Singularities in nonuniform tensor fields

Combinatorial topology, also known as rubber sheet geometry, has extensive applications in geometry and analysis, many of which result from connections with the theory of differential equations. A link between topology and differential equations is vector fields. Recent developments in scientific visualization have shown that vector fields also play an important role in the analysis of second-order tensor fields. A second-order tensor field can be transformed into its eigensystem, namely, eigenvalues and their associated eigenvectors without loss of information content. Eigenvectors behave in a similar fashion to ordinary vectors with even simpler topological structures due to their sign indeterminacy. Incorporating information about eigenvectors and eigenvalues in a display technique known as hyperstreamlines reveals the structure of a tensor field. The simplify and often complex tensor field and to capture its important features, the tensor is decomposed into an isotopic tensor and a deviator. A tensor field and its deviator share the same set of eigenvectors, and therefore they have a similar topological structure. A a deviator determines the properties of a tensor field, while the isotopic part provides a uniform bias. Degenerate points are basic constituents of tensor fields. In 2-D tensor fields, there are only two types of degenerate points; while in 3-D, the degenerate points can be characterized in a Q'-R' plane. Compressible and incompressible flows share similar topological feature due to the similarity of their deviators. In the case of the deformation tensor, the singularities of its deviator represent the area of vortex core in the field. In turbulent flows, the similarities and differences of the topology of the deformation and the Reynolds stress tensors reveal that the basic addie-viscosity assuptions have their validity in turbulence modeling under certain conditions.

Correlation functions of components of second-order tensor fields in isotropic systems can be reduced to an isotropic fourth-order tensor field characterized by a few invariant correlation functions (ICFs). It is emphasized that components of this field depend in general on the coordinates of the field vector variable and thus on the orientation of the coordinate system. These angular dependencies are distinct from those of ordinary anisotropic systems. As a simple example of the procedure to obtain the ICFs we discuss correlations of time-averaged stresses in isotropic glasses where only one ICF in reciprocal space becomes a finite constant e for large sampling times and small wave vectors. It is shown that e is set by the typical size of the frozen-in stress components normal to the wave vectors, i.e., it is caused by the symmetry breaking of the stress for each independent configuration. Using the presented general mathematical formalism for isotropic tensor fields this finding explains in turn the observed long-range stress correlations in real space. Under additional but rather general assumptions e is shown to be given by a thermodynamic quantity, the equilibrium Young modulus E. We thus relate for certain isotropic amorphous bodies the existence of finite Young or shear moduli to the symmetry breaking of a stress component in reciprocal space.

The stress tensor is expressed in terms of an arbitrary symmetric tensor field of second rank and the couple-stress tensor. The stress and couple-stress tensors are represented by arbitrary tensor fields satisfying the homogeneous equilibrium equations. These tensors are also given in the form of the expressions satisfying the inhomogeneous equilibrium equations used in the microcontinuum theory of elasticity. The stress tensor functions are considered.

Some general properties of Eshelby’s tensor fields in transport phenomena and anti-plane elasticity

We present a visualization method for the exploration of three-dimensional tensor fields. The representation of the tensor field on a one-parameter family of two-dimensional surfaces as stretched, compressed and bent piece of fabric reflects the physical properties of stress and strain tensor fields. The texture parameters as the fiber density and fiber direction are controlled by tensor field. The surface family is defined as a set of isosurfaces extracted from an additional scalar field. This field can be a “connected” scalar field, for example, pressure or a scalar field representing some symmetry or inherent structure of the dataset. The texture generation consists basically of three steps. The first is the transformation of the tensor field into a positive definite metric. In the second step, we generate a spot noise texture as input for the final fabric generation. Shape and density of the spots are controlled by the eigenvalues of the tensor field. This spot image incorporates the entire information defined by the three eigenvalue fields. In the third step we use line integral convolution (LIC) to provide a contin- uous representation that enhances the visibility of the eigendirections. This method supports an intuitive distinction between positive and negative eigenvalues and supports the additional visualization of a connected scalar field.

This paper addresses several issues related to topological analysis of 3D second order symmetric tensor fields. First, we show that the degenerate features in such data sets form stable topological lines rather than points, as previously thought. Second, the paper presents two different methods for extracting these features by identifying the individual points on these lines and connecting them. Third, this paper proposes an analytical form of obtaining tangents at the degenerate points along these topological lines. The tangents are derived from a Hessian factorization technique on the tensor discriminant and leads to a fast and stable solution. Together, these three advances allow us to extract the backbone topological lines that form the basis for topological analysis of tensor fields.

Transition points are well defined topological features in 3D tensor fields, which are important for the study of other prominent topological singularities such as wedges and trisectors. In this paper, we study the maximum number of transition points in a linear tensor field, which is important to process wedge and trisector classification along degenerate curves.

We propose a topology-based segmentation of 2D symmetric tensor fields, which results in cells bounded by tensorlines. We are particularly interested in the influence of the interpolation scheme on the topology, considering eigenvector-based and component-wise linear interpolation. When using eigenvector-based interpolation the most significant modification to the standard topology extraction algorithm is the insertion of additional vertices at degenerate points. A subsequent Delaunay re-triangulation leads to connections between close degenerate points. These new connections create degenerate edges and tri angles.When comparing the resulting topology per triangle with the one obtained by component-wise linear interpolation the results are qualitatively similar, but our approach leads to a less “cluttered” segmentation.

