A GENERAL FORMULATION OF EQUILIBRIUM MACRO-ELEMENTS WITH CONTROL OF SPURIOUS KINEMATIC MODES: THE EXORCISM OF AN OLD CURSE

This paper illustrates a method whereby a family of robust equilibrium elements can be formulated in a general manner. The effects of spurious kinematic modes, present to some extent in all primitive equilibrium elements, are eliminated by judicious assembly into macro-equilibrium elements. These macroelements are formulated with sufficient generality so as to retain the polynomial degree of the stress field as a variable. Such a family of macro-elements is a new development, and results for polynomials of degree greater than two have not been seen before. The quality of results for macro-equilibrium elements with varying degrees of polynomial is demonstrated by numerical examples.     

Suppression of spurious intermediate frequency modes in under‐integrated elements by combined stiffness/viscous stabilization

Solutions employing perturbation stiffness or viscous hourglass control with one‐point quadrature finite elements often exhibit spurious modes in the intermediate frequency range. These spurious frequencies are demonstrated in several examples and their origin is explained. Then it is shown that by critically damping the hourglass modes, these spurious mid‐range frequency modes can be suppressed. Estimates of the hourglass frequency and damping coefficients are provided for the plane 4‐node quadrilateral and a 4‐node shell element. Results are presented that show almost complete annihilation of spurious intermediate frequency modes for both linear and non‐linear problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Elimination of spurious static modes in meshes of hybrid compatible triangular and tetrahedral finite elements

In the assumed displacement, or primal, hybrid finite element method, the requirements of continuity of displacements across the sides are regarded as constraints, imposed using Lagrange multipliers. In this paper, such a formulation for linear elasticity, in which the polynomial approximation functions are not associated with nodes, is presented. Elements with any number of sides may be easily used to create meshes with irregular vertices, when performing a non‐uniform h‐refinement. Meshes of non‐uniform degree may be easily created, when performing an hp‐refinement. The occurrence of spurious static modes in meshes of triangular elements, when compatibility is strongly enforced, is discussed. An algorithm for the automatic selection, based on the topology of a mesh of triangular elements, of the sides in which to decrease the degree of the approximation functions, in order to eliminate all these spurious modes and preserve compatibility, is presented. A similar discussion is presented for the occurrence of spurious static modes in meshes of tetrahedral elements. An algorithm, based on heuristic criteria, that succeeded in eliminating these spurious modes and preserving compatibility in all the meshes of tetrahedral elements of uniform degree that were tested, is also presented. Copyright © 2001 John Wiley & Sons, Ltd.     

A SET OF HYBRID EQUILIBRIUM FINITE ELEMENT MODELS FOR THE ANALYSIS OF THREE-DIMENSIONAL SOLIDS

In this paper an approach to the formulation of equilibrium elements for the analysis of three-dimensional elasticity problems is presented. This formulation is an extension of the approach previously proposed for the analysis of two-dimensional elasticity problems. The general aspects of the formulation remain unchanged when applied to the new problem, but new points are considered, namely the way to perform volume integrations for general elements and the techniques used to obtain the self-equilibrated three-dimensional stress approximation functions. The numerical behaviour of such elements is presented and discussed.     

A distributed memory parallel implementation of the multigrid method for solving three‐dimensional implicit solid mechanics problems

We describe the parallel implementation of a multigrid method for unstructured finite element discretizations of solid mechanics problems. We focus on a distributed memory programming model and use the MPI library to perform the required interprocessor communications. We present an algebraic framework for our parallel computations, and describe an object‐based programming methodology using Fortran90. The performance of the implementation is measured by solving both fixed‐ and scaled‐size problems on three different parallel computers (an SGI Origin2000, an IBM SP2 and a Cray T3E). The code performs well in terms of speedup, parallel efficiency and scalability. However, the floating point performance is considerably below the peak values attributed to these machines. Lazy processors are documented on the Origin that produce reduced performance statistics. The solution of two problems on an SGI Origin2000, an IBM PowerPC SMP and a Linux cluster demonstrate that the algorithm performs well when applied to the unstructured meshes required for practical engineering analysis. Copyright © 2004 John Wiley & Sons, Ltd.     

