Scalar Potential Functions Research Articles

Scalar field induced dynamical evolution in teleparallel gravity

In this paper, we investigate the role of scalar field potentials in the dynamical evolution of the Universe. A gravity theory with a non-minimally coupled scalar field with torsion in the geometrical action simulating effective dark energy is considered to study an extended matter bounce scenario. The dynamical behaviour of the equation of state parameter has been studied near the bouncing epoch. Keeping in mind the inflationary behaviour near the bounce, five different scalar field potential functions are explored, and their effect on the equation of state parameter is investigated.

Robust and efficient implementation of finite strain generalized continuum models for material failure: Analytical, numerical, and automatic differentiation with hyper-dual numbers

Generalized continuum models for representing nonlinear material behavior including material failure in the finite strain regime are commonly formulated based on scalar elastic and dissipation potential functions. The evolution of stresses and internal variables, i.e., the material state, is governed by partial derivatives of the potential functions with respect to deformation and stress measures. Furthermore, for application of such models in implicit nonlinear finite element analysis tangent operators, consistent with the numerical integration algorithm, are required. In this work, we study analytical, numerical and automatic differentiation schemes for the implementation of coupled, generalized continuum models considering finite inelastic deformations and fracture. This includes the application of finite difference approximations, complex-step derivative approximations and automatic differentiation based on hyper-dual numbers. For the use of automatic differentiation, a semi-analytical split approach is introduced for increasing the computational efficiency. Based on a comprehensive 2D and 3D finite element study, we demonstrate the superior properties of automatic differentiation compared to numerical differentiation methods with regards to accuracy and robustness. Furthermore, the additional computational cost for automatic differentiation using the proposed split approach becomes negligible for large problems.

Formulation of the beam shape coefficients based on spherical expansion of the scalar function

In solving spherical particle scattering in generalized Lorenz-Mie theory, two sets of beam shape coefficients (BSCs) are calculated, one is for the transverse electric field and another is for the transverse magnetic field. The current work presents an indirect method which introduces the spherical expansion of a scalar potential function. In this way, the BSCs of the beam fields are obtained as combinations of the expansion coefficients of the scalar function, which allows formulating the BSCs more easily and speeding up the numerical calculation. The method is rigorous and is flexible to different shaped beams.

Non-polynomial hybrid models for the bending of magneto-electro-elastic shells

This paper presents different non-polynomial hybrid models in the framework of Carrera’s Unified Formulation for the bending of a magneto-electric shell with variable radii of curvature. The shell’s middle surface is graphed by a parametric surface. Differential Geometry is employed for evaluating the Lamé Parameters and Radius of Curvature. The mechanical displacements are modeled in the context of an equivalent single layer by sinusoidal, hyperbolic, and tangential models. The electrical and magnetic scalar potential functions are written by a polynomial thickness function in the framework of Layerwise theory. The shell panels are subjected to mechanical, electrical, and magnetic loads. The governing equations are obtained by the Principle of Virtual Displacement. The correspondent partial differential equations are discretized by Chebyshev-Gauss-Lobatto grid distribution and solved by the so-called Differential Quadrature Method. The classical Lagrange polynomial is employed as the basis function for the method. The stresses, electrical displacement, and magnetic induction are recovered by the three-dimensional (3D) equilibrium equations. A comparative analysis with 3D solutions provided in the literature is performed for a square plate and a doubly curved shallow shell panel and remarkable results are obtained. So, the validated models are further used to study shells with variable radii of curvature; specifically, for helicoid, ellipsoid, and catenoid panels.

Second-order Stokes wave-induced dynamic response and instantaneous liquefaction in a transversely isotropic and multilayered poroelastic seabed

The ocean waves exhibit obvious non-linearity with asymmetric distribution of wave crests and troughs, which could induce significantly different effect on the seabed compared to the commonly used linear wave theory. In this paper, a semi-analytical solution for a transversely isotropic and multilayered poroelastic seabed under non-linear ocean wave is proposed by virtue of the dual variable and position (DVP) method. The ocean wave and seabed are, respectively, modelled using second-order Stokes theory and Biot’s complete poroelastodynamic theory. Then the established governing equations are decoupled and solved via the powerful scalar potential functions. Making use of the DVP scheme, the layered solutions are finally gained by combining the boundary conditions of the seabed. The developed solutions are verified by comparing with existing solutions. The selected numerical examples are presented to investigate the effect of main parameters on the dynamic response of the seabed and evaluate the corresponding liquefaction potential. The results show that the anisotropic stiffness and permeability, degree of saturation and stratification have remarkable influence on the dynamic response and liquefaction behavior of the seabed. The present solution is a useful tool to estimate the stability of transversely isotropic and layered seabed sediment in the range of non-linear ocean wave.

