Finite element formulations for transient dynamic problems in solids using explicit time integration

This paper presents the displacement, mixed and stress formulations of the finite element method when applied to transient dynamic problems of solids. The formulations are chosen so that explicit time integration may be used. Large deformations are considered for these formulations, and infinitesimal strain assumptions are employed with the stress formulation. Displacement formulations are well-known, but the mixed formulations presented provide a viable alternative. The stress formulation has not proven successful for the large deformation problem, but when infinitesimal strains are assumed, the formulation is attractive. A problem of an internally pressurized ring is solved in order to evaluate the different proposed formulations.     

Mechanics of field-consistency in finite element analysis—a penalty function approach

Although C° continuous beam, plate and shell elements have become very popular, a simple understanding of several difficulties such as shear locking, parasitic shear, stress oscillations, etc. is not yet available in the literature. In displacement type finite element formulations, these difficulties can be easily rationalized using “field-consistency” concepts and a priori quantitative error estimates of these difficulties can also be made using an operational procedure called the “functional re-constitution” technique. In this paper we demonstrate the mechanics of these concepts using the simplest shear flexible beam element (linear Timoshenko beam element) as an example, and verify its a priori projections through digital computations. The rationale behind the use of optimal stress sampling at Gaussian points is also derived directly from these arguments.     

Mixed finite element formulations of strain-gradient elasticity problems

Theories on intrinsic or material length scales find applications in the modeling of size-dependent phenomena. In elasticity, length scales enter the constitutive equations through the elastic strain energy function, which, in this case, depends not only on the strain tensor but also on gradients of the rotation and strain tensors. In the present paper, the strain-gradient elasticity theories developed by Mindlin and co-workers in the 1960s are treated in detail. In such theories, when the problem is formulated in terms of displacements, the governing partial differential equation is of fourth order. If traditional finite elements are used for the numerical solution of such problems, then C¹ displacement continuity is required. An alternative “mixed” finite element formulation is developed, in which both the displacement and the displacement gradients are used as independent unknowns and their relationship is enforced in an“integral-sense”. A variational formulation is developed which can be used for both linear and non-linear strain-gradient elasticity theories. The resulting finite elements require only C⁰ continuity and are simple to formulate. The proposed technique is applied to a number of problems and comparisons with available exact solutions are made.     

A stabilized mixed finite element method for Darcy flow

We develop new stabilized mixed finite element methods for Darcy flow. Stability and an a priori error estimate in the “stability norm” are established. A wide variety of convergent finite elements present themselves, unlike the classical Galerkin formulation which requires highly specialized elements. An interesting feature of the formulation is that there are no mesh-dependent parameters. Numerical tests confirm the theoretical results.     

A feedback finite element method with a posteriori error estimation: Part I. The finite element method and some basic properties of the a posteriori error estimator

This paper is the first in a series of two in which we discuss some theoretical and practical aspects of a feedback finite element method for solving systems of linear second-order elliptic partial differential equations (with particular interest in classical linear elasticity). In this first part we introduce some nonstandard finite element spaces, which, though based on the usual square bilinear elements, permit local mesh refinement. The algebraic structure of these spaces and their approximation properties are analyzed. An “equivalent estimator” for the H¹ finite element error is developed. In the second paper we shall discuss the asymptotic properties of the estimator and computational experience.     

Nonlinear shell models with seven kinematic parameters

We discuss design of nonlinear finite rotation shell model with seven kinematic displacement-like parameters, which are: three displacements of the middle surface, two rotations of the shell director, and two through-the-thickness stretching parameters. From the theoretical side we examine several possibilities for constructing the enriched kinematic field, which leads to different higher-order 7-parameter shell formulations. From the finite element implementation side a shell director interpolation is identified which eliminates the “curvature thickness locking”. Numerical examples are presented in order to compare different formulations and to illustrate the performance of the developed finite elements.     

Non-linear strain-displacement equations exactly representing large rigid-body motions. Part III. Analysis of TM shells with constraints

The precise representation of arbitrarily large rigid-body motions in the displacement patterns of curved Timoshenko-Mindlin-type (TM) shell elements has been considered in Part I of the present work. In Part II it has been developed an enhanced mixed finite element formulation that allows using load increments that are much larger than possible with existing geometrically exact displacement-based shell element formulations. In this paper the developed formulation is employed to solve frictionless contact problems for TM shells undergoing finite deformations and interacting with rigid bodies. The contact conditions are incorporated into the assumed stress-strain TM shell formulation by applying a perturbed Lagrangian procedure with the fundamental unknowns consisting of 6 displacements and 11 strains of the bottom and top surfaces of the shell, 11 conjugate stress resultants and the Lagrange multiplier, associated with a nodal contact force, through using the non-conventional technique. The efficiency and accuracy of the proposed finite element formulation are demonstrated by means of several numerical examples.     

