Reissner-Mindlin Plate Problem Research Articles

A Reissner–Mindlin plate formulation using symmetric Hu-Zhang elements via polytopal transformations

In this work we develop new finite element discretisations of the shear-deformable Reissner–Mindlin plate problem based on the Hellinger–Reissner principle of symmetric stresses. Specifically, we use conforming Hu-Zhang elements to discretise the bending moments in the space of symmetric square integrable fields with a square integrable divergence M∈HZ⊂Hsym(Div). The latter results in highly accurate approximations of the bending moments M and in the rotation field being in the Lebesgue space ϕ∈[L2]2, such that the Kirchhoff–Love constraint can be satisfied for t→0. In order to preserve optimal convergence rates across all variables for the case t→0, we present an extension of the formulation using Raviart–Thomas elements for the shear stress q∈RT⊂H(div).We prove existence and uniqueness in the continuous setting and rely on exact complexes for inheritance of well-posedness in the discrete setting.This work introduces an efficient construction of the Hu-Zhang base functions on the reference element via the polytopal template methodology and Legendre polynomials, making it applicable to hp-FEM. The base functions on the reference element are then mapped to the physical element using novel polytopal transformations, which are suitable also for curved geometries.The robustness of the formulations and the construction of the Hu-Zhang element are tested for shear-locking, curved geometries and an L-shaped domain with a singularity in the bending moments M. Further, we compare the performance of the novel formulations with the primal-, MITC- and recently introduced TDNNS methods.

A New Approach to Reduce Transverse Shear Locking of Reissner–Mindlin Plate Based on The Cell-Centered Finite Element Method

This study investigates a new approach to solve a shear locking free on a low-order cell-centered finite element method (FECC) of the Reissner–Mindlin plate. The proposed method pursues the Lagrange piecewise of linear polynomial functions to attain the transverse displacement and rotations on the dual sub-mesh derived from the general one. The suggested theory is employed in numerical computation to verify the effectiveness and accurate performance. Detailed studies are carried out on several mesh types to compare with the analytical solutions reported by the low-order mimetic finite difference method [Beir ao Da Veiga, L., Lovadina, C. and Mora, D. [2013] “Numerical results for mimetic discretization of Reissner–Mindlin plate problems,” Calcolo 50, 209–237] and high-order mixed finite element methods [Chinosi, C. and Lovadina, C. [1995] “Numerical analysis of some mixed finite element methods for Reissner–Mindlin plates,” Comput. Mech. 16, 36–44]. The numerical results indicated that the present scheme based on the low-order FECC approximation can effectively predict on even the distorted meshes, satisfy the optimized convergence and be uniformly stable to the thin plate thicknesses.

A variational multiscale stabilized and locking-free meshfree formulation for Reissner–Mindlin plate problems

A variational multiscale stabilized and locking-free meshfree formulation for Reissner–Mindlin plate problems

A Robust HDG Method for Reissner-Mindlin Plate Problem

We propose and analyze a hybridizable discontinuous Galerkin (HDG) method for Reissner-Mindlin plate problems. The method uses piecewise-polynomials of degree k(≥ 1) to approximate the transverse displacement and the displacement trace on inter-element boundaries, uses piecewise-polynomial vectors of degrees k and ‘, max(1; k - 1) ≤ ‘ ≤ k, to approximate respectively the rotation and the rotation trace on inter-element boundaries, and uses piecewise-polynomial vectors of degree k and piecewise-polynomial tensors of degree m, k - 1 ≤ m ≤ ‘, to approximate respectively the shear stress and the bending moment. We show that the HDG method is robust in the sense that the derived a priori error estimates are optimal and uniform with respect to the plane thickness t. Numerical experiments are performed to confirm our theoretical results.

