A Posteriori Error Estimation and Adaptivity for Second-Order Optimally Convergent G/XFEM and FEM

M. E. Bento, S. Proença, C. Duarte
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Abstract

The Generalized/eXtended Finite Element Method (G/XFEM) is known to efficiently and accurately solve problems that are challenging for standard methodologies. The method can deliver optimal convergence rates in the energy norm and global matrices with a scaled condition number that has the same order as in the Finite Element Method (FEM). This is achieved even for problems of Linear Elastic Fracture Mechanics (LEFM), which have solutions containing singularities and discontinuities. Despite delivering optimal convergence rates, it has been shown [1], however, that first-order G/XFEM are not competitive with second-order FEM that uses quarter-point elements, especially for three-dimensional (3-D) problems. Because of this, optimally convergent second-order G/XFEM, customized to solve LEFM problems, have been recently proposed [1, 2, 3]. The formulations presented in these works augment both standard lagrangian FEM approximation spaces [3] and p FEM approximation spaces [1, 2] in order to insert into the G/XFEM numerical approximation the discontinuous and singular behaviors of fractures. It is important to note that, in addition to using enrichment functions, G/XFEM still needs local mesh refinement around crack fronts in order to achieve optimal convergence. This must be considered especially for 3-D problems that violate the assumptions of the adopted singular enrichments. While this local mesh refinement can be easily performed for simple cases, the level of refinement
二阶最优收敛G/XFEM和FEM的后验误差估计和自适应
广义/扩展有限元法(G/XFEM)以高效、准确地解决标准方法难以解决的问题而闻名。该方法在能量范数和全局矩阵上具有最优的收敛速度,且条件数与有限元法具有相同的阶数。即使对于具有包含奇异点和不连续点的解的线弹性断裂力学(LEFM)问题,也可以实现这一点。尽管提供了最佳的收敛速度,但已经证明[1],一阶G/XFEM与使用四分之一点单元的二阶FEM没有竞争力,特别是对于三维(3-D)问题。正因为如此,最近提出了最优收敛的二阶G/XFEM,专门用于解决LEFM问题[1,2,3]。这些工作中提出的公式扩充了标准拉格朗日有限元近似空间[3]和p有限元近似空间[1,2],以便在G/XFEM数值近似中插入断裂的不连续和奇异行为。值得注意的是,除了使用富集函数外,G/XFEM还需要在裂缝前缘周围进行局部网格细化,以实现最优收敛。对于违反所采用的奇异富集假设的三维问题,必须特别考虑这一点。虽然这种局部网格细化可以很容易地执行简单的情况下,细化的水平
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