Reissner-Mindlin Plate Bending Research Articles

A Discontinuous Petrov–Galerkin Method for Reissner–Mindlin Plates

We present a discontinuous Petrov–Galerkin method with optimal test functions for the Reissner–Mindlin plate bending model. Our method is based on a variational formulation that utilizes a Helmholtz decomposition of the shear force. It produces approximations of the primitive variables and the bending moments. For any canonical selection of boundary conditions the method converges quasi-optimally. In the case of hard-clamped convex plates, we prove that the lowest-order scheme is locking free. Several numerical experiments confirm our results.

A discrete de Rham method for the Reissner–Mindlin plate bending problem on polygonal meshes

In this work we propose a discretisation method for the Reissner–Mindlin plate bending problem in primitive variables that supports general polygonal meshes and arbitrary order. The method is inspired by a two-dimensional discrete de Rham complex for which key commutation properties hold that enable the cancellation of the contribution to the error linked to the enforcement of the Kirchhoff constraint. Denoting by k≥0 the polynomial degree for the discrete spaces and by h the meshsize, we derive for the proposed method an error estimate in hk+1 for general k, as well as a locking-free error estimate for the lowest-order case k=0. The theoretical results are validated on a complete panel of numerical tests.

Brittle fracture on plates governed by topological derivatives

PurposeIn the paper an approach for crack nucleation and propagation phenomena in brittle plate structures is presented.Design/methodology/approachThe Francfort–Marigo damage theory is adapted to the Kirchhoff and Reissner–Mindlin plate bending models. Then, the topological derivative method is used to minimize the associated Francfort–Marigo shape functional. In particular, the whole damaging process is governed by a threshold approach based on the topological derivative field, leading to a notable simple algorithm.FindingsNumerical simulations are driven in order to verify the applicability of the proposed method in the context of brittle fracture modeling on plates. The obtained results reveal the capability of the method to determine nucleation and propagation including bifurcation of multiple cracks with a minimal number of user-defined algorithmic parameters.Originality/valueThis is the first work concerning brittle fracture modeling of plate structures based on the topological derivative method.

Notable highlights on locking-free techniques of Reissner-Mindlin plate finite elements in elastostatics

In this paper, we discuss about the notable locking-free techniques of several simple plate bending finite elements for the Reissner-Mindlin plate bending theory. The brief background for Reissner-Mindlin plate theory is presented, in which stress and strain derivation are given along with one-field and two-field variational approaches. Afterwards, we classify several efficient robust techniques in a cluster of main categories to present sequentially, which are all able to overcome the locking phenomenon in thick plate bending problems. Only selective algorithms are programmed to conduct numerical simulations. The corresponding results are compared between these elements to show their performances.

Topological Derivative-Based Topology Optimization of Plate Structures Under Bending Effects

In this work, the topological derivatives of L2 and energy norms associated with the solution to Kirchhoff and Reissner-Mindlin plate bending models are introduced. Based on existing theoretical results, closed forms of the sensitivities are presented. A zero-order term is introduced in the equilibrium equations, which allows for adapting the obtained sensitivities to the context of topology optimization of plates under elastic support and free vibration condition as well. The resulting analytical formulae are used together with a level-set domain representation method to devise a simple topology design algorithm. Several finite element-based representative numerical experiments are presented showing its applications to the compliance minimization and eigenvalue maximization of Kirchhoff as well as Reissner-Mindlin plate structures under bending effects.

An ultraweak formulation of the Reissner–Mindlin plate bending model and DPG approximation

We develop and analyze an ultraweak variational formulation of the Reissner–Mindlin plate bending model both for the clamped and the soft simply supported cases. We prove well-posedness of the formulation, uniformly with respect to the plate thickness t. We also prove weak convergence of the Reissner–Mindlin solution to the solution of the corresponding Kirchhoff–Love model when $$t\rightarrow 0$$. Based on the ultraweak formulation, we introduce a discretization of the discontinuous Petrov–Galerkin type with optimal test functions (DPG) and prove its uniform quasi-optimal convergence. Our theory covers the case of non-convex polygonal plates. A numerical experiment for some smooth model solutions with fixed load confirms that our scheme is locking free.

Local radial basis function collocation method for bending analyses of quasicrystal plates

• Meshless formulations are developed for static and dynamic bending of QC plates. • The phonon-phason coupling effects are studied numerically. • The influence of elastic foundation and variable plate thickness is investigated. • The response of QC plates to impact loading is simulated. The local radial basis function collocation method (LRBFCM) is proposed for plate bending analysis in orthorhombic quasicrystals (QCs) under static and transient dynamic loads. Three common types of the plate bending problems are considered: (1) QC plates resting on Winkler foundation (2) QC plates with variable thickness and (3) QC plates under a transient dynamic load. According to the Reissner–Mindlin plate bending theory, there is allowed to simulate the behavior of the two excitations in QC plates, phonon and phason, by 2D strong formulations for the system of governing equations. The governing equations, which describe the phason displacements, are based on Agiasofitou and Lazar elastodynamic model. Numerical results demonstrate the effect of the elastic foundation, as well as plate thickness on the phonon and phason characteristics in this paper. For the transient dynamic analysis, the influence of the phason friction coefficients on the responses of QC plate to transient dynamic loads is also studied.

