The Non-linear Mapping between 3D Surface and 2D Plane by Manifold Learning Algorithm LLE

• Main Author: 闕聖翰
• Other Authors: 楊建成
• Format: Others
• Language: zh-TW
• Published: 2013
• Online Access: http://ndltd.ncl.edu.tw/handle/88793393919934265969

碩士 === 逢甲大學 === 應用數學學系 === 101 === The techniques of adjusting grid are frequently used to obtain the more accurate numerical solutions in computational field simulation. In the past, it is difficult to adjust the surface grid in the three-dimensional space. In this thesis, we use the techniques of dimensionality reduction in manifold learning to develop a strategy of bi-directional nonlinear mapping between three-dimensional surface and two-dimensional plane that can be applied to the study of multigrid, adaptive grid and so on. Both the structured grid and unstructured grid can be mapped from the three-dimensional space into the two-dimensional space via LLE, and the relative position of points in the three-dimensional space will be preserved in the two-dimensional space. As a result, if the data points are changed in the two-dimensional space, these points can be map into the three-dimensional space and the outline of three-dimensional surface is still unchanged. Therefore, we can adjust the grid points directly in the two-dimensional space and map it into the corresponding three-dimensional grid either for the local refinement grid or for the redistributed grid. After adjusting the grid, in order to find the relation between the adjusted grid and the non-adjusted grid, the bilinear interpolation is applied if it is structured grid, and the barycentric interpolation is applied if it is unstructured grid. And then we can map the adjusted grid into the three-dimensional space according to the relation. In addition, we also apply this strategy to transform a quadrilateral grid surface into a triangular grid surface, and vice versa. Therefore, we can change the grid type according to the requirement for a specified case.