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Combination of the finite element method and particle-based methods for predicting the failure of reinforced concrete structures under extreme water forces

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We present a combination of the Finite Element Method (FEM), the Particle Finite Element Method (PFEM), and the Discrete Element Method (DEM) for modeling and analyzing the failure of reinforced concrete structures under impulsive wave forces originating from free-surface flows in critical water hazards. The free-surface water flow is modeled with the PFEM, while the structural behavior and the fractures induced by the water forces in the structure are modeled with a coupled FEM-DEM technique. The concrete material behavior is simulated with a standard isotropic damage model. The reinforcing bars are modeled by a rule of mixtures procedure, for simplicity. The possibilities of the new integrated PDFEM approach for predicting the evolution of free-surface tsunami-type waves and their devastating effect on constructions are validated with experiments on the failure of reinforced concrete plates under large impacting waves, performed in a laboratory facility in Japan.
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Combination of the finite element method and particle-based methods for
predicting the failure of reinforced concrete structures under extreme water
forces
Eugenio O˜natea,b, Alejandro Cornejoa,b,1 , Francisco Z´aratea,b, Kazuo Kashiyamac, Alessandro Francia,b
aCentre Internacional de M`etodes Num`erics en Enginyeria (CIMNE), Campus Norte UPC, 08034 Barcelona, Spain
bUniversitat Polit`ecnica de Catalunya (UPC), Campus Norte UPC, 08034 Barcelona, Spain
cChuo University, 1-13-27 Kasuga, Bunkyo-ku, Tokio, Japan
Abstract
We present a combination of the Finite Element Method (FEM), the Particle Finite Element Method (PFEM),
and the Discrete Element Method (DEM) for modeling and analyzing the failure of reinforced concrete structures
under impulsive wave forces originating from free-surface flows in critical water hazards. The free-surface water
flow is modeled with the PFEM, while the structural behavior and the fractures induced by the water forces in
the structure are modeled with a coupled FEM-DEM technique. The concrete material behavior is simulated
with a standard isotropic damage model. The reinforcing bars are modeled by a rule of mixtures procedure, for
simplicity. The possibilities of the new integrated PDFEM approach for predicting the evolution of free-surface
tsunami-type waves and their devastating effect on constructions are validated with experiments on the failure
of reinforced concrete plates under large impacting waves, performed in a laboratory facility in Japan.
Keywords: Tsunami force, Finite Element Method, Particle Finite Element Method, Discrete Element
Method, Reinforced concrete, Fluid-Structure Interaction, Fracture Mechanics
Dedication: This paper is dedicated to Professor Herbert Mang on the occasion of his 80th birthday.
1. Introduction1
The recent critical weather events around the world are leading to an increasing concern on environmental2
risks connected to the destructive action of water. Apart from the major disasters due to the failure of the3
dykes in New Orleans (2005) and the tsunamis in Japan in 2011 and in the Indonesian region in late 2004,4
other dramatic examples of water-induced hazards are, just to cite a few, the big flooding in parts of Central5
Europe in summer 2021 and in 2005 and the events at Sarno, Italy in 1998 and in Venezuela in 1999 where6
1Corresponding author. E-mail address: acornejo@cimne.upc.edu (A. Cornejo)
1
intense precipitations triggered devastating mud-flows. Yet another example is the complete failure of the Tous7
rockfill dam in Valencia, Spain (1982) due to over-spill in an intense rain with devastating effects on thousands8
of hectares. It is commonly accepted that the risk of extreme climatic events is increasing and will do so in9
the next decades. Considering that hydraulic constructions and standard buildings and infrastructures that10
might be affected by water hazards are designed from scenarios defined from historical data, it is clear that the11
extreme water hazards derived from climate changes might result in an increasing number of structural failures12
in the future.13
Even if in most cases little could be done to minimize the effect of such disasters, the design of new protecting14
structures (dams, dykes, breakwaters, etc.), as well as of new buildings and other constructions adjacent to areas15
that can be affected by water hazards, should be performed so as to minimize the damage induced by critical16
water forces. A first step towards this goal is the possibility to assess in a fast and accurate way the interaction17
between the free-surface fluid in a water hazard and the surrounding infrastructure.18
Several researches have investigated, via ad-hoc mathematical models and experiments, the impulsive fluid19
force induced on structures by tsunami waves [1, 2, 3, 4, 5, 6, 7, 8, 9]. Of particular interest here are the20
laboratory experiments carried out in Japan by Arikawa et al. in the Large Hydro-Geo Flume (LHGF) facility21
[7]. The experiments focused on the study of the collapse mechanisms of concrete plates by an impulsive22
force induced by a large impacting wave. Different concrete plates with columns were tested by changing the23
compressive strength of concrete and the thickness of the plate. From the test results, the fluid forces on24
the plates were measured and the failure modes of the plates against the impulsive tsunami-type forces were25
assessed.26
It is an evidence that laboratory experiments on ad-hoc scale models, such as those defined above, is a27
paramount task for the purpose of assessing the response of structures under tsunami-type forces due to the28
complexity of the experiments and the specific requirements and facilities needed, available only to a few29
specialized organizations. Other problems are the difficulty for the scaling up the experimental results to real-30
life problems, the large execution time of the tests and their very high costs. The alternative is therefore31
numerical modeling.32
Despite the practical importance of the problem and the intensive work in the last decades in the development33
of suitable mathematical and computational models, the study of the failure of a construction by the action34
2
of water forces accounting for coupled fluid-structure interaction (FSI) effects is still a major challenge. Some35
reasons exist and some are connected to the complex mathematical structure of this particular class of FSI36
problem. Others are related to the presence of breaking waves, the high unsteadiness of the flow and the37
difficulties for modeling the multi-cracking pattern in concrete structures and the water flow between the38
fractured parts of the structure.39
The authors have developed in recent years an innovative Lagrangian numerical method called PFEM (for40
Particle Finite Element Method) which combines a particle-based approach and the finite element method41
(FEM). The method is applicable for the modeling and simulation of free-surface particle flows and their42
interactions with structures ([10, 11, 12, 13, 14]). Quite recently, the authors have extended the PFEM to43
account for multi-fractures in concrete structures by using the so-called FEM-DEM technique. This approach44
combines the standard FEM with the well-known Discrete Element Method (DEM) [15, 16] for predicting the45
onset and evolution of multiple cracks in structures under external forces [17, 18], accounting for the separation46
of the fractured part from the main body of the structure. The coupling of the PFEM and the FEM-DEM47
procedure leads to a new Particle-Discrete Finite Element Method (hereafter called PDFEM). The PDFEM48
has the necessary numerical ingredients and tools for modeling the complex interactions between critical free49
surface environmental flows and constructions up to the failure of the structure and beyond. Examples of the50
possibilities of the PDFEM are reported in Cornejo et al. [19, 20].51
In this paper we explore the possibilities of the PDFEM technique for modeling the failure of reinforced52
concrete structures under impulsive wave forces, as those occurring in critical water hazards. The structure is53
modeled with the FEM. The concrete material is modeled with an isotropic damage model. On the other hand,54
the reinforcing bars are modeled by a standard rule of mixtures procedure [21, 22, 23]. The free-surface water55
flow is simulated with the PFEM, while the fractured induced by the water forces in the structure is modeled56
with the FEM-DEM technique. The integrated PDFEM approach has been validated with experiments on57
reinforced concrete plates under a large impacting wave performed in the LHGF facility, mentioned above [7].58
The structure of the paper is as follows. First, we present the basis of the PFEM. Then we briefly describe59
the general features of the FEM-DEM technique for modeling the failure of reinforced concrete structures. The60
possibilities and accuracy of the FEM-DEM approach for predicting the onset and propagation of cracks in61
concrete structures are shown in the study of a standard Brazilian test and the failure of a reinforced concrete62
3
beam under external forces. The new PDFEM procedure is then validated by studying the failure of a reinforced63
