Mesh Deformation Using Radial Basis Function Interpolation With Sliding Boundary Nodes

Abstract

This thesis studies a novel mesh deformation method based on radial basis functions to improve mesh quality in regions with small wall clearances. For fluid-structure interaction problems or imposed motion CFD problems, the usually fixed Eulerian fluid meshes have to undergo deformation to conform to the deforming structure. In such cases, it is important that the fluid mesh retains a reasonably good mesh quality after deformation, since otherwise it can introduce additional errors to the numerical simulation. A suitably robust mesh deformation algorithm is required for this. A variety of mesh deformation methods exist, and the radial basis function (RBF) interpolation method is one of the most robust among these. However, in cases with large deformations in areas with small wall clearances, this method can still fail and the chances of degenerate or low-quality skewed cells occurring is high. If a poor quality mesh results, the domain might have to be re-meshed which is a tedious and computationally expensive procedure.

This thesis aims to combat this potential failure for low wall clearance test cases by modifying the RBF interpolation method to allow for sliding of the usually fixed boundary nodes along the boundary edges/surfaces. The additional degrees of freedom of the boundary nodes reduces skewing in meshes and allows for higher quality meshes in these types of test cases. In this thesis, the sliding method was first created to work on straight edges and planar surfaces, and then further extended to be able to work on any arbitrary surface. Tests were done with both 2D and 3D meshes.

Two sliding algorithm alternatives were tried during this project, with the first one having the sliding conditions built into the RBF interpolation system directly, while the second one performed the sliding in a more indirect manner by using a projection algorithm which ensures that the points remain on the surface. The first direct sliding method results in the interpolation calculations of the spatial directions being coupled to each other, which is not the case with the classical RBF method. This results in an interpolation system that is up to nine times as large (3Nx3N for N boundary points), and much less computationally efficient. The second pseudo-sliding method continues to use the decoupled system and is computationally faster to use. While it initially appeared to be promising, giving good final mesh qualities at low computational speeds for simple 2D cases, this second method was ultimately found to be less robust than the direct coupled method, failing especially with 3D meshes and irregular mesh boundaries.

The direct sliding RBF method was found to be more robust compared to the regular RBF method. The method gives good results with straight edges and planar surfaces, as well as smooth curves. However, when used with non-smooth irregular surfaces, the magnitude of sliding was affected, often resulting in almost no sliding. The method should be avoided in such cases as it is unprofitable. Apart from these cases, the method was able to achieve good improvements in mesh quality compared with the classical RBF method. For applications that have smooth surfaces without large surface irregularities, this method can be used to produce high quality deformed meshes.

Despite its robustness, the direct sliding method is not practical to use due to its large interpolation matrix unless some additional efficiency optimisation method is used. In this case, a control point selection algorithm was used such that fewer points were used to find the interpolation, resulting in a smaller system. With this algorithm, the computational time of the method was considerably reduced to being almost comparable with the classical RBF method (also with the same optimisation applied) particular for large test cases. Another method to reduce the total solution time of the direct sliding method is to use a reduced interpolation system that requires coupling of the interpolation coefficients thereby reducing the size of the interpolation matrix. This has only been briefly investigated but shows large improvements in solution time, while still effectively solving the same system without any reduction in control points.

This means that with the new method, much larger displacement steps can be used to arrive at the same or usually better final quality as obtained with the original RBF method, which results in faster computations. Of course, the time-step size is restricted by the maximum step of the CFD simulation itself. If the magnitude of the deformation is not too high, absolute deformation can also be used instead of relative deformation. This means that the interpolation system will need to be constructed and solved only once at the start of the run based on the initial mesh, resulting in large time savings for the deformation calculation. In conclusion, this method has great potential to be used for mesh deformations, since it can result in an improvement in both computation time and mesh quality.