A new stabilized finite element method for reaction–diffusion problems: The source‐stabilized Petrov–Galerkin method

This paper proposes a new stabilized finite element method to solve singular diffusion problems described by the modified Helmholtz operator. The Galerkin method is known to produce spurious oscillations for low diffusion and various alternatives were proposed to improve the accuracy of the solution. The mostly used methods are the well‐known Galerkin least squares and Galerkin gradient least squares (GGLS). The GGLS method yields the exact nodal solution in the one‐dimensional case and for a uniform mesh. However, the behavior of the method deteriorates slightly in the multi‐dimensional case and for non‐uniform meshes. In this work we propose a new stabilized finite element method that leads to improved accuracy for multi‐dimensional problems. For the one‐dimensional case, the new method leads to the same results as the GGLS method and hence provides exact nodal solutions to the problem on uniform meshes. The proposed method is a Galerkin discretization used to solve a modified equation that includes a term depending on the gradient of the original partial differential equation. Copyright © 2008 John Wiley & Sons, Ltd.     

A residual‐based finite element method for the Helmholtz equation

A new residual‐based finite element method for the scalar Helmholtz equation is developed. This method is obtained from the Galerkin approximation by appending terms that are proportional to residuals on element interiors and inter‐element boundaries. The inclusion of residuals on inter‐element boundaries distinguishes this method from the well‐known Galerkin least‐squares method and is crucial to the resulting accuracy of this method. In two dimensions and for regular bilinear quadrilateral finite elements, it is shown via a dispersion analysis that this method has minimal phase error. Numerical experiments are conducted to verify this claim as well as test and compare the performance of this method on unstructured meshes with other methods. It is found that even for unstructured meshes this method retains a high level of accuracy. Copyright © 2000 John Wiley & Sons, Ltd.     

On stabilized finite element methods for the Reissner–Mindlin plate model

Stabilized finite element formulation for the Reissner–Mindlin plate model is considered. Physical interpretation for the stabilization procedure for low‐order elements is established. Explicit interpolation functions for linear and bilinear stabilized MITC elements are derived. Some numerical examples including buckling and frequency analyses are presented. Copyright © 2007 John Wiley & Sons, Ltd.     

Inf–sup testing of upwind methods

We propose inf–sup testing for finite element methods with upwinding used to solve convection–diffusion problems. The testing evaluates the stability of a method and compactly displays the numerical behaviour as the convection effects increase. Four discretization schemes are considered: the standard Galerkin procedure, the full upwind method, the Galerkin least‐squares scheme and a high‐order derivative artificial diffusion method. The study shows that, as expected, the standard Galerkin method does not pass the inf–sup tests, whereas the other three methods pass the tests. Of these methods, the high‐order derivative artificial diffusion procedure introduces the least amount of artificial diffusion. Copyright © 2000 John Wiley & Sons, Ltd.     

Displacement‐based finite elements with nodal integration for Reissner–Mindlin plates

An assumed‐strain finite element technique is presented for shear‐deformable (Reissner–Mindlin) plates. The weighted residual method (reminiscent of the strain–displacement functional) is used to enforce weakly the balance equation with the natural boundary condition and, separately, the kinematic equation (the strain–displacement relationship). The a priori satisfaction of the kinematic weighted residual serves as a condition from which strain–displacement operators are derived via nodal integration, for linear triangles, and quadrilaterals, and also for quadratic triangles. The degrees of freedom are only the primitive variables: transverse displacements and rotations at the nodes. A straightforward constraint count can partially explain the insensitivity of the resulting finite element models to locking in the thin‐plate limit. We also construct an energy‐based argument for the ability of the present formulation to converge to the correct deflections in the limit of the thickness approaching zero. Examples are used to illustrate the performance with particular attention to the sensitivity to element shape and shear locking. Copyright © 2009 John Wiley & Sons, Ltd.     

Modelling the transient interaction of a thin elastic shell with an exterior acoustic field

Coupled finite and boundary element methods for solving transient fluid–structure interaction problems are developed. The finite element method is used to model the radiating structure, and the boundary element method (BEM) is used to determine the resulting acoustic field. The well‐known stability problems of time domain BEMs are avoided by using a Burton–Miller‐type integral equation. The stability, accuracy and efficiency of two alternative solution methods are compared using an exact solution for the case of a thin spherical elastic shell. The convergence properties of the preferred solution method are then investigated more thoroughly. Copyright © 2007 John Wiley & Sons, Ltd.     

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

In 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.     

Finite element computation of sloshing modes in containers with elastic baffle plates

A finite element method to approximate the vibration modes of a plate in contact with an incompressible fluid is analysed in this paper. The effect of the fluid is taken into account by means of an added mass formulation, discretized by standard piecewise linear tetrahedral finite elements. Gravity waves on the free surface of the liquid are considered in the model. The plate is modelled by Reissner–Mindlin equations discretized by MITC3 locking‐free elements. Implementation issues are discussed and numerical experiments are presented. In particular, the method is compared with analytical approximations and with an experimental study which has been recently reported. Copyright © 2002 John Wiley & Sons, Ltd.     

