Tensor Coefficients Research Articles

Tensorial harmonic bases of arbitrary order with applications in elasticity, elastoviscoplasticity and texture-based modeling

In continuum mechanics, one regularly encounters higher-order tensors that require tensorial bases for theoretical or numerical calculations. By using results from [Formula: see text] representation theory, we present a method to derive tensorial harmonic bases which unifies existing approaches in one framework. Applying this convention to symmetric second-order tensors results in a second-order harmonic basis which, for example, simplifies the depiction and numerical handling of stiffness tensors for most common material symmetries and reduces the computational effort involved in implementations of incompressible elastoviscoplastic material laws. The same convention can be applied to texture analysis to yield tensorial texture coefficients for both polycrystals and fiber-reinforced composites. Rotations of higher-order tensors are particularly efficient in harmonic bases as both material and index symmetries can be exploited. While special attention is given here to the examples of small-strain material laws and texture analysis, the framework is entirely general and can be used to simplify calculations in other physical contexts involving tensors of arbitrary order and symmetry. A Python implementation of the harmonic basis convention defined in this work is available as a Supplementary Material.

Nonlinear Reduced Order Modeling of Heated Structures with Temperature-Dependent Properties

This paper focuses on extending coupled structural–thermal reduced order modeling approaches for the prediction of the nonlinear geometric response of heated structures to efficiently account for temperature-dependent structural and thermal properties. Structurally, a linear dependence of the elasticity tensor and thermal expansion coefficient is assumed consistently with typical material behavior. The resulting structural reduced order model (ROM) equations exhibit polynomials of the temperature generalized coordinates, the coefficients of which are readily identified. A different strategy is proposed for conductance and capacitance properties which typically depend nonlinearly on temperature. It relies on splitting the temperature generalized coordinates into a small set of dominant ones having a nonlinear effect on the ROM conductances and capacitances and a much larger set affecting them linearly. The inclusion of uncertainty in the structural ROM is also addressed extending earlier work limited to temperature independent properties. These strategies are exemplified on a representative hypersonic vehicle panel undergoing a full mission profile and with aero-thermal–structural coupling. The ROM predictions are observed to be very close to the full finite element ones for the mean model. Moreover, an uncertainty analysis demonstrates the strong sensitivity of the response to the structural model in the strongly nonlinear regions of the trajectory.

Investigation of the structural, electronic, magnetic, and optical properties of the double perovskite oxide semiconductor La2NiMnO6: Ab initio calculations by using GGA-PBE and GGA + mBJ approximations

In this paper, we examined the La2NiMnO6 double perovskite oxide semiconductor materials, our treatment included the structural, electronic, magnetic, and optical characteristics by utilizing the density functional theory, which was performed in the Wien2k software. The exchange–correlation potential was carried out in combination with the GGA-PBE (Perdew Burke Ernzerhof Generalized Gradient Approximation), and GGA + mBJ (modified Becke and Johnson) was used for the exchange–correlation potentials. The results suggest that the compound La2NiMnO6 exhibits in a ferrimagnetic state. It was also found that the stable ground state of the compound is ferrimagnetic. The results demonstrate that La2NiMnO6 has a band gap whose spin-up value is 0 eV and spin-down (dn) value is 2.86 eV (GGA-PBE), with up values is 0 eV and dn value being of 3 eV (GGA + mBJ). We also explored different optical properties, among them electron energy loss, absorption coefficient, real and imaginary dielectric tensor, and real and imaginary optical conductivity.

Study of the anisotropy of α-<sup>33</sup>S single crystal thermal expansion

A technique for arbitrary symmetry crystals unit cell parameters refining on modern single-crystal diffractometers is described. The technique is based on the 2D detector position calibration. The elementary orthorhombic unit cell parameters of the α-33S single crystal have been refined. The anisotropy of parameters changes in the range of 90–350 K has been studied. It is shown that the relative increase in parameter c is 6.4%. The obtained dependences are approximated by second–third degree polynomials. The absolute increase in cell volume is 138.4 A3, and the relative increase is 4.3%. The temperature dependencies of the thermal expansion tensor elements has been refined. The coefficients of α-33S thermal expansion tensor at the room temperature are: α11 = 15.35 × 10–5, α22 = 8.56 × 10–5, α33 = 9.12 × 10–5 К–1.

