Terms Of Series Expansion Research Articles

Numerical Contour Integration

We present a straightforward implementation of contour integration by setting options for Integrate and NIntegrate, taking advantage of powerful results in complex analysis. As such, this article can be viewed as documentation to perform numerical contour integration with the existing built-in tools. We provide examples of how this method can be used when integrating analytically and numerically some commonly used distributions, such as Wightman functions in quantum field theory. We also provide an approximating technique when time-ordering is involved, a commonly encountered scenario in quantum field theory for computing second-order terms in Dyson series expansion and Feynman propagators. We believe our implementation will be useful for more general calculations involving advanced or retarded Green’s functions, propagators, kernels and so on.

Stability analysis of a hyperbolic stochastic Galerkin formulation for the Aw-Rascle-Zhang model with relaxation.

We investigate the propagation of uncertainties in the Aw-Rascle-Zhang model, which belongs to a class of second order traffic flow models described by a system of nonlinear hyperbolic equations. The stochastic quantities are expanded in terms of wavelet-based series expansions. Then, they are projected to obtain a deterministic system for the coefficients in the truncated series. Stochastic Galerkin formulations are presented in conservative form and for smooth solutions also in the corresponding non-conservative form. This allows to obtain stabilization results, when the system is relaxed to a first-order model. Computational tests illustrate the theoretical results.

Numerical approach for a continuum theory with higher stress gradients

AbstractWe use an extended balance of linear momentum derived from stress field analysis of higher order terms in power series expansion. Thus, the balance equation accounts for higher gradients of stress in the contiguity of continuum points. Interestingly, it does not coincide with the balance of linear momentum from strain gradient elasticity. As shown in [1], it exhibits an inverse sign for the extended term compared to strain gradient elasticity. We are interested in the mechanical interpretation of this inversed sign since it seems to inverse the stiffening effect of strain gradient elasticity. Therefore, we set up the weak form of our extended balance equation by means of Galerkin's approach. Then, we use the Finite Element Method to approximate the weak form with help of different shape functions. In this context we also use Isogeometric Analysis since it is very promising for a numerical model with higher gradients.

High-Resolution Ionosphere Corrections for Single-Frequency Positioning

The ionosphere is one of the main error sources in positioning and navigation; thus, information about the ionosphere is mandatory for precise modern Global Navigation Satellite System (GNSS) applications. The International GNSS Service (IGS) and its Ionosphere Associated Analysis Centers (IAAC) routinely provide ionospheric information in terms of global ionosphere maps (final GIM). Typically, these products are modeled using series expansion in terms of spherical harmonics (SHs) with a maximum degree of n=15 and are based on post processed observations from Global Navigation Satellite Systems (GNSS), as well as final satellite orbits. However, precise applications such as autonomous driving or precision agriculture require real-time (RT) information about the ionospheric electron content with high spectral and spatial resolution. Ionospheric RT-GIMs are disseminated via Ntrip protocol using the SSR VTEC message of the RTCM. This message can be streamed in RT, but it is limited for the dissemination of coefficients of SHs of lower degrees only. It allows the dissemination of SH coefficients up to a degree of n=16. This suits to most the SH models of the IAACs, but higher spectral degrees or models in terms of B-spline basis functions, voxels, splines and many more cannot be considered. In addition to the SHs, several alternative approaches, e.g., B-splines or Voxels, have proven to be appropriate basis functions for modeling the ionosphere with an enhanced resolution. Providing them using the SSR VTEC message requires a transfer to SHs. In this context, the following questions are discussed based on data of a B-spline model with high spectral resolution; (1) How can the B-spline model be transformed to SHs in order to fit to the RTCM requirements and (2) what is the loss of detail when the B-spline model is converted to SHs of degree of n=16? Furthermore, we discuss (3) what is the maximum necessary SH degree n to convert the given B-spline model and (4) how can the transformation be performed to make it applicable for real-time applications? For a final assessment, we perform both, the dSTEC analysis and a single-frequency positioning in kinematic mode, using the transformed GIMs for correcting the ionospheric delay. The assessment shows that the converted GIMs with degrees n≥30 coincide with the original B-spline model and improve the positioning accuracy significantly.