ADVANCES IN SHAPE-FREE FINITE ELEMENT METHODS: A REVIEW

As the most important tool for simulation and computation, the finite element method has been widely applied in engineering and scientific problems. However, this powerful method still suffers from some inherent deficiencies, such as the sensitivity problem to mesh distortion and so on, which have not been completely solved yet. This paper systematacially introduces some research developments on new finite element methods, i.e., the shape-free finite element methods, including the hybrid stress-function elements for plane and 2D fracture problems, the hybrid displacement-function elements for Reissner-Mindlin plate bending problem, and new unsymmetric finite element for plane and 3D elasticity. Based on the existing hybrid stress and unsymmetric finite element methods, some advanced techniques, including the analytical trial function method, new natural coordinate method, generalized conforming technique, etc., were adopted in the aforementioned new approaches. The resulting new finite element models possess high precision and good robustness. In particular, they can keep their level of performance even in extremely distorted meshes. Furthermore, some historical challenges in the finite element method, such as the limitation defined by the MacNeal's theorem, the edge effect problem of the Resissner-Mindlin plate, were also successfully solved. At the end of this paper, some features of the new methods and further development are discussed.

Numerical analysis of bending and transverse shear properties of plain-weave fabric composite laminates considering intralaminar inhomogeneity

In this study, the bending and transverse shear properties of plain-weave fabric composite laminates are investigated by considering the intralaminar inhomogeneity through finite element analysis. Using homogenization procedures, the effective Reissner–Mindlin plate bending and transverse shear properties of composite laminates are calculated. Assuming the number of plies to be infinite, the effective three-dimensional (3D) continuum properties (elastic properties of the equivalent 3D bulk material) of the composite are also calculated. Then, the effect of the number of plies on the bending stiffness and transverse shear stiffness of the laminate, in other words, the effect of the intralaminar inhomogeneity on the laminate stiffness, is investigated. Through the numerical investigation, it is found that if the number of plies of the laminate is very small, the bending stiffness and transverse shear stiffness of the laminate are significantly lower than those evaluated using effective 3D continuum properties.

Computation of critical loads and buckling modes using hybrid-mixed stress finite element models

This work presents a hybrid mixed stress finite element model for the linear stability analysis of frame structures and Reissner–Mindlin plate bending problems. This model approximates the stress and displacement fields in the domain and the displacement field on the static boundary. Legendre polynomials are used to define all approximation bases, in order to compute analytical solutions for all structural operators. It enables also the implementation of highly efficient p-refinement procedures. A material linear behaviour is assumed and to validate the model and to illustrate its potential, numerical tests are presented and compared with analytical and numerical solutions.

Energy change to insertion of inclusions associated with the Reissner–Mindlin plate bending model

The topological derivative concept has been proved to be useful in many relevant applications such as topology optimization, inverse problems, image processing, multi-scale constitutive modeling, fracture mechanics and damage evolution modeling. The topological asymptotic analysis has been fully developed for a wide range of problems modeled by partial differential equations. On the other hand, the topological derivatives associated with coupled problems have been derived only in their abstract forms. In this paper, therefore, we deal with the Reissner–Mindlin plate bending model, which is written in the form of a coupled system of partial differential equations. In particular, the topological asymptotic analysis of the associated total potential energy is developed and the topological derivative with respect to the nucleation of a circular inclusion is derived in its closed form. Finally, we provide the estimates for the remainders of the topological asymptotic expansion and perform a complete mathematical justification for the derived formulas.

Analysis of the discontinuous Petrov–Galerkin method with optimal test functions for the Reissner–Mindlin plate bending model

We analyze the discontinuous Petrov–Galerkin (DPG) method with optimal test functions when applied to solve the Reissner–Mindlin model of plate bending. We prove that the hybrid variational formulation underlying the DPG method is well-posed (stable) with a thickness-dependent constant in a norm encompassing the L2-norms of the bending moment, the shear force, the transverse deflection and the rotation vector. We then construct a numerical solution scheme based on quadrilateral scalar and vector finite elements of degree p. We show that for affine meshes the discretization inherits the stability of the continuous formulation provided that the optimal test functions are approximated by polynomials of degree p+3. We prove a theoretical error estimate in terms of the mesh size h and polynomial degree p and demonstrate numerical convergence on affine as well as non-affine mesh sequences.

