Concept Of Tensor Research Articles

Finding the Maximum Eigenvalue of Essentially Nonnegative Symmetric Tensors via Sum of Squares Programming

Finding the maximum eigenvalue of a tensor is an important topic in tensor computation and multilinear algebra. Recently, for a tensor with nonnegative entries (which we refer it as a nonnegative tensor), efficient numerical schemes have been proposed to calculate its maximum eigenvalue based on a Perron–Frobenius-type theorem. In this paper, we consider a new class of tensors called essentially nonnegative tensors, which extends the concept of nonnegative tensors, and examine the maximum eigenvalue of an essentially nonnegative tensor using the polynomial optimization techniques. We first establish that finding the maximum eigenvalue of an essentially nonnegative symmetric tensor is equivalent to solving a sum of squares of polynomials (SOS) optimization problem, which, in its turn, can be equivalently rewritten as a semi-definite programming problem. Then, using this sum of squares programming problem, we also provide upper and lower estimates for the maximum eigenvalue of general symmetric tensors. These upper and lower estimates can be calculated in terms of the entries of the tensor. Numerical examples are also presented to illustrate the significance of the results.

Micromechanics-Derived Scaling Relations for Poroelasticity and Strength of Brittle Porous Polycrystals

Abstract There are lots of ceramic geological and biological materials whose microscopic load carrying behavior is not dominated by bending of structural units, but by the three-dimensional interaction of disorderedly arranged single crystals. A particularly interesting solution to capture this so-called polycrystalline behavior has emerged in the form of self-consistent homogenization methods based on an infinite amount of nonspherical (needle or disk-shaped) solid crystal phases and one spherical pore phase. Based on eigenstressed matrix-inclusion problems, together with the concentration and influence tensor concept, we arrive at the following results: Young’s modulus and the poroelastic Biot modulus of the porous polycrystal scale linearly with the Young’s modulus of the single crystals, the former independently of the Poisson’s ratio of the single crystals. Biot coefficients are independent of the single crystals’ Young’s modulus. The uniaxial strength of a pore pressure-free porous polycrystal, as well as the blasting pore pressure of a macroscopic stress-free polycrystal, scale linearly with the tensile strength of the single crystals, independently of all other elastic and strength properties of the single crystals. This is confirmed by experiments on a wide range of bio- and geomaterials, and it is of great interest for numerical simulations of structures built up by such polycrystals.

Log-Gabor Filters and Lda Based Facial Expression Recognition

This project introduces a tensor perceptual color framework (TPCF) for facial expression recognition (FER), which is based on information contained in color facial images. The TPCF enables multilinear image analysis in different color spaces, and demonstrates that color components provide additional information for robust FER. Using this framework, the components (in either RGB, YCbCr, CIELab space) of color images are unfolded to 2-D tensors based on multilinear algebra and tensor concepts, from which the features are extracted by Log-Gabor filters. The mutual information quotient method is employed for feature selection. These features are classified using a multiclass linear discriminant analysis classifier. The effectiveness of color information on FER using low-resolution and facial expression images with illumination variations is assessed for performance evaluation. Experimental results demonstrate that color information has significant potential to improve emotion recognition performance due to the complementary characteristics of image textures. Furthermore, the perceptual color space CIELab are better overall for FER than other color spaces, by providing more efficient and robust performance for FER using facial images with illumination variation.

Three-dimensional inversion of magnetotelluric phase tensor data

SUMMARY Recent increase in the application of 3-D inversion of magnetotelluric (MT) data is facilitated by the availability of several 3-D inversion codes, which led to improved interpretation of MT data. However, still the galvanic effects continue to pose problems in interpretation of the MT data. We have addressed this problem using the phase tensor (PT) concept and developed a scheme based on a modification of the sensitivity matrix in the 3-D inversion code of Siripunvaraporn et al., which enables us to directly invert the phase tensor elements. We have used this modified code for PT inversion of MT data and evaluated its efficacy in reducing the galvanic effects through a few examples of inversion of synthetic data and its application on real data. The synthetic model study suggests that the prior model (mo) setting is important in retrieving the true model. Comparison of results obtained from conventional impedance inversion and the proposed PT inversion method suggests that, even in the presence of the galvanicdistortion(duetonear-surfacecheckerboardanomaliesinourcase),thenewinversion algorithmreliablyretrievestheregionalconductivitystructurewhenthepriormodelorregional resistivity value level can be estimated with sufficient accuracy. Data errors were propagated to the PT elements using delta method while inverting the real data from USArray. The PT results compare very well with those from full tensor inversion published earlier, signifying the efficacy of this new inversion scheme.

