Behaviour of isoparametric quadrilateral family of Lagrangian fluid finite elements

This paper presents the formulation of both the consistent and inconsistent four‐, eight‐ and nine‐noded isoparametric quadrilateral fluid finite elements that are based on Lagrangian frame of reference. The mesh locking phenomenon due to simultaneous enforcement of twin constraints, namely the incompressibility and irrotationality constraints, is studied in detail. The study shows that the characteristic of the locked fluid elements is that it always generates numerous spurious acoustic (volume change) modes upon the enforcement of rotational constraints. That is, the rotational constraints change the character of certain volume change modes. The study further reinforces the necessity of rotational constraints in not only identifying the spurious pressure modes, but also in reducing the computational effort for determining the eigenvalues and eigenvectors. It is found that all fully integrated inconsistent models exhibit locking behaviour. However, the inconsistent eight‐ and nine‐noded elements, integrated with full integration of volumetric stiffness and one point integration of the rotational stiffness matrices, gives excellent performance, although they do not pass the inf–sup test. The four‐ and nine‐noded consistent models are found to give locking free performance while their eight‐noded counterpart exhibited locking behaviour. The study shows that only consistent nine‐noded element models pass the inf–sup test. The utility of these elements in the coupled fluid–structure interaction problem is also demonstrated. Copyright © 2002 John Wiley & Sons, Ltd.     

Stability and accuracy of finite element methods for flow acoustics. II: Two‐dimensional effects

This is the second of two articles that focus on the dispersion properties of finite element models for acoustic propagation on mean flows. We consider finite element methods based on linear potential theory in which the acoustic disturbance is modelled by the convected Helmholtz equation, and also those based on a mixed Galbrun formulation in which acoustic pressure and Lagrangian displacement are used as discrete variables. The current paper focuses on the effects of numerical anisotropy which are associated with the orientation of the propagating wave to the mean flow and to the grid axes. Conditions which produce aliasing error in the Helmholtz formulation are of particular interest. The 9‐noded Lagrangian element is shown to be superior to the more commonly used 8‐noded serendipity element. In the case of the Galbrun elements, the current analysis indicates that isotropic meshes generally reduce numerical error of triangular elements and that higher order mixed quadrilaterals are generally less effective than an equivalent mesh of lower order triangles. Copyright © 2005 John Wiley & Sons, Ltd.     

A two‐noded locking–free shear flexible curved beam element

A new two‐noded shear flexible curved beam element which is impervious to membrane and shear locking is proposed herein. The element with three degrees of freedom at each node is based on curvilinear deep shell theory. Starting with a cubic polynomial representation for radial displacement (w), the displacement field for tangential displacement (u) and section rotation (θ) are determined by employing force‐moment and moment‐shear equilibrium equations. This results in polynomial displacement field whose coefficients are coupled by generalized degrees of freedom and material and geometric properties of the element. The procedure facilitates quartic polynomial representation for both u and θ for curved element configurations, which reduces to linear and quadratic polynomials for u and θ, respectively, for straight element configuration. These coupled polynomial coefficients do not give rise to any spurious constraints even in the extreme thin regimes, in which case, the present element exhibits excellent convergence to the classical thin beam solutions. This simple C⁰ element is validated for beam having straight/curved geometries over a wide range of slenderness ratios. The results indicates that performance of the element is much superior to other elements of the same class. Copyright © 1999 John Wiley & Sons, Ltd.     

Finite element modelling of cracked inelastic shells with large deflections: two‐dimensional and three‐dimensional approaches

Higher utilization of structural materials leads to a need for accurate numerical tools for reliable predictions of structural response. In some instances, both material and geometrical non‐linearities are allowed for, typically in assessments of structural collapse or residual strength in damaged conditions. The present study addresses the performance of surface‐cracked inelastic shells with out‐of‐plane displacements not negligible compared to shell thickness. This situation leads to non‐linear membrane force effects in the shell. Hence, a cracked part of the shell will be subjected to a non‐proportional history of bending moment and membrane force. An important point in the discretization of the problem is whether a two‐dimensional model describes the structural performance sufficiently, or a three‐dimensional model is required. Herein, the two‐dimensional modelling is performed by means of a Mindlin shell finite element. The cracked parts are accounted for by means of inelastic line spring elements. The three‐dimensional models employ eight‐noded solid elements. These models also account for ductile crack growth due to void coalescence by means of a modified Gurson–Tvergaard constitutive model, hence providing detailed solutions that the two‐dimensional simulations can be tested against. Using this, the accuracy of the two‐dimensional approach is checked thoroughly. The analyses show that the two‐dimensional modelling is sufficient as long as the cracks do not grow. Hence, using fracture initiation as a capacity criterion, shell elements and line springs provide acceptable predictions. If significant ductile tearing occurs before final failure, the line spring ligaments have to be updated due to crack growth.     

