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

Author (s): Yolanda Vidal, Pierre Villon and Antonio Huerta
Journal: Communications in Numerical Methods in Engineering

Volume: 19, Issue 9
Pages: 725 – 735
Date: 2003

Locking 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.



  Author   = {Y. Vidal and P. Villon and A. Huerta},
  Title    = {Locking in the incompressible limit: pseudo-divergence-free element-free Galerkin},
  Fjournal = {Communications in Numerical Methods in Engineering},
  Journal  = {Commun. Numer. Methods Eng.},
  Volume   = {19},
  Number   = {9},
  Pages    = {725--735},
  Year     = {2003}}