On evaluating the inf–sup condition for plate bending elements

This paper addresses the evaluation of the inf–sup condition for Reissner–Mindlin plate bending elements. This fundamental condition for stability and optimality of a mixed finite element scheme is, in general, very difficult to evaluate analytically, considering for example distorted meshes. Therefore, we develop a numerical test methodology. To demonstrate the test methodology and to obtain specific results, we apply it to standard displacement-based elements and elements of the MITC family. Whereas the displacement-based elements fail to satisfy the inf–sup condition, we find that the MITC elements pass our numerical test for uniform meshes and a sequence of distorted meshes. © 1997 John Wiley & Sons, Ltd.     

An improved three‐node hybrid‐mixed element for Mindlin/Reissner plates

Based on the mixed shear projected (MiSP) approach [6], an enhanced bending approximation for homogeneous isotropic plates is presented. Some hard benchmark tests, as skew plate (30°) problem for instance, have often shown poor convergence when low order elements (3‐ or 4‐node element) are developed using linear approximations for kinematic variables. To put right this weakness, we propose a high‐order interpolation for rotational dofs which results in more rich bending curvatures. The mid‐side rotational nodes are eliminated using a combination of local discrete kinematic and constitutive Mindlin hypothesises. The derived 3‐node triangular element, called MiSP3+, is free of shear locking and passes all patch‐tests for thick and thin plates in an arbitrary mesh. Copyright © 2001 John Wiley & Sons, Ltd.     

Plate spectral elements based upon Reissner–Mindlin theory

Plate bending spectral elements based upon the Reissner–Mindlin theory is developed for a rectangular plate. Numerical comparison studies for static and eigenvalue problems, demonstrate the excellent performance and robustness of the present method. The improvement in accuracy and computational efficiency in comparison with low order plate bending elements is remarkable and suggests that the proposed approach is an alternative to existing schemes.     

Imposing Dirichlet boundary conditions in the extended finite element method

This paper is devoted to the imposition of Dirichlet‐type conditions within the extended finite element method (X‐FEM). This method allows one to easily model surfaces of discontinuity or domain boundaries on a mesh not necessarily conforming to these surfaces. Imposing Neumann boundary conditions on boundaries running through the elements is straightforward and does preserve the optimal rate of convergence of the background mesh (observed numerically in earlier papers). On the contrary, much less work has been devoted to Dirichlet boundary conditions for the X‐FEM (or the limiting case of stiff boundary conditions). In this paper, we introduce a strategy to impose Dirichlet boundary conditions while preserving the optimal rate of convergence. The key aspect is the construction of the correct Lagrange multiplier space on the boundary. As an application, we suggest to use this new approach to impose precisely zero pressure on the moving resin front in resin transfer moulding (RTM) process while avoiding remeshing. The case of inner conditions is also discussed as well as two important practical cases: material interfaces and phase‐transformation front capturing. Copyright © 2006 John Wiley & Sons, Ltd.     

ASSUMED STRESS C0 QUADRILATERAL/TRIANGULAR PLATE ELEMENTS BY INTERRELATED EDGE DISPLACEMENTS

This paper presents two simple quadrilateral C⁰ plate bending elements with explicit element stiffness matrix. The element formulation is based on assumed element stress fields and the interrelated transverse displacement and rotation along element boundaries. The interrelated edge displacements not only can result in higher-order displacements interpolations for higher accuracy element and overcome the shear locking in thin plate analysis encountered by C⁰ plate elements, but can also unify the four-noded quadrilateral element and its corresponding three-noded triangular element. The latter cannot be achieved by the assumed displacement formulation. The numerical examples demonstrate the accuracy and robustness of the present assumed stress C⁰ plate elements.