Isotropic Function Research Articles

The Role of the Second Law of Thermodynamics in Continuum Physics: A Muschik and Ehrentraut Theorem Revisited

In continuum physics, constitutive equations model the material properties of physical systems. In those equations, material symmetry is taken into account by applying suitable representation theorems for symmetric and/or isotropic functions. Such mathematical representations must be in accordance with the second law of thermodynamics, which imposes that, in any thermodynamic process, the entropy production must be nonnegative. This requirement is fulfilled by assigning the constitutive equations in a form that guaranties that second law of thermodynamics is satisfied along arbitrary processes. Such an approach, in practice regards the second law of thermodynamics as a restriction on the constitutive equations, which must guarantee that any solution of the balance laws also satisfy the entropy inequality. This is a useful operative assumption, but not a consequence of general physical laws. Indeed, a different point of view, which regards the second law of thermodynamics as a restriction on the thermodynamic processes, i.e., on the solutions of the system of balance laws, is possible. This is tantamount to assuming that there are solutions of the balance laws that satisfy the entropy inequality, and solutions that do not satisfy it. In order to decide what is the correct approach, Muschik and Ehrentraut in 1996, postulated an amendment to the second law, which makes explicit the evident (but rather hidden) assumption that, in any point of the body, the entropy production is zero if, and only if, this point is a thermodynamic equilibrium. Then they proved that, given the amendment, the second law of thermodynamics is necessarily a restriction on the constitutive equations and not on the thermodynamic processes. In the present paper, we revisit their proof, lighting up some geometric aspects that were hidden in therein. Moreover, we propose an alternative formulation of the second law of thermodynamics, which incorporates the amendment. In this way we make this important result more intuitive and easily accessible to a wider audience.

A novel method for identifying geomechanical parameters of rock masses based on a PSO and improved GPR hybrid algorithm

In view of the shortcomings of existing artificial neural network (ANN) and support vector regression (SVR) in the application of three-dimensional displacement back analysis, Gaussian process regression (GPR) algorithm is introduced to make up for the shortcomings of existing intelligent inversion methods. In order to improve the generality of the standard GPR algorithm with single kernel function, an improved Gaussian process regression (IGPR) algorithm with combined kernel function is proposed by adding two single kernel functions. In addition, in the training process of IGPR model, the particle swarm optimization (PSO) is combined with the IGPR model (PSO-IGPR) to optimize the parameters of the IGPR model. After the IGPR model can accurately map the relationship between geomechanical parameters and rock mass deformation, the PSO algorithm is directly used to search the best geomechanical parameters to match the deformation calculated by igpr model with the measured deformation of rock mass. The application case of Beikou tunnel shows that the combined kernel function GPR has higher identification accuracy than the single kernel function GPR and SVR model, the IGPR model with automatic correlation determination (ARD) kernel function can obtain higher identification accuracy than the IGPR model with isotropic (ISO) kernel function, and the PSO-IGPR hybrid model based on ARD kernel function has the highest identification accuracy. Therefore, this paper proposes a displacement back analysis method of the PSO-IGPR hybrid algorithm based on ARD kernel function, which can be used to identify the geomechanical parameters of rock mass and solve other engineering problems.

Necessary and sufficient conditions for asymptotically optimal linear prediction of random fields on compact metric spaces

Optimal linear prediction (aka. kriging) of a random field {Z(x)}x∈X indexed by a compact metric space (X,dX) can be obtained if the mean value function m:X→R and the covariance function ϱ:X×X→R of Z are known. We consider the problem of predicting the value of Z(x∗) at some location x∗∈X based on observations at locations {xj}j=1n, which accumulate at x∗ as n→∞ (or, more generally, predicting φ(Z) based on {φj(Z)}j=1n for linear functionals φ,φ1,…,φn). Our main result characterizes the asymptotic performance of linear predictors (as n increases) based on an incorrect second-order structure (m˜,ϱ˜), without any restrictive assumptions on ϱ, ϱ˜ such as stationarity. We, for the first time, provide necessary and sufficient conditions on (m˜,ϱ˜) for asymptotic optimality of the corresponding linear predictor holding uniformly with respect to φ. These general results are illustrated by weakly stationary random fields on X⊂Rd with Matérn or periodic covariance functions, and on the sphere X= S2 for the case of two isotropic covariance functions.

