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Combination of an adaptive remeshing technique with a coupled FEM-DEM approach for analysis of crack propagation problems

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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 using the enhanced FEM-DEM formulation are presented. Accurate results in comparison with analytical and experimental solutions are obtained.
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(will be inserted by the editor)
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
Carrer del Gran Capità, S/N, 08034 Barcelona
Tel.: +34 934 01 74 95
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 +and
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 A1f(¯
σ)
κ0 (5)
where the Aparameter is determined from the energy dissipated in an
uniaxial tension test as[69]
A= GfE
ˆ
lf 2
t1
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,max1)(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(εδεj)
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,i1
kun,i1k!(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 =r2r1
kr2r1k,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˜
3
2
n(17)
10 Alejandro Cornejo et al.
where ˜
R:= (1/R1+ 1/R2)1,˜
Ei:= Ei/(1 ν2),˜
E=1/˜
E1+ 1/˜
E21
.
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=RΛR1, 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=M1
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=Pt
λ10 0
0λ20
0 0 λ3
P1and M2=Pt
µ10 0
0µ20
0 0 µ3
P1(25c)
Title Suppressed Due to Excessive Length 13
The metric intersection can be computed as:
M12=M1M2=Pt
max(λ1, µ1) 0 0
0 max(λ2, µ2) 0
0 0 max(λ3, µ3)
P1
(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
∂x1xn
.
.
.....
.
.
2f
∂xnx1··· 2f
∂x2
n
or just: Hi,j =2f
∂xixj
(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 106. 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(y0.50.25 sin(2πx)))
+ tanh(100(yx)) (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
+1r
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+1r
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=K1∆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:
Title Suppressed Due to Excessive Length 29
Fig. 25: 3-Point bending skew beam experimental results with Plexiglass [68]
Fig. 26: 3-Point bending skew beam force-displacement evolution at the center
of the beam
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
b0,1,2
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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