Finite calculus formulation for incompressible solids using linear triangles and tetrahedra

Abstract

SUMMARYMany 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.

Multiscale modelling of particle debonding in reinforced elastomers subjected to finite deformations

“…Considerable effort has been devoted in recent years to develop novel numerical techniques that give stable solution [30][31][32]. Especially, stabilized theories, where Babuška-Brezzi stability condition is circumvented, have been recently explored [33][34][35].…”

Finite increment calculus (FIC): a framework for deriving enhanced computational methods in mechanics

“…The FIC method can also be applied to derive a modified equation relating the pressure and the volumetric strain change over a finite size domain as [16,18] …”

“…Once more, for consistency, the Neumann boundary conditions should incorporate a FIC stabilization term as in Eq. (24a) [16,18].…”

Performance of mixed formulations for the particle finite element method in soil mechanics problems

“…The reason behind this behaviour is the failure to satisfy the Babuska-Brezzy conditions [2,8] or the equivalent inf-sup condition [3] due to an improper finitedimensional space in the finite element discretization. To avoid this problem two strategies are common: either to use more complex, but stable, finite elements [38,26], or to apply stabilization procedures to originally unstable finite elements [23,39,21]. In the later approach, locking is mitigated at the cost of using a mixed formulation, thereby introducing extra degrees of freedom per node with respect to the primal formulation.…”