Second-order Field Research Articles

A conservative second order phase field model for simulation of N-phase flows

We present a novel N-phase extension to the second-order conservative phase field method, first introduced by Chiu and Lin [1]. The proposed N-phase model is in conservative form and is symmetric with respect to the phases while satisfying volume conservation. The model is reduction consistent, meaning that in the absence of M phases, the equations reduce to the equations for an N−M-phase flow. This eliminates the possibility of fictitious phases appearing. For coupling to momentum transport, we extend the two-phase mass-momentum consistent model to N-phase flows. By adopting second-order central spatial schemes, the boundedness properties of the two-phase model are inherited by the N-phase model, and the coupled mass-momentum consistent N-phase flow solver inherits the conservation properties of its two-phase version, resulting in the first N-phase flow method that analytically and discretely conserves mass, momentum and kinetic energy (in the absence of capillary and viscous effects). A novel surface tension model is proposed for modeling surface tension forces in N-phase flows. The phase field model allows for variable interface thicknesses between different phase pairs. This endows the model with the property of attaining correct equilibirum configurations at triple junction points. Specifically, we analytically demonstrate that by using interfacial thicknesses that are inversely proportional with the pairwise surface tension, the coupled system achieves steady state at the correct configuration. Using several canonical and practical numerical tests, we demonstrate the accuracy of the phase field equation, the surface tension model, and the fully coupled N-phase flow solver.

Conformal gravitational theories in Barthel–Kropina-type Finslerian geometry, and their cosmological implications

We consider dark energy models obtained from the general conformal transformation of the Kropina metric, representing an (alpha , beta )-type Finslerian geometry, constructed as the ratio of the square of a Riemannian metric alpha and the one-form beta . Conformal symmetries appear in many fields of physics, and they may play a fundamental role in our understanding of the Universe. We investigate the possibility of obtaining conformal theories of gravity in the osculating Barthel–Kropina geometric framework, where gravitation is described by an extended Finslerian-type model, with the metric tensor depending on both the base space coordinates and a vector field. We show that it is possible to formulate a family of conformal Barthel–Kropina theories in an osculating geometry with second-order field equations, depending on the properties of the conformal factor, whose presence leads to the appearance of an effective scalar field of geometric origin in the gravitational field equations. The cosmological implications of the theory are investigated in detail by assuming a specific relation between the component of the one-form of the Kropina metric and the conformal factor. The cosmological evolution is thus determined by the initial conditions of the scalar field and a free parameter of the model. We analyze in detail three cosmological models corresponding to different values of the theory parameters. Our results show that the conformal Barthel–Kropina model can provide an acceptable description of the observational data, and may represent a theoretically attractive alternative to the standard Lambda CDM cosmology.

A generally variational phase field model of fracture

Although the phase field method has been widely used in the field of fracture, various models exist, such as the second-order and fourth-order phase field model in the field of fracture mechanics, and the phase field evolution model in the field of physics, and it is not yet possible to establish these models under a unified framework. In this paper, a thermodynamically generalized framework of variational phase field model of fracture is presented, which can effectively unify different models under this framework. Based on this variational framework, not only the current second-order and fourth-order phase field model can be unified, but also new second-order, fourth-order, sixth-order, and eighth-order phase field model can be obtained from this framework. In addition, a method is proposed to convert the lower-order model into a higher-order model, which can convert a phase field distribution function of the lower-order model into a high-order phase field distribution function, and then obtain a higher-order phase field model. Finally, the new model is verified by some representative examples, and the second-order and fourth-order phase field models are compared, which shows the effectiveness of building a new phase field model from the variational framework.

Energy-Preserving AVF Methods for Riesz Space-Fractional Nonlinear KGZ and KGS Equations

The Riesz space-fractional derivative is discretized by the Fourier pseudo-spectral (FPS) method. The Riesz space-fractional nonlinear Klein–Gordon–Zakharov (KGZ) and Klein–Gordon–Schrödinger (KGS) equations are transformed into two infinite-dimensional Hamiltonian systems, which are discretized by the FPS method. Two finite-dimensional Hamiltonian systems are thus obtained and solved by the second-order average vector field (AVF) method. The energy conservation property of these new discrete schemes of the fractional KGZ and KGS equations is proven. These schemes are applied to simulate the evolution of two fractional differential equations. Numerical results show that these schemes can simulate the evolution of these fractional differential equations well and maintain the energy-preserving property.

