Abstract An error-driven grid refinement technique is introduced for 2-D reliable crack analysis by an enriched natural element method (more exactly, Petrov-Galerkin natural element method). A quasi-exact solution for a posteriori error estimation was obtained by enhancing the bare approximation solution of NEM (natural element method) using the enrichment method and the global patch recovery. The proposed method is illustrated through the error-driven grid refinement for a rectangular plate with a slant edge crack. The quantitative error amount is measured in terms of the energy norm, and the accuracy (i.e., the effective index) of the proposed method was evaluated from the comparison with the errors which were obtained by FEM using a very fine mesh. The proposed method provides the effective index which is much improved from that of non-enriched PG-NEM. The NEM grid was non-uniformly refined based on the local error information, and the resulting error distributions were investigated. It has been observed that the difference between the maximum and minimum values in the local error distribution of enriched PG-NEM is larger than that of non-enriched PG-NEM. The reduction of global errors according to the non-uniform grid refinement was also investigated to the grid density, from which it is found that the enriched PF-NEM provides smaller errors than the non-enriched PG-NEM. In addition, the proposed grid refinement provides the 1.676 times higher convergence rate than the uniform grid refinement.

中文翻译：

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用丰富的自然元素法进行二维可靠裂纹分析的误差驱动网格细化

摘要 引入了一种误差驱动网格细化技术，通过丰富的自然元素方法（更准确地说，是Petrov-Galerkin自然元素方法）进行二维可靠裂纹分析。通过使用富集方法和全局补丁恢复增强 NEM（自然元素方法）的裸近似解，获得了后验误差估计的准精确解。通过对具有倾斜边缘裂纹的矩形板进行误差驱动的网格细化来说明所提出的方法。定量误差量是根据能量范数来衡量的，并且通过与使用非常精细的网格的有限元法获得的误差进行比较来评估所提出方法的精度（即有效指标）。所提出的方法提供了比非富集PG-NEM 有很大改进的有效指标。基于局部误差信息对 NEM 网格进行非均匀细化，并对由此产生的误差分布进行了调查。已经观察到，富集的 PG-NEM 的局部误差分布的最大值和最小值之间的差异大于非富集的 PG-NEM。还研究了根据非均匀网格细化减少全局误差的网格密度，从中发现富集 PF-NEM 比非富集 PG-NEM 提供更小的误差。此外，所提出的网格细化提供了比统一网格细化高 1.676 倍的收敛速度。并对由此产生的误差分布进行了调查。已经观察到，富集的 PG-NEM 的局部误差分布的最大值和最小值之间的差异大于非富集的 PG-NEM。还研究了根据非均匀网格细化减少全局误差的网格密度，从中发现富集 PF-NEM 比非富集 PG-NEM 提供更小的误差。此外，所提出的网格细化提供了比统一网格细化高 1.676 倍的收敛速度。并对由此产生的误差分布进行了调查。已经观察到，富集的 PG-NEM 的局部误差分布的最大值和最小值之间的差异大于非富集的 PG-NEM。还研究了根据非均匀网格细化减少全局误差的网格密度，从中发现富集 PF-NEM 比非富集 PG-NEM 提供更小的误差。此外，所提出的网格细化提供了比统一网格细化高 1.676 倍的收敛速度。还研究了根据非均匀网格细化减少全局误差的网格密度，从中发现富集 PF-NEM 比非富集 PG-NEM 提供更小的误差。此外，所提出的网格细化提供了比统一网格细化高 1.676 倍的收敛速度。还研究了根据非均匀网格细化减少全局误差的网格密度，从中发现富集 PF-NEM 比非富集 PG-NEM 提供更小的误差。此外，所提出的网格细化提供了比统一网格细化高 1.676 倍的收敛速度。