Efficient and exact mesh deformation using multiscale RBF interpolation

Abstract

Radial basis function (RBF) interpolation is popular for mesh deformation due to robustness and generality, but the cost scales with the number of surface points sourcing the deformation as O(N 3 s ). Hence, there have been numerous works investigating efficient methods using reduced datasets. However, although reduced-data methods are efficient, they require a secondary method to treat an error vector field to ensure surface points not included in the primary deformation are moved to the correct location, and the volume mesh moved accordingly.A new method is presented which captures global and local motions at multiple scales using all the surface points, and so no correction stage is required; all surface points are used and a single interpolation built, but the cost and conditioning issues associated with RBF methods are eliminated. Moreover, the sparsity introduced is exploited using a wall distance function, to further reduce the cost. The method is compared to an efficient greedy method, and it is shown mesh quality is always comparable with or better than with the greedy method, and cost is comparable or cheaper at all stages. Surface mesh preprocessing is the dominant cost for reduced-data methods and this cost is reduced significantly here: greedy methods select points to minimise interpolation error, requiring repeated system solution and cost O(N 4 red ) to select N red points; the multiscale method has no error, and the problem is transferred to a geometric search, with cost O(N s log(N s )), resulting in an eight orders of magnitude cost reduction for three-dimensional meshes. Furthermore, since the method is dependent on geometry, not deformation, it only needs to be applied once, prior to simulation, as the mesh deformation is decoupled from the point selection process.

Gradient-Limiting Shape Optimisation Applied to AIAA ADODG Test Cases

“…In this work interpolation using multiscale radial basis functions (RBFs) [46] is used. Interpolation using radial basis functions (RBFs) has recently become a prominent mesh deformation method boasting excellent robustness and quality-preserving characteristics [47][48][49].…”

“…due to the localised surface rotation. The multiscale RBF method [46] is particularly effective, both increasing the computational efficiency and improving the system conditioning over conventional or reduced datapoint RBF methods, by using variable length scales depending on boundary point locations.…”

Optimisation of Multi-Modal Aerodynamic Shape and Topology Problems

“…Not only is this computationally cheaper than regenerating a mesh for each geometry iteration but it also maintains consistency of the discretisation error which is highly desirable during iterative numerical optimisation. In this work interpolation using multiscale radial basis functions (RBFs) [50] is used. Interpolation using radial basis functions (RBFs) has recently become a prominent mesh deformation method boasting excellent robustness and quality-preserving characteristics [51][52][53].…”

A balanced load mapping method based on radial basis functions and fuzzy sets

“…Another hot spot when dealing with RBFs is the expensive evaluation of the inverse matrix shown in Equations and . To deal with this problem, algorithms are conceived to reduce the number of source points . The fast multipole method, originally introduced by Greengard and Rokhlin, allows performing fast evaluations of sums of the type of Equation .…”