The stability of stars of triangular equilibrium plate elements

Equilibrium models for finite element analyses are becoming increasingly important in complementary roles to those from conventional conforming models, but when formulating equilibrium models questions of stability, or admissibility of loads, are of major concern. This paper addresses these questions in the context of flat plates modelled with triangular hybrid elements involving membrane and/or flexural actions. Patches of elements that share a common vertex are considered, and such patches are termed stars. Stars may be used in global analyses as assemblies of elements forming macro‐elements, or in local analyses. The conditions for stability, or the existence and number of spurious kinematic modes, are determined in a general algebraic procedure for any degree of the interpolation polynomials and for any geometric configuration. The procedure involves the determination of the rank of a compatibility matrix by its transformation to row echelon form. Examples are presented to illustrate some of the characteristics of spurious kinematic modes when they exist in stars with open or closed links. Copyright © 2008 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.     

An efficient methodology for stress-based finite element approximations in two-dimensional elasticity

We present an efficient approach to compute the matrices used in finite element formulations where self-equilibrated (zero divergence) approximations of the stress field are used. The fundamental aspect of this approach is that it is applicable to polynomial approximations of high degree (it was tested up to degree 10), providing closed-form expressions for the elementary matrices involved. The components of the stress field are defined as a function of the coordinates in the master element, facilitating the postprocessing of the finite element results and allowing for the consideration of geometric parameters in reduced order models.     

Recovering local equilibrium from three-dimensional compatible finite element solutions

We study the application to compressible and incompressible three-dimensional elasticity problems of the technique that we proposed in 2009 for the recovery of equilibrated stresses from compatible finite element solutions. The case of finite elements with linear displacement approximations, for which the partitioned systems of loads are not initially balanced in terms of rotational equilibrium, addressed in 2012, is also considered. The article includes a presentation of the implementation details, a study of the convergence in terms of the energy of the error, and a characterization of the solutions obtained.     

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.     

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.     

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.     

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.     

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.     

Stress and strain‐driven algorithmic formulations for finite strain viscoplasticity for hybrid and standard finite elements

This work deals with the formulation and implementation of finite deformation viscoplasticity within the framework of stress‐based hybrid finite element methods. Hybrid elements, which are based on a two‐field variational formulation, are much less susceptible to locking than conventional displacement‐based elements. The conventional return‐mapping scheme cannot be used in the context of hybrid stress methods since the stress is known, and the strain and the internal plastic variables have to be recovered using this known stress field. We discuss the formulation and implementation of the consistent tangent tensor, and the return‐mapping algorithm within the context of the hybrid method. We demonstrate the efficacy of the algorithm on a wide range of problems. Copyright © 2009 John Wiley & Sons, Ltd.     

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.     

Geodesic finite elements on simplicial grids

We introduce geodesic finite elements as a conforming way to discretize partial differential equations for functions v : Ω → M, where Ω is an open subset of and M is a Riemannian manifold. These geodesic finite elements naturally generalize standard first‐order finite elements for Euclidean spaces. They also generalize the geodesic finite elements proposed for d = 1 in a previous publication of the author. Our formulation is equivariant under isometries of M and, hence, preserves objectivity of continuous problem formulations. We concentrate on partial differential equations that can be formulated as minimization problems. Discretization leads to algebraic minimization problems on product manifolds Mⁿ. These can be solved efficiently using a Riemannian trust‐region method. We propose a monotone multigrid method to solve the constrained inner problems with linear multigrid speed. As an example, we numerically compute harmonic maps from a domain in to S². Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

A general degree hybrid equilibrium finite element for Kirchhoff plates

We present a family of hybrid equilibrium finite elements for the quasistatic linear elastic analysis of plates governed by Kirchhoff theory. The moments are approximated by self‐balanced polynomial fields of general degree, and in order to impose strong codiffusivity, the normal boundary rotations are approximated with complete polynomials of the same degree, whereas the transverse deflections use polynomials one degree lower. Furthermore, it is also necessary to include an independent approximation of the vertex translations. We show that the triangular form of this element is stable, that is, free from spurious kinematic modes, and the formulation that we present allows these elements to be used as a standard displacement element. Examples of computed values and convergence of the solutions are presented, which demonstrate the performance of these elements. Copyright © 2013 John Wiley & Sons, Ltd.     

Classification of stress modes in assumed stress fields of hybrid finite elements

A classification method is presented to classify stress modes in assumed stress fields of hybrid finite element based on the eigenvalue examination and the concept of natural deformation modes. It is assumed that there only exist m (=n−r) natural deformation modes in a hybrid finite element which has n degrees of freedom and r rigid-body modes. For a hybrid element, stress modes in various assumed stress fields proposed by different researchers can be classified into m stress mode groups corresponding to m natural deformation modes and a zero-energy stress mode group corresponding to rigid-body modes by the m natural deformation modes. It is proved that if the flexibility matrix [H] is a diagonal matrix, the classification of stress modes is unique. Each stress mode group, except the zero-energy stress mode group, contains many stress modes that are interchangeable in an assumed stress field and do not cause any kinematic deformation modes in the element. A necessary and sufficient condition for avoiding kinematic deformation modes in a hybrid element is also presented. By means of the m classified stress mode groups and the necessary and sufficient condition, assumed stress fields with the minimum number of stress modes can be constructed and the resulting elements are free from kinematic deformation modes. Moreover, an assumed stress field can be constructed according to the problem to be solved. As examples, 2-D, 4-node plane element and 3-D, 8-node solid element are discussed. © 1997 John Wiley & Sons, Ltd.     

