An edge-based smoothed finite element method (ES-FEM) with stabilized discrete shear gap technique for analysis of Reissner–Mindlin plates

An edge-based smoothed finite element method (ES-FEM) for static, free vibration and buckling analyses of Reissner–Mindlin plates using 3-node triangular elements is studied in this paper. The calculation of the system stiffness matrix is performed by using the strain smoothing technique over the smoothing domains associated with edges of elements. In order to avoid the transverse shear locking and to improve the accuracy of the present formulation, the ES-FEM is incorporated with the discrete shear gap (DSG) method together with a stabilization technique to give a so-called edge-based smoothed stabilized discrete shear gap method (ES-DSG). The numerical examples demonstrated that the present ES-DSG method is free of shear locking and achieves the high accuracy compared to the exact solutions and others existing elements in the literature.     

A finite element method with discontinuous rotations for the Mindlin–Reissner plate model

We present a continuous-discontinuous finite element method for the Mindlin–Reissner plate model based on continuous polynomials of degree k ? 2 for the transverse displacements and discontinuous polynomials of degree k ? 1 for the rotations. We prove a priori convergence estimates, uniformly in the thickness of the plate, and thus show that locking is avoided. We also derive a posteriori error estimates based on duality, together with corresponding adaptive procedures for controlling linear functionals of the error. Finally, we present some numerical results.     

A priori and a posteriori analysis for a locking-free low order quadrilateral hybrid finite element for Reissner–Mindlin plates

This paper proposes a quadrilateral finite element method of the lowest order for Reissner–Mindlin (R–M) plates on the basis of Hellinger–Reissner variational principle, which includes variables of displacements, shear stresses and bending moments. This method uses continuous piecewise isoparametric bilinear interpolation for the approximation of transverse displacement and rotation. The piecewise-independent shear stress/bending moment approximation is constructed by following a self-equilibrium criterion and a shear-stress-enhanced condition. A priori and reliable a posteriori error estimates are derived and shown to be uniform with respect to the plate thickness t. Numerical experiments confirm the theoretical results.     

An improved Reissner–Mindlin triangular element

A modified version of the low-order mixed finite element method proposed by Oñate et al. [Internat. J. Numer. Method. Engng. 37 (1994) 2569] for the Reissner–Mindlin plate model is analyzed. It is proved that the new method is optimal convergent, uniform in the thickness of the plate.     

Shape-free polygonal hybrid displacement-function element method for analyses of Mindlin–Reissner plates

A high-performance shape-free polygonal hybrid displacement-function finite-element method is proposed for analyses of Mindlin–Reissner plates. The analytical solutions of displacement functions are employed to construct element resultant fields, and the three-node Timoshenko’s beam formulae are adopted to simulate the boundary displacements. Then, the element stiffness matrix is obtained by the modified principle of minimum complementary energy. With a simple division, the integration of all the necessary matrices can be performed within polygonal element region. Five new polygonal plate elements containing a mid-side node on each element edge are developed, in which element HDF-PE is for general case, while the other four, HDF-PE-SS1, HDF-PE-Free, IHDF-PE-SS1, and IHDF-PE-Free, are for the edge effects at different boundary types. Furthermore, the shapes of these new elements are quite free, i.e., there is almost no limitation on the element shape and the number of element sides. Numerical examples show that the new elements are insensitive to mesh distortions, possess excellent and much better performance and flexibility in dealing with challenging problems with edge effects, complicated loading, and material distributions.

     

A stabilized MITC element for accurate wave response in Reissner–Mindlin plates

Residual based finite element methods are developed for accurate time-harmonic wave response of the Reissner–Mindlin plate model. The methods are obtained by appending a generalized least-squares term to the mixed variational form for the finite element approximation. Through judicious selection of the design parameters inherent in the least-squares modification, this formulation provides a consistent and general framework for enhancing the wave accuracy of mixed plate elements. In this paper, the mixed interpolation technique of the well-established MITC4 element is used to develop a new mixed least-squares (MLS4) four-node quadrilateral plate element with improved wave accuracy. Complex wave number dispersion analysis is used to design optimal mesh parameters, which for a given wave angle, match both propagating and evanescent analytical wave numbers for Reissner–Mindlin plates. Numerical results demonstrates the significantly improved accuracy of the new MLS4 plate element compared to the underlying MITC4 element.     

