微分法在有限元分析中的应用

Application of Differential Method in Finite Element Analysis

摘要:

基于有限元原理及结构的平衡微分方程，提出了一种求解结构单元形函数的方法。首先，根据结构单元的变形微分方程求解结构变形的常系数解析式；其次，根据单元的边界条件求解相应的系数表达式，并将求出的系数表达式代到解析式中，形成关于单元局部坐标的解析式；最后，提取单元节点广义位移向量，形成单元的形函数矩阵。运用所提出的方法对结构典型杆单元、梁单元的形函数进行求解后发现，所获得的形函数与相关文献提供的形函数完全一致，表明所提方法正确有效，并且同样适用于其他结构单元形函数的求解，从而为有限元的进一步推广和拓展提供了一条行之有效的途径。

Abstract:

Based on the finite element principle and the structural equilibrium differential equation, a method for solving the structural unit shape function is proposed. Firstly, the constant coefficient analytic formula of structural deformation is solved according to the deformation differential equation of the structural unit. Secondly, the corresponding coefficient expression is solved according to the boundary condition of the unit. Then, the obtained coefficient expression is substituted into the analytical expression to form an analytical expression about the local coordinates of the unit. Finally, the generalized displacement vector of the element node is extracted to form the shape function matrix of the unit. It is found that after the shape functions of the typical rod unit and beam unit of the structure are solved by the proposed method, the obtained shape function is completely consistent with the shape function provided by the related literature, which indicates that the proposed method is correct and effective. And the method is also applicable to solve the other structural unit shape functions so as to provide an effective way for the further promotion and expansion of the finite elements.