Possibilities of the particle finite element method for fluid-soil-structure interaction problems

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==POSSIBILITIES OF THE PARTICLE FINITE ELEMENT METHOD FOR FLUID-SOIL-STRUCTURE INTERACTION PROBLEMS==

'''Eugenio Oñate<math>^{1,2}</math>, Miguel Angel Celigueta<math>^{1,2}</math>, Sergio R. Idelsohn<math>^{*}</math>,

Fernando Salazar<math>^{1}</math> and Benjamín Suárez<math>^{2}</math>

'''

{|
|-
|<math>^{1}</math> International Center for Numerical Methods in Engineering (CIMNE) 
|-
| Campus Norte UPC, 08034 Barcelona, Spain 
|-
| [mailto:onate@cimne.upc.edu onate@cimne.upc.edu], [http://www.cimne.com www.cimne.com]
|-
| <math>^*</math> ICREA Research Professor at CIMNE
|-
| <math>^{2}</math> Technical University of Catalonia (UPC), Barcelona, Spain
|}
-->

==Abstract==

We present some developments in the Particle Finite   Element Method (PFEM) for analysis of complex coupled problems in   mechanics involving  fluid-soil-structure interaction (FSSI). The   PFEM uses  an updated Lagrangian description to   model the motion of nodes (particles) in both the fluid and the solid domains (the later including soil/rock and structures).  A mesh connects the particles (nodes) defining the discretized   domain where the governing equations for each of the constituent materials are   solved as in the standard FEM. The stabilization for dealing with   an incompressibility continuum is introduced via the finite calculus   (FIC) method. An incremental iterative scheme for the solution of the non   linear transient coupled FSSI problem is described. The procedure to model frictional contact conditions and material erosion at fluid-solid and   solid-solid interfaces is   described. We present several examples of application of the PFEM to solve FSSI problems such as the motion of rocks by water streams, the erosion of a river bed adjacent to a bridge foundation, the stability of breakwaters and constructions sea waves and the study of  landslides.

==1 Introduction==

The analysis of problems involving the interaction of fluids, soil/rocks and structures  is of  relevance in many areas of engineering. Examples 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.

These studies can be considered as an extension of the so-called fluid-structure interaction (FSI) problems <span id='citeF-46'></span>[[#cite-46|[46]]]. Typical difficulties of FSI  analysis in free surface flows using the FEM  both the Eulerian or ALE formulation include the treatment of the convective terms and the incompressibility constraint in the fluid equations, the modelling and tracking of the free surface in the fluid, the transfer of information between the fluid and the moving solid domains via the contact interfaces, the modeling of wave splashing, the possibility to deal with large  motions of multi-bodies within the fluid domain, the efficient updating of the finite element meshes for both the structure and the fluid, etc. Examples of 3D analysis of FSI problems using ALE and space-time FEM are reported in <span id='citeF-4'></span><span id='citeF-6'></span><span id='citeF-26'></span><span id='citeF-27'></span><span id='citeF-31'></span><span id='citeF-34'></span><span id='citeF-40'></span>[[#cite-4|[4,6,26,27,31,34,40]]].

Most of the above problems disappear if a ''Lagrangian description'' is used to formulate the governing equations of both the solid and the fluid domains. In the Lagrangian formulation the motion of the individual particles are followed and, consequently, nodes in a finite element mesh can be viewed as moving material points (hereforth called “particles”). Hence, the motion of the mesh discretizing the total domain (including both the fluid and solid parts) is followed during the transient solution.

A powerful Lagrangian method for FSI analysis is the so-called Soboran Grid CIP technique, which has been successfully applied to different class of 3D problems <span id='citeF-44'></span>[[#cite-44|[44]]].

