Differing from the finite element method, most of mesh‐free methods are restricted to two‐dimensional (2D) problems owing to the difficulty in constructing three‐dimensional (3D) grids. But, for 3D elastic structures, this limitation could be effectively overcome when hierarchical models are employed. In the hierarchical models of elastic structures, only the in‐plane displacement field is needed to be approximated because the thickness‐wise displacement field is assumed by polynomials. In this context, this paper is concerned with the reliable numerical implementation of hierarchical models for plate‐like structures using 2D natural element method (NEM). However, one critical problem in implementing the hierarchical model is to secure the continuity in strain and stress fields. To resolve this problem, this paper presents a level‐wise strain recovery method. In addition, a recovery‐based error estimation is introduced to predict the modeling error and the numerical approximation error. The proposed methods are illustrated and validated through the numerical experiments. It is confirmed that the level‐wise strain recovery successfully and CPU‐time effectively provides the continuous stress resultants and the proposed error estimator accurately predicts the modeling and approximation errors.

中文翻译：

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自然元素分层板模型的逐级应变恢复和误差估计

与有限元方法不同，由于构建三维（3D）网格的困难，大多数无网格方法都限于二维（2D）问题。但是，对于3D弹性结构，当使用分层模型时，可以有效克服此限制。在弹性结构的分层模型中，仅需要近似平面内位移场，因为厚度方向的位移场是由多项式假设的。在这种情况下，本文关注使用2D自然元法（NEM）的板状结构分层模型的可靠数值实现。然而，实现分层模型的一个关键问题是确保应变和应力场的连续性。为了解决这个问题，本文提出了一种水平应变恢复方法。此外，引入了基于恢复的误差估计以预测建模误差和数值近似误差。通过数值实验对所提出的方法进行了说明和验证。可以确定的是，水平应变恢复成功且CPU时间有效地提供了连续应力结果，并且所提出的误差估计器准确地预测了建模误差和近似误差。