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

Abstract

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.