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,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 Lagrangian formulation is that the convective terms disappear from the fluid equations <span id='citeF-11'></span><span id='citeF-48'></span>[[#cite-11|[11,48]]]. The difficulty is however transferred to the problem of adequately (and efficiently) moving the mesh nodes.  We use a  mesh regeneration procedure blending elements of different shapes using an extended Delaunay tesselation with special shape functions <span id='citeF-17'></span><span id='citeF-19'></span>[[#cite-17|[17,19]]]. The theory and applications of the PFEM are reported in <span id='citeF-2'></span><span id='citeF-7'></span><span id='citeF-10'></span><span id='citeF-18'></span><span id='citeF-20'></span><span id='citeF-21'></span><span id='citeF-23'></span><span id='citeF-26'></span><span id='citeF-32'></span><span id='citeF-34'></span><span id='citeF-35'></span><span id='citeF-36'></span><span id='citeF-37'></span><span id='citeF-38'></span><span id='citeF-39'></span>[[#cite-2|[2,7,10,18,20,21,23,26,32,34,35,36,37,38,39]]].

The FEM solution of  (incompressible) fluid flow problem implies solving the momentum and incompressibility equations. This is not a simple problem as the incompressibility condition limits the choice of the FE approximations for the velocity and pressure to overcome the well known <math display="inline">div</math>-stability condition <span id='citeF-11'></span><span id='citeF-48'></span>[[#cite-11|[11,48]]]. 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-15'></span><span id='citeF-29'></span><span id='citeF-30'></span><span id='citeF-31'></span><span id='citeF-33'></span><span id='citeF-34'></span>[[#cite-15|[15,29,30,31,33,34]]]. Among the other stabilized FEM with similar features we mention the PSPG method <span id='citeF-41'></span>[[#cite-41|[41]]], multiscale methods <span id='citeF-3'></span><span id='citeF-6'></span><span id='citeF-8'></span><span id='citeF-9'></span>[[#cite-3|[3,6,8,9]]] and the CBS method <span id='citeF-9'></span><span id='citeF-48'></span>[[#cite-9|[9,48]]].

The aim of this paper is to describe recent advances of the PFEM for fluid-soil-structure interaction (FSSI) problems. These problems are of relevance in many areas of civil, hydraulic, marine and environmental engineering, among others. It is shown that the PFEM provides a general analysis methodology for treat such complex problems in a simple and efficient manner.

The layout of the paper is the following. In the next section the key ideas of the PFEM are outlined. Next the basic equations for a compressible/incompressible continuum using a Lagrangian description and the FIC formulation are schematically presented. Then an algorithm for the transient solution is briefly described. The treatment of the coupled FSSI problem and the methods for mesh generation and for identification of the free surface nodes are outlined. The procedure for treating the frictional contact interaction between fluid, soil and structure interfaces is explained.  We present several examples of application of the PFEM  to solve FSSI problems such as the motion of rocks by water streams, the erosion of a river bed adjacent to a bridge foundation, the stability of breakwaters and constructions under sea waves and the study of landslides falling into reservoirs.

==2 The basis of the particle finite element method==

Let us consider a domain containing both fluid and solid subdomains (the solid subdomain may include soil/rock materials and/or structural elements). The moving fluid particles interact with the solid boundaries thereby inducing the deformation of the solid which in turn affects the flow motion and, therefore, the problem is  fully coupled.

In the PFEM  both the fluid and the solid domains are modelled using an ''updated'' ''Lagrangian formulation'' <span id='citeF-47'></span>[[#cite-47|[47]]]. That is, all 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>. The finite element method (FEM) is used to solve the  equations of continuum mechanics for each of the subdomains. Hence a mesh discretizing these domains must be generated in order to solve the governing equations for each subdomain in the standard FEM fashion.

The quality of the numerical solution  depends on the discretization chosen as in the standard FEM. Adaptive mesh refinement techniques can be used to improve the solution in zones where large motions of the fluid or the structure occur.

<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_395841693-Figure2_con276.png|600px|Sequence of steps to update a “cloud” of nodes representing a domain containing a fluid and a solid part from time n   (t=tₙ)  to   time n+2 (t=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 part from time <math>n</math>   (<math>t=t_n</math>)  to   time <math>n+2</math> (<math>t=t_n +2\Delta t</math>)  
|}

===2.1 Basic steps of the PFEM===

For clarity purposes we will define the ''collection or cloud of nodes (C)'' pertaining to the fluid and solid domains, the ''volume (V)'' defining the analysis domain for the fluid and the solid and the ''mesh (M)'' discretizing both domains.

A typical solution with the PFEM involves the following steps.

