The Particle Finite Element Method is a numerical tool that has been introduced more than a decade ago for the solution of engineering problems involving large deformations.
The method falls under the category of mesh-based particle methods, meaning that all information is stored on moving particles that represent the domain under analysis and a computational mesh is used for the solution of the governing equations.
Although the method was initially developed for simulating fluid\hyp{}structure interaction problems, owing to its versatility in handling large deformation and constant changes in domain boundaries and contact interfaces, it has been recently employed for solid mechanics applications. However, the lack of a consistent framework for this kind of problems has lead to different implementations of the method presented in the literature, each with its own special features.
The main objective of this thesis is to implement a variation of the Particle Finite Element Method and investigate the efficiency of the different features available in literature.

Initially, the meshing procedure of the method was developed, which consists of a Delaunay triangulation for assessing the connectivity of the particles and the ${\alpha}$-shape method for detection of the boundaries of the different domains. This was followed by an investigation on the influence of the related parameter ${\alpha_{lim}}$ on the outcome of the analysis; it appears that this choice has an impact on the results, in terms of the recovered domain volumes and the simulation response; this parameter has to be selected with care, with respect to the nature of the examined problem. Volume variations are also observed, caused by element deletion and/or addition during remeshing, which, eventually, lead to mass oscillations. These effects can be mitigated by either adjusting the value of the ${\alpha_{lim}}$ parameter, refining the particle distribution or prescribing the boundary surface during remeshing, by using a constrained Delaunay triangulation.

Another important feature of the PFEM is the treatment of contact, which is, typically, done in literature via employment of an interface mesh. This mesh is generated during remeshing, using the same scheme as for the regular domain meshes, i.e. a Delaunay triangulation and the ${\alpha}$-shape method, and the generated contact elements are then used to enforce the contact constraints, with a variety of methods. In this work, a simple algorithm that disallows inter-penetration and allows free separation and free movement perpendicular to the contact surfaces was formulated and validated against benchmark solid mechanics problems. The automatic contact detection and interface mesh generation allows for the incorporation of more advanced contact treatment schemes.

Transference of information between successive meshes is important in PFEM for solid mechanics, especially when the history of elemental variables, e.g. stresses, is required for capturing the solid material behavior accurately. The most popular technique is the nodal smoothing technique, where the values are mapped back and forth between the integration points and the particles at each time step; other schemes have been also presented in literature. This scheme has been shown to introduce some smoothing of information, which can be reduced by refining the particle distribution and, in general, does not seem to affect the overall system response significantly.

The developed method was, finally, compared with the available in-house implicit Material Point Method code, which shares the same formulation, on some benchmark quasi-static and dynamic solid mechanics problems. The PFEM demonstrates a more stable behavior in terms of capturing the evolution of stresses and kinematic variables, despite some inaccuracies caused by the smoothing of information and the use of simple, constant-strain triangles. On the other hand, the MPM -in its standard form- exhibits some instabilities in the assembly of equations and stress recovery, which is intensified when cell-crossing occurs, i.e. jumping of material points between elements.
Regarding the computational cost of the two methods, the MPM seems to be faster and require less computer memory for the same number of information points, i.e. particles, with the simulation times, however, increasing exponentially with the number of degrees of freedom.