基于离散Kirchhoff理论的多边形样条薄板单元

在有限元方法中,采用多边形单元可以有效地模拟材料的力学性能,又使得网格剖分变得灵活方便.此外,允许退化情形的多边形单元可以处理出现悬节点的奇异网格.但目前对多边形薄板单元的研究却不多.多边形单元的研究难点在于插值基函数的构造.本文采用样条和基于三角形面积坐标的B网方法,将多边形进行三角剖分,通过适当选取连续性条件消去内部节点自由度,构造允许1-irregular退化的多边形样条插值基函数,再结合离散Kirchhoff理论得到多边形薄板弯曲单元,记为DKPS单元.该单元的插值自由度为各个顶点处的扰度和两个转角,并对直角坐标具有二次完备性,可以处理凸多边形、凹多边形和退化的网格.而且,采用多项式B网方法,可以方便地进行单元刚度矩阵的计算,无需使用数值积分公式.数值实验显示该单元对畸变网格仍然能保持很好的计算精度,是一种高效的单元.

The polygonal spline thin plate element based on the discrete Kirchhoff theory

As we know, the polygonal elements can do well in simulation of the materials behavior and provide greater flexibility for the meshing of complex geometries. Besides, the hanging nodes can be handled as irregular nodes of polygonal element. Hence, the study on the polygonal element is a very useful and necessary part in the finite element method.However, the research on polygonal plate element is lack. The difficulty of polygonal element is the construction of the interpolation bases. In this paper, by using the spline and B-net method, we construct a polygonal spline thin plate element based on the discrete Kirchhoff theory. The key is to construct a set of spline interpolation bases corresponding to the boundary nodes of the polygonal element including the irregular case. The basic idea is to subdivide the n-sided polygon into n subtriangles. We represent the spline functions in each subtriangle as quadratic polynomials in the B-net form.Then the degree of freedoms of the B-net coefficients corresponding to the interior B-net domain points are eliminated by some proper continuous conditions. As a result, we obtain the spline interpolation bases as the shape functions on the polygonal element. Combined with the discrete Kirchhoff theory, the new polygonal thin plate element is constructed and denoted by DKPS element. The nodal degrees of freedom are the displacement and two rotations at each vertex of the polygonal element. It can possess quadratic completeness in the Cartesian coordinates and is valid for both nonconvex and irregular polygonal elements. The mathematical formulas for the element stiffness matrices are also presented, which can be computed directly by the B-net coefficients of the spline bases. Thus there is no need to use the numerical integration in the computation of the element stiffness matrices. Some numerical experiments show that the new element has efficiency and good accuracy even for some badly distorted meshes.