concrete plate under a large impacting wave generated in the flume channel of the LHGF laboratory. Numerical64
results for the wave evolution and the plate failure are compared with those obtained in the LHGF facility for65
the same problem.66
2. Basis of the PDFEM approach67
In this section, we describe the PDFEM method used for the simulation of the case study. First, we present68
the PFEM formulation used for the free-surface fluid flow solution and for detecting the fluid-solid interface.69
Then, we describe the FEM-DEM procedure to simulate failure of plain and reinforced concrete structures in70
the presence of reinforcing bars and multi-fracturing processes. Finally, we briefly explain the fluid-structure71
interaction (FSI) algorithm.72
We make particular emphasis on the new features of the formulation that has been implemented for simulat-73
ing the LHGF experiment, such as the mesh refinement method used in the PFEM and the mixture rule used74
in the FEM-DEM method to deal with reinforced concrete structures. On the other hand, we briefly describe75
those parts of the formulation that have been presented in previous publications. The interested reader may76
found details of the numerical formulation in the referenced works.77
2.1. PFEM solution of fluid dynamics problem78
The movement of solids in fluids is usually analyzed with the finite element method (FEM) [24] using the79
so-called arbitrary Lagrangian-Eulerian (ALE) formulation [25]. In the ALE approach, the movement of fluid80
and solid particles is decoupled from that of mesh nodes. Hence the relative velocity between mesh nodes and81
particles is used as the convective velocity in the momentum equations. Typical difficulties of FSI analysis82
using the FEM with both the Eulerian and ALE formulations are the treatment of the convective terms and the83
incompressibility constraint in the fluid equations, the tracking of the free surface in the fluid, the transfer of84
information between the fluid and solid domains via the contact interfaces, the modeling of wave splashing, the85
large rigid body motions of the solid within the fluid domain, the efficient updating of the finite element meshes86
for both the structure and the fluid, etc. Most of these problems can be overcome using a Lagrangian description87
is used to formulate the governing equations for both the solid and the fluid domains. In the Lagrangian88
formulation, the nodes in a finite element mesh are viewed as “particles” which motion is followed during the89
4
transient solution. In this work, we have extended the particle finite element method (PFEM) developed by90
the authors’ group in the last years for FSI analysis [26, 11, 12, 13, 27]. The PFEM treats the mesh nodes in91
the fluid and solid domains as Lagrangian particles which can freely move and even separate from the main92
fluid domain representing, for instance, the effect of water splashing. A mesh connects the nodes defining the93
discretized domain where the governing equations for the fluid and the solid are solved using the standard FEM.94
An advantage of Lagrangian formulations, such as the PFEM, is that the convective terms disappear from the95
fluid equations. The difficulty is however transferred to the problem of adequately (and efficiently) moving the96
mesh nodes. Indeed for large mesh motions, as it is the case in free-surface environmental flows, remeshing is a97
frequent necessity along with the solution. In this work, we have used a fast mesh regeneration procedure based98
on a Delaunay tessellation. On the other hand, the free-surface nodes are identified at each time step using an99
Alpha-Shape technique [26, 11]. Satisfaction of the incompressibility condition in the fluid still remains in the100
Lagrangian formulation. Several stabilization procedures aiming to alleviate the volumetric locking problem in101
incompressible fluids have been proposed (see [28] and the references therein]). A general aim is to use low order102
finite elements with equal order interpolation for the velocity and pressure variables. In this work, we have used103
a stabilization procedure based on the Finite Increment Calculus (FIC) approach [29, 30]. Applications of the104
FIC method for incompressible free-surface flow analysis using linear triangles and tetrahedra are reported in105
[31, 32]. A comprehensive description of the PFEM and its many applications in fluid and solid mechanics can106
be found in [27].107
Details about the FEM solution strategy used and the remeshing algorithm are provided below.108
2.1.1. FEM solution of the fluid mechanics equations109
The fluid motion is described by the Navier-Stokes equations, i.e. the linear momentum balance (Eq. 1)
and the mass conservation (Eq. 2). Following the standard PFEM strategy [11], the problem is here solved in
an Updated Lagrangian framework. Moreover, following [33, 34, 35], a reduced compressibility is considered in
the fluid. Basing on this, the fluid governing equations read
ρf
v
∂t −5·σρfg=0in Ωf×(0, T ) (1)
5
5 · v1
κf
∂p
∂t = 0 in f×(0, T ) (2)
where vis the velocity vector, σis the Cauchy stress tensor, gis the gravity acceleration vector, pis the110
pressure, ρfand κfare the fluid density and bulk modulus, respectively, Ωfis the updated fluid domain, and111
tis the time.112
The fluid material considered in this work is water. Consequently, the fluid stress tensor is computed
according to a standard Newtonian law as
σ=pI+ 2 µfd’ (3)
where Iis the 2nd order identity tensor, µfis the fluid dynamic viscosity, and d’ is the deviatoric part of the113
deformation rate tensor.114
Besides the required initial conditions, the following boundary conditions are considered
v=ˆ
von Γv
f
σ·n=ˆ
ton Γt
f
(4)
where nis the normal vector to the fluid boundaries, ˆ
vare the prescribed velocities at the Dirichlet boundaries115
Γv
f, and ˆ
tare the prescribed tractions at the Neumann contours Γt
f.116
In this work, the fluid governing equations (Eqs.1-2) are solved with the stabilized FEM strategy presented117
in [32]. Details about the derivation of the fluid formulation can be found in the mentioned work. Here, we118
recall only the main features of the method.119
The governing equations are solved with an implicit two-step strategy. At each time step, the linear mo-120
mentum equations are solved first for the increments of nodal velocities and the stabilized continuity equation121
is then solved for the nodal pressures. The solution of the two equations is iterated until reaching convergence.122
Since the same linear interpolation is used for both velocity and pressure fields, the so-called inf sup123
condition is not satisfied and the problem needs to be stabilized. In this work, we have the Finite Increment124
Calculus (FIC) method derived in [32] to provide the required stabilization. We remark that due to the125
Lagrangian nature of the method and the consequent absence of the convective term, the FIC stabilization126
6
terms appear only in the mass conservation equation. The resulting FIC-FEM formulation has excellent mass127
conservation and accuracy features [32].128
2.1.2. PFEM remeshing procedure129
In the PFEM the mesh distortion issues arising from the Lagrangian solution of the fluid governing equations130
are circumvented via an efficient remeshing strategy that is performed whenever the discretization overcomes a131
pre-established threshold of deformation.132
Once a converged solution has been obtained at the end of the time step, a new mesh is regenerated according133
to the following three steps:134
1. Erase all elements of the mesh and keep the nodes;135
2. Perform a Delaunay triangulation over the retained nodes;136
3. Apply the Alpha-Shape technique to recognize the physical contours of the domain containing the retained137
nodes.138
Figure 1 shows a graphical representation of this strategy applied to a fluid-structure interaction problem.139
Note that in this FSI method the remeshing step only involves the fluid parts of the computational domain.140
The Delaunay triangulation [36] ensures elements of good quality but does not allow for the recognition of141
the actual contours of the fluid domain. This task is done by the Alpha Shape method [37]. With this technique,142
all those elements that are considered excessively distorted or too large according to a pre-fixed criterion, are143
removed from the Delaunay tessellation.144
We remark that the PFEM remeshing operations also enable the detection of the boundaries of rigid and/or145
deformable solids. In Figure 1 we show two different situations in which the remeshing strategy is applied146
to a FSI problem. As previously mentioned, at the beginning of each remeshing step, the fluid elements of147
the previous mesh are removed while the nodes are maintained (Figure 1a). To enable the detection of the148
fluid-solid interface, the boundary nodes of the solid structures are considered together with the fluid points in149
the generation of the Delaunay triangulation. As shown in Figure 1b, the new tessellation connects the fluid150
and the solid domains but it has many distorted or large elements, above all, close to the fluid-solid interface.151
These undesired triangular elements are removed through the Alpha Shape check. If after this step, no contact152
elements are maintained at the interface, as it occurs in the situation depicted at the left-hand column of Figure153
7
(a) Erase elements
(b) Delaunay triangulation
(c) Alpha Shape method
Figure 1: PFEM remeshing steps in a FSI problem with refined fluid mesh. Left column: non-active FSI. Right column: active
FSI.
