曲面标架场生成的快速算法

Fast Algorithm for Generating Frame Fields on Surfaces

在网格生成、有限元剖分中,标架场起着重要作用.针对已有的生成标架场的算法耗时较多的问题,提出一种新颖的高效生成标架场的算法.给定一个三角网格,首先在每一点上根据曲面的第二基本型定义一个黎曼度量;然后把各向异性标架场看成在该黎曼度量意义下的各向同性场,求出一个在该黎曼度量意义下的各向同性场;最后利用该黎曼度量把各向同性场变回各向异性标架场.此外,在优化过程中应用一种多分辨率层次结构,并使用Gauss-Seidel迭代法来求解每层中的能量优化问题,以提高计算速度.实验结果表明,该算法不仅提高了生成标架场的速度,并且使得生成的标架场与曲率走势更加接近.

The frame field has important applications in mesh generation and finite element analysis.In the past,the method of generating frame field took much time.This paper presents a new technique for generating frame fields fast.Given a triangular mesh,first,Riemann metrics can be defined at each point by the second fundamental tensor of the surface.Then,the anisotropic frame field is regarded as an isotropic field in the sense of this Riemann metrics,and an isotropic field in the sense of the Riemann metric is obtained.In addition,in order to accelerate the solving process,this paper also constructs a multi-resolution hierarchical structure,and using the Gauss-Seidel iterative method to solve the optimization problems in each layer which improves the speed of generating frame field.This algorithm is compared with the previous methods,which show that this algorithm not only improves the speed of generating the frame field,but also makes the generated frame field and curvature closer in trend.