【作者】 刘新儒；

【导师】 韩旭里；

【作者基本信息】 中南大学 ， 应用数学， 2011， 博士

【摘要】 本论文在研究带边界约束的网格和曲面生成理论与方法的基础上,给出并讨论了几种网格和卷积曲面的生成方法。给出了边界形变网格生成方法。该方法先通过边界的四个角点生成初始网格,然后从边界线开始递推计算初始网格到目标网格的形变矩阵,最后将初始网格与形变矩阵相加,即生成给定边界上的目标网格。边界形变方法生成网格速度快,且没有以结点为变量的优化方法那样异常的折线。对于某些复杂边界区域上网格生成质量不佳的问题,可以通过重新定义边界来解决。提出了可生成弹性网格的应力平衡法。在该方法中,结点坐标很容易由解迭代方程得到。针对某一方向发生压缩形变而导致单元格内部不存在稳定平衡点的问题,用对数函数得到了改进的平衡方程。由改进方程,只要单元格在任意一个方向上有拉伸形变,则存在可迅速收敛的稳定平衡点。在实际网格生成中,由于边界的复杂性,可能存在完全压缩的单元格,建议将整张网格设置为拉伸形变,并给出了参数设置公式。数值实例表明,由(改进)应力平衡方法生成的网格符合在边界线内凹附近网格线较密、外凸附近网格线较为稀疏的物理特性。给出了由两个方向上带形状参数曲线簇生成网格的方法。随后的定理保证了该方法同样适用于空间网格的生成。讨论了在各种不同边界条件下,形状参数对生成网格的影响。基于相邻网格线间相对位置关系的定理,构造了目标函数来求解最优形状参数。数值实例表明,在这种指标下的最优网格与应力平衡网格具有同样的疏密特征。更重要的是,该方法生成的网格是可调控形状的。利用指数函数给出了可控密度参数网格生成方法。讨论了该方法中密度参数对各种形状边界网格生成的影响。基于单元格的度量性质,提出了求解最优密度参数的正方形准则。数值实例表明,当两组对边存在一定差异时,最优网格与普通网格存在显著差异。我们提出的两种参数网格中优化变量都仅仅为两个,故算法空间、时间复杂度都很低,更容易收敛。讨论了基于轮廓线约束的卷积曲面造型方法。首先简单介绍了一种基于元球的隐式曲面造型方法。这个方法使用球或者椭球对轮廓线的进行逼近,并使用超二次曲面作为原始体的扩展元球模型构造曲面。然后我们提出了一种卷积曲面造型方法,使用轮廓线的向内等距曲线形成的二维形体作为骨架构造曲面,该曲面能满足逼近给定的轮廓线的条件。最后,我们介绍了两种方法生成轮廓线的等距曲线,推导出场值解析计算公式并进行误差分析。实验结果表明,使用我们的方法,特别适用于生成扁平模型。更多还原

【Abstract】 Based on the study about the theories and methods of grid and surface generation with boundary constrains, several methods of grid and convolution surface were presented and discussed.A method based on boundary deformation for grid generation was presented. Firstly, the initial grid was generated by the four corners of the boundary lines. Then the deformation matrix, between the initial grid and the target grid, could be calculated recursively according to the boundary lines. Finally, the target grid in the domain with boundary constrains generated. The grid was generated very fast by using this method, and there are no such unusual wave curves generated by those optimization methods which using the grid nodes as variables. For the problem of generating poor quality grids on some complex boundary zone by this method, it can be resolved by redefining the boundary.We proposed a stress balance method to generate elastic grid, where the grid nodes was easy to obtain by solving the iterate equations. For there does stable equilibrium position in the cell for compression deformation in one direction at least, we improved the equilibrium equation by using logarithmic function. Then, there must be an equilibrium position which can be quickly reached in the cells, except those completely compression in two directions. Moreover, in practice, for the complexity of boundary shape, there may exist completely compression cells in zone. It was recommended to set the entire zone as tensile deformation, and the formula of parameter setting was proposed. Numerical experiments showed that the grids generated by (improved) stress balance method meet the physical characteristics of the denser grid curves near the concave boundary and the sparser grid curves near the convex boundary.A method to generate the grid by the curve clusters with shape parameters in two directions was proposed. The following theorem ensured that the method can also be used in space grid generation. Then, the influence of shape parameters on the grids was discussed by considering a variety of different shape boundaries. Based on the theorem about the relative positions of the adjacent grid curves, an objective function to solve the optimal shape parameters was constructed. Numerical experiments showed that the grid generated by the method and the objective function owns the same density characteristics of the curves. More importantly, the grid generated by the method was shape-controllable.By using the exponential function, a method to generate density controllable grid is presented. Then we discussed the influence of density parameters on generating the grids with all kinds of shape boundaries. Based on the measurements of cells, a square criteria to obtain the optimal density parameters was proposed. Numerical experiments showed that the optimal grid was significant better than the general, when there exists some difference between the boundary in two direction.The two methods with parameters listed above both owned the characteristics of the lower algorithm space and the lower algorithm complexity for there only existing two optimal variables in objective functions. Moreover, the methods were easy to converge.The method to generate the convolution surface constrained with contour curves was discussed. First, a brief introduction of implicit surface modeling based on the metaball. In this method, the contour curves was approximated by the ball or ellipse, and the surface was constructed by the extended metaball model which using the hyper-quadratic as the primitive. Then a modeling method of convolution surface was proposed. In this method, a two-dimensional surface shape formed by inward offset lines of the contour curves was used as the skeleton to construct the surface, which met the condition of approximating the given contour curves. Finally, we introduced two methods to generate the offset lines of the contour curves, and then derived the analytical formula of field value and analyzed its error. Experiments showed that our methods were especially used to generate flat model.更多还原