Reissner Formulation Research Articles

A new structural model of warping torsion. II: Evaluation for a model problem

This paper presents an evaluation of the mixed formulation of warping torsion developed in the first part of this work. Specifically, we present a comparative study with two widely-used existing formulations, namely, the TWKV formulation of Timoshenko–Wagner–Kappus–Vlasov involving only the twist rotation (thus constraining the twist to the warping) and the RBV formulation of Reissner–Benscoter–Vlasov incorporating an additional warping field, as the newly proposed mixed formulation. However, in big contrast to those two formulations, the new formulation involves an alternative kinematic relation between the twist and warping motivated by a full mixed treatment of the axial strain–stress response. The evaluations are based on the exact closed-form solution for a model problem of a prismatic elastic shaft, with the torsional constants appearing in the different formulations computed with plane finite element approximations of the relevant warping functions on the cross section. The structural models are also compared with full 3D finite element simulations of the elastic solids. The new mixed formulation is found to improve upon the over-stiff constrained TWKV formulation as well as the RBV formulation with its physically incorrect stresses.

Dispersion analysis of displacement-based and TDNNS mixed finite elements for thin-walled elastodynamics

We compare several lowest-order finite element approximations to the problem of elastodynamics of thin-walled structures by means of dispersion analysis, which relates the parameter frequency-times-thickness (fd) and the wave speed. We restrict to analytical theory of harmonic front-crested waves that freely propagate in an infinite plate. Our study is formulated as a quasi-periodic eigenvalue problem on a single tensor–product element, which is eventually layered in the thickness direction. In the first part of the paper it is observed that the displacement-based finite elements align with the theory provided there are sufficiently many layers. In the second part we present novel anisotropic hexahedral tangential-displacement and normal–normal-stress continuous (TDNNS) mixed finite elements for Hellinger–Reissner formulation of elastodynamics. It turns out that one layer of such elements is sufficient for fd up to 2000[kHzmm]. Nevertheless, due to a large amount of TDNNS degrees of freedom the computational complexity is only comparable to the multi-layer displacement-based element. This is not the case at low frequencies, where TDNNS is by far more efficient since it allows for rough anisotropic discretizations, contrary to the displacement-based elements that suffer from the shear locking effect.

Parallel block preconditioners for three-dimensional virtual element discretizations of saddle-point problems

Several physical phenomena are described by systems of partial differential equations (PDEs) that, after space discretization, yield the solution of saddle point algebraic linear systems. In realistic three-dimensional numerical simulations, these linear systems are large scale and ill-conditioned, thus they require the development of effective solvers. The aim of this work is the construction and numerical validation of parallel block preconditioners for a set of three-dimensional saddle point problems discretized by the low order virtual element method (VEM). VEM is a recent numerical technology for the approximation of PDEs on polygonal and polyhedral meshes. We focus on the following systems of PDEs: stationary Maxwell equations in the mixed Kikuchi formulation; elliptic equations in mixed form; Stokes system; linear elasticity in the mixed Hellinger–Reissner formulation. We provide two parallel block preconditioners: one based on the approximate Schur complement and the other on a regularization technique. Several numerical experiments are run in parallel on a Linux cluster. We analyze the performance of the iterative solvers in terms of GMRES iterations and computational time. We verify the robustness of the solvers with respect to different polyhedral meshes and the scalability of both the assembling and solution time by varying the number of processors. The performance of the two iterative solvers is also compared with state-of-the-art parallel direct linear solvers.

Finite strain theories of extensible and shear-flexible planar beams based on three different hypotheses of member forces

This study intends to answer how nonlinear beam theories could be rigorously derived when based on three hypotheses: Haringx/Reissner, Engesser, and Ziegler. For this purpose, Reissner formulation is first summarized for an elastic beam segment undergoing large displacements and strains, and the nonlinear equations obtained are compactly converted into a non-dimensional form using two parameters of shear-flexibility and extensibility. After that, conjugate strain measures corresponding to axial and shear forces based on Engesser and Ziegler's assumptions are consistently derived, and the resulting governing equations are presented in a dimensionless form. Finally, nonlinear problems of extensible and shearable cantilever beams are solved using the 4th order Runge-Kutta method combined with a shooting method. Shear-deformation and extensibility effects are addressed through two examples showing large deflections and post-buckling behaviors of cantilever beams.

