A new hierarchical partition of unity formulation of EFG meshless methods

Meshless methods have shown increased accuracy and better convergence rates compared to other well-known simulation methods in a variety of computational mechanics problems. Their computational cost is relatively higher especially in adaptivity analysis, when new nodes are inserted at each step, where the construction of updated shape functions requires a significant amount of computational effort. In this paper, two hierarchical formulations are proposed in the context of an h-type refinement scheme. The addition of new nodes and subsequently the re-calculation of the influenced moment matrices, that are necessary for obtaining the shape functions and their derivatives and subsequently for the construction of the stiffness matrix, are properly addressed. Both hierarchical schemes do not need the recalculation of the initial stiffness matrix but only the additional node contributions to each shape function field, while the second scheme produces purely hierarchically refined stiffness matrices leaving the initial stiffness matrix unmodified.