Tensor Functions Research Articles

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.

Mathematical Model for Excited State Fluid Dynamics

The rational mechanic’s research method is synthesized with the help of mathematical means such as generalized function analysis and tensor analysis. The basic definition of the fluctuation velocity generation is based on phenomenological physics in this article. The basic control equations of general excited state fluid dynamics applicable to the flowfield are obtained based on the basic principles of quantum mechanical superposition states. The simplified basic equations of the excited State are finally obtained through time and space discretization. The basic theory of excited state fluid dynamics is established, providing new ideas for the innovation and application of flow control, fluid mechanical engineering design, and other aspects of research work.

Curvature effects on the structure of near-wall turbulence

The interaction between near-wall turbulence and wall curvature is described for the incompressible flow in a plane channel with a small concave–convex–concave bump on the bottom wall, with height comparable to the wall-normal location of the main turbulent structures. The analysis starts from a database generated by a direct numerical simulation and hinges upon the anisotropic generalised Kolmogorov equations, i.e. the exact budget equations for the second-order structure function tensor. The influence of the bump on the wall cycle and on the energy production, redistribution and transfers is described in the physical and scale spaces. Over the upstream side of the bump, the energy drained from the mean flow to sustain the streamwise fluctuations decreases, and the streaks of high and low streamwise velocity weaken and are stretched spanwise. After the bump tip, instead, the production of streamwise fluctuations grows and the streaks intensify, progressively recovering their characteristic spanwise scale. The wall-normal fluctuations, and thus the quasi-streamwise vortices, are sustained by the mean flow over the upstream side of the bump, while energy flows from the vertical fluctuations to the mean field over the downstream side. On the concave portion of the upstream side, the near-wall fluctuations form structures of spanwise velocity which are consistent with Taylor–Görtler vortices at an early stage of development. Their evolution is described by analysing the scale-space pressure–strain term. A schematic description of the bump flow is presented, in which various regions are identified according to the signs of curvature and streamwise pressure gradient.

Non-Local Q-Tensor Approach for Description of Elastic Deformations of Nematic Liquid Crystals at Sub-Micron Scale

Within the framework of phenomenological approach, a generalized concept of a non-local order parameter, that have the form of a traceless correlation tensor function or a tensor integral operator, is introduced. The main relationships, which determine the equilibrium orientational states and the phase transitions of a deformed nematic liquid crystal, are described. The resulting expression for the free energy is quadratic with respect to the gradients of the order parameter tensor. The limiting transition of the non-local order parameter to the local one leads to an expression that complements the classical Oseen–Frank theory of elasticity. The considered model includes eight independent elastic constants. Three of them relate to "bulk" constants analogous to Frank constants, another three ones relate to "surface" constants, and the remaining two constants determine the anisotropic effect of the gradient of the scalar order parameter on the director field. The constants are necessary for the consideration of a liquid crystal state near defects, as well as for the description of orientation effects caused by the inhomogeneity of order parameter. It is shown that in the free energy expression, the "surface" terms contribute to the director orientation in the case of a non-constant scalar order parameter, even in the approximation of rigid boundary conditions.

Temperature‐dependent electrical conductivity impact on radiative and dissipative peristaltic transport of boron nitride‐ethylene glycol nanofluid through asymmetric channels

AbstractMathematical simulation of biological fluids is of upmost significance due to its numerous medical uses. Interpreting various biological flows necessitates a thorough knowledge of the peristaltic mechanism. This paper presents a computational study for the peristaltic pumping within vertical asymmetric channels filled with BN‐EG nanofluid under the influence of temperature‐dependent electrical conductivity and thermal radiation. Experimental study showed that the nanofluid created by suspending Boron Nitride particles in a combination of Ethylene Glycol exhibited non‐Newtonian characteristics. Further, the Carreau's fluid model provides accurate predictions about the rheological properties of BN‐EG nanofluid. Various configurations of the outer boundaries are considered, namely, square wave, multi‐sinusoidal wave, trapezoidal wave, and triangular wave. A uniform magnetic field together with nanoparticles and mass concentrations, joule heating, first‐order chemical reaction as well as viscous dissipation are considered. Influences of the Dufour and Soret numbers are examined, and the cases of biological scientific assumptions which is known as low Reynolds number and long wavelength are applied. All the computations are obtained numerically using Mathematica symbolical software (ND‐Solve), and the obtained results are presented in terms of the axial velocity u, heat transfer rate Z, concentration profile Ω, temperature profile θ, extra stress tensor , pressure gradient , pressure rise and stream function ψ. The major outcomes revealed that the maximizing in electrical conductivity coefficient, variable viscosity coefficient and magnetic field parameter is better to obtain a higher rate of the heat transfer while the increase in thermo‐diffusion effects as well as linear thermal radiation coefficient causes a reduction in the rate of heat transfer.

