An augmented Lagrangian approach to essential boundary conditions in meshless methods

The problem of boundary conditions enforcement in meshless methods has been solved in the literature by several approaches. In the present paper, the moving least-squares (MLS) approximation is introduced in the total potential energy functional for the elastic solid problem and an augmented Lagrangian term is added to satisfy essential boundary conditions. The method can be easily extended to any kind of constraint for the approximation variables, The solution is found by iterating alternatively on approximation variables and on Lagrange multipliers. The advantages of the proposed formulation are: (a) the ability to deal with the same approach with any constraint type; (b) the number of the variables is not increased by the Lagrange multipliers; (c) the Hessian of the functional w.r.t. the approximation variables is banded, well conditioned and strictly positive definite and (d) the cost of the augmented Lagrangian iteration is a very small fraction of the global computing time. Therefore, the augmented Lagrangian element-free (ALEF) approach represents an attractive and very efficient numerical tool, not only for the boundary conditions enforcement, but also for the solution of interface and non-linear problems. Copyright (C) 2001 John Wiley Sons, Ltd.