Tensor Functions Research Articles

Robust Computation and Analysis of Vibrational Spectra of Layered Framework Materials Including Host-Guest Interactions.

Layered framework materials, a rapidly advancing class of porous materials, are composed of molecular components stitched together via covalent bonds and are usually synthesized through wet-chemical methods. Computational infrared (IR) and Raman spectra are among the most important characterization tools for this material class. Besides the a priori known spectra of the molecular building blocks and the solvent, they allow for in situ monitoring of the framework formation during synthesis. Therefore, they need to capture the additional peaks from host-guest interactions and the bands from emerging bonds between the molecular building blocks, verifying the successful synthesis of the desired material. In this work, we propose a robust computational framework based on ab initio molecular dynamics (AIMD), where we compute IR and Raman spectra from the time-correlation functions of dipole moments and polarizability tensors, respectively. As a case study, we apply our methodology to a covalent organic framework (COF) material, COF-1, and present its AIMD-computed IR and Raman spectra with and without 1,4-dioxane solvent molecules in its pores. To determine robust settings, we meticulously validate our model and explore how stacking disorder and different methods for computing dipole moments and polarizabilities affect IR and Raman intensities. Using our robust computational protocol, we achieve excellent agreement with experimental data. Furthermore, we illustrate how the computed spectra can be dissected into individual contributions from the solvent molecules, the molecular building blocks of COF-1, and the bonds connecting them.

Exploring a novel feature of ellis spacetime: Insights into scalar field dynamics

We have studied neutral and charged massive particles dynamics in Ellis spacetime in the presence of the external scalar field. Focusing on the circular motion of massive particles, the impact of an external scalar field on the Innermost Stable Circular Orbit (ISCO) position is analyzed, revealing a non-linear relationship with the scalar field parameter. Perturbation techniques are employed to investigate oscillatory motion near stable orbits in the Ellis spacetime, yielding analytical expressions for radial and angular oscillations. The throat of the wormhole has been constrained by comparing theoretical and observational results for fundamental frequencies of particles from quasars. Finally, scalar and gravitational perturbations in the Ellis spacetime have been studied. It is shown that the equation for the scalar profile function is fully independent from the tensor functions and the solution can be represented in terms of the confluent Heun function. However, it has been shown that equations for the tensor profile functions strongly depend on the scalar profile functions in the Ellis spacetime and they are reduced to the Regge–Wheeler–Zerilli equation. Finally, numerical solutions to the Regge–Wheeler–Zerilli equation for the radial functions in the Ellis have been presented.

On sign function of tensors with Einstein product and its application in solving Yang–Baxter tensor equation

On sign function of tensors with Einstein product and its application in solving Yang–Baxter tensor equation

Wigner Function Method for Describing Electromagnetic Field in Plasma-Like Media with Spatial Dispersion and Resonant Dissipation

The paper gives a systematic presentation of the Wigner function method (Weyl formalism) for modeling the propagation and absorption of electromagnetic waves in anisotropic and gyrotropic dissipative media with spatial dispersion. A general kinetic equation for the Wigner function (tensor) is formulated, and its asymptotic expansion up to the second order for smoothly inhomogeneous and weakly dissipative media is constructed. As a result, a modification of the method of the kinetic equation for rays is proposed, based on the stochastic description of rays, making it possible to increase the accuracy of numerical modeling of wave problems with strong transverse inhomogeneity of the absorption coefficient without increasing the amount of calculations. The technique developed can be used to describe the propagation, absorption, and scattering of electron-cyclotron waves in high-temperature plasma of magnetic traps for controlled fusion in cases where standard modeling methods do not provide the necessary accuracy.

Smoothed asymptotics: From number theory to QFT

Inspired by the method of smoothed asymptotics developed by Terence Tao, we introduce a new ultraviolet regularization scheme for loop integrals in quantum field theory which we call η regularization. This reveals a connection between the elimination of divergences in divergent series of powers and the preservation of gauge invariance in the regularization of loop integrals in quantum field theory. In particular, we note that a method for regularizing the series of natural numbers so that it converges to minus one-twelfth inspires a regularization scheme for non-Abelian gauge theories coupled to Dirac fermions that preserves the Ward identity for the vacuum polarization tensor and other higher point functions. We also comment on a possible connection to Schwinger proper time integrals. Published by the American Physical Society 2024

A discrete-to-continuum model for the human cornea with application to keratoconus.