Fast BEM–FEM mortar coupling for acoustic–structure interaction

A coupling algorithm based on Lagrange multipliers is proposed for the simulation of structure–acoustic field interaction. Finite plate elements are coupled to a Galerkin boundary element formulation of the acoustic domain. The interface pressure is interpolated as a Lagrange multiplier, thus, allowing the coupling of non‐matching grids. The resulting saddle‐point problem is solved by an approximate Uzawa‐type scheme in which the matrix–vector products of the boundary element operators are evaluated efficiently by the fast multipole boundary element method. The algorithm is demonstrated on the example of a cavity‐backed elastic panel. Copyright © 2005 John Wiley & Sons, Ltd.     

Active control of sloshing in containers with elastic baffle plates

We consider the finite element approximations of an optimal control problem consisting in the suppression of slosh arising in fluid–structure interaction problems with free surface. The vibration of a plate in contact with an incompressible fluid is considered as state equations in the optimization problem, and distributed controls on the plate are calculated to suppress the slosh. Locking‐free finite elements are used to discretize the plate, which is modeled by Reissner–Mindlin equations. The effect of the fluid is taken into account by means of an added mass formulation, discretized by standard piecewise linear tetrahedral finite elements, and the gravity waves on the free surface of the liquid are considered in the model. The control variable is the amplitude of a secondary force actuating on the structure. Implementation issues are discussed, and numerical experiments are presented. Copyright © 2012 John Wiley & Sons, Ltd.     

Eight‐node Reissner–Mindlin plate element based on boundary interpolation using Timoshenko beam function

A new eight‐node Reissner–Mindlin plate element is developed with a special interpolation within the element. This special interpolation is an extension of the element boundary interpolation that employs Timoshenko beam function for the boundary segment interpolation. The element function can effectively capture the structural behaviours of thick plates and achieve high precision in the analysis of thick plates. Patch tests and numerical investigations are conducted. It can be seen that the proposed element successfully passes all the patch tests. The results of the numerical investigation show that the proposed element is free of the shear locking phenomenon and possesses a higher accuracy in the analyses, as compared to the earlier research in the literature. Copyright © 2006 John Wiley & Sons, Ltd.     

An adaptive non‐conforming finite‐element method for Reissner–Mindlin plates

Adaptive algorithms are important tools for efficient finite‐element mesh design. In this paper, an error controlled adaptive mesh‐refining algorithm is proposed for a non‐conforming low‐order finite‐element method for the Reissner–Mindlin plate model. The algorithm is controlled by a reliable and efficient residual‐based a posteriori error estimate, which is robust with respect to the plate's thickness. Numerical evidence for this and the efficiency of the new algorithm is provided in the sense that non‐optimal convergence rates are optimally improved in our numerical experiments. Copyright © 2003 John Wiley & Sons, Ltd.     

Improved hybrid displacement function (IHDF) element scheme for analysis of Mindlin–Reissner plate with edge effect

For a Mindlin–Reissner plate subjected to transverse loadings, the distributions of the rotations and some resultant forces may vary very sharply within a narrow district near certain boundaries. This edge effect is indeed a great challenge for conventional finite element analysis. Recently, an effective hybrid displacement function (HDF) finite element method was successfully developed for solving such difficulty 1 , 2 . Although good performances can be obtained in most cases, the distribution continuity of some resulting resultants is destroyed when coarse meshes are employed. Moreover, an additional local coordinate system must be used for avoiding a singular problem in matrix inversion, which makes the derivations more complicated. Based on a modified complementary energy functional containing Lagrangian multipliers, an improved HDF (IHDF) element scheme is proposed in this work. And two new special IHDF elements, named by IHDF‐P4‐Free and IHDF‐P4‐SS1, are constructed for modeling plate behaviors near free and soft simply supported boundaries, respectively. The present modeling scheme not only greatly improves the precision of the numerical results but also avoids usage of the additional local Coordinate system. The numerical tests demonstrate that the new IHDF element scheme is an effective way for solving the challenging edge effect problem in Mindlin–Reissner plates. Copyright © 2016 John Wiley & Sons, Ltd.     

Computable exact bounds for linear outputs from stabilized solutions of the advection–diffusion–reaction equation

The paper introduces a methodology to compute strict upper and lower bounds for linear‐functional outputs of the exact solutions of the advection–diffusion–reaction equation. The bounds are computed using implicit a posteriori error estimators from stabilized finite element approximations of the exact solution. The new methodology extends the a posteriori error estimates yielding bounds for the standard Galerkin formulation to be able to obtain bounds for stabilized formulations. This methodology is combined with both hybrid‐flux and flux‐free techniques for error assessment. The application to stabilized formulations provides sharper estimates than when applied to Galerkin methods. The best results are found in combination with the flux‐free technique. Copyright © 2012 John Wiley & Sons, Ltd.     

