Fluid-soil-structure interaction analysis with the particle finite element method

Pulished in ''Bytes and Science'' G. Zavarise and D. Boso (Eds.), pp. 119 - 130, CIMNE, Barcelona Spain, 2012
==Abstract==

We present some developments and applications of the Particle Finite   Element Method (PFEM) for analysis of complex coupled problems in   mechanics involving  fluid-soil-structure interaction.

'''Keywords''' Particle finite element method, Fluid-soil structure interaction.

==1 INTRODUCTION==

The analysis of problems involving the interaction of fluids, soil/rocks and structures  is of  relevance in many areas of engineering. Examples of fluid-soil-structure interaction (FSSI) problems are common in the study of landslides and their effect on reservoirs and adjacent structures, off-shore and harbour structures under large waves, constructions hit by floods and tsunamis, soil erosion and stability of  rock-fill dams in overspill situations, etc.

The authors have successfully developed in previous works a particular class of Lagrangian formulation for solving problems involving complex interactions between (free surface) fluids and solids. The method, called the ''particle finite element method'' (PFEM, http://www.cimne.com/pfem/), treats the mesh nodes in the fluid and solid domains as particles which can freely move and even separate from the main fluid domain representing, for instance, the effect of water drops. A mesh connects the nodes discretizing the domain where the governing equations are solved using a stabilized FEM.

An advantage of the PFEM Lagrangian formulation is that the convective terms disappear from the fluid equations. The difficulty is however transferred to the problem of adequately (and efficiently) moving the mesh nodes.  We use a  mesh regeneration procedure base on an extended Delaunay tesselation <span id='citeF-4'></span>[[#cite-4|[4]]]. The theory and applications of the PFEM are reported in <span id='citeF-1'></span><span id='citeF-5'></span><span id='citeF-6'></span><span id='citeF-7'></span><span id='citeF-8'></span><span id='citeF-10'></span><span id='citeF-11'></span><span id='citeF-12'></span><span id='citeF-13'></span><span id='citeF-14'></span><span id='citeF-15'></span><span id='citeF-16'></span>[[#cite-1|[1]],[[#cite-5|5]],[[#cite-6|6]],[[#cite-7|7]],[[#cite-8|8]],[[#cite-10|10]],[[#cite-11|11]],[[#cite-12|12]],[[#cite-13|13]],[[#cite-14|14]],[[#cite-15|15]],[[#cite-16|16]]].

The FEM solution of  (incompressible) fluid flow problem implies solving the momentum and incompressibility equations. In our work we use a stabilized mixed FEM based on the Finite Calculus (FIC) approach which allows for a linear approximation for the velocity and pressure variables <span id='citeF-9'></span><span id='citeF-11'></span>[[#cite-9|[9]],[[#cite-10|10]]].

In the next section the key ideas of the PFEM are outlined. Next the basic equations for a compressible/incompressible Lagrangian continuum are  presented. An algorithm for the transient solution is briefly described. The methods for mesh generation and for identification of  free surface nodes are outlined. The procedure for treating the frictional contact interaction between  interfaces is explained.  We present several examples of application of the PFEM  to solve FSSI problems such as the erosion of a river bed, the stability of breakwaters and constructions under sea waves and the study of landslides.

==2 THE BASIS OF THE PARTICLE FINITE ELEMENT METHOD==

In the PFEM  both the fluid and the solid domains are modelled using an ''updated'' ''Lagrangian formulation'' <span id='citeF-6'></span><span id='citeF-17'></span>[[#cite-6|[6]],[[#cite-17|17]]]. That is, variables are assumed to be known in the ''   current configuration'' at time <math display="inline">t</math>. The new set of variables in both domains are sought for in the ''next or updated configuration'' at time <math display="inline">t+\Delta t</math>.

<div id='img-1'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Samper_534298620-Figure2_con276.png|400px|Sequence of steps to update a “cloud” of nodes representing a domain containing a fluid and a solid from time n   (tₙ)  to   time n+2 (tₙ+2∆t)  ]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 1:''' Sequence of steps to update a “cloud” of nodes representing a domain containing a fluid and a solid from time <math>n</math>   (<math>t_n</math>)  to   time <math>n+2</math> (<math>t_n +2\Delta t</math>)  
|}

We define the ''collection or cloud of nodes (C)'' pertaining to the fluid and solid domains, the ''volume (V)'' containing these domains and the ''mesh (M)'' discretizing <math display="inline">V</math>.

