Consistency approach and diffuse derivation in element free methods based on moving least squares approximation

Abstract

The paper concerns shape functions formulations in the scope of the recent methods generalizing finite elements and whose common feature is the absence of a mesh. These methods may also be interpreter as a generalization of the finite difference approach for irregular grids. The shape functions obtained by the Moving Least Squares an by the GFDM (Generalized Finite Difference Method) approach exhibit a number of interesting properties, the most interesting being a local character of the approximation, high degree of continuity and the satisfaction of consistencu contraints neccesary for exact reproduction of polynomials. In the present work we formulate the shape functions directly as solutions of the minimization of a weighted quadratic form subjected to the consistency contraints explicity introduced by Lagrange multipliers. This approach gives similar results as the standard moving least squares algorithm applied to the Taylor series expansion where the consistency is automatically satisfied but is more general in the sense, that an explicit specification of wished properties permits an introduction of additional arbitrary conraints other than consistency. It also leads to faster and more robust algorithms by avoiding matrix inversion. On the other hand, the consistency based formulations naturally lead to diffuse (or incomplete) derivatives of the shape functions. They are obtained at a significantly lower cost than full derivatives and their convergence to ext act derivatives is demonstrated.