Superradiant instability of dyonic Reissner–Nordström black holes

We investigate the superradiant instability of dyonic Reissner–Nordström (dRN) black holes with two charges under an electrically charged massive scalar perturbation. Two conditions for possessing a trapping well are obtained from analyzing asymptotic scalar potential and far-region wave functions. It is clear that the superradiant instability is not allowed in the dRN black holes because the conditions for a trapping well are not compatible with the superradiance condition.

Lie Symmetry Classification and Qualitative Analysis for the Fourth-Order Schrödinger Equation

The Lie symmetry analysis for the study of a 1+n fourth-order Schrödinger equation inspired by the modification of the deformation algebra in the presence of a minimum length is applied. Specifically, we perform a detailed classification for the scalar field potential function where non-trivial Lie symmetries exist and simplify the Schrödinger equation. Then, a qualitative analysis allows for the reduced ordinary differential equation to be analysed to understand the asymptotic dynamics.

Superradiant instability in slowly rotating Kerr–Newman black holes

In this paper, we study the superradiant instability of slowly rotating Kerr–Newman (sKN) black holes under a charged massive scalar perturbation. These black holes resemble closer the Reissner–Nordström black holes than the Kerr–Newman black holes. From the scalar potential analysis, we find that the superradiant instability is not allowed in the sKN black holes because the condition for a trapping well is not compatible with the superradiance condition. However, the rate of energy extraction might grow exponentially if the sKN black hole is placed inside a reflecting cavity. Finally, we obtain two conditions for the trapping well to possess quasibound states in the sKN black holes by analyzing asymptotic scalar potential and far-region wave functions.

Conditions for superradiant instability of the Kerr-Newman black holes

We find two conditions for superradiant instability of Kerr-Newman black holes under a charged massive scalar perturbation by analyzing asymptotic scalar potential and far-region wave function. Actually, they correspond to the condition for getting a trapping well. Also, we obtain the conditions for superradiant stability of Kerr-Newman black holes which states that there is no trapping well. The analysis is applied to Kerr black holes to find a condition for superradiant instability.

Towards a smooth unification from an ekpyrotic bounce to the dark energy era

In the context of a ghost free f ( R , G ) model, we present a non-singular cosmological scenario in which the universe initially contracts through an ekpyrotic phase having a bouncing like behaviour, and following the bounce, it smoothly transits to a matter or radiation like deceleration era which is further smoothly connected to the dark energy era at present epoch. The ghost free character of the model is ensured by the presence of a Lagrange multiplier, and we consider the Gauss–Bonnet (GB) coupling function in such a way that it gets compatible with the event GW170817. Using suitable reconstruction technique, we obtain the non-trivial scalar field potential as well as the GB coupling function. Such scalar potential and GB coupling function source a smooth unified scenario from an ekpyrotic bounce to the dark energy era with an intermediate deceleration era. The occurrence of ekpyrotic contraction phase justifies the resolution of the anisotropic problem (also known as BKL instability) in the background evolution. Consequently we determine the background Hubble parameter and the corresponding effective equation of state parameter, and discussed several qualitative features of the model. The Hubble radius shows an asymmetric behaviour around the bounce, in particular, the evolution of the Hubble radius leads to the generation era of the primordial perturbation modes far before the bounce in the deep sub-Hubble regime. Accordingly we perform the scalar and tensor perturbation evolution in the present context, and as a result, the scalar power spectrum at large scale modes is found to be problematic. Thus an extended scenario is proposed where we consider a pre-ekpyrotic phase having the equation of state parameter is less than unity, and re-examine the scalar and tensor power spectra, on large scales that cross the Hubble radius during the pre-ekpyrotic stage. In this regard, the GB coupling function shows considerable effects in reducing the tensor to scalar ratio compared to the case where the GB coupling is absent. Furthermore the dark energy epoch is consistent with the Planck + SNe + BAO data.

