On evaluating the inf–sup condition for plate bending elements

This paper addresses the evaluation of the inf–sup condition for Reissner–Mindlin plate bending elements. This fundamental condition for stability and optimality of a mixed finite element scheme is, in general, very difficult to evaluate analytically, considering for example distorted meshes. Therefore, we develop a numerical test methodology. To demonstrate the test methodology and to obtain specific results, we apply it to standard displacement-based elements and elements of the MITC family. Whereas the displacement-based elements fail to satisfy the inf–sup condition, we find that the MITC elements pass our numerical test for uniform meshes and a sequence of distorted meshes. © 1997 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/pressure mixed interpolation in the method of finite spheres

The displacement‐based formulation of the method of finite spheres is observed to exhibit volumetric ‘locking’ when incompressible or nearly incompressible deformations are encountered. In this paper, we present a displacement/pressure mixed formulation as a solution to this problem. We analyse the stability and optimality of the formulation for several discretization schemes using numerical inf–sup tests. Issues concerning computational efficiency are also discussed. Copyright © 2001 John Wiley & Sons, Ltd.     

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.     

Automatically inf − sup compliant diamond‐mixed finite elements for Kirchhoff plates

We develop a mixed finite‐element approximation scheme for Kirchhoff plate theory based on the reformulation of Kirchhoff plate theory of Ortiz and Morris [1]. In this reformulation the moment‐equilibrium problem for the rotations is in direct analogy to the problem of incompressible two‐dimensional elasticity. This analogy in turn opens the way for the application of diamond approximation schemes (Hauret et al. [2]) to Kirchhoff plate theory. We show that a special class of meshes derived from an arbitrary triangulation of the domain, the diamond meshes, results in the automatic satisfaction of the corresponding inf ? sup condition for Kirchhoff plate theory. The attendant optimal convergence properties of the diamond approximation scheme are demonstrated by means of the several standard benchmark tests. We also provide a reinterpretation of the diamond approximation scheme for Kirchhoff plate theory within the framework of discrete mechanics. In this interpretation, the discrete moment‐equilibrium problem is formally identical to the classical continuous problem, and the two differ only in the choice of differential structures. It also follows that the satisfaction of the inf ? sup condition is a property of the cohomology of a certain discrete transverse differential complex. This close connection between the classical inf ? sup condition and cohomology evinces the important role that the topology of the discretization plays in determining convergence in mixed problems. Copyright © 2013 John Wiley & Sons, Ltd.     

A three‐dimensional hybrid meshfree‐Cartesian scheme for fluid–body interaction

A numerical method based on a hybrid meshfree‐Cartesian grid is developed for solving three‐dimensional fluid–solid interaction (FSI) problems involving solid bodies undergoing large motion. The body is discretized and enveloped by a cloud of meshfree nodes. The motion of the body is tracked by convecting the meshfree nodes against a background of Cartesian grid points. Spatial discretization of second‐order accuracy is accomplished by the combination of a generalized finite difference (GFD) method and conventional finite difference (FD) method, which are applied to the meshfree and Cartesian nodes, respectively. Error minimization in GFD is carried out by singular value decomposition. The discretized equations are integrated in time via a second‐order fractional step projection method. A time‐implicit iterative procedure is employed to compute the new/evolving position of the immersed bodies together with the dynamically coupled solution of the flow field. The present method is applied on problems of free falling spheres and tori in quiescent flow and freely rotating spheres in simple shear flow. Good agreement with published results shows the ability of the present hybrid meshfree‐Cartesian grid scheme to achieve good accuracy. An application of the method to the self‐induced propulsion of a deforming fish‐like swimmer further demonstrates the capability and potential of the present approach for solving complex FSI problems in 3D. Copyright © 2011 John Wiley & Sons, Ltd.     

