6次Serendipity单元Q32的实现

Implementation of sextic Serendipity element Q32

高次Serendipity单元有内部节点,不便于网格划分及实际使用,文献较少涉及其实现和实际性能。本文给出了32节点6次单元Q32的形函数,探讨了单元刚度矩阵的两类积分方案,即高斯积分和完全多项式积分。后者的积分点数少于前者。以悬臂梁的横力弯曲为例,测试表明,6×6(36点)高斯积分或11阶的各种完全多项式积分都可作为Q32的积分方案。实用中可取25点对称积分,其精度与6×6高斯积分相当,但计算量约只有后者的69%。对三次位移场问题,即使网格严重畸变,Q32也能较准确地模拟。

The high-order Serendipity elements have internal nodes,which is not convenient for grid division and practical use.There are few publications discussing their implementation and actual performance.This paper introduces the shape function of the 32-node sextic element Q32,and discusses two integration schemes of element stiffness matrix:Gaussian integration and complete polynomial integration.The latter has fewer integration points than the former.Taking the transverse bending of a cantilever beam as an example,the results show that the integration scheme for Q32 can be 6×6(36-point)Gaussian integration or various 11-order complete polynomial integrations.The 25-point symmetrical integration can be used in practice:its accuracy is comparable to that of 6×6 Gaussian integration,but its calculation workload is about 69% of the latter.Q32 can simulate a cubic displacement field quite accurately even if the mesh is seriously distorted.