Assembling algorithm for Green's tensors and absorbing boundary conditions for Galbrun's equation in radial symmetry

Solar oscillations can be modeled by Galbrun's equation which describes Lagrangian wave displacement in a self-gravitating stratified medium. For spherically symmetric backgrounds, we construct an algorithm to compute efficiently and accurately the coefficients of the Green's tensor of the time-harmonic equation in vector spherical harmonic basis. With only two resolutions, our algorithm provides values of the kernels for all heights of source and receiver, and prescribes analytically the singularities of the kernels. We also derive absorbing boundary conditions (ABC) to model wave propagation in the atmosphere above the cut-off frequency. The construction of ABC, which contains varying gravity terms, is rendered difficult by the complex behavior of the solar potential in low atmosphere and for frequencies below the Lamb frequency. We carry out extensive numerical investigations to compare and evaluate the efficiency of the ABCs in capturing outgoing solutions. Finally, as an application towards helioseismology, we compute synthetic solar power spectra that contain pressure modes as well as internal-gravity (g-) and surface-gravity (f-) ridges which are missing in simpler approximations of the wave equation. For purpose of validation, the locations of the ridges in the synthetic power spectra are compared with observed solar modes.

Heavy-to-light form factors to three loops

We compute three-loop corrections of O(αs3) to form factors with one massive and one massless quark coupling to an external vector, axialvector, scalar, pseudoscalar, or tensor current. We obtain analytic results for the color-planar contributions, for the contributions of light-quark loops, and the contributions with two heavy-quark loops. For the computation of the remaining master integrals we use the “expand and match” approach which leads to semianalytic results for the form factors. We implement our results in a and a ortran code which allows for fast and precise numerical evaluations in the physically relevant phase space. The form factors are used to compute the hard matching coefficients in Soft-Collinear Effective Theory for all currents. The tensor coefficients at lightlike momentum transfer are used to extract the hard function in B¯→Xsγ to three loops. Published by the American Physical Society 2024

Development of a numerical and analytical methodology for analyzing hybrid laminates with multi-oriented piezoelectric and structural layers

Piezoelectric materials can generate an electrical response under mechanical stress, functioning as sensors. Conversely, applying an electrical field to these materials enables precise motion control, making them effective as actuators. Using a proposed methodology, this work focuses on evaluating the effective properties of hybrid multi-oriented composite laminates consisting of structural and piezoelectric layers driven by monoclinic constitutive equations. Square unit cell model was used to calculate all coefficients of the material tensor. The finite element (FE) homogenization method and periodic boundary conditions implemented by node-to-node constraint equations are used to study a representative volume element (RVE) modeled as three-layer unit cell in the mesoscale. The pre-processing, FE analysis, and post-processing are conducted in the FE package ABAQUSⓇ through Python scripts. The computational approach yields results that align closely with the analytical effective coefficients derived from the Asymptotic Homogenization Method (AHM), demonstrating the robustness and accuracy of the proposed methodology.

The generalized method of separation of variables for diffusion-influenced reactions: Irreducible Cartesian tensor technique.

Motivated by the various applications of the trapping diffusion-influenced reaction theory in physics, chemistry, and biology, this paper deals with irreducible Cartesian tensor (ICT) technique within the scope of the generalized method of separation of variables (GMSV). We provide a survey from the basic concepts of the theory and highlight the distinctive features of our approach in contrast to similar techniques documented in the literature. The solution to the stationary diffusion equation under appropriate boundary conditions is represented as a series in terms of ICT. By means of proved translational addition theorem, we straightforwardly reduce the general boundary value diffusion problem for N spherical sinks to the corresponding resolving infinite set of linear algebraic equations with respect to the unknown tensor coefficients. These coefficients exhibit an explicit dependence on the arbitrary three-dimensional configurations of N sinks with different radii and surface reactivities. Our research contains all relevant mathematical details such as terminology, definitions, and geometrical structure, along with a step by step description of the GMSV algorithm with the ICT technique to solve the general diffusion boundary value problem within the scope of Smoluchowski's trapping model.