1, the fluid and solid solutions are decoupled. Otherwise, if there is at least one element connecting the fluid154
and the solid domains, the FSI interaction is active (right-hand column of Figure 1). The pictures also show155
clearly that the fluid and solid nodes of the interface are overlapped. This characteristic feature of the PFEM156
for FSI analysis, has important implications on the solution scheme and the fluid-solid boundary conditions157
transmission. More details about the FSI strategy are provided in Section 2.3.158
Figure 1b also shows that the Delaunay triangulation is performed over a convex hull that includes all the159
nodes of the fluid domain, the rigid contours, and the boundary nodes of the solid domain. This mesh is then160
polished not only by the Alpha-Shape check but also by a further control that removes from the mesh all those161
elements that are composed only by rigid and/or interface nodes. This explains the presence of voids at the162
corners of the fluid domain and the absence of fluid elements inside the solid domain.163
Figure 1 is also helpful to highlight another important aspect of the PFEM strategy here used. With the164
aim of reducing the computational cost of large-scale computations, an improved mesh refinement technique165
has been implemented in the PFEM framework. Two different mesh sizes are clearly distinguishable in the166
discretizations plotted in Figure 1, a coarser one at the left-hand side and a finer one in the part near the167
8
solid obstacle. These mesh sizes and the respective regions are established at the beginning of the analyses168
and are maintained along with the problem solution, also in the presence of large changes in the topology of169
the fluid domain. In particular, the Alpha Shape check and the operations of nodes reallocation are performed170
considering different characteristic mesh sizes for the different part of the computational domain. In order to171
obtain a smooth evolution of the mesh between the coarse and the fine discretizations, a transition zone is172
also maintained along with the analyses. In this zone, the characteristic size of mesh is defined from a linear173
interpolation between the finest and the coarsest mesh sizes.174
This mesh refinement technique is particularly helpful for FSI problems solved with a conforming-mesh175
strategy, as the one followed in this work. In fact, it allows us to set similar mesh sizes for the fluid and solid176
elements at the interface and to use a different one (bigger or smaller) for the rest of the fluid domain, leading177
to an optimized discretization and to a reduced computational cost.178
Finally, we highlight that the solid boundaries may change during the analyses as a consequence of the179
fracturing of the structure. This implies that the solid boundary nodes considered in the PFEM remeshing are180
continuously updated during the analyses.181
Details on the PFEM remeshing procedure and its implications are given in [11, 38, 26, 39].182
2.2. FEM-DEM solution of the non-linear solid mechanics problem183
In this work, the onset and evolution of cracks in the structure are modeled with the so-called FEM-DEM184
approach [17, 40, 18, 20, 19]. The method efficiently combines FEM and DEM schemes to solve solid dynamics185
problems also in the presence of fracturing processes and complex contact interactions. An isotropic damage186
model is used to reproduce the solid material degradation. Those finite elements that exceed a pre-established187
damage limit are removed from the mesh and adequately replaced by a set of particles, modeled with the DEM.188
These particles are overlapped to each node of the removed finite element and are placed closest as possible189
between themselves but avoiding any initial indentation. This operation is performed automatically on the190
run. The contact interaction forces computed through these particles are then transferred to the FEM solution.191
These forces prevent the indentation of different solid blocks interacting with each other.192
In the following sections, we describe the solid governing equations, the coupling between the FEM and the193
DEM, and the treatment for composite materials (such as reinforced concrete). For more details about the194
numerical methodology, the reader is referred to previous works [40, 17, 18, 20].195
9
2.2.1. Governing equations for the structure196
The motion of a solid is governed by the linear momentum balance, formulated in a Total Lagrangian (TL)
framework as
ρs¨
uDivPb0=0in Ωs0×[0, T ] (5)
where ρsis the density of the solid, uis the displacement vector, Pis the first Piola-Kirchhoff stress tensor,197
b0is the external body force per current unit volume according to the initial undeformed solid configuration198
s0, and Tis the total time.199
After considering the required initial conditions, the problem set is closed by the boundary conditions:200
P·n=ˆ
t0in Γσ×[0, T ] (6)
u=ˆ
uin Γu×[0, T ] (7)
where nis the unit normal vector of the face, ˆ
t0are the tractions acting on the Neumann boundary Γσ, and201
ˆ
uare the prescribed displacements on the Dirichlet contour Γu.202
In the proposed coupled approach, the traction vector ˆ
t0is obtained as the sum of the contributions ema-203
nating from external loads, contact forces and fluid dynamic pressures as204
ˆ
t0=tloads +tcontact +tfluid (8)
where tloads are the tractions arising from standard external loads, tcontact are the tractions due to the205
contact with other solid bodies, and tf luid are the tractions induced by the fluid.206
2.2.2. FEM-DEM methodology for concrete material207
In this section, we describe the FEM-DEM methodology used for modeling the mechanical behavior of208
reinforced concrete structures, from the elastic regime to a generalized failure.209
The DEM is a popular method to simulate granular matter and non-continuum media. It is particularly210
useful to model the propagation of initial cracks in concrete structures and geomaterials. The frictional contact211
properties for the interaction between discrete particles are defined at the micro scale, while the material212
10
properties usually refer to experimental results at the macro scale. The step between both scales is not easy213
and requires a calibration task. The FEM otherwise is based on a continuum formulation involving the macro214
properties of the material. The FEM allows one to establish failure criteria compatible with the equilibrium215
equations in continuum mechanics, which makes it consistent and easy to apply for different materials.216
Extensive research has been carried out in the last years to combine FEM and DEM procedures, taking217
profit the advantages of both numerical methods. The authors [40, 17, 18] have developed a coupled FEM-218
DEM formulation for the numerical simulation of cracks starting from a finite element discretization of the219
domain.220
The FEM-DEM formulation used here discretizes the continuum with linear 3-nodded triangles (in 2D) and221
4-nodded tetrahedra (in 3D) whose nodes define the position of a (virtual) discrete element. These particles are222
introduced in the simulation process when cracks appear. The normal contact forces between discrete elements223
are calculated integrating the stiffness matrix of the corresponding linear triangle along the element edges that224
connect the discrete particles. The mechanical problem in the crack-free region is solved using the standard225
FEM and an appropriate constitutive model. In the applications developed by the authors, the onset and226
evolution of damage in each element are governed by a standard isotropic damage model and a Mohr-Coulomb227
failure criterion.228
The onset of a crack at the center of an element side depends on the damage level at that point. The stresses229
over an element edge are computed as the mean of the stresses in the elements sharing that side. Once the230
damage limit is reached, a stiffness loss is induced in the element associated with the area determined by the231
centroid of the triangle and the damaged side. For further details see [40, 17, 18].232
Computation of contact forces with the DEM233
In order to compute the interaction contact forces between colliding solids and to avoid non-physical pen-234
etrations, a set of auxiliary DEM particles (discrete elements) is placed at the boundary nodes of the solid235
domains. These particles are generated at the beginning of the simulation and are used to detect and model236
the contact between the DEM particles placed on different solid blocks boundaries. We remark that during a237
fracturing process additional particles are placed over the new boundary nodes of the propagating crack. Once238
an indentation among particles or between a particle and a solid boundary is detected, a repulsive frictional239
contact force is computed via the DEM and then transmitted to the FEM solution.240
11
The treatment of the contact force in the DEM formulation here used is based on the works of Casas et241
al [41], O˜nate et al [42] and Thornton et al. [43]. A description of the contact force evaluation within the242
FEM-DEM approach is given in [20].243
The contact force between two particles or between a particle and a boundary Fcontact can be decomposed
into its normal and tangential components as:
Fcontact =Fnn+Ftt(9)
The normal component of the contact force Fnis obtained as a combination of an elastic (Fn,el) and a
damping (Fn,damp) parts, i.e.:
Fn=Fn,el +Fn,damp (10)
Analogously, the tangential component of the contact force (Eq. 9) is computed as:
