Efficient and exact mesh deformation using multiscale RBF interpolation

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(Ns^3 ). 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(Nred^4) to select Nred points; the multiscale method has no error, and the problem is transferred to a geometric search, with cost O(Nslog(Ns)), 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.