Efficient Visualization Techniques For Piecewise Linear Vector Fields On Unstructured Grids

The vector field is an important subject in scientific researches and numerical simulations,which involves some very complicated dynamical behaviours,so that there are more and more visualization methods to convert vector fields to more visual-friendly graphical scenes.However,most of these methods heavily depend on numerical integrations.Piecewise vector field defined on unstructured grids is a popular data form for visualization of vector fields,which is flexible but complicated compared to regular meshes.So it is necessary to take piecewise vector fields seriously as the main subject of a thesis.Due to the complexity of vector fields,researchers realize that the complex behaviours in vector fields need to be revealed by enormous integral curves,i.e.we can compute trajectories for as many particles as possible to show the features in vector fields.At the same time,some topological methods also adopt a similar approach to extract features.Their dependency on a huge amount of integral curves contradicts that fact that numerical integration is expensive in computation.To solve this,we use the local linearity to speedup the computation of integral curves.And there are some connections between local linearity and topological features,which are also discussed in this thesis.Our main contributions are listed as follows:(1)To speedup the computation of large amount of integral curves,we propose a batch advection methods to improve the traditional numerical methods,by refactoring the Runge-Kutta method and reusing temporary results.This batch advection method works for both steady and unsteady vector fields with excellent parallel scalability,and it performs even better in dense-seeding situation.Our analysis and experimental results show that our method can greatly improve the efficiency of traditional Runge-Kutta methods.(2)To achieve better efficiency of streamline computation,we propose an approximation methods for vector fields on tetrahedral grids.In this method,the linear vector fields in a tetrahedron is approximated by a series of line bundles,then the streamlines are computed by looking up table and no integration is required,so that the cost of streamline computation is greatly reduced.The experimental results show that we can effectively reduce the time to compute large amount of streamlines without affect the visual effects.(3)To improve the efficiency of critical points detection and to classify them,we propose a method to extract the critical points and to classify them by computing their Poincaré index simultaneously.In this method,we reduce the redundant computation of existing detection method of critical points,and compute their Poincaré index from temporary results,so that we can classify critical points with even less cost of detection.The analysis and experimental results show that our method has better efficiency than existing method and with additional ability of classification.(4)To quickly extract 3D periodic orbits,we propose a method based on vector field approximation.We replace the streamline integration by face-to-face map on tetrahedrons to avoid the inner-cell integration,so that the computation efficiency is improved.The experimental results show that we can extract complex periodic in chaos vector field correctly in the sense of visualization.We have systematically researched the processing of piecewise linear vector fields defined on unstructured grids,which is meaningful in both theoretical and practical senses,and it will an important supplements of vector field visualization.