Ft=Ft,el td+Ft,damptν(11)
where the directions tdand tνare based on the particle kinematics during its tangential deformation [42].244
FEM-DEM solution scheme245
The time integration scheme used in the FEM formulation is implicit whereas the DEM employs an explicit246
integration scheme. This implies that the time step required for each method can be considerably different. In247
order to reduce the computational cost of the analyses, a sub-stepping procedure is used [18, 20]. Thus, for each248
FEM solution of the implicit time step increment ∆t, the DEM problem is solved several times using a smaller249
time increment ∆teuntil the time of both solutions is synchronized again.250
At each explicit time step (et), the kinematics information of each DEM particle is interpolated between251
the FEM results of the previous time step (nt) and the converged values of the current step (n+1t). Then, the252
contact forces contribution Fcontact computed at each DEM solution step for each DEM element is added to253
the accumulated impulse I(et) as [20]254
I(et) = I(e1t)+∆te·Fcontact (12)
12
Once the sub-stepping operations are completed, the contact forces that are transferred to the FEM in the255
next implicit time step are computed as256
fcontact =I(et)
t(13)
This implies that in the FEM solution of a time step, the contact forces considered are those obtained with257
the DEM at the end of the previous time step. We note that the contact forces can be also computed at each258
non-linear iteration for the FEM. Nevertheless, this implicit computation of the contact forces would lead to259
an increase of the problem’s non-linearity and subsequently to a higher computational cost. For this reason,260
in this work, the contact forces are treated explicitly. However, for other applications where the modeling of261
the interaction between deforming solids is crucial for the problem solution accuracy, the implicit version of the262
scheme should be used.263
During the fracturing process and the material degradation, it may happen that some DEM elements become264
isolated. This occurs when all the neighboring FEM elements of some DEM particle have been removed from265
the mesh due to excessive damage. In this case, the motion of these free particles is governed by the DEM and266
not related anymore with the FEM solution of the rest of the structure.267
Validation example for FEM-DEM method. Brazilian tensile strength (BTS) test268
In order to clarify the usefulness and potential of the FEM-DEM technique, we present an example of the269
simulation of a standard BTS test on a cylindrical concrete sample of diameter D= 0.2mand thickness t=270
0.1m. The example has been taken from [40]. The theoretical value of the tensile strength is given by:271
ft=2P
πtD (14)
where Pis the applied load.272
Using the above expression and considering a tensile strength ft= 10KP a, the failure load is P= 314.16N.273
The rest of material properties are: Young modulus E= 21GP a, Poisson ratio ν= 0.2, and fracture energy274
Gf= 10J/m2.275
Three finite element meshes were used for the analysis with 9338, 31455 and 61623 4-noded linear tetrahedra,276
respectively. The crack patterns obtained for each mesh with the FEM-DEM approach are depicted in Figure277
13
2.278
Figure 2: 3D FEM-DEM analysis of BTS test on a concrete specimen. Damage zone and discrete elements generated. (a) Coarse
mesh. (b) intermediate mesh. (c) Fine mesh.
The numerical results for the load-displacement curve are presented in Figure 3. The numerical values279
obtained for the tensile strength were (coarse to fine mesh) 10693P a, 10351P a and 10235Pa which yielded a280
range of 2% to 6% error versus the expected value of ft= 10kP a.281
It is remarkable that the force-displacement results have a little sensibility to mesh refinement. The mesh282
independence of the FEM-DEM methodology is achieved thanks to the smoothing procedure of the stress field283
based on the super-convergence patch proposed by Zienkiewicz and Zhu [44]. In particular, the stresses are284
computed at the edges of the FE as an average of the Gauss point stress of those elements which share that285
edge [18, 20]. This operation can be seen as an enrichment of the stress field through the information derived286
from the elements that share the patch. Once the effective stress tensor is computed at the edge of the finite287
element, the standard procedure of the isotropic damage model is followed. This means that the damage288
criterion is checked and the damage value is computed at each edge. Next, the elemental damage derived from289
the computed damages at the edges is evaluated.290
Other applications of the FEM-DEM procedure can be found in [40, 17, 18, 19].291
14
Figure 3: 3D FEM-DEM analysis of BTS test on a concrete specimen. Force-displacement relationship for the three meshes used.
2.2.3. Composite constitutive model for reinforced concrete292
This work is focused on reinforced concrete structures which are composed of two materials: concrete and293
reinforcing steel bars. An isotropic damage model is used for the material degradation of the concrete, while an294
isotropic plastic model is employed for the steel bars. The compound composite material is modeled following295
the so-called rule of mixtures [45, 21, 22]. In this section, we describe the main features of the composite296
material constitutive model used. More details can be found in the referenced publications.297
Constitutive laws for concrete and steel298
The material degradation of the concrete parts of the structure is predicted according to the isotropic damage299
model developed by Oliver et al. [46].300
In this model, the historical damage variable delem governs the material degradation process. This variable301
ranges from 0 (intact material) to 1 (fully degraded material). The constitutive model relates the Green-302
Lagrange strain tensor Ewith the second Piola-Kirchhoff stress tensor Sas:303
S=CsE= (1 delem )C0E= (1 delem)ˆ
S,(15)
where ˆ
Sis the effective second Piola-Kirchhoff stress tensor, Cs= (1d)C0is the secant constitutive tensor,304
and C0is the standard elastic constitutive tensor.305
The damage parameter for an element delem is computed using the effective stresses sampled at the midpoint306
of the sides shared with adjacent elements. This allows the model to obtain super-convergent values of the307
stresses at these points [47, 48]. Details regarding the damage model and the super-convergent points are given308
in [20].309
15
For the steel rebars, an isotropic plastic model based on the Huber-Von Mises yield surface has been con-310
sidered. Both linear hardening and perfect plasticity situations have been accounted for. In a small strain311
framework, the isotropic plasticity constitutive law used for steel is defined as312
S=C0Ee=C0(EEp),(16)
where C0is the isotropic elastic constitutive tensor and Epis the plastic strain vector.313
Rule of mixtures model314
The reinforced concrete material is modeled according to the so-called classical rule of mixture theory (also315
called parallel mixing theory). This model was proposed by Trusdell and Toupin [49] in the 1960s and further316
generalized by Green and Naghdi [50], Ortiz and Popov [51], Oller et al. [45, 52]. Further developments317
of this theory led to the so-called Serial/Parallel rule of mixtures (SP-RoM) [21, 22, 23, 53]. This rule of318
mixtures distinguishes between the serial and parallel behavior of a composite material and reproduces better319
its anisotropic properties than the standard theory.320
Due to the good balance between accuracy and complexity/computational cost, the classical rule of mixtures321
has been used in this work. Its basic features are as follows.322
1. Each point of the composite material is representative of a specific set of component materials323
2. Each component material contributes to the behavior of the compound proportionally to its volumetric324
participation325
3. All the components have the same strain state (iso-strain condition)326
This implies that, at each material point, different materials can be taken into account, i.e. concrete, steel327
reinforcement and pre-stressed steel, among others. The material components are assumed to be isotropically328
and uniformly distributed over the composite domain. Furthermore, the composite behavior is directly related329
to the volumetric participation and the geometrical distribution of each of the composite constituents. We also330
allow the combination of different constitutive models for each constituent material.331
According to this RoM, the homogenized stress tensor of the composite material can be calculated as332
S=
n
X
i=1
ki·Si.(17)
16
where Siis the stress tensor of material iand ki=Vi/Vtis its volumetric participation, being Viits volume333
inside the total volume Vtof the homogenized domain.334
In an analogous way, the elastic constitutive tensor of the composite is obtained as335
C0=
n
X
i=1
ki·C0,i.(18)
In order to achieve a quadratic convergence rate, a numerical derivation of the tangent constitutive tensor336
is performed. The procedure can be found in [18, 20].337
In this work, the described RoM for reinforced concrete structures has been integrated into the FEM-DEM338
approach presented in Section 2.2.2.339
In Algorithm 1 we summarize the solution scheme of the extended FEM-DEM method used here to model340
the failure mechanisms of reinforced concrete structures.341
Algorithm 1: FEM-DEM solution algorithm for a generic time step nt;n+1 tof duration ∆t.