A typical solution with the PFEM involves the following steps.

<ol>
<div id='step-1'></div>
<li>The starting point at each time step is the cloud of points in the fluid and solid   domains. <math display="inline">{}^nC</math> denotes the cloud at time <math display="inline">t=t_n</math> (Figure [[#img-1|1]]).  </li>

<li>Identify the  boundaries for both the fluid and solid domains. The Alpha Shape method   <span id='citeF-2'></span>[[#cite-2|[2]]] is used for the boundary definition.  </li>

<li>Discretize the fluid and solid domains with a finite element mesh <math display="inline">{}^nM</math>. We use a mesh generation scheme based on the extended Delaunay tesselation <span id='citeF-4'></span><span id='citeF-10'></span>[[#cite-4|[4]],[[#cite-10|10]]].  </li>

<li>Solve the coupled Lagrangian equations of motion for  the fluid and the   solid domains. Compute the state variables in both domains at the   next (updated) configuration for <math display="inline">t+\Delta t</math>.  </li>

<li>Move the mesh nodes to a new position <math display="inline">{}^{n+1} C</math> where <math display="inline">n+1</math> denotes   the time <math display="inline">t_n+\Delta t</math>, in terms of the time increment size.  </li>

<li>Go back to step [[#step-1|1]] and repeat the solution process for the next time step   to obtain <math display="inline">{}^{n+2} C</math> (Figure [[#img-1|1]]). </li>

</ol>

The quality of the numerical solution  depends on the discretization as in the FEM. Adaptive mesh refinement techniques can be used to improve the solution.

==3 PFEM FORMULATION FOR A LAGRANGIAN CONTINUUM==

===3.1 Governing equations===

The equations to be solved are the standard ones in continuum mechanics, written in the Lagrangian frame of reference <span id='citeF-17'></span><span id='citeF-18'></span>[[#cite-17|[17]],[[#cite-18|18]]]:

''Momentum''

<span id="eq-1"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\rho {\partial v_i \over \partial t}={\partial \sigma _{ij} \over \partial x_j}+b_i\qquad \hbox{in } V  </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (1)
|}

''Pressure-velocity relationship''

<span id="eq-2"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\frac{1}{K} {\partial p \over \partial t}-{\partial v_i \over \partial x_i}=0 \qquad \hbox{in }V  </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (2)
|}

<math display="inline">v_{i}</math> is the velocity along the ''i''th global  axis, <math display="inline">p</math> is the pressure (assumed to be positive in tension) <math display="inline">\rho </math> and <math display="inline">K</math> are the density and bulk modulus of the material and <math display="inline">b_i</math> and <math display="inline">\sigma _{ij}</math> are the body forces and the Cauchy stresses. Eqs.([[#eq-1|1]]) and ([[#eq-2|2]]) are completed with the constitutive relationships for the fluid and solid materials:

''Incompressible fluid''

<span id="eq-3"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>{}^{t+1}\sigma _{ij} = 2 \mu \dot \varepsilon _{ij} + {}^{t+1} p \delta _{ij}   </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3)
|}

''Compressible/quasi-incompressible solid''

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>{}^{t+1}\sigma _{ij}= {}^{t}\hat \sigma _{ij} + 2 \mu \dot \varepsilon _{ij} +\lambda \dot \varepsilon _{ii} \delta _{ij} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (4)
|}

where <math display="inline">\hat \sigma _{ij}</math> are the component of the stress tensor <math display="inline"> [\hat \sigma ]= \frac{1}{J} {\boldsymbol F}^T {\boldsymbol S} {\boldsymbol F}</math>, where <math display="inline">{\boldsymbol S}</math> is the second Piola-Kirchhoff stress tensor, <math display="inline">{\boldsymbol F}</math> is the deformation gradient tensor and <math display="inline">J = \det {\boldsymbol F}</math> <span id='citeF-17'></span>[[#cite-17|[17]]]. Parameters <math display="inline">\mu </math> and <math display="inline">\lambda </math>  take the following values <span id='citeF-6'></span>[[#cite-6|[6]]].

''Fluid material:'' <math display="inline">\mu </math>: viscosity 

''Solid material:'' <math display="inline">\displaystyle \mu = \frac{\Delta t G}{J}</math>; <math display="inline"> \displaystyle \lambda =\frac{2G \nu \Delta t}{J(1-2\nu )}</math>, where <math display="inline">\nu </math> is the Poisson ration, <math display="inline">G</math> is the shear modulus and <math display="inline">\Delta t</math> the time increment.

