Lagrangian analysis of multiscale particulate flows with the particle finite element method

''Published at Computational Particle Mechanics , Vol. 1 (1), pp. 85-102, 2014''

==Abstract==

We present a Lagrangian numerical technique for the analysis of flows incorporating physical particles of different sizes. The numerical approach is based on the Particle Finite Element Method (PFEM) which blends concepts from particle-based techniques and the FEM. The basis of the Lagrangian formulation for particulate flows and the procedure for modelling the motion of small and large particles that are submerged in the fluid are described in detail. The numerical technique for analysis of this type of multiscale particulate flows using a stabilized mixed velocity-pressure formulation and the PFEM is also presented. Examples of application of the PFEM to several particulate flows problems are given.

'''keywords''': Lagrangian analysis, Multiscale particulate flows, Particle finite element method

==1 Introduction==

The study of fluid flows containing particles of different sizes (hereafter called ''particulate flows'') is relevant to many areas of engineering and applied sciences. In this work we are concerned with particulate flows containing small to large particles. This type of flows is typical in slurry flows originated by natural hazards such as floods, tsunamis and landslides, as well as in many processes of the bio-medical and pharmaceutical industries, in the manufacturing industry and in the oil and gas industry (i.e. cuttings transport in boreholes), among other applications <span id='citeF-1'></span><span id='citeF-2'></span><span id='citeF-6'></span><span id='citeF-13'></span><span id='citeF-14'></span><span id='citeF-16'></span><span id='citeF-21'></span><span id='citeF-22'></span><span id='citeF-23'></span><span id='citeF-26'></span><span id='citeF-47'></span><span id='citeF-50'></span><span id='citeF-51'></span><span id='citeF-55'></span><span id='citeF-61'></span><span id='citeF-62'></span>[[#cite-1|[1,2,6,13,14,16,21,22,23,26,47,50,51,55,61,62]]].

Our interest in this work is the modelling and simulation of free surface particulate quasi-incompressible flows containing particles of different sizes using a particular class of Lagrangian FEM termed the Particle Finite Element Method (PFEM, www.cimne.com/pfem) [4,5,8,11,17-20,25,27,28,35,36,38,40,42-46]. The 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. A mesh connects the nodes discretizing the domain where the governing equations are solved using a stabilized FEM.

In Lagrangian analysis procedures (such as the PFEM) the motion of fluid particles is tracked during the transient solution. Hence, the convective terms vanish in the momentum equations and no numerical stabilization is needed. Another source of instability, however, remains in the numerical solution of Lagrangian flows, that due to the treatment of the incompressibility constraint which requires using a stabilized numerical method.

In this work we use a stabilized Lagrangian formulation that has excellent mass preservation features. The success of the formulation relies on the consistent derivation of a residual-based stabilized expression of the mass balance equation using the Finite Calculus (FIC) method <span id='citeF-29'></span><span id='citeF-30'></span><span id='citeF-31'></span><span id='citeF-32'></span><span id='citeF-33'></span><span id='citeF-37'></span><span id='citeF-38'></span><span id='citeF-39'></span>[[#cite-29|[29-33,37-39]]].

The lay-out of the paper is the following. In the next section we present the basic equations for conservation of linear momentum and mass for a quasi-incompressible particulate fluid in a Lagrangian framework. The treatment of the different force terms for micro, macro and large particles are explained. Next we derive the stabilized FIC form of the mass balance equation. Then, the finite element discretization using simplicial element with equal order approximation for the velocity and the pressure is presented and the relevant matrices and vectors of the discretized problem are given.  Details of the implicit transient solution of the Lagrangian FEM equations for a particulate flow using a Newton-Raphson type iterative scheme are presented. The basic steps of the PFEM for solving free-surface particulate flow problems are described.

The efficiency and accuracy of the PFEM  for analysis of particulate flows are verified by solving a set of  free surface and confined fluid flow problems incorporating particles of small and large sizes in two (2D) and three (3D) dimensions. The problems include the study of soil erosion, landslide situations, tsunami and flood flows, soil dredging problems and particle filling of fluid containers, among others. The excellent performance of the numerical method proposed for analysis of particulate flows is highlighted.

==2 Modelling of micro, macro and large particles==

Figure [[#img-1|1]] shows a domain containing a fluid and particles of different sizes. Particles will be termed ''microscopic'', ''macroscopic'' and ''large'' in terms of their dimensions. Microscopic and macroscopic particles will be assumed to have a cylindrical (in 2D) or spherical (in 3D) shape.  These particles are modelled as rigid objects that undergo interaction forces computed in terms of the relative distance between particles (for microscropic particles) or via the physical contact   between a particle and its neighbors (for macroscopic particles), as in the standard discrete element method (DEM) <span id='citeF-2'></span><span id='citeF-16'></span><span id='citeF-34'></span>[[#cite-2|[2,16,34]]].

<div id='img-1'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 70%;max-width: 100%;"
|-
|[[Image:draft_Samper_928603926-Imagen1.png|420px|Microscopic, macroscopic and large particles within a fluid domain]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 1:''' Microscopic, macroscopic and large particles within a fluid domain
|}

<div id='img-2'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_928603926-Imagen3.png|400px|]]
|[[Image:draft_Samper_928603926-Imagen4.png|400px|(a) Particles of different sizes surrounding the nodes in a FEM mesh. (b) Representative volume for a node (in shadowed  darker colour)]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 2:''' (a) Particles of different sizes surrounding the nodes in a FEM mesh. (b) Representative volume for a node (in shadowed  darker colour)
|}

In this work microscopic and macroscopic particles are assumed to be spherical and submerged in the fluid (Figure [[#img-2|2]]). Fluid-to-particle forces are transferred to the particles via appropriate drag and buoyancy functions. Particle-to-fluid forces have equal magnitude and opposite direction as the fluid-to-particle ones and are transferred to the fluid points as an additional body force vector in the momentum equation (Figure [[#img-3|3]]). These equations, as well as the mass balance equations account for the percentage of particles in the fluid, similarly at it is done in standard immersed approaches for particulate flows <span id='citeF-53'></span><span id='citeF-54'></span><span id='citeF-56'></span>[[#cite-53|[53,54,56]]].

<div id='img-3'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 90%;max-width: 100%;"
|-
|[[Image:draft_Samper_928603926-Imagen5.png|540px|Immersed approach for treating the motion of physical particles in a fluid <span id='citeF-61'></span>[[#cite-61|[61]]]]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 3:''' Immersed approach for treating the motion of physical particles in a fluid <span id='citeF-61'></span>[[#cite-61|[61]]]
|}

Large particles, on the other hand, can have any arbitrary shape and they can be treated as rigid or deformable bodies. In the later case, they are discretized with the standard FEM. The forces between the fluid and the particles and viceversa are computed via fluid-structure interaction (FSI) procedures <span id='citeF-31'></span><span id='citeF-60'></span>[[#cite-31|[31,60]]].

The following sections describe the basic governing equations for a Lagrangian particulate fluid and the computation of the forces for microscopic, macroscopic and large particles.

==3 Basic governing equations for a Lagrangian particulate fluid <span id='citeF-1'></span><span id='citeF-22'></span><span id='citeF-23'></span><span id='citeF-61'></span>[[#cite-1|[1,22,23,61]]]==

==='''Conservation of linear momentum'''===

<span id="eq-1"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\rho _f \frac{Dv_i}{Dt}=  {\partial \sigma _{ij} \over \partial x_j} +b_i + \frac{1}{n_f}  f_i^{pf}\quad , \quad i,j=1,\cdots ,n_s \quad  \hbox{in } V  </math>
|}
| style="width: 5px;text-align: right;" | (1)
|}

In <math display="inline">V</math> is the analysis domain, <math display="inline">n_s</math> is the number of space dimensions (<math display="inline">n_s=3</math> for 3D problems), <math display="inline">\rho _f </math> is the density of the fluid, <math display="inline">v_i</math> and <math display="inline">b_i</math> are the velocity and body force components along the <math display="inline">i</math>th cartesian axis, respectively, <math display="inline">\sigma _{ij}</math> are the fluid Cauchy stresses, <math display="inline">f_i^{pf}</math> are averaged particle-to-fluid interaction forces for which closure relations must be provided and <math display="inline">n_f</math> is the fluid volume fraction defined for each node <math display="inline">j</math> as

<span id="eq-2"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>n_{f_j} = 1- \frac{1}{V_j}\sum \limits _{i=1}^{n_j}V_j^i  </math>
|}
| style="width: 5px;text-align: right;" | (2)
|}

where <math display="inline">V_j</math> is the volume of the representative domain associated to a fluid node <math display="inline">j</math>, <math display="inline">V_j^i</math> is the volume of the <math display="inline">i</math>th particle belonging to <math display="inline">V_j</math> and <math display="inline">n_j</math> is the number of particles contained in <math display="inline">V_j</math>. Note that <math display="inline">n_{f_j}=1 </math> for a representative fluid domain containing no particles (Figure [[#img-2|2]]).

