ecause scientists don 't have proper tensor-display techniques, they now visualize many physical problems incompletely in terms of vector or scalar data. Scientists could undoubtedly get new insights into these problems if they had a methodology for visualizing 3D second-order tensor fields. We present hyperstreamlines as a way of visualizing these data. Second-order tensor fields are fundamental in engineering and the physical sciences. Stresses and strains in solids, for example, are tensor fields. In fluid flows , stresses" viscous stresses, rate of strain , and momentum transfers are all described in terms of tensor data . In fact , the steady-state Navier-Stb kes equations describe gas flows with only one quantity-m,mentum flux density-which is itself a tensor field. Table 1 lists for reference some common tensor fields that we analyze in more detail later. It shows that tensor data carry a large amount of information. They include diverse physical quantities such as pressure, kinetic energy density, mass density, velocity, and derivatives of the velocity field. Visualizing tensor fields correlates these quantities. (More information about tensor fields can be found in Borisenko and Tarapov 's book. Because of the wealth of multivariate information in tensor fi elds, tensor visualization is a challenge. Indeed, a 3D secondorder tensor field T consists of a 3 x 3 array of scalar functions (7;d , i, k = 1, 2, 3 defined over a 3D domain . Independent visualization of these nine functions is possible but meaningless. To help scientists visualize 3D tensor data, we have developed a methodology based on the concept of a hyperstreamline, the simplest continuous tensor structure that can be extracted from a tensor field (as opposed to many other scalar or vector features) . We first introduce hyperstreamlines for the particular case of symmetric tensor fields U = (U;d, whose individual components are related to each other by U;k = Uk; for i, k = 1, 2, 3. (As seen in Table 1 , symmetric tensor fields are very common in fluid-flow studies.) We later derive a structural depiction of symmetric tensor fields from the representation of many hyperstreamlines. And, finally , we provide a methodology to visualize un symmetric tensor data by encoding an additional vector field along the trajectory of the hyperstreamlines.
Delmarcelle and Hesselink REF provide a comprehensive study on the topology of 2D symmetric tensor fields and define hyperstreamlines, which they use to visualize tensor fields.
Visualizing second-order tensor fields with hyperstreamlines
IEEE Computer Graphics and Applications
IEEE Computer Graphics and Applications
Computer Science