ADVANCES IN POLYGONAL FINITE ELEMENT METHOD
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摘要: 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.
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关键词:
- 有限元法 /
- 多边形单元 /
- Wachspress插值 /
- 多边形Laplace插值 /
- 多边形重心坐标
Abstract: The finite element method is an important method to solveboundary value problems. In two dimensional problems, the constant strain three-nodetriangular element and the bilinear four-node quadrilateral element arewidely used. Irregular polygonal elements can be used not only toconveniently and effectively simulate mechanical properties of materials,but also to enhance flexibility in meshing. For complexgeometries, the polygonal element grid enjoys greater advantages. In the past decade,researchers have shown interestis in the numerical methods based on polygonalelements, and have obtained some new results. In this paper the advancesinpolygonal finite elements are reviewed. The development of polygonalfinite elements is discussed, including Wachspressinterpolation, Laplace interpolation and barycentric coordinates. Unlikethe polynomial form of shapefunctions in the classical finite element, the shape functions of a polygonalelement can take both rational and irrational forms. The shape functionsinterpolate nodal values, satisfy linear completeness, can be used to reconstructthe linear displacement field, and permit the direct imposition ofessential boundary conditions as in the conventional finite element method. Theyare linear on the boundary of a polygonal element, which ensuresautomatically the consistency of inter-elements. The shape functionshave a uniformformulation for different side number elements, so one can convenientlyprogram for a variety of meshes. Some issues for future development ofpolygonal finite elements are also discussed.
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