Initialization of the implicit transient dynamic scheme for the FEM:
k0,¯
u0n¯
u
Apply the DE contact forces obtained at the last time step as equivalent no dal force for the FEM (nfcontact of Eq.(9))
while kreff,dyn k> tol do
for Elements do
Integrate the damage constitutive law Sconcr = (1 delem)C0E
Integrate the plasticity constitutive law Ssteel =C0(EEp)
Calculate the composite stress level Scompo =kconcr Sconcr +ksteel Ssteel
Calculate the elemental tangent stiffness matrix K(e)
Tand internal forces vector f(e)
int
Assemble the elemental contributions: KTK(e)
Tand fint f(e)
int
end
Calculate the displacement increments ∆uk+1 =K1
Trk
k+= 1
end
for Elements do
if Damage 0.98 and norm(Ep)Ep
max then
Erase the finite element
Place discrete elements over the nodes of the removed finite element
end
end
Initialization of the explicit transient dynamic scheme for the DEM solution
etnt
while etn+1tdo
et+= ∆te
Obtain the FEM kinematic information for the discrete particles by interpolating the FEM results between the initial time
(nt) and the final one (n+1t)
Compute the contact forces Fcontact using Eq. (9)
Update the explicit contact impulses at each particle using I(et) = I(e1t)+∆te·Fcontact
Integrate the equations of motion for the free particles
Compute the displacements, velocities, and accelerations of free particles
end
Compute the updated equivalent contact nodal forces n+1fcontact using Eq. (13)
Validation test for the composite material strategy. Failure of a reinforced concrete beam342
We present a validation example of the extended FEM-DEM technique using the constitutive model for343
composite materials presented. The objective is to reproduce the four-point bending test of a reinforced concrete344
17
beam experimentally tested by Al-Mahmoud et al. [54]. The symmetry of the problem (Figure 4.a) allows us345
to study only half of the length and width of the beam. According to [54], a set of steel rebars is placed as346
longitudinal and transversal reinforcement.347
(a) (b)
Figure 4: Reinforced concrete beam. Initial geometry.
The dimensions of the beam and the steel reinforcement are provided in Figure 5. The finite element mesh348
has 2.5 M 4-noded tetrahedra and 442,275 nodes.349
(a) Geometry of half of the reinforced concrete beam (b) Steel reinforcement
Figure 5: Set up of the experiment conducted in Al-Mahmoud et al. [54]. Units in [mm].
In this example, the concrete material exhibits stiffness degradation (isotropic damage), whereas the steel350
undergoes plastic strains (isotropic plasticity). The material properties used are given in Table 1.351
Figure 6 shows the evolution of cracking in the beam modeled with the FEM-DEM. The cracks initiate at352
the tensile zones of the beam and propagate upwards due to the increasing load. At a certain point, cracks start353
developing near the supports (Figure 6.c) and tend to deflect due to shear stresses.354
The equivalent plastic strain distribution in the reinforcing steel in the last stage of the simulation is shown355
in Figure 7. As expected, plasticity is localized in the central part of the beam.356
Figure 8 depicts a comparison of the bending moment-displacement plots obtained in this work and the357
18
Table 1: Material properties of the reinforced concrete four-point bending test.
Material Variable Value Units
Concrete
Young modulus 30.1 GPa
Poisson ratio 0.2 -
Yield stress 3.5 MPa
Fracture energy 80 J/m2
Steel
Young modulus 200 GPa
Poisson ratio 0.3 -
Yield stress 1100 MPa
Hardening modulus 106Pa
Composite Lower steel bars volumetric part. 50% -
Upper steel bars volumetric part. 25% -
(a) t= 1.3s(b) t= 3.1s
(c) t= 4.3s(d) t= 11.9s
(e) Legend
Figure 6: Reinforced concrete beam solved with the FEM-DEM. Fracture path obtained during the simulation.
experiment conducted by Al-Mahmoud et al. [54]. The graphs show a good agreement between the numerical358
and the experimental results, proving the capability of the FEM-DEM approach for capturing accurately the359
three stages of the experiment, namely the elastic loading, the progressive cracks of concrete and the plasticityof360
the steel reinforcement until failure.361
2.3. Fluid-structure interaction362
The FSI problem has been solved by combining the PFEM solution of a free-surface fluid flow dynamics363
problem with the FEM-DEM approach for treating complex non-linear solid mechanics problems. We remark364
that the PFEM, the FEM, and the DEM are all formulated in a Lagrangian framework. This facilitates the365
exchange of information between the different numerical schemes.366
The PFEM and the FEM-DEM solution schemes are combined via a strongly coupled partitioned strategy.367
An Aitken relaxation scheme is used to improve the overall convergence of the FSI scheme.368
As explained in Section 2.1.2, the PFEM takes care of the contact detection between the evolving solid and369
19
Figure 7: Reinforced concrete beam solved with the FEM-DEM. Plastic strain development in the steel.
Figure 8: Bending moment-displacement evolution of the 4 point bending test.
fluid domains. In an active interaction situation (meaning that there is at least one fluid element sharing some370
node with the solid mesh), the FSI solution for a generic time step tn, tn+1 is obtained as follows. First, the371
fluid strategy computes the velocities ¯
vi+1
tn+1 , pressures ¯
pi+1
tn+1 and positions ¯
xi+1
tn+1 of the fluid domain (iteration372
i+ 1), considering the fluid-solid interface kinematics obtained at the end of the previous time step tn. Next,373
the updated fluid pressures at the interface Γiare transferred to the structure as pressure loads ¯
pi+1
tn+1,Γi.374
These pressures are used in the subsequent solid mechanics solution with the FEM-DEM. We remark here that375
this represents a simplification, since the normal component of the whole fluid Cauchy stress tensor should be376
transferred to the solid rather than the pressure only. However, for problems of impact and involving fluids with377
low viscosity, such as the one we are analyzing in this work, the effect of this simplification can be considered378
as reduced.379
The result of the solid strategy is the updated set of displacements ˜
ui+1
tn+1 , velocities ˜
vi+1
tn+1 and accelerations380
20
˜
ai+1
tn+1 . The next step is to relax the velocities ˜
vi+1
tn+1 at the fluid-solid interface via an Aitken scheme as:381
vi+1
tn+1,Γi=ωi˜
vi+1
tn+1,Γi+ (1 ωi)vi
tn+1,Γi(19)
where vi+1
tn+1,Γiis the relaxed velocity at the solid interface Γiand ωiis the Aitken relaxation parameter382
which calculation is:383
ωi+1 =ωi
RiT(RiRi1)
RiRi1
2(20)
where Riis the residual of the velocities at the interface Ri=˜
vi
tn+1,Γivi
tn+1,Γiand the initial value of the384
Aitken parameter has been taken as 0.825 in this work.385
This iterative scheme has to be repeated until the norm of the residual is lower than a certain predefined386
tolerance (tol). Convergence is achieved when:387
Ri
Number of DoF <tol (21)
All the operations realized to couple the PFEM and the FEM-DEM formulations are summarized in Algo-388
rithm 2.389
Algorithm 2: PDFEM coupled solution scheme for a time step.
