Valentin Zobel

Tensor Visualization Driven Mechanical Component Design

This paper is the result of a close collaboration of mechanical engineers and visualization researchers. It showcases how interdisciplinary work can lead to new insight and progress in both fields. Our case is concerned with one step in the product development process. Its goal is the design of mechanical parts that are functional, meet required quality measures and can be manufactured with standard production methods. The collaboration started with unspecific goals and first experiments with the available data and visualization methods. During the course of the collaboration many concrete questions arose and in the end a hypothesis was developed which will be discussed and evaluated in this paper. We facilitate a case study to validate our hypothesis. For the case study we consider the design of a reinforcement structure of a brake lever, a plastic ribbing. Three new lever geometries are developed on basis of our hypothesis and are compared against each other and against a reference model. The validation comprises standard numerical and experimental tests. In our case, all new structures outperform the reference geometry. The results are very promising and suggest potential to impact the product development process also for more complex scenarios.

Feature-based tensor field visualization for fiber reinforced polymers

Virtual testing is an integral part of modern product development in mechanical engineering. Numerical structure simulations allow the computation of local stresses which are given as tensor fields. For homogeneous materials, the tensor information is usually reduced to a scalar field like the von Mises stress. A material-dependent threshold defines the material failure answering the key question of engineers. This leads to a rather simple feature-based visualisation. For composite materials like short fiber reinforced polymers, the situation is much more complex. The material property is determined by the fiber distribution at every position, often described as fiber orientation tensor field. Essentially, the materials ability to cope with stress becomes anisotropic and inhomogeneous. We show how to combine the stress field and the fiber orientation field in such cases, leading to a feature-based visualization of tensor fields for composite materials. The resulting features inform the engineer about potential improvements in the product development.

2D Tensor Field Segmentation

We present a topology-based segmentation as means for visualizing 2D symmetric tensor fields. The segmentation uses directional as well as eigenvalue characteristics of the underlying field to delineate cells of similar (or dissimilar) behavior in the tensor field. A special feature of the resulting cells is that their shape expresses the tensor behavior inside the cells and thus also can be considered as a kind of glyph representation. This allows a qualitative comprehension of important structures of the field. The resulting higher-level abstraction of the field provides valuable analysis. The extraction of the integral topological skeleton using both major and minor eigenvector fields serves as a structural pre-segmentation and renders all directional structures in the field. The resulting curvilinear cells are bounded by tensorlines and already delineate regions of equivalent eigenvector behavior. This pre-segmentation is further adaptively refined to achieve a segmentation reflecting regions of similar eigenvalue and eigenvector characteristics. Cell refinement involves both subdivision and merging of cells achieving a predetermined resolution, accuracy and uniformity of the segmentation. The buildingblocks of the approach can be intuitively customized to meet the demands or different applications. Application to tensor fields from numerical stress simulations demonstrates the effectiveness of our method.

Extremal curves and surfaces in symmetric tensor fields

The visualization of symmetric second-order tensor fields in two or three dimensions is still a challenging task, particularly if global structures of the data are desired. One approach is tensor field topology which provides structures characterizing the behavior of the eigenvector fields. Another widely used approach is analyzing tensor fields by means of scalar invariants, i.e., quantities invariant with respect to changes of the coordinate system. In this case, the selection of the relevant invariants might be difficult. Thus, we propose an approach which analyzes the complete invariant part of the tensor. We define extremal points for tensor fields in a mathematically rigorous way, which form curves for two-dimensional and surfaces for three-dimensional tensor fields. We propose a way to compute extremal curves or surfaces from a suitable set of two or three invariants, respectively. We also show that commonly used sets of invariants lead to the same extremal points. Consequently, extremal points contain minima and maxima of most invariants used in tensor field analysis and they are linked to the tensor field topology by containing the degenerate points. Moreover, we show that each extremal point is an extremum or a saddle of a certain invariant. The method is demonstrated on synthetic datasets as well as on stress tensor fields from structure simulations.

Visualizing Gradients of Stress Tensor Fields

In some applications, it is necessary to look into gradients of (symmetric) second order tensor fields. These tensors are of third order. In three-dimensional space, we have 18 independent coefficients at each position, so the visualization of these fields provides a challenge. A particular case are stress gradients in structural mechanics. There are specific situations where the stress gradient is required together with the stress to study material behavior. Since the visualization community lacks methods to show these fields, we look at some preliminary ideas to design appropriate glyphs. We motivate our glyph designs by typical depictions of stress in engineering textbooks.

