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

Abstract

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 3 b) 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.

A fast semi-discrete Kansa method to solve the two-dimensional spatiotemporal fractional diffusion equation

“…However, the computational cost of the dense discretization matrix remains a critical problem in the presented method. A future investigation in fast matrix-vector multiplication techniques, such as the adaptive cross approximation [60], the fast multipole method [61], and other approaches [33,34], is still needed for real-world applications involving large spatial domains.…”

An efficient preconditioner for the fast simulation of a 2D stokes flow in porous media

“…The IFMM is an inexact fast direct solver for ‐matrices that can be applied as a preconditioner for GMRES. It has successfully been used to precondition matrices arising from integral operators and radial basis interpolation . The main features of the algorithm are concisely described in the following subsections; the reader is referred to Coulier et al for a more detailed description.…”

“…It has successfully been used to precondition matrices arising from integral operators 68 and radial basis interpolation. 75 The main features of the algorithm are concisely described in the following subsections; the reader is referred to Coulier et al 68 for a more detailed description. Although a 2D problem is considered in this paper, the IFMM is applicable to 3D problems as well (see Coulier et al 68 for examples).…”

Parallelization of the inverse fast multipole method with an application to boundary element method

“…This allows the IFMM to solve 1 with the same asymptotic computation and memory complexities as the FMM, i.e., O(N log 2 1 ε ), where ε is a prescribed accuracy. The efficiency of the IFMM has been studied in several previous works for solving 1: Quaife et al [13] and Coulier et al [6] applied the IFMM as preconditioners for the GMRES [14] to solve 1 from the immersed boundary method regarding Stokes flow problems in two and three dimensions (3D); Takahashi et al [15] applied the IFMM together with the low-frequency FMM to accelerate the BEM for the 3D Helmholtz equation; Coulier et al [16] applied the IFMM to reduce the cost of a mesh deformation method which is based on the radial basis function interpolation.…”