On the Meshless Numerical Modelling of Forming Processes
Résumé
The meshfree methods based on Moving Least Squares (MLS) approximation have been confronted to an active research during the last decade. These include Smooth Particle Hydrodynamics, Element Free Galerkin, Diffuse Elements, Reproducing Kernel Particle and other Methods. However, one of the issues is the satisfaction of essential boundary conditions. This is due to the nature of the approximation itself. In fact, the MLS nodal domains of influence are the same as those of the corresponding weighting functions, who generally do not fit the boundary. The choice of neighboring nodes is also an issue. The approach based in considering the $k$ closest nodes from any evaluation point results only in a $C^0$ continuity. Moreover, the geometrical complexity of the shape functions supports induces integration difficulties. Simpler integration and an arbitrary degree of continuity are obtained when nodes are associated with fixed, spherical or hexahedral domains of influence, whose optimal size constitutes the main difficulty of that approach. On the other hand, the Natural Neighbor (NN) approximation and associated family of computational methods [20] do not present these drawbacks. The boundary approximation is obtained naturally due to the fact that NN shape functions of internal nodes vanish at the boundary where only the boundary nodes contribute. The list of connected points - the natural neighbors - is also known in advance. However, the NN do not present all the advantages of the MLS. In particular, the shape function support is geometrically complex. Moreover, the NN shape functions have only $C^0$ continuity at the nodes and only linear consistency is guaranteed. In this paper we summarize the main ideas related to meshless approximations, addressing numerous papers that concern the application of meshless techniques to the simulation of forming processes.
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