论文标题
凝结的广义有限元法（CGFEM）
Condensed Generalized Finite Element Method (CGFEM)
论文作者
论文摘要
由于引入了富集功能，因此普遍或扩展有限元方法（GFEM/XFEM）通常条件不佳，并且与FEM相比具有许多额外的自由度（DOF）。在本文中，我们开发了一种使用Unity（PU）技术和局部最小平方过程的分区来建立常规GFEM/XFEM近似空间的子空间。所提出的GFEM称为冷凝GFEM（CGFEM），（i）具有与初步fem一样多的DOF，（ii）具有与GFEM/XFEM相似的近似属性，并且（iii）具有良好的条件，其条件的条件与FEM的顺序相同。 CGFEM的基本近似特性在数学上被证明。 CGFEM应用于高阶多项式近似和泊松裂纹问题的问题。严格证明了前者的最佳收敛顺序。进行数值实验和与常规GFEM/XFEM和FEM的比较以验证CGFEM的理论和有效性。
Generalized or extended finite element methods (GFEM/XFEM) are in general badly conditioned and have numerous additional degrees of freedom (DOF) compared with the FEM because of introduction of enriched functions. In this paper, we develop an approach to establish a subspace of a conventional GFEM/XFEM approximation space using partition of unity (PU) techniques and local least square procedures. The proposed GFEM is referred to as condensed GFEM (CGFEM), which (i) possesses as many DOFs as the preliminary FEM, (ii) enjoys similar approximation properties with the GFEM/XFEM, and (iii) is well-conditioned in a sense that its conditioning is of the same order as that of the FEM. The fundamental approximation properties of CGFEM is proven mathematically. The CGFEM is applied to a problem of high order polynomial approximations and a Poisson crack problem; optimal convergence orders of the former are proven rigorously. The numerical experiments and comparisons with the conventional GFEM/XFEM and FEM are made to verify the theory and effectiveness of CGFEM.
