Non-symmetric Tensor Research Articles

Tensor gauge boson dark matter extension of the electroweak sector

The existence of dark matter is explained by a new, massive, neutral, non-symmetric, rank-2 tensor gauge boson (hbox {Z}_{{upmu upnu }}-boson). The hbox {Z}_{{upmu upnu }}-boson can be predicted by the tensor gauge boson extension of the Electro Weak (EW) theory, proposed by Savvidy (Phys Lett B 625:341, 2005). The non-symmetric rank-2 tensor hbox {Z}_{{upmu upnu }} can be decomposed into a symmetric (hbox {Z}_{{(upmu upnu )}}) and anti-symmetric (hbox {Z}_{{[upmu upnu ]}}) part. Based on the non-Lagrangian formulation for the free sector of the hbox {R}_{textrm{2}}-theory proposed recently by Criado et al. (Phys Rev D 102:125031, arXiv:2010.02224, 2020), our massive anti-symmetric tensor field hbox {Z}_{{[upmu upnu ]}} corresponds to the massive symmetric spinor field hbox {Z}_{{upalpha upbeta upgamma updelta }} in the (2,0) irrep. For the massive hbox {Z}_{{upalpha upbeta upgamma updelta }} with the hbox {Z}_{textrm{2}}-symmetric Higgs portal couplings to a Standard Model (SM) particle, we compute the self-annihilation cross-section of the hbox {Z}_{{upalpha upbeta upgamma updelta }} dark matter and calculate its relic abundance. We also study the SM-SM particle scattering due to the exchange of the massive-hbox {Z}_{{(upmu upnu )}} symmetric field at a high energy scale. This proposition may have far reaching applications in astrophysics and cosmology.

A structure preserving hybrid finite volume scheme for semiconductor models with magnetic field on general meshes

We are interested in the discretisation of a drift–diffusion system in the framework of hybrid finite volume (HFV) methods on general polygonal/polyhedral meshes. The system under study is composed of two anisotropic and nonlinear convection–diffusion equations with nonsymmetric tensors, coupled with a Poisson equation and describes in particular semiconductor devices immersed in a magnetic field. We introduce a new scheme based on an entropy-dissipation relation and prove that the scheme admits solutions with values in admissible sets – especially, the computed densities remain positive. Moreover, we show that the discrete solutions to the scheme converge exponentially fast in time towards the associated discrete thermal equilibrium. Several numerical tests confirm our theoretical results. Up to our knowledge, this scheme is the first one able to discretise anisotropic drift–diffusion systems while preserving the bounds on the densities.

Cloaking Rayleigh waves via symmetrized elastic tensors

In this work we propose a strategy based on coordinate transformation to cloak Rayleigh waves. Rayleigh waves are in-plane elastic waves which propagate along the free surface of semi-infinite media. They are governed by Navier equations that retain their form for an in-plane arbitrary coordinate transformation x=Ξ(X), upon choosing the specific kinematic relation U(Ξ(X))=u(x) between displacement fields in virtual, i.e. reference, (U) and transformed, i.e. cloaked, (u) domains. However, the elasticity tensor of the transformed domain is no longer fully symmetric, and thus, it is difficult to design with common materials. Motivated by this issue, we propose a symmetrization technique, based on the arithmetic mean, to obtain anisotropic, yet symmetric, elastic tensors for Rayleigh wave near-cloaking. In particular, by means of time-harmonic numerical simulations and dispersion analyses, we compare the efficiency of triangular and semi-circular cloaks designed with the original non-symmetric tensors and the related symmetrized versions. In addition, different coordinate transformations, e.g. linear, quadratic and cubic, are adopted for the semi-circular cloaks. Through the analyses, we show that a symmetrized semi-circular cloak, obtained upon the use of a quadratic transformation, performs better than the other investigated designs. Our study provides a step towards the design of feasible and efficient broadband elastic metamaterial cloaks for surface waves.

