A WEAK–WEAK FORMULATION FOR LARGE DISPLACEMENTS BEAM STATICS: A FINITE VOLUMES APPROXIMATION

This paper addresses the problem of the numerical solution of beam statics undergoing large displacements. A kinematic analysis outlines the beam geometrical model through the definition of its Lagrangian co-ordinate and strain parameters. A definition of the stress parameters, a constitutive law and an expression for the strain energy of the beam are then provided under the hypothesis of small strain. The equations governing the beam equilibrium are introduced and their weak form is derived. These equations are then proved to be equivalent to the primal and mixed form of Principle of Virtual Work. The numerical approximation is introduced by applying the bidiscontinuous finite elements method on the linearized weak form. The weak–weak formulation is attained by using the lowest interpolation order both for test and trial functions on two staggered decompositions of the space domain. Some numerical examples prove the capability of present formulation in handling actual problems.     

A CO-ROTATIONAL FORMULATION FOR 2-D CONTINUA INCLUDING INCOMPATIBLE MODES

The co-rotational finite element method is well known for decomposing the motion of an element into a rigid body motion and a strain-producing deformation. A key feature is the definition of the rotating frame and, partially for this reason, the method has almost exclusively been applied to beams and shells. A novel feature of the current paper is that the method is applied to continua—in the current case, two-dimensional. The main motivation is to allow the analyst to quickly introduce the latest and best linear elements into a non-linear context. To this end, in the current work, a set of ‘incompatible modes’ or ‘enhanced strains’ is added to the conventional four-noded elements. While the main body of the paper considers small strains, as a further novel aspect, it later applies the co-rotational method to problems with large strains (here via hyperelasticity) and, to this end, establishes a link between the co-rotational technique and a Biot stress formulation.     

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.     

An isogeometric locking‐free NURBS‐based solid‐shell element for geometrically nonlinear analysis

In this work, we develop an isogeometric non‐uniform rational B‐spline (NURBS)‐based solid‐shell element for the geometrically nonlinear static analysis of elastic shell structures. A single layer of continuous 3D elements through the thickness of the shell is considered, and the order of approximation in that direction is chosen to be equal to two. A complete 3D constitutive relation is assumed. The objective is to develop a highly accurate low‐order element for coarse meshes. We propose an extension of the mixed method of Bouclier et al. [11] to deal with locking in the context of large rotations and large displacements. The main idea is to modify the interpolation of the average through the thickness of the stress components. It is also necessary to stabilize the element in order to avoid the occurrence of spurious zero‐energy modes. This was achieved, for the quadratic version, through the adjunction of artificial elementary stabilization stiffnesses. The result is an element of order 2, which is at least as accurate as standard NURBS shell elements of order 4. Linear and nonlinear test calculations have been carried out along with comparisons with other published NURBS and classical techniques in order to assess the performance of the element. Copyright © 2014 John Wiley & Sons, Ltd.