多边形有限元研究进展

有限元法是数值求解偏微分方程边值问题的重要方法, 采用 不规则多边形单元网格, 可以方便有效地模拟材料的力学性能, 又使得区域网格剖分变得灵 活方便. 特别是对于复杂的几何形状, 多边形单元网格具有更大的优势. 本文对国内外有关 多边形有限元法的最新进展作了初步的总结和评述, 主要以基于位移法的多边形有限元为主. 论述了多边形有限元的发展历史, 给出了多边形单元上的Wachspress插值、Laplace插值和 重心坐标的一些最新研究成果. 与经典有限元法形函数为多项式形式不同, 多边形单元的形 函数为有理函数或者无理函数形式. 多边形单元插值形函数满足线性完备性, 可以再现线性 位移场, 像经典有限元法一样直接施加本质边界条件; 插值函数在多边形的边界上是线性的, 确保不同单元间的自动协调. 不同单元的插值形函数表达公式形式统一, 方便混合单元网格 计算的程序编写. 提出了多边形有限元法今后需要研究的问题.

ADVANCES IN POLYGONAL FINITE ELEMENT METHOD

The finite element method is an important method to solve boundary value problems. In two dimensional problems, the constant strain three-node triangular element and the bilinear four-node quadrilateral element are widely used. Irregular polygonal elements can be used not only to conveniently and effectively simulate mechanical properties of materials, but also to enhance flexibility in meshing. For complex geometries, the polygonal element grid enjoys greater advantages. In the past decade, researchers have shown interestis in the numerical methods based on polygonal elements, and have obtained some new results. In this paper the advances in polygonal finite elements are reviewed. The development of polygonal finite elements is discussed, including Wachspress interpolation, Laplace interpolation and barycentric coordinates. Unlike the polynomial form of shape functions in the classical finite element, the shape functions of a polygonal element can take both rational and irrational forms. The shape functions interpolate nodal values, satisfy linear completeness, can be used to reconstruct the linear displacement field, and permit the direct imposition of essential boundary conditions as in the conventional finite element method. They are linear on the boundary of a polygonal element, which ensures automatically the consistency of inter-elements. The shape functions have a uniform formulation for different side number elements, so one can conveniently program for a variety of meshes. Some issues for future development of polygonal finite elements are also discussed.