Facial Expression Recognition in Perceptual Color Space

This paper introduces a tensor perceptual color framework (TPCF) for facial expression recognition (FER), which is based on information contained in color facial images. The TPCF enables multi-linear image analysis in different color spaces and demonstrates that color components provide additional information for robust FER. Using this framework, the components (in either RGB, YCbCr, CIELab or CIELuv space) of color images are unfolded to two-dimensional (2- D) tensors based on multi-linear algebra and tensor concepts, from which the features are extracted by Log-Gabor filters. The mutual information quotient (MIQ) method is employed for feature selection. These features are classified using a multi-class linear discriminant analysis (LDA) classifier. The effectiveness of color information on FER using low-resolution and facial expression images with illumination variations is assessed for performance evaluation. Experimental results demonstrate that color information has significant potential to improve emotion recognition performance due to the complementary characteristics of image textures. Furthermore, the perceptual color spaces (CIELab and CIELuv) are better overall for facial expression recognition than other color spaces by providing more efficient and robust performance for facial expression recognition using facial images with illumination variation.

On the formulation of boundary value problems with the incompressible constituents constraint in finite deformation poroelasticity

This paper considers the finite deformation theory of poroelasticity for the case in which a deformable solid constituent and an interpenetrating liquid constituent are each regarded as incompressible, and the mixing itself is regarded as taking place without the creation of voids. The resulting kinematical constraint gives rise to a Lagrange multiplier pressure in the resulting constitutive description. This pressure therefore enters into the separate momentum balance statements for each individual constituent. The formulation of boundary value problems in this context is well known in continuum mechanics. This paper examines how a systematic elimination of the Lagrange multiplier pressure from the mathematical formulation leads to a stress‐like tensor that generalizes a stress tensor concept introduced by Rajagopal and Wineman in the late 1980s, which they called the saturation stress. Here, by providing a rather complete development, it is discussed how boundary value problems are naturally formulated in terms of a single such stress tensor, how the constitutive theory is framed in terms of this stress tensor, and how certain questions concerning the formulation of boundary conditions are naturally addressed. Connections to the small deformation linear poroelastic (biphasic) theory are also provided. Copyright © 2012 John Wiley & Sons, Ltd.

Analytical Nonlinear Propagation of Uncertainty in the Two-Body Problem

One topic of recent interest in the field of space situational awareness is the accurate and consistent representation of an observed object’s uncertainty under nonlinear dynamics. This paper presents amethod of analytical nonlinear propagation of uncertainty under two-bodydynamics. In particular, the probability density function over state space and itsmean and covariancematrix are expressed analytically for all time via a special solution of the Fokker-Planck equations for deterministic Hamiltonian systems. The state transition tensor concept is used to express the solution flow of the dynamics. Some numerical examples, where a second-order state transition tensor is found to sufficiently capture the nonlinear effects, are also discussed.

2216 成分分離による構造体の剛性評価方法の提案(OS6-4 設計と最適化IV)

There are many ways to evaluate stiffness of automobile body in white. Widely used evaluation methods are torsional stiffness and bending stiffness. These methods are effective when comparing each other on the same basis, but the car which has more stiffness does not always have good drivability. In order to optimize stiffness of body in white, allocation of stiffness is important. In this paper, in order to understand the deformation in detail, we propose a method to separate stiffness into three components, which are bending, torsional and strains using concept of deformation gradient tensor. Then using mutual mean compliance, we will analyze which part of the body will reduce amount of these components.

The generalized polarization tensors for resolved imaging Part II: Shape and electromagnetic parameters reconstruction of an electromagnetic inclusion from multistatic measurements

This paper deals with the problem of reconstructing the electromagnetic parameters and the shape of a target from multi-static response matrix measurements at a single frequency. The target is of characteristic size less than the operating wavelength. Using long-wavelength asymptotic expansions of the measurements of high-order, we show how the electromagnetic parameters and the equivalent ellipse of the target can be reconstructed. The asymptotic expansions of the measurements are written in terms of the new concept of frequency dependent polarization tensors. Moreover, we extend the optimization approach proposed in Part I [9] to image geometric details of an electromagnetic target that are finer than the equivalent ellipse. The equivalent ellipse still provides a good initial guess for the optimization procedure. However, compared to the conductivity case in [9], the cost functional measures the discrepancy between the computed and measured high-order frequency dependent polarization tensors rather than between the generalized polarization tensors. The main reason for such a modification of the cost functional is the fact that the (measured) frequency dependent polarization tensors can be easily obtained from multistatic measurements by solving a linear system while the derivation of the generalized polarization tensors from measurements requires more delicate treatment. The proposed methods are numerically implemented to demonstrate their validity and efficiency. Mathematics Subject Classification (MSC2000): 35R30, 35B30.

