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Efficient mesh deformation based on radial basis function interpolation by means of the inverse fast multipole method

机译：逆快速多极点法基于径向基函数插值的高效网格变形

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摘要

Radial basis function interpolation is often employed in mesh deformation algorithms for unstructured meshes, for example in fluid-structure interaction or design optimization problems. This is known to be a robust methodology that results in high quality deformed meshes. The applicability of this method to large problems is currently hampered by its prohibitive computational cost, however, which is caused by the need to solve a dense system of equations. The computation time grows as O(N-b(3)) if a direct solver is employed for the solution (where N-b denotes the number of boundary nodes in the mesh), while alternative iterative solvers often suffer from an unfavorable convergence behavior. In this paper, we present the inverse fast multipole method as a novel fast approximate direct solver with a computational cost scaling as O(N-b). The linear complexity is achieved by transforming the dense system into an extended sparse system, along with the compression of certain matrix blocks into low-rank factorizations. The solver is inexact, although the error can be controlled and made as small as needed; a low accuracy solver can hence be used as an efficient preconditioner in an iterative scheme. Numerical benchmarks are presented, demonstrating that the proposed approach enhances the computational efficiency of mesh deformation algorithms based on radial basis function interpolation significantly, without jeopardizing their robustness and quality. (C) 2016 Elsevier B.V. All rights reserved.
机译：径向基函数插值通常用于非结构化网格的网格变形算法中，例如在流固耦合或设计优化问题中。众所周知，这是一种可靠的方法，可以产生高质量的变形网格。目前，该方法对大问题的适用性由于其过高的计算成本而受到阻碍，这是由于需要求解密集的方程组而引起的。如果将直接求解器用于求解（其中N-b表示网格中边界节点的数量），则计算时间将随着O（N-b（3））的增长而增加，而替代的迭代求解器通常会遇到不利的收敛行为。在本文中，我们将逆快速多极点方法作为一种新颖的快速近似直接求解器，其计算成本定标为O（N-b）。通过将密集系统转换为扩展的稀疏系统，以及将某些矩阵块压缩为低秩分解，可以实现线性复杂度。尽管可以控制误差并使误差尽可能小，但求解器并不精确。因此，低精度求解器可以在迭代方案中用作有效的预处理器。给出了数值基准，表明所提出的方法大大提高了基于径向基函数插值的网格变形算法的计算效率，而不会损害其鲁棒性和质量。 （C）2016 Elsevier B.V.保留所有权利。

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