Visualizing vector field topology in fluid flows

Methods for automating the analysis and display of vector field topology in general, and flow topology in particular, are described. By using techniques to extract and visualize topological information, it is possible to combine the simplicity of schematic depictions with the quantitative accuracy of curves and surfaces computed directly from the data. Two-dimensional vector field topology is discussed, covering critical points and time-dependent flows, to provide a basis for the examination of topology in three-dimensional separated flows. Surface topology and separation structures in three-dimensional flows are then addressed. The construction of representations of tangent surfaces that are accurate, as well as efficient to compute and display, is examined, covering tessellation, clipping, and refinement. Locating, characterizing, and displaying three-dimensional critical points are considered     

Fast combinatorial vector field topology

This paper introduces a novel approximation algorithm for the fundamental graph problem of combinatorial vector field topology (CVT). CVT is a combinatorial approach based on a sound theoretical basis given by Forman's work on a discrete Morse theory for dynamical systems. A computational framework for this mathematical model of vector field topology has been developed recently. The applicability of this framework is however severely limited by the quadratic complexity of its main computational kernel. In this work, we present an approximation algorithm for CVT with a significantly lower complexity. This new algorithm reduces the runtime by several orders of magnitude and maintains the main advantages of CVT over the continuous approach. Due to the simplicity of our algorithm it can be easily parallelized to improve the runtime further.     

Representation and display of vector field topology in fluid flowdata sets

The visualization of physical processes in general and of vector fields in particular is discussed. An approach to visualizing flow topology that is based on the physics and mathematics underlying the physical phenomenon is presented. It involves determining critical points in the flow where the velocity vector vanishes. The critical points, connected by principal lines or planes, determine the topology of the flow. The complexity of the data is reduced without sacrificing the quantitative nature of the data set. By reducing the original vector field to a set of critical points and their connections, a representation of the topology of a two-dimensional vector field is much smaller than the original data set but retains with full precision the information pertinent to the flow topology is obtained. This representation can be displayed as a set of points and tangent curves or as a graph. Analysis (including algorithms), display, interaction, and implementation aspects are discussed     

Comments on "An algorithm for decomposing a two-dimensional nonlinear vector field"

The algorithm is used to derive a closed form for the desired decomposition and potential function.     

Symmetry in scalar field topology

Study of symmetric or repeating patterns in scalar fields is important in scientific data analysis because it gives deep insights into the properties of the underlying phenomenon. Though geometric symmetry has been well studied within areas like shape processing, identifying symmetry in scalar fields has remained largely unexplored due to the high computational cost of the associated algorithms. We propose a computationally efficient algorithm for detecting symmetric patterns in a scalar field distribution by analysing the topology of level sets of the scalar field. Our algorithm computes the contour tree of a given scalar field and identifies subtrees that are similar. We define a robust similarity measure for comparing subtrees of the contour tree and use it to group similar subtrees together. Regions of the domain corresponding to subtrees that belong to a common group are extracted and reported to be symmetric. Identifying symmetry in scalar fields finds applications in visualization, data exploration, and feature detection. We describe two applications in detail: symmetry-aware transfer function design and symmetry-aware isosurface extraction.     

Visualizing the computational intelligence field [Application Notes]

In this paper, we visualize the structure and the evolution of the computational intelligence (CI) field. Based on our visualizations, we analyze the way in which the CI field is divided into several subfields. The visualizations provide insight into the characteristics of each subfield and into the relations between the subfields. By comparing two visualizations, one based on data from 2002 and one based on data from 2006, we examine how the CI field has evolved over the last years. A quantitative analysis of the data further identifies a number of emerging areas within the CI field. The data that we use consist of the abstracts of the papers presented at the IEEE World Congress on Computational Intelligence (WCCI) in 2002 and 2006. Using a fully automatic procedure, so-called concept maps are constructed from the data. These maps visualize the associations between the main concepts in the CI field. Our analysis of the structure and the evolution of the CI field are largely based on the constructed concept maps     

On topology optimization of linear and nonlinear plate problems

In this paper we propose a new restriction method based on employing C ⁰-continuous fields of density defined on a set of meshes different from the one used for the finite element analysis. The optimization procedure starts with using a coarse density-mesh compared to the finite element one. Once the convergence is obtained in the optimization steps, a finer density-mesh is nominated for the further steps. Linear and nonlinear plate behaviors are considered and formulated by Kirchhoff or Mindlin–Reissner hypothesis. Comparison is made with element/nodal based approaches using filter. The results show excellent and robust performance of the proposed method.     

