三角形REISSNER-MINDLIN板元

本文提出构造无自锁现象的Reissuer－Mindlin板元的一个一般性方法.此方法将剪切应变用它的适当的插值多项式代替,当板厚趋于零时这对应于插值点的Kirchhoff条件,因而单元无自锁现象.根据这种方法我们构造两个三角形元──一个3节点元和一个6节点元,并给出数值结果.     

MITC元的分析

1 引言 有限元求解厚薄板通用的R-M(Reissner-Mindlin)模型板问题,单元只需具有C°连续性,这一点优于需具有C~1连续性的Kirchhoff模型薄板单元.但是当板厚趋于零时,通常的低阶C°元却不收敛,这就是所谓的Locking现象.Bathe和Brezzi等将R-M板模型转化成2阶椭园问题与Stokes问题的耦合形式,据此提出求解R-M板问题的混合扦值单元MITC~([1]、[2]、[3]):设挠度ω的形函数空间是W,转角β=(βx,βy)的形空间是B,在计算剪切应变时,分别将βx,βy按不同方式投影到空间和.数值结果表明这类单元具有很好的收敛性.本文分析MITC元,导出投影算子的显表达式,根据[5]关于Locking现象的一个数学分析,证明当板厚趋于零时,投影算子的选取方式使剪切应变部分对应于特定点上的Kirchhoff条件,引起Locking现象的因素被消除,从而显式证明MITC元避免了Locking 现象. 2 MITC元的整体性质 考虑R-M板弯曲问题,求挠度,转角,使下列板的能量泛函达极小: (1) (2) 其中E是杨氏模量,υ是Possion比,0<υ<1/2,t是板厚,k是剪力校正因子,Ω是板的中面占有的平面区域,f是横向荷载.(1)的第一项是弯曲应变能,第二项是剪切应变能. 设有限元空间是W_h×B_h,W_hH_0~1(Ω),B_h[H_0~1(Ω)]~2,J_h是Ω的单元部分,Ω=K,K是单元,对(1)的直接离散是求(     

用于平板和圆柱壳分析的势能—杂交／混合分离位移有限元格式

本文基于势能～杂交/混合有限元格式,导出了具有分离转动变量的4节点四边形Reissner-Mindlin板元MP4、MP4a和圆柱壳元MCS4.所有这些单元都显示了良好的收敛性;不含有多余机动模式;当趋于薄板/壳极限时,不存在“自锁”现象.本文还指明了在C~0和C~1连续的单元列式中使用的修正泛函,存在相互联系.本文的方法可导出Prathap的一致场列式,也可导出RIT/SRIT的位移协调模型.     

Reissner-Mindlin板的Weissman-Taylor有限元的误差分析

对Reissner-Mindlin板的Weissman-Taylor有限元逼近进行了误差分析.得到了与板的厚度一致无关的旋度、挠度和剪切应力的最优误差估计.揭示了Weissman-Taylor元与稳定化方法的关系.提出了另外两种与Weissman-Taylor元类似的元.     

Reissner-Mindlin板问题带约束非协调旋转Q_1有限元方法

本文利用带约束非协调旋转Q_1元逼近Reissner-Mindlin板问题中旋度的两个分量.并分别选择Wilson元、双线性元和带约束非协调旋转Q_1元逼近挠度,相应地选取不连续的矢量值分片线性函数空间、最低阶旋转Raviart-Thomas元空间和矢量值分片常数函数空间为离散的剪应力空间,在矩形网格上构造了三个板元.通过证明一个离散的Korn不等式,并借助MITC4元的解构造了旋度、挠度和剪应力一个具有某种特殊且关键的可交换性的插值.再利用Helmholtz分解分析相容性误差.我们证明了这三个矩形元在能量范数意义下与板厚无关的一致最优收敛性.数值算例验证了我们的理论结果.     

