal variables information has to be recovered in the reﬁned mesh in

order to work with constitutive models that depend on historical values, such

as the damage model used in this work. Fig. 9 shows graphically how each one

of the transfer operators work[56] (all of them are available in Kratos[48]).

16 Alejandro Cornejo et al.

(a)

new mesh

old mesh

ip of new mesh

ip of old mesh

(b)

(c)

new mesh

old mesh

ip of new mesh

ip of old mesh

Fig. 9: Transfer operators: a) Closest Point Transfer b) Shape Function Pro-

jection Transfer c) Least-Square Projection Transfer . Image from [56]

– CPT:Closest Point Transfer.(a). It takes the value from the closest point.

It provides acceptable results at low cost.

– SFT:Shape Function Projection transfer.(b). It interpolates the values

using the standard FEM shape functions. It leads to an artiﬁcial damage

diﬀusion, but preserves the original shape of the damage proﬁle.

– LST:Least-Square Projection transfer.(c). It considers a least-square

transfer across the closest points. Probably it is the most accurate technique

but also the most expensive from a computational point of view.

In our simulations we have used the CPT technique.

Title Suppressed Due to Excessive Length 17

18 Alejandro Cornejo et al.

5 Implemented Algorithm

The FEM-DEM formulation presented can be summarized in the algorithm

below.

Initialization of the implicit transient dynamic scheme for the FEM:

ti=ti+∆ti,k= 0 being tithe current time of the implicit scheme.

Apply the DE contact forces from the previous time step as equivalent

nodal force for the FEM

if It is time to remesh then

Compute nodal indicator Υ=1

2ρε:C0:ε(1 −d)r

gt+1−r

gc

Evaluate the Hessian matrix H

Calculate the metric tensor M

Perform the remeshing

Mapping of the internal variables and nodal values

end

while ∆F =Fint −Fext< tol do

for Elements do

Compute the eﬀective stresses ¯

σ=C0:ε

Smoothing of the eﬀective stress ﬁeld at the FE edges

Compute the damage dat the edges by Eq. (5)

Obtain the elemental damage by Eq. (7)

Calculate the tangent stiﬀness matrix KT

eand the updated

internal forces vector Fint

e

end

Assemble the global expression of KTand Fint

Calculate the displacement increments ∆ut

k=K−1∆F

Check convergence ∆F < tol

k=k+ 1

end

for Elements do

if Damage > 0.98 then

ERASE the FE

Generate the Discrete Elements (DE) at the nodes of the

damaged FE

end

end

Initialization of the explicit transient dynamic scheme for the DEM

Import the kinematic information (displacements and velocities) from

the FEM nodes to the DE as an initial condition

while te=te+∆te<tido

Compute the contact forces between the DE

Integrate the equations of motion

Compute the displacements, velocities and accelerations at the DE

end

Transfer the contact forces as equivalent nodal forces to the FE

Algorithm 1: Enhanced FEM-DEM algorithm

Title Suppressed Due to Excessive Length 19

6 Numerical Examples

Several examples are presented in order to show the accurate results and good

representation of the fracture paths obtained with the enhanced FEM-DEM

formulation developed in this work. The ﬁrst example is the well-known four

point bending test whose fracture path is theoretically known and the force-

displacement evolution has been compared with the results from [46]. The

second example is a tensile test whose analytical solution is trivial, so it is very

useful in order to validate the formulation. Finally, a three-point bending test

on skew notched beam has been performed. The FEM-DEM results have been

compared with those obtained by Cervera et al. [47]. For the 2D examples

(Section 6.1) we have used 3-noded triangles. The 3D problems have been

solved using 4-noded linear tetraedra.

6.1 Four-Point Bending Beam

This example is a plane stress four point supported beam with a double notch.

In the two central supports a vertical displacement is imposed whereas in the

exterior supports only the vertical displacement is enforced to be zero (one of

them must be clamped, as depicted in Fig. 10). The dimensions of the beam are

134 x 30.6 x 30 cm. The yield surface used is the Modiﬁed Mohr-Coulomb[41].

The material properties used are: E = 30 GPa, ν= 0.2, t= 0.3 m, ft= 2

MPa, Gf= 100 J/m2and the friction angle φ=32o. The initial FE mesh

is displayed in Fig. 11. Fig. 12 shows that the remeshing technique and the

Hessian variable indicator deﬁned in the Section 4.2 are performing excellently

as far as capturing the crack path is concerned. Another interesting feature is

that the number of FE does not increase indeﬁnitely. Fig. 12 shows that the

number of FE in the mesh increases with respect to the initial coarse mesh but

during the calculation is bounded up to a reasonable value (even decreasing

at the end of the simulation) so the computational cost is balanced.