Numerical experiment based on non-linear micropolar finite element to study micro-rotations generated by the non-symmetric Maxwell stress tensor

The main aim of the present work is to investigate the role of the Maxwell stress tensor in the study of active materials. Despite the importance of this tensor in modeling mechatronic devices used in sophisticated applications, its non–symmetry still generates controversies in the literature, probably because classical continuum mechanics assumes a symmetric Cauchy stress, although the sum of Cauchy and Maxwell stresses is non–symmetric. In the framework of generalised continuum mechanics–a more advanced formalism than the classical one–, each material point has an associated microstructure so that the micro–rotations of the electric/magnetic dipoles present in real active materials may be simulated. To this end, a modified total stress formulation, including an angular momentum balance, is developed and implemented into a finite element research code using a complex–step formulation. It is concluded that generalised mechanics allows for incorporating both symmetric and non–symmetric contributions of the Maxwell tensor. Consequently, the rotations generated by the electromagnetic field may be analysed. The influence of the complete Maxwell tensor in a magnetostrictive actuator is studied by several magneto–mechanical numerical experiments of a Terfenol–D rod encircled by air, and several conclusions are highlighted.

Ferrofluid droplets in planar extensional flows: Droplet shape and magnetization reveal novel rheological signatures of ferrofluid emulsions

By subjecting a ferrofluid emulsion to an external magnetic field, the flow at the droplet scale is altered, resulting in significant changes in the emulsion's rheological behavior. We present the first study of the rheological response of a dilute ferrofluid emulsion under extensional flow. Our findings reveal that at the microscopic level, the droplets experience a field-induced internal torque that leads to a complex non-symmetric stress tensor. We identify the remarkable presence of shear stresses within a material experiencing a pure extensional flow.

On the Smallest Number of Functions Representing Isotropic Functions of Scalars, Vectors and Tensors

Summary In this article, we prove that for isotropic functions that depend on $P$ vectors, $N$ symmetric tensors and $M$ non-symmetric tensors (a) the minimal number of irreducible invariants for a scalar-valued isotropic function is $3P+9M+6N-3,$ (b) the minimal number of irreducible vectors for a vector-valued isotropic function is $3$ and (c) the minimal number of irreducible tensors for a tensor-valued isotropic function is at most $9$. The minimal irreducible numbers given in (a), (b) and (c) are, in general, much lower than the irreducible numbers obtained in the literature. This significant reduction in the numbers of irreducible isotropic functions has the potential to substantially reduce modelling complexity.

Stress-based dimensional reduction and dual-mixed hp finite elements for elastic plates

Starting from the linearized weak forms of the kinematic equation and the angular momentum balance equation of three-dimensional non-linear elasticity, a stressbased dimensional reduction procedure is presented for elastic plates. After expanding the three-dimensional non-symmetric stress tensor into power series with respect to the thickness coordinate, the translational equilibrium equations, written in terms of the expanded stress coefficients, are satisfied by introducing first-order stress functions. The symmetry of the stress field is satisfied in a weak sense by applying the material rotations as Lagrangian multipliers. The seven-field plate model developed in this way employs unmodified three-dimensional strain-stress relations. On the basis of the dimensionally reduced plate model derived, a new dual-mixed plate bending finite element model is developed and presented. The numerical performance of the hp-version plate elements is investigated through the solutions of standard plate bending problems. It is shown that the modeling error of the stress-based plate model in the energy norm is better than that of the displacement-based Kirchhoff- and Reissner-Mindlin plate models. The numerical solutions and their comparisons to reference solutions indicate that the dual-mixed hp elements are free from locking problems, in either the energy norm or the stress computations, both for h- and p-extensions, and the results obtained for the stresses are accurate and reliable even for extremely thin plates.

