二维三维弹性问题混合元

发布时间：2018-01-11 11:00

  本文关键词：二维三维弹性问题混合元　出处：《郑州大学》2016年博士论文　论文类型：学位论文

  更多相关文章： 线弹性方程 混合有限元 弹性复形 各向异性 仿射等价 误差估计

【摘要】：本文主要讨论线弹性方程,在Hellinger-Reissner变分形式的基础上,系统的构造了二维空间下的矩形和三角形单元,三维空间下的立方体和四面体单元等一系列简单稳定的单元.对单元的适定性,收敛性,误差估计,以及二维三维矩形和立方体单元的各向异性特征进行了深入的分析和系统的研究.并对二维协调的矩形单元和非协调的三角形单元进行了相应的数值实验.在这篇论文中,首先,我们构造了具有最少自由度的协调的二维矩形单元R8-2和三维立方体单元C18-3.即矩形单元的应力和位移空间分别为8个和2个自由度.立方体单元的应力和位移空间中分别为18和3个自由度.由于所构造单元不满足关于散度的投影性质,因此我们采用了构造的方法证明了离散BB条件,即离散混合问题的唯一可解性条件.在此基础上进一步分析,发现了单元的各向异性特征,并由此得到了单元的误差估计.据我们所知这是首次构造的各向异性弹性问题混合元。其次,在构造最简单矩形单元的基础上,进一步构造了一系列矩形高阶单元,当次数大于等于4时满足散度的投影性质,据此得到单元的适定性和离散问题的唯一可解性及误差估计,并得到相应的弹性复形.再次,在构造矩形类单元的基础上,又对三维四面体单元构造过程中的空间Mκ(K)进行了研究和讨论,因为在Hellinge-Reissner变分形式下,应力是属于H(diυ,Ω；S)空间,协调元构造要求应力的法向分量跨过单元边界连续,在协调元构造中计算空间Mk(K)={τ∈Pk(K;S)|divτ=0,τn|(?)K=0)的维数是个难点问题.在本部分中给出了任意阶空间维数计算的一股方法,并且此方法能够很方便的求出相应的显式基,同时证明了κ=3时,集合Mκ(K)的维数为0,并且给出κ=4时空间的一组基.最后,构造了关于线弹性问题的一系列新的从低阶到高阶的三角形和四面体非协调单元及相应刚体运动下的简化单元,这里构造的二维三维单元区别于之前文献中构造的单元,构造简单,自由度少.这些单元定义在参考单元上,形函数空间显式给出,我们严格证明了这类单元的仿射等价性,易于进行数值实验.当κ=1时简化的三角形单元的应力空间和位移空间具有12+3个自由度,简化的四面体单元的应力空间和位移空间分别具有42+12个自由度.这些单元满足散度的投影性质.本文得到了这些单元的适定性,离散问题的唯一可解性及相应的误差估计.
[Abstract]:This paper mainly discusses the linear elastic equations in Hellinger-Reissner based on the form, system structure and rectangular two dimensional triangular element, 3D cube and tetrahedron and a series of simple and stable unit. For unit well posedness, convergence, error estimation, and 2D and 3D rectangular cube unit of the anisotropic characteristics are analyzed and systematically. And the two-dimensional rectangular unit coordination and non coordination triangle unit the corresponding numerical experiments. In this thesis, firstly, we construct a minimum degree of freedom of the coordination of the two-dimensional rectangular unit R8-2 and unit C18-3. rectangular three-dimensional cube the unit of stress and displacement of space are respectively 8 and 2 degrees of freedom. A cubic element of stress and displacement in the space are 18 and 3 degrees of freedom. Because of the single structure Element does not satisfy the projection properties on the divergence, so we adopt the construction method show that the discrete BB condition, namely the discrete mixed problem of unique solvability conditions. On the basis of further analysis, found the anisotropic characteristics of unit, and obtained the error estimation unit. To our knowledge this is the first structure the anisotropic elastic problem of mixed element. Secondly, based on the structure of simple rectangular elements, and further construct a series of high order rectangular unit, when the projection properties of divergence times greater than or equal to 4, then get the well posedness and uniqueness of the solution of the discrete problem and error estimation unit, and the corresponding elastic complex. Thirdly, based on constructing the rectangle class unit, and the three-dimensional tetrahedron structure in the process of space M (K) were studied and discussed, because in Hellinge-Reissner variational form 涓，

本文编号：1409233