Reissner-Mindlin板问题的一种有限元近似解

通过应用对板的厚度做局部修改的混合有限元方法,计算R e issner-M ind lin板问题的近似解,得到横向位移和旋度的误差分别在H1模和L2模意义下的阶都是2,并且它们不依赖于板的厚度.     

基于台劳展式的矩形Reissner-Mindlin板元

1.引言 Rdissner-Mindlin板模型放弃了经典板模型的Kirchhoff假说,考虑了剪切变形,能应用于更广泛的板问题。Reissner-Mindlin板模型的挠度与转角是相互独立的,单元只需具有c~0连续性,这一点优于需要具有c~1连续性的Kirchhoff板元,但一个严重困难是普通c~0元,尤其是低阶c~0元,当板厚趋于零时不收敛,这就是所谓的自锁现象(locking)。近年来,研究避免自锁现象的Reissner-Mindlin模型板元吸引了不少的注     

拟线性奇异摄动问题一致收敛差分格式

1 引言 我们考虑拟线性奇异摄动Dirichlet问题 εy″-(f(y))′-b(x,y)=0,0相似文献   

任意秩多元线性模型中的最优预测

本文研究了任意秩多元线性模型中可预测变量的最优预测,特别地,我们考虑了一类特殊的预测函数,Φ－线性预测函数,给出了Φ－可预测变量和最优Φ线性一无偏预测的定义,得到了Φ－可预测变量的最优Φ－线性无偏预测,并证明了它在几乎处处意主意义下的唯一性。     

带新熵函数的Entropy-Ultra-bee格式计算线性传输方程

本文研究了线性传输方程的数值解法.利用物理的熵函数得到了一种新的Entropy-Ultrabee格式.数值实验表明该格式长时间计算效果非常好,在间断附近有比较高的分辨率.     

解线性抛物方程的一类新格式

解线性抛物方程的一类新格式孙志忠（中国科学院计算中心）ＡＮＥＷＣＬＡＳＳＯＦＤＩＦＦＥＲＥＮＣＥＳＣＨＥＭＥＳＦＯＲＬＩＮＥＡＲＰＡＲＡＢＯＬＩＣＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳ￥ＳｕｎＺｈｉ－ｚｈｏｎｇ（ＣｏｍｐｕｔｉｎｇＣｅｎｔｅｒ，Ａ...     

含双参数的半线性奇异摄动问题的一致收敛差分格式

考虑半线性边值问题: T(y)≡-εy″+μ(a(x)y)′+b(x,y)=0,0相似文献   

Reissner板中的夹杂和缺陷问题研究

基于保角变换技术和Faber级数展开,研究了含任意形状夹杂或缺陷的无限大Reissner板弯曲问题．将变换域中单位圆内、外解析函数分别展开成Faber级数,并将波动函数展开成第一类和第二类修正的n阶Bessel函数;利用边界位移、剪力和弯矩连续性条件得到问题的高阶线性方程组．以含椭圆形夹杂和缺陷的无限大Reissner板柱面弯曲为例,进一步给出了数值算例和理论分析．结果表明,对于软夹杂,板内力矩随夹杂与板厚尺寸比a/h变化非常敏感;在含硬夹杂条件下,板内力矩随夹杂尺寸变化相对不敏感．     

求解线性互补问题的一种新算法

包括数学规划、对策论、经济学和力学等应用领域中的某些问题,都可以转化成如下的线性互补问题:     

解拟线性抛物方程的一类二阶差分格式

解拟线性抛物方程的一类二阶差分格式孙志忠（中国科学院计算中心）ＡＣＬＡＳＳＯＦＳＥＣＯＮＤ－ＯＲＤＥＲＡＣＣＵＲＡＴＥＤＩＦＦＥＲＥＮＣＥＳＣＨＥＭＥＳＦＯＲＱＵＡＳＩ－ＬＩＮＥＡＲＰＡＲＡＢＯＬＩＣＥＱＵＡＴＩＯＮＳ￥ＳｕｎＺｈｉ－ｚｈｏｎｇ（Ｃｏ...     

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.

     

Analysis of some low order quadrilateral Reissner-Mindlin plate elements

Four quadrilateral elements for the Reissner-Mindlin plate model are considered. The elements are the stabilized MITC4 element of Lyly, Stenberg and Vihinen  (1933), the MIN4 element of Tessler and Hughes (1983), the Q4BL element of Zienkiewicz et al. (1993) and the FMIN4 element of Kikuchi and Ishii (1999). For all elements except the Q4BL element, a unifying variational formulation is introduced, and optimal H and L error bounds uniform in the plate thickness are proven. Moreover, we propose a modified Q4BL element and show that it admits the optimal H and L error bounds uniform in the plate thickness. In particular, we study the convergence behavior of all elements regarding the mesh distortion.

