A Refined Error Analysis of (original) (raw)
On Mixed Finite Element Methods for the Reissner-Mindlin Plate Model
Mathematics of Computation, 1992
In this paper we analyze the convergence of mixed finite element approximations to the solution of the Reissner-Mindlin plate problem. We show that several known elements fall into our analysis, thus providing a unified approach. We also introduce a low-order triangular element which is optimalorder convergent uniformly in the plate thickness.
Residual-based a posteriori error estimate for a mixed Reißner-Mindlin plate finite element method
Numerische Mathematik, 2006
Reliable and efficient residual-based a posteriori error estimates are established for the stabilised locking-free finite element methods for the Reissner-Mindlin plate model. The error is estimated by a computable error estimator from above and below up to multiplicative constants that do neither depend on the mesh-size nor on the plate's thickness and are uniform for a wide range of stabilisation parameter. The error is controlled in norms that are known to converge to zero in a quasi-optimal way. An adaptive algorithm is suggested and run for improving the convergence rates in three numerical examples for thicknesses 0.1, .001 and .001.
Mixed-interpolated elements for Reissner-Mindlin plates
International Journal for Numerical Methods in Engineering, 1989
We present in this paper a procedure to establish Reissner-Mindlin plate bending elements. The procedure is based on the idea to combine known resuits on the approximation of Stokes problems with known results on the approximation of elliptic problems. The proposed elements satisfy the mathematical conditions of stability and convergence, and some of them promise to provide efficient elements for practical solutions. *The abbreviation MITC4 is used to designate our clement based on 'mixed interpolated tensorial components with four nodes' and we use similar abbrcviations for the other elements we introduce
Analysis of a Linear–Linear Finite Element for the Reissner–Mindlin Plate Model
Mathematical Models and Methods in Applied Sciences, 1997
An analysis is presented for a recently proposed nite element method for the Reissner{ Mindlin plate problem. The method is based on the standard variational principle, uses nonconforming linear elements to approximate the rotations and conforming linear elements to approximate the transverse displacements, and avoids the usual \locking problem" by interpolating the shear stress into a rotated space of lowest order Raviart-Thomas elements. When the plate thickness t = O(h), it is proved that the method gives optimal order error estimates uniform in t. However, the analysis suggests and numerical calculations con rm that the method can produce poor approximations for moderate sized values of the plate thickness. Indeed, for t xed, the method does not converge as the mesh size h tends to zero.
Hierarchic and mixed-interpolated finite elements for Reissner-Mindlin problems
Communications in Numerical Methods in Engineering, 1995
The focus of the present work is directed towards the construction of a family of special finite elements for the numerical solution of the Reissner-Mindlin plate problem. The model describes the deformation of a plate when transverse shear deformation is taken into account. The model is widely used for thin to moderately thick plates. Despite its simple formulation, the numerical approximation is not straightforward. When standard low-order finite elements are used for the approximation the solution degenerates very rapidly for small thickness (locking phenomenon). To overcome such behaviour, nonstandard formulations of the problem are usually combined with low-order finite elements to weaken or possibly eliminate the locking of the numerical solution. In recent years high-order finite elements have been introduced and successfully applied in several fields. Previously we have constructed a family of hierarchic high-order finite elements to solve the Reissner-Mindlin problem in its plain formulation. The locking was strongly reduced but was still active for very thin plates. Meanwhile some mixed-interpolated finite elements have been developed and shown to be locking free. In this paper we combine the two approaches, namely, the hierarchic high-order elements and the mixed-interpolated elements. The result is a family of special finite elements that exhibits both properties of convergence and robustness.
Least-squares finite element approximations for the Reissner-Mindlin plate
Numerical Linear Algebra with Applications, 1999
Based on the Helmholtz decomposition of the transverse shear strain, Brezzi and Fortin in introduced a three-stage algorithm for approximating the Reissner-Mindlin plate model with clamped boundary conditions and established uniform error estimates in the plate thickness. The first and third stages involve approximating two simple Poisson equations and the second stage approximates a perturbed Stokes equation. Instead of using the mixed finite element method which is subject to the 'infsup' condition, we consider a least-squares finite element approximation to such a perturbed Stokes equation. By introducing a new independent vector variable and associated div equation, we are able to establish the ellipticity and continuity of the homogeneous least-squares functional in an H 1 product norm appropriately weighted by the thickness. This immediately yields optimal discretization error estimates for finite element spaces in this norm which are uniform in the thickness. We show that the resulting algebraic equations can be uniformly well preconditioned by wellknown techniques in the thickness. The Reissner-Mindlin model with pure traction boundary condition is also studied. Finally, we consider an alternative least-squares formulation for the perturbed Stokes equation by introducing an independent scalar variable.