Tensor-based statistical learning methods for diagnosing product quality defects in multistage manufacturing processes

Multistage manufacturing processes with identical stages provide three-dimensional process data in which the first dimension represents the process (control/sensing) variable, the second is the stage, and the third is the measurement/sampling/data acquisition time point. Diagnosing quality faults in such processes often requires the simultaneous identification of crucial process variables and stages associated with product quality anomalies. Most existing diagnosis methods convert 3D data into a 2D matrix, resulting in loss of information and reduced diagnostic accuracy and stability. To address this challenge, we propose a penalized tensor regression model that regresses the product quality index against its 3D process data. For the estimation of high-dimensional regression coefficients with the limited amount of historical data, we apply the CANDECOMP/PARAFAC and Tucker decompositions to the coefficient tensor, which significantly reduces the number of parameters to be estimated. Based on the decompositions, a new regularization term is designed to enable the joint identification of critical process variables and stages. To estimate the parameters, we develop the block coordinate proximal descent algorithm and provide its convergence guarantee. Numerical studies demonstrate that the proposed methods can enhance diagnostic stability and on average improve the diagnostic accuracy by around 20% over existing benchmarks.

Investigation of Piezoelectric Properties of Wurtzite AlN Films under In-Plane Strain: A First-Principles Study

This research article presents a comprehensive first-principles study on the piezoelectric properties of Wurtzite Aluminum Nitride (AlN) films under in-plane strain conditions. By calculating the piezoelectric tensor coefficients (e33, e31, and e15), we investigate the variation patterns of these constants with respect to in-plane strain. Our results indicate significant changes in the piezoelectric constants within the range of in-plane strain considered, exhibiting a linear trend despite opposite trends for e33 compared to e31 and e15. This study highlights the extreme sensitivity of AlN films’ piezoelectric performance to in-plane strain, suggesting its potential as an effective means for tuning and optimizing the piezoelectric properties of AlN-based devices.

Evaluation of a decomposition-based interpolation method for fourth-order fiber-orientation tensors: An eigensystem approach

We propose and assess a new decomposition-based interpolation method on fourth-order fiber-orientation tensors. This method can be used to change the resolution of discretized fields of fiber-orientation tensors, e.g., obtained from flow simulations or computer tomography, which are common in the context of short- and long-fiber–reinforced composites. The proposed interpolation method separates information on structure and orientation using a parametrization which is based on tensor components and a unique eigensystem. To identify this unique eigensystem of a given fourth-order fiber-orientation tensor in the absence of material symmetry, we propose a sign convention on tensor coefficients. We explicitly discuss challenges associated with material symmetries, e.g., non-distinct eigenvalues of the second-order fiber-orientation tensor and propose algorithms to obtain a unique set of parameters combined with a minimal number of eigensystems of a given fourth-order fiber-orientation tensor. As a side product, we specify for the first time, parametrizations and admissible parameter ranges of cubic, tetragonal, and trigonal fiber-orientation tensors. Visualizations in terms of truncated Fourier series, quartic plots, and tensor glyphs are compared.

Machine learning for exoplanet detection in high-contrast spectroscopy: Combining cross-correlation maps and deep learning on medium-resolution integral-field spectra

The advent of high-contrast imaging instruments combined with medium-resolution spectrographs allows spectral and temporal dimensions to be combined with spatial dimensions to detect and potentially characterize exoplanets with higher sensitivity. We developed a new method to effectively leverage the spectral and spatial dimensions in integral-field spectroscopy (IFS) datasets using a supervised deep-learning algorithm to improve the detection sensitivity to high-contrast exoplanets. We began by applying a data transform whereby the four-dimensional (two spatial dimensions, one spectral dimension, and one temporal dimension) IFS datasets are replaced by four-dimensional cross-correlation coefficient tensors obtained by cross-correlating our data with young gas giant spectral template spectra. Thus, the spectral dimension is replaced by a radial velocity dimension and the rest of the dimensions are retained `as is'. This transformed data is then used to train machine learning (ML) algorithms. We trained a 2D convolutional neural network with temporally averaged spectral cubes as input, and a convolutional long short-term memory memory network that uses the temporal data as well. We compared these two models with a purely statistical (non-ML) exoplanet detection algorithm, which we developed specifically for four-dimensional datasets, based on the concept of the standardized trajectory intensity mean (STIM) map. We tested our algorithms on simulated young gas giant s inserted into a SINFONI dataset that contains no known exoplanet, and explored the sensitivity of algorithms to detect these exoplanets at contrasts ranging from $10^ $ to $10^ $ for different radial separations. We quantify the relative sensitivity of the algorithms by using modified receiver operating characteristic curves (mROCs). We discovered that the ML algorithms produce fewer false positives and have a higher true positive rate than the STIM-based algorithm. We also show that the true positive rate of ML algorithms is less impacted by changing radial separation than the STIM-based algorithm. Finally, we show that preserving the velocity dimension of the cross-correlation coefficients in the training and inference plays an important role in ML algorithms being more sensitive to the simulated young gas giant s. black In this paper we demonstrate that ML techniques have the potential to improve the detection limits and reduce false positives for directly imaged planets in IFS datasets, after transforming the spectral dimension into a radial velocity dimension through a cross-correlation operation and that the presence of the temporal dimension does not lead to increased sensitivity.

