Investigation of linear dependence problem of three-dimensional partition of unity-based finite element methods
Investigation of linear dependence problem of three-dimensional partition of unity-based finite element methods
复制标题
基于单位的有限元方法三维划分线性相关问题的研究
DOI：
10.1016/j.cma.2012.04.010
复制
发表时间：
2012-08
影响因子：
7.2
通讯作者：
Li LX
中科院分区：
文献类型：
--
作者：
An XM;Zhao ZY;Zhang HH;Li LX
基于单位元的广义有限元方法的一个已知问题是近似空间的线性相关性，这导致奇异刚度矩阵。到目前为止，线性相关问题尚未得到充分的理解，也没有有效的方法来缓解它。在我们以前的论文“预测秩不足的单位为基础的方法与平面三角形或四边形网格分区”[COMPUT。方法应用机械工程200（2011）665-674]中，首先剖析了线性相关问题的起源，然后提出了一种方法来可靠地预测基于单位元的二维划分的广义有限元方法的线性相关全局近似的秩亏。本文将以前的工作扩展到三维情况。线性相关问题首先在单元级进行研究，然后扩展到整个网格。由于复杂的元件拓扑结构，在三维环境中推导一般公式无疑比二维环境更具挑战性。再次应用秩亏增加的原理。进一步证明了将每个单元的附加秩亏求和为整个网格的附加秩亏的方法在三维情况下是有效的。这一工作与以前的工作被认为是必要的步骤，成功地和完整地解决线性相关问题的分区基于单位的有限元方法。
A known problem of partition of unity-based generalized finite element methods is the linear dependence of the approximation space, which leads to singular stiffness matrix. Up to now, the linear dependence problem has not been fully understood and an efficient way to alleviate it is not available. In our previous paper “Prediction of rank deficiency in partition of unity-based methods with plane triangular or quadrilateral meshes” [Comput. Methods Appl. Mech. Engrg. 200 (2011) 665–674], the origin of the linear dependence problem was first dissected and then a method was proposed to reliably predict the rank deficiency of the linearly dependent global approximations of two-dimensional partition of unity-based generalized finite element methods. This paper extends the previous work to three-dimensional cases. The linear dependence problem is first investigated at an element level and then extended to the whole mesh. Derivation of general formulations in a three-dimensional setting is undoubtedly more challenging than a two-dimensional setting because of the complicated element topology. The principle of the increase of rank deficiency is once more applied. The methodology of summing up the added rank deficiency of each element as that of the whole mesh is further proved to be valid in three-dimensional cases. This work together with the previous work is regarded as the essential step to successfully and completely solve the linear dependence problem in partition of unity-based finite element methods.
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DOI：
10.1016/j.cma.2008.04.026
发表时间：
2009-03
影响因子：
7.2
作者：
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通讯作者：
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影响因子：
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