N-ary relationships, which relate $N$ entities where $N$ is not necessarily two, can be visually represented as polygons whose vertices are the entities of the relationships. Manually generating a high-quality layout using this representation is labor-intensive. In this paper, we provide an automatic polygon layout generation algorithm for the visualization of N-ary relationships. At the core of our algorithm is a set of objective functions motivated by a number of design principles that we have identified. These objective functions are then used in an optimization framework that we develop to achieve high-quality layouts. Recognizing the duality between entities and relationships in the data, we provide a second visualization in which the roles of entities and relationships in the original data are reversed. This can lead to additional insight about the data. Furthermore, we enhance our framework for a joint optimization on the primal layout (original data) and the dual layout (where the roles of entities and relationships are reversed). This allows users to inspect their data using two complementary views. We apply our visualization approach to a number of datasets that include co-authorship data and social contact pattern data.

Mode surfaces are the generalization of degenerate curves and neutral surfaces, which constitute 3D symmetric tensor field topology. Efficient analysis and visualization of mode surfaces can provide additional insight into not only degenerate curves and neutral surfaces, but also how these features transition into each other. Moreover, the geometry and topology of mode surfaces can help domain scientists better understand the tensor fields in their applications. Existing mode surface extraction methods can miss features in the surfaces. Moreover, the mode surfaces extracted from neighboring cells have gaps, which make their subsequent analysis difficult. In this paper, we provide novel analysis on the topological structures of mode surfaces, including a common parameterization of all mode surfaces of a tensor field using 2D asymmetric tensors. This allows us to not only better understand the structures in mode surfaces and their interactions with degenerate curves and neutral surfaces, but also develop an efficient algorithm to seamlessly extract mode surfaces, including neutral surfaces. The seamless mode surfaces enable efficient analysis of their geometric structures, such as the principal curvature directions. We apply our analysis and visualization to a number of solid mechanics data sets.

Asymmetric tensor fields have found applications in many science and engineering domains, such as fluid dynamics. Recent advances in the visualization and analysis of 2D asymmetric tensor fields focus on pointwise analysis of the tensor field and effective visualization metaphors such as colors, glyphs, and hyperstreamlines. In this paper, we provide a novel multi-scale topological analysis framework for asymmetric tensor fields on surfaces. Our multi-scale framework is based on the notions of eigenvalue and eigenvector graphs. At the core of our framework are the identification of atomic operations that modify the graphs and the scale definition that guides the order in which the graphs are simplified to enable clarity and focus for the visualization of topological analysis on data of different sizes. We also provide efficient algorithms to realize these operations. Furthermore, we provide physical interpretation of these graphs. To demonstrate the utility of our system, we apply our multi-scale analysis to data in computational fluid dynamics.