Quantitatively, the force-displacement evolution in one of the central sup-

ports is depicted in Fig. 14. In this ﬁgure the results from [46] and the ones

from the FEM-DEM formulation, with or without remeshing, are compared,

showing a good agreement between them.

Additionally, the comparison between the crack paths using the remeshing

technique and the standard FEM-DEM formulation is depicted in Fig. 15. As

one can see, the quality of the crack path is improved with the inclusion of the

remeshing technique but, as the non-remeshed solution uses a coarser mesh,

the CPU is about 14 min whereas the remeshed solution increases the CPU

time up to 45 min. The main advantage of this methodology lies in obtaining

great quality crack paths without the requirement of a very ﬁne original mesh.

20 Alejandro Cornejo et al.

Fig. 10: Geometry and boundary conditions of the four point bending test

(units in cm)

Fig. 11: Initial FE mesh used in the calculation (2912 3-noded triangles ele-

ments and 1573 nodes)

6.2 Tensile Test

In this example a conventional 3D tensile test has been reproduced. The ge-

ometry of the sample is depicted in Fig. 16 with a thickness equal to 0.2 m.

The left end is clamped and the right one has a monotonic imposed displace-

ment. The Modiﬁed Mohr-Coulomb yield surface has been used. The material

parameters are: E = 35 GPa, ν= 0.2, ft= 1.5 MPa, Gf= 30 J/m2and the

friction angle φ=32o.

Fig. 17 shows that the mesh reﬁnement is concentrated at the center zone,

where all the energy dissipation is taking place due to the damage in the

necking zone. The force-displacement evolution at one of the ends of the sample

is depicted in Fig. 18. The results are in good agreement with the analytical

expected solution (Rmax =Area ∗ft).

In Fig. 19 the ﬁnal fracture of the sample is depicted. As expected, fracture

occurs at the center of the necking. It is important to notice that the remeshing

technique improves the quality of the cracking path (see the comparison in Fig.

20) but quantitatively is always consistent (Fig. 18), even when using coarse

meshes.

Title Suppressed Due to Excessive Length 21

Fig. 12: FE meshes during calculation (a) 5388 FE, b) 6276 FE, c) 8985 FE,

d) 8188 FE, e) 6252 FE and f) Final result without remeshing technique

6.3 Three-Point Bending Skew Notched Beam

In this section, a skew notched beam subjected to three-point bending is anal-

ysed. The same problem was studied by Cervera et al. [47]. The original ex-

periment was performed by Buchholz et al. [68] using Plexiglass in order to

identify the fracture path along the sample. The geometry of the sample is

22 Alejandro Cornejo et al.

Fig. 13: Zoom of the reﬁned FEM mesh in Fig. 12b

Fig. 14: Force-displacement evolution in the four point bending test at one of

the inner supports

shown in Fig. 21 in which the deviation of the notch can be seen. The Rankine

yield surface was used in this test as in [47]. The material parameters are: E

= 28 GPa, ν= 0.38, ft= 40 MPa and Gf= 3000 J/m2. The analysed prob-

lem is symmetric with respect to the notch and it fractures under a mixed

Mode I and Mode III. Initially the crack path twists around the vertical axis

until it is oriented perpendicular to the longitudinal direction of the beam.

The initial mesh is depicted in Fig. 22. The FE meshes generated during the

calculation using the remeshing technique can be analysed in Fig. 23. As it can

be seen, the remeshing technique reﬁnes the elements near the notch due to

Title Suppressed Due to Excessive Length 23

Fig. 15: Crack paths comparison between the remeshed and non-remeshed

solutions

Fig. 16: Tensile test geometry (units in m)

the high dissipation that takes place in these zones. As the crack propagates,

the remeshing follows the expected path by reﬁning the front of the fracture

at each remeshing step.

If one compares the results obtained with the simulation (Fig. 24) with the

experimental results (Fig. 25) it is clear that the crack path follows the pattern

obtained by the experiment accurately. As stated before, the solution obtained

is skew-symmetrical. Also, the crack surface is perpendicular to the longitu-

dinal axis at the end of the propagation as expected. The force-displacement

evolution can be seen in Fig. 26. No numerical results regarding the force-

displacement evolution was provided by the authors of this experiment.