Mixed isoparametric finite element models of laminated composite shells

Mixed shear-flexible isoparametric elements are presented for the stress and free vibration analysis of laminated composite shallow shells. Both triangular and quadrilateral elements are considered. The “generalized” element stiffness, consistent mass, and consistent load coefficients are obtained by using a modified form of the Hellinger-Reissner mixed variational principle. Group-theoretic techniques are used in conjunction with computerized symbolic integration to obtain analytic expressions for the stiffness, mass and load coefficients. A procedure is outlined for efficiently handling the resulting system of algebraic equations.The accuracy of the mixed isoparametric elements developed is demonstrated by means of numerical examples, and their advantages over commonly used displacement elements are discussed.     

T-elements: a finite element approach with advantages of boundary solution methods

Since their introduction in 1977, the so-called T-elements have considerably evolved and have now become a highly efficient and well established tool for the solution of complex boundary value problems. This class of finite elements, associated with the Trefftz method, is based on enforcement of interelement continuity and boundary conditions on assumed displacement fields chosen so as to a priori satisfy the governing differential equations of the problem. Several alternative T-element formulations are available which yield for a particular subdomain the customary force-displacement relationship with a symmetric positive definite stiffness matrix which makes it possible for such elements to be implemented into the standard finite element (FE) codes.

Owing to their nature, the T-elements may either be considered as a new FE model or as a non-conventional symmetric substructure-oriented form of the boundary element method (BEM). From the point of view of the latter, the outstanding features of the T-element approaches are the use of T-complete sets of non-singular solutions (rather than the singular Kelvin's type fundamental solutions) and the replacement of the customary integral equations form by a symmetric variational formulation.

This paper reviews and critically assesses the most important T-element formulations developed over the past years. It shows that such elements not only cumulate the advantages and discard the drawbacks of the conventional finite element and boundary element methods, but also offer additional advantages not available in the standard form of these methods.     

Basis for isoparametric stress elements

The family of the so-called ‘isoparametric strain (displacement) elements’ is restricted to membranes and solids. The reason for this restriction has led to the development of a new family based on stress assumptions; these elements will be referred to as ‘isoparametric stress elements’. This family contains plates and solids but no membranes. The omission of a particular element in each family is consistent with the plate-membrane analogies. The basic flexibility matrix of an isoparametric stress element is singular since the zero stress state is directly included. The rank technique is adopted to automatically extract the zero stress modes such that the element can be completely interchangeable between any finite element system. The theory for stress assumed isoparametric “quadrilateral” plate bending elements with curved boundaries is given. A brief presentation of the theory for isoparametric stress solid elements is also included.     

Gershgorin theory for stiffness and stability of evolution systems and convection-diffusion

An analysis of stiffness and stability based on Gershgorin theorems for eigenvalues is developed for initial value systems. In particular, semidiscrete formulations for evolution problems are analysed. Common techniques such as semidiscrete finite difference and finite element methods are examined using eigenvalue bounds to characterize stiffness and stability of the associated systems. The analysis is applied to a prototype convection-diffusion problem to demonstrate the arguments and clarify several current questions concerning the qualitative nature of the solution and errors, including effects of: “lumped” versus “consistent” finite element formulations; high- or low-degree bases; mesh refinement, dimensionality and differing material properties. To study general initial value systems such as those arising in consistent finite element formulations, a generalized Gershgorin theory and computable bounds in the chordal metric are utilized.     

Alternate stress and conjugate strain measures,and mixed variational formulations involving rigid rotations,for computational analyses of finitely deformed solids,with application to plates and shells—I: Theory

Attention is focused in this paper on: (i) definitions of alternate measures of “stress-resultants” and “stress-couples” in a finitely deformed shell (finite mid-plane stretches as well as finite rotations); (ii) mixed variational principles for shells, undergoing large mid-plane stretches and large rotations, in terms of a stress function vector and the rotation tensor. In doing so, both types of polar decomposition, namely rotation followed by stretch, as well as stretch followed by rotation, of the shell midsurface, are considered; (iii) two alternate bending strain measures which depend on rotation alone for a finitely deformed shell; (iv) objectivity of constitutive relations, in terms of these alternate strain/“stress-resultants”, and “stress-couple” measures, for finitely deformed shells. To motivate these topics, and for added clarity, a discussion of relevant alternate stress measures, work-conjugate strain measures, and mixed variational principles with rotations as variables, is presented first in the context of three-dimensional continuum mechanics.Comments are also made on the use of the presently developed theories in conjunction with mixed-hydrid finite element methods. Discussion of numerical schemes and results is deferred to the Part II of the paper, however.     