For each time step:
Detect the new fluid-solid interface position with the PFEM
while
Ri
/Number of DoF > tol do
Fix the velocity and position of the interface nodes of the solid.
Solve the free-surface fluid flow with the PFEM (Section 2.1)
Free the velo cities and position of the boundary nodes of the solid
Update the values of the fluid pressure loads on the structure (Eq. 8)
Solve the FEM-DEM part of the calculation (do Algorithm 1, Section 2.2)
Relax the interface nodal velocities via Aitken relaxation (Eq. 19)
Check convergence of velo cities at the interface Γi(Eq. 21)
end
3. Experimental campaign390
In this section, we briefly describe the experimental results conducted by Arikawa et al. [7] related to391
the impact of tsunami waves on structures. The tsunami wave is generated inside a Large Hydro-Geo Flume392
(LHGF) installed at the Port and Airport Research Institute (PARI) in Japan.393
The LHGF wave generator consists of a fluid channel of 184mlong, 3.5mwide, and 12mdeep. The394
21
geometry of the LHGF is depicted in Figure 9. The wave paddle is located at the left end of the channel and395
moves rightwards in order to generate the tsunami wave. The maximum displacement of the paddle is 14 m.396
Figure 9: Experimental set up. Geometry of the Large Hydro-Geo Flume (LHGF) used in [7]. Units in [m].
In Figure 10 we plot the time function of the paddle motion as it was considered in [7].397
Figure 10: Paddle position along time according to Arikawa et al. [7].
The paddle motion acts as a piston and generates a solitary wave that propagates along the flume and impacts398
against the reinforced concrete wall located at the end of the channel. The fluid height level is monitored by a399
wave height gauge located in the central part of the geometry (wave gauge in Figure 9).400
The wall is placed 1.82maway from the slope and totally fixed in both lateral buttresses by stiff steel401
elements. Figure 11 depicts schematically a lateral view of the reinforced concrete wall.402
Figure 12.a shows the geometry of the reinforced concrete wall. The width of the wall is 2.7m. Each403
supporting column has a width of 0.3m. The height is 2.45mand its thickness is 0.06min the central part. The404
single set of reinforcement is placed at the middle of the section and consists of steel rebars of 6mm diameter405
22
Figure 11: Detail of the wall position and support. Deduced from Arikawa et al. [7].
at the interval of 20cm. The geometrical distribution of the reinforcement can be seen in Figure 12.b.406
(a) Dimensions and pressure gauges (b) Steel rebars distribution
(c) Steel rebars distribution, top view
Figure 12: Geometry of the reinforced concrete wall and steel reinforcement placed according to Arikawa et al. [7]. Units in [mm]
Arikawa et al. [7] provide the time evolution of the pressure at the wall, the wave height evolution along407
time at a certain location of the channel and a complete photographic report of the different reinforced concrete408
walls studied. The pressure measurements gauges considered in this work are the P1and P2points depicted in409
Figure 12.410
The wave height is experimentally tracked at a distance of 90mfrom the initial position of the paddle and at411
about 32mfrom the wall position. Using the system of coordinates plotted in Figure 9, the wave gauge position412
23
is x=30m.413
Arikawa et al. [7] also provide the final configuration of the reinforced wall after the impact of the water wave.414
An image is shown in Figure 13. The picture shows clearly the damage suffered by the wall as a consequence415
of the violent impact.416
Figure 13: Fractured reinforced concrete wall after the water wave impact. Image from Arikawa et al. [7].
4. Numerical results with PDFEM417
In this section, we present the numerical simulation of the experimental test case described in the previous418
section using the proposed PDFEM formulation.419
First, in Section 4.1, we provide a general description of the numerical model. Then, in Section 4.1.1, we420
present a deep analysis of the fluid dynamics problem. In this study, we verify the convergent behavior of the421
formulation with a two-dimensional (2D) analysis, we assess the good agreement of the numerical results with422
the experimental observations in terms of wave height evolution, and we discuss the PFEM mesh used for the423
simulation of this big scale three-dimensional (3D) problem. A special refinement algorithm has been used in424
this numerical campaign with the aim of reducing the computational costs of the analysis. More details about425
this strategy are provided in Section 4.1.1.426
Finally, in Section 4.1.2, we present and discuss the simulation of the 3D FSI problem. The numerical results427
are compared to the experimental ones provided in [7]. For comparison purposes, we have analyzed the full428
problem considering the concrete wall with or without the steel rebars.429
24
(a) Wall geometry (b) Steel rebars placed (c) FE mesh used, 37,000 elements
Figure 14: Geometrical details of the reinforced concrete wall.
4.1. Numerical set-up430
The flume and wall geometries considered are the same as those of the experimental tests of Arikawa et al.431
[7] described in Section 3.432
Figure 14 shows three different views of the reinforced wall showing its initial configuration, the spatial433
distribution of the reinforcing steel, and the FE mesh, which is initially composed of 33,322 4-noded tetrahedral434
elements.435
As previously mentioned, a specific refinement technique has been used for the fluid domain. This method436
increases the number of fluid elements significantly along with the analysis. Figure 15 shows the tetrahedral437
mesh at two instants of the analysis. Remarkably, at the beginning of the analysis the fluid mesh has 399,000438
tetrahedral elements, but during the wave propagation process, the number of elements increases up to 1.5439
million. More details about the remeshing algorithm used for this test are given in the next section.440
Concerning the material properties, we have used those provided in [7] for the reinforced concrete. Standard441
water properties are considered for the fluid. All the material properties are collected in Table 2. We remark442
that the volumetric participation of the reinforcing steel has been calculated considering the volume of the real443
reinforcement and the volume of the preset homogenization domain that includes the steel part.444
Finally, the paddle motion is the same as the one provided in [7] and given in Figure 10.445
25
Figure 15: PFEM mesh at two time instants.
4.1.1. Wave propagation analysis446
For such a large-scale 3D numerical simulation, it was of primary importance to determine a mesh configu-447
ration capable of giving accurate results with a reasonable computational cost. For this reason, the 2D problem448
was first analyzed with the PFEM for different mesh sizes assuming plane strain conditions and without consid-449
ering the presence of the wall. This 2D fluid dynamics study allowed us to assess the convergent behavior of the450
PFEM formulation, and to obtain useful information to set up the optimum 3D FEM mesh for the simulation451
of the actual experimental test.452
2D convergence analysis453
Five different discretizations were tested for the 2D PFEM analyses, spanning from the coarsest mesh with454
mean size h= 1mand initially formed by 878 3-noded triangular elements, to the finest discretization with455
mean element size h= 0.125mand initially composed by 60,165 linear 3-noded triangles.456
The numerical results for the different discretizations have been compared by analyzing the time evolution457
of the water level measured at the position of the experimental gauge, namely at 90mfrom the initial position of458
26
Table 2: Material properties of reinforced concrete wall and fluid (some properties* were calibrated since no data was given in [7]).