The h–p version of the finite element method for problems with interfaces

Regularities of the solutions of interface problems in two dimensions are described in the frame of the weighted Sobolev spaces and countably normed spaces. Based upon the regularity of solutions the geometric meshes and the distribution of polynomial degrees are properly designed so that the h–p version of the finite element method for interface problems can lead to the exponential rate of convergence. Numerical results on an elliptic equation with interfaces are presented. The optimal mesh factor, optimal degree factors, and optimal layer factors of the geometric mesh in neighbourhoods of singular points having varied intensities are discussed from both theoretical and practical point of view.     

Numerical approximation of a thermally driven interface using finite elements

A two‐dimensional finite element model for dendritic solidification has been developed that is based on the direct solution of the energy equation over a fixed mesh. The model tracks the position of the sharp solid–liquid interface using a set of marker points placed on the interface. The simulations require calculation of the temperature gradients on both sides of the interface in the direction normal to it; at the interface the heat flux is discontinuous due to the release of latent heat during the solidification (melting) process. Two ways to calculate the temperature gradients at the interface, evaluating their interpolants at Gauss points, were proposed. Using known one‐ and two‐dimensional solutions to stable solidification problems (the Stefan problem), it was shown that the method converges with second‐order accuracy. When applied to the unstable solidification of a crystal into an undercooled liquid, it was found that the numerical solution is extremely sensitive to the mesh size and the type of approximation used to calculate the temperature gradients at the interface, i.e. different approximations and different meshes can yield different solutions. The cause of these difficulties is examined, the effect of different types of interpolation on the simulations is investigated, and the necessary criteria to ensure converged solutions are established. Copyright © 2003 John Wiley & Sons, Ltd.     

New superelements for singular derivative problems of arbitrary order

The paper describes a new method for development of singular elements through optimal positioning of offset mid‐side nodes using adjacent six‐ and eight‐noded isoparametric elements. By joining the two elements to form a superelement, an attempt to rectify a known problem with existing approaches proves successful. Calculations are made to optimize the mid‐side node positions and the relative size of the two elements for a wide range of singular orders. By way of example, the approach is applied to two different forms of singular stress field in a bimaterial joint in a three‐point‐bend test. Numerical solutions for the singular fields existing at both ends of the material interface compare favourably with those obtained with the more usual refined mesh approach. Copyright © 2001 John Wiley & Sons, Ltd.     

Hierarchical high‐order conforming C1 bases for quadrangular and triangular finite elements

We present three new sets of C¹ hierarchical high‐order tensor‐product bases for conforming finite elements. The first basis is a high‐order extension of the Bogner–Fox–Schmit basis. The edge and face functions are constructed using a combination of cubic Hermite and Jacobi polynomials with C¹ global continuity on the common edges of elements. The second basis uses the tensor product of fifth‐order Hermite polynomials and high‐order functions and achieves global C¹ continuity for meshes of quadrilaterals and C² continuity on the element vertices. The third basis for triangles is also constructed using the tensor product of one‐dimensional functions defined in barycentric coordinates. It also has global C¹ continuity on edges and C² continuity on vertices. A patch test is applied to the three considered elements. Projection and plate problems with smooth fabricated solutions are solved, and the performance of the h‐ and p‐refinements are evaluated by comparing the approximation errors in the L₂‐ and energy norms. A plate with singularity is then studied, and h‐ and p‐refinements are analysed. Finally, a transient problem with implicit time integration is considered. The results show exponential convergence rates with increasing polynomial order for the triangular and quadrilateral meshes of non‐distorted and distorted elements. Copyright © 2016 John Wiley & Sons, Ltd.     

Stress intensity factors for three‐dimensional surface cracks using enriched finite elements

The analysis of three‐dimensional crack problems using enriched crack tip elements is examined in this paper. It is demonstrated that the enriched finite element approach is a very effective technique for obtaining stress intensity factors for general three‐dimensional crack problems. The influence of compatibility, integration, element shape function order, and mesh refinement on solution convergence is investigated to ascertain the accuracy of the numerical results. It is shown that integration order has the greatest impact on solution accuracy. Sample results are presented for semi‐circular surface cracks and compared with previously obtained solutions available in the literature. Good agreement is obtained between the different numerical solutions, except in the small zone near the free surface where previously published results have often neglected the change in the stress singularity at the free surface. The enriched crack tip element appears to be particularly effective in this region, since boundary conditions can be easily imposed on the stress intensity factors to accurately represent the correct free surface condition. Copyright © 2002 John Wiley & Sons, Ltd.     

