弹性力学问题的一阶有限元解法

求解弹性力学问题的应力时,如果采用常规的位移有限元法,需要先求得单元的节点位移,再经过求导运算得到。为了解决这种求解方式引起的应力精度下降的问题,提出了弹性力学问题的一阶多变量形式,使得应力与位移精度同阶,并推导了弱形式。采用有限元方法,对弹性力学问题给出了一阶解法的二维、三维数值算例,并且将一阶解法的结果与常规位移有限元法的解进行了比较。数值计算结果表明,一阶解法有效提高了应力的精度,并且应力的误差和节点位移的误差具有相同的收敛阶,验证了本文方法的有效性,为提高有限元法的应力精度提供了新的思路。

First-order finite element method for elasticity

In the conventional displacement finite element methods,the stresses are obtained from a diffe-rential operation after the nodal displacements are determined,which is not accurate.In order to improve the stress accuracy,a first-order multi-variable form is proposed to make sure the order of stresses and displacements are equal,and the weak form is then derived.Using the finite element method,a two-dimensional and a three-dimensional elastic problem is solved by first-order method respectively,and the first-order solutions and the solutions of the conventional displacement finite element method were compared.The numerical results show that the first-order solution effectively improves the accuracy of the stresses while the displacement errors and the stress errors have the same convergence order,which verify the effectiveness of the method and provides a new idea of improving the stress accuracy of the finite element method.