Radial Basis Functions Mesh Morphing: A Comparison Between the Bi-harmonic Spline and the Wendland C2 Radial Function

Radial basis functions (RBFs) based mesh morphing allows to adapt the shape of a computational grid onto a new one by updating the position of all its nodes. Usually nodes on surfaces are used as sources to define the interpolation field that is propagated into the volume mesh by the RBF. The method comes with two distinctive advantages that makes it very flexible: it is mesh independent and it allows a node wise precision. There are however two major drawbacks: large data set management and excessive distortion of the morphed mesh that may occur. Two radial kernels are widely adopted to overtake such issues: the bi-harmonic spline (BHS) and the Wendland C2 (WC2). The BHS minimizes the mesh distortion but it is computational intense as a dense linear system has to be solved whilist the WC2 leads to a sparse system easier to solve but which can lack in smoothness. In this paper we compare these two radial kernels with a specific focus on mesh distortion. A detailed insight about RBF fields resulting from BHS and WC2 is first provided by inspecting the intensity and the distribution of the strain for a very simple shape: a square plate with a central circular hole. An aeronautical example, the ice formation onto the leading edge of a wing, is then exposed adopting an industrial software implementation based on the state of the art of RBF solvers.