A Refined Error Analysis of Mitc Plate Elements
Mathematical Models and Methods in Applied Sciences, 2006
We consider the Mixed Interpolated (Tensorial Components) finite element families for the Reissner–Mindlin plate model. For the case of a convex domain with clamped boundary conditions, we prove regularity results and derive new error estimates which are uniformly valid with respect to the thickness parameter.
Guaranteed Functional Error Estimates for the Reissner-Mindlin Plate Problem
Journal of Mathematical Sciences, 2006
We consider a class of explicitly computed majorants that provide accuracy estimates for a solution to the Reissner-Mindlin plate problem. The proof is based on a generalized statement of the problem. It does not require any extra analysis of properties of controlled approximate solutions is independent of features of the finite element method for obtaining approximate solutions. Bibliography: 13 titles. Illustrations: 1
Error estimates for low-order isoparametric quadrilateral finite elements for plates
This paper deals with the numerical approximation of the bending of a plate modeled by 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 based on the family of elements called MITC (mixed interpolation of tensorial components). We consider two lowest-order methods of this family on quadrilateral meshes.
On the uniform approximation of the Reissner-Mindlin plate model by p/hp finite element methods
Proceedings of the Fifth World Congress on Computational Mechanics (WCCM V), 2002
We study the approximation of the Reissner-Mindlin plate using the p/hp version of the finite element method (FEM). Our goal is to identify a method that: (i) is free of shear locking, (ii) approximates the boundary layer independently of the thickness of the plate and (iii) converges exponentially with respect to the number of degrees of freedom. We will consider both standard and reduced constraint/mixed formulations, in the context of the p/hp version of the FEM, and we will give guidelines for the construction of appropriate mesh-degree combinations that accomplish the above three goals, using straight as well as curved sided elements.
Analysis of a mixed finite element method for the Reissner-Mindlin plate problems
Computer Methods in Applied Mechanics and Engineering, 1998
An analysis of a triangular mixed finite element method, proposed by Taylor and Auricchio [13] is presented. The method is based on a linked interpolation between deflections and rotations in order to avoid the locking phenomenon (cf. ). The analysis shows that the approximated deflections and rotations are first order convergent to the exact solution, uniformly in the thickness.
Mathematics of Computation, 1999
We consider mixed finite element discretizations of linear second order elliptic boundary value problems with respect to an adaptively generated hierarchy of possibly highly nonuniform simplicial triangulations. In particular, we present and analyze four different kinds of error estimators: a residual based estimator, a hierarchical one, error estimators relying on the solution of local subproblems and on a superconvergence result, respectively. Finally, we examine the relationship between the presented error estimators and compare their local components.
Interpolation estimate for a finite-element space with embedded discontinuities
IMA Journal of Numerical Analysis, 2012
We consider a recently proposed finite-element space that consists of piecewise affine functions with discontinuities across a smooth given interface Γ (a curve in two dimensions, a surface in three dimensions). Contrary to existing extended finite element methodologies, the space is a variant of the standard conforming P 1 space that can be implemented element by element. Further, it neither introduces new unknowns nor deteriorates the sparsity structure. It is proved that, for u arbitrary in W 1, p (Ω \ Γ) ∩ W 2,s (Ω \ Γ), the interpolant I h u defined by this new space satisfies u − I h u L q (Ω) C h 1+ 1 q − 1 p |u| W 1, p (Ω\Γ) + h 2 |u| W 2,s (Ω\Γ) , where h is the mesh size, Ω ⊂ R d is the domain, p > d, p q, s q and standard notation has been adopted for the function spaces. This result proves the good approximation properties of the finiteelement space as compared to any space consisting of functions that are continuous across Γ , which would yield an error in the L q (Ω)-norm of order h 1 q − 1 p. These properties make this space especially attractive for approximating the pressure in problems with surface tension or other immersed interfaces that lead to discontinuities in the pressure field. Furthermore, the result still holds for interfaces that end within the domain, as happens for example in cracked domains.
