弹性力学平面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,需要先求得单元的结点位移,再经过求导运算来得到。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,并推导弱形式。采用有限元方法,对弹性力学问题给出一阶解法的数值算例,并且将一阶解法的结果与常规位移有限元法的解进行比较。数值计算的结果表明:一阶解法有效地提高了应力的计算精度,并且应力的误差与结点位移的误差具有相同的收敛阶,从而验证所提方法的有效性,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 TWO-DIMENSIONAL ELASTICITY

In the conventional displacement finite element methods,the stresses are obtained from a differential operation after the nodal displacements are determined. In order to improve the accuracy of the stress,a first-order multivariable form was proposed to make sure that the order of stresses and displacements should be equal,and the weak form was then derived. Using the finite element method,a two-dimensional elastic problem was solved by firstorder method,and the first-order solutions and the solutions of the conventional displacement finite element method were compared. The numerical results showed that the first-order solution effectively improved the accuracy of the stress while the displacement errors and the stress errors had the same convergence order,which verified the effectiveness of the method and provided a new idea of improving the accuracy for the finite element method.