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.

     

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…     

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)     

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.     

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.

     

Locking-free nonconforming finite elements for three-dimensional elasticity problem

Two locking-free nonconforming finite elements are presented for three-dimensional elasticity problem with pure displacement boundary condition. Convergence rate of the elements are uniformly optimal with respect to λ. The energy norm and L² norm errors are O(h²) and O(h³), respectively. Lastly, a numerical experiment is carried out, which coincides with the theoretical analysis.     

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.     

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     

Locking-free finite element methods for shells

We propose a new family of finite element methods for the Naghdi shell model, one method associated with each nonnegative integer . The methods are based on a nonstandard mixed formulation, and the th method employs triangular Lagrange finite elements of degree augmented by bubble functions of degree for both the displacement and rotation variables, and discontinuous piecewise polynomials of degree for the shear and membrane stresses. This method can be implemented in terms of the displacement and rotation variables alone, as the minimization of an altered energy functional over the space mentioned. The alteration consists of the introduction of a weighted local projection into part, but not all, of the shear and membrane energy terms of the usual Naghdi energy. The relative error in the method, measured in a norm which combines the norm of the displacement and rotation fields and an appropriate norm of the shear and membrane stress fields, converges to zero with order uniformly with respect to the shell thickness for smooth solutions, at least under the assumption that certain geometrical coefficients in the Nagdhi model are replaced by piecewise constants.