A Galerkin finite element scheme for time–space fractional diffusion equation

In this paper, a Galerkin finite element scheme to approximate the time–space fractional diffusion equation is studied. Firstly, the fractional diffusion equation is transformed into a fractional Volterra integro-differential equation. And a second-order fractional trapezoidal formula is used to approach the time fractional integral. Then a Galerkin finite element method is introduced in space direction, where the semi-discretization scheme and fully discrete scheme are given separately. The stability analysis of semi-discretization scheme is discussed in detail. Furthermore, convergence analysis of semi-discretization scheme and fully discrete scheme are given in details. Finally, two numerical examples are displayed to demonstrate the effectiveness of the proposed method.     

A posteriori error estimates for combined finite volume–finite element discretizations of reactive transport equations on nonmatching grids

We derive in this paper guaranteed and fully computable a posteriori error estimates for vertex-centered finite-volume-type discretizations of transient linear convection–diffusion–reaction equations. Our estimates enable actual control of the error measured either in the energy norm or in the energy norm augmented by a dual norm of the skew-symmetric part of the differential operator. Lower bounds, global-in-space but local-in-time, are also derived. These lower bounds are fully robust with respect to convection or reaction dominance and the final simulation time in the augmented norm setting. On the basis of the derived estimates, we propose an adaptive algorithm which enables to automatically achieve a user-given relative precision. This algorithm also leads to efficient calculations as it balances the time and space error contributions. As an example, we apply our estimates to the combined finite volume–finite element scheme, including such features as use of mass lumping for the time evolution or reaction terms, of upwind weighting for the convection term, and discretization on nonmatching meshes possibly containing nonconvex and non-star-shaped elements. A collection of numerical experiments illustrates the efficiency of our estimates and the use of the space–time adaptive algorithm.     

Refinement of flexible space–time finite element meshes and discontinuous Galerkin methods

In this paper we present an algorithm to refine space–time finite element meshes as needed for the numerical solution of parabolic initial boundary value problems. The approach is based on a decomposition of the space–time cylinder into finite elements, which also allows a rather general and flexible discretization in time. This also includes adaptive finite element meshes which move in time. For the handling of three-dimensional spatial domains, and therefore of a four-dimensional space–time cylinder, we describe a refinement strategy to decompose pentatopes into smaller ones. For the discretization of the initial boundary value problem we use an interior penalty Galerkin approach in space, and an upwind technique in time. A numerical example for the transient heat equation confirms the order of convergence as expected from the theory. First numerical results for the transient Navier–Stokes equations and for an adaptive mesh moving in time underline the applicability and flexibility of the presented approach.     

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.     

Weighted least-squares finite element methods for the linearized Navier–Stokes equations

We implemented weighted least-squares finite element methods for the linearized Navier-Stokes equations based on the velocity–pressure–stress and the velocity–vorticity–pressure formulations. The least-squares functionals involve the L²-norms of the residuals of each equation multiplied by the appropriate weighting functions. The weights included a mass conservation constant, a mesh-dependent weight, a nonlinear weighting function, and Reynolds numbers. A feature of this approach is that the linearized system creates a symmetric and positive-definite linear algebra problem at each Newton iteration. We can prove that least-squares approximations converge with the linearized version solutions of the Navier–Stokes equations at the optimal convergence rate. Model problems considered in this study were the flow past a planar channel and 4-to-1 contraction problems. We presented approximate solutions of the Navier–Stokes problems by solving a sequence of the linearized Navier–Stokes problems arising from Newton iterations, revealing the convergence rates of the proposed schemes, and investigated Reynolds number effects.     

Local and parallel finite element methods for the mixed Navier–Stokes/Darcy model

In this paper, the mixed Navier–Stokes/Darcy problem which describes a fluid flow filtrating through porous media is considered. Based on two-grid discretizations, two local and parallel finite element algorithms for solving this mixed model are proposed. Numerical analysis and experiments are presented to show the efficiency and effectiveness of the local and parallel finite element algorithms.     