<ol>

<li>The starting point at each time step is the cloud of points in the fluid and solid   domains. For instance <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 defining   the analysis domain <math display="inline">{}^nV</math> in the fluid and the solid. This is  an   essential step as some boundaries (such as the free surface in fluids)   may be severely distorted during the solution, including separation   and re-entering of nodes. The Alpha Shape method   <span id='citeF-12'></span>[[#cite-12|[12]]] is used for the boundary definition.  </li>

<li>Discretize the fluid and solid domains with a finite element mesh <math display="inline">{}^nM</math>. In our work we use an innovative mesh generation scheme based on the extended Delaunay tesselation <span id='citeF-17'></span><span id='citeF-19'></span><span id='citeF-20'></span>[[#cite-17|[17,19,20]]].  </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>: velocities, pressure and   viscous stresses in the fluid and   displacements, stresses and strains in the solid.  </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. This step is   typically a consequence of the solution process of step 4.  </li>

<li>Go back to step 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>

==3 FIC/FEM 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:

===='''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)
|}

In above equations <math display="inline">v_{i}</math> is the velocity along the ''i''th global (cartesian) axis, <math display="inline">p</math> is the pressure (assumed to be positive in compression) <math display="inline">\rho </math> and <math display="inline">K</math> are the density and bulk modulus of the material, respectively, <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:

====''Incompressible continuum''====

<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 continuum''====

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>{}^{t+1}s _{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.a)
|}

where <math display="inline">\hat \sigma _{ij}</math> are the component of the stress tensor <math display="inline">[\hat \sigma ]</math>

<span id="eq-4.b"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>[\hat \sigma ]= \frac{1}{J} {F}^T {S} {F}  </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (4.b)
|}

where <math display="inline">{S}</math> is the second Piola-Kirchhoff stress tensor, <math display="inline">{F}</math> is the deformation gradient tensor and <math display="inline">J = \det {F}</math> <span id='citeF-22'></span><span id='citeF-47'></span>[[#cite-22|[22,47]]]. Parameters <math display="inline">\mu </math> and <math display="inline">\lambda </math>  take the following values for a fluid or solid material:

'''Fluid:''' <math display="inline">\mu </math>: viscosity; <math display="inline">\lambda = \Delta t K - \frac{2\mu }{3}</math>      '''Solid:''' <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.(3) and (4), <math display="inline">s _{ij}</math> are the deviatoric stresses, <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]])&#8211;(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 of prescribed velocities and surface tractions in the mechanical problem  <span id='citeF-11'></span><span id='citeF-36'></span><span id='citeF-47'></span><span id='citeF-48'></span>[[#cite-11|[11,36,47,48]]].

===3.2 Discretization of the equations===

A key problem in the numerical solution of Eqs.([[#eq-1|1]])&#8211;([[#''Compressible/quasi-incompressible continuum''|3.1]]) 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]])). A number of procedures to solve his problem exist in the finite element literature <span id='citeF-11'></span><span id='citeF-48'></span>[[#cite-11|[11,48]]]. In our approach we use a stabilized formulation based in the so-called finite calculus procedure <span id='citeF-15'></span><span id='citeF-29'></span><span id='citeF-30'></span><span id='citeF-31'></span><span id='citeF-33'></span><span id='citeF-34'></span>[[#cite-15|[15,29,30,31,33,34]]]. 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 <span id='citeF-34'></span>[[#cite-34|[34]]]

<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>\tau = \left(\frac{2\rho \vert \mathbf{v}\vert }{h}+\frac{8\mu }{3h^2} \right)^{-1}  </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (6)
|}

In the above, <math display="inline">h</math> is 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 and therefore it vanishes for the exact solution. The set of governing equations  is completed by adding the following constraint equation <span id='citeF-32'></span><span id='citeF-36'></span>[[#cite-32|[32,36]]]

<span id="eq-7"></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;" | (7)
|}

where <math display="inline">w_i</math> are arbitrary weighting functions.

The rest of the integral equations are obtained by applying the standard weighted residual technique to the governing equations (1), (2), (3) and (5) and the corresponding boundary conditions <span id='citeF-11'></span><span id='citeF-22'></span><span id='citeF-48'></span>[[#cite-11|[11,22,48]]].

