A stable and efficient meshfree Galerkin method with consistent integration schemes for strain gradient thin beams and plates

Abstract

Abstract The strain gradient (SG) theory, incorporating with thin beam and plate models, can effectively describe size effects of micro- and nano-structures. However, since these models are determined by a sixth-order partial differential equation that requires the C2 continuity of deflection in a Galerkin weak form, it is difficult to make stable and efficient numerical analysis. In this paper, a meshfree Galerkin method is presented for SG thin beams and plates. To satisfy the continuity and convergence requirement, moving least square or reproducing kernel shape functions are employed with cubic approximation bases. To pass the patch test, integration constraints are derived and consistent integration schemes are proposed with nodal smoothed derivatives instead of standard ones on evaluating points. Numerical results show that consistent integration is superior to the standard Gauss integration in convergence, accuracy and efficiency.