Korteweg-Type Fluids and Thermodynamic Modelling via Higher-Order Gradients

This paper investigates the modelling of Korteweg-type fluids and hence the dependence of the stress tensor on gradients of mass density. This topic, originating from the need for describing capillarity effects, is mainly of interest in connection with nanosystems where the mean free path may be comparable with the geometric dimensions of the system. In addition to the Korteweg fluid model, the paper gives a review of the stress tensor function arising in quantum fluid hydrodynamics. Next, thermodynamic consistency is established for a fluid involving first- and second-order density gradients. The modelling investigated is a generalization of the classical Korteweg fluid and allows a better understanding of previous thermodynamic restrictions. The restrictions determined for the general scheme with second-order gradients are applied to the particular cases of the Korteweg fluid and the quantum fluid. Further, to allow for discontinuity wave solutions with finite speed of propagation, a model is established which involves higher-order derivatives and reduces to the Korteweg fluid in stationary conditions.

Divergence-free finite element spaces for stress tensors

We construct finite element spaces of symmetric stress tensors that are exactly divergence-free. Moreover, their basis functions can be chosen so that they have small supports. These properties are highly desired in a number of important applications. Approximation properties of finite element spaces of divergence-free tensor functions are derived from properties of C1 finite elements.

Bootstrapping string dynamics in the 6d \U0001d4a9 = (2, 0) theories

We present two complementary approaches to calculating the 2-point function of stress tensors in the presence of a 1/2 BPS surface defect of the 6d \U0001d4a9 = (2, 0) theories. First, we use analytical bootstrap techniques at large N to obtain the first nontrivial correction to this correlator, from which we extract the defect CFT (dCFT) data characterising the 2d dCFT of the 1/2 BPS plane. Along the way we derive a supersymmetric inversion formula, obtain the relevant superconformal blocks and check that crossing symmetry is satisfied. Notably our result features a holomorphic function whose appearance is related to the chiral algebra construction of Beem, Rastelli and van Rees. Second, we use that chiral algebra description to obtain exact results for the BPS sector of the dCFT, valid at any N and for any choice of surface operator. These results provide a window into the dynamics of strings of the mysterious 6d theories.

The structure of the radiation field of a symmetric slot line perpendicular to the edge of an infinite half-plane, taking into account the cross-polarization component

Background. Symmetric slot lines on a perfectly reflecting half-plane are good mathematical models for slot antennas on an ideal plane of finite dimensions, which in practice have a high gain and operate over a wide frequency band. These antennas are used in ultra-fast information processing systems based on microwave and optical volume integrated circuits. Aim. The work presents a solution to the problem of radiation of a perfectly reflecting half-plane, which was excited by a symmetric slot line with a different law of slot expansion. Methods. The problem was solved by the Fourier variable separation method, by which the Green tensor function was obtained, which connects the two components of the antenna radiation field with the excitation field in the gap. Results. It is shown that in the far zone, the radiation field of an ideally reflecting half-plane has two components: the main and crosspolarization. The Green tensor function has a complex form and consists of two terms. The analysis of the Green tensor function shows the participation of each term in the formation of a field in the far zone, demonstrates the mechanism of radiation and leads to known special cases for slot radiators. Conclusion. The validity of the presented mathematical model is confirmed by experimental research.