7 Conclusions

In this work a coupled FEM-DEM formulation enhanced with a novel adap-

tive remeshing technique has been presented. The proposed methodology has

demonstrated a good performance: quantitatively, when comparing the force

displacement curves obtained with the analytical ones, and qualitatively when

analysing the crack paths obtained versus the expected or experimental results.

24 Alejandro Cornejo et al.

Fig. 17: Tensile test FE meshes during the remeshed FEM-DEM calculation

using 4-noded tetrahedra (a) 12000 FE, b) 8248 FE, c) 14092 FE and d) 70749

FE

Fig. 18: Force-displacement evolution for the tensile test at one of the ends of

the sample

Title Suppressed Due to Excessive Length 25

Fig. 19: Tensile test fracture in the sample at the end of the calculation

Fig. 20: Tensile test comparison of the crack pattern between the solution with

(a) or without (b) the remeshing technique

26 Alejandro Cornejo et al.

Fig. 21: Three point bending skew notched beam geometry (units in m)

Fig. 22: Three point bending skew notched beam initial FE mesh (15546 4-

noded tetrahedra)

The standard FEM-DEM is an accurate numerical procedure due to its in-

trinsic mesh-independence and consistency features[2][3]. However, the adap-

tive remeshing technique here presented improves considerably the crack path

geometry obtained and optimizes the calculation cost, because it only reﬁnes

the zones of interest, where the non-linear dissipation takes place.

Regarding the remeshing technique, the Hessian based methodology com-

bined with the nodal variable indicator developed (normalized free energy)

has behaved very well in all the examples performed, capturing the zones of

interest where the mesh needs to be reﬁned.

In conclusion the FEM-DEM formulation, enhanced with the adaptive

remeshing technique presented, is suitable for simulating complex fracture me-

chanics problems at an aﬀordable computational cost. For instance, the four

Title Suppressed Due to Excessive Length 27

Fig. 23: Adaptive FE meshes of 4-noded tetrahedra during calculation a) 15546

FE, b) 14436 FE, c) 16707 FE, d) 25811 FE, e) 27478 FE and f) 29738 FE

point bending test was run in 50 min, the tensile test in 9 hours and the three

point bending test in 4 hours. All the tests were carried out in a personal

computer (CPU: Intel Core i7-8700, RAM: 16 GB DDR4) using 12 threads.

8 Appendix

8.1 Kratos multiphysics

The FEM-DEM formulation presented has been implemented in the Kratos

multi-physics framework [48] that has been specially designed for helping the

development of multi-disciplinary ﬁnite element programs. We can summarize

the following features:

– Kernel: The kernel and application approach is used to reduce the possible

conﬂicts arising between developers of diﬀerent ﬁelds.

– Object oriented: The modular design, hierarchy and abstraction of these

approaches ﬁts to the generality, ﬂexibility and re-usability required for

the current and future challenges in numerical methods. The main code is

developed in C++ and the Python language is used for scripting

28 Alejandro Cornejo et al.

Fig. 24: 3-Point bending beam test skew fracture path obtained with the sim-

ulation

– Open source: The BSD (Berkeley Software Distribution) licence allows

to use and distribute the existing code without any restriction, but with

the possibility to develop new parts of the code on an open or close basis

depending on the developers. Additionally Kratos can be freely used.

8.2 Mmg library

8.2.1 What is Mmg and how does it work?

Mmg is an open source software for anisotropic automatic remeshing for un-

structured meshes based on Delaunay triangulations. It is licenced under a

LGPL license and it has been integrated in Kratos[48] via the mmg_process.h

in the MeshingApplication. It provides three applications and four libraries:

30 Alejandro Cornejo et al.

Kratos

Multi-Physics

Fig. 27: Kratos Multiphysics logo. https://github.com/KratosMulti-

physics/Kratos

Fig. 28: Mmg logo. Image from Mmg web

–The mmg2d application and the libmmg2d library: adaptation and optimiza-

tion of a two-dimensional triangulation and generation of a triangulation

from a set of points or from given boundary edges.

–The mmgs application and the libmmgs library: adaptation and optimiza-

tion of a surface triangulation and isovalue discretization.

–The mmg3d application and the libmmg3d library: adaptation and optimiza-

tion of a tetrahedral mesh and implicit domain meshing.

–The libmmg library gathering the libmmg2d,libmmgs and libmmg3d li-

braries.

