Abstract 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 in-plane 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.

中文翻译：

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板状弹性结构分层模型的自然元逼近

摘要 由于难以生成 3-D 网格，几乎所有的无网格方法都仅限于 2-D 问题。克服这种限制的一种有效方法是利用弹性结构的分层建模概念。它最初被引入用于 3-D 结构的降维分析，但它可以用于相反的目的。通过多项式假设厚度方向的位移场，平面内位移可以通过二维无网格方法来近似。受此启发，本文旨在通过二维自然单元法 (NEM) 演示板状结构的数值分析。厚度方向的多项式阶数由模型层级定义，通过依次增加模型层级来构建多个层次模型。结构的中面被离散化为 NEM 网格，并根据无锁定选择性减少积分技术来近似相应的面内位移场。进行数值实验以说明和验证所提出的方法。中心偏转和应力合力都在模型水平上进行了研究，并与全 3-D 弹性进行了比较。此外，还从厚度和模型层次上考察了分层模型的建模误差，并将其对网格点总数的收敛特性与有限元法进行了比较。进行数值实验以说明和验证所提出的方法。中心偏转和应力合力都在模型水平上进行了研究，并与全 3-D 弹性进行了比较。此外，还从厚度和模型层次上考察了分层模型的建模误差，并将其对网格点总数的收敛特性与有限元法进行了比较。进行数值实验以说明和验证所提出的方法。中心偏转和应力合力都在模型水平上进行了研究，并与全 3-D 弹性进行了比较。此外，还从厚度和模型层次上考察了分层模型的建模误差，并将其对网格点总数的收敛特性与有限元法进行了比较。