A hybrid method for solving fuzzy Volterra integral equations of separable type kernels

The paper deals with the computation of solutions of fuzzy Volterra integral equations with degenerate kernel by applying a hybrid method. The proposed method is built on Laplace transform coupled with Adomian Decomposition Method; Laplace Adomian Decomposition Method abbreviated as (LADM). In the considered equation the unknown function has a solution in terms of infinite series expansion and hence LADM becomes more accurate to give the exact solution. Firstly, using the fuzzy number in term of parametric form, the fuzzy Volterra integral equation is converted to two crisp integral equations and then LADM is applied to get the exact fuzzy solutions of fuzzy linear Volterra integral equations. For the illustration, some examples of considered equations are solved to highlight the robustness, efficiency and the applicability of the developed scheme. The obtained results play an important role in developing the theory of fuzzy analytical dynamic equations.

Free vibration of fully coupled thermoelastic multilayered composites with imperfect interfaces

A fully coupled thermoelastic framework is formulated to cope with the free vibration response of anisotropic multilayered plates in three dimensions. The laminated structure consists of homogeneous laminae of arbitrary thickness and width under simply supported edge conditions in thermal environment. The general and exact field expressions of the temperature, heat flux, displacement and stress components are expressed in terms of double Fourier series expansions in any rectangular plate, which lead to the extended Stroh formalism with thermomechanical coupling effects in a concise and compact matrix form. Different imperfect interface conditions are introduced to characterize specific structural and thermal contact properties at the bounding interfaces, and further to determine the finite complex valued coefficients in the suitable series relations. The complete time-harmonic solutions in the laminated composites in the presence of perfect/imperfect interfaces are recursively obtained by means of the modified dual variable and position technique with explicit layer-to-layer transfer matrices. Results are obtained for different layups, length-to-thickness ratios and interfacial boundary conditions for two application examples, namely the graphite/epoxy cross-ply composites and the thermal barrier coatings on superalloys, without suffering from numerical exponential instability. These investigations reveal that the natural frequencies and first and higher vibration mode shapes of the multilayered structures can be considerably affected by increasing the environmental temperature and the severity of the interfacial imperfections. Since the through-thickness stress distribution in 2, 5, and 10 layered composites appears to be strongly correlated to the layups, such modal stress analysis could be exploited to locate the fatigue hotspots operated in dynamic structures and to guide the structural design of aircraft and spacecraft composite laminates subjected to residual vibrations.

Discrete Chebyshev polynomials for nonsingular variable‐order fractional KdV Burgers' equation

In this article, nonlinear variable‐order (VO) fractional Korteweg‐de Vries (KdV) Burgers' equation with nonsingular VO time fractional derivative is introduced and discussed. The approximate solution of the expressed problem is obtained in the form of a series expansion in terms of the shifted discrete Chebyshev polynomials (CPs) with great accuracy. The method is a computational procedure based on the collocation technique and the shifted discrete CPs together with their operational matrices (ordinary and VO fractional derivatives). The main advantage of the designed approach is that it provides a global solution for the problem. In order to examine the efficiency of the designed algorithm, some numerical problems have been provided. The obtained solutions confirm that the present method is computationally effective and sufficiently accurate in solving this equation.

Model for Investigating the Physical Properties of a System of Charged Particles Taking into Account the External Magnetic Field and Friction Force

In the present work, a numerical modified Verlet scheme (MVS) is obtained. This scheme is intended to solve the equations of motion of charged particles immersed in an external stationary environment and a uniform magnetic field, for example, charged particles of a condensed substance in a buffer plasma (dusty plasma). The influence of the environment on the particle dynamics is described by the friction force. The particle dynamics are also affected by interparticle interaction and an external uniform magnetic field. To obtain the Verlet scheme (VS), the coordinates and velocities of the particles are decomposed into a Taylor series, taking into account the Lorentz force and the friction force. All Taylor series expansion terms that give the same order of accuracy are taken into account. In the obtained numerical scheme, the time step of modeling does not depend on the magnitude of the magnetic field but is determined only by the internal physical properties of the system under consideration, which is important when modeling an ensemble of charged particles taking electromagnetic fields into account. The paper solves a test problem for which particle trajectories obtained based on the conventional and modified VS for different values, both the friction parameter and the magnetic field parameter, are compared. Based on the analysis of the dependence of the maximum relative deviation of the coordinate on the time step, the time step is independent of the magnetic field in the Taylor expansion scheme, while in the inverse VS it is dependent.