In Eqs.([[#eq-3|3]]) and ([[#eq-4|4]]), <math display="inline">\dot \varepsilon _{ij}</math>  is the rate of deformation, <math display="inline">\mu </math> is the viscosity and <math display="inline">\delta _{ij}</math> is the Kronecker delta. <math display="inline">{}^{t} (\cdot )</math> denotes values at time <math display="inline">t</math>. Indexes in Eqs.([[#eq-1|1]])-([[#eq-4|4]]) range from <math display="inline">i,j=1,n_{d}</math>, where <math display="inline">n_d</math>  is the number of space dimensions of the problem (i.e. <math display="inline">n_{d} = 2</math> for 2D problems). These equations  are completed with the standard boundary conditions <span id='citeF-6'></span><span id='citeF-17'></span><span id='citeF-18'></span>[[#cite-6|[6]],[[#cite-17|17]],[[#cite-18|18]]].

===3.2 Discretization of the equations===

A key problem in the numerical solution of Eqs.([[#eq-1|1]])-([[#eq-2|2]]) is the satisfaction of the mass balance condition for the incompressible case (i.e. <math display="inline">K=\infty </math> in Eq.([[#eq-2|2]])). Many procedures to solve his problem exist in the FEM literature <span id='citeF-18'></span>[[#cite-18|[18]]]. In our approach we use a stabilized formulation based in the so-called finite calculus procedure <span id='citeF-9'></span><span id='citeF-11'></span>[[#cite-9|[9]],[[#cite-11|11]]]. The essence of this method is the solution of a ''modified mass balance'' equation which is written as

<span id="eq-5"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\frac{1}{K} {\partial p \over \partial t} - {\partial v_i \over \partial x_i} -\sum \limits _{i=1}^{3}\tau {\partial q \over \partial x_i}\left[{\partial p \over \partial x_i} +\pi _i\right]=0  </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (5)
|}

where <math display="inline">q</math> are weighting functions, <math display="inline">\tau </math> is a stabilization parameter given by <math display="inline">\tau = \left(\frac{2\rho \vert \mathbf{v}\vert }{h}+\frac{8\mu }{3h^2} \right)^{-1}</math> <span id='citeF-9'></span><span id='citeF-11'></span>[[#cite-9|[9]],[[#cite-11|11]]] with <math display="inline">h</math> being a characteristic length of each finite element  and <math display="inline">\vert \mathbf{v}\vert </math> is the modulus of the velocity vector.

In Eq.([[#eq-5|5]]) <math display="inline">\pi _i</math>   are auxiliary pressure projection variables chosen so as to ensure that the second term in Eq.([[#eq-5|5]]) can be interpreted as weighted sum of the residuals of the momentum equations. The set of governing equations  is completed by adding the following constraint equation <span id='citeF-11'></span>[[#cite-11|[11]]]

<span id="eq-6"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\int _V \tau w_i\left({\partial p \over \partial x_i} +\pi _i\right)dV=0 \quad i=1,n_d \quad \hbox{(no sum in )} i  </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (6)
|}

where <math display="inline">w_i</math> are arbitrary weighting functions. The rest of  integral equations are obtained by applying the standard weighted residual technique to the governing equations ([[#eq-1|1]])-([[#eq-5|5]]) and the  boundary conditions <span id='citeF-15'></span><span id='citeF-17'></span><span id='citeF-18'></span>[[#cite-15|[15]],[[#cite-17|17]],[[#cite-18|18]]].

We interpolate next in the standard finite element fashion the set of problem variables. For 3D problems these are the three velocities <math display="inline">v_i</math>, the pressure <math display="inline">p</math>, the temperature <math display="inline">T</math> and the three pressure gradient projections <math display="inline">\pi _i</math>.  In our work we use equal order ''linear interpolation'' for all variables over meshes of 3-noded triangles (in 2D) and 4-noded tetrahedra (in 3D). The set of discretized equations using the Galerkin technique has the following form

''Momentum''

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\mathbf{M} \dot{\bar{\boldsymbol v}} + \mathbf{K} \bar {\boldsymbol v} + \mathbf{G} \bar {\boldsymbol p}= {\boldsymbol f} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (7)
|}

''Pressure-velocity relationship''

