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In this paper, an unconditionally stable mesheless method is proposed with the implementation of the leapfrog alternating-direction-implicit scheme in the 3-D radial point interpolation meshless method. The unconditional stability of the proposed method is analytically proven and numerically verified. The accuracy and efficiency of the method are assessed through experiments. Compared with the conventional radial interpolation method, the computational cost can be saved by 85% with little sacrifice of accuracy. The principle presented in this paper can be extended to other existing meshless methods in developing other types of the unconditionally stable meshless methods.