We develop an interactive approach for analyzing multi-field tensor data from simulations in close collaboration with domain scientists. Our approach is based on extensive application analysis and built around a multi-field clustering addressing multiple user-defined quantities which were required by the domain scientists. Established techniques like linked views complement the approach to support reasoning while offering an overview and detailed insight into the multi-field tensor data. Further, we include an evaluation containing a real-world use case and a user study with domain scientists to demonstrate the usefulness compared to existing tools.

Abstract Tensors are used to describe complex physical processes in many applications. Examples include the distribution of stresses in technical materials, acting forces during seismic events, or remodeling of biological tissues. While tensors encode such complex information mathematically precisely, the semantic interpretation of a tensor is challenging. Visualization can be beneficial here and is frequently used by domain experts. Typical strategies include the use of glyphs, color plots, lines, and isosurfaces. However, data complexity is nowadays accompanied by the sheer amount of data produced by large‐scale simulations and adds another level of obstruction between user and data. Given the limitations of traditional methods, and the extra cognitive effort of simple methods, more advanced tensor field visualization approaches have been the focus of this work. This survey aims to provide an overview of recent research results with a strong application‐oriented focus, targeting applications based on continuum mechanics, namely the fields of structural, bio‐, and geomechanics. As such, the survey is complementing and extending previously published surveys. Its utility is twofold: (i) It serves as basis for the visualization community to get an overview of recent visualization techniques. (ii) It emphasizes and explains the necessity for further research for visualizations in this context.

Modern engineering uses optimization to design long-living and robust components. One part of this process is to find the optimal stress-aware design under given geometric constraints and loading conditions. Tensor visualization provides techniques to show and evaluate the stress distribution based on finite element method simulations. One such technique are hyperstreamlines. They allow us to explore the stress along a line following one principal stress direction while showing the other principal stress directions and their values. In this paper, we show shortcomings of this approach from an engineer’s point of view and propose a variant called tensor spines. It provides an improved perception of the relation between the principal stresses helping engineers to optimize their designs further.

Scientific visualization developed successful methods for scalar and vector fields. For tensor fields, however, effective, interactive visualizations are still missing despite progress over the last decades. We present a general approach for the generation of separating surfaces in symmetric, second-order, three-dimensional tensor fields. These surfaces are defined as fiber surfaces of the invariant space, i.e. as pre-images of surfaces in the range of a complete set of invariants. This approach leads to a generalization of the fiber surface algorithm by Klacansky et al. [16] to three dimensions in the range. This is due to the fact that the invariant space is three-dimensional for symmetric second-order tensors over a spatial domain. We present an algorithm for surface construction for simplicial grids in the domain and simplicial surfaces in the invariant space. We demonstrate our approach by applying it to stress fields from component design in mechanical engineering.

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.

Abstract Tensors provide a powerful language to describe physical phenomena. Consequently, they have a long tradition in physics and appear in various application areas, either as the final result of simulations or as intermediate product. Due to their complexity, tensors are hard to interpret. This motivates the development of well‐conceived visualization methods. As a sub‐branch of scientific visualization, tensor field visualization has been especially pushed forward by diffusion tensor imaging. In this review, we focus on second‐order tensors that are not diffusion tensors. Until now, these tensors, which might be neither positive‐definite nor symmetric, are under‐represented in visualization and existing visualization tools are often not appropriate for these tensors. Hence, we discuss the strengths and limitations of existing methods when dealing with such tensors as well as challenges introduced by them. The goal of this paper is to reveal the importance of the field and to encourage the development of new visualization methods for tensors from various application fields.