Integral representations of isotropic semiclassical functions and applications

In a previous paper, we introduced a class of “semiclassical functions of isotropic type,” starting with a model case and applying Fourier integral operators associated with canonical transformations. These functions are a substantial generalization of the “oscillatory functions of Lagrangian type” that have played major role in semiclassical and microlocal analysis. In this paper we exhibit more clearly the nature of these isotropic functions by obtaining oscillatory integral expressions for them. Then, we use these to prove that the classes of isotropic functions are equivariant with respect to the action of general FIOs (under the usual cleanintersection hypothesis). The simplest examples of isotropic states are the “coherent states,” a class of oscillatory functions that has played a pivotal role in mathematics and theoretical physics beginning with their introduction by of Schrödinger in the 1920’s. We prove that every oscillatory function of isotropic type can be expressed as a superposition of coherent states, and examine some implications of that fact.We also show that certain functions of elliptic operators have isotropic functions for Schwartz kernels. This lead us to a result on an eigenvalue counting function that appears to be new (Corollary 4.5).

AnMas: Anisotropic strength domain for masonry

A new orthotropic limit function is developed to describe the failure of masonry under in-plane loads. The formulation starts from the isotropic smooth 3D yield function proposed by Bigoni and Piccolroaz (2004) to provide a strength condition for masonry through a single analytical function including orthotropy and different tensile and compressive strengths. The material parameters are calibrated by means of a simple optimization procedure based on available experimental and numerical data. Finally, the limit function is implemented in a finite element code in the framework of a perfect plasticity, through a numerical procedure based on a predictor–corrector algorithm. The effectiveness of proposed model is verified through the comparison of numerical simulations with experimental data available in the literature.

The Kinetic Expansion of Solar-wind Electrons: Transport Theory and Predictions for the Very Inner Heliosphere

Abstract We propose a transport theory for the kinetic evolution of solar-wind electrons in the heliosphere. We derive a gyro-averaged kinetic transport equation that accounts for the spherical expansion of the solar wind and the geometry of the Parker spiral magnetic field. To solve our three-dimensional kinetic equation, we develop a mathematical approach that combines the Crank–Nicolson scheme in velocity space and a finite-difference Euler scheme in configuration space. We initialize our model with isotropic electron distribution functions and calculate the kinetic expansion at heliocentric distances from 5 to 20 solar radii. In our kinetic model, the electrons evolve mainly through the combination of ballistic particle streaming, the magnetic mirror force, and the electric field. By applying fits to our numerical results, we quantify the parameters of the electron strahl and the core part of the electron velocity distributions. The strahl fit parameters show that the density of the electron strahl is around 7% of the total electron density at a distance of 20 solar radii, the strahl bulk velocity and strahl temperature parallel to the background magnetic field stay approximately constant beyond a distance of 15 solar radii, and β ∥s (i.e., the ratio of the strahl parallel thermal pressure to the magnetic pressure) is approximately constant with heliocentric distance at a value of about 0.02. We compare our results with data measured by the Parker Solar Probe. Furthermore, we provide theoretical evidence that the electron strahl is not scattered by the oblique fast-magnetosonic/whistler instability in the near-Sun environment.

A rank-one convex, nonpolyconvex isotropic function on with compact connected sublevel sets

According to a 2002 theorem by Cardaliaguet and Tahraoui, an isotropic, compact and connected subset of the group $\textrm {GL}^{\!+}(2)$ of invertible $2\times 2$ - - matrices is rank-one convex if and only if it is polyconvex. In a 2005 Journal of Convex Analysis article by Alexander Mielke, it has been conjectured that the equivalence of rank-one convexity and polyconvexity holds for isotropic functions on $\textrm {GL}^{\!+}(2)$ as well, provided their sublevel sets satisfy the corresponding requirements. We negatively answer this conjecture by giving an explicit example of a function $W\colon \textrm {GL}^{\!+}(2)\to \mathbb {R}$ which is not polyconvex, but rank-one convex as well as isotropic with compact and connected sublevel sets.