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\bar {\boldsymbol M} \dot{\bar{\boldsymbol p}} - \mathbf{G}^T \bar{\boldsymbol v} - \mathbf{L}\bar {\boldsymbol p} - \mathbf{Q} \bar {\boldsymbol \pi }=\mathbf{0} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (8)
|}

''Pressure gradient projection''

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\hat {\boldsymbol M} \bar {\boldsymbol \pi }+\mathbf{Q}^T\bar {\boldsymbol p}=\mathbf{0} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (9)
|}

In Eqs.([[#eq-7|7]])-([[#eq-9|9]]) <math display="inline">\bar{(\cdot )}</math> denotes nodal variables, <math display="inline">\dot{\bar{(\cdot )}}=  {\partial  \over \partial t}\bar{(\cdot )}</math>. The different matrices and vectors  are given in  <span id='citeF-6'></span><span id='citeF-11'></span><span id='citeF-13'></span>[[#cite-6|[6]],[[#cite-11|11]],[[#cite-13|13]]].

The solution in time of Eqs.([[#eq-7|7]])-([[#eq-9|9]]) can be performed using any time integration scheme typical of the updated Lagrangian FEM <span id='citeF-17'></span>[[#cite-17|[17]]]. A basic algorithm following the  process described in Section [[#section-2|2]] is presented in Box [[#Box-1|I]].

<span id="Box-1"></span>
'''Box I'''. Basic PFEM algorithm for a Lagrangian continuum

{|  class="floating_tableSCP wikitable" style="text-align: right; margin: 1em auto;min-width:50%;"

|- style="border-top: 2px solid;"
| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | 1. LOOP OVER TIME STEPS, <math display="inline">t=1</math>, NTIME

Known values

<math display="inline">^{t} \bar{\boldsymbol x},{}^{t} \bar{\boldsymbol v},{}^{t} \bar{\boldsymbol p},{}^{t} \bar{\boldsymbol \pi },{}^{t} \bar{T},{}^{t} \mu ,{}^{t}{\boldsymbol f},{}^{t}\mathbf{ q},{}^{t} C,{}^{t} V,{}^{t} M</math> 

<span id='section-2'></span>
2. LOOP OVER NUMBER OF ITERATIONS, <math display="inline">i=1</math>, NITER 

<math display="inline">\bullet </math>  Compute  nodal velocities by solving Eq.(8)

<math display="inline">\displaystyle \left[\frac{1}{\Delta t} \mathbf{M}+\mathbf{K}\right]{}^{t+1} \bar{\boldsymbol v}^{i+1} ={}^{t+1} \mathbf{f} - {\boldsymbol G}^{t+1} {\boldsymbol p}^{i} +\frac{1}{\Delta t} \mathbf{M} {}^{t} \bar{\boldsymbol v}</math>

<math display="inline">\bullet </math>   Compute nodal pressures from Eq.(9)

<math display="inline">\displaystyle \left[\frac{1}{\Delta t}\bar {\boldsymbol M}-{\boldsymbol L}\right]{}^{t+1} \bar{\boldsymbol p}^{i+1} =\mathbf{G}{}^{T} {}^{t+1} \bar{\boldsymbol v}^{i+1} + \mathbf{Q}{}^{t+1} \bar{\boldsymbol \pi }^{i} +\frac{1}{\Delta t}\bar {\boldsymbol M} {}^t\bar{\boldsymbol p} </math> 

<math display="inline">\bullet </math> Compute nodal pressure gradient projections from Eq.(10)

<math display="inline">{}^{n+1} \bar{\boldsymbol \pi }^{i+1} =-\hat{\boldsymbol M}^{-1}_{D} \left[\mathbf{Q}^{T} \right]{}^{t+1} \bar{\boldsymbol p}^{i+1} \begin{array}{ccc} {} & {,} & {\hat{\boldsymbol M}_{D} =diag\left[\hat{\boldsymbol M}_{D} \right]} \end{array}</math> 

<math display="inline">\bullet </math>   Update position of analysis domain nodes:

<math display="inline">{}^{t+\Delta t} \bar{\boldsymbol x}^{i+1} ={}^{t} \mathbf{x}^{i} +{}^{t+\Delta t} \mathbf{v}^{i+1} \Delta t</math>

