Tensor Representation, Contraction and Visualization

Tensors are multivariate data embedded with complex information in time and space. Tensors of practical interests may conveniently be represented in matrix form with respect to a reference coordinate system. These include for example strain tensors, stress tensors and convective momentum flux tensors. In this chapter, tensor rank reduction techniques are applied. These include: contractions, tensor invariants, principal states and principal axes trajectories combined with evolving tensor ellipses. Study in strain/stress analysis has contributed significantly to the development of tensor visualization techniques both experimentally and theoretically. For example, the two-dimensional photo-elasticity technique in stress analysis has provided experimental means of tensor visualization. Several graphical plots have been developed from tensor/matrix theory to aid visualization of tensor fields. Vorticity tensor is a typical example of antisymmetric tensors and because it plays an important part in hydrodynamics and has analogy in other fields. Controlled Vocabulary Terms hydrodynamics; photoelasticity; stress-strain relations; tensors