Mesh Deformation Using Radial Basis Function Interpolation for Numerical Optimisation of Internal Flow Applications

Abstract

Developments in computational capabilities and the always increasing demand for higher performance of internal flow applications has meant that Computational Fluid Dynamics (CFD) has become an essential tool within the design process. A key point of interest is to couple the fluid solver with numerical optimisation techniques in order to obtain a more automated design process that can handle a vast amount of different design aspects. The open source CFD suite SU2 has emerged as an enabler for aerodynamic shape optimisation involving a large number of design variables, due to its efficient, accurate and flexible discrete adjoint solver. 
For the adjoint-based aerodynamic design optimisation of internal flow applications the deformation of the volumetric mesh has to be performed in an robust and efficient manner. Often small wall clearance gaps and periodic domains are encountered in internal flow domains, which could potentially lead to the deterioration of the mesh. Sliding boundary node methods can be applied in order to maintain the mesh quality in case of small wall clearance gaps. Additionally, periodic boundaries can be displaced in a periodic manner following the applied deformation in order to prevent low quality cells near the periodic interface. Therefore, it would be of interest to implement the sliding boundary node methods and periodic conditions in a Radial Basis Function (RBF) interpolation method, one of the most robust mesh deformation methods available. 
Additionally, the computational efficiency should be considered, since high computational times should be prevented for large and complex three-dimensional cases with a high number of design variables. 
The aim of this thesis project is therefore to develop a robust and computationally efficient mesh deformation method suitable within the discrete adjoint optimisation framework of SU2 for internal flow applications by means of developing an implementation of the RBF interpolation method including sliding boundary node algorithms, periodic boundary conditions and data reductions methods.
The sliding is achieved by replacing the interpolation condition for the sliding nodes with a planar slip condition. Or alternatively, by freely displacing the sliding nodes based on the known deformation and subsequently projecting the nodes back onto the boundary. The periodic displacement of the boundaries is ensured by making the distance function of the RBF periodic. The periodic nodes are then treated as internal nodes to allow them to move.
The developed RBF-SliDe tool is able to generate higher minimum mesh qualities compared with the regular RBF interpolation method. The sliding of the boundary nodes reduces the degree of skewing of the mesh elements in case of drastic deformations, resulting in a higher minimum mesh quality. Furthermore, the introduction of the periodic displacement prevents low quality skewed or compressed mesh elements, as the periodic boundaries move along with the deformation. 
The Aachen turbine stator blade is considered as a realistic three-dimensional test case. For this stator blade an optimised geometry was available, which was obtained with an adjoint-based aerodynamic optimisation performed with SU2. Therefore, the resulting minimum mesh quality is compared to the one obtained with the more conventional linear elasticity equation method as used in SU2. The minimum mesh quality obtained with the RBF-SliDe tool is nearly three times higher compared to the minimum mesh quality of the linear elasticity equations methods. This highlights the potential of the periodic sliding RBF interpolation method in terms of preserving the mesh quality.