基于改进位移模式的二维有限元线法超收敛算法

Algorithm of Super-Convergent in Two-Dimensional Finite Element of Lines Based on Improved Displacement Mode

提出了基于改进位移模式的二维有限元线法超收敛算法.利用单元内部需满足平衡方程的条件,推导了超收敛计算的解析公式的显式,即将高阶有限元线法解的位移模式用常规有限元线法解的位移模式表示.用常规有限元线法解的位移模式与高阶有限元线法解的位移模式之和构造新的位移模式,基于线性形函数,采用变分形式推导了有限元线法求解的修正的常微分方程组.该算法在前处理和后处理中同时使用超收敛计算公式,在原有试函数的基础上,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.对于二维Poisson方程问题,给出了有代表性的算例,节点和单元内的位移、导数的收敛精度得到了极大的提高.

Algorithm of super-convergent in two-dimensional finite element method of lines（FEMOL） based on improved displacement mode was presented.An explicit analytical formula of super-convergent calculating was derived with the conditions of equilibrium equations strictly met within the element,of which the displacement mode of high-order finite element of lines solution was expressed with that of a conventional finite element of lines solution.The new displacement mode was constructed with the sum of the displacement mode of conventional finite element of lines solution and that of high-order finite element of lines solution.Based on the linear shape function,the improved ordinary differential equations for FEMOL solution were derived in the variation form.The super-convergent formula was used for this algorithm in both the pre-processing and post-processing to improve the accuracy of the solution and reduce the residual of balance equation,with the higher-order trial function added to the original trial function.A calculation example is presented for Poisson＇s equation of a two-dimensional problem,the convergence accuracy of the displacement and derivative at nodes and in elements is greatly improved.