We introduce a discrete mathematical model for the mechanical behaviour of a planar slice of human corneal tissue, in equilibrium under the action of physiological intraocular pressure (IOP). The model considers a regular (two-dimensional) network of structural elements mimicking a discrete number of parallel collagen lamellae connected by proteoglycan-based chemical bonds (crosslinks). Since the thickness of each collagen lamella is small compared to the overall corneal thickness, we upscale the discrete force balance into a continuum system of partial differential equations and deduce the corresponding macroscopic stress tensor and strain energy function for the micro-structured corneal tissue. We demonstrate that, for physiological values of the IOP, the predictions of the discrete model converge to those of the continuum model. We use the continuum model to simulate the progression of the degenerative disease known as keratoconus, characterized by a localized bulging of the corneal shell. We assign a spatial distribution of damage (i.e. reduction of the stiffness) to the mechanical properties of the structural elements and predict the resulting macroscopic shape of the cornea, showing that a large reduction in the element stiffness results in substantial corneal thinning and a significant increase in the curvature of both the anterior and posterior surfaces.

Prediction of the in-plane permeability and air evacuation time of fiber-placed thermoplastic composite preforms with engineered intertape channels

The two main void removal mechanisms during vacuum-bag-only (VBO) consolidation of thermoplastic composites are through-thickness diffusion and in-plane air evacuation. The automated fiber placement (AFP) process allows for the creation of preforms with an engineered intertape channel network by deliberately introducing spacing between the tapes that can facilitate air evacuation during the VBO consolidation. However, it is unclear what dimensions of the intertape channels allow for effective in-plane air evacuation. The current research presents a predictive simulation tool to optimize the intertape channel dimensions for effective in-plane air evacuation. An analytical method is developed to calculate the in-plane permeability tensors of the composite preform with uniform intertape channel dimensions. In addition, a more elaborate mesoscale method is developed that estimates the in-plane permeability tensor of the engineered intertape channel network based on the distributions in intertape channel dimensions. Finally, a finite difference model is implemented to calculate the time required for air evacuation as a function of the in-plane permeability tensors of the preform and its in-plane dimensions. The results from the models indicated that in-plane air evacuation through the engineered intertape channel network is quick, and it takes only a few minutes to evacuate 99% of the air from large preforms like fuselage panels.

On Itskov (2024) counterexample to the functional basis of isotropic vector and tensor functions by Shariff (2023)

On Itskov (2024) counterexample to the functional basis of isotropic vector and tensor functions by Shariff (2023)

Low-Rank Tensor Function Representation for Multi-Dimensional Data Recovery.

Since higher-order tensors are naturally suitable for representing multi-dimensional data in real-world, e.g., color images and videos, low-rank tensor representation has become one of the emerging areas in machine learning and computer vision. However, classical low-rank tensor representations can solely represent multi-dimensional discrete data on meshgrid, which hinders their potential applicability in many scenarios beyond meshgrid. To break this barrier, we propose a low-rank tensor function representation (LRTFR) parameterized by multilayer perceptrons (MLPs), which can continuously represent data beyond meshgrid with powerful representation abilities. Specifically, the suggested tensor function, which maps an arbitrary coordinate to the corresponding value, can continuously represent data in an infinite real space. Parallel to discrete tensors, we develop two fundamental concepts for tensor functions, i.e., the tensor function rank and low-rank tensor function factorization, and utilize MLPs to paramterize factor functions of the tensor function factorization. We theoretically justify that both low-rank and smooth regularizations are harmoniously unified in LRTFR, which leads to high effectiveness and efficiency for data continuous representation. Extensive multi-dimensional data recovery applications arising from image processing (image inpainting and denoising), machine learning (hyperparameter optimization), and computer graphics (point cloud upsampling) substantiate the superiority and versatility of our method as compared with state-of-the-art methods. Especially, the experiments beyond the original meshgrid resolution (hyperparameter optimization) or even beyond meshgrid (point cloud upsampling) validate the favorable performances of our method for continuous representation.

