薄板弯曲分析的高阶高效无网格法

与传统有限元法相比，无网格法具有节点形函数高度光滑、易于形成高阶近似等优势，更适合于以薄板弯曲问题为代表的高阶偏微分方程的数值求解。然而，高阶无网格法的形函数是非多项式的有理函数，导致弱形式的区域积分难以得到精确计算，通常采用的高阶高斯积分方法需使用大量积分点，计算效率低且精度不高。本文针对薄板弯曲问题的高阶（三阶）无网格法分析，首次发展了与该高阶近似相一致的曲率光顺方案，并基于背景三角形积分单元建立了相应的数值积分格式，大幅度减少了所需的积分点数目。所发展方法的关键在于计算刚度阵所需的形函数的二阶导数由形函数及其一阶导数通过散度定理确定，而非对形函数直接求导获得。数值结果表明，基于标准的高斯积分方案的高阶无网格法精度不高，不能精确再现纯弯曲和线性弯曲模式，且得到的弯矩场分布存在严重的虚假数值振荡。而本文所建议的基于曲率光顺方案的高阶无网格法能够方便高效地求解薄板弯曲问题，尤其是它能精确反映纯弯曲和线性弯曲模式。与标准的高斯积分方法和目前主流的常曲率光顺方法相比，本文方法在计算效率、精度、弯矩分布等方面均展现出显著优势，因而具有较好的应用价值。

An Efficient High-order Meshfree Method for Thin Plate Bending Analysis

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.