Application of radial basis functions to linear and nonlinear structural analysis problems

Abstract

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 structural analysis problems. The technique used here consists of 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 (RBFs). In this work various types of one-dimensional problems are analyzed, ranging from the static linear elastic case, free vibration and linear stability analysis (for a beam on elastic foundation), to physically nonlinear (damage models) problems. To further complement the range of problems analysed, the static analysis of a plate on elastic foundation was also addressed. Several error measures are used to numerically establish the performance of both symmetric and nonsymmetric approaches for several global RBFs. The results obtained show that RBF collocation leads to good approximations and very high convergence rates.