Variational consistent one-point integration with Taylor's expansion-based stabilization in the second-order meshfree Galerkin method for strain gradient elasticity

A generalized variational principle with five independent variables is proposed for strain gradient elasticity, including displacement, strain, strain gradient, stress, and double stress. Based on the principle, a one-point integration scheme is designed for the second order meshfree Galerkin method through nodal smoothed derivatives and their high order derivatives by Taylor's expansion. Since the proposed integration scheme meets the orthogonality conditions, it is variational consistent. The weak form expanded with Taylor's polynomials can be well evaluated by nodal smoothed derivatives and their high order derivatives on one quadrature point. Numerical one- and two-dimensional case studies show that the proposed integration scheme performs better than the standard Gaussian integration method in terms of accuracy, convergence, efficiency, and stability.