Locking in the incompressible limit: pseudo-divergence-free element free Galerkin

Abstract

SUMMARYLocking in finite elements has been a major concern since its early developments. It appears because poor numerical interpolation leads to an over-constrained system. This paper proposes a new formulation that asymptotically suppresses locking for the Element Free Galerkin (EFG) method in incompressible limit, i.e. the so-called volumetric locking. Originally it was claimed that EFG did not present volumetric locking. However, recently, performing a modal analysis, the senior author has shown that EFG presents volumetric locking. In fact, it is concluded that an increase of the dilation parameter attenuates, but never suppresses, the volumetric locking and that, as in standard finite elements, an increase in the order of reproducibility (interpolation degree) reduces the relative number of locking modes. Here an improved formulation of the Element Free Galerkin method is proposed in order to alleviate volumetric locking.Diffuse derivatives are defined in the thesis of the second author and their convergence to the derivatives of the exact solution, when the radius of the support goes to zero (for a fixed dilation parameter), it's proved. Therefore diffuse divergence converges to the exact divergence. Since the diffuse divergence-free condition can be imposed a priori, new interpolation functions are defined that asymptotically verify the incompressibility condition. Modal analysis and numerical results for classical benchmark tests in solids corroborate this issue.

A meshfree‐enriched finite element method for compressible and near‐incompressible elasticity

“…The influence of nodal supports and approximation order on the locking behavior was also discussed. A pseudodivergence‐free interpolation for EFG method was proposed by Vidal et al to impose the divergence‐free constraint a priori in a displacement‐based Galerkin meshfree formulation. Another Galerkin meshfree formulation was presented by Chen et al for the rubber‐like incompressible materials, where a pressure projection approach was introduced and is applicable to general large deformation.…”

On the Analysis of an EFG Method Under Large Deformations and Volumetric Locking

“…Wells et al [29] propose that a local extrinsic enrichment be accomplished just where the plastic flow takes place. Vidal et al [31] proposes a pseudo-divergence-free approach, which consists in using shape functions that verify approximately the divergence-free constraint. In Askes et al [4], it is numerically shown, for near incompressible solids, that the volumetric locking problem is not evidenced if a sufficiently large support of the global shape function is used.…”

Bubble‐enhanced smoothed finite element formulation: a variational multi‐scale approach for volume‐constrained problems in two‐dimensional linear elasticity

“…Several non‐conventional methods also have been developed to solve the volume‐constrained problems. Examples are the pseudo‐divergence‐free interpolation for element‐free Galerkin method , the pressure‐projection method for reproducing kernel particle formulation , the iso‐geometric approach , the mixed maximum‐entropy meshfree method , the meshfree‐enriched FEM , and so on. (see for a more complete overview).…”