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SUMMARY An element-free Galerkin (EFG) method with linear, quadratic and cubic approximations, which can exactly, in a numerical sense, pass the corresponding patch tests is proposed and is named as consistent EFG (CEFG) method. The development of this method is based on the Hu-Washizu three-field variational principle. Numerical integration schemes with corrected nodal derivatives at quadrature points are proposed according to the satisfaction of the orthogonality condition between stress and strain difference. Thus, the method is variationally consistent. The consistency of the corrected nodal derivatives and the satisfaction of patch test conditions are theoretically proved and also numerically validated. Numerical results show that the proposed CEFG method greatly improves the numerical performance of the EFG method in terms of accuracy, convergence, efficiency and stability, especially the proposed cubic CEFG method, which shows exceptional accuracy and convergence. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Consistent element-free Galerkin (CEFG) method for phase-field model of 3D cracks with adaptive nodal refinement. Element-free Galerkin (EFG) method for three-dimensional phase-field model of cracks is described. Quadratic moving least-squares (MLS) approximation is constructed for both the phase field and the displacement. The quadratically consistent 4-point (QC4) integration scheme using background tetrahedral elements (cells) is employed to efficiently compute the domain integrals in weak forms. As cracks propagate, adaptive refinement of the approximation nodes around crack fronts is implemented with the use of background integration cells. The refinement is triggered by the maximum residual strain energy history. New approximation nodes are inserted at the midpoints of the element edges. Several benchmark examples are investigated. Numerical results show that three-dimensional crack propagation is successfully modeled by the proposed method and correct crack paths and load-displacement responses are obtained. Due to the implemented h-adaptivity, lots of approximation nodes are saved and the computational scale is significantly reduced. Improved accuracy of the proposed method in stress distribution and crack width is also demonstrated. [ABSTRACT FROM AUTHOR]