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 resulting set of discretized equations using the standard Galerkin technique has the following form

===='''Momentum'''====

<span id="eq-8"></span>
{| 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{v}} + \mathbf{K} \bar {v} - \mathbf{G} \bar  {p}= {f}  </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (8)
|}


<div class="center" style="font-size: 75%;">'''Box I'''. Basic PFEM algorithm for a Lagrangian continuum
</div>

{|  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{x},{}^{t} \bar{v},{}^{t} \bar{p},{}^{t} \bar{\boldsymbol \pi },{}^{t} \bar{T},{}^{t} \mu ,{}^{t}{f},{}^{t}\mathbf{ q},{}^{t} C,{}^{t} V,{}^{t} M</math> 
|-
| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | 2. LOOP OVER NUMBER OF ITERATIONS, <math display="inline">i=1</math>, NITER 
|-
| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | [.15cm] <math display="inline">\bullet </math>  Compute  nodal velocities by solving Eq.(8)
|-
| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | <math display="inline">\displaystyle \left[\frac{1}{\Delta t} \mathbf{M}+\mathbf{K}\right]{}^{t+1} \bar{v}^{i+1} ={}^{t+1} \mathbf{f}+G{}^{t+1} \bar{p}^{i} +\frac{1}{\Delta t} \mathbf{M} {}^{t} \bar{v}</math> 
|-
| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | <math display="inline">\bullet </math>   Compute nodal pressures from Eq.(9)
|-
| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | <math display="inline">\displaystyle \left[\frac{1}{\Delta t}-{L}\bar {M}\right]{}^{t+1} \bar{p}^{i+1} =\mathbf{G}{}^{T} {}^{t+1} \bar{v}^{i+1} + \mathbf{Q}{}^{t+1} \bar{\boldsymbol \pi }^{i} +\frac{1}{\Delta t}\bar {M} {}^t\bar{p} </math> 
|-
| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | <math display="inline">\bullet </math> Compute nodal pressure gradient projections from Eq.(10)
|-
| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | <math display="inline">{}^{n+1} \bar{\boldsymbol \pi }^{i+1} =-\hat{M}^{-1}_{D} \left[\mathbf{Q}^{T} \right]{}^{t+1} \bar{p}^{i+1} \begin{array}{ccc} {} & {,} & {\hat{M}_{D} =diag\left[\hat{M}_{D} \right]} \end{array}</math> 
|-
| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | <math display="inline">\bullet </math>   Update position of analysis domain nodes:
|-
| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | <math display="inline">{}^{t+\Delta t} \bar{x}^{i+1} ={}^{t} \mathbf{x}^{i} +{}^{t+\Delta t} \mathbf{v}^{i+1} \Delta t</math>
|-
| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | Define new “cloud” of nodes <math display="inline">{}^{t+1} C^{i+1} </math> 
|-
| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | <math display="inline">\bullet </math>   Update strain rate and strain values
|-
| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | <math display="inline">\bullet </math>   Update stress values 
|-
| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | Check convergence <math display="inline">\rightarrow </math> NO <math display="inline">\rightarrow </math> Next iteration <math display="inline">i\to i+1</math>
|-
| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | <math display="inline">\downarrow </math> YES
|-
| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | Next time step <math display="inline">t\to t+1</math> 
|-
| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | [.15cm] <math display="inline">\bullet </math> Identify new analysis domain boundary: <math display="inline">{}^{t+1} V</math> 
|-
| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | [.15cm] <math display="inline">\bullet </math> Generate mesh:<math display="inline">{}^{t+1} M</math> 
|- style="border-bottom: 2px solid;"
| style="text-align: left;border-left: 2px solid;border-right: 2px solid;" | [.15cm] Go to 1 

|}

===='''Pressure-velocity relationship'''====

<span id="eq-9"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\bar {M} \dot{\bar{p}} - \mathbf{G}\bar{v} - \mathbf{L}\bar {p} - \mathbf{Q} \bar {\boldsymbol \pi }=\mathbf{0}   </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (9)
|}

===='''Pressure gradient projection'''====

<span id="eq-10"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\hat {M} \bar {\boldsymbol \pi }+\mathbf{Q}^T\bar {p}=\mathbf{0}  </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (10)
|}

In Eqs.([[#eq-8|8]])&#8211;([[#eq-10|10]]) <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-22'></span><span id='citeF-34'></span><span id='citeF-36'></span>[[#cite-22|[22,34,36]]].