The fluid volume fraction <math display="inline">n_f</math> in Eq.([[#eq-1|1]]) is a continuous function that is interpolated from the nodal values in the finite element fashion <span id='citeF-22'></span>[[#cite-22|[22]]].

Summation of terms with repeated indices  is assumed in Eq.([[#eq-1|1]]) and in  the following, unless otherwise specified.

'''Remark 1.''' The term <math display="inline">\frac{Dv_i}{Dt}</math> in Eq.([[#eq-1|1]]) is the ''material derivative'' of the velocity <math display="inline">v_i</math>. This term is typically computed in a Lagrangian framework as

<span id="eq-3a"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\frac{Dv_i}{Dt} = \frac{{}^{n+1} v_i - {}^n v_i }{\Delta t}     </math>
|}
| style="width: 5px;text-align: right;" | (3a)
|}

with

<span id="eq-3b"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{}^{n+1} v_i:=v_i ({}^{n+1} {\bf x},{}^{n+1} t)\quad ,\quad {}^n v_i:=v_i ({}^n{\bf x},{}^n t)       </math>
|}
| style="width: 5px;text-align: right;" | (3b)
|}

where <math display="inline">{}^n v_i({}^n{\bf x},{}^nt)</math> is the velocity of the material point that has the position <math display="inline">{}^n{\bf x}</math> at time <math display="inline">t={}^nt</math>, where <math display="inline">{\bf x}=[x_1,x_2,x_3]^T</math> is the coordinates vector of a point  in a fixed Cartesian  system. Note that the convective term, typical of Eulerian formulations, does not appear in the definition of the material derivative <span id='citeF-3'></span><span id='citeF-9'></span><span id='citeF-60'></span>[[#cite-3|[3,9,60]]].

==='''Constitutive equations'''===

The Cauchy stresses <math display="inline">\sigma _{ij}</math>  in the fluid are split in the deviatoric (<math display="inline">s_{ij}</math>) and pressure (<math display="inline">p</math>) components as

<span id="eq-4"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\sigma _{ij} = s_{ij}+ p \delta _{ij} </math>
|}
| style="width: 5px;text-align: right;" | (4)
|}

where <math display="inline">\delta _{ij}</math> is the Kronecker delta. In this work the pressure is assumed to be positive for a tension state.

The relationship between the deviatoric stresses and the strain rates has the standard form for a Newtonian fluid <span id='citeF-9'></span>[[#cite-9|[9]]],

<span id="eq-5"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>s_{ij} = 2\mu \varepsilon '_{ij} \quad \hbox{with } \varepsilon '_{ij}= \varepsilon _{ij} - {1 \over 3}\varepsilon _{v} \delta _{ij} \quad \hbox{and}\quad \varepsilon _{v} = \varepsilon _{ii} </math>
|}
| style="width: 5px;text-align: right;" | (5)
|}

In Eq.([[#eq-5|5]]) <math display="inline">\mu </math> is the viscosity, <math display="inline">\varepsilon '_{ij}</math> and <math display="inline">\varepsilon _{v}</math> are the deviatoric and volumetric strain rates, respectively. The strain rates <math display="inline">\varepsilon _{ij}</math> are related to the velocities by

<span id="eq-6"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\varepsilon _{ij} = {1 \over 2}\left({\partial v_i \over \partial x_j} +  {\partial v_j \over \partial x_i}  \right) </math>
|}
| style="width: 5px;text-align: right;" | (6)
|}

==='''Mass conservation equation'''===

The  mass conservation equation for a particulate flow is written as

<span id="eq-8"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>r_v =0  </math>
|}
| style="width: 5px;text-align: right;" | (7a)
|}

with

<span id="eq-9"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>r_v:=\frac{D (n_f \rho _f)}{Dt}+n_f \rho _f \varepsilon _{v} </math>
|}
| style="width: 5px;text-align: right;" | (7b)
|}

Expanding the material derivative and dividing Eq.([[#eq-8|7a]]) by <math display="inline">n_f</math> the expression of <math display="inline">r_v</math> can be rewritten as

<span id="eq-10"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>r_v:= \frac{1}{\kappa } \frac{Dp}{Dt}+ \frac{1}{n_f} \frac{Dn_f}{Dt} +\varepsilon _{v}  </math>
|}
| style="width: 5px;text-align: right;" | (8)
|}

where <math display="inline">\kappa = \rho _f c^2</math> and <math display="inline">c= - {\partial p \over \partial \rho }</math> is the speed of sound in the fluid.

'''Remark 2.''' For <math display="inline">n_f =1</math> (i.e. no particles are contained in the fluid) the standard momentum and mass conservation equations for the fluid are recovered.

==='''Boundary conditions'''===

The boundary conditions at the Dirichlet (<math display="inline">\Gamma _v</math>) and Neumann (<math display="inline">\Gamma _t</math>) boundaries with <math display="inline">\Gamma  = \Gamma _v \cup \Gamma _t</math> are

<span id="eq-11"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>v_i -v_i^p =0 \qquad \hbox{on }\Gamma _v </math>
|}
| style="width: 5px;text-align: right;" | (9)
|}

<span id="eq-12"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\sigma _{ij}n_j -t_i^p =0 \qquad \hbox{on }\Gamma _t \quad i,j=1,\cdots ,n_s  </math>
|}
| style="width: 5px;text-align: right;" | (10)
|}

where <math display="inline">v_i^p</math> and <math display="inline">t_i^p</math> are the prescribed velocities and prescribed tractions on the <math display="inline">\Gamma _v</math> and <math display="inline">\Gamma _t</math> boundaries, respectively and <math display="inline">n_j</math> are the components of the unit normal vector to the boundary <span id='citeF-3'></span><span id='citeF-9'></span><span id='citeF-60'></span>[[#cite-3|[3,9,60]]].

At a free surface the Neumann boundary conditions typically apply.

==4 Motion of microscopic and macroscopic particles==

As mentioned early, microscopic and macroscopic particles are assumed to be rigid spheres (in 3D). Their motion follows the standard law for Lagrangian particles. For the <math display="inline">i</math>th spherical particle

<span id="eq-13"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>m_i \dot {\bf u}_i = {\bf F}_i\quad ,\quad J_i \dot {\bf w}_i = {\bf T}_i  </math>
|}
| style="width: 5px;text-align: right;" | (11)
|}

where <math display="inline">{\bf u}_i</math> and <math display="inline">{\bf w}_i</math> are the velocity and rotation vector of the center of gravity of the particle, <math display="inline">m_i</math> and <math display="inline">J_i </math> are the mass and rotational inertia of the particle, respectively and <math display="inline">{\bf F}_i</math> and <math display="inline">{\bf T}_i</math> are the vectors containing the forces and torques acting at the gravity center of the particle <span id='citeF-34'></span>[[#cite-34|[34]]].

The forces <math display="inline">{\bf F}_i</math> acting on the <math display="inline">i</math>th particle are computed as

<span id="eq-14"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{\bf F}_i = {\bf F}_i^w +  {\bf F}_{i}^c + {\bf F}_i^{fp}  </math>
|}
| style="width: 5px;text-align: right;" | (12)
|}

<math display="inline">{\bf F}_i^w</math>, <math display="inline">{\bf F}_{i}^c</math> and <math display="inline">{\bf F}_i^{fp}</math> are the forces on the particle due to self-weight, contact interactions and fluid effects. These forces are computed as follows.