A meshfree‐enriched finite element method for compressible and near‐incompressible elasticity

In this paper, a two‐dimensional displacement‐based meshfree‐enriched FEM (ME‐FEM) is presented for the linear analysis of compressible and near‐incompressible planar elasticity. The ME‐FEM element is established by injecting a first‐order convex meshfree approximation into a low‐order finite element with an additional node. The convex meshfree approximation is constructed using the generalized meshfree approximation method and it possesses the Kronecker‐delta property on the element boundaries. The gradient matrix of ME‐FEM element satisfies the integration constraint for nodal integration and the resultant ME‐FEM formulation is shown to pass the constant stress test for the compressible media. The ME‐FEM interpolation is an element‐wise meshfree interpolation and is proven to be discrete divergence‐free in the incompressible limit. To prevent possible pressure oscillation in the near‐incompressible problems, an area‐weighted strain smoothing scheme incorporated with the divergence‐free ME‐FEM interpolation is introduced to provide the smoothing on strains and pressure. With this smoothed strain field, the discrete equations are derived based on a modified Hu–Washizu variational principle. Several numerical examples are presented to demonstrate the effectiveness of the proposed method for the compressible and near‐incompressible problems. Copyright © 2012 John Wiley & Sons, Ltd.     

On the modelling of incompressibility in linear and non‐linear elasticity with the master–slave approach

The master–slave approach is adapted to model the kinematic constraints encountered in incompressibility. The method presented here allows us to obtain discrete displacement and pressure fields for arbitrary finite element formulations that have discontinuous pressure interpolations. The resulting displacements satisfy exactly the incompressibility constraints in a weak sense, and are obtained by solving a system of equations with the minimum (independent) degrees of freedom. In linear analysis, the method reproduces the well‐known stability results for inf–sup compliant elements, and permits to compute the pressure modes (physical or spurious) when they exist. By rewriting the equilibrium equations of a hyperelastic material, the method is extended to non‐linear elasticity, while retaining the exact fulfilment of the incompressibility constraints in a weak sense. Problems with analytical solution in two and three dimensions are tested and compared with other solution methods. Copyright © 2007 John Wiley & Sons, Ltd.     

Energy–momentum consistent finite element discretization of dynamic finite viscoelasticity

This paper is concerned with energy–momentum consistent time discretizations of dynamic finite viscoelasticity. Energy consistency means that the total energy is conserved or dissipated by the fully discretized system in agreement with the laws of thermodynamics. The discretization is energy–momentum consistent if also momentum maps are conserved when group motions are superimposed to deformations. The performed approximation is based on a three‐field formulation, in which the deformation field, the velocity field and a strain‐like viscous internal variable field are treated as independent quantities. The new non‐linear viscous evolution equation satisfies a non‐negative viscous dissipation not only in the continuous case, but also in the fully discretized system. The initial boundary value problem is discretized by using finite elements in space and time. Thereby, the temporal approximation is performed prior to the spatial approximation in order to preserve the stress objectivity for finite rotation increments (incremental objectivity). Although the present approach makes possible to design schemes of arbitrary order, the focus is on finite elements relying on linear Lagrange polynomials for the sake of clearness. The discrete energy–momentum consistency is based on the collocation property and an enhanced second Piola–Kirchhoff stress tensor. The obtained coupled non‐linear algebraic equations are consistently linearized. The corresponding iterative solution procedure is associated with newly proposed convergence criteria, which take the discrete energy consistency into account. The iterative solution procedure is therefore not complicated by different scalings in the independent variables, since the motion of the element is taken into account for solving the viscous evolution equation. Representative numerical simulations with various boundary conditions show the superior stability of the new time‐integration algorithm in comparison with the ordinary midpoint rule. Both the quasi‐rigid deformations during a free flight, and large deformations arising in a dynamic tensile test are considered. Copyright © 2009 John Wiley & Sons, Ltd.     

Mixed isogeometric analysis of strongly coupled diffusion in porous materials

In this paper, we develop a mixed isogeometric analysis approach based on subdivision stabilization to study strongly coupled diffusion in solids in both small and large deformation ranges. Coupling the fluid pressure and the solid deformation, the mixed formulation suffers from numerical instabilities in the incompressible and the nearly incompressible limit due to the violation of the inf‐sup condition. We investigate this issue using subdivision‐stabilized nonuniform rational B‐spline (NURBS) elements, as well as different families of mixed isogeometric analysis techniques, and assess their stability through a numerical inf‐sup test. Furthermore, the validity of the inf‐sup stability test in poromechanics is supported by a mathematical proof concerning the corresponding stability estimate. Finally, two numerical examples involving a rigid strip foundation on saturated soil and a swelling hydrogel structure are presented to validate the stability and to demonstrate the robustness of the proposed approach.     

