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.

Abstract Scientific visualization deals with increasingly complex data consisting of multiple fields. Typical disciplines generating multivariate data are fluid dynamics, structural mechanics, geology, bioengineering, and climate research. Quite often, scientists are interested in the relation between some of these variables. A popular visualization technique for a single scalar field is the extraction and rendering of isosurfaces. With this technique, the domain can be split into two parts, i.e. a volume with higher values and one with lower values than the selected isovalue. Fiber surfaces generalize this concept to two or three scalar variables up to now. This article extends the notion further to potentially any finite number of scalar fields. We generalize the fiber surface extraction algorithm of Raith et al. [RBN∗19] from 3 to d dimensions and demonstrate the technique using two examples from geology and climate research. The first application concerns a generic model of a nuclear waste repository and the second one an atmospheric simulation over central Europe. Both require complex simulations which involve multiple physical processes. In both cases, the new extended fiber surfaces helps us finding regions of interest like the nuclear waste repository or the power supply of a storm due to their characteristic properties.

The use of clean and renewable energy and the abandoning of fossil energy have become goals of many national and international energy policies. But even when once accomplished, mankind has to take charge of the relics of the current energy supply system. For example, due to its harmful effects, nuclear waste has to be isolated from the biosphere safely and for sufficiently long times. The geological subsurface is considered as a promising option for the deposition of such by-or end products. In order to investigate the long-term evolution of a repository system, a multiphysics simulation was performed. It combines the structural mechanics of the host rock, the fluid dynamics of formation fluids, and the thermodynamics of all materials resulting in a highly multivariate data set. A visualization of such multiphysics data challenges the current methodology. In this article, we demonstrate how an analysis of a carefully selected subset of the variables in attribute space allows to visualize and interpret the simulation data. We apply a fiber surface extraction algorithm to explore the relationships between these variables. Studying the temporal evolution in attribute space, we found a regionally bulge that could be identified as an effect of the nuclear waste repository because it can be clearly separated from the natural geophysical state prior to waste disposal. Furthermore, we used the extracted fiber surface as a starting point to examine the distribution of other variables inside this area of the physical domain. We conclude this case study with lessons learned from the visualization as well as the geotechnical side.

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.