Determining the normalization of the quantum field theory vacuum, with implications for quantum gravity

In a standard quantum field theory the norm of the vacuum state is taken to be finite. In this paper we provide a procedure, based on constructing an equivalent wave mechanics, for determining whether or not actually is finite. We provide an example based on a second-order plus fourth-order scalar field theory, a prototype for quantum gravity, in which it is not. In this example the Lorentzian path integral with a real measure diverges though the Euclidean path integral does not. Thus in this example contributions from the Wick rotation contour cannot be ignored. Since is not finite, use of the standard Feynman rules is not valid. And while these rules not only lead to states with negative norm, they in fact lead to states with infinite negative norm. However, if the fields in that theory are continued into the complex plane, we show that then there is a domain in the complex plane known as a Stokes wedge in which one can define an appropriate time-independent, positive and finite inner product, viz. the overlap of left-eigenstates and right-eigenstates of the Hamiltonian; with the vacuum state then being normalizable, and with there being no states with negative or infinite norm. In this Stokes wedge it is the Euclidean path integral that diverges while the Lorentzian path integral does not. The concerns that we raise in this paper only apply to bosons since the matrices associated with their creation and annihilation operators are infinite dimensional. Since the ones associated with fermions are finite dimensional, the fermion theory vacuum is automatically normalizable. We discuss some general implications of our results for quantum gravity studies, and show that they are relevant to the construction of a consistent, unitary and renormalizable quantum theory of gravity.

Adaptive stochastic isogeometric analysis for nonlinear bending of thin functionally graded shells with material uncertainties

This paper presents an adaptive stochastic isogeometric method to incorporate material uncertainties in the nonlinear bending analysis of thin functionally graded material (FGM) shells. The gradient index is modeled as a second-order random field to describe the spatial randomness of material properties. An adaptive, nested, and non-intrusive Chebyshev interpolation process based on Leja sequences is formulated to approximate the response surface concerning random inputs, in which the discrete responses are parallelly obtained and incrementally augmented to save computational effort. The response surface is taken as a surrogate model to promote the efficiency of the stochastic analysis. In particular, a new probabilistic post-processing technique is proposed to accelerate the evaluation of the probability characteristics of the response. Isogeometric analysis is applied to solve the discrete responses of FGM shells, eliminating geometric errors and simplifying the implementation. Through three examples with different computational costs, the effectiveness of the present method is demonstrated by comparing it with two forms of Monte Carlo simulation methods. The influence of the statistical characteristics of the gradient index field on the response surface analysis and the random response is investigated.

Upward continuation of total magnetic field intensity data over Sokoto basin, north-western Nigeria

High-resolution aeromagnetic data covering the Sokoto basin, north-western Nigeria were acquired and analyzed using regional enhancement techniques, upward continuation and second order regional. The objective was to delineate major geological structures. The data, which are in form of a grid, were knitted using Oasis Montaj software to obtain the magnetic total field intensity map. The composite grid was reduced to a magnetic equator to align the peaks of magnetic anomalies over the center of their causative bodies. The reduced grid was continued at various planes of observation to understate anomalies due to shallow features. The planes were at 10 km, 20 km, 30 km, 40 km, 50 km, 70 km, and 100 km above the surface. Upward continuation maps produced greatly enhanced the long-wavelength (low wavenumber) anomalies associated with deeper subsurface regional structures. The continuation revealed a high magnetic intensity anomaly in western parts of the area and low magnetic intensity anomalies in the eastern parts. The low magnetic intensity in the eastern part is interpreted as a sedimentary deposit. The residual field was removed from the total magnetic field data using the second-order least square method to get the second-order regional field. The regional field analysis agrees with the upward continuation result.

Correlations of tensor field components in isotropic systems with an application to stress correlations in elastic bodies.

