Robustness for 2D Symmetric Tensor Field Topology

2017 ◽  
pp. 3-27 ◽  
Author(s):  
Bei Wang ◽  
Ingrid Hotz
Keyword(s):  
Tensor Field ◽  
Symmetric Tensor

scholarly journals Spherically Symmetric Tensor Fields and Their Application in Nonlinear Theory of Dislocations

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 830
Author(s):  
Evgeniya V. Goloveshkina ◽  
Leonid M. Zubov
Keyword(s):  
Nonlinear Theory ◽  
Hollow Sphere ◽  
Tensor Field ◽  
Exact Analytical Solution ◽  
Symmetric Tensor ◽  
Symmetric Distribution ◽  
Spherically Symmetric ◽  
Tensor Fields ◽  
Spherically Symmetric Distribution ◽  
Nonlinear Elastic

The concept of a spherically symmetric second-rank tensor field is formulated. A general representation of such a tensor field is derived. Results related to tensor analysis of spherically symmetric fields and their geometric properties are presented. Using these results, a formulation of the spherically symmetric problem of the nonlinear theory of dislocations is given. For an isotropic nonlinear elastic material with an arbitrary spherically symmetric distribution of dislocations, this problem is reduced to a nonlinear boundary value problem for a system of ordinary differential equations. In the case of an incompressible isotropic material and a spherically symmetric distribution of screw dislocations in the radial direction, an exact analytical solution is found for the equilibrium of a hollow sphere loaded from the outside and from the inside by hydrostatic pressures. This solution is suitable for any models of an isotropic incompressible body, i. e., universal in the specified class of materials. Based on the obtained solution, numerical calculations on the effect of dislocations on the stress state of an elastic hollow sphere at large deformations are carried out.

scholarly journals A Note On Umbilics of a Closed Surface

1959 ◽  
Vol 15 ◽  
pp. 219-223
Author(s):  
Minoru Kurita
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Tensor Field ◽  
Symmetric Tensor ◽  
Closed Surface ◽  
Dimension 2 ◽  
Direction Field

In this paper we investigate indices of umbilics of a closed surface in the euclidean space. Most part of the discussion is concerned with a symmetric tensor field of degree 2, or rather a direction field, on a Riemannian manifold of dimension 2.

scholarly journals A Yang–Mills Theory in Loop Space and Chapline–Manton Coupling

1997 ◽  
Vol 12 (02) ◽  
pp. 111-119 ◽  
Author(s):  
Shinichi Deguchi ◽  
Tadahito Nakajima
Keyword(s):  
Field Theory ◽  
Local Field ◽  
Loop Space ◽  
Tensor Field ◽  
Symmetric Tensor ◽  
Tensor Fields ◽  
Yang Mills ◽  
Local Field Theory ◽  
Antisymmetric Tensor Field ◽  
Chern Simons

We consider a Yang–Mills theory in loop space with the affine gauge group. From this theory, we derive a local field theory with Yang–Mills fields and Abelian antisymmetric and symmetric tensor fields of the second rank. The Chapline–Manton coupling, i.e. coupling of Yang–Mills fields and a second-rank antisymmetric tensor field via the Chern–Simons three-form is obtained systematically.

scholarly journals Robust Extraction and Simplification of 2D Symmetric Tensor Field Topology

2021 ◽  
Author(s):  
Jochen Jankowai ◽  
Bei Wang ◽  
Ingrid Hotz
Keyword(s):  
Structural Stability ◽  
Real World ◽  
Tensor Field ◽  
Symmetric Tensor ◽  
Tensor Fields ◽  
Real World Data ◽  
World Data ◽  
Labeling Algorithm ◽  
Edge Labeling

In this work, we propose a controlled simplification strategy for degenerated points in symmetric 2D tensor fields that is based on the topological notion of robustness. Robustness measures the structural stability of the degenerate points with respect to variation in the underlying field. We consider an entire pipeline for generating a hierarchical set of degenerate points based on their robustness values. Such a pipeline includes the following steps: the stable extraction and classification of degenerate points using an edge labeling algorithm, the computation and assignment of robustness values to the degenerate points, and the construction of a simplification hierarchy. We also discuss the challenges that arise from the discretization and interpolation of real world data.