Anisotropic Phase Stretch Transform-Based Algorithm for Segmentation of Activated Sludge Phase-Contrast Microscopic Image

The activated sludge (AS) process is a biological treatment process of wastewater used in sewage treatment plants, in which settling of AS is vitally important for wastewater treatment. In AS, however, bulking caused by filamentous bacteria will significantly reduce the settling capacity of activated sludge. Traditionally, the physicochemical method has been used to monitor the status or performance of filamentous bacteria in activated sludge, while it is a very compromising means in modern digital quality control for wastewater treatment when digital image processing and analysis technology is used to determine the status of filamentous bacteria in activated sludge. In order to avoid the disadvantages of deficient direction selectivity in the translation vector field for isotropic phase stretch transform (PST) and high noise susceptibility in determining the status of filamentous bacteria in AS and low accuracy, an anisotropic PST (APST) method for filamentous bacteria test in AS was proposed in this paper; specifically speaking, by analyzing and deriving the disadvantages of the traditional isotropic PST kernel function, we designed an APST kernel function, put forward a non-maximum suppression strategy for processing of images from different aspects and combined with the relative total variation theory to form APST-based algorithm for segmentation of activated sludge phase-contrast microscopic image. According to a large number of experimental results, in terms of the overall segmentation effect or the subjective and objective evaluation indicators, the resultant algorithm in this paper is superior to the latest traditional PST segmentation algorithm, Canny image segmentation algorithm and Sobel image segmentation algorithm. Also, the segmentation results for filamentous bacteria showed that probability rand index (PRI) and global consistency error (GCE) indicators were all improved by about 30%, supporting the effectiveness and superiority of the algorithm in this paper.

The power loss requirement in the non-thermal ³He-³He fusion plasma

The control of power loss is the most important issue in the plasma and it should be minimised by suitable selection of design parameters. This research is done with the aim of determining power loss and the parameters affecting the ³He-³He plasma with isotropic non-Maxwellian electron velocity distribution function (EVDF). In order to investigate the power loss requirement, bi-Maxwellian EVDF is considered. The results show that the rate of entropy production decreases as the electron density decreases and collision time increases. The ratio of the minimum power loss to the fusion power decreases as the electron temperature, and electron and helium-3 densities decrease, and the ion temperature increases. It is favourable to decrease the electron temperature lower than the thermal limit. Moreover, decreasing the electron temperature increases the fusion power.

The influence of phonon boundary scattering on the thermal conductivity of a two-dimensional noninteractive phonon system of nanosized structures

One of the problems that arise when studying the thermal conductivity of low-dimensional phonon systems at low temperatures is the appearance of differences in expressions for the thermal conductivity as a function of sample size, as well as the appearance of unusual dependences of heat fluxes on temperature gradients. For example, in the generally accepted Casimir – Zaiman model, it is assumed that a linear temperature gradient is created on the lateral surface by external sources. Moreover, the Casimir model requires two conditions at the border. This is a diffuse reflection in which the phonon is reflected with an isotropic angular distribution function. The second condition is the presence of redistribution of phonons by energy, so that the distribution of reflected phonons corresponds to the radiation of an absolutely black body - that is, the reflection of phonons must be inelastic. And if the first condition can be achieved, for example, by boundaries with a certain degree of roughness, the second condition can be achieved only in the presence of thermal contact between the side edges of the sample and the thermal medium at a certain temperature distribution. In the case of thermally insulated sample boundaries (for example, when the sample is in vacuum) or at least with imperfect thermal contact, the fulfillment of the second condition is practically impossible.In this paper, we consider the problem of thermal conductivity of two-dimensional nanostructures - nanobands - in the temperature range, when the interaction between phonons can be neglected. In this ballistic mode, heat fluxes can be limited only by the interaction of phonons with the sample boundaries. A number of types of interaction of phonons with the boundaries of two-dimensional samples are considered: absorption at the boundary, finite number of reflections, absorption inside the sample on defects, impurities, etc. Explicit expressions of thermal conductivity in these cases are derived. Interpolation relations are obtained, which generalize the existing expressions of thermal conductivity in the case of mirror reflection and reflection with losses.

