基于分组圆的贪婪算法的径向基函数网格变形
Efficient mesh deformation using radial basis functions with a grouping-circular-based greedy algorithm
摘要：
为了提高网格变形的效率，提出了一种基于分组圆（GCB）的贪婪算法。该算法引入多重网格思想，将细网格上的计算误差近似于粗网格上的计算误差，随机将所有边界节点划分为$m$组，并在每次迭代过程中使用每组局部最大径向基函数（RBF）插值误差近似于所有边界结点中的全局最大值，以减少RBF支持节点。为此，它避免了在所有边界结点处进行插值，从而将相应的计算复杂性从$O\left({N_c^2{N_b}} \right)$降低到$O\left( {N_c^3} \right)$。此外，在$m$迭代之后，所有边界结点的插值误差被计算一次，从而允许所有边界结点都可以贡献于误差控制。对ONERA M6机翼和DLR-F6机翼-机身-短舱-挂架组合体的典型变形问题进行了计算，验证了GCB贪婪算法的有效性。计算结果表明，GCB贪婪算法能使数据约简过程中插值误差的计算效率显著提高几十倍。由于$m$的增加导致$N_c$的增加，因此建议$m$的$\left[ {{N_b}/{N_c},{\rm{ }}2{N_b}/{N_c}}\right]$的适当范围，以避免因$N_c $的增加而导致求解线性代数系统和计算体积节点位移的额外计算量过大。结果还表明，当网格规模较大时，GCB贪婪算法对网格变形的效率有更大的提高。
英文摘要：
A grouping-circular-based (GCB) greedy algorithm is proposed to promote 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 each group as an approximation to the globally maximum one of all boundary nodes in each iterative procedure for reducing the RBF support nodes. For this reason, it avoids the interpolation conducted at all boundary nodes and thus reduces the corresponding computational complexity from $O\left({N_c^2{N_b}} \right)$ to $O\left( {N_c^3} \right)$. Besides, after $m$ iterations, the interpolation errors of all boundary nodes are computed once, thus allowing all boundary nodes can contribute to error control. Two canonical deformation problems of the ONERA M6 wing and the DLR-F6 Wing-Body-Nacelle-Pylon configuration are computed to validate the GCB greedy algorithm. The computational results show that the GCB greedy algorithm is able to remarkably promote the efficiency of computing the interpolation errors in the data reducing procedure by dozens of times. Because an increase of $m$ results in an increase of $N_c$, an appropriate range of $\left[ {{N_b}/{N_c},{\rm{ }}2{N_b}/{N_c}}\right]$ for $m$ is suggested to prevent too much additional computations for solving the linear algebraic system and computing the displacements of volume nodes induced by the increase of $N_c $. The results also show that the GCB greedy algorithm tends to generate a more significant efficiency improvement for mesh deformation when a larger-scale mesh is applied.
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