Visualization and exploration of nematic liquid crystal (NLC) data is a challenging task due to the multidimensional and multivariate nature of the data. Simulation study of an NLC consists of multiple timesteps, where each timestep computes scalar, vector, and tensor parameters on a geometrical mesh. Scientists developing an understanding of liquid crystal interaction and physics require tools and techniques for effective exploration, visualization, and analysis of these data sets. Traditionally, scientists have used a combination of different tools and techniques like 2D plots, histograms, cut views, etc. for data visualization and analysis. However, such an environment does not provide the required insight into NLC datasets. This thesis addresses two areas of the study of NLC data---understanding of the tensor order field (the Q-tensor) and defect detection in this field. Tensor field understanding is enhanced by using a new glyph (NLCGlyph) based on a new design metric which is closely related to the underlying physical properties of an NLC, described using the Q-tensor. A new defect detection algorithm for 3D unstructured grids based on the orientation change of the director is developed. This method has been used successfully in detecting defects for both structured and unstructured models with varying grid complexity.

Tensors are of great interest to many applications in engineering and in medical imaging, but a proper analysis and visualization remains challenging. Physics-based visualization of tensor fields has proven to show the main features of symmetric second-order tensor fields, while still displaying the most important information of the data, namely the main directions in medical diffusion tensor data using texture and additional attributes using color-coding, in a continuous representation. Nevertheless, its application and usability remains limited due to its computational expensive and sensitive nature. We introduce a novel approach to compute a fabric-like texture pattern from tensor fields on arbitrary non-selfintersecting surfaces that is motivated by image space line integral convolution (LIC). Our main focus lies on regaining three-dimensionality of the data under user interaction, such as rotation and scaling. We employ a multi-pass rendering approach to estimate proper modification of the LIC noise input texture to support the three-dimensional perception during user interactions.

This work presents four new methods for the analysis and visualization of tensor fields. The focus is on tensor fields which arise in the context of structural mechanics simulations. The first method deals with the design of components made of short fiber reinforced polymers using injection molding. The stability of such components depends on the fiber orientations, which are affected by the production process. For this reason, the stresses under load as well as the fiber orientations are analyzed. The stresses and fiber orientations are each given as tensor fields. For the analysis four features are defined. The features indicate if the component will resist the load or not, and if the respective behavior depends on the fiber orientation or not. For an in depth analysis a glyph was developed, which shows the admissible fiber orientations as well as the given fiber orientation. With these visualizations the engineer can rate a given fiber orientation and gets hints for improving the fiber orientation. The second method depicts gradients of stress tensors using glyphs. A thorough understanding of the stress gradient is desirable, since there is some evidence that not only the stress but also its gradient influences the stability of a material. Gradients of stress tensors are third order tensors, the visualization is therefore a great challenge and there is very little research on this subject so far. The objective of the third method is to analyse the complete invariant part of the tensor field. Scalar invariants play an important role in many applications, but proper selection of such invariants is often difficult. For the analysis of the complete invariant part the notion of 'extremal point' is introduced. An extremal point is characterized by the fact that there is a scalar invariant which has a critical point at this position. Moreover it will be shown that the extrema of several common invariants are contained in the set of critical points. The fourth method presented in this work uses the Heat Kernel Signature (HKS) for the visualization of tensor fields. The HKS is computed from the heat kernel and was originally developed for surfaces. It characterizes the metric of the surface under weak assumptions. i.e. the shape of the surfaces is determined up to isometric deformations. The fact that every positive definite tensor field can be considered as the metric of a Riemannian manifold allows to apply the HKS on tensor fields.

Materialien wie Kristalle, biologisches Gewebe oder elektroaktive Polymere kommen häufig in verschiedenen Anwendung, wie dem Prothesenbau oder der Simulation von künstlicher Muskulatur vor. Diese und viele weitere Materialien haben gemeinsam, dass sie unter gewissen Umständen ihre Form und andere Materialeigenschaften ändern. Um diese Veränderung beschreiben zu können, werden, abhängig von der Anwendung, verschiedene Tensoren unterschiedlicher Ordnung benutzt. Durch die Komplexität und die starke Abhängigkeit der Tensorbedeutung von der Anwendung, gibt es bisher kein Verfahren Tensoren höherer Ordnung darzustellen, welches standardmäßig benutzt wird. Auch bezogen auf einzelne Anwendungen gibt es nur sehr wenig Arbeiten, die sich mit der visuellen Darstellung dieser Tensoren auseinandersetzt. Diese Arbeit beschäftigt sich mit diesem Problem. Es werden drei verschiedene Methoden präsentiert, Tensoren höherer Ordnung zu analysieren und zu visualisieren. Alle drei Methoden basieren auf der sogenannte deviatorischen Zerlegung und der Multipoldarstellung. Mit Hilfe der Multipole können die Symmetrien des Tensors und damit des beschriebenen Materials bestimmt werden. Diese Eigenschaft wird in für die Visualisierung des Steifigkeitstensors benutzt. Die zweite Methode basiert direkt auf den Multipolen und kann damit beliebige Tensoren in drei Dimensionen darstellen. Dieses Verfahren wird anhand des Kopplungs Tensors, ein Tensor dritter Ordnung, vorgestellt. Die ersten zwei Verfahren sind lokale Glyph-basierte Verfahren. Das dritte Verfahren ist ein erstes globales Tensorvisualisierungsverfahren, welches Tensoren beliebiger Ordnung und Symmetry in drei Dimensionen mit Hilfe eines linienbasierten Verfahrens darstellt. / Materials like crystals, biological tissue or electroactive polymers are frequently used in applications like prosthesis construction or the simulation of artificial musculature. These and many other materials have in common that they change their shape and other material properties under certain circumstances. To describe these changes, different tensors of different order, dependent of the application, are used. Due to the complexity and the strong dependency of the tensor meaning of the application, there is, by now, no visualization method that is used by default. Also for specific applications there are only a few methods that address the visual analysis of higher-order tensors. This work adresses this problem. Three different methods to analyse and visualize tensors of higher order will be provided. All three methods are based on the so called deviatoric decomposition and the multipole representation. Using the multipoles the symmetries of a tensor and, therefore, of the described material, can be calculated. This property is used to visualize the stiffness tensor. The second method uses the multipoles directly and can be used for each tensor of any order in three dimensions. This method is presented by analysing the third-order coupling tensor. These two techniques are glyph-based visualization methods. The third one, a line-based method, is, according to our knowledge, a first global visualization method that can be used for an arbitrary tensor in three dimensions.

Tensors occur in many areas of science and engineering. Especially, they are used to describe charge, mass and energy transport (i.e. electrical conductivity tensor, diffusion tensor, thermal conduction tensor resp.) If the locale transport pattern is complicated, usual second order tensor representation is not sufficient. So far, there are no appropriate visualization methods for this case. We point out similarities of symmetric higher order tensors and spherical harmonics. A spherical harmonic representation is used to improve tensor glyphs. This paper unites the definition of streamlines and tensor lines and generalizes tensor lines to those applications where second order tensors representations fail. The algorithm is tested on the tractography problem in diffusion tensor magnetic resonance imaging (DT-MRI) and improved for this special application.