An improved hypoplastic model incorporating intergranular strain for overconsolidated clay

Hypoplastic constitutive models are able to describe overconsolidated behaviors of clay with a single nonlinear tensor function using a set of parameters. For instance, an advanced hypoplastic model by Wang and Wu (2020) has successfully reproduced overconsolidated history by incorporating a structure tensor. To simulate the recent stress history effects and the stiffness variations at small strains of clay, the intergranular strain tensor is introduced by Niemunis and Herle (1997) to reproduce stress-path dependent stiffness of clay. The improved hypoplastic model is examined against experiments on overconsolidated Chicago clay. It is evidently concluded that the improved model is capable of replicating stress-path dependency on the small-strain behavior of overconsolidated clay.

On almost quotient Yamabe solitons

Abstract In this paper, we investigate the structure of certain solutions of the fully nonlinear Yamabe flow, which we call almost quotient Yamabe solitons as they extend quite naturally those already called quotient Yamabe solitons. We present sufficient conditions for a compact almost quotient Yamabe soliton to be either trivial or isometric with an Euclidean sphere. We also characterize noncompact almost gradient quotient Yamabe solitons satisfying certain conditions on both its Ricci tensor and potential function.

Solver-free reduced order homogenization for nonlinear periodic heterogeneous media

Reduced-order homogenization (ROH) and related methods are important computational tools for simulating the material behavior of composites. These methods generally sacrifice accuracy in exchange for superior computational efficiency, relative to methods such as classical computational homogenization (CCH). In this study, building on the recently developed solver-free CCH, we propose a fine-scale solver-free reduced order homogenization approach that avoids solving the fine-scale equilibrium equations and approximates the phase-average eigenstrains by sampling the fine-scale eigenstrain at a small number of points. The proposed method, which we call solver-free ROH, works by pre-computing history-dependent eigenstrain influence function tensors and sampling point contribution factors based on training data from a small set of CCH simulations. Then, during the online stage of the computation, phase-average eigenstrains are computed from sampling point eigenstrains and used to compute homogenized stresses and strains. Focusing on small-deformation problems, this paper formulates and verifies the solver-free ROH approach for nonlinear periodic heterogeneous media. First, in the formulation, we delineate a sampling point approximation of the phase eigenstrains and describe the use of this approximation within the coarse-scale stress update. Next, we verify the solver-free ROH using loading cases outside the training data set. Finally, we use the proposed method to simulate a multilayer composite plate in three point bending (3pt-bend) and open hole tension (OHT), demonstrating the method’s efficiency and accuracy relative to the CCH.

Amplitude basis for conformal correlators

We present a classification of conformally-invariant three-point tensor structures in d dimensions that parallels the classification of three-particle scattering amplitudes in d + 1 dimensions. Using a set of canonically-normalized weight-shifting operators, we construct a basis of three-point structures involving conserved currents or stress tensors and non-conserved spinning operators, directly from their amplitude counterparts. As an application, we also examine the conformal block expansion of the four-point functions of external currents and stress tensors in this amplitude basis. Our results can be useful for conformal bootstrap applications involving spinning correlators as well as Witten diagram computations in anti-de Sitter space.

Higher-order spectral representation method: New algorithmic framework for simulating multi-dimensional non-Gaussian random physical fields

This study derives a novel higher-order spectral representation method (HOSRM) to represent and simulate multi-dimensional fourth-order non-Gaussian random physical fields. The method primarily extends the traditional second-order spectral representation method (SRM) for simulating non-Gaussian random physical fields by introducing higher-order cumulant function tensors and trispectrum tensors, thereby accomplishing the modeling of fourth-order non-Gaussian random physical fields (symmetric nonlinear physical fields) from a frequency domain perspective. In an endeavor to enhance the simulation efficiency of this theoretical framework, the Fast Fourier Transform (FFT) algorithm is astutely amalgamated into the simulation. This integration contributes significantly to the augmentation of computational efficiency. Furthermore, exhaustive derivations and proofs are presented for the first-order, second-order, and fourth-order ensemble properties of simulated fourth-order non-Gaussian random physical fields. Subsequently, the reliability and accuracy of the proposed algorithm framework are validated through numerical simulations of two two-dimensional and two three-dimensional non-Gaussian random physical fields. The findings demonstrate that the simulated sample function effectively captures the probability characteristics of the random field, including mean, variance, and kurtosis. Finally, an in-depth analysis of numerical simulation results highlights the difference between the proposed method and the traditional second-order spectral representation method, which further underscores the distinctive attributes and potential superiority of the proposed method.