==='''Self-weight forces'''===

<span id="eq-15"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{\bf F}_i^w =-\rho _i \Omega _i {\bf g}  </math>
|}
| style="width: 5px;text-align: right;" | (13)
|}

where <math display="inline">\rho _i</math> and <math display="inline">\Omega _i</math> are the density and the volume of the <math display="inline">i</math>th particle, respectively and <math display="inline">{g}</math> is the gravity acceleration vector.

==='''Contact forces'''===

<span id="eq-16"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{\bf F}_{i}^c = \sum \limits _{j=1}^{n_i} {\bf F}_{ij}^c   </math>
|}
| style="width: 5px;text-align: right;" | (14)
|}

where <math display="inline">n_i</math> is the number of contact interfaces for the <math display="inline">i</math>th particle.

<span id="eq-17"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{\bf F}_{ij}^c = {\bf F}_n^{ij} + {\bf F}_s^{ij} = {\bf  F}_n^{ij}n_i + {\bf F}_s^{ij}  </math>
|}
| style="width: 5px;text-align: right;" | (15)
|}

where <math display="inline">{\bf F}_n^{ij}</math> and <math display="inline">{\bf F}_s^{ij}</math> are the normal and tangential forces acting at the <math display="inline">i</math>th interface connecting particles <math display="inline">i</math> and <math display="inline">j</math> (Figure [[#img-4|4]]). These forces are computed in terms of the relative motion of the interacting particles as in the standard DEM <span id='citeF-2'></span><span id='citeF-16'></span><span id='citeF-34'></span>[[#cite-2|[2,16,34]]].

<div id='img-4'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_928603926-contact-particles.png|400px|]]
|[[Image:draft_Samper_928603926-contact.png|400px|]]
|-
|[[Image:draft_Samper_928603926-Fig3.png|400px|]]
|[[Image:draft_Samper_928603926-Fig5.png|400px|Interacting forces between microscopic (a) and macroscopic (b) particles]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 4:''' Interacting forces between microscopic (a) and macroscopic (b) particles
|}

For microscopic particles the tangential forces <math display="inline">{\bf F}_s^{ij}</math> are neglected in Eq.([[#eq-17|15]]).

<br/>

'''Fluid-to-particle forces:''' <math display="inline">{\bf F}_i^{fp} = {\bf F}_i^d + {\bf F}_i^b</math>, where <math display="inline">{\bf F}_i^b</math> and <math display="inline">{\bf F}_i^d</math> are, respectively, the buoyancy and drag forces on the <math display="inline">i</math>th particle. These forces are computed as:

==='''Buoyancy forces'''===

<span id="eq-18"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{\bf F}_i^b = {\Omega }_i {\boldsymbol \nabla } p  </math>
|}
| style="width: 5px;text-align: right;" | (16)
|}

==='''Drag forces'''===

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{\bf F}_i^d = {\bf f}_i^d n_f^{-(\chi +1)}</math>
|}
|}

where <math display="inline">\chi = \chi (Re)</math> is a coefficient that depends on the local Reynolds number for the particle <math display="inline">Re</math> <span id='citeF-1'></span><span id='citeF-6'></span><span id='citeF-15'></span><span id='citeF-22'></span><span id='citeF-23'></span>[[#cite-1|[1,6,15,22,23]]] and

<span id="eq-19"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{\bf f}_i^d = \frac{1}{2} \rho _f A_i C_d \Vert {\bf v}_{f_i} - {\bf v}_i\Vert ({\bf v}_{f_i} - {\bf v}_i)  </math>
|}
| style="width: 5px;text-align: right;" | (17)
|}

In Eq.([[#eq-19|17]]) <math display="inline">{\bf v}_{fi}</math> and <math display="inline">{\bf v}_i</math> are respectively the velocity of the fluid and of the particle center, <math display="inline">A_i</math> is the area of the particle surface with radius <math display="inline">r_i</math> (<math display="inline">2\pi r_i</math> or <math display="inline">4\pi r_i^2</math> for a circle or a sphere, respectively) and <math display="inline">C_d</math> is a drag coefficient that depends on the particle geometry and the rugosity of its surface <span id='citeF-22'></span><span id='citeF-23'></span>[[#cite-22|[22,23]]].

The force term <math display="inline"> {f}^{pf}_i</math> in the r.h.s. of the momentum equations (Eq.([[#eq-1|1]])) is computed for each particle (in vector form) as <math display="inline"> {\bf f}^{pf}=-{\bf f}^{fp}</math> with vector <math display="inline"> {\bf f}^{fp}</math> computed ''at each node'' in the fluid mesh         from the drag forces <math display="inline">{\bf F}_i^{d}</math> as

<span id="eq-20"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{\bf f}^{fp}_j = \frac{1}{V_j} \sum \limits _{i=1}^{n_j} {\bf F}_i^d~~~,~~ j=1,N  </math>
|}
| style="width: 5px;text-align: right;" | (18)
|}

A continuum distribution of <math display="inline">{\bf f}^{fp}</math> is obtained by interpolating its nodal values over each element in the FEM fashion.

We note that the forces on the particles due to lift effects have been neglected in the present analysis. These forces can be accounted for as explained in <span id='citeF-22'></span>[[#cite-22|[22]]].

==5 Motion of large particles==

As mentioned earlier, large particles may be considered as rigid or deformable bodies. In the first case the motion follows the rules of Eq.([[#eq-13|13]]) for rigid Lagrangian particles. The contact forces at the particle surface due to the adjacent interacting particles are computed using a frictional contact interface layer between particles as in the standard PFEM (Section 10.2).

The fluid forces on the particles are computing by integrating the tangential (viscous) and normal (pressure) forces at the edges of the fluid elements surrounding the particles.

Large deformable particles, on the other hand, behave as deformable bodies immersed in the fluid which are discretized via the standard FEM. Their motion can be followed using a staggered FSI scheme, or else by treating the deformable bodies and the fluid as a single continuum with different material properties. Details of this unified treatment of the interaction between fluids and deformable solids can be found in <span id='citeF-12'></span><span id='citeF-18'></span><span id='citeF-46'></span>[[#cite-12|[12,18,46]]].

==6 Stabilized FIC form of the mass balance equation==

The modelling of incompressible fluids with a mixed finite element method using an equal order interpolation for the velocities and the pressure requires introducing a stabilized formulation for the mass balance equation.

In our work we use a stabilized form of the mass balance equation obtained via the Finite Calculus (FIC) approach <span id='citeF-29'></span><span id='citeF-30'></span><span id='citeF-31'></span><span id='citeF-32'></span><span id='citeF-33'></span><span id='citeF-37'></span><span id='citeF-38'></span><span id='citeF-39'></span>[[#cite-29|[29-33,37-39]]]. The FIC stabilized mass balance equation is written as

<span id="eq-21"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>r_v - \tau {\partial \bar r_{m_i} \over \partial x_i}=0 \quad \hbox{in } V  </math>
|}
| style="width: 5px;text-align: right;" | (19)
|}

where

<span id="eq-22"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\bar r_{m_i} = {\partial \sigma _{ij} \over \partial x_j}+b_i   + \frac{1}{n_f} f_i^{pf}  </math>
|}
| style="width: 5px;text-align: right;" | (20)
|}

is a static momentum term and <math display="inline">\tau </math> is a stabilization parameter computed as

<span id="eq-23"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\tau = \left(\frac{8\mu }{h^2}+\frac{2\rho _f}{\delta }   \right)^{-1}  </math>
|}
| style="width: 5px;text-align: right;" | (21)
|}

where <math display="inline">h</math> is a characteristic distance of each finite element and <math display="inline">\delta </math> is a time parameter.

The derivation of Eq.([[#eq-21|19]]) for an homogeneous quasi-incompressible fluid is presented in <span id='citeF-45'></span>[[#cite-45|[45]]].