MODELLING AND ANALYSIS OF INITIAL STRESS STATE IN A THIN-WALL CYLINDER WITH VARIABLE CHARACTERISTICS

A linearized model of prestressed elastic body vibrations is described in terms of the nonsymmetric Piola stress tensor, and a weak statement of the original problem is given. With the help of the Lagrange variational principle, we construct a statement of the problem of steady axisymmetric oscillations of a thin-walled (shell-like) cylinder with variable material properties in the presence of an inhomogeneous prestress field. The motion equations, essential and natural boundary conditions are formulated. Integral characteristics are introduced that average the change in prestresses and material properties over the shell thickness. On the basis of the model presented, the problem on steady-state vibrations of a functionally graded prestressed cylindrical shell is numerically investigated via the shooting method; the influence of the laws of change in material modules and prestress components on dynamic characteristics (e.g., amplitude-frequency characteristics, natural vibration frequencies) is analyzed, a comparative analysis of the solutions is carried out depending on the thin-walled parameter. For the considered cylindrical shell with axial inhomogeneity, based on the acoustic sounding data, we study the inverse problem of restoring two laws of prestress inhomogeneity characterizing the nonuniform initial stress-strain state of tension or inflation. As additional information, we consider the measurement data of the displacement components on the cylinder's surface, given in a set of points at a certain frequency of steady-state oscillations; in such a case, the material characteristics are considered to be known and may depend on the axial coordinate, which makes it possible to set an arbitrary functional gradient along the axis of the cylinder. The mode of probing loading is determined by specifying the traction type and oscillation frequency. Explicit formulas for the desired laws of prestress inhomogeneity are obtained, and computational experiments are carried out to restore them. A high accuracy of reconstruction is revealed far from the ends of the cylinder when probing at a frequency in the vicinity of the first resonance.

Chiral nonreciprocal elasticity and mechanical activity

There has been significant recent interest in creating, modeling, and exploiting the novel functionality afforded by odd elastic solids, which are a specific class of active matter whose behavior cannot be described by a free energy function. As a result, the mechanical behavior of such solids can be described by a non-symmetric elasticity tensor which means they can be mechanically active, and thus do work on their surroundings through quasistatic deformation cycles without energetic gain or loss terms explicitly appearing in the solid’s equation of state. However, previous incarnations of such solids have required the usage of active elements coupled with robotic machinery powered by independent external energy sources to operate. As such, it is unclear whether the non-symmetric elasticity of these solids can be developed using only passive elements that do not require the usage of energy sources, and furthermore how nonreciprocity in elastic media enables non-symmetric elasticity in elastic solids or mechanical activity. In this work, we propose the notion of chiral, nonreciprocal elasticity, which represents a generic route to enabling 2D, isotropic elastic solids exhibiting non-symmetric elasticity. Chiral, nonreciprocal elasticity describes elastic behaviors that result from coupling chirality with nonreciprocity—specifically, (1) the modulation of the elastic properties depending on the mode and direction of deformation and (2) the nonreciprocal coupling of different deformation fields, both of which enable the solid to exhibit a non-symmetric elasticity tensor. To motivate this, we introduce an isotropic 2D chiral metamaterial made of passive chiral elements that, by exploiting local geometric asymmetry, behaves in a chiral, nonreciprocal elastic fashion. We derive, based on the mechanics of a discrete model of this chiral element, the resulting continuum field equations and constitutive relationships that capture the chiral, nonreciprocal elastic behavior. Then, we establish a thermodynamic framework of energy balance and conservation of chiral, nonreciprocal elastic solids, based on which we demonstrate the ability of the proposed chiral metamaterial to act as a source of mechanical work when used in specific quasistatic deformation cycles, though no energy is dissipated by its passive elements. Finally, we demonstrate through numerical finite element simulations the practical implementation of the deformation cycles, while elucidating the specific conditions needed for the chiral metamaterial to exhibit linear, chiral nonreciprocal elastic behavior throughout the deformation cycle, and thus reveal mechanical activity.