A Stable Node-Based Smoothed Finite Element Method with Transparent Boundary Conditions for the Elastic Wave Scattering by Obstacles

In this work, a stable node-based smoothed finite element method with TBC (SNS-FEM-TBC) is proposed to solve the scattering of elastic plane waves by a two-dimensional (2D) homogeneous isotropic elastic medium. First, using Helmholtz decomposition, two scalar potential functions are introduced to divide the Navier equation into Helmholtz equations with the coupled boundary conditions for the elastic scattering problem. Second, based on the analytical solutions of Helmholtz equations, TBC operators are deduced. Then, the gradient Taylor expansion is used to construct the stability term to deal with the instability of the original NS-FEM, the gradient of the solution is expressed into a linear formulation through approximating the node-based smoothing domain as a circle. Finally, based on smoothed Galerkin weak formula, the SNS-FEM-TBC formula of linear algebraic system with linear smoothing gradient is derived. Numerical examples show that SNS-FEM can obtain more stable and accurate solutions than standard FEM. Moreover, the convergence rates of [Formula: see text] and [Formula: see text] semi-norm errors of SNS-FEM are faster.

Local-flux vectors of conserved quantities in wavenumber space: Anisotropic structures in Charney–Hasegawa–Mima turbulence

Heterogeneous coexistence turbulence (HCT) with anisotropic wave turbulence and isotropic vortex turbulence includes a transition range beyond the application limits of cascade theory and weak turbulence theory. Local-flux vectors, formulated based on scalar potential functions, of conserved quantities in the wavenumber space quantify their directional properties. Two balance types, time and term, make up the idea of critical balance, which the proposed local-flux vectors can quantitatively evaluate. The local-flux vectors reveal anisotropic fluxes along the wavenumbers deduced from the critical balance as well as energy-enstrophy double cascade in HCT of Charney-Hasegawa-Mima model.

A study of anisotropic compact stars in f(R,ϕ,X) theory of gravity

In this paper, we investigate the behavior of anisotropic compact stars in generalized modified gravity, namely [Formula: see text] gravity, where [Formula: see text] represents the Ricci scalar, [Formula: see text] is the scalar potential function and [Formula: see text] is a kinetic term of [Formula: see text]. We consider the spherically symmetric spacetime to analyze the feasible exposure of compact stars. We observe the behavior of anisotropic compact stars which includes Her X1, SAX J 1808.4-3658 and 4U 1820-30. From the graphical evaluation of energy density, tangential pressure, radial pressure, equilibrium conditions, energy conditions, mass–radius relationship, compactness and stability analysis of compact stars, it is concluded that the behavior of candidates of compact stars is regular in [Formula: see text] gravity for the considered parameter.

3D asymmetric dynamic Green’s functions of a thermoelastic transversely isotropic solid by a method of potentials

A theoretical formulation is presented for the displacements, temperature and stresses in a 3 D transversely isotropic thermoelastic solid under arbitrary mechanical loads and heat flux. By using two scalar potential functions, 3 D patch-load, patch-heat source, point-load and point-heat source Green’s functions are given for the displacements, temperature and stresses. Basic governing equations of coupled thermoelasticity in transversely isotropic solids are presented in the cylindrical coordinates system. By means of two scalar potential functions, the four coupled governing equations for this class of problems are uncoupled into a 6th order and a 2nd order partial differential equations. Applying the Fourier expansions and Hankel transforms, the angular and radial variables are suppressed and 6th and 2nd order ordinary differential equations are obtained. By solving the ordinary differential equations, the potential functions, displacements, temperature and stresses are obtained in the Fourier-Hankel transformed space. By applying the Hankel inversion theorem, all responses are presented in the physical time-spatial space. To prove the validity of the analytical and numerical solutions, our results are compared with existing solutions reported in the literature. Numerical results are also included to show the effects of material anisotropy, elastic and thermal properties; and frequency of excitation on the responses.

Propagation and reflection of plane waves in biological tissue based on nonlocal TPL thermoelasticity

The investigation for propagation and reflection of plane waves is carried out based on three-phase-lag (TPL) thermoelasticity with the aids of Eringen's nonlocal theory in a special medium - biological tissue. The scalar and vector potential functions of Helmhotz decomposition are adopted to obtain the propagation speeds, attenuation coefficients and reflection coefficients of plane waves analytically. There are two sets of dispersive attenuated longitudinal waves whose speeds of propagation are associated with angular frequency, and independent transverse elastic shear wave which exhibits no attenuation. It is found that the blood perfusion rate has few effects on wave propagation speed and attenuation coefficient and the phase lag of temperature gradient is negligible in present configuration. The effects of working angular frequency, nonlocal parameters, phase lag times, reflecting thermal boundaries and the type of incident waves on the propagation speeds, attenuation coefficients and reflection coefficients of harmonic waves are discussed and investigated graphically.