First-principles calculations to investigate structural, elastic and electronic properties of chloro-perovskite NaMgCl3

In this work, we have investigated the structural, electronic, and optical properties of the halide-perovskite NaMgCl3 by performing first-principles calculations using a mixed basis of the full-potential linearized augmented plane wave (FP-LAPW) approach under the generalized gradient approximation (GGA-PBEsol), a revised Perdew-Burke-Ernzerhof, and the modified Becke-Johnson (TB-mBJ) implemented on the Wien2k package. The obtained results show that NaMgCl3 is more stable in the nonmagnetic phase.The compound is found to be thermodynamically and elastically stable. It is found that there is a good agreement between the calculated lattice constants and the values published in previous studies. The complete set of first-order elastic constants tensor has been calculated, implying mechanical stability. The Young's modulus, shear modulus, Poisson's ratio,Pugh's ratio, and universal anisotropy index have been additionally calculated. The atomic bonds along both x and y axis are stronger than those along the z-axis. The Debye temperature is calculated using the average sound velocity. From electronic band structures, we have found that NaMgCl3 is an insulator with a direct wide band gap ΓV→ΓC. The electronic charge densities were plotted.Analysis of the charge density distribution map reveals that the bondings in NaMgCl3 are ionic. The calculated density of states (DOS) indicates that the valence band is mainly dominated by Cl− anions, the conduction band is mainly dominated by Mg2+ cations, whereas Na+ has a negligible contribution. The optical characteristics, notably the dielectric function tensor, refractive function, reflectivity, absorption coefficient, photo-conductivity and transmittance were plotted and discussed. The optical properties confirm that NaMgCl3 is optically anisotropic in 10–24 eV range .The low transmittance and high absorption coefficient in the far ultra-violet region, between 90 and 250 nm, make this compound a good candidate for use as an ultraviolet shield, in vacuum ultraviolet lenses and in thermoelectric promising features due to his direct wide band gap.Experimental and theoretical data are scarce for this compound in an ilmenite-type structure.

Biaxial Testing of EPDM Rubbers for Automotive Applications Using a Uniaxial Testing Machine

The FE-based structural design of components made of rubber materials requires an experimental characterization to calibrate a proper hyperelastic material model. A strain energy density (SED) parameter is typically adopted to describe the hyperelastic behavior of isotropic rubbers. This can be written as a function of the strain tensors, the strain invariants or the stretch ratios and proper material parameters. The latter must be calibrated on experimental data generated by testing rubber materials under different loading conditions, typically two among uniaxial tension, pure shear and biaxial tension. In this context, the mechanical behavior of Ethylene–Propylene–Diene Monomer (EPDM) rubbers, both in the dense and expanded configurations, which are typically employed to manufacture automotive seals, has been investigated. First, uniaxial tension tests have been performed. Afterwards, a new fixture has been developed to perform equi-biaxial tension tests using a uniaxial testing machine. The fixture has been adopted to test EPDM cruciform specimens under equi-biaxial tension loading. Finally, both uniaxial and equi-biaxial data have been employed to fit a Mooney–Rivlin material model for each tested EPDM rubber, which has been validated by comparing the results of FE simulations with the experimental ones.

