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.

     

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)     

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.

     

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…     

A locking-free Reissner-Mindlin quadrilateral element

On arbitrary regular quadrilaterals, a new finite element method for the Reissner-Mindlin plate is proposed, where both transverse displacement and rotation are approximated by isoparametric bilinear elements, with local bubbles enriching rotation, and a local reduction operator is applied to the shear energy term. This new method gives optimal error bounds, uniform in the thickness of the plate, for both transverse displacement and rotation with respect to and norms.

     

Approximation of the vibration modes of a plate by Reissner-Mindlin equations

This paper deals with the approximation of the vibration modes of a plate modelled by the Reissner-Mindlin equations. It is well known that, in order to avoid locking, some kind of reduced integration or mixed interpolation has to be used when solving these equations by finite element methods. In particular, one of the most widely used procedures is the mixed interpolation tensorial components, based on the family of elements called MITC. We use the lowest order method of this family. Applying a general approximation theory for spectral problems, we obtain optimal order error estimates for the eigenvectors and the eigenvalues. Under mild assumptions, these estimates are valid with constants independent of the plate thickness. The optimal double order for the eigenvalues is derived from a corresponding -estimate for a load problem which is proven here. This optimal order -estimate is of interest in itself. Finally, we present several numerical examples showing the very good behavior of the numerical procedure even in some cases not covered by our theory.

     

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.     

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.     

关于Reissner-Mindlin板问题的线性格式

本文给出Ｒｅｉｓｓｎｅｒ－Ｍｉｎｄｌｉｎ板问题的线性格式[1]中的汽泡函数的最优选取。     

The longtime behavior of a nonlinear Reissner-Mindlin plate with exponentially decreasing memory kernels

In this paper we investigate the longtime behavior of the mathematical model of a homogeneous viscoelastic plate based on Reissner-Mindlin deformation shear assumptions. According to the approximation procedure due to Lagnese for the Kirchhoff viscoelastic plate, the resulting motion equations for the vertical displacement and the angle deflection of vertical fibers are derived in the framework of the theory of linear viscoelasticity. Assuming that in general both Lame's functions, λ and μ, depend on time, the coupling terms between the equations of displacement and deflection depend on hereditary contributions. We associate to the model a nonlinear semigroup and show the behavior of the energy when time goes on. In particular, assuming that the kernels λ and μ decay exponentially, and not too weakly with respect to the physical properties considered in the model, then the energy decays uniformly with respect to the initial conditions; i.e., we prove the existence of an absorbing set for the semigroup associated to the model.     

On the connection between some linear triangular Reissner-Mindlin plate bending elements

Summary. We consider three triangular plate bending elements for the Reissner-Mindlin model. The elements are the MIN3 element of Tessler and Hughes [19], the stabilized MITC3 element of Brezzi, Fortin and Stenberg [5] and the T3BL element of Xu, Auricchio and Taylor [2, 17, 20]. We show that the bilinear forms of the stabilized MITC3 and MIN3 elements are equivalent and that their implementation may be simplified by using numerical integration of reduced order. The T3BL element is shown to be essentially the same as the MIN3 and stabilized MITC3 elements with reduced integration. We finally introduce a general stabilized finite element formulation which covers all three methods. For this class of methods we prove the stability and optimal convergence properties. Received November 4, 1996 / Revised version received May 29, 1997 / Published online January 27, 2000     

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.

     

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.     

Unilateral dynamic contact problem for viscoelastic Reissner-Mindlin plates

The existence of solutions is proved for systems of dynamic Reissner-Mindlin equations expressing vibrations of viscoelastic plates. We consider the cases of short memory and singular memory material. Contact with the rigid support is considered.     

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     

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.     

Development of a higher order finite element on a Winkler foundation

Analyzing thick plates as a construction component has been of interest to structural engineering research for several decades. In particular, thick plates resting on elastic foundations are more specific. Mindlin's plate theory for thick plate analysis and the Winkler theory for elastic foundation analyses have wide applications. The current research considers analysis of isotropic plates on a Winkler foundation according to Mindlin's plate theory. The analysis uses a higher order plate element to avoid shear locking phenomena in the plate. The main features of this element are representation of real displacement functions of the plate perfect and shear locking do not occur at the plates modeled with this element. Derivation of the equations for finite element formulation for thick plate theory uses fourth-order displacement shape functions. A computer program using the finite element method, coded in C++, analyzes the plates resting on an elastic foundation. The analysis involves a 17-noded finite element. The study's graphs and tables assist engineers' designs of thick plates resting on elastic foundations. The study concludes with the computer-coded program, which allows effective use for the shear locking-free analysis of thick Mindlin plates resting on elastic foundations.     

一类Kirchhoff 板弯问题自适应高阶混合元方法及理论分析

本文针对Kirchhoff 板弯问题提出了一个基于高阶Hellan-Herrmann-Johnson (简记为H-H-J)方法的自适应有限元算法, 分析了它的收敛性和计算复杂度. 证明了算法在执行过程中, 相应的拟能量误差会以几何级数单调衰减, 从而得到收敛性. 利用此单调下降性质, 进一步给出了算法的计算复杂度. 推导过程中的一个关键步骤是建立基于平衡方程的单元误差表示(error indicator) 与平衡方程右端载荷震荡项(data oscillation) 的局部等价关系.     

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.