A general arbitrary order recursive gradient formulation is presented for meshfree approximation. According to this method, an n th order recursive meshfree gradient is formulated as an interpolation of the ( n − 1)th order gradients by standard first order meshfree gradients, which finally can be expressed as a successive multiplication of standard first order meshfree gradients. This formulation avoids the complex and costly computation of conventional high order derivatives of meshfree shape functions. One crucial ingredient of the proposed methodology is that the resulting recursive meshfree gradients with a p th degree basis function not only meet the conventional p th order consistency conditions for standard gradients, but also satisfy ( p + 1)th to ( p + n − 1)th extra high order consistency conditions. This important property leads to superconvergent meshfree collocation algorithms and here we focus on the classical fourth order Kirchhoff plate problems. An accuracy analysis of the proposed recursive gradient meshfree collocation formulation for Kirchhoff plates reveals that superconvergence is simultaneously achieved for both even and odd degrees of basis functions. More specifically, two and four additional orders of accuracy are respectively gained by the proposed method for even and odd degree basis functions, compared with the standard meshfree collocation scheme. Furthermore, the extra high order consistency conditions of recursive meshfree gradients enable superconvergent meshfree collocation analysis of Kirchhoff plates using low order basis functions of less than 4th degree, while the standard meshfree collocation approach requires at least a 4th degree basis function to maintain convergence. The accuracy and efficiency of the proposed methodology are holistically demonstrated by numerical results.

中文翻译：

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无网格梯度的任意阶递归公式，适用于基尔霍夫板的超收敛搭配分析

为无网格近似提供了一种通用的任意阶递归梯度公式。根据该方法，n阶递归无网格梯度被公式化为标准一阶无网格梯度对(n-1)阶梯度的插值，最终可以表示为标准一阶无网格梯度的连续乘法。该公式避免了对无网格形状函数的常规高阶导数进行复杂且昂贵的计算。所提出的方法的一个关键要素是，得到的具有 ap 次基函数的递归无网格梯度不仅满足标准梯度的常规 p 阶一致性条件，而且满足 (p + 1)th 到 (p + n − 1 ) 次超高阶一致性条件。这个重要的性质导致了超收敛无网格搭配算法，这里我们关注经典的四阶基尔霍夫板问题。对基尔霍夫板提出的递归梯度无网格配置公式的准确性分析表明，对于偶数和奇数的基函数，同时实现了超收敛。更具体地说，与标准无网格配置方案相比，所提出的偶数和奇数基函数方法分别获得了两个和四个额外的精度数量级。此外，递归无网格梯度的超高阶一致性条件能够使用小于 4 阶的低阶基函数对基尔霍夫板进行超收敛无网格搭配分析，而标准的无网格搭配方法至少需要一个四阶基函数来保持收敛。数值结果全面证明了所提出方法的准确性和效率。