Quadratically consistent one-point (QC1) integration for three-dimensional element-free Galerkin method

• One-point integration scheme, named as QC1, is developed for 3D EFG method.
• The consistency of the corrected nodal derivatives are theoretically proved.
• The superiority of the QC1 scheme is sufficiently demonstrated.

A stable and efficient integration scheme which evaluates the Galerkin weak form only at the centers of background tetrahedral elements (cells) for three-dimensional element-free Galerkin method with quadratic approximation is proposed. The derivation of the method is based on the Hu–Washizu three-field variational principle and the orthogonality condition between stress and strain difference is satisfied by correcting the nodal derivatives at quadrature points with Taylor series expansion technique. The consistency of such corrected derivatives is theoretically proved. Numerical experiments validate that the proposed method can exactly pass linear and quadratic patch tests. Therefore, it is named as quadratically consistent one-point (QC1) integration. The superiority of the proposed QC1 than other integration schemes for three-dimensional element-free Galerkin methods in accuracy, convergence, efficiency and stability is sufficiently demonstrated by several 3D examples.