Weak-form collocation – A local meshless method in linear elasticity

Abstract

This paper is concerned with the formulation of local meshfree methods, for the solution of two-dimensional problems in linear elasticity, in the framework of the theory of structures.Local meshfree methods are derived through a weighted-residual formulation which leads to a local weak form that is the well known work theorem of the theory of structures. In an arbitrary local region, the work theorem establishes an energy relationship between a statically-admissible stress field and an independent kinematically-admissible strain field. Based on the independence of these two fields, this paper presents two new meshless formulations that aim a reduction of the computational effort. While in the first formulation the local form of the work theorem is reduced to regular boundary terms only, in the second formulation the local form of the work theorem is simply an integration-free formula.The moving least squares (MLS) approximation of the elastic field is used in this paper to implement both local meshless formulations.Several problems were analyzed with these techniques, in order to assess the accuracy and efficiency of the formulations. The results obtained in this work are in perfect agreement with those of the available analytical solutions. The accuracy and efficiency of the integration-free formulation make this a reliable and robust local meshfree method, generated in the framework of the theory of structures.