Enriched contact finite elements for stable peeling computations

During peeling of a soft elastic strip from a substrate, strong adhesional forces act locally inside the peeling zone. It is shown here that when a standard contact finite element (FE) formulation is used to compute the peeling process, a large mesh refinement is required since the numerical solution procedure becomes unstable otherwise. To improve this situation, several different efficient enrichment strategies are presented that provide stable solution algorithms for comparably coarse meshes. The enrichment is based on the introduction of additional unknowns inside the contact elements discretizing the slave surface. These are chosen in order to improve the approximation of the peeling forces, while keeping the overall number of degrees of freedom low. If needed, these additional unknowns can be condensed out locally. The enrichment formulation is developed for both 2D and 3D nonlinear FE formulations. The new enrichment technique is applied to the peeling computation of a gecko spatula. The proposed enriched contact element formulations are also investigated in sliding computations. Copyright © 2011 John Wiley & Sons, Ltd.     

Modeling of cement hydration in high performance concrete structures with hybrid finite elements

A hybrid finite element formulation is used to model the hygro‐thermo‐chemical process of cement hydration in high performance concrete. The temperature and the relative humidity fields are directly approximated in the domain of the element using naturally hierarchical bases independent of the mapping used to define its geometry. This added flexibility in modeling implies the independent approximation of the heat and moisture flux fields on the boundary of the element, the typical feature of hybrid finite element formulations. The formulation can be implemented using coarse and, eventually, unstructured meshes, which may contain elements with high aspect ratios, an option that can be advantageously used in the simulation of the casting of concrete structural elements. The resulting solving system is sparse and well suited to adaptive refinement and parallelization. It is solved coupling a trapezoidal time integration rule with an adaptation of the Newton–Raphson method designed to preserve symmetry. The relative performance of the formulation is assessed using a set of testing problems supported by experimental data and results obtained with conventional (conform) finite elements. Copyright © 2015 John Wiley & Sons, Ltd.     

Stabilized low‐order finite elements for strongly coupled poromechanical problems

Finite element solutions of poromechanical problems often exhibit oscillating pore pressures in the limits of low permeability, fast loading rates, coarse meshes, and/or small time step sizes. To suppress completely the pore pressure oscillations, a stabilized finite element scheme with a better performance on monotonicity is proposed for modeling compressible fluid‐saturated porous media. This method, based on the polynomial pressure projection technique, allows the use of linear equal‐order interpolation for both displacement and pore pressure fields, which is more straightforward for both code development and maintenance compared to others. By employing the discrete maximum principle, a proper stabilization parameter is deduced, which is efficient to guarantee the monotonicity and optimal in theory in the 1‐dimensional case. An appealing feature of the method is that the stabilization parameter is evaluated in terms of the properties of porous material only, while no mesh or time step size is involved. Through comparing the numerical simulations with the analytical benchmarks, the efficiency of the proposed stabilization scheme is confirmed.     

A vertex‐based finite volume method applied to non‐linear material problems in computational solid mechanics

A vertex‐based finite volume (FV) method is presented for the computational solution of quasi‐static solid mechanics problems involving material non‐linearity and infinitesimal strains. The problems are analysed numerically with fully unstructured meshes that consist of a variety of two‐ and three‐dimensional element types. A detailed comparison between the vertex‐based FV and the standard Galerkin FE methods is provided with regard to discretization, solution accuracy and computational efficiency. For some problem classes a direct equivalence of the two methods is demonstrated, both theoretically and numerically. However, for other problems some interesting advantages and disadvantages of the FV formulation over the Galerkin FE method are highlighted. Copyright © 2002 John Wiley & Sons, Ltd.     

Enriching the finite element method with meshfree particles in structural mechanics

The automatic generation of meshes for the finite element (FE) method can be an expensive computational burden, especially in structural problems with localized stress peaks. The use of meshless methods can address such an issue, as these techniques do not require the existence of an underlying connection among the particles selected in a general domain. This study advances a numerical strategy that blends the FE method with the meshless local Petrov–Galerkin technique in structural mechanics, with the aim at exploiting the most attractive features of each procedure. The idea relies on the use of FEs to compute a background solution that is locally improved by enriching the approximation space with the basis functions associated to a few meshless points, thus taking advantage of the flexibility ensured by the use of particles disconnected from an underlying grid. Adding the meshless particles only where needed avoids the cost of mesh refining, or even of remeshing, without the prohibitive computational cost of a thoroughly meshfree approach. In the present implementation, an efficient integration strategy for the computation of the coefficients taking into account the mutual FE–meshless local Petrov–Galerkin interactions is introduced. Moreover, essential boundary conditions are enforced separately on both FEs and meshless particles, thus allowing for an overall accuracy improvement also when the enriched region is close to the domain boundary. Numerical examples in structural problems show that the proposed approach can significantly improve the solution accuracy at a local level, with no remeshing effort, and at a low computational cost. Copyright © 2016 John Wiley & Sons, Ltd.