In this paper, a novel weak-form meshless method, Galerkin Free Element Collocation Method (GFECM), is proposed for the mechanical analysis of thin plates. This method assimilates the benefits of establishing spatial partial derivatives by isoparametric elements and forming coefficient matrices node by node, which makes the calculation more convenient and stable. The pivotal aspect of GFECM is that the surrounding nodes can be freely chosen as collocation elements, which can adapt to irregular node distribution and suitable for complex models. Meanwhile, each collocation element is used as a Lagrange isoparametric element individually, which can easily construct high-order elements and improve the calculation accuracy, especially for high-order partial differential equations such as the Kirchhoff plate bending problem. In order to obtain the weak-form of the governing equation, the Galerkin form of the governing equation is constructed based on the virtual work principle and variational method. In addition, due to the Lagrange polynomials possessing the Kronecker delta property as shape functions, it can accurately impose boundary conditions compared with traditional meshless methods that use rational functions. Several numerical examples are proposed to verify the correctness and effectiveness of the proposed method in thin plate bending problems.

中文翻译：

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一种基于拉格朗日插值的弱型无网格复杂薄板力学分析方法


本文提出了一种新颖的弱型无网格方法，即伽辽金自由元素搭配法 （GFECM），用于薄板的力学分析。该方法吸收了通过等参元建立空间偏导数和逐节点形成系数矩阵的好处，使计算更加方便和稳定。GFECM 的关键之处在于可以自由选择周围的节点作为搭配元素，可以适应不规则的节点分布，适用于复杂的模型。同时，每个搭配元单独用作拉格朗日等参元，可以很容易地构造高阶单元并提高计算精度，特别是对于基尔霍夫板弯曲问题等高阶偏微分方程。为了获得控制方程的弱形式，基于虚拟工作原理和变分方法构建了控制方程的 Galerkin 形式。此外，由于拉格朗日多项式具有作为形状函数的 Kronecker delta 属性，因此与使用有理函数的传统无网格方法相比，它可以准确地施加边界条件。提出了几个数值算例，以验证所提方法在薄板弯曲问题中的正确性和有效性。