The partition of unity finite element method for elastic wave propagation in Reissner–Mindlin plates

This paper reports a numerical method for modelling the elastic wave propagation in plates. The method is based on the partition of unity approach, in which the approximate spectral properties of the infinite dimensional system are embedded within the space of a conventional finite element method through a consistent technique of waveform enrichment. The technique is general, such that it can be applied to the Lagrangian family of finite elements with specific waveform enrichment schemes, depending on the dominant modes of wave propagation in the physical system. A four‐noded element for the Reissner–Mindlin plate is derived in this paper, which is free of shear locking. Such a locking‐free property is achieved by removing the transverse displacement degrees of freedom from the element nodal variables and by recovering the same through a line integral and a weak constraint in the frequency domain. As a result, the frequency‐dependent stiffness matrix and the mass matrix are obtained, which capture the higher frequency response with even coarse meshes, accurately. The steps involved in the numerical implementation of such element are discussed in details. Numerical studies on the performance of the proposed element are reported by considering a number of cases, which show very good accuracy and low computational cost. Copyright © 2006 John Wiley & Sons, Ltd.     

On the accuracy of p-version elements for the Reissner–Mindlin plate problem

This paper addresses the question of accuracy of p-version finite element formulations for Reissner–Mindlin plate problems. Three model problems, a circular arc, a rhombic plate and a geometrically complex structure are investigated. Whereas displacements and bending moments turn out to be very accurate without any post-processing even for very coarse meshes, the quality of shear forces computed from constitutive equations is poor. It is shown that significantly improved results can be obtained, if shear forces are computed from equilibrium equations instead. A consistent computation of second derivatives of the shape functions is derived. © 1998 John Wiley & Sons, Ltd.     

A stabilized discrete shear gap finite element for adaptive limit analysis of Mindlin–Reissner plates

This paper presents a numerical formulation for computation of collapse load of Mindlin–Reissner plates that uses stabilized discrete shear gap finite elements and second‐order cone programming. Displacement fields are approximated using the discrete shear gap in combination with a stabilized strain smoothing technique, ensuring that shear‐locking problem can be avoided and that accurate solutions can be obtained. The underlying optimization problem is formulated in the form of a standard second‐order cone programming, so that it can be solved using highly efficient primal‐dual interior‐point algorithm. An error indicator based on plastic dissipation will be used in the adaptive refinement scheme. Various plates with arbitrary geometries and boundary conditions are examined to illustrate the performance of the proposed procedure. Copyright © 2013 John Wiley & Sons, Ltd.     

An element‐free method for in‐plane notch problems with anisotropic materials

This study developed an element‐free Galerkin method (EFGM) to simulate notched anisotropic plates containing stress singularities at the notch tip. Two‐dimensional theoretical complex displacement functions are first deduced into the moving least‐squares interpolation. The interpolation functions and their derivatives are then determined to calculate the nodal stiffness using the Galerkin method. In the numerical validation, an interface layer of the EFGM is used to combine the mesh between the traditional finite elements and the proposed singular notch EFGM. The H‐integral determined from finite element analyses with a very fine mesh is used to validate the numerical results of the proposed method. The comparisons indicate that the proposed method obtains more accurate results for the displacement, stress, and energy fields than those determined from the standard finite element method. Copyright © 2013 John Wiley & Sons, Ltd.     

Stabilized finite elements for time-harmonic waves in incompressible and nearly incompressible elastic solids

The propagation of waves in elastic solids at or near the incompressible limit is of interest in many current and emerging applications. Standard low-order Galerkin finite element discretization struggles with both incompressibility and wave dispersion. Galerkin least squares stabilization is known to improve computational performance of each of these ingredients separately. A novel approach of combined pressure-curl stabilization is presented, facilitating the use of continuous, equal-order interpolation of displacements and pressure. The pressure stabilization parameter is determined by stability considerations, while the curl stabilization parameter is determined by dispersion considerations. The proposed pressure-curl–stabilized scheme provides stable and accurate results on a variety of numerical tests for incompressible and nearly incompressible elastic waves computed with linear elements.     

Limit analysis of plates using the EFG method and second‐order cone programming

The meshless element‐free Galerkin (EFG) method is extended to allow computation of the limit load of plates. A kinematic formulation that involves approximating the displacement field using the moving least‐squares technique is developed. Only one displacement variable is required for each EFG node, ensuring that the total number of variables in the resulting optimization problem is kept to a minimum, with far fewer variables being required compared with finite element formulations using compatible elements. A stabilized conforming nodal integration scheme is extended to plastic plate bending problems. The evaluation of integrals at nodal points using curvature smoothing stabilization both keeps the size of the optimization problem small and also results in stable and accurate solutions. Difficulties imposing essential boundary conditions are overcome by enforcing displacements at the nodes directly. The formulation can be expressed as the problem of minimizing a sum of Euclidean norms subject to a set of equality constraints. This non‐smooth minimization problem can be transformed into a form suitable for solution using second‐order cone programming. The procedure is applied to several benchmark beam and plate problems and is found in practice to generate good upper‐bound solutions for benchmark problems. Copyright © 2009 John Wiley & Sons, Ltd.