A penalty finite element approach for couple stress elasticity

There is mounting evidence for size dependent elastic deformation at micron and submicron length scales. Material formulations incorporating higher order gradients in displacements have been successful in modeling such size dependent deformation behavior. A couple stress theory without micro-rotation is considered here as micro-rotations increase complexity and necessitate parameters that are difficult to determine. Higher order gradient theories require continuity in both displacements and their derivatives and direct approaches with both displacements and their derivatives as nodal variables results in a large number of degrees of freedom. Here nodal rotations are applied along with nodal displacements to obtain a simpler finite element formulation with fewer degrees of freedom. The difference in rotation gradients determined with nodal displacements and rotations are minimized by a penalty term. To assess the suggested approach simulations are presented and discussed, where the material parameters have been obtained from experiments of epoxy microbeams in the literature.     

A general finite element approach for contact stress analysis

This paper presents a general finite element approach for the treatment of contact stress problems. Stanctard shape function routines are used for the detection of contact between previously separate meshes and for the application of displacement constraints where contact has been identified. The mesh contact routines are installed in an incremental approach whereby the contact constraints are imposed by using either penalty functions or Lagrange multipliers.     

Alternate hybrid stress finite element models

Alternate hybrid stress finite element models in which the internal equilibrium equations are satisfied on the average only, while the equilibrium equations along the interelement boundaries and the static boundary conditions are adhered to exactly a priori, are developed. The variational principle and the corresponding finite element formulation, which allows the standard direct stiffness method of structural analysis to be used, are discussed. Triangular elements for a moderately thick plate and a doubly-curved shallow thin shell are developed. Kinematic displacement modes, convergence criteria and bounds for the direct flexibility-influence coefficient are examined.     

A new hybrid finite element approach for plane piezoelectricity with defects

In this paper, a new type of hybrid fundamental solution-based finite element method (HFS-FEM) is developed for analyzing plane piezoelectric problems with defects by employing fundamental solutions (or Green’s functions) as internal interpolation functions. The hybrid method is formulated based on two independent assumptions: an intra-element field covering the element domain and an inter-element frame field along the element boundary. Both general elements and a special element with a central elliptical hole or crack are developed in this work. The fundamental solutions of piezoelectricity derived from the elegant Stroh formalism are employed to approximate the intra-element displacement field of the elements, while the polynomial shape functions used in traditional FEM are utilized to interpolate the frame field. By using Stroh formalism, the computation and implementation of the method are considerably simplified in comparison with methods using Lekhnitskii’s formalism. The special-purpose hole element developed in this work can be used efficiently to model defects such as voids or cracks embedded in piezoelectric materials. Numerical examples are presented to assess the performance of the new method by comparing it with analytical or numerical results from the literature.     

A hybrid finite element of stress analysis of fastener details

Conventional finite element analysis of plates with fastener holes requires many degrees of freedom in the assembled structure model. The assumed-stress hybrid method is proposed as an alternative. The formulation is presented for a new hybrid element which simulates elastic behavior near a fastener hole, permitting greater efficiency in concentration of mesh refinement in regions where stress gradients are high and resulting in a significant reduction of total degrees of freedom. Performance tests demonstrate that the new element is capable of simulating both open and loaded fastener holes. Some recent example applications are mentioned.     

A general framework for interpreting time finite element formulations

In this work, an attempt is made at filling the apparent gap existing between the two major approaches evolved in the literature towards formulating space-time finite element methods. The first assumes Hamilton's Law as underlying concept, while the second performs a weighted residual approach on the ordinary differential equations emanating from the semidiscretization in the space dimension.A general framework is proposed in the following pages, where the configuration space and the phase space forms of Hamilton's Law provide the general statements of the problem of motion. Within this framework, different families of integration algorithms are derived, according to different interpretations of the boundary terms. The bi-discontinuous form is obtained as the consequence of a consistent impulsive formulation of dynamics, while the discontinuous Galerkin form is obtained when the boundary terms at the end of the time interval are appropriately approximated.     