Two improvements in formulation of nine‐node element MITC9

The paper concerns a well‐known two‐dimensional nine‐node quadrilateral element MITC9, which is based on two‐level approximations of strains (assumed strain method). The element has good accuracy, but does not pass the patch test. As the first improvement, we propose a modification of the element's transformations, partly resolving the problem with the patch test. The source of the problem is the use of covariant components in a (local) natural co‐basis, different at each sampling point. As the second improvement, we use the corrected shape functions of Celia MA, Gray WG. An improved isoparametric transformation for finite element analysis. International Journal for Numerical Methods in Engineering 1984; 20 :1447–1459, extending their applicability to the nine‐node element for plane elasticity and the 3 × 3 integration. Originally, they are tested for an eight‐node element for the heat conduction equation and the 4 × 4 integration. The improved element, designated as MITC9i, is based on the Green strain and derived from the potential energy for the plane stress condition. It is subjected to a range of tests, to confirm that it passes the patch test for several types of mesh distortions, to prove its coarse mesh accuracy and the absence of locking as well as to establish its sensitivity to mesh distortions. The improved element MITC9i performs substantially better than the MITC9 element, QUAD9** element, and our previous 9‐AS element.Copyright © 2012 John Wiley & Sons, Ltd.     

Classical and advanced multilayered plate elements based upon PVD and RMVT. Part 2: Numerical implementations

This paper presents numerical evaluations related to the multilayered plate elements which were proposed in the companion paper (Part 1). Two‐dimensional modellings with linear and higher‐order (up to fourth order) expansion in the z‐plate/layer thickness direction have been implemented for both displacements and transverse stresses. Layer‐wise as well as equivalent single‐layer modellings are considered on both frameworks of the principle of virtual displacements and Reissner mixed variational theorem. Such a variety has led to the implementation of 22 plate theories. As far as finite element approximation is concerned, three quadrilaters have been considered (four‐, eight‐ and nine‐noded plate elements). As a result, 22×3 different finite plate elements have been compared in the present analysis. The automatic procedure described in Part 1, which made extensive use of indicial notations, has herein been referred to in the considered computer implementations. An assessment has been made as far as convergence rates, numerical integrations and comparison to correspondent closed‐form solutions are concerned. Extensive comparison to early and recently available results has been made for sample problems related to laminated and sandwich structures. Classical formulations, full mixed, hybrid, as well as three‐dimensional solutions have been considered in such a comparison. Numerical substantiation of the importance of the fulfilment of zig‐zag effects and interlaminar equilibria is given. The superiority of RMVT formulated finite elements over those related to PVD has been concluded. Two test cases are proposed as ‘desk‐beds’ to establish the accuracy of the several theories. Results related to all the developed theories are presented for the first test case. The second test case, which is related to sandwich plates, restricts the comparison to the most significant implemented finite elements. It is proposed to refer to these test cases to establish the accuracy of existing or new higher‐order, refined or improved finite elements for multilayered plate analyses. Copyright © 2002 John Wiley & Sons, Ltd.     

Smooth finite element methods: Convergence,accuracy and properties

A stabilized conforming nodal integration finite element method based on strain smoothing stabilization is presented. The integration of the stiffness matrix is performed on the boundaries of the finite elements. A rigorous variational framework based on the Hu–Washizu assumed strain variational form is developed. We prove that solutions yielded by the proposed method are in a space bounded by the standard, finite element solution (infinite number of subcells) and a quasi‐equilibrium finite element solution (a single subcell). We show elsewhere the equivalence of the one‐subcell element with a quasi‐equilibrium finite element, leading to a global a posteriori error estimate. We apply the method to compressible and incompressible linear elasticity problems. The method can always achieve higher accuracy and convergence rates than the standard finite element method, especially in the presence of incompressibility, singularities or distorted meshes, for a slightly smaller computational cost. It is shown numerically that the one‐cell smoothed four‐noded quadrilateral finite element has a convergence rate of 2.0 in the energy norm for problems with smooth solutions, which is remarkable. For problems with rough solutions, this element always converges faster than the standard finite element and is free of volumetric locking without any modification of integration scheme. Copyright © 2007 John Wiley & Sons, Ltd.     

