一维C~1有限元后处理超收敛计算的新型算法

发布时间：2018-11-05 13:53

【摘要】：该文针对一维C~1有限元提出一种新型后处理超收敛算法,由该法可求得全域超收敛的位移和内力。该法在单个单元上逐单元实施,通过将单元端部结点位移有限元解设为本质边界条件,在单元域上建立单元位移恢复的局部边值问题。对该局部边值问题,以单元内任一点为结点将单元划分为两个子单元进行有限元求解,子单元次数与原单元相同,由此获得该点位移的超收敛解。对单元内所有点均作这样的超收敛求解,可获得整个单元上位移的超收敛解。该位移超收敛解光滑、连续,通过对该位移超收敛解求导可获得转角和内力的超收敛解。数值结果表明,对于m次元,该法得到的挠度和转角具备与结点位移相同的h~(2m-2)阶的最佳收敛阶;弯矩和剪力则分别具备h~(2m-3)、h~(2m-4)阶的收敛阶,均比相应有限元解高出m-2阶。该法可靠、高效、易于实施,是一种颇具潜力的后处理超收敛算法。
[Abstract]:In this paper, a new post-processing superconvergence algorithm is proposed for one dimensional C ~ 1 finite element method, from which the displacement and internal force of global superconvergence can be obtained. The method is implemented on a single element by element. The local boundary value problem of element displacement recovery is established in the element domain by setting the finite element solution of the end node displacement of the element as the essential boundary condition. For the local boundary value problem, the element is divided into two subelements at any point in the element. The number of subelements is the same as that of the original element, and the superconvergence solution of the displacement of the point is obtained. The superconvergence solution of the displacement on the whole element can be obtained by using the superconvergence solution for all points in the element. The displacement superconvergence solution is smooth and continuous. The superconvergence solution of rotation angle and internal force can be obtained by the derivation of the displacement superconvergence solution. The numerical results show that the deflection and rotation angle obtained by this method have the best convergence order of 2m-2 order which is the same as the node displacement for m-dimensional element. The bending moment and shear force have the convergence order of order H ~ (2m-3) and h ~ (2m-4) respectively, which are higher than the corresponding finite element solution by order m ~ (-2). The method is reliable, efficient and easy to implement. It is a potential post-processing superconvergence algorithm.
【作者单位】：清华大学土木工程系土木工程安全与耐久教育部重点实验室;
【基金】：清华大学自主科研计划项目(2011THZ03)
【分类号】：O241.82

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