The stabilization parameter <math display="inline">\tau </math>  is computed in practice for each element <math display="inline">e</math> using <math display="inline">h=l^e</math> and <math display="inline">\delta = \Delta t</math> as

<span id="eq-24"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\tau = \left(\frac{8\mu }{(l^e)^2} + \frac{2\rho }{\Delta t}  \right)^{-1} </math>
|}
| style="width: 5px;text-align: right;" | (22)
|}

where <math display="inline">\Delta t</math> is the time step used for the transient solution and <math display="inline">l^e</math> is a characteristic element length computed as <math display="inline">l^e = 2(V^e)^{1/n_s}</math> where <math display="inline">V^e</math> is the element area (for 3-noded triangles) or volume (for 4-noded tetrahedra). For fluids with heterogeneous material properties the values of <math display="inline">\mu </math> and <math display="inline">\rho </math> in Eq.([[#eq-24|22]]) are computed at the element center.

==7 Variational equations for the fluid==

The variational form of the momentum and mass balance equations is obtained via the standard weighted residual approach <span id='citeF-9'></span><span id='citeF-60'></span>[[#cite-9|[9,60]]]. The resulting integral expressions after integration by parts and some algebra are:

==='''Momentum equations'''===

<span id="eq-25"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\int _V \! w_i \rho \frac{Dv_i}{Dt} dV + \!\int _V \left[\delta \varepsilon _{ij} 2\mu \varepsilon '_{ij} + \delta \varepsilon _{v}p\right]dV - \!\int _V w_i \left(b_i   + \frac{1}{n_f} f_i^{pf}\right)dV - \!\int _{\Gamma _t} w_i t_i^p d\Gamma =0  </math>
|}
| style="width: 5px;text-align: right;" | (23)
|}

==='''Mass balance equation'''===

<span id="eq-26"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\begin{array}{r}\displaystyle \int _V \frac{q}{\kappa } \frac{Dp}{Dt}dV - \int _V q \left(\frac{1}{n_f} \frac{Dn_f}{Dt} +\varepsilon _{v}\right)dV + \int _V \tau {\partial q \over \partial x_i} \left({\partial  \over \partial x_i} (2\mu \varepsilon _{ij})+ {\partial p \over \partial x_i}+b_i\right)dV\\ \displaystyle - \int _{\Gamma _t} q \tau \left[\rho \frac{Dv_n}{Dt}-\frac{2}{h_n} (2\mu {\partial v_n \over \partial n} + p -t_n)\right]d\Gamma =0 \end{array}   </math>
|}
| style="width: 5px;text-align: right;" | (24)
|}

The derivation of Eqs.([[#eq-25|23]]) and ([[#eq-26|24]]) for homogeneous fluids can be found in <span id='citeF-45'></span>[[#cite-45|[45]]].

==8 FEM discretization==

We discretize  the analysis domain containing <math display="inline">N_p</math> microscopic and macroscopic particles and a number of large particles into finite elements with <math display="inline">n</math> nodes in the standard manner leading to a mesh with a total number of <math display="inline">N_e</math> elements and <math display="inline">N</math> nodes. In our work we will choose simple 3-noded linear triangles (<math display="inline">n=3</math>) for 2D problems and 4-noded tetrahedra (<math display="inline">n=4</math>) for 3D problems with local linear shape functions <math display="inline">N_i^e</math> defined for each node <math display="inline">i</math> of element <math display="inline">e</math> <span id='citeF-41'></span><span id='citeF-58'></span>[[#cite-41|[41,58]]]. The velocity components, the weighting functions and the pressure are interpolated over the mesh in terms of their nodal values in the same manner using the  global linear shape functions <math display="inline">N_j</math> spanning over the elements sharing node <math display="inline">j</math> (<math display="inline">j=1,N</math>)  <span id='citeF-41'></span><span id='citeF-58'></span>[[#cite-41|[41,58]]].

The finite element interpolation over the fluid domain can be written in matrix form as

<span id="eq-27"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{\bf v} = {\bf N}_v\bar {\bf v} ~,~{\bf w} = {\bf N}_v\bar {\bf w} ~,~p = {\bf N}_p\bar {\bf p}~,~ q= {\bf N}_p \bar {\bf q}  </math>
|}
| style="width: 5px;text-align: right;" | (25)
|}

where

<span id="eq-28"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\begin{array}{c}\displaystyle \bar {\bf v} = \left\{\begin{matrix}\bar{\bf v}^1\\\bar{\bf v}^2\\\vdots \\ \bar{\bf v}^N\end{matrix}  \right\}\quad \hbox{with}~ \bar{\bf v}^i = \left\{\begin{matrix}\bar{v}^i_1\\\bar{v}^i_2\\\bar{v}^i_3\end{matrix}  \right\}~, ~ \bar {\bf w}= \left\{\begin{matrix}\bar{\bf w}^1\\\bar{\bf w}^2\\\vdots \\ \bar{\bf w}^N\end{matrix}  \right\}\quad \hbox{with}~ \bar{\bf w}^i = \left\{\begin{matrix}\bar{w}^i_1\\\bar{w}^i_2\\\bar{w}^i_3\end{matrix}  \right\}~,~\bar {\bf p} =\left\{\begin{matrix}\bar{p}^1\\\bar{p}^2\\\vdots \\ \bar{p}^{N}\end{matrix}  \right\}\quad \hbox{and}~ \bar {\bf q} =\left\{\begin{matrix}\bar{q}^1\\\bar{q}^2\\\vdots \\ \bar{q}^{N}\end{matrix}  \right\}\\ \displaystyle {\bf N}_v = [{\bf N}_1, {\bf N}_2,\cdots , {\bf N}_N ]^T ~,~ {\bf N}_p = [{N}_1, { N}_2,\cdots , {N}_N ] \end{array} </math>
|}
| style="width: 5px;text-align: right;" | (26)
|}

with <math display="inline">{\bf N}_j = N_j {\bf I}_{n_s}</math> where <math display="inline">{I}_{n_s}</math> is the <math display="inline">n_s\times n_s</math> unit matrix.

In Eq.([[#eq-28|26]]) vectors <math display="inline">\bar{\bf v}</math>, <math display="inline">(\bar{\bf w},\bar {\bf q})</math> and <math display="inline">\bar {\bf p}</math> contain the nodal velocities, the nodal weighting functions and the nodal pressures for the whole mesh, respectively and the upperindex denotes the nodal value for each vector or scalar magnitude.

Substituting the approximation ([[#eq-27|25]]) into the variational forms ([[#eq-25|23]]) and ([[#eq-26|24]]) gives the system of algebraic equations for the particulate fluid in matrix form as

<span id="eq-29"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{\bf M}_0 {\dot{\bar{\bf v}}} + {\bf K}\bar{\bf v}+{\bf Q}\bar {\bf p}- {\bf f}_v={\bf 0} </math>
|}
| style="width: 5px;text-align: right;" | (27a)
|}

<span id="eq-30"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{\bf M}_1 {\dot{\bar{\bf p}}}-{\bf Q}^T \bar{\bf v} + ({\bf L}+{\bf M}_b)\bar {\bf p}- {\bf f}_p={\bf 0} </math>
|}
| style="width: 5px;text-align: right;" | (27b)
|}

The different matrices and vectors in Eqs.([[#8 FEM discretization|27]]) are shown in Box 1 for 2D problems.

<div class="center" style="font-size: 75%;">
[[File:Draft_Samper_928603926_2153_Box1.png|500px]]

'''Box 1'''. Element form of the matrices and vectors in Eqs.(27) for 2D problems</div>


'''Remark 3.''' The boundary terms of vector <math display="inline">{\bf f}_p</math> can be incorporated in the matrices of Eq.([[#eq-30|27b]]). This, however, leads to a non symmetrical set of equations. For this reason we have chosen to compute these boundary terms iteratively within the incremental solution scheme.

'''Remark 4.''' Matrix <math display="inline">{\bf M}_b</math> in Eq.([[#eq-30|27b]]) allows us to compute the pressure without the need of prescribing its value at the free surface. This eliminates the error introduced when the pressure is prescribed to zero in free boundaries, which may lead to considerable mass losses <span id='citeF-20'></span><span id='citeF-45'></span>[[#cite-20|[20,45]]].