Fractional description of creep behavior for fiber reinforced concrete: Simulation and parameter study

Fiber reinforced concrete (FRC) exhibits enhanced strength and excellent durability, and the creep behaviors of FRC are major concerns for its application in structural analysis and design process. In this work, a novel fractional creep model for FRC is developed by introducing the fractional flow rule into the classical elastoplastic constitutive model. Specifically, a hierarchical single-surface (HISS) type yield function is adopted and an isotropic hardening function is introduced as hardening variables. Without using plastic potential, the proposed model exhibits higher accuracy than several conventional creep models with relatively simple form which is validated by comparing the numerical predictions with experimental results from short-term and long-term creep tests. Furthermore, a parameter study on the fractional order suggests that the order can adjust the magnitude and rate of the creep deformation, and the consistency between macroscopic deformation and the material formulation of FRC could be reflected by the tendency of the fractional order.

Tempered fractional LES modeling

The presence of non-local interactions and intermittent signals in the homogeneous isotropic turbulence grant multi-point statistical functions a key role in formulating a new generation of large-eddy simulation (LES) models of higher fidelity. We establish a tempered fractional-order modelling framework for developing non-local LES subgrid-scale models, starting from the kinetic transport. We employ a tempered Lévy-stable distribution to represent the source of turbulent effects at the kinetic level, and we rigorously show that the corresponding turbulence closure term emerges as the tempered fractional Laplacian, $(\varDelta +\lambda )^{\alpha } (\cdot )$ , for $\alpha \in (0,1)$ , $\alpha \neq \frac {1}{2}$ and $\lambda >0$ in the filtered Navier–Stokes equations. Moreover, we prove the frame invariant properties of the proposed model, complying with the subgrid-scale stresses. To characterize the optimum values of model parameters and infer the enhanced efficiency of the tempered fractional subgrid-scale model, we develop a robust algorithm, involving two-point structure functions and conventional correlation coefficients. In an a priori statistical study, we evaluate the capabilities of the developed model in fulfilling the closed essential requirements, obtained for a weaker sense of the ideal LES model (Meneveau, Phys. Fluids, vol. 6, issue 2, 1994, pp. 815–833). Finally, the model undergoes the a posteriori analysis to ensure the numerical stability and pragmatic efficiency of the model.

The isotropic energy function and formation rate of short gamma-ray bursts

Gamma-ray bursts (GRBs) are brief, intense, gamma-ray flashes in the universe, lasting from a few milliseconds to a few thousand seconds. For short gamma-ray bursts (sGRBs) with duration less than 2 seconds, the isotropic energy (E iso) function may be more scientifically meaningful and accurately measured than the luminosity (L p) function. In this work we construct, for the first time, the isotropic energy function of sGRBs and estimate their formation rate. First, we derive the L p – E p correlation using 22 sGRBs with known redshifts and well-measured spectra and estimate the pseduo redshifts of 334 Fermi sGRBs. Then, we adopt the Lynden-Bell c − method to study isotropic energy functions and formation rate of sGRBs without any assumption. A strong evolution of isotropic energy E iso ∝ (1+z)5.79 is found, which is comparable to that between L p and z. After removing effect of the cosmic evolution, the isotropic energy function can be reasonably fitted by a broken power law, which is for dim sGRBs and for bright sGRBs, with the break energy 4.92 × 1049 erg. We obtain the local formation rate of sGRBs is about 17.43 events Gpc−3 yr−1. If assuming a beaming angle is 6° to 26°, the local formation rate including off-axis sGRBs is estimated as ρ 0,all = 155.79 – 3202.35 events Gpc−3 yr−1.

Patch-Based U-Net Model for Isotropic Quantitative Differential Phase Contrast Imaging.