Smoothing Technique Based Beta FEM (βFEM) for Static and Free Vibration Analyses of Reissner–Mindlin Plates

Recently, the edge-based and node-based smoothed finite element method (ES-FEM and NS-FEM) has been proposed for Reissner–Mindlin plate problems. In this work, in order to utilize the numerical advantages of both ES-FEM and NS-FEM for static and vibration analysis, a hybrid smoothing technique based beta FEM ([Formula: see text]FEM) is presented for Reissner–Mindlin plate problems. A tunable parameter [Formula: see text] is introduced to tune the proportion of smoothing domains calculated by ES-FEM or NS-FEM, which controlled the accuracy of the results. Numerical illustrations in both static and free vibration analysis are conducted. The shear locking free property, converge property and dynamic stability are carefully examined via several well-known benchmark examples. Moreover, an experimental test is carefully designed and conducted for validations, in which the mode values and shape of a rectangular steel plate is tested. Numerical examples demonstrate the advantages of [Formula: see text]FEM, in comparison with the standard FEM, ES-FEM and NS-FEM using the same meshes. The numerical and experimental results are in good agreement with each other and the [Formula: see text]FEM achieves the best accuracy among all the methods for the static or free vibration analysis of plates.

Fracture analysis in plane structures with the two-scale G/XFEM method

Generalized or extended finite element method (G/XFEM) uses enrichment functions that holds a priori knowledge about the problem solution. These enrichment functions are mostly limited to two-dimensional problems. A well-established solution for problems without any specific types of analytically derived enrichment functions is to use numerically-build functions in which they are called global-local enrichment functions. These functions are extracted from the solution of boundary value problems defined around the region of interest discretized by a fine mesh. Such solution is used to enrich the global solution space through the partition of unity framework of the G/XFEM. Here it is presented a two-scale/global-local G/XFEM approach to model crack propagation in plane stress/strain and Reissner–Mindlin plate problems. Discontinuous functions along with the asymptotic crack-tip displacement fields are used to represent the crack without explicitly represent its geometries. Under the linear elastic fracture mechanics approach, the stress intensity factor (obtained from a domain-based interaction energy integral) can be used to either determine the crack propagation direction or propagation status, i.e., the crack can start to propagate or not. The proposed approach is presented in detail and validated by solving several linear elastic fracture mechanics problems for both plane stress/strain and Reissner–Mindlin plate cases to demonstrate its the robustness and accuracy.

A volume-averaged nodal projection method for the Reissner–Mindlin plate model

We introduce a novel meshfree Galerkin method for the solution of Reissner–Mindlin plate problems that is written in terms of the primitive variables only (i.e., rotations and transverse displacement) and is devoid of shear-locking. The proposed approach uses linear maximum-entropy basis functions for field variables approximation and is built variationally on a two-field potential energy functional wherein the shear strain, written in terms of the primitive variables, is computed via a volume-averaged nodal projection operator that is constructed from the Kirchhoff constraint of the three-field mixed weak form. The meshfree approximation is constructed over a set of scattered nodes that are obtained from an integration mesh of three-node triangles on which the meshfree stiffness matrix and nodal force vector are numerically integrated. The stability of the method is rendered by adding bubble-like enrichment to the rotation degrees of freedom. Some benchmark problems are presented to demonstrate the accuracy and performance of the proposed method for a wide range of plate thicknesses.

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.

Numerical analysis of a main crack interactions with micro-defects/inhomogeneities using two-scale generalized/extended finite element method

Generalized or extended finite element method (G/XFEM) models the crack by enriching functions of partition of unity type with discontinuous functions that represent well the physical behavior of the problem. However, this enrichment functions are not available for all problem types. Thus, one can use numerically-built (global-local) enrichment functions to have a better approximate procedure. This paper investigates the effects of micro-defects/inhomogeneities on a main crack behavior by modeling the micro-defects/inhomogeneities in the local problem using a two-scale G/XFEM. The global-local enrichment functions are influenced by the micro-defects/inhomogeneities from the local problem and thus change the approximate solution of the global problem with the main crack. This approach is presented in detail by solving three different linear elastic fracture mechanics problems for different cases: two plane stress and a Reissner–Mindlin plate problems. The numerical results obtained with the two-scale G/XFEM are compared with the reference solutions from the analytical, numerical solution using standard G/XFEM method and ABAQUS as well, and from the literature.