Object-oriented finite element programming-the importance of data modelling

A C++ implementation of a finite element class system and its links to a graphical model of a structure are described. The principles underlying the finite element and graphical class systems are outlined, together with the reasoning behind the design. Two of the key points are (i) the finite element classes have a “lean” interface; (ii) the finite element objects (e.g. nodes and elements) are distributed around the graphical model objects (e.g. points, lines, sub-structures). Some of the advantages of adopting such an approach are outlined with reference to user interaction, mesh generation, and sub-structuring.     

An analysis of,and remedies for,kinematic modes in hybrid-stress finite elements: selection of stable,invariant stress fields

In this paper a study of the existence of spurious kinematic modes in hybrid-stress finite elements, based on assumed equilibrated stresses and compatible boundary displacements, and the resulting rank-deficiency of the element stiffness matrix, is presented. A method of selection of least-order, stable, invariant, stress fields is developed so as to ensure the prevention of kinematic modes. A 20-node cubic element, a 8-node cubic element and a 4-node square, based on assumed equilibrated stresses within the element and compatible displacements at the boundary of the element, are discussed for purposes of illustration. Comments are made on the generality of the present method, which is based on group theoretical arguments.     

A computer algebra based finite element development environment

A finite element development environment based on the technical computing program Mathematica is described. The environment is used to automatically program standard element formulations and develop new elements with novel features. Source code can also be exported in a format compatible with commercial finite element program user-element facilities. The development environment is demonstrated for three mixed Petrov–Galerkin plane stress elements: a standard formulation, an advanced formulation incorporating rotational degrees of freedom and a standard formulation in which the stiffness matrix is integrated analytically, before being exported as ANSYS user elements. The results presented illustrate the accuracy of the standard mixed formulation element and the enhancement of performance when rotational degrees of freedom are added. Further, the analytically integrated element shows that computational requirements can be greatly reduced when analytical integration schemes are used in the formation.     

Error estimates of fully discrete mixed finite element methods for semilinear quadratic parabolic optimal control problem

In this paper we study the fully discrete mixed finite element methods for quadratic convex optimal control problem governed by semilinear parabolic equations. The space discretization of the state variable is done using usual mixed finite elements, whereas the time discretization is based on difference methods. The state and the co-state are approximated by the lowest order Raviart–Thomas mixed finite element spaces and the control is approximated by piecewise constant elements. By applying some error estimates techniques of mixed finite element methods, we derive a priori error estimates both for the coupled state and the control approximation. Finally, we present a numerical example which confirms our theoretical results.     

Frame element with lateral deformable supports: Formulations and numerical validation

This paper presents the theory and the numerical validation of three different formulations of nonlinear frame elements with nonlinear lateral deformable supports. The governing differential equations of the problem are derived first and the three different finite element formulations are then presented. The first model follows a displacement-based formulation, which is based on the virtual displacement principle. The second one follows the force-based formulation, which is based on the virtual force principle. The third model follows the Hellinger–Reissner mixed formulation, which is based on the two-field mixed variational principle. The selection of the displacement and force interpolation functions for the different formulations is discussed. Tonti’s diagrams are used to conveniently represent the equations governing both the strong and the weak forms of the problem. The general matrix equations of the three formulations are presented, with some details on the issues regarding the elements’ implementations in a general-purpose finite element program. The convergence, accuracy, and computational times of the three elements are studied through a numerical example. The distinctive element characteristics in terms of force and deformation discontinuities between adjacent elements are discussed. The capability of the proposed frame models to trace the softening response due to softening of the foundation is also investigated. Overall, the force-based and the mixed models are much more accurate than the displacement-based model and require very few elements to reach the converged solution. The force-based element is slightly more accurate than the mixed model, but it is more prone to numerical instabilities as it involves inverting the element flexibility matrix.     

Alternative FEM formulations for the inelastic large deflection analysis of plane frames

The present paper is concerened with the analysis of plane frames of arbitrary geometry undergoing large elasto-plastic deformations. On the basis of so-called mixed-hybrid variational theorems wherein displacements and stress resultants represent independent variables, three alternative finite element schemes are proposed. Their theoretical and computational advantages in comparison with the standard assumed displacement formulation are pointed out and discussed in detail.