Purpose

It has been usual to prefer an enrichment pattern independent of the mesh when applying singular functions in the Generalized/eXtended finite element method (G/XFEM). This choice, when modeling crack tip singularities through extrinsic enrichment, has been understood as the only way to surpass the typical poor convergence rate obtained with the finite element method (FEM), on uniform or quasi-uniform meshes conforming to the crack. Then, the purpose of this study is to revisit the topological enrichment strategy in the light of a higher-order continuity obtained with a smooth partition of unity (PoU). Aiming to verify the smoothness' impacts on the blending phenomenon, a series of numerical experiments is conceived to compare the two GFEM versions: the conventional one, based on piecewise continuous PoU's, and another which considers PoU's with high-regularity.

Design/methodology/approach

The stress approximations right at the crack tip vicinity are qualified by focusing on crack severity parameters. For this purpose, the material forces method originated from the configurational mechanics is employed. Some attempts to improve solution using different polynomial enrichment schemes, besides the singular one, are discussed aiming to verify the transition/blending effects. A classical two-dimensional problem of the linear elastic fracture mechanics (LEFM) is solved, considering the pure mode I and the mixed-mode loadings.

Findings

The results reveal that, in the presence of smooth PoU's, the topological enrichment can still be considered as a suitable strategy for extrinsic enrichment. First, because such an enrichment pattern still can treat the crack independently of the mesh and deliver some advantage in terms of convergence rates, under certain conditions, when compared to the conventional FEM. Second, because the topological pattern demands fewer degrees of freedom and impacts conditioning less than the geometrical strategy.

Originality/value

Several outputs are presented, considering estimations for the J–integral and the angle of probable crack advance, this last computed from two different strategies to monitoring blending/transition effects, besides some comments about conditioning. Both h- and p-behaviors are displayed to allow a discussion from different points of view concerning the topological enrichment in smooth GFEM.

中文翻译：

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关于通过广义/扩展有限元法进行裂纹建模的拓扑丰富：考虑统一平滑划分的新讨论

目的

在广义/扩展有限元方法 (G/XFEM) 中应用奇异函数时，通常更喜欢独立于网格的富集模式。在通过外在富集对裂纹尖端奇异点进行建模时，这种选择被认为是在符合裂纹的均匀或准均匀网格上超越使用有限元方法 (FEM) 获得的典型较差收敛速度的唯一方法。然后，本研究的目的是根据通过统一划分（PoU）获得的高阶连续性重新审视拓扑富集策略。为了验证平滑度对混合现象的影响，设想了一系列数值实验来比较两个 GFEM 版本：传统的基于分段连续 PoU 的版本，

设计/方法/方法

裂纹尖端附近的应力近似值通过关注裂纹严重性参数来限定。为此，采用了源自配置力学的材料力方法。除了单一的多项式丰富方案外，还讨论了使用不同的多项式丰富方案来改进解决方案的一些尝试，旨在验证过渡/混合效果。解决了线性弹性断裂力学 (LEFM) 的经典二维问题，考虑了纯模式 I 和混合模式载荷。

发现

结果表明，在存在光滑 PoU 的情况下，拓扑富集仍然可以被认为是外在富集的合适策略。首先，因为在某些条件下，与传统 FEM 相比，这种富集模式仍然可以独立于网格处理裂缝，并在收敛速度方面提供一些优势。其次，因为与几何策略相比，拓扑模式需要更少的自由度和影响条件。

原创性/价值

考虑到对 J– 积分和可能的裂纹推进角度，这最后是从两种不同的策略计算出来的，以监测混合/过渡效果，除了一些关于调节的评论。双方^ h -和p -behaviors显示允许但从关于光滑GFEM拓扑富集不同点的讨论。