A generalized probabilistic edge-based smoothed finite element method for elastostatic analysis of Reissner–Mindlin plates

Probabilistic analysis is becoming more important in mechanical science and real-world engineering applications. In this work, a novel generalized stochastic edge-based smoothed finite element method is proposed for Reissner–Mindlin plate problems. The edge-based smoothing technique is applied in the standard FEM to soften the over-stiff behavior of Reissner–Mindlin plate system, aiming to improve the accuracy of predictions for deterministic response. Then, the generalized nth order stochastic perturbation technique is incorporated with the edge-based S-FEM to formulate a generalized probabilistic ES-FEM framework (GP_ES-FEM). Based upon a general order Taylor expansion with random variables of input, it is able to determine higher order probabilistic moments and characteristics of the response of Reissner–Mindlin plates. The significant feature of the proposed approach is that it not only improves the numerical accuracy of deterministic output quantities with respect to a given random variable, but also overcomes the inherent drawbacks of conventional second-order perturbation approach, which is satisfactory only for small coefficients of variation of the stochastic input field. Two numerical examples for static analysis of Reissner–Mindlin plates are presented and verified by Monte Carlo simulations to demonstrate the effectiveness of the present method.

A polygonal finite element method for plate analysis

A Reissner-Mindlin plate formulation on arbitrary polygonal meshes is proposed for plate analysis. We consider four barycentric shape function types named Wachspress, mean-value, Laplace and piecewise-linear and show its properties in numerical computation for Reissner-Mindlin plate problems. We then generalize an assumed strain field along sides of polygons under the enforcement of the Timoshenko’s beam assumption. The present approach is numerically verified by the bending patch test. It offers a general yet simple form for Reissner-Mindlin plate elements that is not only implementable for arbitrary polygonal meshes but also avoids transverse shear locking phenomenon at thin plate limit. The performance of the proposed elements is found through numerical examples. Our key contribution to this work is that the present formulation is established in a compact (or unified) form which is valid for triangular, quadrilateral and arbitrary polygonal meshes.

Isogeometric collocation methods for the Reissner–Mindlin plate problem

Within the general framework of isogeometric methods, collocation schemes have been recently proposed as a viable and promising low-cost alternative to standard isogeometric Galerkin approaches. In this paper, isogeometric collocation methods for the numerical approximation of Reissner–Mindlin plate problems are proposed for the first time. Locking-free primal and mixed formulations are herein considered, and the potential of isogeometric collocation as a geometrically flexible and computationally efficient simulation tool for shear deformable plates is shown through the solution of several numerical tests.

Numerical results for mimetic discretization of Reissner–Mindlin plate problems

A low-order mimetic finite difference method for Reissner–Mindlin plate problems is considered. Together with the source problem, the free vibration and the buckling problems are investigated. Details about the scheme implementation are provided, and the numerical results on several different types of meshes are reported.

A Priori and a Posteriori Error Analysis of the Discontinuous Galerkin Methods for Reissner-Mindlin Plates

AbstractIn this paper, we apply an a posteriori error control theory that we develop in a very recent paper to three families of the discontinuous Galerkin methods for the Reissner-Mindlin plate problem. We derive robust a posteriori error estimators for them and prove their reliability and efficiency.

Robust BDDC Preconditioners for Reissner–Mindlin Plate Bending Problems and MITC Elements

A Balancing Domain Decomposition Method by Constraints (BDDC) is constructed and analyzed for the Reissner–Mindlin plate bending problem discretized with Mixed Interpolation of Tensorial Components (MITC) finite elements. This BDDC algorithm is based on selecting the plate rotations and deflection degrees of freedom at the subdomain vertices as primal continuity constraints. After the implicit elimination of the interior degrees of freedom in each subdomain, the resulting plate Schur complement is solved by the preconditioned conjugate gradient method. The preconditioner is based on the solution of local Reissner–Mindlin plate problems on each subdomain with clamping conditions at the primal degrees of freedom and on the solution of a coarse Reissner–Mindlin plate problem for the primal degrees of freedom. The main results of the paper are the proof and numerical verification that the proposed BDDC plate algorithm is scalable, quasi-optimal, and, most important, robust with respect to the plate thickness. While this result is due to an underlying mixed formulation of the problem, both the interface plate problem and the preconditioner are positive definite. The numerical results also show that the proposed algorithm is robust with respect to discontinuities of the material properties.