Counting Tensor Rank Decompositions

Tensor rank decomposition is a useful tool for geometric interpretation of the tensors in the canonical tensor model (CTM) of quantum gravity. In order to understand the stability of this interpretation, it is important to be able to estimate how many tensor rank decompositions can approximate a given tensor. More precisely, finding an approximate symmetric tensor rank decomposition of a symmetric tensor Q with an error allowance Δ is to find vectors ϕi satisfying ∥Q−∑i=1Rϕi⊗ϕi⋯⊗ϕi∥2≤Δ. The volume of all such possible ϕi is an interesting quantity which measures the amount of possible decompositions for a tensor Q within an allowance. While it would be difficult to evaluate this quantity for each Q, we find an explicit formula for a similar quantity by integrating over all Q of unit norm. The expression as a function of Δ is given by the product of a hypergeometric function and a power function. By combining new numerical analysis and previous results, we conjecture a formula for the critical rank, yielding an estimate for the spacetime degrees of freedom of the CTM. We also extend the formula to generic decompositions of non-symmetric tensors in order to make our results more broadly applicable. Interestingly, the derivation depends on the existence (convergence) of the partition function of a matrix model which previously appeared in the context of the CTM.

Symmetric tensor decomposition by alternating gradient descent

AbstractThe symmetric tensor decomposition problem is a fundamental problem in many fields, which appealing for investigation. In general, greedy algorithm is used for tensor decomposition. That is, we first find the largest singular value and singular vector and subtract the corresponding component from tensor, then repeat the process. In this article, we focus on designing one effective algorithm and giving its convergence analysis. We introduce an exceedingly simple and fast algorithm for rank‐one approximation of symmetric tensor decomposition. Throughout variable splitting, we solve symmetric tensor decomposition problem by minimizing a multiconvex optimization problem. We use alternating gradient descent algorithm to solve. Although we focus on symmetric tensors in this article, the method can be extended to nonsymmetric tensors in some cases. Additionally, we also give some theoretical analysis about our alternating gradient descent algorithm. We prove that alternating gradient descent algorithm converges linearly to global minimizer. We also provide numerical results to show the effectiveness of the algorithm.

Field-Induced Transversely Isotropic Shear Response of Ellipsoidal Magnetoactive Elastomers.

Magnetoactive elastomers (MAEs) claim a vital place in the class of field-controllable materials due to their tunable stiffness and the ability to change their macroscopic shape in the presence of an external magnetic field. In the present work, three principal geometries of shear deformation were investigated with respect to the applied magnetic field. The physical model that considers dipole-dipole interactions between magnetized particles was used to study the stress-strain behavior of ellipsoidal MAEs. The magneto-rheological effect for different shapes of the MAE sample ranging from disc-like (highly oblate) to rod-like (highly prolate) samples was investigated along and transverse to the field direction. The rotation of the MAE during the shear deformation leads to a non-symmetric Cauchy stress tensor due to a field-induced magnetic torque. We show that the external magnetic field induces a mechanical anisotropy along the field direction by determining the distinct magneto-mechanical behavior of MAEs with respect to the orientation of the magnetic field to shear deformation.

Energy and flux budget closure theory for passive scalar in stably stratified turbulence

The energy and flux budget (EFB) closure theory for a passive scalar (non-buoyant and non-inertial particles or gaseous admixtures) is developed for stably stratified turbulence. The physical background of the EFB turbulence closures is based on the budget equations for the turbulent kinetic and potential energies and turbulent fluxes of momentum and buoyancy as well as the turbulent flux of particles. The EFB turbulence closure is designed for stratified geophysical flows from neutral to very stable stratification, and it implies that turbulence is maintained by the velocity shear at any stratification. In a steady-state, expressions for the turbulent flux of the passive scalar and the anisotropic non-symmetric turbulent diffusion tensor are derived, and universal flux Richardson number dependencies of the components of this tensor are obtained. The diagonal component in the vertical direction of the turbulent diffusion tensor is suppressed by strong stratification, while the diagonal components in the horizontal directions are not suppressed, but they are dominant in comparison with the other components of the turbulent diffusion tensor. This implies that any initially created strongly inhomogeneous particle cloud is evolved into a thin pancake in a horizontal plane with very slow increase in its thickness in the vertical direction. The turbulent Schmidt number (the ratio of the eddy viscosity and the vertical turbulent diffusivity of the passive scalar) linearly increases with the gradient Richardson number. The physics of such a behavior is related to the buoyancy force that causes a correlation between fluctuations of the potential temperature and the particle number density. This correlation that is proportional to the product of the vertical turbulent particle flux and the vertical gradient of the mean potential temperature reduces the vertical turbulent particle flux. Considering the applications of these results to the atmospheric boundary-layer turbulence, the theoretical relationships are derived, which allows us to determine the turbulent diffusion tensor as a function of the vertical coordinate measured in the units of the local Obukhov length scale. The obtained relations are potentially useful in modeling applications of particle dispersion in the atmospheric boundary-layer turbulence and free atmosphere turbulence.