On hybrid stress,hybrid strain and enhanced strain finite element formulations for a geometrically exact shell theory with drilling degrees of freedom

The paper is concerned with the finite element formulation of a recently proposed geometrically exact shell theory with natural inclusion of drilling degrees of freedom. Stress hybrid finite elements are contrasted by strain hybrid elements as well as enhanced strain elements. Numerical investigations and comparison is carried out for a four-node element as well as a nine-node one. As far as the four-node element is concerned it is shown that the stress hybrid element and the enhanced strain one are equivalent. The hybrid strain formulation corresponds to the hybrid stress formulation only in shear dominated problems, that is the case of the plate. © 1998 John Wiley & Sons, Ltd.     

An assumed stress hybrid curvilinear triangular finite element for plate bending

An assumed stress hybrid curvilinear triangular finite element is described which is based upon the Kirchhoff theory of plate bending. The derivation extends the assumed stress hybrid technique to curvilinear boundaries where the twelve connectors are related to those of an equilibrium rectilinear element and to Semiloof. The solution process demands only first derivatives of the shape functions. The element is subjected to various patch tests for constant bending, e.g. where the central element is in close approximation to a circle. All tests are passed for stress couples and vertex displacements, but values of the remaining connectors do not resemble exact results. Patch tests for rigid-body movements are passed exactly in every respect.     

A hybrid finite element model for multilayered plates

We present a finite element model for multilayered plates, based on a primal-hybrid variational formulation. Namely, each layer is analyzed as it were a lonely structure, and the displacement continuity is imposed from one layer to the other by means of Lagrange multipliers. Then, a Mindlin-like displacement field is assumed for any layer; the resulting continuous problem is proven to be well-posed under rather general hypotheses. Finally, a finite element model is deduced, using a very simple scheme (piecewise linear approximation for the displacement components and piecewise constant Lagrange multipliers). The numerical results assess the good performance of the proposed model.     

A restricted hybrid stress formulation based on a direct finite element method

A direct method, which uses stress and displacement modes obtained from the governing equations of a problem, is adopted for finite element formulation. It is shown that this method actually leads to a restricted hybrid stress formulation if the displacement modes are changed to ensure symmetry of the stiffness matrix. Through this direct method, however, the problem of selecting the appropriate number of stress modes in the regular hybrid stress model is bypassed. Only the minimum number of modes that are compatible with the number of nodal degrees-of-freedom of an element is needed in the formulation. Using more modes only leads to a combination of stress modes, and will not improve the order of performance of the element. It is shown through numerical examples that the restricted hybrid stress formulation leads to well-balanced elements.     

A polygonal finite element for plate bending problems using the assumed stress approach

The assumed stress distribution approach is used to derive the stiffness matrix of a plate-bending element of general polygonal shape having any number of nodes. The effect of assuming various numbers of unknown coefficients in the stress distributions is examined and the convergence properties of the resulting elements compared with others derived form assumed displacements.     

Penalty-function iterative procedures for mixed finite element formulations

Iterative methods for solving mixed finite element equations that correct displacement and stress unknowns in ‘staggered’ fashion are attracting increased attention. This paper looks at the problem from the standpoint of allowing fairly arbitrary approximations to be made on both the stiffness and compliance matrices used in solving for the corrections. The resulting iterative processes usually diverge unless stabilized with Courant penalty terms. An iterative procedure previously constructed for equality-constrained displacement models is recast to fit the mixed finite element formulation in which displacements play the role of Lagrange multipliers. The penalty function iteration is shown to reduce to an ordinary staggered stress-displacement iteration if the weight is set to zero. Convergence conditions for these procedures are stated and the potentially troublesome effect of prestress modes noted.     

Dual finite element formulations for lumped reluctances coupling

A method for coupling magnetostatic and magnetodynamic finite element formulations with lumped reluctances is developed. Two dual h and b-conform formulations are extended to define the necessary magnetic relations between the magnetic fluxes and the magnetomotive forces. Adequate surface scalar potentials are defined and adequate boundary terms in the weak formulations are used. The magnetic relations can then be included in a reluctance network, in which the coupling of finite element regions and lumped regions aims at higher computational efficiency. The methods are developed in three dimensions, using a coupling of nodal and edge finite element approximations for the unknowns, and can easily be particularized in two dimensions.     