The solution in time of Eqs.([[#eq-8|8]])&#8211;([[#eq-10|10]]) can be performed using any time integration scheme typical of the updated Lagrangian FEM <span id='citeF-36'></span><span id='citeF-47'></span>[[#cite-36|[36,47]]]. A basic algorithm following the conceptual process described in Section 2 is presented in Box I.

==4 Generation of a new mesh==

One of the key points for the success of the PFEM  is the fast regeneration of a mesh at every time step on the basis of the position of the nodes in the space domain. Indeed, any fast meshing algorithm can be used for this purpose. In our work the mesh is generated at each time step using the so called extended Delaunay tesselation (EDT) presented in <span id='citeF-17'></span><span id='citeF-19'></span>[[#cite-17|[17,19]]].

The CPU time required for meshing grows linearly with the number of nodes. The CPU time for solving the equations exceeds that required for meshing as the number of nodes increases. This situation has been found in all the problems solved with the PFEM. As a general rule for large 3D problems meshing consumes around 15% of the total CPU time for each time step, while the solution of the equations (with typically 3 iterations to reach convergence within a time step) and the assembling of the system consume approximately 70% and 15% of the  CPU time for each time step, respectively. These figures refer to solutions obtained in a standard single processor Pentium IV PC for all the computations and prove that the generation of the mesh has an acceptable cost in the PFEM. The cost of remeshing is similar to that reported in <span id='citeF-24'></span>[[#cite-24|[24]]]. Indeed considerable speed can be gained using parallel computation techniques.

==5 Identification of boundary surfaces==

One of the main tasks  in the PFEM is the correct definition of the boundary domain. Boundary nodes are sometimes explicitly identified. In other cases, the total set of nodes is the only information available and the algorithm must recognize the boundary nodes.

In our work we use an extended Delaunay partition for recognizing boundary nodes <span id='citeF-19'></span>[[#cite-19|[19]]]. 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''. In practice <math display="inline">\alpha </math>  is a parameter close to, but greater than one. Values of <math display="inline">\alpha </math> ranging between 1.3 and 1.5 have been found to be optimal in all examples analyzed. This criterion is coincident with the Alpha Shape concept <span id='citeF-12'></span>[[#cite-12|[12]]].

Once a decision has been made concerning which  nodes are on the boundaries, the boundary surface is defined by  all the polyhedral surfaces (or polygons in 2D) having all their nodes on the boundary and belonging to just one polyhedron.

The method described also allows one to identify isolated fluid particles outside the main fluid domain. These particles are treated as part of the external boundary where the pressure is fixed to the atmospheric value. We recall that each particle is a material point characterized by the density of the solid or fluid domain to which it belongs. The mass which is lost when a boundary element is eliminated due to departure of a node from the main analysis domain is again regained when the “flying” node falls down and a new boundary element is created by the Alpha Shape algorithm.

The boundary recognition method  is also useful for detecting contact conditions between the fluid domain and a fixed boundary, as well as between different solids interacting with each other as detailed in the next section.

We emphasize that the main difference between the PFEM and the classical FEM is just the remeshing technique and the identification of the domain boundary at each time step.

==6 Treatment of contact  conditions in the PFEM==

===6.1 Contact  between  the fluid and a fixed boundary===

The  condition of prescribed velocities  at the fixed boundaries in the PFEM is  applied in strong form to the boundary nodes. These nodes might belong to fixed external boundaries or to moving boundaries linked to the interacting solids. Contact between the fluid particles and the fixed  boundaries is accounted for by the incompressibility condition which ''naturally prevents    the fluid nodes to penetrate into the solid boundaries'' <span id='citeF-32'></span><span id='citeF-36'></span>[[#cite-32|[32,36]]].


===6.2 Contact between solid-solid interfaces===

The contact between two solid interfaces is simply treated by introducing a layer of ''contact elements'' between the two interacting solid interfaces. This layer is ''automatically 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 contact elements where a relationship between the tangential and normal forces and the corresponding displacement is introduced (Figure [[#img-2|2]]).