非协调板元的一般性误差估计式

１．引言 薄板弯曲问题对应于４阶椭圆边值问题,协调有限元求解此问题需要单元具有Ｃ~１连续性,这难于构造且应用不便,因此求解该问题主要应用非协调元．当非协调板元不具有 Ｃ０连续性时,标准能量模误差估计是 ,这一结果不理想,因为对一般的区域,甚至是凸多边形区域,真解只能属于 Ｈ３．近年来,企图将真解属于 Ｈ４的假定改为真解属于 Ｈ３的研究引起重视．针对最简单的三角形非协调板元－Ｍｏｒｌｅｙ元,石钟慈［２］在 Ｈ３假定下,直接利用非协调元分析技巧得到弯距和转角的误差估计式．本文将［２］的结果一般化,同时通过修…     

板的低阶间断与连续有限元法统一分析

基于Reissner-Mindlin板问题的间断Galerkin有限元逼近, 建立了一个对挠度空间和角位移空间取连续或间断元都适用的低阶有限元离散格式. 取剪切力空间为分片常数元, 挠度空间和角位移空间无论取间断元还是连续元, 格式都是一致稳定的, 并给出了H¹范数估计及L²范数估计. 作为应用,对几类低阶有限元空间讨论. 结果表明, 该格式对常见的低阶有限元空间都适用, 并且若至少有一个元连续时, 该格式需要的空间比[1,2]中的都要简单.       

一个改进的Reissner-Mindlin矩形元

本文提出了一个改进的Reissner-Mindlin矩形非协调元方法：旋度用连续双线性元逼近，横向位移用旋转矩形非协调元逼近，而作为中间变量的剪切力用增广的分片常数元逼近，我们证明：该方法具有关于板厚一致稳定性和一致最优收敛性。     

三边夹紧一边自由的矩形厚板的弯曲

利用厚板的Reissner理论中的广义简支边概念[1]得到了三边夹紧一边自由受均布横向载荷作用的矩形厚板的精确解.研究和考察了板的厚度对弯曲的影响及薄板弯曲的Kirchhoff理论的适用范围.     

具有几何对称性的12参数矩形板元

1 引言 三角形板元中,形式最简单的是九参数元,节点参数是单元三个顶点上的函数值和两个一阶偏导数值,非协调九参三角形板元的研究取得了丰硕成果,根据不同方法已构造出多种收敛性能好的单元.相比之下,矩形板元的研究较少见报道.矩形板元中形式最简单的是12参元,节点参数是单元4个顶点上的函数值和两个一阶偏导数值,这类似于九参三角形板元.常见的12参矩形板元是ACM元,其形函数空间是完整3次多项式空间加上两个4次多项式的基函数,ACM元是C°元,但位移形函数的外法向导数平均值在单元间不连续,这类似于Zienkiewicz九参三角形板元,但由于矩形单元的特殊形状,ACM元是收敛的.龙驭球教授等在[1]中提出一种12参矩形广义协调元,其位移形函数的外法向导数平均值在     

NONCONFORMING QUADRILATERAL ROTATED ELEMENT FOR REISSNER-MINDLIN PLATE

In this paper, we extend two rectangular elements for Reissner-Mindlin plate [9] to the quadrilateral case. Optimal H and L error bounds independent of the plate hickness are derived under a mild assumption on the mesh partition.     

NONCONFORMING QUADRILATERAL ROTATED Q1 ELEMENT FOR REISSNER－MINDLIN PLATE

In this paper,we extend two rectangular elements for Reissner-Mindlin plate[9] to the quadrilateral case,Optimal H^1 and L^2 error bounds independent of the plate hickness are derived under a mild assumption on the mesh partition.     

Finite element analysis of the vibration problem of a plate coupled with a fluid

Summary. We consider the approximation of the vibration modes of an elastic plate in contact with a compressible fluid. The plate is modelled by Reissner-Mindlin equations while the fluid is described in terms of displacement variables. This formulation leads to a symmetric eigenvalue problem. Reissner-Mindlin equations are discretized by a mixed method, the equations for the fluid with Raviart-Thomas elements and a non conforming coupling is used on the interface. In order to prove that the method is locking free we consider a family of problems, one for each thickness , and introduce appropriate scalings for the physical parameters so that these problems attain a limit when . We prove that spurious eigenvalues do not arise with this discretization and we obtain optimal order error estimates for the eigenvalues and eigenvectors valid uniformly on the thickness parameter t. Finally we present numerical results confirming the good performance of the method. Received February 4, 1998 / Revised version received May 26, 1999 / Published online June 21, 2000     

Design principles and error analysis for reduced-shear plate-bending finite elements