Material Variable Value Units
Concrete
Young modulus 21 GPa
Poisson ratio 0.2 -
Yield stress compression 33 MPa
Fracture energy* 100 J/m2
Density 2400 kg/m3
Steel
Young modulus 210 GPa
Poisson ratio 0.3 -
Yield stress 500 MPa
Hardening modulus 0 Pa
Composite Steel volumetric part. 25% -
Fluid
Density 1000 kg/m3
Dynamic viscosity 0.001 P a ·s
the paddle (x=30maccording to the coordinates system of Figure 16). The measurement point is placed at a459
crucial position since the wave breaks very close to the gauge position. This is also confirmed by the snapshots460
collected in Figure 16 showing the wave configuration obtained with the three finest meshes (h= 0.5m, 0.25m,461
0.125m) at the moment when the wave passes through the measurement point (t= 13.55s). For the three cases,462
the gauge is reached practically at the same time. It can be appreciated from these pictures that the gauge463
position is almost coincident with the break point of the wave.464
The pictures also show a good agreement between the results for the two finest meshes (h= 0.25mand465
h= 0.125m) which exhibit a very similar velocity field and water wave configuration. The results of the mesh466
for h= 0.5mcan be also considered acceptable, although the wave is slightly smaller and is moving a bit slower467
than for the finest meshes.468
The convergent behavior of the PFEM solution is also confirmed by the graphs of Figure 17 showing the time469
evolution of the water level measured at the gauge position (x=30m) and obtained with the five different470
meshes. The results show that the coarser the mesh is, the lower is the wave height and the slower is its motion.471
We believe that this behavior is due to the effect of the no-slip boundary conditions considered at the rigid472
contours of the fluid domain. Figure 17b shows a comparison between the two finest meshes confirming that a473
convergent solution is reached for h= 0.125m.474
3D PFEM mesh design475
As mentioned earlier, the 2D analyses were carried out to design the best mesh to be used for the 3D PFEM476
simulation. The 2D study showed that, on the one hand, the mesh size should not exceed h= 0.5mto yield477
reasonably accurate results. On the other hand, a mesh with size h= 0.125mensures converged results.478
27
(a) h= 0.5m
(b) h= 0.25m
(c) h= 0.125m
Figure 16: 2D PFEM results. Water wave obtained at t= 13.55swith three different element sizes (h).
Nevertheless, using a mesh size h= 0.5min the 3D simulations would have led to significant topological479
drawbacks. First of all, it would have induced an excessive artificial volume gain as a consequence of the480
remeshing operations and, in particular, due to the creation of contact elements between the fluid free surface and481
the lateral boundaries (this interaction is absent in the 2D plane strain analyses). In this test, this inconvenience482
is accentuated by the narrowness of the channel which is only 3.5m wide. Furthermore, using an element size483
of h= 0.5min the 3D mesh would have created very skewed tetrahedral fluid elements in the zone at the484
end of the slope (close to x= 0m). These elements would have been inevitably removed by the Alpha-Shape485
technique (Section 2.1.2), producing an artificial loss of computational domain and also the risk of fluid leakage486
through the boundaries. Finally, since a conforming mesh FSI algorithm is used, the fluid and solid meshes must487
have a similar size in the interface zone to avoid topological problems and to transfer accurately the boundary488
conditions. Using a fluid mesh with elements size h= 0.5mwould have forced us to use an excessive coarse489
discretization for the structure. In this respect, we remark that the wall width is only 0.06m.490
28
(a) All meshes (b) Finest meshes
Figure 17: Time evolution of the water level at x=-30m obtained the PFEM 2D model with different meshes.
All these drawbacks would have been significantly limited by using a mesh size of h= 0.125m. However,491
this choice would have led us to use a mesh of around 7.5 million linear tetrahedra and so to an excessive492
computational cost.493
All these considerations brought us to avoid using a standard homogeneous mesh, as it is generally done in494
the PFEM, and to design a special discretization for this test. In particular, the finest mesh size tested in the495
2D problems (h= 0.125m) has been used in the free-surface zone (y > 0.125m) and in the critical zone at496
the top of the slope described before (x > 5m). Instead, in the rest of the discretization (the inner part of497
the fluid domain) a mesh size h= 0.5mhas been employed. Figure 18 shows two views of the initial 3D mesh498
where it can be appreciated the different refinement within the fluid domain.499
(a) Back zone (b) Front zone
Figure 18: Details of the initial 3D mesh.
This configuration allowed us to reduce significantly the total amount of elements (399,010 tetrahedra in the500
initial PFEM mesh) while maintaining good performances of the finite element discretization. We remark that,501
as the refined zone is defined by a pre-fixed geometrical border between the two mesh regions (y=0.125m),502
29
the number of elements increases as soon as the volume above this limit grows. Thus, in this specific problem,503
the number of linear tetrahedra increases progressively with the motion of the paddle and the consequent rise of504
the water level and the wave formation. For example, at t= 13.5s, the number of 4-noded tetrahedral elements505
in the fluid mesh is around 1,300,000.506
Validation of the PFEM results507
The results obtained with the described 3D PFEM mesh were compared to the experimental observations508
in terms of the time evolution of the water level at the measuring gauge (x=30m). Figure 19 shows the509
experimental values together with the 3D PFEM results. For comparison purposes, the 2D results obtained510
with the finest mesh are also depicted in the picture.511
The graphs show a very good agreement between the experimental and the numerical results. The numerical512
model is shown to be capable of capturing well the duration and amplitude of the wave, although it slightly513
overestimates the first peak.514
It is also interesting to compare the 2D and the 3D solutions. The 2D wave propagates slightly faster and515
with a higher amplitude than in the 3D case. This result is compatible with the plane strain assumption that516
does not take into account the resistance exerted by the lateral walls.517
Figure 19: Time evolution of the water level at x=-30m obtained experimentally and with the PFEM 2D and 3D models.
In Figure 20, we report the velocity field obtained close to the measurement point obtained with the 3D518
mesh. The pictures show that the wave breaks almost at that position, some instants before than in the519
2D simulation (Figure 16). These results also evidence the capability of the PFEM for dealing with complex520
problems involving large changes of topology, such as the formation of breaking waves and water splashes.521
30
(a) t=13.20s
(b) t=13.35s
(c) t=13.50s
Figure 20: 3D PFEM results at three time instants. Central section and view from the top.
31
4.1.2. Structural collapse of the reinforced concrete wall under a tsunami wave impact522
We discuss here the numerical simulation of the full FSI problem using the PDFEM approach described in523
this paper.524
In Figure 21, we provide a lateral view of the 3D simulation, while in Figure 22 we show the velocity field525
results obtained at the central section of the flume.526
(a) t= 2.5s
(b) t= 5.5s
(c) t= 11.5s
(d) t= 14s
(e) t= 15.6s
Figure 21: Time lapse of the tsunami wave generation, movement and impact against the reinforced concrete wall.
The pictures show clearly the dynamics of the problem, namely the motion of the paddle, the generation and527
propagation of a solitary wave, its breakage in the final slope of the flume, the violent impact of the water mass528
against the solid barrier, and the consequent huge waves and splashes. From these views, it can be appreciated529
that the reinforced concrete wall stays at its place, although, as it will be shown later, it suffered huge damages530
as a consequence of the impact. The overall time duration of the numerical test is 17sand the impact occurs531
after around 14sfrom the onset of the paddle motion.532
The overall dynamics of the test obtained agree well with the experimental work of Arikawa et al. [7].533
The accuracy of the numerical results has been quantitatively proved for the water wave propagation in the534
32
(a) t= 2.5s
(b) t= 5.5s
(c) t= 11.5s
(d) t= 14s
(e) t= 15.6s
(f) Velocity legend [m/s].
