III: Small: Three-Dimensional Visualization and Analysis of Complex Dynamic Physical Phenomena

拟议的研究将开发在三维空间中可视化复杂动力学现象的技术，该空间可以表示为非对称张量场。向量场出现在许多科学和工程应用中。传统的矢量场可视化研究集中在流场粒子的运动轨迹，即，他们的运动。这项研究制定了一个可视化框架，提供额外的信息，基本流粒子运动，包括体积膨胀和收缩，旋转和角变形。这些运动对于许多现象具有物理意义，例如流体的可压缩性、流动分离、流体混合速率、环流（例如，漩涡、龙卷风、飓风等）和能量耗散。有效的可视化和分析动态复杂的物理将能够改善工业过程，节能，天气建模，预测和预警。参与该研究项目的学生将获得计算机科学，科学可视化，微分几何，抽象代数和代数拓扑学方面的知识，他们的研究经验将包括与其他科学和工程学科的合作者一起工作。该项目的目的是为3D非对称张量场的分析奠定基础，并将其纳入矢量场可视化的多场框架。该项目将通过利用矢量场分析和对称张量场分析的知识，以及应用和增强张量演算，不同几何，代数拓扑和动力系统的结果，来理解3D非对称张量场的拓扑和几何结构。为了从非对称张量场中可靠地提取拓扑特征，这些特征的数学公式将构成曲线和曲面提取技术的基础。有效的三维非对称张量场可视化技术的研究将探索和提高三维对称张量场可视化技术以及二维非对称张量场可视化方法。通过扩展标量场和矢量场的多场可视化技术，将开发新的可视化技术用于矢量场及其梯度张量场的多场可视化。研究结果将在一系列应用的数据上进行测试，包括流体和固体力学，长期气候学和地震学。非对称张量场分析的预期结果将增强张量微积分、微分几何、代数拓扑和动力系统的领域，并可能促进科学和工程学科的发展。项目结果，包括理论分析、数值算法、可视化技术、数据集和软件，可通过项目网站（http：//web.engr.oregonstate.edu/publications_NSF_asymetric_tensor.html）查阅。

The proposed research will develop techniques for visualizing complex dynamic phenomena in a 3-dimentional space that can be represented as asymmetric tensor fields. Vector fields appear in many science and engineering applications. Traditional vector field visualization research focuses on the trajectory of flow particles, i.e., their movement. This research formulates a visualization framework that provides additional information on fundamental flow particle motions including volumetric expansion and contraction, rotation, and angular deformation. These motions have physical significance for many phenomena, such as compressibility of the fluids, flow separation, rate of fluid mixing, circular currents (e.g., eddies, tornadoes, hurricanes, etc.) and energy dissipation. Effective visualization and analysis of dynamic complex physical will enable improved industrial processes, energy conservation, weather modeling, prediction and warnings. Students involved in the research project will gain knowledge in computer science, scientific visualization, differential geometry, abstract algebra, and algebraic topology and their research experience will include working with collaborators from other science and engineering disciplinesThe aim of this project is to develop foundations for analysis of 3D asymmetric tensor fields, and incorporate it into a multi-field framework for vector field visualization. This project will lead to understanding the topological and geometric structures of 3D asymmetric tensor fields by leveraging knowledge from vector field analysis and symmetric tensor field analysis as well as applying and enhancing results from tensor calculus, different geometry, algebraic topology, and dynamical systems. To extract topological features reliably from asymmetric tensor fields, mathematical formulations of these features will form the basis for curve and surface extraction techniques. Research on effective 3D asymmetric tensor field visualization techniques will explore and enhance 3D symmetric tensor field visualization techniques as well as 2D asymmetric tensor field visualization methods. Novel visualization techniques will be developed for multi-field visualization for both a vector field and its gradient tensor field by extending multi-field visualization techniques from scalar and vector fields. Research results will be tested on data from a range of applications, including fluid and solid mechanics, long-term climatology, and seismology. The expected results in asymmetric tensor field analysis will enhance the fields of tensor calculus, differential geometry, algebraic topology, and dynamical systems and may scientific and engineering disciplines. Project results, including theoretical analyses, numerical algorithms, visualization techniques, data sets and software will accessible via the project web site (http://web.engr.oregonstate.edu/~zhange/publications_NSF_asymmetric_tensor.html).