<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_395841693-Contact_conditions.png|570px|Modelling of contact conditions at a solid-solid interface with the PFEM]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 2:''' Modelling of contact conditions at a solid-solid interface with the PFEM
|}

This algorithm has proven to be very effective and it allows to identifying and modeling complex frictional contact conditions between two or more interacting bodies moving in water in an extremely simple manner.

This  algorithm can also be used effectively to model frictional contact conditions between rigid or elastic solids in  structural mechanics applications <span id='citeF-7'></span><span id='citeF-36'></span>[[#cite-7|[7,36]]].

==7 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 thereby favouring structural failure. Modeling of bed erosion is also relevant for predicting the evolution of surface material dragged in earth dams in overspill situations. Bed erosion is one of the main causes of environmental damage in floods.

Bed erosion models are traditionally based on a relationship between the rate of erosion and the shear stress level <span id='citeF-25'></span>[[#cite-25|[25]]]. In a recent work we have proposed an extension of the PFEM to model bed erosion <span id='citeF-35'></span><span id='citeF-36'></span>[[#cite-35|[35,36]]]. The erosion model is based on the frictional work  at the bed surface originated by the shear stresses in the fluid. The resulting erosion model resembles Archard law typically used for modeling abrasive wear in surfaces under frictional contact conditions <span id='citeF-1'></span>[[#cite-1|[1]]].

The algorithm for modeling the  erosion of soil/rock particles at the fluid bed is the following:

<ol>

<li>Compute at every point of the bed surface the resultant tangential   stress <math display="inline">\tau </math> induced by the fluid motion. In 3D problems <math display="inline">\tau =   (\tau _{s}^2 + \tau _{t})^2</math> where <math display="inline">\tau _s</math> and <math display="inline">\tau _t</math> are the tangential stresses   in the plane defined by the normal direction <math display="inline">{n}</math> at the bed   node. The value of <math display="inline">\tau </math> for 2D problems can be estimated as follows:

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>

\tau _t =\mu \gamma _t \quad \hbox{with} \quad  \gamma _t ={1\over  2}{\partial v_t\over \partial n} ={v_t^k\over 2h_k} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (11)
|}</li>

where <math display="inline">v_t^k</math> is the modulus of the tangential velocity at the node <math display="inline">k</math>  and <math display="inline">h_k</math> is a prescribed distance along the normal of the bed node <math display="inline">k</math>. Typically <math display="inline">h_k</math> is of the order of magnitude of the smallest fluid element adjacent to node <math display="inline">k</math> (Figure [[#img-3|3]]).

<li>Compute the frictional work originated by the tangential stresses at the   bed surface as

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>

W_f =\int _\circ ^t \tau _t \gamma _t\, dt = \int _\circ ^t {\mu \over 4} \left({v_t^k\over h_k}\right)^2 dt </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (12)
|}</li>

Eq.(12) is integrated in time as

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>

{}^n W_f ={}^{n-1} W_f + \tau _t \gamma _t\, \Delta t </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (13)
|}

<li>The onset of erosion at a bed point occurs when <math display="inline">{}^nW_f</math> exceeds a critical   threshold value <math display="inline">W_c</math>.  </li>

</ol>

<ol>

<li>If <math display="inline">{}^nW_f > W_c</math> at a bed node, then the node is detached from the bed   region and it is allowed to move with the fluid flow. As a consequence, the   mass of the patch of bed elements surrounding the bed node vanishes in the   bed domain and it is transferred to the new   fluid node. This mass is subsequently transported with the fluid.  </li>

<li>Sediment deposition can be modeled by an inverse process to that described   in the previous step. Hence, a suspended node adjacent to the bed surface with a   velocity below a threshold value is attached to the bed surface.  </li>

</ol>

Figure [[#img-3|3]] shows an schematic view of the bed erosion algorithm described.

<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_395841693-Bed_erosion.png|600px|Modeling of bed erosion with the PFEM by dragging of bed material]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 3:''' Modeling of bed erosion with the PFEM by dragging of bed material
|}

==8 Examples==

===8.1 Dragging of rocks by a water stream===

Predicting the critical speed at which a rock will be dragged by a water stream is of great importance in many problems in hydraulic, harbour, civil and environmental engineering.

