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Second-order accurate derivatives and integration schemes for meshfree methods. (English) Zbl 1352.65390
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.

MSC:
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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References:
[1] Belytschko, Element-free Galerkin methods, International Journal for Numerical Methods in Engineering 37 (2) pp 229– (1994) · Zbl 0796.73077
[2] Beissel, Nodal integration of the element-free Galerkin method, Computer Methods in Applied Mechanics and Engineering 139 (1) pp 49– (1996) · Zbl 0918.73329
[3] Chen, A stabilized conforming nodal integration for Galerkin mesh-free methods, International Journal for Numerical Methods in Engineering 50 (2) pp 435– (2001) · Zbl 1011.74081
[4] Chen, Non-linear version of stabilized conforming nodal integration for Galerkin mesh-free methods, International Journal for Numerical Methods in Engineering 53 (12) pp 2587– (2002) · Zbl 1098.74732
[5] Wang, A hermite reproducing kernel approximation for thin-plate analysis with sub-domain stabilized conforming integration, International Journal for Numerical Methods in Engineering 74 (3) pp 368– (2008) · Zbl 1159.74460
[6] Puso, Meshfree and finite element nodal integration methods, International Journal for Numerical Methods in Engineering 74 (3) pp 416– (2008) · Zbl 1159.74456
[7] Liu, A nodal integration technique for meshfree radial point interpolation method (NI-RPIM), International Journal of Solids and Structures 44 (11) pp 3840– (2007) · Zbl 1135.74050
[8] Bonet, Correction and stabilization of smooth particle hydrodynamics methods with applications in metal forming simulations, International Journal for Numerical Methods in Engineering 47 (6) pp 1189– (2000) · Zbl 0964.76071
[9] Dyka, An approach for tension instability in smoothed particle hydrodynamics (SPH), Computers & Structures 57 (4) pp 573– (1995) · Zbl 0900.73945
[10] Dyka, Stress points for tension instability in SPH, International Journal for Numerical Methods in Engineering 40 (13) pp 2325– (1997) · Zbl 0890.73077
[11] Rabczuk, Stable particle methods based on Lagrangian kernels, Computer Methods in Applied Mechanics and Engineering 193 (12) pp 1035– (2004) · Zbl 1060.74672
[12] Fries, Convergence and stabilization of stress-point integration in mesh-free and particle methods, International Journal for Numerical Methods in Engineering 74 (7) pp 1067– (2008) · Zbl 1158.74525
[13] Duan, Gradient and dilatational stabilizations for stress-point integration in the element-free Galerkin method, International Journal for Numerical Methods in Engineering 77 (6) pp 776– (2009) · Zbl 1156.74393
[14] Atluri, The Meshless Local Petrov-Galerkin (MLPG) Method (2002) · Zbl 1012.65116
[15] De, The method of finite spheres with improved numerical integration, Computers & Structures 79 (22) pp 2183– (2001)
[16] Kwon, The support integration scheme in the least-squares mesh-free method, Finite Elements in Analysis and Design 43 (2) pp 127– (2006)
[17] Liu, A new support integration scheme for the weakform in mesh-free methods, International Journal for Numerical Methods in Engineering 82 (2) pp 699– (2010) · Zbl 1188.74084
[18] Krongauz, Consistent pseudo-derivatives in meshless methods, Computer Methods in Applied Mechanics and Engineering 146 (3) pp 371– (1997) · Zbl 0894.73156
[19] Liu, A smoothed finite element method for mechanics problems, Computational Mechanics 39 (6) pp 859– (2007) · Zbl 1169.74047
[20] Liu, Theoretical aspects of the smoothed finite element method (SFEM), International Journal for Numerical Methods in Engineering 71 (8) pp 902– (2007) · Zbl 1194.74432
[21] Nguyen-Xuan, Smooth finite element methods: convergence, accuracy and properties, International Journal for Numerical Methods in Engineering 74 (2) pp 175– (2008) · Zbl 1159.74435
[22] Tang, Quasi-conforming elements for finite element analysis, Journal of Dalian University of Technology 19 (2) pp 19– (1980)
[23] Tang, Formulation of quasi-conforming element and Hu-Washizu principle, Computers & Structures 19 (1) pp 247– (1984) · Zbl 0548.73051
[24] Tang, Quasi-conforming element techniques for penalty finite element methods, Finite Elements in Analysis and Design 1 (1) pp 25– (1985) · Zbl 0573.73084
[25] Fernández-Méndez, Imposing essential boundary conditions in mesh-free methods, Computer Methods in Applied Mechanics and Engineering 193 (12) pp 1257– (2004) · Zbl 1060.74665
[26] Dunavant, High degree efficient symmetrical Gaussian quadrature rules for the triangle, International Journal for Numerical Methods in Engineering 21 (6) pp 1129– (1985) · Zbl 0589.65021
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