引用本文: 齐栋梁. 超收敛光滑再生梯度无网格配点法. 力学与实践, 2024, 46(4): 1-10.
Qi Dongliang. Superconvergent smoothed reproducing gradient meshfree collocation method. Mechanics in Engineering, 2024, 46(4): 1-10.
 Citation: Qi Dongliang. Superconvergent smoothed reproducing gradient meshfree collocation method. Mechanics in Engineering, 2024, 46(4): 1-10.

## SUPERCONVERGENT SMOOTHED REPRODUCING GRADIENT MESHFREE COLLOCATION METHOD

• 摘要: 无网格配点(meshfree collocation, MC )法易于实现，但形函数高阶梯度的计算限制了其计算效率。为了提高MC法的梯度计算效率和收敛精度，本文结合无网格再生梯度理论与梯度光滑方法，提出了一种超收敛光滑再生梯度无网格配点(smoothed reproducing gradient meshfree collocation, SRGMC)法。所提方法以一阶再生梯度为基础递推构造二阶光滑再生梯度，避免了形函数中矩量矩阵的逆矩阵求导运算，数值实现便捷且计算效率高。文中通过典型数值算例验证了SRGMC法的精度和收敛性，结果表明，本文所提SRGMC法具有超收敛特性，且精度明显优越于MC法。

Abstract: The meshfree collocation (MC) method is straightforward to implement, but the computation efficiency is limited by the calculation of high order gradients of shape functions. In order to enhance the computation efficiency and convergence rate of the meshfree collocation method, this work proposes a superconvergent smoothed reproducing gradient meshfree collocation (SRGMC) method through combining meshfree gradient reproducing theory with gradient smoothing operations. In the proposed method, the first order reproduced gradients of meshfree shape functions are utilizing to iteratively construct the second order smoothed reproducing gradients, which further avoids the direct differentiation of the inverse matrix of the moment matrix in the shape functions. Furthermore, the present formulation is more concise and significantly improves computational efficiency. Numerical examples well demonstrate the accuracy and convergence of the presented SRGMC method, which is significantly superior to MC method.

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