Anisotropic excitonic photocurrent in β−Ga2O3

Polarization-dependent photocurrent spectra are measured on a (001) $\ensuremath{\beta}\text{\ensuremath{-}}{\mathrm{Ga}}_{2}{\mathrm{O}}_{3}$ Schottky photodetector, where the linear polarization of light is rotated within the ab plane. Three spectral peaks at 4.92, 5.15, and 5.44 eV are observed that vary in intensity with the optical polarization direction. The peak transition energies are consistent with excitons previously reported in $\ensuremath{\beta}\text{\ensuremath{-}}{\mathrm{Ga}}_{2}{\mathrm{O}}_{3}$ due to interband transitions modified by the valence-band $p$-orbital anisotropy and the electron-hole Coulombic attraction. The measured polarization dependence of the photocurrent matches our predictions based on electromagnetic simulations of anisotropic absorption using the complex dielectric function tensor extracted from previous ellipsometry studies. These results illustrate the dominance of excitonic absorption and photocurrent in $\ensuremath{\beta}\text{\ensuremath{-}}{\mathrm{Ga}}_{2}{\mathrm{O}}_{3}$ both below and above the band gap, demonstrate a combined theoretical/experimental understanding of anisotropic photocarrier generation, and validate previous atomistic band-structure calculations in this low-symmetry ultrawide-band-gap semiconductor.

On the Kelvin–Voigt model in anisotropic viscoelasticity

We propose an anisotropic and nonlinear generalization of the Kelvin–Voigt viscoelastic model obtained considering the additive splitting of the Cauchy stress tensor in an elastic and a dissipative part. The former one corresponds to a fiber-reinforced hyperelastic material while the dissipative effect is described by the most general linear transverse-isotropic tensorial function of symmetric part of the velocity gradient. In a such a way we characterize the dissipative contribution via three viscoelastic moduli. We then show, by a detailed analysis of the simple shear quasistatic motion and the corresponding creep phenomena, that this motion may be used to determine experimentally the viscoelastic parameters.

Tensor decomposition for modified quasi-linear viscoelastic models: Towards a fully non-linear theory

We discuss the decomposition of the tensorial relaxation function for isotropic and transversely isotropic (TI) modified quasi-linear viscoelastic (MQLV) models. We show how to formulate the constitutive equation using a convenient decomposition of the relaxation tensor into scalar components and tensorial bases. We show that the bases must be symmetrically additive, i.e., they must sum up to the symmetric fourth-order identity tensor. This is a fundamental property both for isotropic and anisotropic bases that ensures the constitutive equation is consistent with the elastic limit. We provide two robust methods to obtain such bases. Furthermore, we show that, in the TI case, the bases are naturally deformation-dependent for deformation modes that induce rotation or stretching of the fibres. Therefore, the MQLV framework allows to capture the non-linear phenomenon of strain-dependent relaxation, which has always been a criticised limitation of the original quasi-linear viscoelastic theory. We illustrate this intrinsic non-linear feature, unique to the MQLV model, with two examples (uni-axial extension and perpendicular shear).

Bootstrapping traceless symmetric $O(N)$ scalars

We use numerical bootstrap techniques to study correlation functions of traceless symmetric tensors of O(N)O(N) with two indices t_{ij}tij. We obtain upper bounds on operator dimensions for all the relevant representations and several values of NN. We discover several families of kinks, which do not correspond to any known model and we discuss possible candidates. We then specialize to the case N=4N=4, which has been conjectured to describe a phase transition in the antiferromagnetic real projective model ARP^{3}ARP3. Lattice simulations provide strong evidence for the existence of a second order phase transition, while an effective field theory approach does not predict any fixed point. We identify a set of assumptions that constrain operator dimensions to a closed region overlapping with the lattice prediction. The region is still present after pushing the numerics in the single correlator case or when considering a mixed system involving tt and the lowest dimension scalar singlet.