The PFEM has been successfully applied to the study of the motion of a 1Tn quasi-spherical rock due to a water stream. The rock lays on a collection of rocks that are kept rigid. Frictional conditions between the analyzed rock and the rest of the rocks have been assumed. Figure [[#img-4|4]]a shows that a water stream of 1m/s is not able to displace the individual rock. An increase of the water speed to 2m/s induces the motion of the rock as shown in Figure [[#img-4|4]]b.

<div id='img-4a'></div>
<div id='img-4b'></div>
<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_395841693-Roca-1ms_1.png|420px|Water speed of 1m/s. The individual rock can not be dragged by the stream]]
|[[Image:Draft_Samper_395841693-Roca-2ms_2.png|420px|Water speed of 2m/s. The individual rock is dragged by the stream]]
|- style="text-align: center; font-size: 75%;"
| (a) Water speed of 1m/s. The individual rock can not be dragged by the stream
| (b) Water speed of 2m/s. The individual rock is dragged by the stream
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 4:''' Study of the drag of an individual rock of 1Tn under a water stream at  speeds of (a) 1m/s and (b) 2m/s
|}

===8.2 Impact of sea waves on piers and breakwaters===

Figures [[#img-5|5]] and [[#img-6|6]] show the analysis of the effect of breaking waves on two different sites of a breakwater containing reinforced concrete blocks (each one of <math display="inline">4\times 4</math> mts). The figures correspond to the study of Langosteira harbour in A Coruña, Spain using PFEM.


<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_395841693-Bloques.png|600px|Breaking waves on breakwater slope containing reinforced concrete blocks. Mesh of 4-noded tetrahedra near the slope]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 5:''' Breaking waves on breakwater slope containing reinforced concrete blocks. Mesh of 4-noded tetrahedra near the slope
|}

<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_395841693-dique-invernada.png|600px|Study of breaking waves on the edge of a breakwater structure formed by reinforced concrete blocks]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 6:''' Study of breaking waves on the edge of a breakwater structure formed by reinforced concrete blocks
|}

===8.3 Soil erosion problems===

Figure [[#img-7|7]] 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 into the bed surface at a downstream point.

Figure [[#img-8|8]] 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.

Figure [[#img-9|9]] displays the progressive erosion and dragging of soil particles in a river  bed adjacent to the foot of bridge pile due to the water stream (water is not shown in the figure). Note the disclosure of the bridge foundation as the adjacent soil particles are removed due to erosion.

Other applications of the PFEM to bed erosion problems can be found in <span id='citeF-35'></span><span id='citeF-36'></span>[[#cite-35|[35,36]]].

<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_395841693-Erosion-transport.png|600px|Erosion, transport and deposition of soil particles at a river bed due to an impacting jet stream]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 7:''' Erosion, transport and deposition of soil particles at a river bed due to an impacting jet stream
|}

<div id='img-8'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Samper_395841693-Erosion-breakwater.png|486px|Erosion of an unprotected shoulder of a breakwater  due to sea waves]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 8:''' Erosion of an unprotected shoulder of a breakwater  due to sea waves
|}

<div id='img-9'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Samper_395841693-Erosionpila.png|486px|Erosion and dragging of soil particles in a river bed adjacent to the foot of a bridge pile due to a water stream. Water is not shown]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 9:''' Erosion and dragging of soil particles in a river bed adjacent to the foot of a bridge pile due to a water stream. Water is not shown
|}

===8.4 Falling of a lorry into the sea by sea wave erosion of the road slope===

Figure [[#img-10|10]] 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 object representing the section of a lorry. The object has been modeled as a rigid solid.

This example, although still quite simple and schematic, shows the possibility of the PFEM  for modeling complex FSSI problems involving soil erosion, free surface waves and rigid/deformable structures.

