An error estimate in the EFG method

Abstract

In this paper, local and global error estimates for the element-free Galerkin (EFG) method are proposed. The essence of proposed error estimates is to use the difference between the values of the projected stress and these given directly by the EFG solution. The stress projection can be obtained simply by taking product of shape function based on a different domain of in¯uence with the stresses at nodes. In this study, it was found that the effectivity index is optimized if the domain of in¯uence in stress projection procedure is the smallest that retains regularity of the matrices in EFG. Numerical tests are shown for various 1D and 2D examples illustrating the good effectiveness of the proposed error estimator in the global energy norm and in the local error estimates.

An adaptive finite element/meshless coupled method based on local maximum entropy shape functions for linear and nonlinear problems

“…The error estimation procedure proposed for the EFGM in [27] is used, in which projected stresses are calculated from the nodal stresses based on reduced domains of influence. It was also shown in [27] that the error estimator performs well when the domain of influence for stress projection is as small as possible but at the same time must be large enough to have sufficient nodes in support of each Gauss point required for the shape functions calculation. In the case of the EFGM, the stress and strain vector returned at any point x is written as…”

“…The Zienkiewicz & Zhu [25] error estimator with the SPR method for recovery of stresses and strain are used in the FE region of the problem domain, while the Chung & Belytschko [27] error estimation is used in the EFG region of the problem domain. The basic idea of both of these error estimation procedures are to use the difference between projected and the direct numerical solutions and are similar to the conventional recovery type error estimation method in the FEM [26].…”

“…After this point, error is only calculated for the EFG region, as FEs with high errors already converted into EFG background cells in the first iteration. After solution of the final system of equations, nodal stresses, local and global relative errors are calculated for the EFG region of the problem domain using the approach of Chung & Belytschko [27]. The global relative error is then used to decide further adaptive refinement in the EFG region, and those background cells, where error is more than the permissible error are automatically refined.…”

“…For linear elastic problems, the well-established error estimation procedure of Zienkiewicz & Zhu [24] with the superconvergent convergent patch recovery (SPR) method for recovery of the nodal stresses [25,26] is used in the FE region to determine the elements requiring conversion to EFG regions. The Chung & Belytschko [27] error estimator is used in the EFG region for further adaptive refinement. For nonlinear problems, incremental forms of the Zienkiewicz & Zhu error estimator [28] and the Chung & Belytschko error estimator [29] are used.…”

Goal Oriented Error Estimation for the Element Free Galerkin Method

“…Liu et al [15] used a wavelet solution as an error indicator in an algorithm where multiplescale adaptive refinement had been introduced. Chung and Belytschko [5] adapted the FEM stress projection technique for error analysis in Element Free Galerkin (EFG). Gavete et al [10] proposed a sort of recovery-based error estimate, which presents the standard drawbacks of these methods.…”