Virtual elements for a shear-deflection formulation of Reissner–Mindlin plates

We present a virtual element method for the Reissner–Mindlin plate bending problem which uses shear strain and deflection as discrete variables without the need of any reduction operator. The proposed method is conforming in $[H^{1}(\Omega )]^2 \times H^2(\Omega )$ and has the advantages of using general polygonal meshes and yielding a direct approximation of the shear strains. The rotations are then obtained by a simple postprocess from the shear strain and deflection. We prove convergence estimates with involved constants that are uniform in the thickness $t$ of the plate. Finally, we report numerical experiments which allow us to assess the performance of the method.

Reissner mixed variational theorem-based finite cylindrical layer methods for the static analysis of functionally graded piezoelectric circular hollow cylinders under electro-mechanical loads

ABSTRACTA unified formulation of Reissner's mixed variational theorem-based finite cylindrical layer methods is developed for the static analysis of simply-supported, multilayered functionally graded piezoelectric material (FGPM) circular hollow cylinders. The material properties of the cylinder are assumed to obey an exponent-law exponentially varying through the thickness coordinate of this. The trigonometric functions and Lagrange polynomials are used to interpolate the in-surface and thickness variations of the primary variables of each individual layer, respectively. The coupled electro-elastic effects on the static behaviors of multilayered FGPM cylinders are closely examined.

On displacement-based and mixed-variational equivalent single layer theories for modelling highly heterogeneous laminated beams

The flexural response of laminated composite and sandwich beams is analysed using the notion of modelling the transverse shear mechanics with an analogous mechanical system of springs in series combined with a system of springs in parallel. In this manner a zig-zag function is derived that accounts for the geometric and constitutive heterogeneity of the multi-layered beam similar to the zig-zag function in the refined zig-zag theory (RZT) developed by Tessler et al. (2007). Based on this insight a new equivalent single layer formulation is developed using the principle of virtual displacements. The theory overcomes the problem in the RZT framework of modelling laminates with Externally Weak Layers but is restricted to laminates with zero B-matrix terms. Second, the RZT zig-zag function is implemented in a third-order theory based on the Hellinger–Reissner mixed variational framework. The advantage of the Hellinger–Reissner formulation is that both in-plane and transverse stress fields are captured to within 1% of Pagano’s 3D elasticity solution without the need for additional stress recovery steps, even for highly heterogeneous laminates. A variant of the Hellinger–Reissner formulation with Murakami’s zig-zag function increases the percentage error by an order of magnitude for highly heterogeneous laminates. Corresponding formulations using the Reissner Mixed Variational Theory (RMVT) show that the independent model assumptions for transverse shear stresses in this theory may be highly inaccurate when the number of layers exceeds three. As a result, the RMVT formulations require extra post-processing steps to accurately capture the transverse stresses. Finally, the relative influence of the zig-zag effect on different laminates is quantified using two non-dimensional parameters.

Numerical model of elastic laminated glass beams under finite strain

Laminated glass structures are formed by stiff layers of glass connected with a compliant plastic interlayer. Due to their slenderness and heterogeneity, they exhibit a complex mechanical response that is difficult to capture by single-layer models even in the elastic range. The purpose of this paper is to introduce an efficient and reliable finite element approach to the simulation of the immediate response of laminated glass beams. It proceeds from a refined plate theory due to Mau (1973), as we treat each layer independently and enforce the compatibility by the Lagrange multipliers. At the layer level, we adopt the finite-strain shear deformable formulation of Reissner (1972) and the numerical framework by Ibrahimbegović and Frey (1993). The resulting system is solved by the Newton method with consistent linearization. By comparing the model predictions against available experimental data, analytical methods and two-dimensional finite element simulations, we demonstrate that the proposed formulation is reliable and provides accuracy comparable to the detailed two-dimensional finite element analyzes. As such, it offers a convenient basis to incorporate more refined constitutive description of the interlayer.

Convergence analysis of a new mixed finite element method for Biot's consolidation model

AbstractIn this article, we propose a mixed finite element method for the two‐dimensional Biot's consolidation model of poroelasticity. The new mixed formulation presented herein uses the total stress tensor and fluid flux as primary unknown variables as well as the displacement and pore pressure. This method is based on coupling two mixed finite element methods for each subproblem: the standard mixed finite element method for the flow subproblem and the Hellinger–Reissner formulation for the mechanical subproblem. Optimal a‐priori error estimates are proved for both semidiscrete and fully discrete problems when the Raviart–Thomas space for the flow problem and the Arnold–Winther space for the elasticity problem are used. In particular, optimality in the stress, displacement, and pressure has been proved in when the constrained‐specific storage coefficient is strictly positive and in the weaker norm when is nonnegative. We also present some of our numerical results.Copyright © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1189–1210, 2014

