A nodal integration scheme for meshfree Galerkin methods using the virtual element decomposition
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In this paper, we present a novel nodal integration scheme for meshfree Galerkin methods that draws on the mathematical framework of the virtual element method. We adopt the linear maximum-entropy basis functions for discretization of the field variables, although the proposed scheme is applicable to any linear meshfree approximant. In our approach, the weak form integrals are nodally integrated using nodal representative cells that carry the nodal displacements and state variables such as strains and stresses. The nodal integration is performed using the virtual element decomposition, wherein the bilinear form is decomposed into a consistency part and a stability part that ensure consistency and stability of the method. The performance of the proposed nodal integration scheme is assessed through various examples in linear elastostatics and linear elastodynamics. We demonstrate that the proposed nodally integrated meshfree method is accurate, converges optimally, and is more efficient than a standard cell-based Gauss integrated meshfree method.
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