Homology and symmetry breaking in Rayleigh-Benard convection: Experiments and simulations

Algebraic topology (homology) is used to analyze the weakly turbulent state of spiral defect chaos in both laboratory experiments and numerical simulations of Rayleigh-Benard convection.The analysis reveals topological asymmetries that arise when non-Boussinesq effects are present.Comment: 21 pages with 6 figure

Efficient Morse decompositions of vector fields

Existing topology-based vector field analysis techniques rely on the ability to extract the individual trajectories such as fixed points, periodic orbits and separatrices which are sensitive to noise and errors introduced by simulation and interpolation. This can make such vector field analysis unsuitable for rigorous interpretations. We advocate the use of Morse decompositions, which are robust with respect to perturbations, to encode the topological structures of a vector field in the form of a directed graph, called a Morse connection graph (MCG). While an MCG exists for every vector field, it need not be unique. Previous techniques for computing MCG's, while fast, are overly conservative and usually results in MCG's that are too coarse to be useful for the applications. To address this issue, we present a new technique for performing Morse decomposition based on the concept of τ-maps, which typically provides finer MCG's than existing techniques. Furthermore, the choice of τ provides a natural tradeoff between the fineness of the MCG's and the computational costs. We provide efficient implementations of Morse decomposition based on τ-maps, which include the use of forward and backward mapping techniques and an adaptive approach in constructing better approximations of the images of the triangles in the meshes used for simulation. Furthermore, we propose the use of spatial τ-maps in addition to the original temporal τ-maps. These techniques provide additional tradeoffs between the quality of the MCG's and the speed of computation. We demonstrate the utility of our technique with various examples in plane and on surfaces including engine simulation datasets.Keywords: Vector field topology, Flow combinatorialization, Morse connection graph, Morse decomposition, tau mapsKeywords: Vector field topology, Flow combinatorialization, Morse connection graph, Morse decomposition, tau map

Efficient Morse decompositions of vector fields

Vector field analysis plays a crucial role in many engineering applications, such as weather prediction, tsunami and hurricane study, and airplane and automotive design. Existing vector field analysis techniques focus on individual trajectories such as fixed points, periodic orbits and separatrices which are sensitive to noise and errors introduced by simulation and interpolation. This can make such vector field analysis unsuitable for rigorous interpretations. In this paper, we advocate the use of Morse decompositions, which are robust with respect to perturbations, to encode the topological structures of the vector field in the form of a directed graph, called a Morse decomposition connection graph (MCG). While an MCG exists for every vector field, it need not be unique. We develop the idea of a [tau] map, which decouples the MCG construction process and the configuration of the underlying mesh. This, in general, results in finer MCGs than mesh-dependent approaches. To compute MCGs effectively, we present an adaptive approach in constructing better approximations of the images of triangles in the meshes used for simulation. These techniques result in fast and efficient MCG construction. We demonstrate the efficacy of our technique on various examples in planar fields and on surfaces including engine simulation data.Keywords: Morse decomposition, connection graph, Vector field topology, multi-valued mapKeywords: Morse decomposition, connection graph, Vector field topology, multi-valued ma

Vector field editing and periodic orbit extraction using Morse decomposition

Design and control of vector fields is critical for many visualization and graphics tasks such as vector field visualization, fluid simulation, and texture synthesis. The fundamental qualitative structures associated with vector fields are fixed points, periodic orbits, and separatrices. In this paper we provide a new technique that allows for the systematic creation and cancellation of fixed points and periodic orbits. This technique enables vector field design and editing on the plane and surfaces with desired qualitative properties. The technique is based on Conley theory which provides a unified framework that supports the cancellation of fixed points and periodic orbits. We also introduce a novel periodic orbit extraction and visualization algorithm that detects, for the first time, periodic orbits on surfaces. Furthermore, we describe the application of our periodic orbit detection and vector field simplification algorithm to engine simulation data demonstrating the utility of the approach. We apply our design system to vector field visualization by creating datasets containing periodic orbits. This helps us understand the effectiveness of existing visualization techniques. Finally, we propose a new streamline-based technique that allows vector field topology to be easily identified.Keywords: Conley index, Morse decomposition, Vector field visualization, Vector field topology, periodic orbit detection, vector field simplification, Vector field design, connection graph