To a dispersion law of an electromagnetic wave in a medium having a double anisotropy

Dispersion relations for a transverse electromagnetic wave in a hypothetical medium having a low crystalline symmetry and double anisotropy are formulated phenomenological based on the general concepts of Maxwell's equations. Formalism for electrical and magnetic ordering of a medium is represented by tensor coefficients of the second rank in a general coordinate system corresponding to an external problem of incidence, reflection and refraction of a transverse wave at the interface. Based on the obtained dispersion law, the Snell law of refraction for the wave vector is displayed, as well as, some special cases for a section of a wave surface and a ray surface in the plane of incidence. Following Fresnel approach as usual, two characteristic types of linear polarization of an electric induction vector of a propagating wave are considered for a wave entering a double anisotropic medium at an arbitrary angle to the interface. Based on a character of a cross-section of a wave surface and of a ray surface for the plane of incidence, the degree of influence of anisotropy on a propagating field is discussed in terms of external/internal conic refraction of ordinary/extraordinary waves in a transparent medium. Following to initially general form of the permittivity ε^ and the magnetic permeability μ^ tensors some particular cases of a more favorable choice of coordinate system are analyzed to decrease a number of non-zero components in mentioned tensors, as well as to decrease in the number of elements in vector relations.

Comparative Analysis of the Apparent Diffusion Coefficient and Diffusion Tensor Imaging in the Diagnosis of Prostate Cancer.

The aim of this work was to demonstrate capabilities of diffusion tensor imaging as a diagnostic tool for prostate cancer in comparison with the apparent diffusion coefficient. 364 patients with suspected prostate cancer underwent multiparametric magnetic resonance imaging including diffusion tensor imaging. The anatomical structure of the prostate obtained on T2-weighted imaging was compared with the apparent diffusion coefficient and diffusion tensor imaging maps. The rest of the gland (central and peripheral regions) were used as healthy areas. The apparent diffusion coefficient at diffusion-weighted imaging, fractional anisotropy and mean diffusivity at diffusion tensor imaging were evaluated in pathological zones. Cancer-suspicious areas of the prostate had high fractional anisotropy fractional anisotropy and low mean diffusivity compared to unaltered areas. Fractional anisotropy values were significantly elevated in central gland cancer, compared to normal tissue, and slightly elevated in peripheral zone cancer. Diffusion tensor imaging has the potential to identify prostate cancer with high accuracy and specificity. The combination of standard magnetic resonance imaging and diffusion tensor imaging can significantly improve the prognosis of the disease during active surveillance. The fractional anisotropy and mean diffusivity values can be useful in assessing the grade of malignancy and the radiolopathological correlation of the lesion.

Derivation, characterization, and application of complete orthonormal sequences for representing general three-dimensional states of residual stress

Residual stresses are self-equilibrated stresses on unloaded bodies. Owing to their complex origins, it is useful to develop functions that can be linearly combined to represent any sufficiently regular residual stress field. In this work, we develop orthonormal sequences that span the set of all square-integrable residual stress fields on a given three-dimensional region. These sequences are obtained by extremizing the most general quadratic, positive-definite functional of the stress gradient on the set of all sufficiently regular residual stress fields subject to a prescribed normalization condition; each such functional yields a sequence. For the special case where the sixth-order coefficient tensor in the functional is homogeneous and isotropic and the fourth-order coefficient tensor in the normalization condition is proportional to the identity tensor, we obtain a three-parameter subfamily of sequences. Upon a suitable parameter normalization, we find that the viable parameter space corresponds to a semi-infinite strip. For a further specialized spherically symmetric case, we obtain analytical expressions for the sequences and the associated Lagrange multipliers. Remarkably, these sequences change little across the entire parameter strip. To illustrate the applicability of our theoretical findings, we employ three such spherically symmetric sequences to accurately approximate two standard residual stress fields. Our work opens avenues for future exploration into the implications of different sequences, achieved by altering both the spatial distribution and the material symmetry class of the coefficient tensors, toward specific objectives.

