The particle-based meshfree methods offer a straightforward and easy construction of arbitrary order smooth approximants, for example, the moving least squares and reproducing kernel meshfree shape functions, which are ideal for the efficient collocation formulation without the need of costly weak form computation [1]. Nonetheless, the well-known basis degree discrepancy and low accuracy are two major issues associated with collocation methods with monomial basis functions [2, 3]. In this work, a unified recursive meshfree gradient formalism is presented to develop superconvergent collocation methods [2, 4, 5]. In this approach, the first order smoothed gradients of meshfree shape functions are constructed through a meshfree interpolation of the standard derivatives of meshfree shape functions and the second order smoothed gradients are obtained by directly performing differentiations to the first order smoothed gradients. A repeating of this procedure yields arbitrary order recursive meshfree gradients, which greatly simplifies the computation of complex conventional meshfree gradient and improves the efficiency dramatically. More importantly, it is shown that the proposed recursive meshfree gradients meet certain extra high order consistency conditions which go beyond the conventional consistency conditions corresponding to the order of basis function. An employment of these recursive meshfree gradients in the collocation formulation immediately leads to superconvergent meshfree colocation methods, for both second order potential and elasticity, and fourth order thin beam and plate problems. A detailed theoretical analysis is carried out to prove the superconvergence of the proposed algorithm, which can effectively resolve the basis degree discrepancy issue. Meanwhile, an accuracy analysis framework is also provided for general meshfree collocation methods [6]. Numerical results clearly demonstrate the accuracy and efficiency of the proposed unified superconvergent meshfree collocation formulation with recursive gradients.

Acknowledgements: The support of this work by the National Natural Science Foundation of China (12072302) and the Natural Science Foundation of Fujian Province of China (2021J02003) is gratefully acknowledged.

References

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