一维C1有限元后处理超收敛计算的新型算法

A NOVEL POST-PROCESSING SUPER-CONVERGENT RECOVERY SCHEME FOR ONE-DEMENSIONAL C1 FE

  • 摘要: 该文针对一维 C1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h2m-2阶的最佳收敛阶;弯矩和剪力则分别具备h2m-3h2m-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: A novel post-processing super-convergent recovery scheme for one-dimensional C1 FE is presented, from which super-convergent displacements and forces on the whole domain can be obtained. The scheme is conducted on each element. On each element a local boundary value problem (BVP) is set up by setting the FE solutions of this element's end nodal displacements as essential boundary conditions. Then for each point on this element, a local mesh is set up via dividing the element into two sub-elements by this point and the displacement at this point is recovered from the FE solution of this local BVP on the local mesh with sub-element's degree the same as original element. The displacements at any other points are recovered in the same way. Thus the displacement on the whole domain is recovered. The recovered displacement is continuous and smooth, and from its derivatives the super-convergent rotations, bending moments and shear forces are derived. Numerical examples show that for sufficient smooth solutions with element of degree m, the convergence order of the recovered displacements and rotations is optimal as h2m-2, while the order of recovered bending moments and shear forces is h2m-3 and h2m-4 respectively, which is (m-2)-th order higher than those corresponding FE solutions. The proposed post-processing super-convergent recovery scheme is reliable, efficient and potential.

     

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