Interactive vector field feature identification

We introduce a flexible technique for interactive exploration of vector field data through classification derived from user-specified feature templates. Our method is founded on the observation that, while similar features within the vector field may be spatially disparate, they share similar neighborhood characteristics. Users generate feature-based visualizations by interactively highlighting well-accepted and domain specific representative feature points. Feature exploration begins with the computation of attributes that describe the neighborhood of each sample within the input vector field. Compilation of these attributes forms a representation of the vector field samples in the attribute space. We project the attribute points onto the canonical 2D plane to enable interactive exploration of the vector field using a painting interface. The projection encodes the similarities between vector field points within the distances computed between their associated attribute points. The proposed method is performed at interactive rates for enhanced user experience and is completely flexible as showcased by the simultaneous identification of diverse feature types.     

A 213-line topology optimization code for geometrically nonlinear structures

Structural and Multidisciplinary Optimization - This paper presents a 213-line MATLAB code for topology optimization of geometrically nonlinear structures. It is developed based on the density...     

Shape preserving design of geometrically nonlinear structures using topology optimization

Structural and Multidisciplinary Optimization - Subparts of load carrying structures like airplane windows or doors must be isolated from distortions and hence structural optimization needs to take...     

Robust structural topology optimization under random field loading uncertainty

A new approach to solving the robust topology optimization problem considering random field loading uncertainty was developed. The Karhunen-Loeve expansion was employed to characterize the random field as a reduced set of random variables. Efficient method of sensitivity analysis was developed and integrated into the density based topology optimization approach. The numerical example demonstrated the efficiency of the proposed approach and the effect of loading uncertainty on the robust design results.     

The multiresolution gradient vector field skeleton

Many algorithms suppress skeleton associated with boundary perturbation by preventing their formation or by costly branch pruning. This work proposes a novel concept of structural and textural skeletons. The former is associated with the general shape structure and the latter with boundary perturbations. These skeletons remain disconnected to facilitate gross shape matching without the need for branch pruning. They are extracted from a multiresolution gradient vector field that is efficiently generated within a pyramidal framework. Experimental results show that these skeletons are scale and rotation invariant. They are less affected by boundary noise compared to skeletons extracted by popular iterative and non-iterative techniques.     

Three-dimensional deformation in curl vector field

Deformation is an important research topic in graphics.There are two key issues in mesh deformation:(1) selfintersection and(2) volume preserving.In this paper,we present a new method to construct a vector field for volume-preserving mesh deformation of free-form objects.Volume-preserving is an inherent feature of a curl vector field.Since the field lines of the curl vector field will never intersect with each other,a mesh deformed under a curl vector field can avoid self-intersection between field lines.Designing the vector field based on curl is useful in preserving graphic features and preventing self-intersection.Our proposed algorithm introduces distance field into vector field construction;as a result,the shape of the curl vector field is closely related to the object shape.We define the construction of the curl vector field for translation and rotation and provide some special effects such as twisting and bending.Taking into account the information of the object,this approach can provide easy and intuitive construction for free-form objects.Experimental results show that the approach works effectively in real-time animation.     

Volume data mining using 3D field topology analysis

This article proposes a novel approach to automating the settings of visualization parameter values for volume data mining. To this end, we extended the conventional Reeb graph-based approach to topological modeling of 3D surfaces to capture the topological skeleton of a volumetric field. The analyzed results take the form of hyper Reeb graphs which give the basic reference structure for designing comprehensible volume visualization     

Continuous force field analysis for generalized gradient vector flow field

We propose a modification of the generalized gradient vector flow field techniques based on a continuous force field analysis. At every iteration the generalized gradient vector flow method obtains a new, improved vector field. However, the numerical procedure always employs the original image to calculate the gradients used in the source term. The basic idea developed in this paper is to use the resulting vector field to obtain an improved edge map and use it to calculate a new gradient based source term. The improved edge map is evaluated by new continuous force field analysis techniques inspired by a preceding discrete version. The approach leads to a better convergence and better segmentation accuracy as compared to several conventional gradient vector flow type methods.     

A discontinuous phase field approach to variational growth-based topology optimization

Numerical instabilities cause the well-known problem of checkerboarding during topology optimization: elements that possess material are periodically neighbored to elements that are material-free. Furthermore, such numerical solutions depend on the finite element mesh and no reasonable processing techniques exist for manufacture. Thus, integral- or gradient-based regularization techniques are usually applied during topology optimization. In this paper, a novel approach to regularization is derived for a recently published variational approach to topology optimization that is based on material growth. The presented approach shares some similarities with the discontinuous Galerkin method and completely removes consideration of additional nodal quantities or complex integration schemes. The derivation and numerical treatment of the resulting phase field equation as well as exemplary numerical results are presented.