==9 Incremental solution of the discretized equations==

Eqs.([[#8 FEM discretization|8]]) are solved in time with an implicit Newton-Raphson type iterative scheme <span id='citeF-3'></span><span id='citeF-9'></span><span id='citeF-58'></span><span id='citeF-60'></span>[[#cite-3|[3,9,58,60]]]. The basic steps within a time interval <math display="inline">[n,n+1]</math> are:<br/>

* Initialize variables: <math display="inline">({}^{n+1}\hbox{x}^1,{}^{n+1}\bar {v}^1,{}^{n+1}\bar{p}^1,{}^{n+1}n_f^i,   {}^{n+1}\bar{r}^1_m)\equiv  \left\{{}^{n}{x},{}^{n}\bar{v},{}^{n}\bar{p},{}^{n} n_f ,{}^{n}\bar{r}_m \right\}</math>.
* Iteration loop: <math display="inline">k=1,\cdots , NITER</math>.<p> For each iteration. 

</p>

<br/>

===''Step 1. Compute the nodal velocity increments <math>\Delta \bar {v}</math>''===

From Eq.([[#8 FEM discretization|8]]a), we deduce

<span id="eq-31"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{}^{n+1}{H}_v^i \Delta \bar {v} = - {}^{n+1}\bar{\hbox{r}}_m^k \rightarrow  \Delta \bar{v} </math>
|}
| style="width: 5px;text-align: right;" | (31)
|}

with the momentum residual <math display="inline">\bar{r}_m</math> and the iteration matrix <math display="inline">{H}_v</math>  given by

<span id="eq-32"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\bar{\hbox{r}}_m = \hbox{M}_0 {\dot{\bar{v}}} + \hbox{K}\bar{v}+\hbox{Q}\bar {p}-\hbox{f}_v\quad , \quad \hbox{H}_v = \frac{1}{\Delta t} \hbox{M}_0 + \hbox{K} + \hbox{K}_v  </math>
|}
| style="width: 5px;text-align: right;" | (32)
|}

with

<span id="eq-33"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{K}_v^e = \int _{{}^nV^e} \hbox{B}^T \hbox{m} \theta \Delta t\kappa \hbox{m}^T \hbox{B}dV </math>
|}
| style="width: 5px;text-align: right;" | (33)
|}

===''Step 2. Update the  velocities''===

<span id="eq-34"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{Fluid nodes:}~~{}^{n+1}\bar{v}^{k+1}= {}^{n+1}\bar{v}^k + \Delta \bar{v} </math>
|}
| style="width: 5px;text-align: right;" | (34)
|}

<span id="eq-35"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{Rigid particles:}~~\left\{\begin{array}{l}{}^{n+1/2}\dot{u}_{j} = {}^{n-1/2}\dot{u}_{j} + {}^{n}\ddot{u}_{j}^{k+1} \Delta t\\ \dot{u}_{j} = \frac{1}{m_j}  {}^{n}{F}_{j}^{k+1} ~~,~~j=1,N_p \end{array}\right.  </math>
|}
| style="width: 5px;text-align: right;" | (35)
|}

===''Step 3. Compute the nodal pressures'' <math> {}^{n+1}\bar{p}^{k+1}</math>===

From Eq.([[#eq-30|30]]) we obtain

<span id="eq-36"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>{}^{n+1} \hbox{H}_p^i  {}^{n+1}\bar{p}^{k+1}= \frac{1}{\Delta t} \hbox{M}_1 {}^{n+1}\bar{p}^i + {Q}^T {}^{n+1}\bar{v}^{k+1}+ {}^{n+1}\bar{f}^{i}_p  \rightarrow  {}^{n+1}\bar{p}^{k+1}   </math>
|}
| style="width: 5px;text-align: right;" | (36)
|}

with

<span id="eq-37"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{H}_p = \frac{1}{\Delta t} \hbox{M}_1 + \hbox{L} + \hbox{M}_b </math>
|}
| style="width: 5px;text-align: right;" | (37)
|}

===''Step 4. Update the coordinates of the fluid nodes and particles''===

<span id="eq-38"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{Fluid nodes:}~~{}^{n+1}{x}^{k+1}_i= {}^{n+1}{x}^k_i + \frac{1}{2} ({}^{n+1}\bar{v}^{k+1}_i + {}^{n}\bar{v}_i)\Delta t \quad ,~~ i=1,N </math>
|}
| style="width: 5px;text-align: right;" | (38)
|}

<span id="eq-39"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\hbox{Rigid particles:}~~\left\{\begin{array}{l}{}^{n+1}{u}_{i}^{k+1} = {}^{n}{u}_{i}^{k+1} + {}^{n+1/2}\dot{u}_{i}^{k+1}\Delta t\\[.4cm] {}^{n+1}{x}^{k+1}_i = {}^{n}{x}_i + {}^{n+1}{u}^{k+1}_i \quad ,~~ i=1,N_p \end{array}\right.  </math>
|}
| style="width: 5px;text-align: right;" | (39)
|}

===''Step 5. Compute the fluid volume fractions for each node <math>{}^{n+1}{n}^{k+1}_{f_i}</math> via Eq.(2)''===

===''Step 6. Compute forces and torques on particles:'' <math>{}^{n+1}{F}^{k+1}_i,{}^{n+1}{T}^{k+1}_i~ ,~i=1,N_p</math>===

===''Step 7. Compute particle-to-fluid nodes:'' <math>({}^{n+1}{f}^{pf}_i)^{k+1} = - ({}^{n+1}{f}^{fp}_i)^{k+1} ~ ,~ i=1,N</math> with <math>{f}^{fp}_i</math> computed by Eq.(18)===

===''Step 8. Check convergence''===

Verify the following conditions:

<span id="eq-40"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;" 
|-
| style="text-align: center;" | <math>\begin{array}{c}\displaystyle \Vert {}^{n+1}\bar {v}^{k+1}- {}^{n+1}\bar{v}^{k}\Vert \le e_v \Vert {}^{n}\bar {v}\Vert \\ \displaystyle \Vert {}^{n+1}\bar {p}^{k+1}-{}^{n+1}\bar {p}^k\Vert \le e_p  \Vert {}^{n}\bar{p}\Vert  \end{array}   </math>
|}
| style="width: 5px;text-align: right;" | (40)
|}

where <math display="inline">e_v</math> and <math display="inline">e_p</math> are prescribed error norms for the nodal velocities and the nodal pressures, respectively. In the examples solved in this work we have set <math display="inline">e_v = e_p=10^{-3}</math>.

If both conditions ([[#eq-40|40]]) are satisfied then make <math display="inline">n \leftarrow n+1 </math> and proceed to the next  time step.

Otherwise, make the iteration counter <math display="inline">k \leftarrow k+1 </math> and repeat Steps 1&#8211;8.

'''Remark 5.''' In Eqs.([[#''Step 1. Compute the nodal velocity increments <math>\Delta \bar {v}</math>''|9]])&#8211;([[#eq-40|40]]) <math display="inline">{}^{n+1}(\cdot )</math> denotes the values of a matrix or a vector computed using the nodal unknowns at time <math display="inline">n+1</math>. In our work the derivatives and integrals in the iteration matrices <math display="inline">{H}_v</math> and <math display="inline">{H}_p</math> and the residual vector <math display="inline">\bar{r}_m</math>  are computed on the ''discretized geometry at time'' <math display="inline">n</math> (i.e. <math display="inline">V^e = {}^{n}V^e</math>) while the nodal force vectors <math display="inline">{f}_v</math> and <math display="inline">{f}_p</math> are computed on the current configuration at time <math display="inline">n+1</math>. This is equivalent to using an ''updated Lagrangian formulation'' <span id='citeF-3'></span><span id='citeF-12'></span><span id='citeF-44'></span><span id='citeF-59'></span>[[#cite-3|[3,12,44,59]]].

'''Remark 6.''' The tangent “bulk” stiffness matrix <math display="inline">\hbox{K}_v</math> in the iteration matrix <math display="inline">\hbox{H}_v</math> of Eq.([[#eq-32|32]]) accounts for the changes of the pressure due to the velocity. Including matrix <math display="inline">\hbox{K}_v</math> in <math display="inline">\hbox{H}_v</math> has proven to be essential for the fast convergence, mass preservation and overall accuracy of the iterative solution <span id='citeF-11'></span><span id='citeF-45'></span>[[#cite-11|[11,45]]].