A hybrid discontinuous in space and time Galerkin method for wave propagation problems

Medium‐frequency regime and multi‐scale wave propagation problems have been a subject of active research in computational acoustics recently. New techniques have attempted to overcome the limitations of existing discretization methods that tend to suffer from dispersion. One such technique, the discontinuous enrichment method, incorporates features of the governing partial differential equation in the approximation, in particular, the solutions of the homogeneous form of the equation. Here, based on this concept and by extension of a conventional space–time finite element method, a hybrid discontinuous Galerkin method (DGM) for the numerical solution of transient problems governed by the wave equation in two and three spatial dimensions is described. The discontinuous formulation in both space and time enables the use of solutions to the homogeneous wave equation in the approximation. In this contribution, within each finite element, the solutions in the form of polynomial waves are employed. The continuity of these polynomial waves is weakly enforced through suitably chosen Lagrange multipliers. Results for two‐dimensional and three‐dimensional problems, in both low‐frequency and medium‐frequency regimes, show that the proposed DGM outperforms the conventional space–time finite element method. Copyright © 2014 John Wiley & Sons, Ltd.     

Unstructured,curved elements for the two‐dimensional high order discontinuous control‐volume/finite‐element method

Quadrilateral and triangular elements with curved edges are developed in the framework of spectral, discontinuous, hybrid control‐volume/finite‐element method for elliptic problems. In order to accommodate hybrid meshes, encompassing both triangular and quadrilateral elements, one single mapping is used. The scheme is applied to two‐dimensional problems with discontinuous, anisotropic diffusion coefficients, and the exponential convergence of the method is verified in the presence of curved geometries. Copyright © 2014 John Wiley & Sons, Ltd.     

Polyhedral elements by means of node/edge‐based smoothed finite element method

The node‐based or edge‐based smoothed finite element method is extended to develop polyhedral elements that are allowed to have an arbitrary number of nodes or faces, and so retain a good geometric adaptability. The strain smoothing technique and implicit shape functions based on the linear point interpolation make the element formulation simple and straightforward. The resulting polyhedral elements are free from the excessive zero‐energy modes and yield a robust solution very much insensitive to mesh distortion. Several numerical examples within the framework of linear elasticity demonstrate the accuracy and convergence behavior. The smoothed finite element method‐based polyhedral elements in general yield solutions of better accuracy and faster convergence rate than those of the conventional finite element methods. Copyright © 2016 John Wiley & Sons, Ltd.     

Patch‐averaged assumed strain finite elements for stress analysis

A finite element model for linear‐elastic small deformation problems is presented. The formulation is based on a weighted residual that requires a priori the satisfaction of the kinematic equation. In this approach, an averaged strain‐displacement matrix is constructed for each node of the mesh by defining an appropriate patch of elements, yielding a smooth representation of strain and stress fields. Connections with traditional and similar procedure are explored. Linear quadrilateral four‐node and linear hexahedral eight‐node elements are derived. Various numerical tests show the accuracy and convergence properties of the proposed elements in comparison with extant finite elements and analytic solutions. Specific examples are also included to illustrate the ability to resist numerical locking in the incompressible limit and insensitive response in the presence of shape distortion. Furthermore, the numerical inf‐sup test is applied to a selection of problems to show the stability of the present formulation. Copyright © 2012 John Wiley & Sons, Ltd.     

Numerical stabilization of Biot's consolidation model by a perturbation on the flow equation

In this paper a stabilized finite element scheme for the poroelasticity equations is proposed. This method, based on the perturbation of the flow equation, allows us to use continuous piecewise linear approximation spaces for both displacements and pressure, obtaining solutions without oscillations independently of the chosen discretization parameters. The perturbation term depends on a parameter which is established in terms of the mesh size and the properties of the material. In the one‐dimensional case, this parameter is shown to be optimal. Some numerical experiments are presented indicating the efficiency of the proposed stabilization technique. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

Three‐dimensional adaptive meshing by subdivision and edge‐collapse in finite‐deformation dynamic–plasticity problems with application to adiabatic shear banding

This paper is concerned with the development of a general framework for adaptive mesh refinement and coarsening in three‐dimensional finite‐deformation dynamic–plasticity problems. Mesh adaption is driven by a posteriori global error bounds derived on the basis of a variational formulation of the incremental problem. The particular mesh‐refinement strategy adopted is based on Rivara's longest‐edge propagation path (LEPP) bisection algorithm. Our strategy for mesh coarsening, or unrefinement, is based on the elimination of elements by edge‐collapse. The convergence characteristics of the method in the presence of strong elastic singularities are tested numerically. An application to the three‐dimensional simulation of adiabatic shear bands in dynamically loaded tantalum is also presented which demonstrates the robustness and versatility of the method. Copyright © 2001 John Wiley & Sons, Ltd.