Error evaluation in approximation of derivatives using Smoothed Particle Hydrodynamics method

Nowadays, events like severe earthquakes or man-made malicious actions are often taken into account in structural design of critical infrastructures and consequently high dynamic loads are considered in structural analyses. Classical numerical methods, like Finite Element Method (FEM), may be inadequate to model the mechanical behavior of structural elements under such actions. In fact high deformation gradients and unforeseeable failure mechanisms can represent critical aspects for FEM methods. As a consequence, several meshless methods, originally developed for fluid-dynamics, have been recently investigated in order to adapt them to solid continuum mechanics. Smoothed Particle Hydrodynamcs (SPH) method, belonging to meshless methods, is here described. Classical numerical formulations are presented and the basic idea of the SPH approach is described. Then, the attention is focused on the expressions used to approximate derivatives, since these formulations play a fundamental role in developing numerical framework to reproduce dynamic problems. Deficiencies and criticalities related to such a point are described and the most common improvements proposed in literature are summarized. Then, an original approach is presented, based on a direct control of the convergence error. Performances of the proposed expressions are outlined via numerical tests. In particular second order of convergence in treating second derivatives is discussed.