Mesh deformation using radial basis interpolation on periodic domains with small clearance gaps

Abstract

In this thesis report, a new radial basis function (RBF) interpolation mesh deformation method is investigated that is applicable on periodic domains that contain small clearance gaps. In the field of fluid structure interaction, there is a strong interaction between a structure and the fluid in which this structure is immersed. When performing simulations of this interaction, the fluid mesh has to adapt to the changing domain which occurs due to the structure deforming. The adaption of the fluid mesh can be done with an RBF interpolation method. Although this is considered to be one of the most robust methods available, there are specific cases in which the method can still yield some poor quality deformed meshes. This is for example the case when small clearance gaps are present such as the region between a blade and the shroud in turbomachinery. Then it becomes beneficial to allow boundary nodes to slide to improve the mesh quality.
In turbomachinery, periodic domains are also often encountered, for example in a gas turbine. Here, the same blade is constantly repeated, allowing to use one single blade in computations which will have a domain with periodic boundaries. In general, the larger the number of blades, the smaller the distance between the blade and the periodic mesh boundary will be. Hence, also here small clearance gaps are present. As the blade might undergo a large deformation, it is possible for the blade to even intersect the mesh boundary if this boundary is kept fixed, which would lead to a degenerate mesh.
The aim of this thesis is to avoid this scenario by making adaptions to the RBF interpolation method that allow the periodic boundaries of the mesh to be displaced with the deforming blade. Evidently, this has to be done in a periodic manner, keeping both periodic boundaries equal. This will yield a higher mesh quality of the deformed mesh as it will reduce the skewness and avoid the occurrence of degenerate cells. Two different methods for periodic boundary displacement are investigated in this report. Both methods work for both translational and rotational periodic boundaries and several two-dimensional (2D) and three-dimensional (3D) meshes are used as test cases.
The first method is the periodic distance method which is based on making the distance between the boundary nodes periodic. This method is very similar to the regular RBF method and only requires a regular Euclidean distance function to be replaced by a periodic distance function. This means that the computational cost is also similar to an interpolation matrix of size N_m x N_m. This method shows good results for both 2D and 3D meshes and both types of periodic boundaries. Apart from giving the periodic boundaries an identical displacement, it also allows for a smooth transition from one mesh to another.
The second method is based on equal displacement. Hence, the RBF interpolation matrix is adapted to include conditions imposing equal displacement of the periodic boundary nodes. These extra conditions yield a larger interpolation matrix of size N_m+N_p x N_m+N_p. This means the computational cost for this method is higher than for the periodic distance or the regular RBF method. This method also shows good results in terms of deformed mesh quality but does not feature the same smoothness over the interface between two meshes and creates small kinks at the interface.
Even though RBF interpolation can be relatively low in computational cost compared to other mesh deformation methods that require mesh connectivity information, it still becomes computationally expensive for large 3D problems with a high number of nodes. Therefore, a greedy algorithm is also used for larger problems which allows for a reduced amount of nodes selected to perform the interpolation.
In conclusion, the periodic displacement method used with RBF interpolation shows great potential for different types of problem sizes and periodic boundaries and can potentially make the simulation process of periodic problems much more efficient.