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 [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 [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 [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 [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 [17,19]. The theory and applications of the PFEM are reported in [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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle div} -stability condition [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 [15,29,30,31,33,34]. Among the other stabilized FEM with similar features we mention the PSPG method [41], multiscale methods [3,6,8,9] and the CBS method [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 [47]. That is, all variables are assumed to be known in the current configuration at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t} . The new set of variables in both domains are sought for in the next or updated configuration at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t+\Delta t} . 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.

Sequence of steps to update a “cloud” of nodes representing a domain containing a fluid and a solid part from time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): n 

  (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): t=t_n

) to time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): n+2 

(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): t=t_n +2\Delta t

)

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.

  1. The starting point at each time step is the cloud of points in the fluid and solid domains. For instance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {}^nC} denotes the cloud at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t=t_n} (Figure 2).
  2. Identify the boundaries for both the fluid and solid domains defining the analysis domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {}^nV} 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 [12] is used for the boundary definition.
  3. Discretize the fluid and solid domains with a finite element mesh Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {}^nM} . In our work we use an innovative mesh generation scheme based on the extended Delaunay tesselation [17,19,20].
  4. 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t+\Delta t}
    velocities, pressure and viscous stresses in the fluid and displacements, stresses and strains in the solid.
  5. Move the mesh nodes to a new position Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {}^{n+1} C} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n+1} denotes the time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t_n+\Delta t} , in terms of the time increment size. This step is typically a consequence of the solution process of step 4.
  6. Go back to step 1 and repeat the solution process for the next time step to obtain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {}^{n+2} C} (Figure 2).

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \rho {\partial v_i \over \partial t}={\partial \sigma _{ij} \over \partial x_j}+b_i\qquad \hbox{in } V
(1)

Pressure-velocity relationship

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{1}{K} {\partial p \over \partial t}-{\partial v_i \over \partial x_i}=0 \qquad \hbox{in }V
(2)

In above equations Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle v_{i}} 

is the velocity along the ith global (cartesian) axis, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle p}
is the pressure (assumed to be positive in compression) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \rho }
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle K}
are the density and bulk modulus of the material, respectively, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle b_i}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \sigma _{ij}}
are the body forces and the (Cauchy) stresses. Eqs.(1) and (2) are completed with the constitutive relationships:

Incompressible continuum

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {}^{t+1}\sigma _{ij} = 2 \mu \dot \varepsilon _{ij} - {}^{t+1} p \delta _{ij}
(3)

Compressible/quasi-incompressible continuum

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {}^{t+1}s _{ij}= {}^{t}\hat \sigma _{ij} + 2 \mu \dot \varepsilon _{ij} +\lambda \dot \varepsilon _{ii} \delta _{ij}
(4.a)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \hat \sigma _{ij}} 

are the component of the stress tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle [\hat \sigma ]}


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): [\hat \sigma ]= \frac{1}{J} {F}^T {S} {F}
(4.b)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {S}} 

is the second Piola-Kirchhoff stress tensor, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {F}}
is the deformation gradient tensor and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle J = \det {F}}
[22,47]. Parameters Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mu }
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lambda }
 take the following values for a fluid or solid material:

Fluid: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mu }

viscosity; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lambda = \Delta t K - \frac{2\mu }{3}}
     Solid: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \displaystyle \mu = \frac{\Delta t G}{J}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \displaystyle \lambda =\frac{2G \nu \Delta t}{J(1-2\nu )}}

, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \nu } 

is the Poisson ration, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle G}
is the shear modulus and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Delta t}
the time increment.

In Eqs.(3) and (4), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle s _{ij}} 

are the deviatoric stresses, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \dot \varepsilon _{ij}}
 is the rate of deformation, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \mu }
is the viscosity and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \delta _{ij}}
is the Kronecker delta. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {}^{t} (\cdot )}
denotes values at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t}

.

Indexes in Eqs.(1)–(4) range from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i,j=1,n_{d}} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n_d} 

 is the number of space dimensions of the problem (i.e. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n_{d} = 2}
for 2D problems). These equations  are completed with the standard boundary conditions of prescribed velocities and surface tractions in the mechanical problem  [11,36,47,48].

3.2 Discretization of the equations

A key problem in the numerical solution of Eqs.(1)–(3.1) is the satisfaction of the mass balance condition for the incompressible case (i.e. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle K=\infty } 

in Eq.(2)). A number of procedures to solve his problem exist in the finite element literature [11,48]. In our approach we use a stabilized formulation based in the so-called finite calculus procedure [15,29,30,31,33,34]. The essence of this method is the solution of a modified mass balance equation which is written as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \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
(5)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle q} 

are weighting functions, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau }
is a stabilization parameter given by [34]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \tau = \left(\frac{2\rho \vert \mathbf{v}\vert }{h}+\frac{8\mu }{3h^2} \right)^{-1}
(6)

In the above, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h} 

is a characteristic length of each finite element  and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \vert \mathbf{v}\vert }
is the modulus of the velocity vector. In Eq.(5) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \pi _i}
  are auxiliary pressure projection variables chosen so as to ensure that the second term in Eq.(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 [32,36]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \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
(7)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle w_i} 

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 [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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle v_i} , the pressure Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle p} , the temperature Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle T} 

and the three pressure gradient projections Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \pi _i}

. 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \mathbf{M} \dot{\bar{v}} + \mathbf{K} \bar {v} - \mathbf{G} \bar {p}= {f}
(8)

Box I. Basic PFEM algorithm for a Lagrangian continuum


1. LOOP OVER TIME STEPS, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t=1}

, NTIME

Known values

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle ^{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}

