高性能六面体组合杂交元研究

【作者】 袁占斌；

【导师】 聂玉峰；

【作者基本信息】 西北工业大学 ， 计算数学， 2005， 硕士

【摘要】 尽管计算机的计算速度在不断的加快,但仅靠高次元或加密网格的方法依然难以解决实际工程中的大规模计算问题,因此低阶高性能元一直是计算力学和数学工作者们孜孜追求的目标。 本文首先对高性能四边形元的发展历程作了回顾,再一次明确在等参坐标体系下要想得到高性能的单元必须使用应变增强位移,并需要基于两场或三场变分原理。众所周知Inf-Sup条件是横在混合元或杂交元面前的一道障碍,且满足了此条件的单元并不标志着就有好的性能,在位移场一定的前提下,用什么样的条件来限制应力场以期获得最佳的搭配就成了设计高性能有限元的关键。在二维情况下,CH(0-1)元的成功揭示了能量协调条件是提高单元性能的关键,而且二维情况下,能量协调条件与正交条件是完全等价的。 按二维四边形组合杂交元的思路,本文向三维六面体单元作了推广,分别用能量协调条件、正交条件和弱平衡条件限制等参坐标系下的完全线性应力场,得到了约束应力矩阵显式表达。对于一般的六面体网格,能量协调条件和正交条件约束下的应力场不相同;只有当网格规则时,两种约束等效,进而弱平衡条件自动满足;当网格满足B条件时,随着网格尺寸的减小,这两种应力模式相差高阶无穷小。这些结论是在对三线性等参变换中的几何参数做了详尽分析之后得到的。基于以上的应力空间和不同的位移空间搭配构造了几种不同的组合杂交元,并通过数值试验研究了其性能,发现当组合参数在0-1之间变化时,由非协调位移与约束应力构造的组合杂交元对数值结果的调节能力有限;而由协调位移与约束应力格式构造的组合杂交元最好的结果在调节参数为1时获得,这时组合杂交变分原理退化为Hellinger-Ressiner原理。更多还原

【Abstract】 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.更多还原