The use of radial basis functions for one-dimensional structural analysis problems

The basic characteristic of the techniques generally known as "meshless methods" is the attempt to reduce or even to eliminate the need for a discretization (at least, not in the way normally associated with traditional finite element techniques) in the context of numerical solutions for boundary and/or initial value problems. The interest in meshless methods is relatively new and this is why, despite the existence of various applications of meshless techniques to several problems of mechanics (as well as to other fields), these techniques are still relatively unknown to engineers. Furthermore, and compared to traditional finite elements, it may be difficult to understand the physical meaning of the variables involved in the formulations. As an attempt to clarify some aspects of the meshless techniques, and simultaneously to highlight the ease of use and the ease of implementation of the algorithms, applications are made, in this work, to one dimensional structural problems. The technique used here consists in the definition of a global approximation for a given variable of interest (in this case, components of the displacement field) by means of a superposition of a set of conveniently placed (in the domain and on the boundary) radial basis functions. Applications to beams on elastic foundation subject to static loads, the identification of critical (instability) modes and the identification of free vibration frequencies and modes are carried out.