Keywords: mesh deformation, radial basis functions, unstructured hybrid meshes, Laplacian smoothing.

The motivation behind the work described in this paper is the evaluation of the mesh deformation methodology based on radial basis functions (RBFs) in cases involving large deformations of unstructured hybrid meshes, used for the solution of viscous flow equations. Mesh deformation is a key element in various aerodynamic shape optimization methodologies using computational flow dynamics (CFD), as well as in fluid-structure interaction (FSI) methodologies, where the coupled calculations between computational structural mechanics (CSM) and CFD algorithms require the repeated deformation of the computational mesh.

Several methods are currently in use, such as the method of spring analogy, simple algebraic methods, transfinite interpolation (TFI), the solution of partial differential equations, the moving submesh approach (MSA) and methods based on radial basis functions. The methods based on RBFs exhibit small computational cost while maintain the overall quality of the original grid after the deformation; they are independent of the mesh connectivities and, therefore, structured, unstructured and hybrid meshes can be treated in the same manner.

RBF interpolation is used to derive the displacement of the internal fluid nodes given the displacement of the nodes on the deformed boundary. There are several types of RBF functions, which are suitable for interpolating multivariate data. They can be divided in two basic groups: functions with compact and functions with global support. In the first case mesh nodes inside a circle (two-dimensional) or sphere (three-dimensional) with support radius around a center are influenced by the movement of this center. On the contrary, functions with global support are not equal to zero outside a certain radius but cover the whole interpolation space. In this work several types of RBFs are utilized, while the complete deformation procedure is presented in detail.

The RBF mesh deformation methodology was used to impose large deformations on two-dimensional unstructured hybrid meshes around airfoils, including translation and rotation of the airfoil with respect to its original position. It was realized that, for such large deformations, some of the adopted RBFs failed to produce acceptable deformed meshes: the resulting meshes included large areas with overlapping elements. In order to improve the resulting mesh quality the RBF mesh deformation methodology was combined with a Laplacian smoothing operator, which is repeatedly called after the completion of the basic RBF deformation procedure. A widely adopted metric for mesh quality is used to evaluate the quality of the deformed mesh, with and without Laplacian smoothing, for various radial basis functions and different deformations. Results for unstructured hybrid meshes are presented, demonstrating the effect of the various RBFs as well as the application of Laplacian smoothing, in terms of grid quality metric values.

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