The investigation of stability of brane induced gravity model in the ‎presence of a scalar field with various kinds of potential‎

In this manuscript we consider the normal branch of DGP brane cosmological model in the presence of a ‎scalar field on the brane as the dark energy component. Using dynamical system approach we investigate ‎the stability properties of the system. We will see that one of the defined new variables lambda, that ‎depends on the potential function of the scalar field has a significant role in the evolution of the universe. ‎Therefore, we divide our discussion into two parts, the constant and the varying lambda. With a good ‎approximation we consider all the obtained critical points in the case of the constant lambda, as the ‎instantaneous critical points for the case of a varying lambda. We investigate two special scalar field ‎potential function: a double exponential potential and an inverse power law potential. In both of the cases we ‎study the evolution of the system and discuss the fate of the universe. We find that the inverse power law ‎potential function is in a good agreement with the current universe, but the double exponential potential ‎function can predict different future for the universe.‎

Some dark energy cosmological models in [formula omitted] gravity

In this paper, locally rotationally symmetric Bianchi type-I cosmological models in f(R,ϕ) theory of gravity have been investigated, where R and ϕ are Ricci scalar and scalar potential function, respectively. Some exact solutions pertaining to the dark energy and the cosmic expansion have been found by implementing the power law model f(R,ϕ)=λRm+1+βϕl+1, with λ,m, β and l being the arbitrary real constants, such that m,β≠−1 for the sake of non-triviality. Some important physical attributes like the pressure, energy density, equation of the state parameter, skewness parameter and energy conditions have been studied and analysed graphically. It has been concluded that both of the models are favorable for cosmic expansion. Furthermore, expansion of universe may be justified in f(R,ϕ) gravity under these models.

Rigid disc vibration in a multi-layered transversely isotropic poroelastic half-space underlying a liquid layer

Vertical vibration of a rigid disk embedded in a multi-layered poroelastic medium is considered in this paper, for the first time. The whole medium is underlying a liquid layer with finite depth to evaluate the submarine structures. Extended reflection and transmission matrix are introduced for layered porous media. The governing differential equations are simplified by virtue of Fourier and Hankel integrals transformations. Scalar potential functions are used to decouple the complex equations. By the definition of the suitable boundary conditions between layers, and solving the equations, the Green's functions of the medium can be reached. The disk surface of the rigid disk is assumed as a set of the concentric rings with equal displacements. The Green's functions are applied to generate the set of equations of the rings. The accuracy of the proposed method is sensitive to determine the number of the rings which make the surface of the disk.

A quasi-exact solution for the analysis of smart multilayered simply supported shallow shell panels

A quasi-three-dimensional solution for the bending analysis of simply supported orthotropic piezoelectric shallow shell panels subjected to thermal, mechanical and electrical potential is introduced. The mechanical governing equations are derived in term of three-dimensional equilibrium relations and the classical Maxwell’s equations. The trough-the-thickness temperature is modeled by the Fourier’s heat conduction equation. The coupled partial differential equations are solved by Navier closed form solutions. The trough-the-thickness profile for electrical potential, temperature profile and displacements is obtained by using a quasi-exact method so-called the differential quadrature method (DQM). Chebyshev polynomials of the third kind are used as the basis functions and the grid thickness domain discretization for the DQM. The interlaminar conditions for transverse stresses, temperature and electrical potential are imposed. The correct traction conditions for transverse stresses and scalar potential function at the top and the bottom are applied. The results for cylindrical, spherical and rectangular plates are presented. The excellent obtained results are compared with layerwise and three-dimensional solutions reported in the literature.

Modeling of 3-D elastic waves in nonlinear elastic solids using the iterative nonlinear contrast source method

In nondestructive testing, the generation of higher harmonics and the mixing of elastic waves are used to measure the nonlinearity parameters, which in turn are closely related to the microscopic state of the material. A 3-D numerical tool that can simulate large scale, four-dimensional, nonlinear elastic wavefields would be very useful for the reliable interpretation of experimental results. Here, a method is presented that can efficiently compute the nonlinear displacement fields in a homogeneous, isotropic elastic medium, taking into account its third-order elastic constants (A, B, C). The method is based on the Neumann iterative solution of an integral equation involving a Green's function of the linear `background' medium, and a contrast source representing the nonlinearity of the medium. The integral equation is solved iteratively, and the computations are based on Fast Fourier Transforms using a sampling rate close to the Nyquist limit, i.e., two grid points of the shortest wavelength. The displacement fields are evaluated using the scalar and vector potential functions representing the compressional and shear displacements. In this presentation, the suitability of the INCS method for modeling the harmonics of 3-D nonlinear elastic waves and its validation by comparison with an analytical benchmark solution, will be presented.