Define new “cloud” of nodes <math display="inline">{}^{t+1} C^{i+1} </math> 

<math display="inline">\bullet </math>   Update strain rate and strain values

<math display="inline">\bullet </math>   Update stress values 

Check convergence <math display="inline">\rightarrow </math> NO <math display="inline">\rightarrow </math> Next iteration <math display="inline">i\to i+1</math>

             <math display="inline">\downarrow </math> YES

Next time step <math display="inline">t\to t+1</math> 

<math display="inline">\bullet </math> Identify new analysis domain boundary: <math display="inline">{}^{t+1} V</math> 

<math display="inline">\bullet </math> Generate mesh:<math display="inline">{}^{t+1} M</math> 

Go to 1
|- style="border-bottom: 2px solid;"
| style="text-align: left;border-left: 2px solid;border-right: 2px solid;"|
|}

==4 MESH GENERATION AND BOUNDARY IDENTIFICATION==

The success of the PFEM  relies on the fast regeneration of a mesh at every time step. Any fast meshing algorithm can be used. In our work the mesh is generated at each time step using an extended Delaunay tesselation <span id='citeF-4'></span>[[#cite-4|[4]]].

For large 3D problems solved in a single processor Pentium IV PC, meshing consumes around 15% of the total CPU time for  time step, while the solution of the equations (with typically 3 iterations to reach convergence per time step) and the assembly of the system consume approximately 70% and 15% of the  CPU time per time step, respectively. Considerable speed can be gained using parallel computing techniques.

Boundary nodes are recognized at mesh generation level <span id='citeF-5'></span><span id='citeF-10'></span>[[#cite-5|[5]],[[#cite-10|10]]]. Considering that the nodes follow a variable <math display="inline">h(x)</math> distribution, where <math display="inline">h(x)</math> is typically the minimum distance between two nodes. ''All nodes on an empty sphere with a radius greater than <math>\alpha h</math>, are considered as boundary nodes''. Values of <math display="inline">\alpha </math> ranging of 1.4 have been found to be optimal. This criterion is coincident with the Alpha Shape  <span id='citeF-2'></span>[[#cite-2|[2]]].

The boundary recognition method  is useful for detecting contact conditions between any fluid/solid or solid/solid interface.

==5 TREATMENT OF CONTACT  CONDITIONS==

Prescribed velocities  at the fixed boundaries in the PFEM are applied in strong form to the boundary nodes. Contact between the fluid particles and the fixed  boundaries is accounted for by the incompressibility condition which '' prevents    the fluid nodes to penetrate into the solid boundaries'' <span id='citeF-9'></span>[[#cite-9|[9]]].

<div id='img-2'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Samper_534298620-Contact_conditions.png|340px|Modelling  contact conditions at a solid-solid interface with  PFEM]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 2:''' Modelling  contact conditions at a solid-solid interface with  PFEM
|}

The contact between two solid interfaces is  treated by introducing a layer of ''contact elements'' between the interacting  interfaces. This layer is ''created during the mesh   generation step'' by prescribing a minimum distance (<math display="inline">h_c</math>) between two solid boundaries. If the distance exceeds the minimum value (<math display="inline">h_c</math>) then the generated elements are treated as fluid elements. Otherwise the elements are treated as frictional contact elements  (Figure [[#img-2|2]]).

The  algorithm can also be used effectively to model frictional contact conditions between rigid or elastic solids in  structural mechanics applications <span id='citeF-1'></span><span id='citeF-15'></span>[[#cite-1|[1]],[[#cite-15|15]]].

==6 MODELING OF BED EROSION==

Prediction of bed erosion and sediment transport in open channel flows are  important tasks in many areas of river and environmental engineering. Bed erosion can lead to instabilities of the river basin slopes. It can also undermine the foundation of bridge piles. Modeling of bed erosion is also relevant for predicting the dragging of surface material  in earth dams in overspill situations.

The erosion model used in this work is based on the frictional work  at the bed surface originated by the shear stresses in the fluid <span id='citeF-12'></span>[[#cite-12|[12]]].

==7 EXAMPLES==

Figure [[#img-3|3]] shows the capacity of the PFEM for modelling soil erosion, sediment transport and material deposition in a river bed. The soil particles are first detached from the bed surface under the action of the jet stream. Then they are transported by the flow and eventually  fall down due to gravity forces.

Figure [[#img-4|4]] shows the progressive erosion of the unprotected part of a breakwater slope in the Langosteira harbour in A Coruña, Spain. The non protected upper shoulder zone is progressively eroded under the  sea waves.

