# Introduction to the PFEM

There is an increasing interest in the development of robust and efficient numerical methods for the analysis of engineering problems involving the interaction of fluids and structures accounting for large motions of the fluid free surface and the existence of fully or partially submerged bodies.

Examples of this kind are common in ship hydrodynamics, off-shore and harbour structures, ocean engineering, modelling of tsunamis, dam spillways, free surface channel flows, liquid containers, stirring reactors, mould filling processes, etc.

The analysis of fluid-structure interaction (FSI) problems using the finite element method (FEM) with either the Eulerian formulation or the Arbitrary Lagrangian Eulerian (ALE) formulation encounters a number of serious problems. Among these we list 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 solid domains via the contact interfaces, the modelling of wave splashing, the possibility to deal with large rigid body motions of the structure within the fluid domain, the efficient updating of the finite element meshes for both the structure and the fluid, etc.

Most of these problems disappear if a Lagrangian description is used to formulate the governing equations of both the solid and the fluid domain. 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 "particles". The motion of the mesh discretizing the total domain (including both the fluid and solid parts) is also followed during the transient solution.

The Particle Finite Element Method (PFEM) is a particular class of Lagrangian formulation aiming to solve problems involving the interaction between fluids and solids in an unified manner.

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 representing, for instance, the effect of water drops.

A finite element mesh connects the nodes defining the discretized domain where the governing equations are solved in the standard FEM fashion.

An obvious advantage of the Lagrangian formulation is that the convective terms disappear from the fluid equations. The difficulty is however transferred to the problem of adequately (and efficiently) moving the mesh nodes.

The information in the PFEM is typically nodal-based, i.e. the element mesh is mainly used to obtain the values of the state variables (i.e. velocities, pressure, etc.) at the nodes. The Alpha Shape technique is used to identify the boundary nodes. Indeed the "boundary" can include the free surface in the fluid and the individual particles moving outside the fluid domain. For large mesh motions remeshing may be a frequent necessity along the time solution. We use an innovative mesh regeneration procedure based on an extension of the standard Delaunay tesselation.

The need to properly treat the incompressibility condition in the fluid still remains in the Lagrangian formulation. The use of standard finite element interpolations may lead to a volumetric locking effect unless some precautions are taken. A number of stabilization finite element procedures aiming to alleviate the locking problem in incompressible fluids have been proposed and some stabilization techniques are used. In our work the stabilization via a finite calculus (FIC) procedure has been chosen. Recent applications of the FIC method for incompressible flow analysis using linear triangles and tetrahedra are reported in the references listed.