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     

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     

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.     

On the derivation of an asymptotically correct shear correction factor for the Reissner-Mindlin plate model

In the framework of thin linear elastic plates it is known that the solutions of both the three-dimensional problem and the Reissner-Mindlin plate model can be developed into asymptotic expansions. By comparing the particular asymptotic expansions with respect to the half-thickness ɛ of the plate in the case of periodic boundary conditions on the lateral side, the shear correction factor in the Reissner-Mindlin plate model can be determined in such a way that this model approximates the three-dimensional solution with one order of the plate thickness better than the classical Kirchhoff model. This fails for hard clamped lateral boundary conditions so that the Reissner-Mindlin model is in this case asymptotically as good as the Kirchhoff model.     

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.     

Continuous finite element methods for Reissner-Mindlin plate problem

On triangle or quadrilateral meshes, two finite element methods are proposed for solving the Reissner-Mindlin plate problem either by augmenting the Galerkin formulation or modifying the plate-thickness. In these methods, the transverse displacement is approximated by conforming (bi)linear macroelements or (bi)quadratic elements, and the rotation by conforming (bi)linear elements. The shear stress can be locally computed from transverse displacement and rotation. Uniform in plate thickness, optimal error bounds are obtained for the transverse displacement, rotation, and shear stress in their natural norms. Numerical results are presented to illustrate the theoretical results.     

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)     

The boundary element method in stress-state problems for an anisotropic plate with holes

We propose a method of solving the problem of the stress state of an anisotropic plate with holes of arbitrary shape. The method is based on approximating the boundary of a region by curved boundary elements. These elements are taken to be a family of semi-ellipses. To satisfy the boundary conditions we use the pointwise least-square method. Numerical experiments showed good agreement of the computations with results known earlier. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 24, 1993, pp. 44–49.     

Magnetohydrodynamic shear flow along a flat plate with uniform suction or blowing

An analysis is made of the steady shear flow of an incompressible viscous electrically conducting fluid past an electrically insulating porous flat plate in the presence of an applied uniform transverse magnetic field. It is shown that steady shear flow exists for suction at the plate only when the square of the suction parameter S is less than the magnetic parameter Q. In this case the velocity at a given point increases with increase in either the magnetic field or suction velocity. The shear stress at the plate increases with increase in either S or the free-stream shear-rate parameter σ₁ or Q. The analysis further reveals that solution exists for steady shear flow past a porous flat plate subject to blowing only when the square of the blowing parameter S₁ is less than Q. It is found that the induced magnetic field at a given location decreases with increase in Q. Further the wall shear stress decreases with increase in S₁. No steady shear flow is possible for blowing at the plate when S₁² > Q. Received: June 16, 2004; revised: October 24, 2004     

Estimation of Deviations from the Exact Solution for the Reissner-Mindlin Plate Problem

In this paper, we obtain a majorant of the difference between the exact solution and any conforming approximate solution of the Reissner-Mindlin plate problem. This majorant is explicitly computable and involves constants that depend only on given data of the problem. The majorant allows us to compute guaranteed upper bounds of errors with any desired accuracy and vanishes if and only if the approximate solution coincides with the exact one. Bibliography: 12 titles. To N. N. Uraltseva with gratitude __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 310, 2004, pp. 145–157.     

A posteriori error analysis for conforming MITC elements for Reissner-Mindlin plates

This paper establishes a unified a posteriori error estimator for a large class of conforming finite element methods for the Reissner-Mindlin plate problem. The analysis is based on some assumption (H) on the consistency of the reduction integration to avoid shear locking. The reliable and efficient a posteriori error estimator is robust in the sense that the reliability and efficiency constants are independent of the plate thickness . The presented analysis applies to all conforming MITC elements and all conforming finite element methods without reduced integration from the literature.

     

Numerical analysis of a locking‐free mixed finite element method for a bending moment formulation of Reissner‐Mindlin plate model

This article deals with the approximation of the bending of a clamped plate, modeled by Reissner‐Mindlin equations. It is known that standard finite element methods applied to this model lead to wrong results when the thickness t is small. Here, we propose a mixed formulation based on the Hellinger‐Reissner principle which is written in terms of the bending moments, the shear stress, the rotations and the transverse displacement. To prove that the resulting variational formulation is well posed, we use the Babu?ka‐Brezzi theory with appropriate t ‐dependent norms. The problem is discretized by standard mixed finite elements without the need of any reduction operator. Error estimates are proved. These estimates have an optimal dependence on the mesh size h and a mild dependence on the plate thickness t. This allows us to conclude that the method is locking‐free. The proposed method yields direct approximation of the bending moments and the shear stress. A local postprocessing leading to H¹ ‐type approximations of transverse displacement and rotations is introduced. Moreover, we propose a hybridization procedure, which leads to solving a significantly smaller positive definite system. Finally, we report numerical experiments which allow us to assess the performance of the method. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Adaptive FE analysis of plates by shear-flexible quadrilateral Reissner-Mindlin elements