Stabilized finite element methods with shock capturing for advection–diffusion problems

Stabilized FEM of streamline-diffusion type for advection–diffusion problems may exhibit local oscillations in crosswind direction(s). As a remedy, a shock-capturing variant of such stabilized schemes is considered as an additional consistent (but nonlinear) stabilization. We prove existence of discrete solutions. Then we present some a priori and a posteriori estimates. Finally we address the efficient solution of the arising nonlinear discrete problems.     

A sixth-order finite difference WENO scheme for Hamilton–Jacobi equations

In this paper, a sixth-order finite difference weighted essentially non-oscillatory (WENO) scheme is developed to approximate the viscosity solution of the Hamilton–Jacobi equations. This new WENO scheme has the same spatial nodes as the classical fifth-order WENO scheme proposed by Jiang and Peng [Weighted ENO schemes for Hamilton–Jacobi equations, SIAM. J. Sci. Comput. 21 (2000), pp. 2126–2143] but can be as high as sixth-order accurate in smooth region while keeping sharp discontinuous transitions with no spurious oscillations near discontinuities. Extensive numerical experiments in one- and two-dimensional cases are carried out to illustrate the capability of the proposed scheme.     

A two-level finite element method for the Allen–Cahn equation

We consider the fully implicit treatment for the nonlinear term of the Allen–Cahn equation. To solve the nonlinear problem efficiently, the two-level scheme is employed. We obtain the discrete energy law of the fully implicit scheme and two-level scheme with finite element method. Also, the convergence of the two-level method is presented. Finally, some numerical experiments are provided to confirm the theoretical analysis.     

A Weak Galerkin Method for the Reissner–Mindlin Plate in Primary Form

A new finite element method is developed for the Reissner–Mindlin equations in its primary form by using the weak Galerkin approach. Like other weak Galerkin finite element methods, this one is highly flexible and robust by allowing the use of discontinuous approximating functions on arbitrary shape of polygons and, at the same time, is parameter independent on its stability and convergence. Error estimates of optimal order in mesh size h are established for the corresponding weak Galerkin approximations. Numerical experiments are conducted for verifying the convergence theory, as well as suggesting some superconvergence and a uniform convergence of the method with respect to the plate thickness.     

A low order Galerkin finite element method for the Navier–Stokes equations of steady incompressible flow: a stabilization issue and iterative methods

A Galerkin finite element method is considered to approximate the incompressible Navier–Stokes equations together with iterative methods to solve a resulting system of algebraic equations. This system couples velocity and pressure unknowns, thus requiring a special technique for handling. We consider the Navier–Stokes equations in velocity––kinematic pressure variables as well as in velocity––Bernoulli pressure variables. The latter leads to the rotation form of nonlinear terms. This form of the equations plays an important role in our studies. A consistent stabilization method is considered from a new view point. Theory and numerical results in the paper address both the accuracy of the discrete solutions and the effectiveness of solvers and a mutual interplay between these issues when particular stabilization techniques are applied.     

Review and comparison of finite element flexural–torsional models for non-linear behaviour of thin-walled beams

The authors have developed a beam finite element model in large torsion context for thin-walled beams with arbitrary cross sections [1]. In the model, the trigonometric functions of the twist angle θ_x (c = cos θ_x − 1 and s = sin θ_x) were included as additional variables in the whole model without any assumption. In the present paper, three other 3D finite element beams are derived according to three approximations based on truncated Taylor expansions of the functions c and s (cubic, quadratic and linear). A finite element approach of these approximations is carried out. Finally, it is worth mentioning that the promising results obtained in [1], [2] encourage the authors to extend the formulation of the model in order to include load eccentricity effects. Solution of the non-linear equations is made possible by Asymptotic Numerical Method (ANM) [3]. This method is used as an alternative to the classical incremental iterative methods. Many comparison examples are considered. They concern the non-linear behaviour of beams under twist moment and the post buckling behaviour of struts under axial loads or the beam lateral buckling under eccentric bending loads. The obtained results highlight the discrepancies between the various approximations often employed in thin-walled beams literature for the geometrically non-linear analysis of beams in flexural–torsional behaviour.