<div id='img-3'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Samper_534298620-Erosion-transport.png|600px|Erosion, transport and deposition of soil particles due to jet stream]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 3:''' Erosion, transport and deposition of soil particles due to jet stream
|}

<div id='img-4'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Samper_534298620-Erosion-breakwater.png|400px|Erosion of an unprotected shoulder of a breakwater  due to sea waves]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 4:''' Erosion of an unprotected shoulder of a breakwater  due to sea waves
|}

<div id='img-5'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Samper_534298620-erosion-camion2.png|600px|Erosion of soil mass due to  waves and  subsequent falling of   lorry]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 5:''' Erosion of soil mass due to  waves and  subsequent falling of   lorry
|}

<div id='img-6'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Samper_534298620-Arrastre-casas1.png|300px|]]
|[[Image:Draft_Samper_534298620-Arrastre-casas3.png|300px|]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 6:''' Simulation of landslide falling on constructions using PFEM
|}

Figure [[#img-5|5]] shows a representative example of the progressive erosion of a soil mass adjacent to the shore due to sea waves and the subsequent falling into the sea of a 2D rigid object representing the section of a lorry.

<div id='img-7'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Samper_534298620-fig_9_01.png|400px|]]
|-
|[[Image:Draft_Samper_534298620-fig_10_02.png|400px|]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 7:''' Lituya Bay landslide. (a) Left: Geometry. Right: Landslide direction and maximum wave level <span id='citeF-3'></span>[[#cite-3|[3]]]. (b) Landslide motion into  reservoir obtained with  PFEM. Maximum level of  generated wave (551 mts) in  north slope
|}

For other applications of the PFEM to bed erosion problems see <span id='citeF-12'></span><span id='citeF-16'></span>[[#cite-12|[12]],[[#cite-16|16]]].

The PFEM is particularly suited for modelling and simulation of landslides and their effect in the surrounding structures. Figure [[#img-6|6]] shows an schematic 3D simulation of a landslide falling on  adjacent constructions. The landslide material has been modelled as a viscous incompressible fluid.

In Figure [[#img-7|7]] we present some results of the 3D analysis of the landslide produced in Lituya Bay (Alaska) on July 9th 1958. The landslide was originated by an earthquake and movilized 90 millions tons of rocks that fell on the bay creating a large wave that reached a hight on the opposed slope of 524 mts.

The sliding mass has been modelled as a quasi-incompressible  continuum with a prescribed shear modulus. No frictional or erosion effects with the underneath soil have been considered.

The maximum water level in north hill obtained with PFEM was 551 mts. This is 5% higher than the value of 524 mts. observed experimental  <span id='citeF-3'></span>[[#cite-3|[3]]]. The maximum height location differs in 300 mts from the observed value. In the south slope the maximum water height observed was 208 mts, while the PFEM result (not shown here) was 195 mts (6% error). For more information on the PFEM solutions of this example see <span id='citeF-15'></span><span id='citeF-16'></span>[[#cite-15|[15]],[[#cite-16|16]]].

==8 CONCLUSIONS==

The PFEM is a promising numerical technique for solving  fluid-soil-structure interaction (FSSI) problems involving large motion of  fluid and solid particles, surface waves, frictional  contact situations between fluid-solid and solid-solid interfaces and bed erosion, among other complex phenomena. The success of the PFEM lies in the accurate and efficient solution of the equations of a continuum containing fluid and solid domains using an updated Lagrangian formulation and a stabilized FEM with low order elements. Other essential solution ingredients are the efficient regeneration of the finite element mesh, the fast identification of the boundaries and the simple algorithm to treat frictional contact conditions and erosion at the interfaces. The examples show the  potential of the PFEM for solving a wide class of practical FSSI problems.

==ACKNOWLEDGEMENTS==

This research was  supported by project SEDUREC of the Consolider Programme of the Ministerio de Educación y Ciencia  of Spain and  projects SAFECON and REALTIME of the European Research Council.

