MLS Shape Functions Research Articles

Integration constraint in diffuse element method

The “patch test” verifies whether a linear solution is reproduced exactly in an elasticity problem. This approach to test the numerical formulation and the code itself is standard in the finite element method. The MLS shape functions do not have a polynomial form. Therefore, the integration is not well performed by the classical Gauss–Legendre scheme and the patch test is only satisfied asymptotically at convergence. In this paper, we propose a custom quadrature scheme for MLS shape functions in order to ensure the properties needed for an exact verification of the patch test.

Explicit form and efficient computation of MLS shape functions and their derivatives

Explicit form and efficient computation of MLS shape functions and their derivatives

Explicit form and efficient computation of MLS shape functions and their derivatives

This work presents a general and efficient way of computing both diffuse and full derivatives of shape functions for meshless methods based on moving least-squares approximation (MLS) and interpolation. It is an extension of the recently introduced consistency approach based on Lagrange multipliers which provides a general framework for constrained MLS along with robust algorithms for the computation of shape functions and their diffuse derivatives. The particularity of the proposed algorithms is that they do not involve matrix inversion or linear system solving. The previous approach is limited to diffuse derivatives of the shape functions and not their full derivatives which are usually much more expensive to obtain. In the present paper we propose to efficiently compute the full derivatives by a new algorithm based on the formal differentiation of the previous one. In this way, we obtain a unified low-cost consistent methodology for evaluating the shape functions and both their diffuse and full derivatives. In the second part of the paper we introduce explicit forms of MLS shape functions in 1D, 2D and 3D for an arbitrary number of nodes. These forms are especially useful for comparing finite element and MLS approximations. Finally we present a general architecture of an MLS program. Copyright © 2000 John Wiley & Sons, Ltd.