Particle Methods for a 1D Elastic Model Problem: Error Analysis and Development of a Second-Order Accurate Formulation

Particle methods represent some of the most investigated meshless approaches, applied to numerical problems, ranging from solid mechanics to fluiddynamics and thermo-dynamics. The objective of the present paper is to analyze some of the proposed particle formulations in one dimension, investigating in particular how the different approaches address second derivative approximation. With respect to this issue, a rigorous analysis of the error is conducted and a novel second-order accurate formulation is proposed. Hence, as a benchmark, three numerical experiments are carried out on the investigated formulations, dealing respectively with the approximation of the second derivative of given functions, as well as with the numerical solution of the static problem and with the approximation of the vibration frequencies for an elastic rod. In each test, the obtained numerical results are compared with exact solutions and the main criticalities of each formulation are addressed.

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