Abstract

Many finite elements exhibit the so‐called ‘volumetric locking’ in the analysis of incompressible or quasi‐incompressible problems.In this paper, a new approach is taken to overcome this undesirable effect. The starting point is a new setting of the governing differential equations using a finite calculus (FIC) formulation. The basis of the FIC method is the satisfaction of the standard equations for balance of momentum (equilibrium of forces) and mass conservation in a domain of finite size and retaining higher order terms in the Taylor expansions used to express the different terms of the differential equations over the balance domain. The modified differential equations contain additional terms which introduce the necessary stability in the equations to overcome the volumetric locking problem. The FIC approach has been successfully used for deriving stabilized finite element and meshless methods for a wide range of advective–diffusive and fluid flow problems. The same ideas are applied in this paper to derive a stabilized formulation for static and dynamic finite element analysis of incompressible solids using linear triangles and tetrahedra. Examples of application of the new stabilized formulation to linear static problems as well as to the semi‐implicit and explicit 2D and 3D non‐linear transient dynamic analysis of an impact problem and a bulk forming process are presented.

Abstract

The characteristic‐based split (CBS) stabilization procedure developed originally in fluid mechanics has been adapted successfully to solid mechanics problems. The CBS algorithm has been implemented within a finite element program using an explicit time integration scheme. Volumetric locking of linear triangular and tetrahedral elements has been successfully eliminated. The performance of the numerical algorithm is illustrated with numerical results. Comparisons with an alternative stabilization technique based on the finite calculus method also are given. Copyright © 2006 John Wiley & Sons, Ltd.

Abstract

Volumetric locking (locking in the incompressible limit) for linear elastic isotropic materials is studied in the context of the element-free Galerkin method. The modal analysis developed here shows that the number of non-physical locking modes is independent of the dilation parameter (support of the interpolation functions). Thus increasing the dilation parameter does not suppress locking. Nevertheless, an increase in the dilation parameter does reduce the energy associated with the non-physical locking modes; thus, in part, it alleviates the locking phenomena. This is shown for linear and quadratic orders of consistency. Moreover, the biquadratic order of consistency, as in finite elements, improves the locking behaviour. Although more locking modes are present in the element-free Galerkin method with quadratic consistency than with standard biquadratic finite elements. Finally, numerical examples are shown to validate the modal analysis. In particular, the conclusions of the modal analysis are also confirmed in an elastoplastic example.

Abstract

In this paper, we firstly introduce a nodal-integration-based finite element method. The method allows the use of first-order tetrahedral elements without suffering from the volumetric locking problem. The most important advantage of tetrahedral meshes is that they can be automatically generated for complex geometries using existing reliable meshing tools. The method is then applied to 3 types of applications. The first application is a large displacement, large strains elastic-plastic simulation on a notched specimen. The second application is an elastic-plastic bending problem. And the last example concerns the numerical simulation of the thermo-mechanical problem. In all the cases, the solution given by the nodal-integration-based FEM is compared to more classical FEM results.