This paper is concerned with developing constitutive equations for the thermoelastic analysis of composites consisting of an isotropic matrix reinforced by independent and inextensible two families of fibers having an arbitrary distribution. The composite medium is assumed to be incompressible, dependent on temperature gradient, and showing linear elastic behavior. The reaction of the composite material subject to external loads is expressed in stress tensor and heat flux vector. The matrix material made of elastic material involving an artificial anisotropy due to fibers reinforcing by arbitrary distributions has been assumed as an isotropic medium. The theory is formulated within the scope of continuum mechanics. As a result of thermodynamic constraints, it has been determined that the stress potential function is dependent on the deformation tensor, the fiber fields vectors and the temperature, while the heat flux vector function is dependent on the deformation tensor, the fiber fields vectors, the temperature and temperature gradient. To determine arguments of the constitutive functionals, findings relating to the theory of invariants have been used as a method because of that isotropy constraint is imposed on the material. The constitutive equations of stress and heat flux vector have been written in terms of different coordinate descriptions.

Summary In this study, we demonstrate geomechanical modeling with fully automatic parameter calibration to estimate the full geomechanical stress fields of a prospective US CO2 storage site, based on sparse measurement data. The goal is to compute full stress tensor field estimates (principal stresses and orientations) that are maximally compatible with observations within the constraints of the model assumptions, thereby extending point-wise, incomplete partial stress measurement to a simulated full formation stress field, as well as a rough assessment of the associated error. We use the Perch site, located in Otsego Country, Michigan, as our case study. Input data consists of partial stress tensor information inferred from in-situ borehole tests, geophysical well logs and processing of seismic data. A static earth model of the site was developed, and geomechanical simulation functionality of the open-source MATLAB Reservoir Simulation Toolbox (MRST) used to model the stress field. Adjoint-based nonlinear optimization was used to adjust boundary conditions and material properties to calibrate simulated results to observations. Results were interpreted through a Bayesian framework. The focus of this article is to demonstrate how the fully automatic calibration procedure works and discuss the results obtained but does not attempt a detailed analysis of the stress field in the context of the proposed CO2 storage initiatives. Our work is part of a larger effort to non-invasively determine in-situ stresses in deep formations considered for CO2 storage. Guided by previously published research on geomechanical model calibration, our work presents a novel calibration approach supporting a potentially large number of linear or nonlinear calibration parameters, in order to produce results optimally agreeing with available measurements and thus extend partial point-wise estimates to full tensor fields compatible with the physics of the site.

This paper introduces a method to synthesize a 3D tensor field within a constrained geometric domain represented as a tetrahedral mesh. Whereas previous techniques optimize for isotropic fields, we focus on anisotropic tensor fields that are smooth and aligned with the domain boundary or user guidance. The key ingredient of our method is a novel computational design framework, built on top of the symmetric orthogonally decomposable (odeco) tensor representation, to optimize the stretching ratios and orientations for each tensor in the domain. In contrast to past techniques designed only for isotropic tensors, we demonstrate the efficacy of our approach in generating smooth volumetric tensor fields with high anisotropy and shape conformity, especially for the domain with complex shapes. We apply these anisotropic tensor fields to various applications, such as anisotropic meshing, structural mechanics, and fabrication.

It is challenging to automatically explore an unknown 3D environment with a robot only equipped with depth sensors due to the limited field of view. We introduce THP, a tensor field-based framework for efficient environment exploration which can better utilize the encoded depth information through the geometric characteristics of tensor fields. Specifically, a corresponding tensor field is constructed incrementally and guides the robot to formulate optimal global exploration paths and a collision-free local movement strategy. Degenerate points generated during the exploration are adopted as anchors to formulate a hierarchical TSP for global path optimization. This novel strategy can help the robot avoid long-distance round trips more effectively while maintaining scanning completeness. Furthermore, the tensor field also enables a local movement strategy to avoid collision based on particle advection. As a result, the framework can eliminate massive, time-consuming recalculations of local movement paths. We have experimentally evaluate our method with a ground robot in 8 complex indoor scenes. Our method can on average achieve 14% better exploration efficiency and 21% better exploration completeness than state-of-the-art alternatives using LiDAR scans. Moreover, compared to similar methods, our method makes path decisions 39% faster due to our hierarchical exploration strategy.

Kinematics, stress, deformation and material behavior

Application of an evolutionary algorithm to LES modelling of turbulent transport in premixed flames

This is a victory declaration in the theory of a quantized field propagating in a given curved background space-time. We now have an unambiguous, internally consistent quantum theory of such a system, in which any physical quantity can in principle be calculated. Whether this kind of model is relevant to the real world is a separate question. I shall present the theory rather dogmatically. Doctrine i: The physical interpretation of a quantum field theory in curved space must be sought in the stress-energy-momentum tensor and other local field observables. (Particle observables are not meaningful, in general.) As a concrete example, let's keep in mind the minimally coupled neutral scalar field, whose stress tensor is