An explicit finite element approach with patch projection technique for strain gradient plasticity formulations

Implicit and explicit finite element approaches are frequently applied in real problems. Explicit finite element approaches exhibit several advantages over implicit method for problems which include dynamic effects and instability. Such problems also arise for materials and structures at small length scales and here length scales at the micro and sub-micron scales are considered. At these length scales size effects can be present which are often treated with strain gradient plasticity formulations. Numerical treatments for strain gradient plasticity applying the explicit finite element approach appear however to be absent in the scientific literature. Here such a numerical approach is suggested which is based on patch recovery techniques which have their origin in error indication procedures and adaptive finite element approaches. Along with the proposed explicit finite element procedure for a strain gradient plasticity formulation some numerical examples are discussed to assess the suggested approach.     

Two hybrid stress membrane finite element families with drilling rotations

Two new assumed stress membrane finite element families with two translational and one rotational degree of freedom per node are presented. The families, denoted by 8β(M) and 8β(D), are rank sufficient and invariant, and are derived using a unified Hu–Washizu like variational formulation. A recent stress mode classification method is used to illustrate the construction of the stress interpolation matrices. In each family, a number of new formulations are derived. Numerical results are presented. Copyright © 2001 John Wiley & Sons, Ltd.     

Combined hybrid approach to finite element schemes of high performance

Based on a combination of Hellinger–Reissner principle with its dual, a mixed/hybrid method of non‐saddle point type has been presented in the previous papers. In the paper, an application of the method to two‐dimensional elasticity problems is considered in a manner of combination of mathematical analysis with numerical investigation. How to construct a combined hybrid scheme of high performance is discussed, and several four‐node quadrilateral CH‐elements are systematically verified. In brief, this paper shows that for the hybrid element scheme, the energy compatibility condition, introduced in Zhou (Combined hybrid finite element method without requirement for Babuska‐Brezzi condition, to appear) is the ultimate key to the achievement of high performance. Copyright © 2001 John Wiley & Sons, Ltd.     

A new approach for the hybrid element method

A new functional which forms the basis of an improved hybrid element formulation is proposed. The variables for the functional include stresses, strains and displacements, and the displacements and stresses are further decomposed into two parts respectively. The proposed new formulation appears to be particularly suitable for improving conforming models. Based on this formulation, a new four-node plane hybrid element Q_cs6 can be developed, and the conforming element Q₄ and the non-conforming Wilson element Q and modified Wilson element Qm₆ can also be derived directly by this hybrid approach. It should be noted that more accurate stresses can be obtained from this element which utilizes the concept of two stress components.     

Primitive variable finite element formulations for steady viscous flows

The solution of incompressible, viscous flow problems using the primitive variable finite element approach is formulated in a coherent and physically reasoned exposition. In particular, it is shown that terms associated with changes in momentum caused by lack of mass conservation should logically be included in the governing equations. Previous formulations have not considered this possibility. Numerical results are presented to show the effect of inclusion of these extra terms; however, it is concluded from the numerical results for the specific case which was tested that there is no obvious benefit obtained by including these terms.     

On finite element formulations for nearly incompressible linear elasticity

In this paper we present a mixed stabilized finite element formulation that does not lock and also does not exhibit unphysical oscillations near the incompressible limit. The new mixed formulation is based on a multiscale variational principle and is presented in two different forms. In the first form the displacement field is decomposed into two scales, coarse-scale and fine-scale, and the fine-scale variables are eliminated at the element level by the static condensation technique. The second form is obtained by simplifying the first form, and eliminating the fine-scale variables analytically yet retaining their effect that results with additional (stabilization) terms. We also derive, in a consistent manner, an expression for the stabilization parameter. This derivation also proves the equivalence between the classical mixed formulation with bubbles and the Galerkin least-squares type formulations for the equations of linear elasticity. We also compare the performance of this new mixed stabilized formulation with other popular finite element formulations by performing numerical simulations on three well known test problems.