A natural neighbour method based on Fraeijs de Veubeke variational principle for materially non-linear problems

Abstract

The natural neighbour method can be considered as one of many variants of the meshless methods. In the present paper, a new approach based on the Fraeijs de Veubeke (FdV) functional, which is initially developed for linear elasticity, is extended to the case of geometrically linear but materially non-linear solids. The new approach provides an original treatment to two classical problems: the numerical evaluation of the integrals over the domain A and the enforcement of boundary conditions of the type ui = ũi on Su. In the absence of body forces (Fi = 0), it will be shown that the calculation of integrals of the type \({\int\nolimits_A \cdot {\rm d}A}\) can be avoided and that boundary conditions of the type ui = ũi on Su can be imposed in the average sense in general and exactly if ũi is linear between two contour nodes, which is obviously the case for ũi = 0.