Efficient Extraction Of Periodic Orbits In Large-scale Vector Fields

Topological visualization is an important research content of vector field visualization.It mainly studies singular points,periodic orbits and separation lines/surfaces.Separation lines/surfaces are determined by both periodic orbits and singular points;singular points are local characteristics of vector fields,and their research progress is relatively mature;and periodic orbits,which involve large-scale dynamic properties of vector fields,are the important and difficult problems in topological visualization,and the research progress is relatively slow.The existing methods of extracting periodic orbits have the shortcomings of low efficiency and poor accuracy.To solve these problems,this paper proposes corresponding solutions.The specific work is summarized as follows:1.Streamline is the basis of extracting periodic orbits.Aiming at the low efficiency of streamline generation,this paper proposes an efficient algorithm for streamline generation.Firstly,the corresponding matrices of all grids are obtained,and the grids are divided into three types according to the different Jordan standard forms of the matrices.Then,the corresponding streamline expressions are obtained for different types of grids,and the streamline expressions of different types of grids are unified.The experimental results show that the efficiency of the algorithm can be increased by about20% compared with the existing methods.2.Aiming at the low efficiency of current two-dimensional periodic orbit extraction methods,this paper proposes an efficient periodic orbit extraction algorithm.Based on the Wischgoll et al algorithm,the algorithm effectively shortens the generation time of alternative streamlines by reducing the number of candidate seed points that may generate closed streamlines;and then further reduces the number of grid points that need to be tested around alternative streamlines according to the disjoint nature of streamlines,thus effectively improving the efficiency of periodic orbit extraction.Rate.Experimental results show that this method can save more than 90%computation time compared with existing methods.3.Aiming at the problem that the accuracy of 3-D periodic orbit extraction is poor at present,this paper proposes an accurate 3-D periodic orbit extraction algorithm based on the ring region theorem.Firstly,the algorithm finds the region where there may be periodic orbits,and then finds the periodic orbits by judging whether there is a positive invariant surface in the region.The method ensures the accuracy of extracting periodic orbits theoretically,and preliminary experimental results show that the proposed method can extract three-dimensional periodic orbits effectively.