Correlation functions of components of second-order tensor fields in isotropic systems can be reduced to an isotropic fourth-order tensor field characterized by a few invariant correlation functions (ICFs). It is emphasized that components of this field depend in general on the coordinates of the field vector variable and thus on the orientation of the coordinate system. These angular dependencies are distinct from those of ordinary anisotropic systems. As a simple example of the procedure to obtain the ICFs we discuss correlations of time-averaged stresses in isotropic glasses where only one ICF in reciprocal space becomes a finite constant e for large sampling times and small wave vectors. It is shown that e is set by the typical size of the frozen-in stress components normal to the wave vectors, i.e., it is caused by the symmetry breaking of the stress for each independent configuration. Using the presented general mathematical formalism for isotropic tensor fields this finding explains in turn the observed long-range stress correlations in real space. Under additional but rather general assumptions e is shown to be given by a thermodynamic quantity, the equilibrium Young modulus E. We thus relate for certain isotropic amorphous bodies the existence of finite Young or shear moduli to the symmetry breaking of a stress component in reciprocal space.

Optimized high-order finite-difference modeling of second-order strain gradient wave field effects

Optimized high-order finite-difference modeling of second-order strain gradient wave field effects

Visualizing Higher-Order 3D Tensors by Multipole Lines.

Applications like physics, medicine, earth sciences, mechanical engineering, geo-engineering, bio-engineering use tensorial data. For example, tensors are used in formulating the balance equations of charge, mass, momentum, or energy as well as the constitutive relations that complement them. Some of these tensors (i.e. stiffness tensor, strain gradient, photo-elastic tensor) are of order higher than two. Currently, there are nearly no visualization techniques for such data beyond glyphs. An important reason for this is the limit of currently used tensor decomposition techniques. In this article, we propose to use the deviatoric decomposition to draw lines describing tensors of arbitrary order in three dimensions. The deviatoric decomposition splits a three-dimensional tensor of any order with any type of index symmetry into totally symmetric, traceless tensors. These tensors, called deviators, can be described by a unique set of directions (called multipoles by J. C. Maxwell) and scalars. These multipoles allow the definition of multipole lines which can be computed in a similar fashion to tensor lines and allow a line-based visualization of three-dimensional tensors of any order. We give examples for the visualization of symmetric, second-order tensor fields as well as fourth-order tensor fields. To allow an interpretation of the multipole lines, we analyze the connection between the multipoles and the eigenvectors/eigenvalues in the second-order case. For the fourth-order stiffness tensor, we prove relations between multipoles and the eigenvectors of the second-order right Cauchy-Green tensor and present different interpretations of the multipole lines.

A Well-Posed Multidimensional Rational Covariance and Generalized Cepstral Extension Problem

In the present paper we consider the problem of estimating the multidimensional power spectral density which describes a second-order stationary random field from a finite number of covariance and generalized cepstral coefficients. The latter can be framed as an optimization problem subject to multidimensional moment constraints, i.e., to search a spectral density maximizing an entropic index and matching the moments. In connection with systems and control, such a problem can also be posed as finding a multidimensional shaping filter (i.e., a linear time-invariant system) which can output a random field that has identical moments with the given data when fed with a white noise, a fundamental problem in system identification. In particular, we consider the case where the dimension of the random field is greater than two for which a satisfying theory is still missing. We propose a multidimensional moment problem which takes into account a generalized definition of the cepstral moments, together with a consistent definition of the entropy. We show that it is always possible to find a rational power spectral density matching exactly the covariances and approximately the generalized cepstral coefficients, from which a shaping filter can be constructed via spectral factorization. In plain words, our theory allows us to construct a well-posed spectral estimator for any finite dimension.

Absolute Stability and Synchronization in Second-Order Neural Fields with Conduction Delays

We consider a class of neural field models represented by a second-order nonlinear system of integro-differential equations with space-dependent delays. Such a system models interaction of populations of neurons, each with a continuum description. We justify the existence and uniqueness of solution for the system in a suitable function space. Global existence and boundedness of solutions for the system are confirmed. Two methodologies, the comparison argument and sequential contracting, are developed to establish sufficient conditions for absolute stability and synchronization among different populations of the system. Finally, we present some numerical examples to demonstrate the theoretical results.

