A four-point integration scheme with quadratic exactness for three-dimensional element-free Galerkin method based on variationally consistent formulation

The formulation of three-dimensional element-free Galerkin (EFG) method based on the Hu–Washizu three-field variational principle is described. The orthogonality condition between stress and strain difference is satisfied by correcting the derivatives of the nodal shape functions. This leads to a variationally consistent formulation which has a similar form as the formulation of standard Galerkin weak form. Based on this formulation, an integration scheme which employs only four cubature points in each background tetrahedral element (cell) is rationally developed for three-dimensional EFG with quadratic approximation. The consistency of the corrected nodal derivatives and the satisfaction of patch test conditions for the developed integration scheme are theoretically proved. Extension of the proposed method to small strain elastoplasticity is also presented. The proposed method can exactly pass quadratic patch test, that is, quadratic exactness is achieved, and thus it is named as quadratically consistent 4-point (QC4) integration method. In contrast, EFG with standard tetrahedral cubature and the existing linearly consistent 1-point (LC1) integration fail to exactly pass quadratic patch test. Numerical results of elastic examples demonstrate the superiority of the proposed method in accuracy, convergence, efficiency and stability. The capability of the proposed QC4 scheme in solving elastoplastic problems is also demonstrated by numerical examples.