<div id='img-10'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Samper_395841693-erosion-camion2.png|600px|Erosion of a soil mass due to sea waves and the subsequent falling into the sea of an adjacent lorry]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 10:''' Erosion of a soil mass due to sea waves and the subsequent falling into the sea of an adjacent lorry
|}

===8.5 Simulation of  landslides===

The PFEM is particularly suited for modelling and simulation of landslides and their effect in the surrounding structures. Figure [[#img-11|11]] shows an schematic 2D simulation of a landslide falling on two adjacent constructions. The landslide material has been modelled as a viscous incompressible fluid.

<div id='img-11'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Samper_395841693-Invasion-casas.png|600px|Landslide falling on two constructions 2D simulation using PFEM]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 11:''' Landslide falling on two constructions 2D simulation using PFEM
|}

===8.6 The landslide in Lituya Bay===

A case of much interest is when the landslide occurs in the vicinity of a reservoir <span id='citeF-43'></span>[[#cite-43|[43]]]. The fall of debris material into the reservoir typically induces large waves that can overtop the dam originating an unexpected flooding that can cause severe damage to the constructions and population in the downstream area.

In the example, we present some results of the 3D analysis of the landslide produced in Lituya Bay (Alaska) on July 9th 1958 (Figure [[#img-12|12]]). The landslide was originated by an earthquake and movilized 90 millions tons of rocks that fell on the bay originating a large wave that reached a hight on the opposed slope of 524 mts.

<div id='img-12'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Samper_395841693-fig_9_01.png|600px|Lituya Bay landslide. Left: Geometry  for the simulation. Right: Landslide direction and maximum wave level <span id='citeF-13'></span><span id='citeF-14'></span>[[#cite-13|[13,14]]]]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 12:''' Lituya Bay landslide. Left: Geometry  for the simulation. Right: Landslide direction and maximum wave level <span id='citeF-13'></span><span id='citeF-14'></span>[[#cite-13|[13,14]]]
|}

<div id='img-13'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Samper_395841693-fig_10_02.png|551px|Lituya Bay landslide. Evolution of the landslide into the reservoir obtained with the PFEM. Maximum level of  generated wave (551 mts) in the north slope]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 13:''' Lituya Bay landslide. Evolution of the landslide into the reservoir obtained with the PFEM. Maximum level of  generated wave (551 mts) in the north slope
|}

Figure [[#img-13|13]] shows  images of the simulation of the landslide with  PFEM. The sliding mass has been modelled as a quasi-incompressible  continuum with a prescribed shear modulus. No frictional effect between the sliding mass and the underneath soil has been considered. Also the analysis has not taken into account the erosion and  dragging of soil material induced by the landslide mass during  motion.

PFEM results have been compared with observed values of the maximum water level in the north hill adjacent to the reservoir. The maximum water level in this hill obtained with PFEM was 551 mts. This is 5% higher than the value of 524 mts. observed experimental by <span id='citeF-13'></span><span id='citeF-14'></span>[[#cite-13|[13,14]]]. The maximum height location differs in 300 mts from the observed value <span id='citeF-13'></span><span id='citeF-14'></span>[[#cite-13|[13,14]]]. In the south slope the maximum water height observed was 208 mts, while the PFEM result (not shown here) was 195 mts (6% error).

More information on the PFEM solutions of this example can be found in <span id='citeF-38'></span><span id='citeF-39'></span>[[#cite-38|[38,39]]].

==9 Conclusions==

The particle finite element method (PFEM) is a promising numerical technique for solving  fluid-soil-structure interaction (FSSI) problems involving large motion of  fluid and solid particles, surface waves, water splashing, 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 an incompressible continuum using an updated Lagrangian formulation and a stabilized finite element method allowing the use of low order elements with equal order interpolation for all the variables. Other essential solution ingredients are the efficient regeneration of the finite element mesh, the identification of the boundary nodes using the Alpha-Shape technique and the simple algorithm to treat frictional contact conditions and erosion/wear at fluid-solid and solid-solid interfaces via mesh generation. The examples presented have shown the  potential of the PFEM for solving a wide class of practical FSSI problems in engineering.

==Acknowledgements==

This research was partially supported by project SEDUREC of the Consolider Programme of the Ministerio de Educación y Ciencia (MEC) of Spain and the projects SAFECON and REALTIME of the European Research Council of the European Commission (EC). Thanks are also given to the Spanish construction company Dragados for financial support for the study of harbour engineering problems with the PFEM .

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