Consistent integration schemes for meshfree analysis of strain gradient elasticity

Integration schemes with nodal smoothed derivatives, which meet integration constraint conditions, are robust and efficient for use in meshfree Galerkin methods, however, most of them are focussed on the classical elasticity determined by a second-order partial differential equation. In this paper, arbitrary-order integration constraint conditions are derived for strain gradient elasticity in a fourth-order partial differential equation. These integration constraint conditions provide the discrete forms of nodal shape functions and their first- and second-order derivatives. Furthermore, to meet the integration constraint conditions, consistent integration schemes are designed with nodal smoothed (but not standard) derivatives at evaluating points. It is shown that such nodal smoothed derivatives are able to satisfy the differentiation of approximation consistency. Finally, several case studies are given and the results demonstrate that, based on convergence, accuracy and efficiency, the numerical performance of consistent integration in meshfree analysis of strain gradient elasticity is superior to the standard Gaussian one