Legendre scalarization in gravity and cosmology

We propose a new formulation of f(R) gravity, dubbed scalarized f(R) gravity, in which the Legendre transform is included as a dynamical term. This leads to a theory with second-order field equations that describes general relativity with a self-interacting scalar field, without requiring the introduction of conformal frames. We demonstrate that the quadratic version of scalarized f(R) gravity reduces to general relativity with a massive scalar field, and we explore its implications for Friedmann cosmology. Our findings suggest that scalarized f(R) gravity may lead to simplified descriptions of cosmological applications, while the proposed formulation could offer a new perspective on the relationship between f(R) gravity and scalar–tensor theories.

New constraints on cosmological modified gravity theories from anisotropic three-point correlation functions of BOSS DR12 galaxies

ABSTRACT We report a new test of modified gravity theories using the large-scale structure of the Universe. This paper is the first attempt to (1) apply a joint analysis of the anisotropic components of galaxy two- and three-point correlation functions (2 and 3PCFs) to actual galaxy data and (2) constrain the non-linear effects of degenerate higher-order scalar-tensor (DHOST) theories on cosmological scales. Applying this analysis to the Baryon Oscillation Spectroscopic Survey (BOSS) data release 12, we obtain the lower bounds of −1.655 < ξt and −0.504 < ξs at the $95{{\ \rm per\ cent}}$ confidence level on the parameters characterizing the time evolution of the tidal and shift terms of the second-order velocity field. These constraints are consistent with GR predictions of ξt = 15/1144 and ξs = 0. Moreover, they represent a 35-fold and 20-fold improvement, respectively, over the joint analysis with only the isotropic 3PCF. We ensure the validity of our results by investigating various quantities, including theoretical models of the 3PCF, window function corrections, cumulative S/N, Fisher matrices, and statistical scattering effects of mock simulation data. We also find statistically significant discrepancies between the BOSS data and the Patchy mocks for the 3PCF measurement. Finally, we package all of our 3PCF analysis codes under the name hitomi and make them publicly available so that readers can reproduce all the results of this paper and easily apply them to ongoing future galaxy surveys.

Linear Peridynamics Fourier Multipliers and Eigenvalues

A characterization for the Fourier multipliers and eigenvalues of linear peridynamic operators is provided. The analysis is presented for state-based peridynamic operators for isotropic homogeneous media in any spatial dimension. We provide explicit formulas for the eigenvalues in terms of the space dimension, the nonlocal parameters, and the material properties. The approach we follow is based on the Fourier multiplier analysis developed by Alali and Albin (Applicable Analysis 2526–2546, 1). The Fourier multipliers of linear peridynamic operators are second-order tensor fields, which are given through integral representations. It is shown that the eigenvalues of the peridynamic operators can be derived directly from the eigenvalues of the Fourier multiplier tensors. We reveal a simple structure for the Fourier multipliers in terms of hypergeometric functions, which allows for providing integral representations as well as hypergeometric representations of the eigenvalues. These representations are utilized to show the convergence of the eigenvalues of linear peridynamics to the eigenvalues of the Navier operator of linear elasticity in the limit of vanishing nonlocality. Moreover, the hypergeometric representation of the eigenvalues is utilized to compute the spectrum of linear peridynamic operators.

Optical magnetic combination method for suppressing the Rb polarization-induced magnetic gradient in Rb-Xe NMR co-magnetometers.

The Rb polarization-induced magnetic field gradient affects the Xe nuclear spin relaxation properties and degrades the long-term stability of the NMR co-magnetometers. This paper proposes a combination suppressing scheme, which uses the second-order magnetic field gradient coils to compensate for the Rb polarization-induced magnetic gradient under counter-propagating pump beams. Based on the theoretical simulation, we find that the spatial distribution of the Rb polarization-induced magnetic gradient and the magnetic field distribution generated by gradient coils are complementary. The experimental results indicate that the compensation effect is 10% higher under the counter-propagating pump beams scheme compared with the compensation effect under the conventional single beam. Besides, due to a more uniform spatial distribution of electronic spin polarization, the Xe nuclear spin polarizability is improved and the signal-to-noise ratio (SNR) of NMR co-magnetometers could be further enhanced. The study provides an ingenious method for suppressing magnetic gradient in the optically polarized Rb-Xe ensemble, which is expected to improve the performance of the atomic spin co-magnetometers.