Vector Field Design on Surfaces

Vector field design on surfaces is necessary for many graphics applications: example-based texture synthesis, non-photorealistic rendering, and fluid simulation. A vector field design system should allow a user to create a large variety of complex vector fields with relatively little effort. In this paper, we present a vector field design system for surfaces that allows the user to control the number of singularities in the vector field and their placement. Our system combines basis vector fields to make an initial vector field that meets the user's specifications. The initial vector field often contains unwanted singularities. Such singularities cannot always be eliminated, due to the Poincar'e-Hopf index theorem. To reduce the effect caused by these singularities, our system allows a user to move a singularity to a more favorable location or to cancel a pair of singularities. These operations provide topological guarantees for the vector field in that they only affect the user-specified singularities. Other editing operations are also provided so that the user may change the topological and geometric characteristics of the vector field. We demonstrate our vector field design system for several applications: example-based texture synthesis, painterly rendering of images, and pencil sketch illustrations of smooth surfaces

Feature-Based Surface Parameterization and Texture Mapping

Surface parameterization is necessary for many graphics tasks: texture-preserving simplification, remeshing, surface painting, and pre-computation of solid textures. The stretch caused by a given parameterization determines the sampling rate on the surface. In this paper, we propose an automatic parameterization method that segments a surface into patches that are then flattened with little stretch. We observe that many objects consist of regions of relative simple shapes, each of which has a natural parameterization. Therefore, we propose a three-stage feature based patch creation method for manifold mesh surfaces. The first two stages, genus reduction and feature identification, are performed with the help of distance-based Morse functions. In the last stage, we create one or two patches for each feature region based on a covariance matrix of the feature's surface points. To reduce the stretch during patch unfolding, we notice that the stretch is a 2x2 tensor which in ideal situations is the identity. Therefore, we propose to use the Green-Lagrange tensor to measure and to guide the optimization process. Furthermore, we allow the boundary vertices of a patch to be optimized by adding scaffold triangles. We demonstrate our feature identification and patch unfolding methods for several textured models. Finally, to evaluate the quality of a given parameterization, we propose an image-based error measure that takes into account stretch, seams, smoothness, packing efficiency and visibility

Multiscale analysis of nonlinear systems using computational homology

This is a collaborative project between the principal investigators. However, as is to be expected, different PIs have greater focus on different aspects of the project. This report lists these major directions of research which were pursued during the funding period: (1) Computational Homology in Fluids - For the computational homology effort in thermal convection, the focus of the work during the first two years of the funding period included: (1) A clear demonstration that homology can sensitively detect the presence or absence of an important flow symmetry, (2) An investigation of homology as a probe for flow dynamics, and (3) The construction of a new convection apparatus for probing the effects of large-aspect-ratio. (2) Computational Homology in Cardiac Dynamics - We have initiated an effort to test the use of homology in characterizing data from both laboratory experiments and numerical simulations of arrhythmia in the heart. Recently, the use of high speed, high sensitivity digital imaging in conjunction with voltage sensitive fluorescent dyes has enabled researchers to visualize electrical activity on the surface of cardiac tissue, both in vitro and in vivo. (3) Magnetohydrodynamics - A new research direction is to use computational homology to analyze results of large scale simulations of 2D turbulence in the presence of magnetic fields. Such simulations are relevant to the dynamics of black hole accretion disks. The complex flow patterns from simulations exhibit strong qualitative changes as a function of magnetic field strength. Efforts to characterize the pattern changes using Fourier methods and wavelet analysis have been unsuccessful. (4) Granular Flow - two experts in the area of granular media are studying 2D model experiments of earthquake dynamics where the stress fields can be measured; these stress fields from complex patterns of 'force chains' that may be amenable to analysis using computational homology. (5) Microstructure Characterization - We extended our previous work on studying the time evolution of patterns associated with phase separation in conserved concentration fields. (6) Probabilistic Homology Validation - work on microstructure characterization is based on numerically studying the homology of certain sublevel sets of a function, whose evolution is described by deterministic or stochastic evolution equations. (7) Computational Homology and Dynamics - Topological methods can be used to rigorously describe the dynamics of nonlinear systems. We are approaching this problem from several perspectives and through a variety of systems. (8) Stress Networks in Polycrystals - we have characterized stress networks in polycrystals. This part of the project is aimed at developing homological metrics which can aid in distinguishing not only microstructures, but also derived mechanical response fields. (9) Microstructure-Controlled Drug Release - This part of the project is concerned with the development of topological metrics in the context of controlled drug delivery systems, such as drug-eluting stents. We are particularly interested in developing metrics which can be used to link the processing stage to the resulting microstructure, and ultimately to the achieved system response in terms of drug release profiles. (10) Microstructure of Fuel Cells - we have been using our computational homology software to analyze the topological structure of the void, metal and ceramic components of a Solid Oxide Fuel Cell