The Particle Finite Element Method (PFEM) is an innovative numerical tool able to solve complex non-linear problems in significantly evolving domains [1,2]. Although the PFEM was initially conceived for fluid dynamics and fluid-structure interaction problems, the method soon has been extended to the solution of non-linear solid and contact mechanics problems and to varied types of multiphysics problems of interest for varied fields of engineering and technology [3].

Tsunami scenario modeled with the PFEM

The PFEM combines the accuracy and robustness of mesh-based techniques with the advantages of particle-based methods. The PFEM discretizes the physical domain with a mesh on which the differential governing equations are solved with a standard finite element approach. Following a Lagrangian description, the mesh nodes move according to the equations of motion, behaving like particles and transporting their momentum together with all their physical properties. In this sense, the PFEM can be seen as both a FEM-based and a particle method.

Interaction of an icebreaker with multiple ice blocks

In the PFEM, mesh distortion issues typical of Lagrangian mesh-based solvers, are overcome by generating a new mesh when the current one gets too distorted. To avoid remapping from the old mesh to the new one, the nodes of the previous mesh are maintained during the remeshing step.  The new connectivity is built using the Delaunay Tessellation and the updated boundaries are identified with the so-called Alpha-Shape method. The obtained mesh is then used as the support over which the differential equations are solved in a standard FEM fashion [3]. More details about the remeshing strategy and methods are given in the other sections of the webpage.

To summarize, the fundamental features of the PFEM are the following:

1. Lagrangian framework for the description of motion.
2. Mesh nodes are treated as physical particles.
3. All information is stored at the mesh nodes.
4. The FEM is used to solve the governing equations.
5. Mesh connectivity is regenerated with a Delaunay Tessellation.
6. Boundaries are recovered through the Alpha-Shape method.

Propagating wave in a channel breaking a concrete wall [4]


[1] Idelsohn SR, 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 
[2] Oñate E, Idelsohn SR, Pin FD, Aubry R (2004) The particle finite element method. an overview. Int J Comput Methods 1:267–307
[3] 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
[4] Oñate E, Cornejo A, Zárate F, Kashiyama K, Franci A (2022) Combination of the finite element method and particle-based methods for predicting the failure of reinforced concrete structures under extreme water forces. Engineering Structures, 251B, 113510