A traction‐based equilibrium finite element free from spurious kinematic modes for linear elasticity problems

This paper presents equilibrium elements for dual analysis. A traction‐based equilibrium element is proposed in which tractions of an element instead of stresses are chosen as DOFs, and therefore, the interelement continuity and the Neumann boundary balance are directly satisfied. To be solvable, equilibrated tractions with respect to the space of rigid body motion are required for each element. As a result, spurious kinematic modes that may inflict troubles on stress‐based equilibrium elements do not appear in the element because only equilibrium constraints on tractions are required. An admissible stress field is eventually constructed in terms of the equilibrated tractions for the element, and hence, equilibrium finite element procedures can proceed. The element is also generalized to accommodate non‐zero body forces, nonlinear boundary tractions and curved Neumann boundaries. Numerical tests including a single equilibrium element, error estimation of a cantilever beam and an infinite plate with a circular hole are conducted, displaying excellent convergence and effectiveness of the element for error estimation. Copyright © 2014 John Wiley & Sons, Ltd.     

A triangular hybrid equilibrium plate element of general degree

Hybrid stress‐based finite elements with side displacement fields have been used to generate equilibrium models having the property of equilibrium in a strong form. This paper establishes the static and kinematic characteristics of a flat triangular hybrid equilibrium element with both membrane and plate bending actions of general polynomial degree p. The principal characteristics concern the existence of hyperstatic stress fields and spurious kinematic modes. The former are shown to exist for p>3, and their significance to finite element analysis is reviewed. Knowledge of the latter is crucial to the determination of the stability of a mesh of triangular elements, and to the choice of procedure adopted for the solution of the system of equations. Both types of characteristic are dependent on p, and are established as regards their numbers and general algebraic forms. Graphical illustrations of these forms are included in the paper. Copyright © 2005 John Wiley & Sons, Ltd.     

FINITE ELEMENT SOLUTION OF INCOMPRESSIBLE FLUID–STRUCTURE VIBRATION PROBLEMS

In this paper we solve an eigenvalue problem arising from the computation of the vibrations of a coupled system, incompressible fluid – elastic structure, in absence of external forces. We use displacement variables for both the solid and the fluid but the fluid displacements are written as curls of a stream function. Classical linear triangular finite elements are used for the solid displacements and for the stream function in the fluid. The kinematic transmission conditions at the fluid–solid interface are taken into account in a weak sense by means of a Lagrange multiplier. The method does not present spurious or circulation modes for non-zero frequencies. Numerical results are given for some test cases. © 1997 by John Wiley & Sons, Ltd.     

Edge‐based finite element techniques for non‐linear solid mechanics problems

Edge‐based data structures are used to improve computational efficiency of inexact Newton methods for solving finite element non‐linear solid mechanics problems on unstructured meshes. Edge‐based data structures are employed to store the stiffness matrix coefficients and to compute sparse matrix–vector products needed in the inner iterative driver of the inexact Newton method. Numerical experiments on three‐dimensional plasticity problems have shown that memory and computer time are reduced, respectively, by factors of 4 and 6, compared with solutions using element‐by‐element storage and matrix–vector products. Copyright © 2001 John Wiley & Sons, Ltd.     

Recovery of equilibrium on star patches from conforming finite elements with a linear basis

A previous technique for recovering equilibrated stresses from compatible finite element models of structural mechanics problems is extended to cover those cases where the partitioned loads applied to star patches are not initially balanced, regarding rotational equilibrium. The residual moments are removed by adding a suitable corrective stress field to the compatible one before deriving the fictitious body forces. Corrective stress fields are determined by solving another set of local problems based on subdomains that each contain elements forming a neighbourhood of a loaded kernel element. The conditions for the existence of a solution of these problems are studied for simplicial elements. The parameters that control the extended technique are assessed from numerical tests on a variety of two‐dimensional linear elastic problems based on constant strain elements for the compatible solutions. These tests are presented in the context of computing bounds on quantities of interest such as total strain energy and local reactions and displacements. Copyright © 2011 John Wiley & Sons, Ltd.     

