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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]

SUMMARY The consistency condition for the nodal derivatives in traditional meshfree Galerkin methods is only the differentiation of the approximation consistency (DAC). One missing part is the consistency between a nodal shape function and its derivatives in terms of the divergence theorem in numerical forms. In this paper, a consistency framework for the meshfree nodal derivatives including the DAC and the discrete divergence consistency (DDC) is proposed. The summation of the linear DDC over the whole computational domain leads to the so-called integration constraint in the literature. A three-point integration scheme using background triangle elements is developed, in which the corrected derivatives are computed by the satisfaction of the quadratic DDC. We prove that such smoothed derivatives also meet the quadratic DAC, and therefore, the proposed scheme possesses the quadratic consistency that leads to its name QC3. Numerical results show that QC3 is the only method that can pass both the linear and the quadratic patch tests and achieves the best performances for all the four examples in terms of stability, convergence, accuracy, and efficiency among all the tested methods. Particularly, it shows a huge improvement for the existing linearly consistent one-point integration method in some examples. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: A robust and efficient integration method, named quadratically consistent one-point (QC1) scheme, which evaluates the Galerkin weak form only at the centers of background triangle elements (cells) is proposed for meshfree methods using quadratic basis. The strain at the evaluation points is approximated by corrected (smoothed) nodal derivatives which are determined by a discrete form of the divergence theorem between nodal shape functions and their derivatives in Taylor’s expansion. We prove that such smoothed nodal derivatives also meet the differentiation of the approximation consistency (DAC). The same Taylor’s expansion is applied to the weak form and the smoothed nodal derivatives are used to compute the stiffness matrix. The proposed QC1 scheme can pass both the linear and the quadratic patch tests exactly in a numerical sense. Several examples are provided to demonstrate its better numerical performance in terms of convergence, accuracy, efficiency and stability over other one-point integration methods in the meshfree literature, especially its superiority over the existing linearly consistent one-point (LC1) quadratures. [Copyright &y& Elsevier]

Abstract: This paper presents two stabilized schemes of the fractional step algorithm using characteristic-based split and Taylor–Galerkin like based split respectively in the meshfree context. An iterative procedure which makes the convective term satisfy the momentum conservation equation in an implicit sense is introduced into the two schemes to allow much larger time step sizes. A modified version of finite increment calculus process for incompressibility stabilization is developed to avoid the calculation of high order spatial derivatives. Numerical experiments demonstrate the capability and stability of the proposed schemes of the meshfree-based fractional step method in the simulation of incompressible flows. [Copyright &y& Elsevier]

Element-free Galerkin (EFG) method is applied for the first time to numerical modeling of additive manufacturing (AM) process, in which the superiorities of the EFG method in implementing adaptive computation and in constructing high order approximation are exploited. Second-order moving-least squares (MLS) approximation is constructed for both temperature and displacement. A non-linear coupled thermo-mechanical model considering moving heat sources, radiation boundaries, elastoplastic materials and temperature-dependent parameters is adopted. Adaptive coarsening based on the background integration mesh is developed to efficiently model the layered deposition process. In addition, an efficient integration scheme using background hexahedral elements is proposed to evaluate the domain integrals of the weak forms. Numerical results show that the developed method is able to effectively model the AM process with substantially reduced number of approximation nodes. The effects of laser power, scan speed and scan path on temperature and distortion are investigated. The improved accuracies due to the proposed integration scheme and the adopted second-order approximation are also demonstrated. [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]