Tensor Lines in Engineering: Success, Failure, and Open Questions

Today, product development processes in mechanical engineering are almost entirely carried out via computer-aided simulations. One essential output of these simulations are stress tensors, which are the basis for the dimensioning of the technical parts. The tensors contain information about the strength of internal stresses as well as their principal directions. However, for the analysis they are mostly reduced to scalar key metrics. The motivation of this work is to put the tensorial data more into focus of the analysis and demonstrate its potential for the product development process. In this context we resume a visualization method that has been introduced many years ago, tensor lines. Since tensor lines have been rarely used in visualization applications, they are mostly considered as physically not relevant in the visualization community. In this paper we challenge this point of view by reporting two case studies where tensor lines have been applied in the process of the design of a technical part. While the first case was a real success, we could not reach similar results for the second case. It became clear that the first case cannot be fully generalized to arbitrary settings and there are many more questions to be answered before the full potential of tensor lines can be realized. In this chapter, we review our success story and our failure case and discuss some directions of further research.

Visualizing Symmetric Indefinite 2D Tensor Fields Using the Heat Kernel Signature

The Heat Kernel Signature (HKS) is a scalar quantity which is derived from the heat kernel of a given shape. Due to its robustness, isometry invariance, and multiscale nature, it has been successfully applied in many geometric applications. From a more general point of view, the HKS can be considered as a descriptor of the metric of a Riemannian manifold. Given a symmetric positive definite tensor field we may interpret it as the metric of some Riemannian manifold and thereby apply the HKS to visualize and analyze the given tensor data. In this paper, we propose a generalization of this approach that enables the treatment of indefinite tensor fields, like the stress tensor, by interpreting them as a generator of a positive definite tensor field. To investigate the usefulness of this approach we consider the stress tensor from the two-point-load model example and from a mechanical work piece.

Visualization of Two-Dimensional Symmetric Positive Definite Tensor Fields Using the Heat Kernel Signature

We propose a method for visualizing two-dimensional symmetric positive definite tensor fields using the Heat Kernel Signature (HKS). The HKS is derived from the heat kernel and was originally introduced as an isometry invariant shape signature. Each positive definite tensor field defines a Riemannian manifold by considering the tensor field as a Riemannian metric. On this Riemmanian manifold we can apply the definition of the HKS. The resulting scalar quantity is used for the visualization of tensor fields. The HKS is closely related to the Gaussian curvature of the Riemannian manifold and the time parameter of the heat kernel allows a multiscale analysis in a natural way. In this way, the HKS represents field related scale space properties, enabling a level of detail analysis of tensor fields. This makes the HKS an interesting new scalar quantity for tensor fields, which differs significantly from usual tensor invariants like the trace or the determinant. A method for visualization and a numerical realization of the HKS for tensor fields is proposed in this chapter. To validate the approach we apply it to some illustrating simple examples as isolated critical points and to a medical diffusion tensor data set.

Visualization of Two-Dimensional Symmetric Tensor Fields Using the Heat Kernel Signature

We propose a method for visualizing two-dimensional symmetric tensor fields using the Heat Kernel Signature (HKS). The HKS is derived from the heat kernel and was originally introduced as an isometry invariant shape signature. The time parameter of the heat kernel allows a multiscale analysis in a natural way. By considering a positive definite tensor field as a Riemannian metric the definition of the HKS can be applied directly. To investigate how this measure can be used to visualize more general tensor fields we apply mappings to obtain positive definite tensor fields. The resulting scalar quantity is used for the visualization of tensor fields. For short times it is closely related to Gaussian curvature, i. e. it is quite different to usual tensor invariants like the trace or the determinant.When scientists analyze datasets in a search for underlying phenomena, patterns or causal factors, their first step is often an automatic or semi-automatic search for structures in the data. Of these feature-extraction methods, topological ones stand out due to their solid mathematical foundation. Topologically defined structuresas found in scalar, vector and tensor fieldshave proven their merit in a wide range of scientific domains, and scientists have found them to be revealing in subjects such as physics, engineering, and medicine. Full of state-of-the-art research and contemporary hot topics in the subject, this volume is a selection of peer-reviewed papers originally presented at the fourth Workshop on Topology-Based Methods in Data Analysis and Visualization, TopoInVis 2011, held in Zurich, Switzerland. The workshop brought together many of the leading lights in the field for a mixture of formal presentations and discussion. One topic currently generating a great deal of interest, and explored in several chapters here, is the search for topological structures in time-dependent flows, and their relationship with Lagrangian coherent structures. Contributors also focus on discrete topologies of scalar and vector fields, and on persistence-based simplification, among other issues of note. The new research results included in this volume relate to all three key areas in data analysistheory, algorithms and applications.