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Abstract In comparison with the traditional finite element method, meshfree methods possess several appealing advantages such as the high-order smoothness of the nodal shape functions, the convenience to construct high-order approximation, etc. However, the nodal shape functions of high-order meshfree methods are non-polynomial rational functions, and this leads to the difficulty to accurately evaluate the domain integration of the weak form. The high-order Gauss integration commonly used in meshfree analysis requires a lot of integration points, and thus it is computationally inefficient. Besides, it is also not accurate enough. In this paper, a curvature smoothing scheme which is consistent to high-order (cubic) approximation is first proposed for the meshfree analysis of thin plate bending problems. Accordingly, a numerical integration scheme based on the curvature smoothing is developed for background triangular integration cells, and the number of quadrature points is dramatically reduced. The key of the developed method is that the second-order derivatives of nodal shape functions used in the computation of the stiffness matrix are obtained by using the divergence theorem among the shape functions, the first- and the second-order derivatives, instead of directly taking derivatives of nodal shape functions. Numerical results show that the high-order meshfree method based on the standard Gauss integration scheme is not accurate enough. It cannot reproduce the pure bending and linear bending modes exactly and leads to severe numerical oscillation in the resulting bending moment contours. The proposed high-order meshfree method based on consistent curvature smoothing technique is capable of efficiently and conveniently solving thin plate bending problems. Especially, it can exactly reproduce the pure bending and linear bending modes. Compared with the standard Gauss integration and the dominated constant curvature smoothing methods, the proposed method possesses remarkable superiorities in computational efficiency, accuracy and bending moment distributions, so it is recommended in meshfree analysis of thin plate bending.
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Received: 08 May 2017
Published: 12 April 2018
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2018, Vol. 39 
