Analysis of the band structure of elastic waves in 3D nano phononic crystals based on the local radial basis function collocation method

The nonlocal elastic theory is applied to describe the size effect of nano-materials by introducing an integration form involved in the stress. Thus, the traditional numerical methods, such as the finite element method (FEM), are quite complicated to be applied to the nonlocal theory phononic crystals, especially for the 3D problem. Previous works show that the strong-form local radial basis function collocation method (LRBFCM) can be used to the 2D nano phononic crystals with an efficient summation approach. However, the 3D integration involved in the nonlocal stress or traction forces is still a challenge in the numerical method. By combining the direct method for computing derivatives and the summation approach for the 3D nonlocal integration, this paper makes the f irst attempt to study 3D nano phononic crystals described by the nonlocal theory based on the LRBFCM. The efficiency of the LRBFCM for 3D nano phononic crystals is validated by a frequency response of periodical structure since few works of 3D nano phononic crystals can be found. Different acoustic impedance ratio, filling ratio, number of constituents, and scatterer geometry are considered. Finally, the influences of size effects on the band structure of 3D nano phononic crystals are fully studied. © 2025 Elsevier Ltd.