Summary. In this paper, we consider the problem of designing plate-bending elements which are free of shear locking. This phenomenon is known to afflict several elements for the Reissner-Mindlin plate model when the thickness of the plate is small, due to the inability of the approximating subspaces to satisfy the Kirchhoff constraint. To avoid locking, a “reduction operator” is often applied to the stress, to modify the variational formulation and reduce the effect of this constraint. We investigate the conditions required on such reduction operators to ensure that the approximability and consistency errors are of the right order. A set of sufficient conditions is presented, under which optimal errors can be obtained – these are derived directly, without transforming the problem via a Hemholtz decomposition, or considering it as a mixed method. Our analysis explicitly takes into account boundary layers and their resolution, and we prove, via an asymptotic analysis, that convergence of the finite element approximations will occur uniformly as , even on quasiuniform meshes. The analysis is carried out in the case of a free boundary, where the boundary layer is known to be strong. We also propose and analyze a simple post-processing scheme for the shear stress. Our general theory is used to analyze the well-known MITC elements for the Reissner-Mindlin plate. As we show, the theory makes it possible to analyze both straight and curved elements. We also analyze some other elements. Received June 19, 1995     

Two lower order nonconforming quadrilateral elements for the Reissner-Mindlin plate

This paper generalizes two nonconforming rectangular elements of the Reissner-Mindlin plate to the quadrilateral mesh. The first quadrilateral element uses the usual conforming bilinear element to approximate both components of the rotation, and the modified nonconforming rotated Q ₁ element enriched with the intersected term on each element to approximate the displacement, whereas the second one uses the enriched modified nonconforming rotated Q ₁ element to approximate both the rotation and the displacement. Both elements employ a more complicated shear force space to overcome the shear force locking, which will be described in detail in the introduction. We prove that both methods converge at optimal rates uniformly in the plate thickness t and the mesh distortion parameter in both the H ¹-and the L ²-norms, and consequently they are locking free. This work was supported by the National Natural Science Foundation of China (Grant No. 10601003) and National Excellent Doctoral Dissertation of China (Grant No. 200718)     

Isogeometric Reissner-Mindlin shell analysis - employing different control meshes for displacements and rotations

The main purpose of this paper is the development and implementation of a method for the reduction of the so-called locking effect in the isogeometric Reissner-Mindlin shell formulation. In [1] an isogeometric Reissner-Mindlin shell formulation with an exact interpolation of the director vector based on continuum mechanics was introduced. The numerical examples showed that the accuracy and efficiency increased. However, there are only few effective concepts for the prevention of locking effects for low polynomial degrees. In the work of Beirão da Veiga [2], shear locking is prevented for a Reissner-Mindlin plate formulation by using suitable solution spaces. Here, the method is extended to the Reissner-Mindlin shell formulation. Different control meshes are used for displacements and rotations. Furthermore, the basis functions in the direction of the relevant rotation are one degree less than the ones which are chosen for the displacements. That leads to control meshes with different number and location of the control points. The aim is to avoid shear locking due to the coupling of shear strains and curvature since the compatibility requirement for pure bending is then fulfilled. The accuracy and efficiency of this method are investigated for different examples. In addition, the results are compared to the analytical solutions. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite element methods for Timoshenko beam,circular arch and Reissner-Mindlin plate problems

In this paper some finite element methods for Timoshenko beam, circular arch and Reissner-Mindlin plate problems are discussed. To avoid locking phenomenon, the reduced integration technique is used and a bubble function space is added to increase the solution accuracy. The method for Timoshenko beam is aligned with the Petrov-Galerkin formulation derived in Loula et al. (1987) and can be naturally extended to solve the circular arch and the Reissner-Mindlin plate problems. Optimal order error estimates are proved, uniform with respect to the small parameters. Numerical examples for the circular arch problem shows that the proposed method compares favorably with the conventional reduced integration method.     

Two lower order nonconforming rectangular elements for the Reissner-Mindlin plate

In this paper, we propose two lower order nonconforming rectangular elements for the Reissner-Mindlin plate. The first one uses the conforming bilinear element to approximate both components of the rotation, and the modified nonconforming rotated element to approximate the displacement, whereas the second one uses the modified nonconforming rotated element to approximate both the rotation and the displacement. Both elements employ a projection operator to overcome the shear force locking. We prove that both methods converge at optimal rates uniformly in the plate thickness in both the - and -norms, and consequently they are locking free.