多边形有限单元形函数的比较研究

多边形有限单元形函数有wachspress插值、Laplace插值和平均值插值三种类型.本文对三种多边形有限单元形函数的性质作了比较研究,给出了三种形函数各自的优点和局限性.Waclaspress和Laplace形函数是有理函数形式,而平均值形函数是无理函数形式.三种形函数均满足单位分解性、线性完备性,且在单元边界上呈线性.在三角形单元上,它们都等价于三角形面积坐标插值.在矩形单元上,Wachspress和Laplace形函数等价于双线性多项式插值形函数.Wachspress和平均值形函数适用于任意凸多边形单元,Laplace形函数更适用于圆内接多边形单元.Wachspress形函数不能推广到含有边节点的单元,平均值形函数可以直接推广到含有边节点的单元.数值试验,验证了本文理论分析的结论.

Some Remarks on Shape Functions of Polygonal Finite Element

Three types of shape functions of polygonal element,Wachspress shape functions,Laplace shape functions and Mean value shape functions,are comparatively investigated.The individual advantages and limitations of the above shape functions of polygonal finite elements are discussed.These shape functions of polygonal finite element all meet the requirements of partition of unity and linear completeness and get linear on the boundaries of polygonal elements,so equivalent to triangular area coordinates in triangular elements.Wachspress and Laplace shape functions are equivalent to bilinear interpolation shape functions in rectangular finite element.Wachspress and Mean value shape functions are suitable for arbitrary convex polygonal finite elements.Wachspress shape functions are unable to extend to the element including side-nodes,but Mean value shape functions are able.The numerical examples for elasticity problems are presented to support the conclusions.