A Study Of Smoothing Method For Surface Meshes

Mesh Smoothing is an important research topic for generation of finite element meshes. Based on the investigation of some current algorithms, the first chapter introduces some basic knowledge and current status on mesh generation, post-processing methods, and mesh smoothing.Among averaging smoothing methods, two new algorithms are proposed, CLABM and MSCLABM. CLABM is based on the mechanical models of Laplacian Method (LM) and Angle-Based Method (ABM), it adjusts length of the edges to be rotated according to the coordinates calculated by Laplacian method, and thus it performs simultaneous improvement of both kinds of angles in a placet while smoothing. Experimental results show its stability and effectiveness. Like Smart Laplacian methods, MSCLABM employs a new smart frame, which integrates LM, ABM and CLABM methods. Numerical experiments results reveal its superiority to the LM, ABM and CLABM with traditional smart frames, respectively.We also propose two optimization-based mesh-smoothing methods, namely AD AW and LMA, which are analogues of the steepest decent method. AD AW solves a multi-objective optimization problem of local mesh smoothing by designing a pair of objective functions and properly combining the optimization of them. The first function calculates the average value of element qualities, and the second is used to measure the disparity between the worst element's quality and the average value for the local mesh. In the course of the steepest decent method, the former one, which has derivative function, is used to get the directions of the optimal points, and when it comes to computing steps, which only requires the continuity of functions, both functions are used to get the final result. The result of numerical experiments shows that the ADAW method can improve the quality of the local mesh as well as the quality of the worst element, and it outperforms the existing methods significantly. The LMA method takes the function of calculating the minimal angle of a local mesh, as the objective function, and solves the non-smooth optimization problem by an analogue process. In each iteration of process, the quality function gradients of the elements containing the minimal angle are used to get the steepest decent direction, and the piecewise, function which calculates the minimal angle of the local mesh, is used in the course of linear search. Compared with traditional methods, LMA algorithm reduces the operation complexity, speeds up the smoothing process, and makes use of the operation more effectively. The results of numerical experiments show that the LMA can improve the quality of meshes significantly by enlarging the minimal angles.Based on the experiments and analyses of some existing smart strategies designed for Laplacian and other averaging methods, we designed our own condition set that is more concise and feasible. The way of combining the optimization-based methods with averaging methods is also carefully analyzed, and realized as a feasible and efficient system.In the forth chapter, we also made an elementary research on finding and markingthe cone vertices in accordance with the demand of improving finite element mesh. We make use of the algorithm to simulate the normal of vertices. To calculate the curvature of local meshes, a measurement by using tangent plane is proposed, in order to recognize cones in more cases.The conclusion part of this thesis lies in the last chapter, lists some existing problems and some future work of our four new methods.