Locking‐free continuum displacement finite elements with nodal integration

An assumed‐strain finite element technique is presented for linear, elastic small‐deformation models. Weighted residual method (reminiscent of the strain–displacement functional) is used to weakly enforce the balance equation with the natural boundary condition and the kinematic equation (the strain–displacement relationship). A priori satisfaction of the kinematic weighted residual serves as a condition from which strain–displacement operators are derived via nodal integration. A variety of element shapes is treated: linear triangles, quadrilaterals, tetrahedra, hexahedra, and quadratic (six‐node) triangles and (27‐node) hexahedra. The degrees of freedom are only the primitive variables (displacements at the nodes). The formulation allows for general anisotropic materials. A straightforward constraint count can partially explain the insensitivity of the resulting finite element models to locking in the incompressible limit. Furthermore, the numerical inf–sup test is applied in select problems and several variants of the proposed formulations (linear triangles, quadrilaterals, tetrahedra, hexahedra, and 27‐node hexahedra) pass the test. Examples are used to illustrate the performance with respect to sensitivity to shape distortion and the ability to resist volumetric locking. Copyright © 2008 John Wiley & Sons, Ltd.     

Refined 15‐DOF triangular discrete degenerated shell element with high performances

A refined discrete degenerated 15‐DOF triangular shell element RDTS15 with high performances is proposed. For constructing the element displacement function, the exact displacement function of the Timoshenko's beam is used as the displacement on the element boundary, and the re‐constitute method for shear strain matrix is adopted. The proposed element can be used in the analysis of both moderate thick and thin plates/shells. Numerical examples presented show that the new model indeed possesses higher accuracy in the analysis of thin and thick plates/shells, and that it can pass the patch test required for the Kirchhoff thin plate elements, and also passed the inf–sup test for free cylindrical shell problems and satisfied both the bending‐ and membrane‐dominated test. Copyright © 2004 John Wiley Sons, Ltd.     

A stabilized meshfree reproducing kernel‐based method for convection–diffusion problems

In this paper, we develop a meshfree particle‐based method for convection–diffusion problems. Discretization is performed by using piecewise constant kernels. The stabilized scheme is based on a new upwind kernel. We show that accurate and stable scheme can be obtained by using purpose‐built kernels. It also shown that under some conditions the classical optimal finite difference scheme can be derived by the new method. Several numerical tests validate the method. Copyright © 2011 John Wiley & Sons, Ltd.     

Three‐dimensional meshfree‐enriched finite element formulation for micromechanical hyperelastic modeling of particulate rubber composites

A three‐dimensional microstructure‐based finite element framework is presented for modeling the mechanical response of rubber composites in the microscopic level. This framework introduces a novel finite element formulation, the meshfree‐enriched FEM, to overcome the volumetric locking and pressure oscillation problems that normally arise in the numerical simulation of rubber composites using conventional displacement‐based FEM. The three‐dimensional meshfree‐enriched FEM is composed of five‐noded tetrahedral elements with a volume‐weighted smoothing of deformation gradient between neighboring elements. The L₂‐orthogonality property of the smoothing operator enables the employed Hu–Washizu–de Veubeke functional to be degenerated to an assumed strain method, which leads to a displacement‐based formulation that is easily incorporated with the periodic boundary conditions imposed on the unit cell. Two numerical examples are analyzed to demonstrate the effectiveness of the proposed approach. Copyright © 2012 John Wiley & Sons, Ltd.     

Interface problems with quadratic X‐FEM: design of a stable multiplier space and error analysis

The aim of this paper is to propose a procedure to accurately compute curved interfaces problems within the extended finite element method and with quadratic elements. It is dedicated to gradient discontinuous problems, which cover the case of bimaterials as the main application. We focus on the use of Lagrange multipliers to enforce adherence at the interface, which makes this strategy applicable to cohesive laws or unilateral contact. Convergence then occurs under the condition that a discrete inf‐sup condition is passed. A dedicated P1 multiplier space intended for use with P2 displacements is introduced. Analytical proof that it passes the inf‐sup condition is presented in the two‐dimensional case. Under the assumption that this inf‐sup condition holds, a priori error estimates are derived for linear or quadratic elements as functions of the curved interface resolution and of the interpolation properties of the discrete Lagrange multipliers space. The estimates are successfully checked against several numerical experiments: disparities, when they occur, are explained in the literature. Besides, the new multiplier space is able to produce quadratic convergence from P2 displacements and quadratic geometry resolution. Copyright © 2014 John Wiley & Sons, Ltd.     