'''Remark 7.'''  The parameter <math display="inline">\theta </math> in <math display="inline">{K}_v</math> (<math display="inline">0<   \theta \le 1</math>)  has the role of preventing the ill-conditioning of the iteration matrix <math display="inline">{H}_v</math> for very large values of the speed of sound in the fluid that lead to a dominant role of the terms of the tangent bulk stiffness matrix <math display="inline">{K}_v</math>. An adequate selection of <math display="inline">\theta </math> also improves the overall accuracy of the numerical solution and the preservation of mass for large time steps. Details of the derivation of Eq.(28c) can be found in <span id='citeF-45'></span>[[#cite-45|[45]]].

'''Remark 8.''' The iteration matrix <math display="inline">\hbox{H}_v</math> in Eq.([[#eq-31|31]]) is an ''approximation of the exact tangent matrix'' in the updated Lagrangian formulation for a quasi/fully incompressible fluid <span id='citeF-44'></span>[[#cite-44|[44]]]. The simplified form of <math display="inline">\hbox{H}_v</math>  used in this work has yielded very good results  with convergence achieved for the nodal velocities and pressure in 3&#8211;4 iterations in all the problems analyzed.

'''Remark 9.''' The time step within a time interval <math display="inline">[n,n+1]</math> has been chosen as <math display="inline">\Delta t =\min \left(\frac{^n l_{\min }^e}{\vert {}^n{v}\vert _{\max }},\Delta t_b\right)</math> where <math display="inline">^n l_{\min }^e</math> is the minimum characteristic distance  of all elements in the mesh, with <math display="inline">l^e</math> computed as explained in Section 6, <math display="inline">\vert ^n{v}\vert _{\max }</math> is the maximum value of the modulus of the velocity of all nodes in the mesh  and <math display="inline">\Delta t_b</math> is the critical time step of all nodes approaching a solid boundary <span id='citeF-45'></span>[[#cite-45|[45]]].

==10 About the particle finite element method (PFEM)==

===10.1 The basis of the PFEM===

Let us consider a domain <math display="inline">V</math> containing  fluid and solid subdomains. Each subdomain is characterized by a set of points, hereafter termed ''virtual particles''. The virtual particles contain all the information  for defining the geometry and the material and mechanical properties of the underlying subdomain. In the PFEM both subdomains are modelled using an ''updated'' ''Lagrangian formulation'' <span id='citeF-3'></span><span id='citeF-44'></span><span id='citeF-59'></span>[[#cite-3|[3,44,59]]].

The solution steps within a time step in the PFEM are as follows:

<ol>

<li>The starting point at each time step is the cloud of points <math display="inline">C</math> in the fluid and solid domains. For instance <math display="inline">^{n} C</math> denotes the cloud at time <math display="inline">t={}^n t </math> (Figure [[#img-5|5]]). </li>

<li>Identify the boundaries defining the analysis domain <math display="inline">^{n} V</math>, as well as the subdomains 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-10'></span>[[#cite-10|[10]]] is used for the boundary definition. Clearly, the accuracy in the reconstruction of the boundaries depends on the number of points in the vicinity of each boundary and on the Alpha Shape parameter. In the problems solved in this work the Alpha Shape method has been implementation as described in <span id='citeF-17'></span><span id='citeF-35'></span>[[#cite-17|[17,35]]]. </li>

<li> Discretize the the analysis domain <math display="inline">^{n} V</math> with a finite element mesh <math display="inline">{}^{n} M.</math>We use an efficient mesh generation scheme based on an enhanced Delaunay tesselation <span id='citeF-17'></span><span id='citeF-35'></span>[[#cite-17|[17,35]]]. </li>

<li> Solve the Lagrangian equations of motion for the overall continuum using the standard FEM. Compute the state variables in at the next (updated) configuration for <math display="inline">^n 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 ''n''+1 denotes the time <math display="inline">{}^n t +\Delta t</math>, in terms of the time increment size. </li>

<li> Go back to step 1 and repeat the solution for the next time step to obtain <math display="inline">{}^{n+2} C</math>. </li>

</ol>

<div id='img-5'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_928603926-Figure2a.png|600px|Sequence of steps in the PFEM to update a “cloud” of nodes representing a domain containing a fluid and a solid part (in darker shadow) from time n   (t=ⁿt)  to   time n+2 (t=ⁿt +2∆t)  ]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 5:''' Sequence of steps in the PFEM to update a “cloud” of nodes representing a domain containing a fluid and a solid part (in darker shadow) from time <math>n</math>   (<math>t=^n t</math>)  to   time <math>n+2</math> (<math>t=^n t +2\Delta t</math>)  
|}

Note that the key differences between the PFEM and the classical FEM are the remeshing technique and the identification of the domain boundary at each time step.

The CPU time required for meshing grows linearly with the number of nodes. As a general rule,  meshing consumes for 3D problems around 15% of the total CPU time per time step <span id='citeF-43'></span>[[#cite-43|[43]]].

Application of the PFEM in fluid and solid mechanics and in fluid-structure interaction problems can be found in [CaOnSu10,CaOnSu13,CrFrPe11,FrOnCa13,IdOnPi04,IdMaLiOn08,IdMiOn09a,IdOn10,
Laetal08,Oliver07,Oliver09,
Onetal04b,OnCeId06a,Onetal06c,Onetal08,Onetal10,Onetal11,OnCa13,OnFrCa14a,OnFrCa14b], as well in www.cimne.com/pfem.

===10.2 Treatment of contact conditions===

Known velocities at boundaries in the PFEM are prescribed in strong form to the boundary nodes. These nodes might belong to fixed external boundaries or to moving boundaries linked to the interacting solids. Surface tractions are applied to the Neumann part of the boundary, as usual in the FEM.

Contact between fluid particles and fixed boundaries is accounted for by adjusting the time step so that fluid nodes do not penetrate into the solid boundaries <span id='citeF-45'></span>[[#cite-45|[45]]].

The contact between two large particles (and between two bodies, in general) is treated by introducing a layer of ''contact elements'' between the two interacting particles. This contact interface layer is ''automatically created during the mesh generation step'' by prescribing a minimum distance <math display="inline">\left(h_{c} \right)</math> between two interacting particles. If the distance exceeds the minimum value <math display="inline">\left(h_{c} \right)</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  <span id='citeF-35'></span><span id='citeF-40'></span><span id='citeF-43'></span>[[#cite-35|[35,40,43]]] (Figure [[#img-6|6]]).

<div id='img-6'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_928603926-Imagen2.png|400px|]]
|[[Image:draft_Samper_928603926-contact-PFEM-particles.png|400px|(a) Large particles (in dark shadow) surrounded by a finite element mesh. The contact interface is shown in lighter shadow. (b) Contact interface between two objects treated as large particles and between an object and a wall]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 6:''' (a) Large particles (in dark shadow) surrounded by a finite element mesh. The contact interface is shown in lighter shadow. (b) Contact interface between two objects treated as large particles and between an object and a wall
|}

This algorithm allows us to model complex frictional contact conditions between two or more interacting bodies moving in water in an a simple manner. The algorithm has  been used to model frictional contact situations between rigid and elastic solids in structural mechanics applications, such as soil/rock excavation problems <span id='citeF-4'></span><span id='citeF-5'></span>[[#cite-4|[4,5]]]. The frictional contact algorithm described above has been extended by Oliver ''et al.'' <span id='citeF-27'></span><span id='citeF-28'></span>[[#cite-27|[27,28]]] for analysis of metal cutting and machining problems.

Figure [[#img-7|7]] shows an example of the analysis with the PFEM of a collection of large particles submerged in a tank containing water in sloshing motion.

<div id='img-7'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 70%;max-width: 100%;"
|-
|[[Image:draft_Samper_928603926-vlcsnap-2014-02-25-16h00m21s64.png|420px|PFEM results for the motion of large particles submerged  in a tank containing water in sloshing motion]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 7:''' PFEM results for the motion of large particles submerged  in a tank containing water in sloshing motion
|}

===10.3 Treatment of surface 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.

