Natural element approximation of hierarchical models of plate-like elastic structures

Almost all the mesh-free methods are restricted to 2-D problems owing to the difficulty in generating 3-D grids. One effective way to overcome this limitation is to utilize the concept of hierarchical modeling for elastic structures. It was introduced originally for the dimensionally-reduced analysis of 3-D structures, but it can be used for the reverse purpose. By assuming the displacement field in the thickness direction with polynomials, the inplane displacement could be approximated by 2-D mesh-free method. Inspired this motivation, this paper intends to demonstrate the numerical analysis of plate-like structures by 2-D natural element method (NEM). The polynomial order in the thickness direction is defined by the model level, and a number of hierarchical models are constructed by sequentially increasing the model level. The mid-surface of structure is discretized into a NEM grid and the corresponding in-plane displacement field is approximated according to a locking-free selectively reduced integration technique. The numerical experiments are performed to illustrate and validate the proposed method. Both the central deflection and the stress resultants are investigated with respect to the model level and compared with the full 3-D elasticity. In addition, the modeling error of hierarchical models is examined with respect to the thickness and the model level, and the convergence characteristic to the total number of grid points is compared with FEM.