The generalized polarization tensors for resolved imaging. Part I: Shape reconstruction of a conductivity inclusion

With each Lipschitz domain and material parameter, an infinite number of tensors, called the Generalized Polarization Tensors (GPTs), is associated. The GPTs contain significant information on the shape of the domain and its material parameter. They generalize the concept of Polarization Tensor (PT), which can be seen as the first-order GPT. It is known that given an arbitrary shape, one can find an equivalent ellipse or ellipsoid with the same PT. In this paper we consider the problem of recovering finer details of the shape of a given domain using higher-order polarization tensors. We design an optimization approach which solves the problem by minimizing a weighted discrepancy functional. In order to compute the shape derivative of this functional, we rigorously derive an asymptotic expansion of the perturbations of the GPTs that are due to a small deformation of the boundary of the domain. Our derivations are based on the theory of layer potentials. We perform some numerical experiments to demonstrate the validity and the limitations of the proposed method. The results clearly show that our approach is very promising in recovering fine shape details. Mathematics Subject Classification (MSC2000): 35R30, 35B30

Diagonalization of tensors with circulant structure

The concepts of tensors with diagonal and circulant structure are defined and a framework is developed for the analysis of such tensors. It is shown a tensor of arbitrary order, which is circulant with respect to two particular modes, can be diagonalized in those modes by discrete Fourier transforms. This property can be used in the efficient solution of linear systems involving contractive products of tensors with circulant structure. Tensors with circulant structure occur in models for image blurring with periodic boundary conditions. It is shown that the new framework can be applied to such problems.

Metric construction by length distribution tensor and edge based error for anisotropic adaptive meshing

Metric tensors play a key role to control the generation of unstructured anisotropic meshes. In practice, the most well established error analysis enables to calculate a metric tensor on an element basis. In this paper, we propose to build a metric field directly at the nodes of the mesh for a direct use in the meshing tools. First, the unit mesh metric is defined and well justified on a node basis, by using the statistical concept of length distribution tensors. Then, the interpolation error analysis is performed on the projected approximate scalar field along the edges. The error estimate is established on each edge whatever the dimension is. It enables to calculate a stretching factor providing a new edge length distribution, its associated tensor and the corresponding metric. The optimal stretching factor field is obtained by solving an optimization problem under the constraint of a fixed number of edges in the mesh. Several examples of interpolation error are proposed as well as preliminary results of anisotropic adaptation for interface and free surface problem using a level set method.

On Choosing Effective Symmetry Classes for Elasticity Tensors

We formulate a method of representing a generally anisotropic elasticity tensor by an elasticity tensor exhibiting a material symmetry: an effective tensor. The method for choosing the effective tensor is based on examining the features of the plot of the monoclinic-distance function of a given tensor, choosing an appropriate symmetry class, and then finding the closest tensor in that class. The concept of the effective tensor is not tantamount to the closest tensor since one always obtains a closer approximation using a monoclinic tensor than a tensor of any other nontrivial symmetry. Hence, we use qualitative features of the plot of the monoclinic-distance function to choose an effective symmetry class within which the closest tensor can be computed.

Tensor approach to averaging Jacobi fields along geodesicswith random curvature

The paper considers the Jacobi field along a geodesic on a Riemannian manifold on which the curvature is a stochastic process. The author introduces the concept of a linearizing tensor forming the base for obtaining equations for the 2-, 3- and 4-order moments. A theorem on the general form of the moment equation is proved.

Viscoelastic flow analysis using the software OpenFOAM and differential constitutive equations

Abstract Viscoelastic fluids are of great importance in many industrial sectors, such as in food and synthetic polymers industries. The rheological response of viscoelastic fluids is quite complex, including combination of viscous and elastic effects and non-linear phenomena. This work presents a numerical methodology based on the split-stress tensor approach and the concept of equilibrium stress tensor to treat high Weissenberg number problems using any differential constitutive equations. The proposed methodology was implemented in a new computational fluid dynamics (CFD) tool and consists of a viscoelastic fluid module included in the OpenFOAM, a flexible open source CFD package. Oldroyd-B/UCM, Giesekus, Phan-Thien–Tanner (PTT), Finitely Extensible Nonlinear Elastic (FENE-P and FENE-CR), and Pom–Pom based constitutive equations were implemented, in single and multimode forms. The proposed methodology was evaluated by comparing its predictions with experimental and numerical data from the literature for the analysis of a planar 4:1 contraction flow, showing to be stable and efficient.