TANGENTIAL-DISPLACEMENT AND NORMAL–NORMAL-STRESS CONTINUOUS MIXED FINITE ELEMENTS FOR ELASTICITY

In this paper, we introduce new finite elements to approximate the Hellinger Reissner formulation of elasticity. The elements are the vector-valued tangential continuous Nédélec elements for the displacements, and symmetric tensor-valued, normal–normal continuous elements for the stresses. These elements do neither suffer from volume locking as the Poisson ratio approaches ½, nor suffer from shear locking when anisotropic elements are used for thin structures. We present the analysis of the new elements, discuss their implementation, and give numerical results.

Stable hp mixed finite elements based on the Hellinger–Reissner principle

In the Hellinger–Reissner formulation for linear elasticity, both the displacement u and the stress σ are taken as unknowns, giving rise to a saddle point problem. We present new pairings of quadrilateral ‘trunk’ finite element spaces for this method and prove stability (and optimality) in terms of both h and p. The effect of mesh shape regularity on the stability constant is explicitly tracked. Our results provide a theoretical basis for recent numerical experiments (in the context of a mixed p formulation for viscoelasticity) that showed these spaces worked well computationally.

Application of linear shallow shell theory of Reissner to frequency response of thin cylindrical panels with arbitrary lamination

A free vibration response of arbitrarily laminated – crafted with advanced fiber reinforced composite materials – thin and shallow cylindrical panels on rectangular planform with simply supported boundary conditions is analytically investigated. The thin shallow shell formulation of Reissner that follows the classical Kirchhoff–Love hypothesis of small displacements is considered to include arbitrarily laminated behaviors. The simply supported boundary condition having normal, tangential, and transverse displacements restrained at the edges is implemented. The solution functions in a form of mixed boundary continuous and discontinuous double Fourier series are assumed into the analytical solution formulation. The efficiency and accuracy of the solution methodology are determined by studying numerically the convergence behavior of frequencies for various parametric effects. The analytically obtained numerical results and mode shapes are compared with available finite element solutions, and the analytical results can be capitalized as base-line solutions for future comparisons.

On the local stability condition in the planar beam finite element

In standard finite element algorithms, the local stability conditions are not accounted for in the formulation of the tangent stiffness matrix. As a result, the loss of the local stability is not adequately related to the onset of the global instability. The phenomenon typically arises with material-type localizations, such as shear bands and plastic hinges. This paper addresses the problem in the context of the planar, finite-strain, rate-independent, materially non-linear beam theory, although the proposed technology is in principle not limited to beam structures. A weak formulation of Reissner's finite-strain beam theory is first presented, where the pseudocurvature of the deformed axis is the only unknown function. We further derive the local stability conditions for the large deformation case, and suggest various possible combinations of the interpolation and numerical integration schemes that trigger the simultaneous loss of the local and global instabilities of a statically determined beam. For practical applications, we advice on a procedure that uses a special numerical integration rule, where interpolation nodes and integration points are equal in number, but not in locations, except for the point of the local instability, where the interpolation node and the integration point coalesce. Provided that the point of instability is an end-point of the beam-a condition often met in engineering practice-the procedure simplifies substantially; one of such algorithms uses the combination of the Lagrangian interpolation and Lobatto's integration. The present paper uses the Galerkin finite element discretization, but a conceptually similar technology could be extended to other discretization methods.

Elasto-plastic analysis of Kirchhoff plates by high simplicity finite elements

This paper presents a mixed triangular finite element named High Simplicity (HS) element. The HS element is designed to analyze elasto-plastic Kirchhoff plates and is characterized by the linear assumption of the displacement field which makes it rigid in bending, and by the hypothesis of constant moments on the area surrounding each node. The additional hypothesis of continuity of the bending moment acting on the mesh sides allows the use of the standard Hellinger–Reissner formulation. The HS element is framed within an incremental iterative algorithm of initial stress type which uses arc-length strategy to reconstruct the whole equilibrium path. This introduces and highlights the advantages of the HS element which are its reasonable accuracy and very simple element algebra. Some numerical results relative to plates of simple form allow an analysis of the element's performance in the case of elastic-perfectly plastic behavior governed by Von Mises yield criterion.

