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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
##### Keywords:
meshfree; integration; Galerkin; finite element method; elasticity; solids
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