     

Locking-free finite elements for the Reissner-Mindlin plate

Two new families of Reissner-Mindlin triangular finite elements are analyzed. One family, generalizing an element proposed by Zienkiewicz and Lefebvre, approximates (for the transverse displacement by continuous piecewise polynomials of degree , the rotation by continuous piecewise polynomials of degree plus bubble functions of degree , and projects the shear stress into the space of discontinuous piecewise polynomials of degree . The second family is similar to the first, but uses degree rather than degree continuous piecewise polynomials to approximate the rotation. We prove that for , the errors in the derivatives of the transverse displacement are bounded by and the errors in the rotation and its derivatives are bounded by and , respectively, for the first family, and by and , respectively, for the second family (with independent of the mesh size and plate thickness . These estimates are of optimal order for the second family, and so it is locking-free. For the first family, while the estimates for the derivatives of the transverse displacement are of optimal order, there is a deterioration of order in the approximation of the rotation and its derivatives for small, demonstrating locking of order . Numerical experiments using the lowest order elements of each family are presented to show their performance and the sharpness of the estimates. Additional experiments show the negative effects of eliminating the projection of the shear stress.

     

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。     

THE EFFECT OF BOUNDARY SHAPE ON BOUNDARY LAYER OF P-MODEL PLATE PROBLEMS WITH HARD SIMPLY SUPPORT

§1.Ｉｎｔｒｏｄｕｃｔｉｏｎ　　Ｍａｎｙｐａｐｅｒｓ(ｓｅｅ[14,6])ｐａｙｍｕｃｈａｔｔｅｎｔｉｏｎｔｏｔｈｅｂｏｕｎｄａｒｙｌａｙｅｒｆｏｒｐｌａｔｅｍｏｄｅｌｐｒｏｂｌｅｍ.Ｂｏｕｎｄａｒｙｌａｙｅｒｍｅａｎｓｔｈａｔｔｈｅｓｏｌｕｔｉｏｎｃｈａｎｇｅｓｓｈａｒｐｌｙａｌｏｎｇｔｈｅｎｏｒｍａｌｄｉｒｅｃｔｉｏｎｏｆｔｈｅｂｏｕｎｄａｒｙ,ｉｔｃａｕｓｅｓｔｈｅｄｉｆｆｅｒｅｎｃｅａｍｏｎｇｖａｒｉｏｕｓｋｉｎｄｓｏｆｐｌａｔｅｍｏｄｅｌｓ,ａｎｄａｌｓｏｉｔｂｒｉｎｇｓｄｉｆｆｉｃｕｌｔｉｅｓ…     

Superconvergence in the projected-shear plate-bending finite element method

A projected-shear finite element method for periodic Reissner–Mindlin plate model are analyzed for rectangular meshes. A projection operator is applied to the shear stress term in the bilinear form. Optimal error estimates in the L²-norm, the H¹-norm, and the energy norm for both displacement and rotations are established and gradient superconvergence along the Gauss lines is justified in some weak senses. All the convergence and superconvergence results are uniform with respect to the thickness parameter t. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 367–386, 1998     

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(     

On Choices of Stress Modes for Lower Order Quadrilateral Reissner-Mindlin Plate Elements

1 Introduction In recent years, a lot of work for the Mindlin-Reissner (R-M) plate model has been done in the engineering and mathematical literatures (see [1-5, 7-15, 17] and references therein). As one knows, one of the most important problems is how to…     

On a structural acoustic model with interface a Reissner-Mindlin plate or a Timoshenko beam

This paper is concerned with a model which describes the interaction of sound and elastic waves in a structural acoustic chamber in which one “wall” is flexible and flat. The model is new in the sense that the composite dynamics of the three-dimensional structure is described by the linearized equations for a gas defined on the interior of the chamber and the Reissner-Mindlin plate equations on the two-dimensional flat wall of the chamber, while, if a two-dimensional acoustic chamber is considered, the Timoshenko beam equations describe the deflections of the one-dimensional “wall.” With a view to achieving uniform stabilization of the structure linear feedback boundary damping is incorporated in the model, viz. in the wave equation for the gas and in the system of equations for the vibrations of the elastic medium. We present the uniform stability result for the case of a two-dimensional chamber and outline the method for the three-dimensional model which shows strong resemblance with the system of dynamic plane elasticity.     

Second-order two-scale method for bending behavior analysis of composite plate with 3-D periodic configuration and its approximation

This paper considers the bending behaviors of composite plate with 3-D periodic configuration.A second-order two-scale(SOTS)computational method is designed by means of construction way.First,by 3-D elastic composite plate model,the cell functions which are defined on the reference cell are constructed.Then the effective homogenization parameters of composites are calculated,and the homogenized plate problem on original domain is defined.Based on the Reissner-Mindlin deformation pattern,the homogenization solution is obtained.And then the SOTS’s approximate solution is obtained by the cell functions and the homogenization solution.Second,the approximation of the SOTS’s solution in energy norm is analyzed and the residual of SOTS’s solution for 3-D original in the pointwise sense is investigated.Finally,the procedure of SOTS’s method is given.A set of numerical results are demonstrated for predicting the effective parameters and the displacement and strains of composite plate.It shows that SOTS’s method can capture the 3-D local behaviors caused by3-D micro-structures well.