Study on Reference Displacement Method Based on Radial Basis Functions With Boundary Orthogonality Correction and Spatial Multiple Point Selection

Abstract

The process of elastic deformation is performed using the radial basis functions (RBF) method for the background mesh and the volume-weighted interpolation method for the computational mesh. The RBF method performs interpolation independently in different directions and does not ensure boundary orthogonality during elastic deformation. This study proposes a strategy to improve the quality and efficiency of the dynamic mesh method. Based on the automatically generated adaptive background mesh, a reference displacement method for spatial points is first developed to obtain reasonable reference displacements with rotational correction for spatial control points. The RBF is then used to interpolate the rotation angle of the boundary represented by the quaternion. Therefore, the large angle deformation can be more smoothly propagated to the spatial region. In the background mesh, the translational deformation is obtained by the damping function based on the minimum distance, the rotational deformation is calculated by rotating around a small number of nodes with the minimum distance from the wall surface, and the final rotational displacement is obtained by inverse distance weighting. This method ensures boundary orthogonality. Afterwards, the one-step reference displacement method is developed. A subset of mesh nodes on the boundary surface and one layer outside the boundary are automatically selected using a greedy algorithm and considered as control points. The spatial point is assigned a value using a reference displacement. The deformation displacement of the background mesh is calculated using the RBF method, and then interpolated into the computational grid. The obtained results show that this method can significantly improve the orthogonality of the boundary layer and retain the high accuracy of the RBF method for elastic deformation. In addition, a two-step spatial multi-point selection method is developed. Using a spatial peak selection method of high flexibility based on the quality change before and after mesh deformation, multiple spatial points are selected and considered as control points at a time, which allows for increased stability and robustness of the dynamic mesh method for the one-step reference displacement method. The deformation ability of the two-step spatial multi-point selection method is increased by 20% compared with that of the one-step reference displacement method in a single deformation step. The criterion for point selection is general, and the algorithm has high efficiency, which allows spatial control points to be selected in multiple regions. The two-step spatial multi-point selection method provides a smooth mesh, stretching the overlapped and squeezed mesh. The CPU cost is comparable to the RBF, and iteration is not required. The time complexity of the proposed multiple point selection method is reduced to 2/(M + 1) times that of the single-point selection method, which adds one spatial point at a time until M points are added. Several typical examples show that the proposed methods improve the mesh quality by about 30% for two-dimensional flow and 17% for three-dimensional flow while ensuring the computational efficiency.