Quantitative differential phase-contrast (qDPC) imaging is a label-free phase retrieval method for weak phase objects using asymmetric illumination. However, qDPC imaging with fewer intensity measurements leads to anisotropic phase distribution in reconstructed images. In order to obtain isotropic phase transfer function, multiple measurements are required; thus, it is a time-consuming process. Here, we propose the feasibility of using deep learning (DL) method for isotropic qDPC microscopy from the least number of measurements. We utilize a commonly used convolutional neural network namely U-net architecture, trained to generate 12-axis isotropic reconstructed cell images (i.e. output) from 1-axis anisotropic cell images (i.e. input). To further extend the number of images for training, the U-net model is trained with a patch-wise approach. In this work, seven different types of living cell images were used for training, validation, and testing datasets. The results obtained from testing datasets show that our proposed DL-based method generates 1-axis qDPC images of similar accuracy to 12-axis measurements. The quantitative phase value in the region of interest is recovered from 66% up to 97%, compared to ground-truth values, providing solid evidence for improved phase uniformity, as well as retrieved missing spatial frequencies in 1-axis reconstructed images. In addition, results from our model are compared with paired and unpaired CycleGANs. Higher PSNR and SSIM values show the advantage of using the U-net model for isotropic qDPC microscopy. The proposed DL-based method may help in performing high-resolution quantitative studies for cell biology.

Encore: an O (Ng2) estimator for galaxy N-point correlation functions

ABSTRACT We present a new algorithm for efficiently computing the N-point correlation functions (NPCFs) of a 3D density field for arbitrary N. This can be applied both to a discrete spectroscopic galaxy survey and a continuous field. By expanding the statistics in a separable basis of isotropic functions built from spherical harmonics, the NPCFs can be estimated by counting pairs of particles in space, leading to an algorithm with complexity $\mathcal {O}(N_\mathrm{g}^2)$ for Ng particles, or $\mathcal {O}(N_\mathrm{FFT}\log N_\mathrm{FFT})$ when using a Fast Fourier Transform with NFFT grid-points. In practice, the rate-limiting step for N > 3 will often be the summation of the histogrammed spherical harmonic coefficients, particularly if the number of radial and angular bins is large. In this case, the algorithm scales linearly with Ng. The approach is implemented in the encore code, which can compute the 3PCF, 4PCF, 5PCF, and 6PCF of a BOSS-like galaxy survey in ${\sim}100$ CPU-hours, including the corrections necessary for non-uniform survey geometries. We discuss the implementation in depth, along with its GPU acceleration, and provide practical demonstration on realistic galaxy catalogues. Our approach can be straightforwardly applied to current and future data sets to unlock the potential of constraining cosmology from the higher point functions.

An invariant-based elasto-visco-plastic model for unidirectional polymer composites at finite strains

An invariant-based constitutive model for unidirectional composites, including viscous effects in the elastic and plastic regimes at finite strains, is proposed. The model is based on the multiplicative decomposition of the deformation gradient, using an isoclinic configuration to avoid the non-uniqueness of intermediate configurations. The visco-elastic behaviour is represented by the generalised Maxwell model, combined with a transversely isotropic yield function and a non-associative plastic potential. Visco-plastic effects are introduced through the Perzyna overstress function. The theoretical basis of the material model and the computational treatment within the finite element solution procedure are presented in detail. Two algorithms are proposed for an implicit integration of the evolution equations: (i) a semi-implicit stress update, and (ii) a fixed-point iterative fully implicit stress update. A comparison of their performance reveals that the former becomes faster, even though the latter guarantees the quadratic convergence rate in the Newton–Raphson scheme. The predicted stress–strain response for different fibre orientation angles is in close agreement with experimental data obtained under quasi-static and dynamic loading. The model can also capture the important re-orientation of the fibres due to external loading.