Balance laws for mass, forces, and moments

2020 ◽  
pp. 69-84
Author(s):  
Lallit Anand ◽  
Sanjay Govindjee
Keyword(s):  
Stress Tensor ◽  
Tensor Field ◽  
Mass Density ◽  
Symmetric Tensor ◽  
The Body ◽  
Surface Element ◽  
Cauchy Stress ◽  
Balance Laws ◽  
Balance Of Forces ◽  
Outward Unit

This chapter discusses the law of balance of mass, as well as the laws of balance of forces and moments. The important related concept of stress is this then presented as formalized by Cauchy in terms of a central theorem of Continuum Mechanics, which asserts that satisfaction of global balance of forces and moments is equivalent to the existence of a symmetric tensor field in the deformed body called the Cauchy stress, such that the traction vector acting across each oriented surface element at a point in the body is given by the Cauchy stress tensor operating linearly on the outward unit normal to the surface at that point. In addition, the stress tensor must satisfy a partial differential equation, known as the equation of motion, which asserts that the divergence of the stress tensor plus a body force per unit volume, is equal to the mass density times the acceleration.

scholarly journals Visualizing High-Order Symmetric Tensor Field Structure with Differential Operators

2011 ◽  
Vol 2011 ◽  
pp. 1-27 ◽  
Author(s):  
Tim McGraw ◽  
Takamitsu Kawai ◽  
Inas Yassine ◽  
Lierong Zhu
Keyword(s):  
Feature Extraction ◽  
Vector Field ◽  
Tensor Field ◽  
Differential Operators ◽  
Synthetic Data ◽  
Symmetric Tensor ◽  
Field Structure ◽  
Data Sets ◽  
Tensor Fields ◽  
New Techniques

The challenge of tensor field visualization is to provide simple and comprehensible representations of data which vary both directionallyandspatially. We explore the use of differential operators to extract features from tensor fields. These features can be used to generate skeleton representations of the data that accurately characterize the global field structure. Previously, vector field operators such as gradient, divergence, and curl have previously been used to visualize of flow fields. In this paper, we use generalizations of these operators to locate and classify tensor field degenerate points and to partition the field into regions of homogeneous behavior. We describe the implementation of our feature extraction and demonstrate our new techniques on synthetic data sets of order 2, 3 and 4.

scholarly journals Robust Extraction and Simplification of 2D Symmetric Tensor Field Topology

2019 ◽  
Vol 38 (3) ◽  
pp. 337-349 ◽  
Author(s):  
Jochen Jankowai ◽  
Bei Wang ◽  
Ingrid Hotz
Keyword(s):  
Tensor Field ◽  
Symmetric Tensor

scholarly journals Lie Derivatives of the Shape Operator of a Real Hypersurface in a Complex Projective Space

2021 ◽  
Vol 18 (5) ◽  
Author(s):  
Juan de Dios Pérez ◽  
David Pérez-López
Keyword(s):  
Differential Operator ◽  
Projective Space ◽  
Tensor Field ◽  
Complex Projective Space ◽  
Shape Operator ◽  
Symmetric Tensor ◽  
Real Hypersurfaces ◽  
Lie Derivative ◽  
Complex Projective ◽  
Derivatives Of

AbstractWe consider real hypersurfaces M in complex projective space equipped with both the Levi-Civita and generalized Tanaka–Webster connections. Associated with the generalized Tanaka–Webster connection we can define a differential operator of first order. For any nonnull real number k and any symmetric tensor field of type (1,1) B on M, we can define a tensor field of type (1,2) on M, $$B^{(k)}_T$$ B T ( k ) , related to Lie derivative and such a differential operator. We study symmetry and skew symmetry of the tensor $$A^{(k)}_T$$ A T ( k ) associated with the shape operator A of M.

Sign in / Sign up

Export Citation Format

Share Document