  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 linear maximum-entropy basis functions for the
discretization of 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 benchmark problems in linear and
nonlinear analyses of solids for small displacements and small-strain
kinematics. Numerical results are presented for linear elastostatics and linear
elastodynamics, and viscoelasticity. We demonstrate that the proposed nodally
integrated meshfree method is accurate, converges optimally, and is more
reliable and robust than a standard cell-based Gauss integrated meshfree
method.
