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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·σ−ρfg=0in Ωf×(0, T ) (1)

5

5 · v−1

κf

∂p

∂t = 0 in Ωf×(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¨

u−DivP−b0=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

Ωs0, 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(e−1t)+∆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= (1−d)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(E−Ep),(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:

k←0,¯

u0←n¯

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(E−Ep)

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: KT←K(e)

Tand fint ←f(e)

int

end

Calculate the displacement increments ∆uk+1 =K−1

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

et←nt

while et≤n+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(e−1t)+∆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(Ri−Ri−1)

Ri−Ri−1

2(20)

where Riis the residual of the velocities at the interface Ri=˜

vi

tn+1,Γi−vi

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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