# zbMATH — the first resource for mathematics

Locking in the incompressible limit: pseudo-divergence-free element free Galerkin. (English) Zbl 1112.74545
Summary: 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 EFG 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 is 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.

##### MSC:
 74S05 Finite element methods applied to problems in solid mechanics
Full Text:
##### References:
 [1] Huerta, Locking in the incompressible limit for the Element free Galerkin method, International Journal for Numerical Methods in Engineering 51 (11) pp 1361– (2001) · Zbl 1065.74635 [2] Hughes, The Finite Element Method, Linear Static and Dynamic Analysis (1987) · Zbl 0634.73056 [3] Zhu, A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method, Computational Mechanics 21 pp 211– (1998) · Zbl 0947.74080 [4] Dolbow, Volumetric locking in the element free Galerkin method, International Journal for Numerical Methods in Engineering 46 pp 925– (1999) · Zbl 0967.74079 [5] Askes, Conditions for locking-free elasto-plastic analyses in the element free Galerkin method, Computer Methods in Applied Mechanics and Engineering 173 pp 99– (1999) · Zbl 0962.74076 [6] Chen, Computer Methods in Applied Mechanics and Engineering 181 pp 117– (2000) [7] Nayroles, Generating the finite element method: diffuse approximation and diffuse elements, Computational Mechanics 10 pp 307– (1992) · Zbl 0764.65068 [8] Villon P Contribution à l’optimisation 1991 [9] Belytschko, Meshless methods: an overview and recent developments, Computer Methods in Applied Mechanics and Engineering 139 pp 3– (1996) · Zbl 0891.73075 [10] Liu, Meshless methods, Computer Methods in Applied Mechanics and Engineering 139 pp 1– (1996) · Zbl 0880.00031 [11] Liu, Overview and applications of the reproducing kernel particle methods, Archives of Computational Methods in Engineering, State of the Art Reviews 3 pp 3– (1996) [12] Liu, Moving least square reproducing kernel methods. (I) Methodology and convergence, Computer Methods in Applied Mechanics and Engineering 143 pp 113– (2000) [13] Huerta, Enrichment and coupling of the finite element and meshless methods, International Journal for Numerical Methods in Engineering 48 pp 1615– (2000) · Zbl 0976.74067 [14] Belytschko, Smoothing and accelerated computations in the element free Galerkin method, Journal of Computational and Applied Mathematics 74 pp 111– (1996) · Zbl 0862.73058
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.