Oñate et al. <span id='citeF-36'></span>[[#cite-36|[36]]] have proposed an extension of the PFEM to model bed erosion. The erosion model is based on the frictional work  at the bed surface originated by the shear stresses in the fluid.

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

<ol>

<li>Compute at every point of the bed surface the  tangential   stress <math display="inline">\tau </math> induced by the fluid motion. </li>

<li>Compute the frictional work <math display="inline">W_f</math> originated by the tangential stress at the   bed surface. </li>

<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>

<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 adjacent   fluid node. This mass is subsequently transported with the fluid as an immersed macroscopic particle. </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-8|8]] shows an schematic view of the bed erosion algorithm described. Details of the algorithm can be found in <span id='citeF-36'></span>[[#cite-36|[36]]].

<div id='img-8'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 90%;max-width: 100%;"
|-
|[[Image:draft_Samper_928603926-bed-erosion-2.png|540px|Modeling of bed erosion with the PFEM. The mass of the eroded domain is assigned to the fluid node k]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 8:''' Modeling of bed erosion with the PFEM. The mass of the eroded domain is assigned to the fluid node <math>k</math>
|}

==11 A nodal  algorithm for transporting  microscopic and macroscopic particles within a finite element mesh==

The fact that in the PFEM a new mesh is regenerated at each time step allows us to make microscopic and macroscopic particles to be coincident with fluid nodes. An advantage of this procedure is that it provides an accurate definition of the particles at each time step which is useful for FSI problems.

The algorithm to compute the position of the particles using the nodal algorithm is as follows.

At each time step <math display="inline">{}^n t</math>:

<ol>

<li>Compute the converged value of the position of the fluid nodes (<math display="inline">{}^{n+1}{x}_i,~i = 1,\cdots ,N</math>) and the particles (<math display="inline">{}^{n+1}{x}_j,~j = 1,\cdots ,N_p</math>) using the algorithm of Section 9. The <math display="inline">N_p</math> particles coinciding with <math display="inline">N_p</math> fluid nodes (<math display="inline">N_p \le N</math>) will typically move to a different position than that of the corresponding fluid nodes (Figure [[#img-9|9]]).   </li>
<li>Regenerate the mesh discretizing the fluid domain treating the position of the <math display="inline">N_p</math> particles as fluid nodes and ignore the fluid nodes originally coinciding with the <math display="inline">N_p</math> particles at <math display="inline">{}^{n+1} t</math>.   </li>
<li>Interpolate the velocity of the fluid nodes at the position of the <math display="inline">N_p</math> particles surrounding the fluid nodes. </li>

</ol>

The algorithm is schematically described in Figure [[#img-9|9]].

Figures [[#img-10|10]] show an example of the application of the nodal algorithm to the study of  the motion of an individual particle  within a rectangular domain filled with water. The correct end velocity for the individual particle is obtained as shown in Figure [[#img-10|10]]c. Figures [[#img-11|11]]&#8211;[[#img-13|13]] show other examples of application of the nodal algorithm to the motion of macro-particles in water containers.

Other examples of application of this algorithm are shown in the next section.

<div id='img-9'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_928603926-motion-particle-node.png|600px|Nodal algorithm for tracking the motion of particles submerged in a fluid. (a) Particle i is coincident with a fluid node. (b) Update the position of the particle and the adjacent nodes. (c) Regeneration of the fluid mesh consistent with the new particle position]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 9:''' Nodal algorithm for tracking the motion of particles submerged in a fluid. (a) Particle <math>i</math> is coincident with a fluid node. (b) Update the position of the particle and the adjacent nodes. (c) Regeneration of the fluid mesh consistent with the new particle position
|}

<div id='img-10'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_928603926-one-particle1.png|400px|]]
|[[Image:draft_Samper_928603926-one-particle-contours.png|400px|]]
|-
| colspan="2"|[[Image:draft_Samper_928603926-Grafica-velocity.png|360px|Cylindrical particles falling in a water container. 2D PFEM solution using the nodal algorithm for tracking the particle motion. (a) Mesh and particle at a certain instant. (b) Contours of the vertical velocity module. (c) Evolution of the vertical velocity of the particle until a steady state solution is found <span id='citeF-6'></span><span id='citeF-15'></span>[[#cite-6|[6,15]]]]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 10:''' Cylindrical particles falling in a water container. 2D PFEM solution using the nodal algorithm for tracking the particle motion. (a) Mesh and particle at a certain instant. (b) Contours of the vertical velocity module. (c) Evolution of the vertical velocity of the particle until a steady state solution is found <span id='citeF-6'></span><span id='citeF-15'></span>[[#cite-6|[6,15]]]
|}

<div id='img-11'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_928603926-particles-crossing1.png|400px|]]
|[[Image:draft_Samper_928603926-particles-crossing2.png|400px|Motion of ascending and descending particles of different density in a fluid domain. PFEM results using the nodal algorithm for tracking the particles motion]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 11:''' Motion of ascending and descending particles of different density in a fluid domain. PFEM results using the nodal algorithm for tracking the particles motion
|}

<div id='img-12'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_928603926-three-particles2.png|600px|Motion of three macroscopic particles in a water sloshing problem within a tank. PFEM results obtained using the nodal algorithm for particle tracking]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 12:''' Motion of three macroscopic particles in a water sloshing problem within a tank. PFEM results obtained using the nodal algorithm for particle tracking
|}

<div id='img-13'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_928603926-tanque-particles-entrada_clara.png|400px|]]
|[[Image:draft_Samper_928603926-tanque-particles-clara_clara.png|400px|PFEM analysis of the penetration of a collection of spherical (macroscopic) particles into a water container]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 13:''' PFEM analysis of the penetration of a collection of spherical (macroscopic) particles into a water container
|}

<div id='img-14'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_928603926-Fig-soilmass1.png|400px|]]
|[[Image:draft_Samper_928603926-Fig-soilmass2.png|400px|]]
|-
|[[Image:draft_Samper_928603926-Fig-soilmass3.png|400px|]]
|[[Image:draft_Samper_928603926-Fig-soilmass4.png|400px|Falling of a lorry into the sea by erosion of the underlying soil mass due to the action of waves]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 14:''' Falling of a lorry into the sea by erosion of the underlying soil mass due to the action of waves
|}

<div id='img-15'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_928603926-caida-particulas1.png|400px|]]
|[[Image:draft_Samper_928603926-caida-particulas2.png|400px|Sliding of macroscopic particles over an inclined wall entering a container with water]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 15:''' Sliding of macroscopic particles over an inclined wall entering a container with water
|}

<div id='img-16'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_928603926-landslide-falling1.png|400px|]]
|[[Image:draft_Samper_928603926-landslide-falling2.png|400px|3D PFEM simulation of a landslide falling on four houses]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 16:''' 3D PFEM simulation of a landslide falling on four houses
|}

<div id='img-17'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_928603926-picture_1_of_4.png|400px|]]
|[[Image:draft_Samper_928603926-picture_2_of_4.png|400px|]]
|-
|[[Image:draft_Samper_928603926-picture_3_of_4.png|400px|]]
|[[Image:draft_Samper_928603926-picture_4_of_4.png|400px|3D PFEM results for the dragging of a collection of large rocks by a water stream]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 17:''' 3D PFEM results for the dragging of a collection of large rocks by a water stream
|}

==12 Examples==

We present the study of a several problems involving the transport of macroscopic and large particles in fluid flows. The problems are schematic representations of particulate flows occurring in practical problems of civil and environmental engineering and industrial problems.

Most problems studied have been solved with the PFEM using the  nodal algorithm for the transport of macroscopic particles described in the previous section. An exception are the problems in Section 12.6 dealing with the vertical  transport of spherical particles in a cylinder where  the standard immersed  approach for the transport macroparticles described in Sections 1&#8211;4 was used and the fluid equations were solved with an Eulerian flow solver implemented in the Kratos open-source software platform of CIMNE <span id='citeF-24'></span>[[#cite-24|[24]]].

===12.1 Erosion of a slope adjacent to the shore due to sea waves===

Figure [[#img-14|14]] 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. Note that the eroded soil particles accumulate at the sea bottom.

