Introduction to the numerical homogenization by means of the Meshless Finite Difference Method with the Higher Order Approximation

Paper focuses on application of the Meshless Finite Difference Method (MFDM) solution approach and its selected extensions to the numerical homogenization of the heterogeneous material. The most commonly used method of computer modeling for the multiscale problem (at both the macro and micro (RVE) levels) is the Finite Element Method (FEM). However, this fact does not mean that one should not search for the alternative, perhaps more efficient approaches, especially based on the meshless discretization. The aim of this paper is to present a problem formulation for numerical homogenization based on the extended algorithm of the Meshless (Generalized) FDM. The Higher Order Approximation (HOA), which is applied here, is based on the correction terms of the simple meshless difference operator. Those terms produce the improved higher order solution, without necessity of providing additional unknowns to both cloud of nodes and MFD operator. The HO solution may be used e.g. for a-posteriori error estimation of superior quality, when compared with the existing estimation techniques. The paper is illustrated with simple tasks of 2D linear theory of elasticity, which highlight benefits of the considered approach towards more complex problems of mechanics.