有限元复合单元的构造方法研究

传统有限单元法中每一种单元都由单一的材料组成,虽然只考虑一种材料的性质给软件编程带来了极大的方便,但是这比利用复合单元进行建模,需要更多的工作量。为此,笔者提出从4节点矩形单元与2节点线单元的虚功原理出发,运用能量方程,来整合两种不同单元。以VC++作为开发工具,利用等参元的特性,根据逆矩阵形函数构造方法编写平面单元的形函数程序,采用高斯数值积分方法,形成普通单元的单元刚度矩阵。然后根据线单元与面单元的节点位移编号,进行单元刚度的整合,进而得到一种包含面单元和线单元的复合单元。同时,通过对平面矩形单元和平面三角形单元与线单元进行复合,用得到的两种复合单元分别对悬臂梁作用端部集中荷载的简单算例进行位移求解,得到的结果与同等条件下ANSYS计算结果进行对比验证。结果表明:两种情况的计算结果都非常接近,复合单元的构造方法正确、可靠,同时也能够减少单元个数。

Study on the construction method of composite finite element

In traditional finite element method,each unit is composed of a single material.While only considering the properties of a material will bring great convenience to the software programming,it also leads to a complex finite element modeling process.Therefore,in this paper,we discussed the principle of virtual work from the four nodes rectangular element and the two nodes linear element and used the energy equation to integrate the two different elements.Using VC++as the development tool,taking advantage of the parametric element characteristics and using Gaussian numerical integral method,we formed the element stiffness matrix of the common elements,according to the inverse matrix construction method of functions written plane element shape function program.Then,the element stiffness was integrated according to the node displacement number of the element,and a new type of finite elements containing plane unit and line unit was obtained.At the same time,two kinds of composite elements were applied to a simple example of calculation,in which the concentrated load was placed in the end of the cantilever beam.Through the calculation of the example,the results obtained by using the composite element method and ANSYS in the same conditions were compared.The results of the two cases are very close,the composite element construction method is correct and reliable,and at the same time,it can reduce the number of elements.