基于改进位移模式的一维有限元超收敛算法

将多尺度方法的思想与超收敛计算的解析公式结合起来，提出了改进有限元位移模式的算法。利用超收敛计算的解析公式，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。该算法在前处理和后处理两个阶段都使用超收敛计算公式，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。对于线性单元，本文结点和单元的位移、导数都达到了h4阶的超收敛精度。

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

An algorithm of improved displacement mode is presented by combining the thought of multiple scales method and calculation formula of super convergence.Using analytical formula of super-convergent calculating the displacement mode of high-order finite element solution is expressed with a conventional finite element solution,the new displacement model is constructed with the sum of the displacement model of conventional finite element and the high-order displacement model of finite element solution.The element stiffness matrix is derived using the integral form,the super-convergent formula is used for this algorithm at the stage of pre-processing and the post-processing,for improving 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 presented for the linear element in this paper,the displacement and the derivative accuracy of nodes and elements have reached h⁴ order.