High-performance Hexahedral Combined Hybrid Research

When we solve practical problems it is not the ideal ways to use high order element or use very fine grids. Not only the computer can't bear the burden but also the error will be greater after long time's transmitting. So engineers and mathematicians never give up efforts to find the best scheme whose precision is high but cost is low.In this thesis, firstly the development procedure of quadrilateral high performance element is reviewed, and the conclusions are recalled that the enforced strain scheme and multi-field variational principle must be used in order to gain high performance and it's known that the Babuska-Brezzi condition is an obstacle for mixed and hybrid elements, avoiding this condition is not a sigh of a high performance element method. For a fixed displacement field how to construct a corresponding stress field is the key to improve the performance. For a two-dimensional elasticity problem energy compatible condition is found as the crucial reason from CH(O-l). And later the equivalence relationship between energy compatible condition and orthogonal condition in CH(O-l) is proved.Then along the apporach of quadrilateral combined hybrid element CH(O-l), the stress modes constrained by so-called energy compatible condition, orthogonality condition and weak-equilibrium condition respectively with Wilson bubbles are explicit expressed. It is obtained that two stress modes restricted by energy compatible condition and orthogonality condition are not the same for general grids. Only when regular meshes are used the two conditions are the same and weak- equilibrium condition can be satisfied automatic. As the meshes satisfy B condition the difference between them is a higher order infinitesimal. All of these conclusions are based on a detail analysis of the geometric parameters used by the tri-linear transformation. A system of 8 nodes hexahedron elements based on above stress space and different displacement space are constructed to solve 3-dimensional problem. And their performances are tested through some numerical examples. The data show that the effect of the adjust of combined parameter α among interval (0-1) is not notable for the numerical results for elements constructed by incompatible displacement and above constrained stress and the best result is gained as α equals to one for elements constrained by compatible displacement and constrained stress.