The PFEM was originally conceived for the solution of free-surface fluid flow problems . The method is designed to deal with the large motion of free-surface fluids, whose boundaries could freely fold, break and reconnect. The range of possible applications of such a technology is wide and interests several branches of engineering and applied sciences.

Simulation of a dam spillway with the PFEM

In fluid dynamics, the automatic capability of tracking the evolving free-surface is not the only benefit of the PFEM. Thanks to its Lagrangian description of the motion, the convective terms are not included in PFEM governing equations. This is a great advantage of the PFEM versus standard Eulerian formulations because the convective terms are responsible for non-linearity, non-symmetry and non-self-adjoin operators, and require the introduction of stabilization terms to avoid numerical oscillations .

On the other hand, we have to remark that the continuous remeshing done in the PFEM has important implications on several aspects of the fluid solver (e.g. time integration and spatial discretization, the imposition of boundary conditions, or mass conservation) that must be taken into account. More details about these key issues can be found in  and in the cited references.

PFEM mesh in a free-surface fluid dynamics problem

In the PFEM, a standard Galerkin approach is used for the fluid flow governing equations. In a standard PFEM framework, only linear shape functions are used to approximate the unknown variables and it is well known that in a Navier-Stokes problem, using equal order interpolation for both velocity and pressure unknowns makes it necessary to stabilize the formulation. In the PFEM literature, different kinds of stabilization procedures have been applied. For instance, the Finite Increment Calculus (FIC) formulation has been frequently used to stabilize the PFEM equations (see e.g. [1, 2, 6]). Alternatively, the Pressure Stabilizing Petrov-Galerkin technique has been used (see e.g. [4, 7]). In , the Algebraic Sub-Grid Scale stabilization technique is introduced to stabilize mixed pressure velocity PFEM formulation. Examples of other stabilization techniques can be found in [5, 9]. In , the authors propose to make use of stable elements belonging to the bubble family .

Thanks to its FEM nature, the PFEM allows for an accurate modelling of fluids having complex non-Netwonian constitutive laws. In the next animation, the simulation of a fresh concrete slump test is shown.

Fresh concrete slump test obtained with PFEM and a non-Newtonian Bingham model

The PFEM also allows for the accurate simulation of multi-fluid flows. An example of this problem can be found in , where a PFEM coupled approach was used to model the 3D air-water interaction occurring at the bottom outlet of a dam in order to estimate the air demand in that zone.

Coupled air-water simulation of bottom outlets of dams done with the PFEM . Velocity vectors of the water phase

References

 Idelsohn S, Oñate E, Pin FD (2004) The particle finite element method: a powerful tool to solve incompressible flows with free-surfaces and breaking waves. Int J Numer Methods Eng 61(7):964–989
 Idelsohn S, Oñate E, Pin FD, Calvo N (2006) Fluid–structure interaction using the particle finite element method. Comput Methods Appl Mech Eng 195(17–18):2100–2113
 Aubry R, Oñate E, Idelsohn S (2006) Fractional step like schemes for free surface problems with thermal coupling using the Lagrangian PFEM. Comput Mech 38(4–5):294–309
 Cremonesi M, Frangi A, Perego U (2010) A Lagrangian finite element approach for the analysis of fluid–structure interaction problems. Int J Numer Methods Eng 84(5):610–630
 Ryzhakov P, Rossi R, Idelsohn S, Oñate E (2010) A monolithic Lagrangian approach for fluid–structure interaction problems. Comput Mech 46(6):883–899
 Oñate E, Franci A, Carbonell J (2014) Lagrangian formulation for finite element analysis of quasi-incompressible fluids with reduced mass losses. Int J Numer Methods Fluid, 74(10):699–731
 Cerquaglia M, Deliége G, Boman R, Terrapon V, Ponthot J (2017) Free-slip boundary conditions for simulating free-surface incompressible flows through the particle finite element method. Int J Numer Methods Eng 110:921–946
 Larese A (2017) A Lagrangian PFEM approach for non-Newtonian viscoplastic materials. Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería 33(3):307–317
 Ryzhakov P, Marti J, Idelsohn S, Oñate E (2017) Fast fluid–structure interaction simulations using a displacement-based finite element model equipped with an explicit streamline integration prediction. Comput Methods Appl Mech Eng 315:1080–1097
 Salazar F, San-Mauro J, Celigueta M, Oñate E (2017) Air demand estimation in bottom outlets with the particle finite element method. Susqueda dam case study. Computat Part Mech 4(3):345–356
 Cremonesi M, Franci A, Idelsohn SR, Oñate E (2020) A state of the art review of the Particle Finite Element Method (PFEM). Archives of Computational Methods in Engineering, 17, 1709-1735