A least squares recursive gradient meshfree collocation method is proposed for the superconvergent computation of structural vibration frequencies. The proposed approach employs the recursive gradients of meshfree shape functions together with smoothed shape functions in the context of least squares formulation, where both meshfree nodes and auxiliary points are taken as the collocation points. It turns out that this least squares formulation can effectively suppress the spurious modes arising from a direct meshfree collocation formulation using recursive gradients. Meanwhile, a detailed theoretical analysis with explicit frequency error measure is presented for the least squares recursive gradient meshfree collocation method in order to assess the frequency accuracy of structural vibrations. This analysis discloses the salient basis degree discrepancy issue regarding the frequency accuracy for the least squares meshfree collocation formulation, and it is shown that this issue can be essentially resolved by the proposed least squares recursive gradient meshfree collocation method. In fact, the proposed method leads to superconvergent vibration frequencies when odd degree basis functions are used, i.e., the frequency convergence rate is improved from \((p - 1)\) for the standard least squares meshfree collocation to \((p + 1)\) for the proposed approach in case of an odd pth degree basis function. This desirable frequency superconvergence of the proposed least squares recursive gradient meshfree collocation method is congruously demonstrated by numerical results.

针对结构振动频率的超收敛计算，提出了一种最小二乘递推梯度无网格配置方法。所提出的方法在最小二乘公式的上下文中使用无网格形状函数的递归梯度和平滑形状函数，其中无网格节点和辅助点都被作为搭配点。事实证明，这种最小二乘公式可以有效地抑制由使用递归梯度的直接无网格搭配公式产生的伪模式。同时，对最小二乘递推梯度无网格配置方法进行了详细的带有显式频率误差测度的理论分析，以评估结构振动的频率精度。该分析揭示了关于最小二乘无网格配置公式的频率精度的显着基度差异问题，并且表明该问题可以通过所提出的最小二乘递归梯度无网格配置方法基本解决。事实上，当使用奇次基函数时，所提出的方法导致超收敛振动频率，即频率收敛速度从\((p - 1)\)用于标准最小二乘无网格搭配到\((p + 1)\)用于在奇数p次基函数的情况下提出的方法。数值结果一致地证明了所提出的最小二乘递归梯度无网格搭配方法的这种理想的频率超收敛性。