Recovery of equilibrium on star patches using a partition of unity technique

A technique for recovering equilibrated element stresses is developed for finite element models of structural mechanics problems. The data for the method consist of the prescribed loading and the stress and displacement fields resulting from a conventional compatible finite element model. Local problems are defined for each star of elements, via the introduction of fictitious body forces and strains. These problems can be solved independently for equilibrium in a process that can be easily parallelized. The quality of the solutions is assessed, for two‐dimensional linear elastic problems, by using them to compute bounds of the error of the finite element solutions, in terms of both global and local quantities of interest. Copyright © 2009 John Wiley & Sons, Ltd.     

A finite element formulation for sliding beams,Part I

We use the updated Lagrangian and the co-rotational finite element methods to obtain solutions for geometrically non-linear flexible sliding beams. Finite element formulations are normally carried out for fixed domains. Since the sliding beam is a system of changing mass, first we discretize the system by introducing a variable-domain beam element and model the sliding beam by a fixed number of elements with changing length. Second, we transform the system governing equations of motion to a fixed domain and use conventional finite elements (fixed size and number) to discretize the system. Then our investigation is followed by a comparison between two formulations. Finally, we use the co-rotational method in conjunction with a variable domain beam element to obtain the discretized system equations. To do so, we consider the beam to slide with respect to a fixed channel and later we consider a formulation in which the beam remains at rest and the channel slides with a prescribed velocity. We show that both formulations end up with identical discretized equations of motion. © 1998 John Wiley & Sons, Ltd.     

A BRIEF HISTORY OF THE BEGINNING OF THE FINITE ELEMENT METHOD

This paper presents summaries of the works of several authors associated with the invention of the analysis technique now referred to as the finite element method. It stresses the notion of first development from which subsequent ideas evolved and gives what is believed to be an accurate record of the historical sequence of published papers in the international literature.     

Finite element interface models for the delamination analysis of laminated composites: mechanical and computational issues

The finite element analysis of delamination in laminated composites is addressed using interface elements and an interface damage law. The principles of linear elastic fracture mechanics are indirectly used by equating, in the case of single‐mode delamination, the area underneath the traction/relative displacement curve to the critical energy release rate of the mode under examination. For mixed‐mode delamination an interaction model is used which can fulfil various fracture criteria proposed in the literature. It is then shown that the model can be recast in the framework of a more general damage mechanics theory. Numerical results are presented for the analyses of a double cantilever beam specimen and for a problem involving multiple delamination for which comparisons are made with experimental results. Issues related with the numerical solution of the non‐linear problem of the delamination are discussed, such as the influence of the interface strength on the convergence properties and the final results, the optimal choice of the iterative matrix in the predictor and the number of integration points in the interface elements. Copyright © 2001 John Wiley & Sons, Ltd.     

Elimination of spurious pressure and kinematic modes in biquadratic nine-node plane element

In two-dimensional penalty finite element analysis of incompressible materials, Q9/3P, the 9-node quadrilateral with three assumed quasi-pressure modes, is the most popular element not bothered by spurious pressure. In constructing the penalty matrix of Q9/3P, it is necessary to form a 3×18 matrix and a 3×3 symmetric matrix. The inverse of the symmetric matrix is then post- and pre-multiplied by the 3×18 matrix and its transpose, respectively. By employing a rank subtraction technique, a new and more efficient implementation scheme is devised for the penalty matrix. Same as the conventional Q9 element, when Q9/3P is fully integrated, it becomes expensive and too stiff. On the other hand, there are two spurious kinematic modes should the element be sub-integrated. In the proposed Q9/3P element, the two mechanisms will be annihilated by judiciously chosen higher-order assumed stress modes in conjunction with a modified Hellinger-Reissner functional. It will be demonstrated that the element is of good accuracy and high efficiency.