On the functional basis of isotropic vector and tensor functions by Shariff (2023)

In the paper Shariff (Q. J. Mech. Appl. Math. 76:143–161, 2023) a functional basis of a system of vectors and symmetric tensors is proposed. The functional basis is expressed in terms of eigenvalues and eigenvectors of the first tensor and includes a smaller number of terms in comparison to the classical irreducible representation (see, e.g., Boehler, J. Appl. Math. Mech. 57:323–327, 1977; Pennisi and Trovato, Int. J. Eng. Sci. 25:1059–1065, 1987). In the present contribution, we show that elements of the functional basis by Shariff (Q. J. Mech. Appl. Math. 76:143–161, 2023) do not represent isotropic invariants of the vector and tensor arguments and cannot thus be referred to as the functional basis. To this end, a counterexample with two symmetric tensors is considered. Under an arbitrary orthogonal transformation the functional basis (Shariff, Q. J. Mech. Appl. Math. 76:143–161, 2023) of these two tensors should remain constant but it does change in contrast to the classical representation.

Fourth-grade nanofluid model with dissipative and nonlinear radiative properties transported peristaltically via a flexible diverging duct holding a porous media under the influence of concentration and heat convection in an induced magnetic field

The major goal of this work is to give a thorough examination of the effect of double diffusion convection (DDC) in addition to a generated magnetic field on peristaltic movement of fourth-grade nanofluid across a vertical sophisticated asymmetrical microchannel using a nondeformable porous media as a basis for intricate pumping systems inspired by biological processes for hazardous waste. The mathematical formulas pertaining to flow, heat/mass transfer under the influence of viscous dissipation, nonlinear heat radiation, and Joule heating were developed using Buongiorno’s framework for nanofluids with combining the thermophoresis and Brownian motion characteristics. Mathematical analysis has been conducted under the suppositions of an extended wavelength and a relatively small Reynolds number. Magnetic field induced axially, density of current, magnetic force function, thermal characteristics, nanoparticles proportion gradient, an additional stress tensor, pressure gradient, and stream function are all given explicit formulas. The constructed function (ND Solve function) within the Wolfram software (Mathematica) is employed to computationally resolve the ensuing system of coupled nonlinear differential equations. Numerical and pictorial evidence is presented to highlight the significance of different physiological characteristics of flow volumes. Further, contour visualizations and circulation bolus have been used to highlight the trapping phenomena, one of among the most noteworthy peristaltic motion occurrences. The main results showed that, despite the dissolvent concentration and the volume percentage of nanoparticles having the opposite effects, the resistance of a substance to heat is shown to climb as the Soret and Dufour numbers rise. At larger levels of the electromagnetic Reynolds number, Strommer’s number, electric field parameter, and thermal Grashof number, stronger axial induced magnetic fields (IMFs) are also provided.

Inverse scattering problem for the tensor fields in a coordinate-free representation

The paper is devoted to the inverse scattering problem in order to study the complex media whose electrical and magnetic properties are the tensor functions. The tensor Fourier diffraction theorem, derived in the paper, gives the relations connecting the scattered field and the inherent properties of the scattering medium. The method is based on the first Born approximation. The tensor (dyadic) analysis for derivation of the tensor Fourier diffraction theorem in the coordinate-free form is used.

Solver‐free classical computational homogenization for nonlinear periodic heterogeneous media

AbstractModeling the behavior of composite materials is an important application of computational homogenization methods. Classical computational homogenization (CCH), based on asymptotic analysis, is such a method. In CCH, equilibrium equations are separated into two length scales and solved numerically. Solving the fine‐scale equilibrium equations at every coarse‐scale Gauss point, in every iteration of a Newton–Raphson loop, is often too computationally expensive for real engineering applications. In this study, we propose a modified CCH approach that avoids solving the fine‐scale equilibrium equations. The proposed method, which we call solver‐free CCH, works by pre‐computing a set of eigenstrain influence function tensors based on data from a small number of numerical experiments. Then, during the online stage of the computation, these eigenstrain influence tensors are used in the fine‐scale problem to evaluate and homogenize strains and stresses. This article begins by formulating the solver‐free CCH approach for small‐deformation problems involving composites with nonlinear constituent phase material models, including computation of the eigenstrain influence tensors and their application within the online stage. To verify the proposed approach, we consider loading cases outside the training set used to derive the eigenstrain influence tensors. The combinational efficiency and accuracy of the solver‐free CCH in comparison to the conventional CCH is studied on a multilayer composite plate in three point bending (3pt‐bend) and open hole tension.