2. LOOP OVER NUMBER OF ITERATIONS, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i=1}

, NITER

[.15cm] Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bullet } 
 Compute  nodal velocities by solving Eq.(8)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \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}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bullet } 
  Compute nodal pressures from Eq.(9)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \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} }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bullet } 
Compute nodal pressure gradient projections from Eq.(10)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {}^{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}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bullet } 
  Update position of analysis domain nodes:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {}^{t+\Delta t} \bar{x}^{i+1} ={}^{t} \mathbf{x}^{i} +{}^{t+\Delta t} \mathbf{v}^{i+1} \Delta t}
Define new “cloud” of nodes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {}^{t+1} C^{i+1} }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bullet } 
  Update strain rate and strain values
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bullet } 
  Update stress values
Check convergence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \rightarrow } 
NO Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \rightarrow }
Next iteration Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i\to i+1}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \downarrow } 
YES
Next time step Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle t\to t+1}
[.15cm] Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bullet } 
Identify new analysis domain boundary: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {}^{t+1} V}
[.15cm] Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bullet } 
Generate mesh:Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {}^{t+1} M}
[.15cm] Go to 1

Pressure-velocity relationship

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \bar {M} \dot{\bar{p}} - \mathbf{G}\bar{v} - \mathbf{L}\bar {p} - \mathbf{Q} \bar {\boldsymbol \pi }=\mathbf{0}
(9)

Pressure gradient projection

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \hat {M} \bar {\boldsymbol \pi }+\mathbf{Q}^T\bar {p}=\mathbf{0}
(10)

In Eqs.(8)–(10) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \bar{(\cdot )}} 

denotes nodal variables, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \dot{\bar{(\cdot )}}=  {\partial  \over \partial t}\bar{(\cdot )}}

. The different matrices and vectors are given in [22,34,36].

The solution in time of Eqs.(8)–(10) can be performed using any time integration scheme typical of the updated Lagrangian FEM [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 [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 [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 [19]. Considering that the nodes follow a variable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h(x)} 

distribution, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h(x)}
is typically the minimum distance between two nodes. All nodes on an empty sphere with a radius greater than Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \alpha h

, are considered as boundary nodes. In practice Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha } 

 is a parameter close to, but greater than one. Values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha }
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 [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 [32,36].


Modelling of contact conditions at a solid-solid interface with the PFEM

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 (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h_c} ) between two solid boundaries. If the distance exceeds the minimum value (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h_c} ) 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 6.1).

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 [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 [25]. In a recent work we have proposed an extension of the PFEM to model bed erosion [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 [1].

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

Modeling of bed erosion with the PFEM by dragging of bed material

  1. Compute at every point of the bed surface the resultant tangential stress Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau } induced by the fluid motion. In 3D problems Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau = (\tau _{s}^2 + \tau _{t})^2} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _s} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau _t} are the tangential stresses in the plane defined by the normal direction Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {n}} at the bed node. The value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \tau } for 2D problems can be estimated as follows:
    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \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}
    		(11)

    2. where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle v_t^k} 

    is the modulus of the tangential velocity at the node Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle k}
 and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h_k}
is a prescribed distance along the normal of the bed node Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle k}

    . Typically Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h_k} 

    is of the order of magnitude of the smallest fluid element adjacent to node Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle k}
(Figure 7).
  2. Compute the frictional work originated by the tangential stresses at the bed surface as
    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 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
    		(12)

    3. Eq.(12) is integrated in time as

    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {}^n W_f ={}^{n-1} W_f + \tau _t \gamma _t\, \Delta t
    		(13)
  3. The onset of erosion at a bed point occurs when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {}^nW_f} exceeds a critical threshold value Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle W_c} .
  1. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {}^nW_f > W_c} 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.
  2. 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.

Figure 7 shows an schematic view of the bed erosion algorithm described.

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 8.2a 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 8.2b.

8.2 Impact of sea waves on piers and breakwaters

Figures 8.2 and 8.2 show the analysis of the effect of breaking waves on two different sites of a breakwater containing reinforced concrete blocks (each one of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 4\times 4} 

mts). The figures correspond to the study of Langosteira harbour in A Coruña, Spain using PFEM.

Study of the drag of an individual rock of 1Tn under a water stream at speeds of (a) 1m/s and (b) 2m/s

Breaking waves on breakwater slope containing reinforced concrete blocks. Mesh of 4-noded tetrahedra near the slope

Study of breaking waves on the edge of a breakwater structure formed by reinforced concrete blocks

Erosion, transport and deposition of soil particles at a river bed due to an impacting jet stream

8.3 Soil erosion problems

Figure 8.2 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 1 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 2 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 [35,36].

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Figure 1: Erosion of an unprotected shoulder of a breakwater due to sea waves
Error creating thumbnail: File missing
Figure 2: 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 8.4 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.

Erosion of a soil mass due to sea waves and the subsequent falling into the sea of an adjacent lorry

Landslide falling on two constructions 2D simulation using PFEM

8.5 Simulation of landslides

The PFEM is particularly suited for modelling and simulation of landslides and their effect in the surrounding structures. Figure 8.4 shows an schematic 2D simulation of a landslide falling on two adjacent constructions. The landslide material has been modelled as a viscous incompressible fluid.

8.6 The landslide in Lituya Bay

A case of much interest is when the landslide occurs in the vicinity of a reservoir [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 8.6). 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.

Lituya Bay landslide. Left: Geometry for the simulation. Right: Landslide direction and maximum wave level [13,14]

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 8.6 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 [13,14]. The maximum height location differs in 300 mts from the observed value [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 [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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