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

===12.2 Landslide falling on houses===

Figure [[#img-15|15]] shows two instants of the 2D simulation with PFEM of the motion of a collection of macroscopic particles as they slide over an inclined wall and fall into a water container.

The PFEM is particularly suited for the modelling and simulation of landslides and their effect in the surrounding structures. Figure [[#img-16|16]] shows an schematic 2D simulation of a landslide falling on two adjacent constructions. The landslide material has been assumed to behave as a pure viscoplastic material modelled as a non-Newtonian viscous incompressible fluid <span id='citeF-57'></span>[[#cite-57|[57]]]. Other applications of the PFEM to the modelling of landslides can be found in <span id='citeF-8'></span><span id='citeF-49'></span>[[#cite-8|[8,49]]].

===12.3 Dragging of rocks by a water stream===

Figure [[#img-17|17]] shows the dragging of a collection of rocks modelled as large rigid particles of arbitrary shape by the action of a water stream. The particles move due to the action of the water forces on the particles computed by integrating the pressure and the viscous stresses in the elements surrounding each particle.

===12.4 Suction of cohesive material submerged in water===

Figures [[#img-18|18]] and [[#img-19|19]] show two examples of the detachment, suction and transport of particles of a cohesive material submerged on water. Figure [[#img-18|18]] shows how the particles detatch  from the cohesive soil bed and are transported within the suctioning tube (modelled as a 2D problem). The last picture shows the erosion in the soil as the mixture of water and eroded particles falls down from within the tube and hits the soil surface due to a stop in the suction pressure.

Figure [[#img-19|19]] shows a similar 3D problem. The suction in the tube erodes the surface of the soil bed in the right hand container. The mixture of water and eroded particles is transported to the adjacent containers.

<div id='img-18'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_928603926-suction1.png|400px|]]
|[[Image:draft_Samper_928603926-suction2.png|400px|]]
|-
|[[Image:draft_Samper_928603926-suction3.png|400px|]]
|[[Image:draft_Samper_928603926-suction4.png|400px|2D PFEM analysis of the detachment and suction of cohesive material submerged in water. The last picture shows the erosion of the bed material after the impact of the mixture of water and eroded particles falling from within  the tube]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 18:''' 2D PFEM analysis of the detachment and suction of cohesive material submerged in water. The last picture shows the erosion of the bed material after the impact of the mixture of water and eroded particles falling from within  the tube
|}

<div id='img-19'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_928603926-picture1_of_4.png|400px|]]
|[[Image:draft_Samper_928603926-picture2_of_4.png|400px|]]
|-
|[[Image:draft_Samper_928603926-picture3_of_4.png|400px|]]
|[[Image:draft_Samper_928603926-picture4_of_4.png|400px|3D PFEM simulation of the detachment, suction and transport of submerged cohesive material from one recipient to another]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 19:''' 3D PFEM simulation of the detachment, suction and transport of submerged cohesive material from one recipient to another
|}

===12.5 Filling of a water container with particles===

Figure [[#img-20|20]] shows a 3D example of the filling of a cylindrical container  with water containing macroscopic spherical particles. Water is allowed to exit the cylinder by a lateral hole while particles enter from two other holes at different height and fall down by gravity until they progressively fill the cylinder. The figures show different instants of the filling process.

<div id='img-20'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_928603926-diposit_2_of_6.png|400px|]]
|[[Image:draft_Samper_928603926-diposit_3_of_6.png|400px|]]
|-
|[[Image:draft_Samper_928603926-diposit_5_of_6.png|400px|]]
|[[Image:draft_Samper_928603926-diposit_6_of_6.png|400px|Filling of a container by injecting water containing macroscopic particles from two holes. Water is allowed to exit through a third hole at the upper right hand side of the cylinder. 3D PFEM results at four instants ]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 20:''' Filling of a container by injecting water containing macroscopic particles from two holes. Water is allowed to exit through a third hole at the upper right hand side of the cylinder. 3D PFEM results at four instants 
|}

===12.6 Transport of spherical particles in a tube filled with water===

The example in Figure [[#img-21|21]] models the vertical transport of some 120.000 spherical particles to the surface of a tube filled with water and the subsequent deposition of the particles on the free water surface at the top of the tube. Particles move upwards within the tube due to a fluid velocity of 1 m/s. The average size of the particles radius is 2 mm and their density is 2300 Kg/m<math display="inline">^3</math>. Particles move vertically until they reach the top of the fluid domain and accumulate there due to the combined effect of their weight and the effect of the interaction force from the fluid. Figure [[#img-21|21]] shows two instants of the particles ascending process. The accumulation of particles in the water free surface  at the  top of the tube is clearly seen.

Figure [[#img-22|22]] shows the interaction of eigth jets of ascending air bubbles with 200.000 spherical solid particles that fall down within a cylinder filled with water.

<div id='img-21'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_928603926-ascending-particles1b.png|400px|]]
|[[Image:draft_Samper_928603926-ascending-particles2.png|400px|Transport of spherical macroparticles up to the free surface of a tube filled with water. Particles move up with a prescribed velocity until they accumulate on the free surface. Results obtained with a coupled DEM-Eulerian CFD code <span id='citeF-24'></span>[[#cite-24|[24]]]]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 21:''' Transport of spherical macroparticles up to the free surface of a tube filled with water. Particles move up with a prescribed velocity until they accumulate on the free surface. Results obtained with a coupled DEM-Eulerian CFD code <span id='citeF-24'></span>[[#cite-24|[24]]]
|}

<div id='img-22'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_928603926-interaction-six-jets1.png|400px|]]
|[[Image:draft_Samper_928603926-interaction-six-jets2.png|400px|]]
|-
|[[Image:draft_Samper_928603926-interaction-six-jets3.png|400px|]]
|[[Image:draft_Samper_928603926-interaction-six-jets4.png|400px|Interaction of eight jets of ascending air bubbles with a thick layer of 200.000 spherical particles that fall down within a cylinder filled with water. Numerical results obtained with a coupled DEM-Eulerian CFD code <span id='citeF-24'></span>[[#cite-24|[24]]]]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 22:''' Interaction of eight jets of ascending air bubbles with a thick layer of 200.000 spherical particles that fall down within a cylinder filled with water. Numerical results obtained with a coupled DEM-Eulerian CFD code <span id='citeF-24'></span>[[#cite-24|[24]]]
|}

===12.7 Dragging of large objects and small particles in a tsunami type flow===

The last example is the dragging of cars, barrels and debris  (modelled as macroscopic particles) by a water stream that flows over a vertical wall. The problem represents an schematic study of a real situation corresponding to the tsunami in Fukushima (Japan) on March 2011 (Figure [[#img-23|23]]). Figures [[#img-24|24]] show two snapshots of the PFEM solution of this complex problem.

<div id='img-23'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 75%;max-width: 100%;"
|-
|[[Image:draft_Samper_928603926-tsunami-japon.png|450px|Dragging of cars and large and small objects in the Fukushima tsunami (Japan) ]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 23:''' Dragging of cars and large and small objects in the Fukushima tsunami (Japan) 
|}

<div id='img-24'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_Samper_928603926-vlcsnap-2013-05-27-13h19m40s59.png|600px|]]
|-
|[[Image:draft_Samper_928603926-vlcsnap-2013-05-27-13h21m48s219.png|510px|Dragging of large objects and macroscopic particles in a tsunami type flow passing over a vertical wall. 3D PFEM results ]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 24:''' Dragging of large objects and macroscopic particles in a tsunami type flow passing over a vertical wall. 3D PFEM results 
|}

==13 Concluding remarks==

We have presented a Lagrangian numerical technique for analysis of flows incorporating physical particles of different sizes. The numerical approach is based on the Particle Finite Element Method (PFEM) and a stabilized Lagrangian mixed velocity-pressure formulation. The examples presented in the paper evidence the possibilities of the PFEM for analysis of practical multiscale particulate flows in industrial and environmental problems.

==Acknowledgements==

This research was  supported by the Advanced Grant  project SAFECON of the European Research Council.

===BIBLIOGRAPHY===

<div id="cite-1"></div>
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