The Mmg remeshing process modiﬁes the mesh[63][64] iteratively until it

is in agreement with the prescribed sizes on the idealized (Fig. 29) contour

(and directions in case of anisotropic mesh). The software reads the mesh and

the metric, then the mesh is modiﬁed using local mesh modiﬁcations of which

an intersection procedure based on anisotropic Delaunay kernel.

We can resume the remeshing algorithm in the following steps:

1. Mmg tries to have a good approximation of the surface (with respect to

the Hausdorﬀ parameter).

2. It remeshes according to a geometric criterion. Mmg scans the surface tetra-

hedra and splits the tetrahedra using predeﬁned patterns if the Hausdorﬀ

distance[65] between the surface triangle of the tetrahedra and its curve

representation does not respect the Hausdorﬀ parameter.

3. The library scans again the surface tetrahedra and collapse all the edges

at a Hausdorﬀ distance smaller than a threshold deﬁned in terms of the

Hausdorﬀ parameter.

4. Next it intersects the provided metric and a surface metric computed at

each point from the Hausdorﬀ parameter and the curvature tensor at the

point.

Title Suppressed Due to Excessive Length 31

T

a0=b3,0,0

b1,2,0

a1=b0,3,0

a2=b0,0,3

b1,0,2

b2,0,1

b2,1,0

b0,1,2

b0,2,1

b1,1,1

•

•

•

•

•

•

•

•

•

•

S

T

(0,0)

(0,1)

(1,0)

(a) A piece of parametric Bézier cubic surface,

associated to triangle T

•

p

TpS

∂Ω

p

•

(b) The resulting conﬁguration of the vertex

relocation procedure

Fig. 29: Mmg idealized geometry. Image from [64]

5. Then Mmg smooths the metric to respect the gradation parameter. The

metrics are iteratively propagated until the respect of the gradation every-

where.

6. Next it remeshes the surface tetrahedra in order to respect the new metric.

7. Finally it remeshes both the volume and surface to have edges between

0.6 and 1.3 (in the metric). The long edges are cutted and short ones are

deleted (collapsed).

8.2.2 Integration between Mmg and Kratos

In order to understand the integration between Kratos and Mmg is important

to understand the data structure of Kratos. On Fig. 30 an example of the data

structure of the Model can be analysed. The Model stores the whole model to

be analyzed, and manages the diﬀerent ModePart used on the simulation. The

ModelPart holds all data related to an arbitrary part of model. It stores all

existing components and data like Nodes,Properties,Elements,Conditions

and solution data related to a part of the Model.

Submodelparts used for

differents pourposes

Domains

Whole problem Model

Fluid MP

Main Fluid

Nodes Elements Conditions

BC Fluid

Nodes

Structure MP

Main

Structure

Nodes Elements

Fig. 30: Model data structure

The entities stored on the ModelPart are:

–Node It is a point with additional facilities. Stores the nodal data, historical

nodal data, and list of DoF.

32 Alejandro Cornejo et al.

–Condition encapsulates data and operations necessary for calculating the

local contributions of Condition to the global system of equations.

–Element encapsulates the elemental formulation in one object and pro-

vides an interface for calculating the local matrices and vectors necessary

for assembling the global system of equations. It holds its geometry that

meanwhile is its array of Nodes.

–Properties encapsulates data shared by diﬀerent Elements or Conditions.

It can store any type of data.

In our implementations we used a process to set the BC (both Neumann

or Dirichlet). In order to preserve that information after remeshing we need

to create an identiﬁcation system, so we are able to create an unique ID that

will allow us to reconstruct the submodelpart structure after remeshing, this

methodologies are commonly called colour identiﬁcation. Fig. 31 shows the

concept of this idea.

Colour 1

SelfWeight

_Part

Automati

cInlet2D_

Inlet

Parts_Fluid

Colour 1

Node 1

Node 2

Cond. 1

Cond. 3

Elem. 1

Elem. 4

Colour 2

Node 3

Node 4

Cond. 2

Cond. 3

Elem. 3

Elem. 4

Fig. 31: Concept of colours

Acknowledgements This work has been supported by the Spanish Government program

FPU: FPU16/02697. The authors gratefully acknowledge the received support.

9 Compliance with Ethical Standards

This study was funded by the Spanish Government program FPU: FPU16/02697.

The authors declare that they have no conﬂict of interest.

Title Suppressed Due to Excessive Length 33

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