Interaction of local and global nonlinearities of elastic rotating structures

Geometrically nonlinear effects like buckling due to large acceleration of lightweight and thin walled structures often play an important role in flexible multibody dynamics. To account for the interaction between local and global nonlinearities, a generalized method to model the motion of large flexible elastic structures is applied. The structure is decomposed into substructures, which are discretized using the finite element method. To reduce the number of degrees of freedom, a set of well-suited modes for the description of the displacement field on substructure level is used. The formulation introduced here combines the advantages of the corotational finite element formulation and the component mode synthesis method, resulting in a reduced set of degrees of freedom including local nonlinearities on substructure level. NVESTIGATIONS into the dynamic behavior of large structures undergoing finite displacements are expensive due to the large amount of computing time required. To reduce the numerical effort, several methods based on substructuring and natural mode decom- position of the displacement field have been developed.1'6'9'10) 12 Cur- rent applications are primarily limited to linear oscillation behavior of general elastic structures. In most cases nonlinearities are incor- porated only for beam-like structures having rigid cross sections. To consider both global (finite rotations) and local (for example, dynamic buckling) nonlinearities, the proposed procedure is based on the following steps: 1) Decomposition of the structure into elastic substructures and discretization of the substructures (for example, a robot manipula- tor) is accomplished using a finite element formulation applied to a shell theory of moderately large (linearized) rotations. To simplify the nonlinearity with respect to the deformation, the derivation of the Euler-Lagrange equations and their consistent linearization is performed by using a mixed Hellinger—Reissner formulation, where the nonlinearities are of second order. 2) Reduction of the number of degrees of freedom is achieved by means of component modes and the subsequent transformation of the inertia and nonlinear stiffness matrices into the reduced sub- space on substructure level. The component modes for the conden- sation satisfy the continuity between adjacent substructures using unit displacement modes and describe the dynamic behavior within the substructures by hand of internal vibration modes. To account for the local buckling phenomena, additional static modes are used, which are adapted to the nonlinear deformation behavior. 3) The reduced substructure matrices are embedded in a coro- tational formulation for handling the finite rotations of the overall structure.

Iterative Solution Methods for Plate Bending Problems: Multigrid and Preconditionedcgalgorithm

The numerical solution of the linear equations arising from a discretization of the plate bending problem based on the Zienkiewicz triangle is studied. The finite element scheme proceeds from the Mindlin–Reissner formulation with modified shear energy. However, the Kirchhoff condition is imposed on discrete points. A multigrid algorithm is analyzed and numerical examples are provided. Furthermore, a suitable preconditioning is established for the use of conjugate gradients.

Some thoughts on shell modelling for crash analysis

This paper presents a mixed triangular finite element named High Simplicity (HS) element. The HS element is designed to analyze elasto-plastic Kirchhoff plates and is characterized by the linear assumption of the displacement field which makes it rigid in bending, and by the hypothesis of constant moments on the area surrounding each node. The additional hypothesis of continuity of the bending moment acting on the mesh sides allows the use of the standard Hellinger–Reissner formulation. The HS element is framed within an incremental iterative algorithm of initial stress type which uses arc-length strategy to reconstruct the whole equilibrium path. This introduces and highlights the advantages of the HS element which are its reasonable accuracy and very simple element algebra. Some numerical results relative to plates of simple form allow an analysis of the element's performance in the case of elastic-perfectly plastic behavior governed by Von Mises yield criterion.

A new family of stable elements for nearly incompressible elasticity based on a mixed Petrov-Galerkin finite element formulation

Adding to the classical Hellinger Reissner formulation another residual form of the equilibrium equation, a new Petrov-Galerkin finite element method is derived. It fits within the framework of a mixed finite element method and is proved to be stable for rather general combinations of stress and displacement interpolations, including equal-order discontinuous stress and continuous displacement interpolations which are unstable within the Galerkin approach. Error estimates are presented using the Babuska-Brezzi theory and numerical results confirm these estimates as well as the good accuracy and stability of the method.

A mixed finite element family in plane elasticity

The mixed Reissner formulation in plane elasticity is presented. A family of triangular elements using continuous linear approximations for the displacements and the stress tensor is built. The Brezzi-Babuska condition is satisfied; the existence and unicity of the solution of the discretized problem is proved. An estimate of the error due to approximation is given.

Thick Plates on Elastic Foundations: One-Variable Formulation

Reissner formulated the problem of the bending of plates, including the effect of shear and transverse normal stress on the behavior of such plates. Frederick extended the Reissner formulation to the case of plates on elastic foundations and used a two .variable approach.