Natural frequency analysis of shells of revolution based on hybrid dual-mixed [formula omitted]-finite element formulation

A newly-developed, dimensionally reduced, hp-type axisymmetric shell finite element model is extended to linear elastodynamic problems of thin shells of revolution. The hp-shell finite element relies on the hybridized version of a three-field dual-mixed variational formulation, the application of which dictates the obligate usage of the inverse three-dimensional constitutive relation for homogeneous and isotropic materials, thereby ensuring the volumetric locking-free characteristic of the shell model at theory level. The fundamental fields are the a priori non-symmetric stress tensor, the displacement vector, the infinitesimal rotation vector and the hybrid variable defined on the element interfaces. Since the dimensional reduction process guided by this hybrid-mixed formulation does not necessitate the use of any classical kinematic assumptions appearing in the scientific literature, the inverse 3D Hooke’s law does not have to be modified. The numerical performance of the shell finite element is analyzed comprehensively for natural frequency computations of clamped-free and simply supported, silicone, conoid, spherical and hyperboloid shells of revolution. From their relative error convergence behaviors it follows that the extended hybrid-mixed shell finite element is not sensitive to the decrease of the slenderness ratio, namely providing reliable, uniformly stable numerical results for both h- and p-approximation. From theoretical point of view, the beneficial properties of the hybrid-mixed hp shell finite element model are as follows: (i) this is effectively applicable to modeling not only extremely thin but also moderately thick shell structures with transverse shear deformations, as well as (ii) both the through-the-thickness variation and the membrane stress normal to the shell mid-surface are retained as independent variables, making it much easier to upbuild shell model for contact problems. From numerical point of view, the nice feature of the hybrid-mixed hp shell finite element is that the global flexibility matrix of the system can be inverted block-wise at element level at cheap computational cost during the assembling procedure because of the hybridization technique.

Angular position of singular optic axes for arbitrary dielectric tensors

We deduce formulas for the angular position of singular optic axes in homogeneous optical media. The angular position of singular optic axes for a given general (complex, nonsymmetric) dielectric tensor is the same for its transposed and complex conjugated tensors as well as for a (complex) multiple. For the case of a symmetric dielectric tensor, the four axes fulfill a connecting relation, meaning that the singular optic axes of a nonsymmetric dielectric tensor do not relate to those of any (effective) symmetric tensor. For the symmetric case, we provide analytical solutions for the angular positions of singular optic axes of triclinic, monoclinic, and orthorhombic crystals.

Scanning electron diffraction tomography of strain

Strain engineering is used to obtain desirable materials properties in a range of modern technologies. Direct nanoscale measurement of the three-dimensional strain tensor field within these materials has however been limited by a lack of suitable experimental techniques and data analysis tools. Scanning electron diffraction has emerged as a powerful tool for obtaining two-dimensional maps of strain components perpendicular to the incident electron beam direction. Extension of this method to recover the full three-dimensional strain tensor field has been restricted though by the absence of a formal framework for tensor tomography using such data. Here, we show that it is possible to reconstruct the full non-symmetric strain tensor field as the solution to an ill-posed tensor tomography inverse problem. We then demonstrate the properties of this tomography problem both analytically and computationally, highlighting why incorporating precession to perform scanning precession electron diffraction (SPED) may be important. We establish a general framework for non-symmetric tensor tomography and demonstrate computationally its applicability for achieving strain tomography with SPED data.