Time-dependent lepto-hadronic modelling of the emission from blazar jets with SOPRANO: the case of TXS 0506 + 056, 3HSP J095507.9 + 355101, and 3C 279

ABSTRACT The observation of a very high-energy neutrino by IceCube (IceCube-170922A) and its association with the flaring blazar TXS 0506 + 056 provided the first multimessenger observations of blazar jets, demonstrating the important role of protons in their dynamics and emission. In this paper, we present SOPRANO (https://www.amsdc.am/soprano), a new conservative implicit kinetic code that follows the time evolution of the isotropic distribution functions of protons, neutrons, and the secondaries produced in photo-pion and photo-pair interactions, alongside with the evolution of photon and electron/positron distribution functions. SOPRANO is designed to study leptonic and hadronic processes in relativistic sources such as blazars and gamma-ray bursts. Here, we use SOPRANO to model the broadband spectrum of TXS 0506 + 056 and 3HSP J095507.9 + 355101, which are associated with neutrino events, and of the extreme flaring blazar 3C 279. The SEDs are interpreted within the guise of both a hadronic and a hybrid model. We discuss the implications of our assumptions in terms of jet power and neutrino flux.

A 3D BEM-FEM approach using layered transversely isotropic half-space Green's functions in the frequency domain for SSI analyses

In this paper, a numerical approach based on a three-dimensional (3D) standard coupled boundary element method-finite element method (BEM-FEM) formulation in the frequency domain is presented. The approach allows studying the dynamics response of a structure bonded to the surface of a layered transversely isotropic half-space subjected to time-harmonic loading. The FEM is used to model the structure, and the BEM is employed for the soil, using layered half-space Green's functions, by which the radiation condition is implicitly satisfied. The BEM procedure is improved by implementing exponentially gradient (EG) elements to capture the singular behavior of tractions at the edges and corners of the soil-structure interface. The proposed approach is verified by comparison with existing semi-analytical and numerical results. Afterward, the dynamic response of a building on a layered transversely isotropic half-space subjected to lateral loading is studied, and the effect of soil layering on the response of the structure and interface tractions is demonstrated.

Limit load instabilities of anisotropic tubes under combined tension and torsion

Under tensile loadings, thin-walled structures such as sheet metal and tubes develop load maxima, or limit loads, beyond which deformation localizes leading to rupture. For uniform stress states, Considère-type estimates of limit loads can serve as forming limits in applications. The paper explores the effect of anisotropy on such estimates for thin-walled Al-alloy tubes under combined tension and torsion. Anisotropy is modeled using Yld04-3D with an exponent of 8. Material hardening originates from a simple shear test using this yield function, taking into account material axes rotation caused by the shearing. A Considère formulation is developed for the problem, which also incorporates the effect of material frame rotation. The analysis is used to establish limit loads for a set of circumferentially constrained tension–torsion experiments tested under radial nominal tension-shear stress paths. The predictions reproduce the strains measured at the limit loads for the range of biaxiality ratios considered. By contrast, corresponding results produced using the isotropic yield functions of von Mises and Hosford(8) increasingly deviate from the measured results as the shear stress increases. Considère-type formulation is also developed for the same tension–torsion loadings for a uniform thickness tube. The results exhibit a similar trend but the limit strains for shear dominant paths are significantly lower, demonstrating the stabilizing effect of the circumferential constraint used in the experiments.

Generalized string-net models: A thorough exposition

We describe how to construct generalized string-net models, a class of exactly solvable lattice models that realize a large family of 2D topologically ordered phases of matter. The ground states of these models can be thought of as superpositions of different "string-net configurations", where each string-net configuration is a trivalent graph with labeled edges, drawn in the $xy$ plane. What makes this construction more general than the original string-net construction is that, unlike the original construction, tetrahedral reflection symmetry is not assumed, nor is it assumed that the ground state wave function $\Phi$ is "isotropic": i.e. in the generalized setup, two string-net configurations $X_1, X_2$ that can be continuously deformed into one another can have different ground state amplitudes, $\Phi(X_1) \neq \Phi(X_2)$. As a result, generalized string-net models can realize topological phases that are inaccessible to the original construction. In this paper, we provide a more detailed discussion of ground state wave functions, Hamiltonians, and minimal self-consistency conditions for generalized string-net models than what exists in the previous literature. We also show how to construct string operators that create anyon excitations in these models, and we show how to compute the braiding statistics of these excitations. Finally, we derive necessary and sufficient conditions for generalized string-net models to have isotropic ground state wave functions on the plane or the sphere -- a property that may be useful in some applications.