LAUSR

Abstract Strong-form direct collocation method based on the reproducing kernel (RK) approximation has been criticized frequently by the complexity and time-consuming computation of the derivatives, and it is rather hard to use the low-order basis functions to improve the efficiency. In this paper, we propose an efficient meshfree gradient smoothing collocation method (GSCM) based on the RK approximation which adopts the gradient smoothing technique for the calculation of the RK derivatives. Low-order basis functions such as constant functions can be utilized for the approximation in the numerical solutions of the elasticity problems which can greatly improve the efficiency. Conforming and nonconforming meshes can be constructed for the numerical integration of the gradient smoothing. Constraints of the numerical integration are established where the proper positions of the integration points and the corresponding weighs for the integration are derived to keep the consistency of the RK shape function and its smoothed gradients. These guarantee the accuracy and convergence of the proposed method. Numerical results demonstrate that the presented method can outmatch the conventional direct collocation method (DCM) in accuracy, stability and computational efficiency, and generally the GSCM-II using a double gradient smoothing performs better than the GSCM-I and the super-convergent gradient smoothing meshfree collocation method (SGSMC) using a single gradient smoothing.