Efficient mesh deformation using radial basis functions with a grouping-circular-based greedy algorithm

Abstract

A grouping-circular-based (GCB) greedy algorithm is proposed to improve the efficiency of mesh deformation. By incorporating the multigrid concept that the computational errors on the fine mesh can be approximated with those on the coarse mesh, this algorithm stochastically divides all boundary nodes into m groups and uses the locally maximum radial basis functions (RBF) interpolation error of the active group as an approximation to the globally maximum one of all boundary nodes for reducing the RBF support nodes. For this reason, it avoids the interpolation conducted at all boundary nodes in each iterative procedure. After m iterations, the interpolation errors of all boundary nodes are computed once, thus allowing all boundary nodes can contribute to error control. A theoretical analysis reveals that this algorithm can make the computational complexity for computing the interpolation errors reduced from O(Nc2Nb) to O(Nc3), where Nb and Nc denote the numbers of boundary nodes and support nodes, respectively. Two deformation problems of the ONERA M6 wing and the DLR-F6 Wing-Body-Nacelle-Pylon configuration are computed to validate the GCB greedy algorithm. It is shown that this algorithm is able to remarkably promote the efficiency of computing the interpolation errors by dozens of times. It is also shown that the convergence indicated by the variation of the globally interpolation error of this algorithm is consistent with that of the traditional greedy algorithm. It is indicated by the Kullback-Leibler (KL) divergence that it can generate a reasonable set of support nodes. Besides, it ensures similar statistical property to the traditional greedy algorithm. The theoretical analysis also reveals that if m>2.25Nb/Nc, the amount of computation for computing the interpolation errors will be lower than that for solving the linear algebraic system. However, it is found that an increase of m results in an increase of Nc, indicating that m cannot be too large, otherwise it will generate too much additional computations for solving the linear algebraic system and computing the displacements of volume nodes since the amounts of computation for these two processes increase as functions of Nc3 and Nc, respectively. For this reason, there is an appropriate value for m. It is also found that this algorithm tends to generate a more significant efficiency improvement for mesh deformation when a larger-scale mesh is applied. Furthermore, it can produce a deformed mesh with a comparable quality to the undeformed one for both structured and unstructured meshes.