The tensor-based model for growth and cell divisions of the root apex. II. Lateral root formation

In this work, the formation of the virtual lateral root (VLR) is shown. The VLR is formed using the 2D simulation model of growth and cell divisions based on the concept of growth tensor, specified for radish. Growth is generated by the field of growth rates of an unsteady type (GT field). Principal directions of growth (PDGs) are assumed to define the orientation of cell divisions. Temporal sequences of the VLR formation are a result of an application of the GT field to the polygon meshwork representing cell pattern of already initiated primordium. The computer-generated lateral root (LR) develops realistically, and its cell pattern is vivid and similar to that observed in anatomical sections. The real and virtual LRs show similar cellular organization, both originate from a small group of cells situated in two-cell layers of the pericycle and both layers are engaged in the LR development. The LR formation seems to be controlled at the tensor level and individual cells presumably detect PDGs and obey them in the course of the cell divisions. PDGs are postulated to affect the cellular organization of the LR. Using the method of computer simulations, cellular aspects of the LR morphogenesis are discussed.

Tensor Presentation of Algorithmic Tangent Modulus for Plastic-Damage Models

For a class of plastic-damage models, spectral return-mapping algorithm performed upon the principal effective stresses is efficient, while algorithmic tangent modulus formulated upon the general effective stress components is much simpler. In order to facilitate the transformation between these two sets of stress components in the formulation of algorithmic tangent modulus, a new concept of fictitious stress tensor, which is orthogonally similar to the real stress tensor, is proposed in this paper. The rank four Jacobian tensor of the fictitious effective stress is calculated from spectral decomposition; and a clear tensor formulation of algorithmic tangent modulus is presented.

A SANISAND model with anisotropic elasticity

Numerous experimental studies indicate that as a result of shear stress, the elastic behavior of granular media becomes both non-linear and anisotropic. This paper presents a simple constitutive model for sands with respect to anisotropic elasticity. To this aim, using the concept of second order fabric tensor, a simplified elasticity theory is presented which is capable of considering the effect of induced anisotropy on the elastic response. SANISAND is the name used for a family of simple anisotropic sand models developed in the framework of critical state soil mechanics and bounding surface plasticity. An existing SANISAND model is modified in order to include the proposed anisotropic elasticity. The modified model simulations are compared with those obtained from the other members of this family. It is shown that considering anisotropic elasticity can take part in explanation of drastic loss of mean principal stress when sand is subjected to reverse loading in dilative branch of behavior and as a result, improve the liquefaction simulations.

Magneto-telluric impedance: fundamental models and possibilities of their generalization

A problem of impedance description of magnetic field on the surface of heterogenic Earth is discussed in the article. The concepts of impedance tensor, Vise - Parkinson tensor, generalized impedance tensor, Leontovich's impedance approximation and its generalization are analyzed. It is noticed that these models are of particular character and are real only in cases of special forms of giving primary field and do not always correspond to conditions of conducting real magnet ot elluric experiment. Generalized system of vector identities of impedance type is given for harmonic electromagnetic field on the closed interfaces and generated on their base generalized accurate differential equation of impedances and the system of scalar equations which determine this surface. It is noticed that in case of necessity of using their incomplete, truncated forms these models in contrast to classical ones, have interior criteria of their application and do not demand for their substantiation any additional empirical or heuristic considerations.

Revisiting quantum notions of stress

An important aspect of multi-scale modelling of materials is to link continuum concepts, such as fields, to the underlying discrete microscopic behaviour in a seamless manner. With the growing importance of atomistic calculations to understand material behaviour, reconciling continuum and discrete concepts is necessary to interpret molecular and quantum-mechanical simulations. In this work, we provide a quantum-mechanical framework to a distinctly continuum quantity: mechanical stress. While the concept of the global macroscopic stress tensor in quantum mechanics has been well established, there still exist open issues when it comes to a spatially varying local quantum stress tensor. We attempt to shed some light on this topic by establishing a general quantum-mechanical operator-based approach to continuity equations and from those, introduce a local quantum-mechanical stress tensor. Further, we elucidate the analogies that exist between the (classical) molecular-dynamics-based stress definition and the quantum stress. Our derivations appear to suggest that the local quantum-mechanical stress may not be an observable in quantum mechanics and therefore traces the non-uniqueness of the atomistic stress tensor to the gauge arbitrariness of the quantum-mechanical state function. Lastly, the virial stress theorem (of empirical molecular dynamics) is re-derived in a transparent manner that elucidates the analogy between quantum-mechanical global stresses.