Topology based methods for vector field comparisons.

摘要

Vector fields are commonly found in almost all branches of the physical sciences. Aerodynamics, dynamical systems, electromagnetism, and global climate modeling are a few examples. These multivariate data fields are often large, and no general, automated method exists for comparing these fields. Existing methods require either subjective visual judgments, or data interface compatibility, or domain specific knowledge. A topology based method intrinsically eliminates all of the above limitations and has the additional advantage of significantly compressing the vector field by representing only key features of the flow. Therefore, large databases are compactly represented and quickly searched.; Topology is a natural framework for the study of many vector fields. It provides rules of an organizing principle, a flow grammar, that can describe and connect together the properties common to flows. Helman and Hesselink first introduced automated methods to extract and visualize this grammar. This work extends their method by introducing automated methods for vector topology comparison. Basic two-dimensional flows are first compared. The theory is extended to compare three-dimensional flow fields and the topology on no-slip surfaces. Concepts from graph theory and linear programming are utilized to solve these problems. Finally, the first automated method for higher order singularity comparisons is introduced using mathematical theories from geometric (Clifford) algebra.