基于改进位移模式的一维C^1有限元超收敛算法

Algorithm of super-convergent in one-dimensional C^1 FEM based on improved displacement mode

提出了基于改进位移模式的一维C1有限元超收敛算法。利用单元内部需满足平衡方程的条件,推导了超收敛计算解析公式的显式,即将高阶有限元解的位移模式用常规有限元解的位移模式表示。用常规有限元解的位移模式与高阶有限元解的位移模式之和构造新的位移模式。采用积分形式推导了单元刚度矩阵。该算法在前处理阶段使用了超收敛计算公式,在常规试函数的基础上,增加了高阶试函数,使得单元内平衡方程的残差减少,从而达到提高精度的目标。对于Hermite单元,本文的结点和单元的位移、导数都达到了h4阶的超收敛精度。

Algorithm of super-convergent in one-dimensional C1 FEM based on an improved displacement mode is presented, an explicit analytical formula of super-convergent calculating is derived with the con- ditions of equilibrium equations must be meet within the element, the displacement mode of high-order fi- nite element solution is expressed with a conventional finite element solution,the new displacement mod- el is constructed with the sum of the displacement model of conventional finite element and the high-or- der displacement model of finite element solution. The element stiffness matrix is derived using the inte- gral form,the super-convergent formula is used for this algorithm at the stage of pre-processing for im- proving the accuracy of the solution and reducing the residual of balance equation, the higher-order trial functions is added at the foundation of the original trial function, the representative example is present for the Hermite element in this paper, the displacement and the derivative accuracy of nodes and elements have reached h4 order.