Visualizing Second-Order Tensor Fields

ecause scientists don't have proper tensor-display techB niques, 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-Stokes equations describe gas flows with only one quantity-momentum 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 fields, tensor visualization is a challenge. Indeed, a 3D secondorder tensor field T consists of a 3 x 3 array of scalar functions [ Tk), i, k = 1 ,2 ,3 defined over a 3D domain. Independent visualization of these nine functions is possible but meaningless. Thierry Delmarcelle and Lambertus Hesselink