Physics, medicine, earth sciences, mechanical engineering, geo-engineering, bio-engineering and many more application areas use tensorial data. For example, tensors are used in formulating the balance equations of charge, mass, momentum, or energy as well as the constitutive relations that complement them. Some of these tensors (i.e., stiffness tensor, strain gradient, photo-elastic tensor) are of order higher than two. Currently, there are nearly no visualization techniques for such data beyond glyphs. An important reason for this is the limit of currently used tensor decomposition techniques. In this article, we propose to use the deviatoric decomposition to draw lines describing tensors of arbitrary order in three dimensions. The deviatoric decomposition splits a three-dimensional tensor of any order with any type of index symmetry into totally symmetric, traceless tensors. These tensors, called deviators, can be described by a unique set of directions (called multipoles by J. C. Maxwell) and scalars. These multipoles allow the definition of multipole lines which can be computed in a similar fashion to tensor lines and allow a line-based visualization of three-dimensional tensors of any order. We give examples for the visualization of symmetric, second-order tensor fields as well as fourth-order tensor fields. To allow an interpretation of the multipole lines, we analyze the connection between the multipoles and the eigenvectors/eigenvalues in the second-order case. For the fourth-order stiffness tensor, we prove relations between multipoles and important physical quantities such as shear moduli as well as the eigenvectors of the second-order right Cauchy-Green tensor.

Electroactive polymers are frequently used in engineering applications due to their ability to change their shape and properties under the influence of an electric field. This process also works vice versa, such that mechanical deformation of the material induces an electric field in the EAP device. This specific behavior makes such materials highly attractive for the construction of actuators and sensors in various application areas. The electromechanical behaviour of electroactive polymers can be described by a third-order coupling tensor, which represents the sensitivity of mechanical stresses concerning the electric field, i.e., it establishes a relation between a second-order and a first-order tensor field. Due to this coupling tensor's complexity and the lack of meaningful visualization methods for third-order tensors in general, an interpretation of the tensor is rather difficult. Thus, the central engineering research question that this contribution deals with is a deeper understanding of electromechanical coupling by analyzing the third-order coupling tensor with the help of specific visualization methods. Starting with a deviatoric decomposition of the tensor, the multipoles of each deviator are visualized, which allows a first insight into this highly complex third-order tensor. In the present contribution, four examples, including electromechanical coupling, are simulated within a finite element framework and subsequently analyzed using the tensor visualization method.

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.