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.

     

Conservation integrals and determination of HRR singularity fields

The angular distribution functions of HRR singularity fields are analyzed via conservation integrals. Two functional equations are proved for the angular distribution functions and can be used for their solutions. The detailed forms of the functional equations and the final governing equations for solutions are given for the cases of plane strain and plane stress. Accurate numerical results are also given for some typical parameters and the equivalence of different governing equations is proved.     

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.

     

Nonconforming finite elements for the equation of planar elasticity

Two new locking-free nonconforming finite elements for the pure displacement planar elasticity problem are presented. Convergence rates of the elements are uniformly optimal with respect to A. The energy norm and L2 norm errors are proved to be O（h2） and O（h3）, respectively. Numerical tests confirm the theoretical analysis.     

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.     

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.

     

平面弹性问题及Stokes问题的一个矩形有限元方法

In this paper,a locking-free nonconforming rectangular finite element scheme is presented for the planar elasticity problem with pure displacement boundary condition.Meanwhile,we prove that this element is also convergent for stationary Stokes problem.     

A quadrilateral nonconforming finite element for linear elasticity problem

In this paper, a four-parameter quadrilateral nonconforming finite element with DSP (double set parameters) is presented. Then we discuss the quadrilateral nonconforming finite element approximation to the linear elastic equations with pure displacement boundary. The optimal convergence rate of the method is established in the broken energy and -norms, and in particular, the convergence is uniform with respect to the Lamé parameter . Also the performance of the scheme does not deteriorate as the material becomes nearly incompressible. Lastly, a numerical test is carried out, which coincides with our theoretical analysis. The research is supported by NSF of China (No. 10471133) and the project of Creative Engineering of Henan Province of China.     

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)     

A locking-free anisotropic nonconforming rectangular finite element approximation for the planar elasticity problem

This paper deals with a new nonconforming anisotropic rectangular finite element approximation for the planar elasticity problem with pure displacement boundary condition. By use of the special properties of this element, and by introducing the complementary space and a series of novel techniques, the optimal error estimates of the energy norm and the L^2-norm are obtained. The restrictions of regularity assumption and quasi-uniform assumption or the inverse assumption on the meshes required in the conventional finite element methods analysis are to be got rid of and the applicable scope of the nonconforming finite elements is extended.