三维自然单元法插值形函数的导数计算

Calculation of the derivative of interpolation shape function for three dimension natural element method

根据Voronoi胞的几何性质,获得了积分点的二阶Voronoi胞顶点的表达式,并对各邻近结点相关的顶点进行排序以使其生成的二阶Voronoi胞切割面为凸多边形,从而获得各切割凸多边形面域的面积表达式;最后,基于复合函数链式求导法则,获得了三维自然单元法non-Sibson插值形函数导数的显式格式。相比Lasserre算法,该方法具有直观、便于编程且计算量小的特点。悬臂梁的算例结果进一步说明了该方法的可靠性,证实了文献[2,7,8]关于自然单元法具有比有限元中常应变单元更高的精度,理论上和双线性单元的精度同阶的结论。

For an integral point x within three-dimension model,the vertex coordinate expression of its second-order Voronoi cell is deduced firstly by using the geometric properties of Voronoi diagram.And then,those second-order Voronoi cell vertexes related to some a neighbor node is further reordered to make the second-order Voronoi cell segment area domain generated become a protrusive polygon so that the segment area expression can be conveniently obtained.Based on the expression of segment area domain and the definition of non-Sibson shape function,the derivative expression of the shape function for three-dimension natural element methods is deduced by making use of the chain derivative rule of compound function.Compared with the Lasserre algorithm,this algorithm is more of intuitionistic characteristic and can be conveniently programmed.Finally,through a cantilever beam case,the reliability of computer results by NEM is further verified.And the precision of NEM is higher than that of the tetrahedron element with FEM and the same as that of the hexahedron with FEM in theory,which discussed in detail in reference[2-4].