==References==

<div id="cite-1"></div>
[[#citeF-1|[1]]]  Carbonell JM, Oñate E, Suárez B (2010) Modeling of ground excavation with the Particle Finite Element Method. Journal of Engineering Mechanics (ASCE) 136(4):455&#8211;463

<div id="cite-2"></div>
[[#citeF-2|[2]]]  Edelsbrunner H,  Mucke EP (1999) Three dimensional alpha shapes. ACM   Trans. Graphics  13:43&#8211;72

<div id="cite-3"></div>
[[#citeF-3|[3]]] Fritz, HM, Hager WH, Minor HE (2001) Lituya Bay Case: Rockslide impact and wave run-up. Science of Tsunami Hazards 19(1):3&#8211;22

<div id="cite-4"></div>
[[#citeF-4|[4]]]  Idelsohn SR, Calvo N, Oñate E (2003c) Polyhedrization of an arbitrary point set. Comput. Method Appl. Mech. Engng. 192(22-24):2649&#8211;2668

<div id="cite-5"></div>
[[#citeF-5|[5]]]  Idelsohn SR, Oñate E, Del Pin F, Calvo N (2006) Fluid-structure   interaction using the particle finite element   method. Comput. Meth. Appl. Mech. Engng. 195:2100&#8211;2113

<div id="cite-6"></div>
[[#citeF-6|[6]]]  Idelsohn SR, Marti J, Limache A, Oñate E (2008) Unified Lagrangian formulation for elastic solids and incompressible fluids: Application to fluid-structure interaction problems via the PFEM. Comput Methods Appl Mech Engrg.  197:1762&#8211;1776

<div id="cite-7"></div>
[[#citeF-7|[7]]]  Idelsohn SR, Mier-Torrecilla M, Oñate E  (2009) Multi-fluid flows with the Particle Finite Element Method. Comput Methods Appl Mech Engrg. 198:2750&#8211;2767

<div id="cite-8"></div>
[[#citeF-8|[8]]]  Larese A,  Rossi R, Oñate E, Idelsohn SR (2008)  Validation of the Particle Finite Element Method (PFEM) for simulation of free surface flows. Engineering Computations 25(4):385&#8211;425

<div id="cite-9"></div>
[[#citeF-9|[9]]]   Oñate E (2004) Possibilities of finite calculus in computational mechanics. Int. J. Num. Meth. Engng. 60(1):255&#8211;281

<div id="cite-10"></div>
[[#citeF-10|[10]]]  Oñate E, Idelsohn SR, Del Pin F, Aubry R (2004b) The particle   finite element method. An overview. Int. J. Comput. Methods    1(2):267&#8211;307

<div id="cite-11"></div>
[[#citeF-11|[11]]]  Oñate E, García J,  SR Idelsohn, F. Del Pin (2006b)  FIC   formulations for finite element analysis of incompressible flows. Eulerian,   ALE and Lagrangian approaches.  Comput. Meth. Appl. Mech. Engng.   195(23-24):3001&#8211;3037

<div id="cite-12"></div>
[[#citeF-12|[12]]]  Oñate E, M.A. Celigueta, Idelsohn SR (2006c) Modeling bed erosion in   free surface flows by the Particle Finite Element Method, Acta   Geotechnia 1(4):237&#8211;252

<div id="cite-13"></div>
[[#citeF-13|[13]]]  Oñate E, Idelsohn SR,  Celigueta MA, Rossi R (2008) Advances in the particle finite element method for the analysis of fluid-multibody interaction and bed erosion in free surface flows. Comput. Meth. Appl. Mech. Engng. 197(19-20):1777&#8211;1800

<div id="cite-14"></div>
[[#citeF-14|[14]]]  Oñate E, Rossi R,  Idelsohn SR, Butler K (2010) Melting and spread of polymers in fire with the particle finite element method. Int. J. Numerical Methods in Engineering, 81(8):1046&#8211;1072

<div id="cite-15"></div>
[[#citeF-15|[15]]]  Oñate E, Celigueta MA, Idelsohn SR,  Salazar F, Suárez B (2011) Possibilities of the particle finite element method for fluid-soil-structure interaction problems. Computational Mechanics, 48(3):307&#8211;318.

<div id="cite-16"></div>
[[#citeF-16|[16]]]  Oñate, E, Salazar F, Morán R (2011) Modeling of landslides into reservoir with the Particle Finite Element Method. Research Report CIMNE No. PI355. Submitted to Int. J. Numerical Methods in Geomechanics

<div id="cite-17"></div>
[[#citeF-17|[17]]]  Zienkiewicz OC,  Taylor RL (2005) The finite element method for   solid and structural mechanics,  Elsevier

<div id="cite-18"></div>
[[#citeF-18|[18]]]  Zienkiewicz OC,  Taylor RL, Nithiarasu P (2006)  The finite element   method for fluid dynamics,   Elsevier