Figure 22: Time lapse of the tsunami wave velocity at the central section of the PFEM domain.
previous section. So here we focus only on the analysis of the response of the reinforced concrete wall.535
In Figure 23 we report two detailed views of the wall hit by the water wave at the final instant of the536
numerical analysis. The pictures show clearly the damages suffered by the structure and the huge wave created537
by the impact.538
Figure 24 shows the graphs of the time evolution of the pressure obtained numerically and experimentally539
at the two gauges placed in the front of the wall (Figure 12). The numerical results obtained at the lowest540
gauge (P1) capture well the impact pressure duration and the peak value obtained experimentally. More541
discrepancies are exhibited in the second graph referring to P2. Here, the numerical solution overestimates the542
experimental observations. In particular, we observe that while the pressure peak values obtained numerically543
at the measurement points are similar, the pressure magnitudes obtained in the experiment at P2are much544
33
(a) View from the back
(b) View from the front
Figure 23: 3D PDFEM results. Fractured geometry of the reinforced concrete wall.
34
(a) P1 pressure gauge (b) P2 pressure gauge
Figure 24: Time evolution of pressure at two pressure gauges. Experimental data from [7] and numerical results obtained with the
proposed method.
smaller than the ones measured at P1, even though the distance between the two pressure gauges is only 35cm.545
We also note that in both experimental curves of the pressure gauges some traction peaks appear, whereas in546
the numerical tests the pressure is always negative (compression).547
For comparison purposes and to show the important effect of the steel reinforcement, the same problem has548
been simulated considering the same wall but without the steel rebars. The concrete material properties are549
the same as those of the previous study (Table 2).550
Figure 25 shows two different views of the impact of the water wave against the concrete structure. The551
pictures show that the wall is completely broken. Several cracks have propagated very quickly through the552
non-reinforced wall showing a clear brittle behavior of the structure.553
The solid debris created by the generalized failure of the non-reinforced wall are clearly visible in Figure 26.554
The isolated parts of the structure are dragged out by the fluid flow proving the capability of the FSI algorithm555
for dealing with non-constrained solid bodies immersed in a fluid flow. We highlight the importance in this test556
of reproducing accurately the interaction between different solid bodies to avoid non-physical indentations.557
It is interesting to compare the failure process obtained with the two different structures. Figures 27-28558
show the fracture paths in three different views obtained for the non-reinforced and reinforced concrete walls,559
respectively.560
Differently from the non-reinforced wall, which presents a diffuse failure, the structure with the steel rebars561
can partially withstand the violent wave impact. The reinforced concrete structure shows a much more ductile562
35
Figure 25: 3D PDFEM results. Fractured geometry of the non-reinforced concrete wall.
36
Figure 26: 3D PDFEM results. Motion of the broken solids created by the collapse of the plain concrete wall.
(a) Front part (b) Rear part
(c) vertical view
Figure 27: Front and rear parts of the non-reinforced wall. Fracture path obtained in the 3D PDFEM analysis.
37
(a) Front part (b) Rear part
(c) vertical view
Figure 28: Front and rear parts of the reinforced concrete wall. Fracture path obtained in the 3D PDFEM analysis.
behavior than the non-reinforced one exhibiting significant plastic deformations of the reinforcing steel, as it563
can be perceived from the top view of Figure 28.564
The numerical results obtained with the reinforced concrete structure have also been compared to the failure565
configuration obtained experimentally and plotted in Figure 13. In both the experimental test and the numerical566
model, the wall collapses under the so-called pushing shear failure mode [7]. The wall withstands the fluid impact567
and is not completely washed out, as it occurs with the non-reinforced structure. In both cases, the concrete is568
removed by the fluid flow in several parts of the wall. Note that there is a clear concentration of concrete and569
steel damages in the lower part of the structure [7], which is in agreement with the results obtained numerically.570
In conclusion, we consider that the differences between the numerical results and the experimental observa-571
tions regarding the structural response can be considered as reasonable for such a complex and computationally572
demanding 3D analysis. The PDFEM numerical tool has shown to be capable of reproducing the general behav-573
ior of the structural failure as observed in the experiment with an acceptable accuracy, both in terms of fracture574
path and pressure measurements. A finer mesh in the solid structure would certainly improve the quality of575
the numerical results. From this point of view, the use of a conforming-mesh FSI algorithm is a bit limiting.576
For this reason, new techniques enabling the use of significantly different mesh sizes for the fluid and the solid577
domains will be explored in following works.578
38
5. Conclusions579
In this work, a coupled numerical approach combining a PFEM model for fluid dynamics with a FEM-DEM580
formulation for non-linear solid mechanics, has been used to reproduce the propagation of a large solitary wave581
in a channel and the structural failure of plain and reinforced concrete walls submitted to high forces originated582
by the impacting wave. The numerical model used is an enhanced version of the FSI method recently proposed583
in [20]. The major improvements of the PDFEM formulation here presented are the inclusion of the effect of584
steel reinforcement (and its plastic behavior) in the solid domain and the use of refining boxes in the fluid part585
of the analysis domain. Both enhancements have been validated and successfully used in this work.586
The PDFEM formulation has been applied to the simulation of the laboratory experiments carried out at the587
Large Hydro-Geo Flume (LHGF) facility [7] aimed at studying the collapse mechanisms of plain and reinforced588
concrete plates by an impulsive force induced by a large impacting wave. The numerical results obtained with589
the PFEM have been compared to the experimental ones focusing on three main aspects: the water wave height,590
the pressure field measured at the wall and the collapse mechanism of the structure.591
Regarding the wave height study, accurate and convergent results using several FE meshes have been ob-592
tained, both in 2D and in 3D geometries. The efficiency of the refining algorithm used in the PFEM framework593
has been also demonstrated.594
In terms of pressure measurements, the magnitude and duration of the fluid impact, as well as the asymptotic595
trend of the numerical results, have an acceptable agreement with those obtained in [7]. More specifically, the596
numerical results at the lowest gauge (P1) are very close to the experimental ones, whereas the numerical model597
overestimates the pressure measured experimentally at the highest gauge (P2). Nevertheless, considering the598
high complexity of the problem, the numerical results can be considered satisfactory and capable of giving useful599
insights for engineering considerations.600
With respect to the fracture path, the numerical results also show a good agreement with the experimental601
test. Remarkably, in both the experimental test and the numerical model, the wall collapses under the so-called602
pushing shear failure mode [7]. In particular, the main damages are located at the lower part of the wall where603
both the concrete and the steel bars are seriously damaged.604
For comparison purposes, the same problem has been modeled for a plain concrete wall. This study has605
shown the crucial effect of steel reinforcement. In fact, while the reinforced structure partially withstands the606
39
fluid impact, the non-reinforced wall is completely washed out by the water wave.607
In conclusion, this work has shown clearly the suitability of the PDFEM procedure for the analysis of608
complex FSI problems involving the collapse of reinforced concrete structures, and its potential for the study609
of complex engineering problems, such as the effects of extreme water events on civil constructions.610
6. Acknowledgments611
The authors are grateful to Prof. T. Arikawa for his advice in the definition of the laboratory test in the612
LHGF facility. This research was partially funded by the project PARAFLUIDS (PID2019-104528RB-I00) of613
the National Research Plan of the Spanish Government. The authors also acknowledge the financial support614
from the CERCA programme of the Generalitat de Catalunya, and from the Spanish Ministry of Economy and615
Competitiveness through the ”Severo Ochoa Programme for Centres of Excellence in R&D” (CEX2018-000797-616
S).617
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