A Modified Finite Particle Method: Multi-dimensional elasto-statics and dynamics

Particle methods are a class of numerical methods that belong to the family of meshless methods and are not based on an underlying mesh or grid, but rather on any general distribution of particles. They are nowadays widely applied in many fields, including, for example, solid mechanics, fluid dynamics, and thermodynamics. In this paper, we start from the original formulation of the so-called modified finite particle method (MFPM) and develop a novel formulation. In particular, after discussing the position of the MFPM in the context of the existing literature on meshless methods, and recalling and discussing the 1D formulation and its properties, we introduce the novel formulation along with its extension to the approximation of 2D and 3D differential operators. We then propose applications of the discussed methods to some elastostatic and elastodynamic problems. The obtained results confirm the potential and the flexibility of the considered methods, as well as their second-order accuracy, proposing MFPM as a viable alternative for the simulation of solids and structures.