应用于弹性问题的重心有限元法

摘要：通过几何的方法构造了在任意多边形上的具有重心型格式的平均值插值函数，并利用Galerkin法提出了应用于弹性问题的重心有限元法。重心有限元法的插值函数在多边形单元间是协调的，能够方便的施加本质边界条件。重心有限元法的插值函数对于不同边数的多边形单元具有统一的表达形式，编程实现简便易行，能够方便的应用于复杂几何区域的求解。通过重心有限元法分别进行了小片试验、悬臂梁和复合材料的有效模量的数值模拟。小片试验的计算精度达到了机器精度；悬臂梁的计算结果与解析解的吻合程度较高；复合材料的有效模量的数值模拟结果与传统有限元和解析解吻合得较好，变化趋势合理。

Barycentric finite element method and its application in elasticity theory

Abstract The basis functions of mean value interpolation on arbitrary polygonal element are constructed by geometric method. The formations of the basis functions were identical. The barycentric finite element method (BFEM) was brought by the Galerkin method. And the BFEM was applied in elasticity problems. The functions of BFEM were conforming on irregular polygons. Because of the property, essential boundary conditions can be imposed exactly. The functions have uniform expression for different edge number polygons, consequently the programs could be written conveniently. This provides greater flexibility to solve partial differential equations on complicated geometries. In this paper, BFEM was used in elasticity problems — the patch test, cantilever beam and the effective moduli of composite material. In the patch test, the error was realized the machine precision accuracy. The numerical results using the BFEM were in good agreement with beam theory predictions. The moduli of the composite material were simulated by BFEM. Compared the theory predictions and solution of FEM, the results showed the good consistency and had reasonable changing trend.