Structure function tensor equations with triple decomposition

Exact budget equations are derived for the coherent and stochastic contributions to the second-order structure function tensor. They extend the anisotropic generalised Kolmogorov equations (AGKE) by considering the coherent and stochastic parts of the Reynolds stress tensor, and are useful for the statistical description of turbulent flows with periodic or quasi-periodic features, like, for example, the alternate shedding after a bluff body. While the original AGKE describe production, transport, inter-component redistribution and dissipation of the Reynolds stresses in the combined space of scales and positions, the new equations, called $\varphi$ AGKE, contain the phase $\varphi$ as an additional independent variable, and describe the interplay among the mean, coherent and stochastic fields at the various phases. The newly derived $\varphi$ AGKE are then applied to a case where an exactly periodic external forcing drives the flow: a turbulent plane channel flow modified by harmonic spanwise oscillations of the wall to reduce drag. The phase-by-phase action of the oscillating transversal Stokes layer generated by the forcing on the near-wall turbulent structures is observed, and a detailed description of the scale-space interaction among mean, coherent and stochastic fields is provided thanks to the $\varphi$ AGKE.

A novel framework for in-vivo diffusion tensor distribution MRI of the human brain

Neural tissue microstructure plays an important role in developmental, physiological and pathophysiological processes. Diffusion tensor distribution (DTD) MRI helps probe subvoxel heterogeneity by describing water diffusion within a voxel using an ensemble of non-exchanging compartments characterized by a probability density function of diffusion tensors. In this study, we provide a new framework for acquiring multiple diffusion encoding (MDE) images and estimating DTD from them in the human brain in vivo. We interfused pulsed field gradients (iPFG) in a single spin echo to generate arbitrary b-tensors of rank one, two, or three without introducing concomitant gradient artifacts. Employing well-defined diffusion encoding parameters we show that iPFG retains salient features of a traditional multiple-PFG (mPFG/MDE) sequence while reducing the echo time and coherence pathway artifacts thereby extending its applications beyond DTD MRI. Our DTD is a maximum entropy tensor-variate normal distribution whose tensor random variables are constrained to be positive definite to ensure their physicality. In each voxel, the second-order mean and fourth-order covariance tensors of the DTD are estimated using a Monte Carlo method that synthesizes micro-diffusion tensors with corresponding size, shape, and orientation distributions to best fit the measured MDE images. From these tensors we obtain the spectrum of diffusion tensor ellipsoid sizes and shapes, and the microscopic orientation distribution function (μODF) and microscopic fractional anisotropy (μFA) that disentangle the underlying heterogeneity within a voxel. Using the DTD-derived μODF, we introduce a new method to perform fiber tractography capable of resolving complex fiber configurations. The results revealed microscopic anisotropy in various gray and white matter regions and skewed MD distributions in cerebellar gray matter not observed previously. DTD MRI tractography captured complex white matter fiber organization consistent with known anatomy. DTD MRI also resolved some degeneracies associated with diffusion tensor imaging (DTI) and elucidated the source of diffusion heterogeneity which may help improve the diagnosis of various neurological diseases and disorders.

Modeling, Analysis, and Computational Design of Muscle-driven Soft Robots.

Muscle driving is a critical actuation mode of soft or flexible robots and plays a key role in the motion of most animals. Although the system development of soft robots has been extensively investigated, the general kinematic modeling of soft bodies and the design methods used for muscle-driven soft robots (MDSRs) are inadequate. With a focus on homogeneous MDSRs, this article presents a framework for kinematic modeling and computational design. Based on continuum mechanics theory, the mechanical characteristics of soft bodies were first described using a deformation gradient tensor and energy density function. The discretized deformation was then depicted using a triangular meshing tool according to the piecewise linear hypothesis. Deformation models of MDSRs caused by external driving points or internal muscle units were established by the constitutive modeling of hyperelastic materials. The computational design of the MDSR was then addressed based on kinematic models and deformation analysis. Algorithms were proposed to infer the design parameters from the target deformation and to determine the optimal muscles. Several MDSRs were developed, and experiments were conducted to verify the effectiveness of the presented models and design algorithms. The computational and experimental results were compared and evaluated using a quantitative index. The presented framework of deformation modeling and computational design of MDSRs can facilitate the design of soft robots with complex deformations, such as humanoid faces.

Addendum to “fourth-order tensor calculus operations and application to continuum mechanics”

The authors provided a novel explicit expression for the dot derivative of an isotropic tensor function in the work by Kellermann et al. An alternate proof is provided in this addendum without the power series assumption.

