扩展有限元法(XFEM)及其应用

THE EXTENDED FINITE ELEMENT METHOD AND ITS APPLICATIONS--A REVIEW

扩展有限元法(extendedfiniteelementmethod,XFEM)是1999年提出的一种求解不连续力学问题的数值方法,它继承了常规有限元法(CFEM)的所有优点,在模拟界面、裂纹生长、复杂流体等不连续问题时特别有效,短短几年间得到了快速发展与应用.XFEM与CFEM的最根本区别在于,它所使用的网格与结构内部的几何或物理界面无关,从而克服了在诸如裂纹尖端等高应力和变形集中区进行高密度网格剖分所带来的困难,模拟裂纹生长时也无需对网格进行重新剖分.重点介绍XFEM的基本原理、实施步骤及应用实例等,并进行必要的评述.单位分解概念保证了XFEM的收敛,基于此,XFEM通过改进单元的形状函数使之包含问题不连续性的基本成分,从而放松对网格密度的过分要求.水平集法是XFEM中常用的确定内部界面位置和跟踪其生长的数值技术,任何内部界面可用它的零水平集函数表示.第2和第3节分别简要介绍单位分解法和水平集法;第4节和第5节介绍XFEM的基本思想、详细实施步骤和若干应用实例,同时修正了以往文献中的一些不妥之处;最后,初步展望了该领域尚需进一步研究的课题.

The extended finite element method (XFEM) originally proposed in 1999 is very powerful for discontinuous problems in mechanics, such as crack growth, complex fluid, interface, and so on. The major difference between the XFEM and the conventional finite element method (CFEM) is that the mesh in XFEM is independent of the internal geometry and physical interfaces, such that meshing and re-meshing difficulties in discontinuous problems can be overcome. Based on the partition of unity concept, the XFEM relaxes the prohibitive requirements for mesh density by improving the shape functions with the basic knowledge of discontinuous problems. The XFEM retains all advantages of the CFEM, such as the single-field variational principle. symmetric banded and sparse system matrices, the ease of application to non-linear problems, anisotropic materials and arbitrary geometries. This paper presents an overview and comments on the XFEM, and is organized as follows. The partition of unity method (PUM) and Level Set Method (LSM) are briefly introduced in sections 2 and 3, respectively. Basic theory, implementation procedures and formulations of the XFEM are described in detail in sections 4 and 5, together with correction to several inaccurated points in literature The future investigations on XFEM are finally recommended in section 6.