Tailored finite point method for the approximation of diffusion operators with non-symmetric diffusion tensor

We present a tailored finite point method (TFPM) for anisotropic diffusion equations with a non-symmetric diffusion tensor on Cartesian grids. The fluxes on each edge are discretized by using a linear combination of the local basis functions, which come from the exact solution of the diffusion equation with constant coefficients on the local cell. In this way, the scheme is fully consistent and the flux is naturally continuous across the interfaces between the subdomains with a non-symmetric diffusion tensor. Additionally, it is convenient to handle the Neumann boundary condition or a variant of Neumann boundary condition. Numerical results obtained from solving different anisotropic diffusion problems including problems with sharp discontinuity near the interface and boundary, show that this approach is efficient. Second order convergence rate can be obtained with the numerical examples. The new scheme is also tested on a time-dependent problem modeling the Hall effect in the resistive magnetohydrodynamics, the results further show the robustness of this new method.

Non-Abelian gauged fracton matter field theory: Sigma models, superfluids, and vortices

By gauging a higher-moment polynomial degree global symmetry and a discrete charge conjugation (i.e., particle-hole) symmetry coupled to matter fields (two symmetries mutually non-commutative), we derive a new class of higher-rank tensor non-abelian gauge field theory with dynamically gauged fractonic matter fields: Non-abelian gauged fractons interact with a hybrid class of higher-rank (symmetric or generic non-symmetric) tensor gauge theory and anti-symmetric tensor topological field theory, generalizing [arXiv:1909.13879, 1911.01804]. We also apply a quantum phase transition similar to that between insulator v.s. superfluid/superconductivity (U(1) symmetry disordered phase described by a topological gauge theory or a disordered Sigma model v.s. U(1) global/gauge symmetry-breaking ordered phase described by a Sigma model with a U(1) target space underlying Goldstone modes): We can regard our tensor gauge theories as disordered phases, and we transient to their new ordered phases by deriving new Sigma models in continuum field theories. While one low energy theory is captured by degrees of freedom of rotor or scalar modes, another side of low energy theory has vortices and superfluids -- we explore non-abelian vortices (two types of vortices mutually interacting non-commutatively beyond an ordinary group structure) and their Cauchy-Riemann relation.

Gravitational dipole moment from an exact solution with torsion

Dipole fields are common in electromagnetism and may be viewed as the result of a positive and negative charge (or pole) which are close together. A dipole field in gravity is not expected to exist because negative mass has never been observed. However, an exact solution to the gravitational field equations with torsion is presented that does correspond to a dipole gravitational field, even though there is no negative mass. The theory is based on a non-symmetric metric tensor, and it is found there is a singularity in the metric tensor at the origin but no event horizon. Gravitational dipoles have been used to solve the dark matter problem.

Dual-mixed hp-version axisymmetric shell finite element using NURBS mid-surface interpolation

A new, general hp-version axisymmetric finite element is derived for the boundary value problems of thin linearly elastic shells of revolution, applying a complementary strain energy-based three-field dual-mixed variational principle. For the interpolation of the mid-surface geometry, non-uniform rational B-splines—NURBS—is used. The independent field variables of the weak formulation are the a priori non-symmetric stress tensor, the displacement vector, and the infinitesimal skew-symmetric rotation tensor. The theoretical model of the shell formulation is based on a consistent dimensional reduction process and a systematic variable-number reduction procedure. The inverse of the unvaried three-dimensional constitutive equation is employed since neither the classical kinematical assumptions nor the stress hypotheses are built in the mathematical model; namely, both the through-the-thickness variation and the normal stress to the shell mid-surface are not excluded. The new hp axisymmetric shell finite element is tested by a representative model problem for extremely thin and moderately thick, singly and doubly curved shells of negative and positive Gaussian curvature. Following from the numerical experiments, the constructed hp-shell finite element gives locking-free results not only for the displacement but also for the stresses.