An enhanced cell‐based smoothed finite element method for the analysis of Reissner–Mindlin plate bending problems involving distorted mesh

SUMMARYIn this work, an enhanced cell‐based smoothed finite element method (FEM) is presented for the Reissner–Mindlin plate bending analysis. The smoothed curvature computed by a boundary integral along the boundaries of smoothing cells in original smoothed FEM is reformulated, and the relationship between the original approach and the present method in curvature smoothing is established. To improve the accuracy of shear strain in a distorted mesh, we span the shear strain space over the adjacent element. This is performed by employing an edge‐based smoothing technique through a simple area‐weighted smoothing procedure on MITC4 assumed shear strain field. A three‐field variational principle is utilized to develop the mixed formulation. The resultant element formulation is further reduced to a displacement‐based formulation via an assumed strain method defined by the edge‐smoothing technique. As the result, a new formulation consisting of smoothed curvature and smoothed shear strain interpolated by the standard transverse displacement/rotation fields and smoothing operators can be shown to improve the solution accuracy in cell‐based smoothed FEM for Reissner–Mindlin plate bending analysis. Several numerical examples are presented to demonstrate the accuracy of the proposed formulation.Copyright © 2013 John Wiley & Sons, Ltd.

Natural element approximation of Reissner–Mindlin plate for locking-free numerical analysis of plate-like thin elastic structures

A rotation-free mixed natural element approximation of Reissner–Mindlin plate for the locking-free numerical analysis of plate-like thin elastic structures is presented, motivated by the rotation-free approximation for plate- and shell-like elastic structures and the high smoothness of Laplace interpolation functions used in natural element method. The mid-surface deflection and the rotation of shear are directly approximated using Laplace interpolation functions, while the Voronoi polygon-wise constant curvatures and bending moments are indirectly computed by area-averaging the boundary integrals of deflection derivatives and rotation of shear. The present approximation of the deflection and rotation of shear of discretized Reissner–Mindlin plate bending problem is formulated according to the modified mixed Hu-Washizu principle. Through the numerical results, it is verified that the rotation-free mixed natural element approximation of only the deflection and rotation of shear successfully prevents shear locking in the numerical analysis of plate-like thin elastic structures.

Reissner–Mindlin Legendre spectral finite elements with mixed reduced quadrature

Legendre spectral finite elements (LSFEs) are examined through numerical experiments for static and dynamic Reissner–Mindlin plate bending and a mixed-quadrature scheme is proposed. LSFEs are high-order Lagrangian-interpolant finite elements with nodes located at the Gauss–Lobatto–Legendre quadrature points. Solutions on unstructured meshes are examined in terms of accuracy as a function of the number of model nodes and total operations. While nodal-quadrature LSFEs have been shown elsewhere to be free of shear locking on structured grids, locking is demonstrated here on unstructured grids. LSFEs with mixed quadrature are, however, locking free and are significantly more accurate than low-order finite-elements for a given model size or total computation time.

Structural dynamic analysis using hybrid and mixed finite element models

This paper presents and discusses a hybrid-mixed stress finite element model for the dynamic analysis of structures, assuming a physically and geometrically linear behavior. In this model, both the stress and the displacement fields are approximated in the domain of each element. The displacements along the static boundary are also independently approximated. Orthonormal Legendre polynomials are used as approximation functions. The use of these functions enables the use of analytical closed form solutions for the computation of all structural operators and leads to the development of very effective p-refinement procedures. The linear dynamic analysis is performed using time integration procedures. For this purpose, the classical Newmark, Wilson-θ and α-HHT methods are implemented and tested. A recent and promising alternative approach, based on the definition of a mixed approximation in the time domain is also implemented and assessed. The model being discussed is applied to the solution of plane elasticity and Reissner–Mindlin plate bending problems. To validate the model, to illustrate its potential and to assess its accuracy and efficiency, several numerical examples are discussed and comparisons are made with analytical solutions and solutions provided by other numerical techniques.

Two efficient hybrid-trefftz elements for plate bending analysis

This study is devoted to the analysis of the Reissner-Mindlin plate bending. In this paper, the hybrid-Trefftz strategy will be utilized. Two novel and efficient elements are formulated in details. They are a Triangular element (THT) and a quadrilateral element (QHT), which have 9 and 12 degrees of freedom, respectively. In this approach, two independent displacement fields are defined; one within the element and the other on the edges of the element. The internal field is selected in such a manner that the governing equation of thick plates could be satisfied. Boundary field is related to the nodal degree of freedoms by the boundary interpolation functions. To calculate these functions, the edges of the element are assumed to behave like a Timoshenko beam. The high accuracy and efficiency of the proposed elements and absence of the shear locking in these formulations are all proven, using various numerical tests.

An isogeometric method for the Reissner–Mindlin plate bending problem

We present a new isogeometric method for the discretization of the Reissner–Mindlin plate bending problem. The proposed scheme follows a recent theoretical framework that makes possible the construction of a space of smooth discrete deflections Wh and a space of smooth discrete rotations Θh such that the Kirchhoff constraint is exactly satisfied at the limit. Therefore we obtain a formulation which is natural from the theoretical/mechanical viewpoint and locking-free by construction. We prove that the method is uniformly stable and satisfies optimal convergence estimates. Finally, the theoretical results are fully supported by numerical tests.

A mimetic discretization of the Reissner–Mindlin plate bending problem

We present a mimetic approximation of the Reissner–Mindlin plate bending problem which uses deflections and rotations as discrete variables. The method applies to very general polygonal meshes, even with non matching or non convex elements. We prove linear convergence for the method uniformly in the plate thickness.

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.