Keywords: reproducing kernels, point collocation, meshless methods, bending, stresses, moments, beams, plates.

Proposing an efficient meshless method in computational mechanics has considerably attracted the researchers' attention for recent decades. This motivation is based on the reports indicating that the conventional computational methods, such as the finite element and finite difference methods of which formulations strongly rely on a grid mesh, may not be suitable for the treatment of discontinuities, moving boundaries and large deformations [1,2]. Hence, the unknown approximants in the meshless methods have been entirely constructed in terms of nodes which are randomly scattered to overcome the previous drawbacks. A comprehensive literature survey on meshless methods has been made by Belytschko et al. [3].

A differential reproducing kernel particle (DRKP) method is proposed and developed for solving partial differential equations governing a certain physical problem by following up the consistent concepts of the reproducing kernel particle (RKP) method [2]. A novel idea is proposed for determining the shape functions for derivatives of the reproducing kernel (RK) approximants. In the conventional RKP method, they have been obtained by directly taking the differential operations toward the shape functions of the RK approximants. That results in the lengthy expressions and complicated computation, especially for the calculation involving the higher-order derivatives of the RK approximants. Contrary to the manipulation in the RKP method, we construct a set of differential reproducing conditions to determine the shape functions for the derivatives of the RK approximants. The proposed formulation is simple and easy for calculation. A point collocation approach based on the present DRKP approximations for multi-dimensional problems is formulated and applied to the static analysis of single-layer and multilayered beams and plates. It is shown that the present method is indeed a fully meshless approach with excellent accuracy and a fast rate of convergence. The highest order of the basis functions is suggested to be one order or two orders higher than the highest order of the governing equations. The suitable support size taken is suggested to include the number of nodes larger than twice the number of the selected basis functions for each reference node.

1

J.S. Chen, C. Pan, C.T. Wu, W.K. Liu, "Reproducing kernel particle methods for large deformation analysis of non-linear structures", Computer Methods in Applied Mechanics and Engineering, 139, 195-227, 1996. doi:10.1016/S0045-7825(96)01083-3

2

W.K. Liu, S. Jun, Y.F. Zhang, "Reproducing kernel particle methods", International Journal for Numerical Methods in Engineering, 20, 1081-1106, 1995. doi:10.1002/fld.1650200824

3

T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, P. Krysl, "Meshless methods: An overview and recent developments", Computer Methods in Applied Mechanics and Engineering, 139, 3-47, 1996. doi:10.1016/S0045-7825(96)01078-X

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