Applied Numerical Mathematics, 2003
We consider the approximation of the Reissner–Mindlin plate model by the standard Galerkin p version finite element method. Under the assumption of sufficient smoothness on the solution, we illustrate that the method is asymptotically free of locking even when certain curvilinear elements are used. The amount of preasymptotic locking is also identified and is shown to depend on the element mappings. We identify which mappings will result in asymptotically locking free methods and through numerical computations we verify the results for various mappings used in practice.
Finite Elements for the One Variable Version of Mindlin-Reissner Plate
Latin American Journal of Solids and Structures, 2020
To analyze thin and thick plates, the paper presents two rectangular finite elements with high accuracy. In these elements, the proposed formulations of the displacement field utilize the Bergan-Wang approach, which depends only on one variable: the plate lateral deflection. This approach ensures that shear-locking problem will not happen as thickness decreases. The degrees of freedom of the proposed elements are twenty-four for the first element and it is named BWRE24, while the second one has thirty-six degrees of freedom and is named BWRE36. To evidence the efficiency of the two elements, a series of numerical examples for an isotropic plate subjected to various loadings and with different boundary conditions have been analyzed. Very good results are obtained suffering no numerical difficulties in case of very thin plates.
Reissner’s mixed variational theorem toward MITC finite elements for multilayered plates
Composite Structures, 2013
In this paper, we analyze a two dimensional model of multilayered plates for which the main interest is to study the mechanical response, that may change in the thickness direction. The finite element method showed successful performances to approximate the solutions of the advanced structures. In this regard, two variational formulations are available to reach the stiffness matrices, the principle of virtual displacement (PVD) and the Reissner mixed variational theorem (RMVT). Here we introduce a strategy similar to MITC (Mixed Interpolated of Tensorial Components) approach, in the RMVT formulation, in order to construct an advanced locking-free finite element. Assuming the transverse stresses as independent variables, the continuity at the interfaces between layers is easily imposed. It is known that unless the combination of finite element spaces for displacement and stresses is chosen carefully, the problem of locking is likely to occur. Following this suggestion, we propose a finite element scheme that it is known to be robust with respect to the locking phenomenon in the classical PVD approach. We show that in the RMVT context, the element exhibits both properties of convergence and robustness when comparing the numerical results with benchmark solutions from literature.
Recovery based error estimation for plate problems
Communications to SIMAI …, 2007
A posteriori error estimation is an important tool in finite element software development, since it allows to verify and validate the finite element simulations, as well as to improve results and control the error, when combined with adaptivity. An efficient and practical way to derive a posteriori error estimators is offered by recovery procedures. The error, generally in stress based norms, is estimated by comparing the original finite element solution with the recovered one. The major steps forward in using recovery procedures were made with the Superconvergent Patch Recovery (SPR) and the Recovery by Equilibrium in Patches (REP), both successfully applied to plate problems, see for example . Recently, a new superconvergent procedure called Recovery by Compatibility in Patches (RCP) has been proposed and shown to provide an excellent basis for error estimation in 2D problems [2,3]. The present paper aims at presenting an extension of the RCP-based error estimation to Reissner-Mindlin plates finite element analysis. The basic idea of the procedure is to recover stress resultants by enforcing compatibility over patches of elements. Displacements computed by the finite element analysis are prescribed on the boundary of the patch, and improved stress resultants are computed by minimizing the complementary energy of such a sub-model. The resulting procedure is simple, efficient, numerically stable and does not need any knowledge of superconvergent points. Its performance is evaluated in a numerical test, using both displacement elements and the new 9βQ4 hybrid stress elements recently proposed in . PLATE EQUATIONS AND FINITE ELEMENT ANALYSIS Consider a plate referred to a Cartesian reference frame (O, x, y, z) with the origin O on the mid-surface Ω and the z-axis in the thickness direction, −h/2 ≤ z ≤ h/2 where h is the thickness. Let ∂Ω be the boundary of Ω. The Reissner-Mindlin theory is employed. The compatibility equations are