A stabilized formulation for incompressible elasticity using linear displacement and pressure interpolations

Abstract

In this paper a stabilized finite element method to deal with incompressibility in solid mechanics is presented. A mixed formulation involving pressure and displacement fields is used and a continuous linear interpolation is considered for both fields. To overcome the Babu s ska-Brezzi condition, a stabilization technique based on the orthogonal sub-scale method is introduced. The main advantage of the method is the possibility of using linear triangular or tetrahedral finite elements, which are easy to generate for real industrial applications. Results are compared with standard Galerkin and Q1P0 mixed formulations for nearly incompressible problems in the context of linear elasticity.

Finite calculus formulation for incompressible solids using linear triangles and tetrahedra

“…Note that the value of i in Equation (24) deduced from the FIC formulation basically coincides for h i = h j = h with that of = h 2 /2G heuristically chosen in other works [6,[14][15][16][17][18].…”

“…As noted in Reference [17] the computational cost due to the iterative algorithm is negligible in a non-linear context where the projected pressure gradient can be computed within the equilibrium iterations induced by the non-linearity. Note that the algorithm of Equation (39a) is also applicable for the full incompressible case when K = ∞ and C = 0.…”

“…where again advantage can be taken of a diagonal form of M. This algorithm has been suggested by Chiumenti et al [17,18], who have proposed a similar formulation for elastic and elastoplastic problems based on the SGS method. As noted in Reference [17] the computational cost due to the iterative algorithm is negligible in a non-linear context where the projected pressure gradient can be computed within the equilibrium iterations induced by the non-linearity.…”

“…Extensions of the CBS algorithm to solve bulk metal forming problems have been recently reported by Rojek et al [11]. Other methods to overcome volumetric locking are based on mixed displacement (or velocity)-pressure formulations using the Galerkin-least-square (GLS) method [12], average nodal pressure [13] and average nodal deformation [14] techniques and sub-grid scale (SGS) methods [15][16][17][18].…”

Explicit mixed strain–displacement finite elements for compressible and quasi-incompressible elasticity and plasticity

“…Using displacements u as primary variables, the first proposals were based on reduced integration [30,38,39], and extended to the use of assumed deformations [47] and the B-bar method [31]. Mixed pressuredisplacement u − p approaches were introduced in the 90's and used thenceforth to address the incompressible limit [18,29,40,49]. The reason for using the pressure as independent variable is to gain control on it and ensure stability; this results in an overall satisfactory behavior of strains and stresses in quasi-incompressible situations.…”