Experimental and numerical analysis of hyperelastic plates using Mooney-Rivlin strain energy function and meshless collocation method

In this research, the flexural behavior of a hyperelastic plate made of silicone under uniformly distributed loading was investigated. The displacements and strains were formulated using the first-order shear deformation plate theory (FSDPT). The right Cauchy-Green tensor and Mooney-Rivlin strain energy function were employed to formulate hyperelastic silicone plate governing equations. The strong-form of the governing equations of the silicone plate using Mooney-Rivlin strain energy function and Euler-Lagrange relations were derived for the first time. In the governing equations, the effects of geometrical and material nonlinearity effects were considered simultaneously. The meshless collocation method (MCM) and thin-plate spline radial basis function were utilized to discretize the governing equations. The arc-length continuation algorithm was also used for analyzing the system of non-linear equations. MCM results were compared to those of the experimental test to validate the results of the meshless method. For this purpose, a square silicone plate with simply supported boundary conditions under uniformly distributed loading was investigated via the digital image correlation (DIC) technique. According to the obtained results, the MCM based on radial basis function is an adequate method for non-linear analysis of hyperelastic plates with Mooney-Rivlin strain energy function under uniformly distributed loading.

Fourth-order tensor calculus operations and application to continuum mechanics

This paper provides a presentation of differential calculus involving fourth-order tensors in consideration of the cyclic class-based tensor operations introduced in an earlier paper on fourth-order tensor algebraic operations. Three classes of cyclic second-order tensor operators, denoted by dot, cross, and star, with separate definitions for double contraction, quadruple contraction, and tensor products were recognized. The present paper examines the differentiation of second-order isotropic tensor functions with respect to a second-order tensor, taking into account the three classes of second-order tensor operators. Product rules, chain rules, and identities necessary to maintain consistency are defined. In combining the proposed calculus rules with the second-order tensor operators, a comprehensive set of proofs for the power derivatives is first presented. A closed-form solution for the second-order tensor derivative of an isotropic tensor function is also detailed. Some of the new solutions are an outcome of capabilities offered by new operators and identities provided in the literature. The paper is arranged so as to be a useful reference work with regard to the mathematics of fourth-order tensors in absolute notation, particularly for application in the field of continuum mechanics in engineering.

Segmental Abnormalities of White Matter Microstructure in Primary Hypothyroidism Identified by Automated Fiber Quantification

Introduction: Hypothyroidism leads to impaired white matter (WM) integrity, associated with cognitive/neuropsychiatric dysfunction. However, the specific segmental abnormalities of the fibers remain unexplored. Therefore, this study aimed to investigate whether the damage of the WM is limited to a specific segment or the entire bundle via diffusion metrics using automated fiber quantification. Methods: A cross-sectional study was conducted on 31 hypothyroid patients and 28 healthy controls. Thyroid-related hormone levels, cognitive/neuropsychiatric function, and diffusion tensor image data were collected and analyzed. Correlation and random forest analyses were also performed. Results: The mean fractional anisotropy (FA) values were reduced at the fiber tract level. The mean diffusivity (MD), axial diffusivity (AD), and radial diffusivity (RD) values were increased in several fiber tracts, i.e., cingulum cingulate (CC), anterior forceps of corpus callosum (CCF_A). Significant correlations were found between cognitive function and diffusion indicators such as the FA value of the left corticospinal tract and arcuate fasciculus (AF), the MD value of left CC, the RD value of left AF, the AD value of left CC, and CCF_A. The widespread microstructure disruption was spread on multiple specific segments of different tracts at the point-wise level. The random forest revealed that the accuracy of recognizing hypothyroid patients was 82.5%, with the anterior component of CCF_A having the most significant contribution. Conclusion: WM microstructural integrity impairments were found in multi-segments of the multiple fiber bundles in hypothyroidism, which might be a potential mechanism of the underlying neurocognitive decline and cerebral impairment. The CCF_A might serve as a neuro biomarker for early warning of cerebral impairment in hypothyroidism.