The enriched space–time finite element method (EST) for simultaneous solution of fluid–structure interaction

The paper introduces a weighted residual‐based approach for the numerical investigation of the interaction of fluid flow and thin flexible structures. The presented method enables one to treat strongly coupled systems involving large structural motion and deformation of multiple‐flow‐immersed solid objects. The fluid flow is described by the incompressible Navier–Stokes equations. The current configuration of the thin structure of linear elastic material with non‐linear kinematics is mapped to the flow using the zero iso‐contour of an updated level set function. The formulation of fluid, structure and coupling conditions uniformly uses velocities as unknowns. The integration of the weak form is performed on a space–time finite element discretization of the domain. Interfacial constraints of the multi‐field problem are ensured by distributed Lagrange multipliers. The proposed formulation and discretization techniques lead to a monolithic algebraic system, well suited for strongly coupled fluid–structure systems. Embedding a thin structure into a flow results in non‐smooth fields for the fluid. Based on the concept of the extended finite element method, the space–time approximations of fluid pressure and velocity are properly enriched to capture weakly and strongly discontinuous solutions. This leads to the present enriched space–time (EST) method. Numerical examples of fluid–structure interaction show the eligibility of the developed numerical approach in order to describe the behavior of such coupled systems. The test cases demonstrate the application of the proposed technique to problems where mesh moving strategies often fail. Copyright © 2007 John Wiley & Sons, Ltd.     

A partition of unity‐based ‘FE–Meshfree’ QUAD4 element for geometric non‐linear analysis

The recently published ‘FE–Meshfree’ QUAD4 element is extended to geometrical non‐linear analysis. The shape functions for this element are obtained by combining meshfree and finite element shape functions. The concept of partition of unity (PU) is employed for the purpose. The new shape functions inherit their higher order completeness properties from the meshfree shape functions and the mesh‐distortion tolerant compatibility properties from the finite element (FE) shape functions. Updated Lagrangian formulation is adopted for the non‐linear solution. Several numerical example problems are solved and the performance of the element is compared with that of the well‐known Q4, QM6 and Q8 elements. The results show that, for regular meshes, the performance of the element is comparable to that of QM6 and Q8 elements, and superior to that of Q4 element. For distorted meshes, the present element has better mesh‐distortion tolerance than Q4, QM6 and Q8 elements. Copyright © 2009 John Wiley & Sons, Ltd.     

A locally anisotropic fluid–structure interaction remeshing strategy for thin structures with application to a hinged rigid leaflet

An immersed finite element fluid–structure interaction algorithm with an anisotropic remeshing strategy for thin rigid structures is presented in two dimensions. One specific feature of the algorithm consists of remeshing only the fluid elements that are cut by the solid such that they fit the solid geometry. This approach allows to keep the initial (given) fluid mesh during the entire simulation while remeshing is performed locally. Furthermore, constraints between the fluid and the solid may be directly enforced with both an essential treatment and elements allowing the stress to be discontinuous across the structure. Remeshed elements may be strongly anisotropic. Classical interpolation schemes – inf–sup stable on isotropic meshes – may be unstable on anisotropic ones. We specifically focus on a proper finite element pair choice. As for the time advancing of the fluid–structure interaction solver, we perform a geometrical linearization with a sequential solution of fluid and structure in a backward Euler framework. Using the proposed methodology, we extensively address the motion of a hinged rigid leaflet. Numerical tests demonstrate that some finite element pairs are inf–sup unstable with our algorithm, in particular with a discontinuous pressure. Copyright © 2015 John Wiley & Sons, Ltd.     

On the enhancement of low‐order mixed finite element methods for the large deformation analysis of diffusion in solids

A stabilized scheme is developed for mixed finite element methods for strongly coupled diffusion problems in solids capable of large deformations. Enhanced assumed strain techniques are employed to cure spurious oscillation patterns of low‐order displacement/pressure mixed formulations in the incompressible limit for quadrilateral elements and brick elements. A study is presented that shows how hourglass instabilities resulting from geometrically nonlinear enhanced assumed strain methods have to be distinguished from pressure oscillation patterns due to the violation of the inf‐sup condition. Moreover, an element formulation is proposed that provides stable results with respect to both types of instabilities. Comparisons are drawn between material models for incompressible solids of Mooney–Rivlin type and models for standard diffusion in solids with incompressible matrices such as polymeric gels. Representative numerical examples underline the ability of the proposed element formulation to cure instabilities of low‐order mixed formulations. Copyright © 2015 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.