Symmetric teleparallel Horndeski gravity

Horndeski gravity is the most general scalar-tensor theory with one scalar field leading to second-order Euler-Lagrange field equations for the metric and scalar field, and it is based on Riemannian geometry. In this paper, we formulate an analog version of Horndeski gravity in a symmetric teleparallel geometry which assumes that both the curvature (general) and torsion are vanishing and gravity is only related to nonmetricity. Our setup requires that the Euler-Lagrange equations for not only metric and scalar field but also connection should be at most second order. We find that the theory can be always recast as a sum of the Riemannian-Horndeski theory and new terms that are purely teleparallel. Due to the nature of nonmetricity, there are many more possible ways of constructing second-order theories of gravity. In this regard, up to some assumptions, we find the most general $k$-essence extension of symmetric teleparallel Horndeski gravity. We also formulate a novel theory containing higher-order derivatives acting on nonmetricity while still respecting the second-order conditions, which can be recast as an extension of kinetic gravity braiding. We finish our study by presenting the FLRW cosmological equations for our model.

Application of differential equation in the field of acoustic

The current study examines a classification of physical problems involving the attenuation and propagation of structure and fluid-coupled acoustic waves in a discontinuous waveguide. In acoustics, the response of sound to boundaries is important. Therefore, it is expected that all of the discontinuous waveguide’s boundaries have the same walls, which can be either hard or impedance. The impedance and hard walls of the waveguide are mathematically modeled with respective Robin and Neumann boundary conditions together with the second-order field differential equation. The physical challenge is solved using the mode-matching (MM) approach, which also matches the continuity criteria for the acoustic pressure and normal velocities at matching connections. Transmission loss and powers scattering graphs against various frequencies and waveguides dimension parameters are shown to evaluate how well the waveguide predicts the sound to enhance or attenuate for both fluid and structure-borne modes incidents. By reconstructing the matching continuity requirements at matching junctions and using the conserved power identity, the accuracy of the derived algebra is confirmed. The current study has significant implications for improving sound quality for audible applications.

Numerical simulation of energy transfer in radiative hybrid nanofluids flow influenced by second-order chemical reaction and magnetic field

In this mathematical model, the consequences of Soret and Dufour’s effects on energy and mass transport have been reported. The hybrid nanofluid (hnf) comprised of silver (Ag) and magnesium oxide (MgO) nps is studied over a stretching rotating sheet. The influence of thermal radiation, second-order chemical reaction, activation energy, and the magnetic field is also applied to the fluid flow. The phenomena are designed in form of nonlinear Partial differential equations (PDEs), which are reduced to dimensionless ordinary differential equations through similarity replacement. Furthermore, the parametric continuation method (PCM) has been employed to estimate the reduced form of differential equations. The graphical results are obtained for velocity, mass, and energy profiles vs different physical interest entities. It has been noted that the velocity curve drops vs the variation of stretching sheet rotation factor, Hartmann number, and the mounting number of MgO and Ag nano particulates. Furthermore, the hnf (MgO–Ag/water) has greater efficiency for momentum and energy dissemination rate as compared to nanofluid.

Nonperturbative gedanken experiments in Einstein-dilaton-Gauss-Bonnet gravity: Nonlinear transitions and tests of the cosmic censorship beyond general relativity

As the only gravity theory with quadratic curvature terms and second-order field equations, Einstein-dilaton-Gauss-Bonnet gravity is a natural testbed to probe the high-curvature regime beyond General Relativity in a fully nonperturbative way. Due to nonperturbative effects of the dilatonic coupling, black holes in this theory have a minimum mass which separates a stable branch from an unstable one. The minimum mass solution is a double point in the phase diagram of the theory, wherein the critical black hole and a wormhole solution coexist. We perform extensive nonlinear simulations of the spherical collapse onto black holes with scalar hair in this theory, especially focusing on the region near the minimum mass. We study the nonlinear transition from the unstable to the stable branch and assess the nonlinear stability of the latter. Furthermore, motivated by modeling the mass loss induced by Hawking radiation near the minimum mass at the classical level, we study the collapse of a phantom field onto the black hole. When the black-hole mass decreases past the critical value, the apparent horizon shrinks significantly, eventually unveiling a high-curvature elliptic region. We argue that evaporation in this theory is bound to either violate the weak cosmic censorship or produce horizonless remnants. Addressing the end-state might require a different evolution scheme.