The Moving Least Squares method (MLS) provides an approximation û of a function u based solely on values u(xj) of u on scattered ”meshless” nodes xj . Derivatives of u are usually approximated by derivatives of û. In contrast to this, we directly estimate derivatives of u from the data, without any detour via derivatives of û. This is a generalized Moving Least Squares technique, and we prove t...

A Meshless Local Petrov-Galerkin (MLPG) method has been developed for solving 3D elastodynamic problems. It is derived from the local weak form of the equilibrium equations by using the general MLPG concept. By incorporating the moving least squares (MLS) approximations for trial and test functions, the local weak form is discretized, and is integrated over the local sub-domain for the transien...

Introduction Moving least squares approximation is one kind of meshless methods in which the approximation is built from nodes only. Comparing to mesh-based methods, meshless methods have advantages on (1) refining is simpler to incorporate, (2) moving discontinuities can be treated easily, (3) large deformation can be handled robustly, (4) higher-order continuous shape functions, (5) non-local...

We describe two experiments recently conducted with the approximate moving least squares (MLS) approximation method. On the one hand, the NFFT library of Kunis, Potts, and Steidl is coupled with the approximate MLS method to obtain a fast and accurate multivariate approximation method. The second experiment uses approximate MLS approximation in combination with a multilevel approximation algori...

a truly meshless local petrov-galerkin (mlpg) method is developed for solving 3d elasto-static problems. using the general mlpg concept, this method is derived through the local weak forms of the equilibrium equations, by using test functions, namely, the heaviside function. the moving least squares (mls) are chosen to construct the shape functions, for the mlpg method. the penalty approximatio...

Moving least squares (MLS) and radial basis function (RBF) methods play a central role in multivariate approximation theory. In this paper we provide a unified framework for both RBF and MLS approximation. This framework turns out to be a linearly constrained quadratic minimization problem. We show that RBF approximation can be considered as a special case of MLS approximation. This sheds new l...

A meshless local Petrov-Galerkin (MLPG) method is applied to solve problems of Reissner-Mindlin shells under thermal loading. Both stationary and thermal shock loads are analyzed here. Functionally graded materials with a continuous variation of properties in the shell thickness direction are considered here. A weak formulation for the set of governing equations in the Reissner-Mindlin theory i...

For multivariate problems with many scattered data locations the use of radial functions has proven to be advantageous. However, using the usual radial basis function approach one needs to solve a large (possibly dense) linear system. In the moving least squares (MLS) method one obtains a best approximation of the given data in a (moving) weighted least-squares sense. The computational burden i...