Spatially-Periodic Solutions for Evolution Anisotropic Variable-Coefficient Navier–Stokes Equations: I. Weak Solution Existence

We consider evolution (non-stationary) spatially-periodic solutions to the n-dimensional non-linear Navier–Stokes equations of anisotropic fluids with the viscosity coefficient tensor variable in spatial coordinates and time and satisfying the relaxed ellipticity condition. Employing the Galerkin algorithm with the basis constituted by the eigenfunctions of the periodic Bessel-potential operator, we prove the existence of a global weak solution.

TENSOR QUANTILE REGRESSION WITH LOW-RANK TENSOR TRAIN ESTIMATION.

Neuroimaging studies often involve predicting a scalar outcome from an array of images collectively called tensor. The use of magnetic resonance imaging (MRI) provides a unique opportunity to investigate the structures of the brain. To learn the association between MRI images and human intelligence, we formulate a scalar-on-image quantile regression framework. However, the high dimensionality of the tensor makes estimating the coefficients for all elements computationally challenging. To address this, we propose a low-rank coefficient array estimation algorithm based on tensor train (TT) decomposition which we demonstrate can effectively reduce the dimensionality of the coefficient tensor to a feasible level while ensuring adequacy to the data. Our method is more stable and efficient compared to the commonly used, Canonic Polyadic rank approximation-based method. We also propose a generalized Lasso penalty on the coefficient tensor to take advantage of the spatial structure of the tensor, further reduce the dimensionality of the coefficient tensor, and improve the interpretability of the model. The consistency and asymptotic normality of the TT estimator are established under some mild conditions on the covariates and random errors in quantile regression models. The rate of convergence is obtained with regularization under the total variation penalty. Extensive numerical studies, including both synthetic and real MRI imaging data, are conducted to examine the empirical performance of the proposed method and its competitors.

Nonlinear Schrödinger equation for integrated photonics

The foundations of nonlinear optics are revisited, and the formalism is applied to waveguide modes. The effects of loss and dispersion are included rigorously along with the vectorial nature of the modes, and a full derivation of a new version of the nonlinear Schrödinger (NLS) equation is presented. This leads to more general expressions for the group index, for the group-index dispersion (GVD), and for the Kerr coefficient. These quantities are essential for the design of waveguides suitable for, e.g., the generation of optical frequency combs and all-optical switches. Examples are given using the silicon nitride material platform. Specifically, values are extracted for the coefficients of the chi-3 tensor based on measurements of Kerr coefficients and mode simulations.

Dequantizing quantum machine learning models using tensor networks

Ascertaining whether a classical model can efficiently replace a given quantum model——is crucial in assessing the true potential of quantum algorithms. In this work, we introduced the dequantizability of the function class of variational quantum-machine-learning (VQML) models by employing the tensor network formalism, effectively identifying every VQML model as a subclass of matrix product state (MPS) model characterized by constrained coefficient MPS and tensor product-based feature maps. From this formalism, we identify the conditions for which a VQML model's function class is dequantizable or not. Furthermore, we introduce an efficient quantum kernel-induced classical kernel which is as expressive as given any quantum kernel, hinting at a possible way to dequantize quantum kernel methods. This presents a thorough analysis of VQML models and demonstrates the versatility of our tensor-network formalism to properly distinguish VQML models according to their genuine quantum characteristics, thereby unifying classical and quantum machine-learning models within a single framework. Published by the American Physical Society 2024

Nonlinear optical, dielectric, and piezoelectric properties of hexagonal fluorocarbonates ABCO3F (A: K, Rb, Cs; B: Mg, Ca, Sr, Zn, Cd, Pb) from first principles

Within the density functional theory, the Hartree–Fock/Kohn–Sham-coupled perturbation method using the basis of localized orbitals, the gradient PBE functional with dispersion correction D3(BJ) and hybrid PBE0 and B3LYP functionals, the structural, dielectric, and piezoelectric properties of hexagonal alkali fluorocarbonates ABCO3F (A: K, Rb, Cs; B: alkaline earth (Mg, Ca, Sr) and [Formula: see text] (Zn, Cd, Pb)) were evaluated. In the series of crystals, fluorocarbonates with lead have the highest values of the calculated components of the dielectric constant tensor, piezoelectric constants, and nonlinear optical (NLO) coefficients.