h-version adaptive finite element analysis of plates using QUAD4 R-M elements is discussed. The necessity of using shear flexible formulations for plate elements in adaptive FEA is explained. Two shear flexible QUAD4 plate elements formulated by reconstituted shear strain fields are selected from literature and are used to solve both thin and thick plates with various geometries and boundary conditions. The effect of boundary layers in plates with soft or free boundaries is discussed and is shown in various examples. A powerful and versatile quadrilateral automatic mesh generator (MSD) is used for the discretization of the plate domain. The error norms are computed by the Z² as well as the superconvergence theory. The global convergence rates of the adaptive solutions with respect to the energy norms are demonstrated and it is seen that the theoretical rates of convergence are exceeded in several cases.     

Closed-form solutions for stress singularities at bi-material wedges in Reissner-Mindlin plates

In this work, stress singularities in isotropic bi-material junctions are investigated using Reissner-Mindlin plate theory by means of a complex potential formalism. The governing system of partial differential equations is solved employing methods of asymptotic analysis. The resulting asymptotic near-fields including the singularity exponent λ are obtained in a closed-form analytical manner as solutions of a corresponding eigenvalue problem. The singular solution character is discussed for different geometrical configurations. In particular, the present study investigates the influence of the material constants on the singularity exponent. It is shown, that the Reissner-Mindlin theory allows for distinguishing between singularities of the bending moments and the transverse shear forces. Further, stronger singularities than the classical crack-tip singularity are observed. The results allow for further application such as a combination with numerical methods. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Reissner-Mindlin plate is the Γ-limit of Cosserat elasticity

The linear Reissner-Mindlin membrane-bending plate model is the rigorous Γ-limit for zero thickness of a linear isotropic Cosserat bulk model with symmetric curvature. For this result we use the natural nonlinear scaling for the displacements u and the linear scaling for the infinitesimal microrotations Ā ∈ so(3) affecting the in-plane drill rotations. No boundary conditions on the microrotations are prescribed. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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。     

hp-Finite element for free vibration analysis of the orthotropic triangular and rectangular plates

The hp-version of the finite element method based on a triangular p-element is applied to free vibration of the orthotropic triangular and rectangular plates. The element's hierarchical shape functions, expressed in terms of shifted Legendre orthogonal polynomials, is developed for orthotropic plate analysis by taking into account shear deformation, rotary inertia, and other kinematics effects. Numerical results of frequency calculations are found for the free vibration of the orthotropic triangular and rectangular plates with the effect of the fiber orientation and plate boundary conditions. The results are very well compared to those presented in the literature.     

On corner singularities in Reissner-Mindlin's theory of elastic plates

In the framework of linear elasticity, singularities occur in domains with non-smooth boundaries. Particularly in Fracture Mechanics, the local stress field near stress concentrations is of interest. In this work, singularities at re-entrant corners or sharp notches in Reissner-Mindlin plates are studied. Therefore, an asymptotic solution of the governing system of partial differential equations is obtained by using a complex potential approach which allows for an efficient calculation of the singularity exponent λ. The effect of the notch opening angle and the boundary conditions on the singularity exponent is discussed. The results show, that it can be distinguished between singularities for symmetric and antisymmetric loading and between singularities of the bending moments and the transverse shear forces. Also, stronger singularities than the classical crack tip singularity with free crack faces are observed. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Uniform analysis of a stabilized hybrid finite element method for Reissner-Mindlin plates

This paper presents a low order stabilized hybrid quadrilateral finite element method for ReissnerMindlin plates based on Hellinger-Reissner variational principle,which includes variables of displacements,shear stresses and bending moments.The approach uses continuous piecewise isoparametric bilinear interpolations for the approximations of the transverse displacement and rotation.The stabilization achieved by adding a stabilization term of least-squares to the original hybrid scheme,allows independent approximations of the stresses and moments.The stress approximation adopts a piecewise independent 4-parameter mode satisfying an accuracy-enhanced condition.The approximation of moments employs a piecewise-independent 5-parameter mode.This method can be viewed as a stabilized version of the hybrid finite element scheme proposed in [Carstensen C,Xie X,Yu G,et al.A priori and a posteriori analysis for a locking-free low order quadrilateral hybrid finite element for Reissner-Mindlin plates.Comput Methods Appl Mech Engrg,2011,200:1161-1175],where the approximations of stresses and moments are required to satisfy an equilibrium criterion.A priori error analysis shows that the method is uniform with respect to the plate thickness t.Numerical experiments confirm the theoretical results.     

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.