The generalized finite element method
The generalized finite element method
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DOI：
10.1016/s0045-7825(01)00188-8
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发表时间：
2001-05
影响因子：
7.2
通讯作者：
T. Strouboulis;K. Copps;I. Babuska
中科院分区：
文献类型：
--
作者：
T. Strouboulis;K. Copps;I. Babuska
本文介绍了一个试点设计和实施的广义有限元法（GFEM），作为一个直接扩展的标准有限元法（SFEM，或FEM），这使得可能的精确解决工程问题，在复杂的领域，可能实际上是不可能解决的有限元。的GFEM的发展示出了拉普拉斯算子在两个空间维度中的域，其中可能包括数百个空隙，和/或裂纹，其中所使用的有限元网格的建设实际上是不可能的。它的两个主要功能是：（1）它可以使用可能与域边界部分或全部重叠的网格来构造近似。(2)它可以纳入近似手册的功能，这是已知的解析，或数值生成的，以及近似的解决方案的边界值问题的附近的角点，空隙，裂缝等的主要工具是一个特殊的集成算法，我们称之为快速重网格方法，这是强大的，适用于任何域的任意复杂性。采用单位分割法（PUM）将手册函数合并到广义有限元中。所提出的公式和实现方式可以很容易地扩展到多材料介质，其中空隙被各种形状和尺寸的夹杂物所取代，以及弹性问题的情况。这项工作也可以被理解为一个试点研究的可行性和示范的GFEM的能力，这是需要类似的实施之前，试图在三维和非线性的情况下，这是未来工作的主要兴趣的情况下。
This paper describes a pilot design and implementation of the generalized finite element method (GFEM), as a direct extension of the standard finite element method (SFEM, or FEM), which makes possible the accurate solution of engineering problems in complex domains which may be practically impossible to solve using the FEM. The development of the GFEM is illustrated for the Laplacian in two space dimensions in domains which may include several hundreds of voids, and/or cracks, for which the construction of meshes used by the FEM is practically impossible. The two main capabilities are: (1) It can construct the approximation using meshes which may overlap part, or all, of the domain boundary. (2) It can incorporate into the approximation handbook functions, which are known analytically, or are generated numerically, and approximate well the solution of the boundary value problem in the neighborhood of corner points, voids, cracks, etc. The main tool is a special integration algorithm, which we call the Fast Remeshing approach, which is robust and works for any domain with arbitrary complexity. The incorporation of the handbook functions into the GFEM is done by employing the partition of unity method (PUM). The presented formulations and implementations can be easily extended to the multi-material medium where the voids are replaced by inclusions of various shapes and sizes, and to the case of the elasticity problem. This work can also be understood as a pilot study for the feasibility and demonstration of the capabilities of the GFEM, which is needed before analogous implementations are attempted in the three-dmensional and nonlinear cases, which are the cases of main interest for future work.