Analysis for quadrilateral MITC elements for the Reissner-Mindlin plate problem

The present paper is made up of two parts. In the first part, we study the mathematical stability and convergence of the quadrilateral MITC elements for the Reissner-Mindlin plate problem in an abstract setting. We generalize the Brezzi-Bathe-Fortin conditions to the quadrilateral MITC elements by weakening the second and fourth conditions. Under these conditions, we show the well-posedness of the discrete problem and establish an abstract error estimate in the energy norm. The conclusion of this part is sparsity in the mathematical research of the quadrilateral MITC elements in the sense that one only needs to check these five conditions. In the second part, we extend four families of rectangular MITC elements of Stenberg and Suri to the quadrilateral meshes. We prove that these quadrilateral elements satisfy the generalized Brezzi-Bathe-Fortin conditions from the first part. We develop the h-p error estimates in both energy and L 2 norm for these quadrilateral elements. For the first three families of quadrilateral elements, the error estimates indicate that their convergent rates in both energy and L 2 norm depend on the mesh distortion parameter a. We can get optimal error estimates for them provided that a = 1. In addition, we show the optimal convergence rates in energy norm uniformly in a for the fourth family of quadrilateral elements. Like their rectangular counterparts, these quadrilateral elements are locking-free.

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 t. The presented analysis applies to all conforming MITC elements and all conforming finite element methods without reduced integration from the literature.

Error analysis of quadrilateral Wilson element for Reissner–Mindlin plate

In this paper, we generalize the rectangular nonconforming Wilson element method proposed in [Z. Zhang, S. Zhang, Wilson element for the Reissner–Mindlin plate,Comput. Methods Appl. Mech. Engrg. 113 (1994) 55–65] for the Reissner–Mindlin plate problem to the general quadrilateral mesh and analyze the error. It is proved that this method converges at uniformly optimal rates with respect to both the energy and L2 norms. These estimates improve those of [Z. Zhang, S. Zhang, Wilson element for the Reissner–Mindlin plate, Comput. Methods Appl. Mech. Engrg. 113 (1994) 55–65] in the sense that the requirement of H3(Ω) regularity on the solution is dropped, and that the L2 error estimate of this scheme is analyzed. The numerical examples at the end of this paper demonstrate the superiority of this method over the MITC4 [K.J. Bathe, E. Dvorkin, A four-node plate bending element based on Mindlin–Reissner plate theory and a mixed interpolation, Int. J. Numer. Methods Engrg. 21 (1985) 367–383].

Locking-free Reissner–Mindlin elements without reduced integration

In a recent paper of Arnold et al. [D.N. Arnold, F. Brezzi, L.D. Marini, A family of discontinuous Galerkin finite elements for the Reissner–Mindlin plate, J. Sci. Comput. 22 (2005) 25–45], the ideas of discontinuous Galerkin methods were used to obtain and analyze two new families of locking free finite element methods for the approximation of the Reissner–Mindlin plate problem. By following their basic approach, but making different choices of finite element spaces, we develop and analyze other families of locking free finite elements that eliminate the need for the introduction of a reduction operator, which has been a central feature of many locking-free methods. For k ⩾ 2 , all the methods use piecewise polynomials of degree k to approximate the transverse displacement and (possibly subsets) of piecewise polynomials of degree k − 1 to approximate both the rotation and shear stress vectors. The approximation spaces for the rotation and the shear stress are always identical. The methods vary in the amount of interelement continuity required. In terms of smallest number of degrees of freedom, the simplest method approximates the transverse displacement with continuous, piecewise quadratics and both the rotation and shear stress with rotated linear Brezzi–Douglas–Marini elements.

On the evaluation of the method of finite spheres for the solution of Reissner–Mindlin plate problems using the numerical inf–sup test

AbstractIn this paper we use the numerical inf–sup test to evaluate both displacement‐based and mixed discretization schemes for the solution of Reissner–Mindlin plate problems using the meshfree method of finite spheres. While an analytical proof of whether a discretization scheme passes the inf–sup condition is most desirable, such a proof is usually out of reach due to the complexity of the meshfree approximation spaces involved. The numerical inf–sup test (Int. J. Numer. Meth. Engng 1997; 40:3639–3663), developed to test finite element discretization spaces, has therefore been adopted in this paper. Tests have been performed for both regular and irregular nodal configurations. While, like linear finite elements, pure displacement‐based approximation spaces with linear consistency do not pass the inf–sup test and exhibit shear locking, quadratic discretizations, unlike quadratic finite elements, pass the test. Pure displacement‐based and mixed approximation spaces that pass the numerical inf–sup test exhibit optimal or near optimal convergence behaviour. Copyright © 2007 John Wiley & Sons, Ltd.