Low‐frequency dipolar electromagnetic scattering by a solid ellipsoid in lossless environment

AbstractElectromagnetic wave scattering phenomena for target identification are important in many applications related to fundamental science and engineering. Here, we present an analytical formulation for the calculation of the magnetic and electric fields that scatter off a highly conductive ellipsoidal body, located within an otherwise homogeneous and isotropic lossless medium. The primary excitation source assumes a time‐harmonic magnetic dipole, precisely fixed and arbitrarily orientated that operates at low frequencies and produces the incident fields. The scattering problem itself is modeled with respect to rigorous expansions of the electromagnetic fields at the low‐frequency regime in terms of positive integral powers of the real wave number of the ambient. Obviously, the Rayleigh static term and a few dynamic terms are sufficient for the purpose of the present work, as the additional terms are neglected due to their minor contribution. Therein, the classical Maxwell's theory is suitably modified, leading to intertwined either Laplace's or Poisson's equations, accompanied by the impenetrable boundary conditions for the total fields and the limiting behavior at infinity. On the other hand, the complete spatial anisotropy of the three‐dimensional space is secured via the introduction of the genuine ellipsoidal coordinate system, being appropriate for tackling incrementally such scattering boundary value problems. The nonaxisymmetric fields are obtained via infinite series expansions in terms of ellipsoidal harmonic eigenfunctions, providing handy closed‐form solutions in a compact fashion, whose validity is verified by a straightforward reduction to simpler geometries of the metal object. The main idea is to demonstrate an efficient methodology, according to which the constructed analytical formulae can offer the appropriate environment for a fast numerical estimation of the scattered electromagnetic fields that could be useful for real data inversion.

Analytical approximations for the inverse Langevin function via linearization, error approximation, and iteration

This paper details an analytical framework, based on an intermediate function, which facilitates analytical approximations for the inverse Langevin function—a function without an explicit analytical form. The approximations have relative error bounds that are typically much lower than those reported in the literature and which can be made arbitrarily small. Results include convergent series expansions in terms of polynomials and sinusoids which have modest relative error bounds and convergence properties but are convergent over the domain of the inverse Langevin function. An important advance is to use error approximations, and then iterative relationships, which allow simple initial approximations for the inverse Langevin function, with modest relative errors, to generate approximations with arbitrarily low relative errors. One example is that of an initial approximating function, with a relative error bound of 0.00969, which yields relative error bounds of 2.77 × 10−6 and 2.66 × 10−16 after the use of first-order error approximation and then first-order iteration. Functions with much lower error bounds are possible and are detailed. First- and second-order Taylor series can be used to simplify the error- and iteration-based approximations.

Adaptive Modeling of the Global Ionosphere Vertical Total Electron Content

The Kalman filter (KF) is widely applied in (ultra) rapid and (near) real-time ionosphere modeling to meet the demand on ionosphere products required in many applications extending from navigation and positioning to monitoring space weather events and naturals disasters. The requirement of a prior definition of the stochastic models attached to the measurements and the dynamic models of the KF is a drawback associated with its standard implementation since model uncertainties can exhibit temporal variations or the time span of a given test data set would not be large enough. Adaptive methods can mitigate these problems by tuning the stochastic model parameters during the filter run-time. Accordingly, one of the primary objectives of our study is to apply an adaptive KF based on variance component estimation to compute the global Vertical Total Electron Content (VTEC) of the ionosphere by assimilating different ionospheric GNSS measurements. Secondly, the derived VTEC representation is based on a series expansion in terms of compactly supported B-spline functions. We highlight the morphological similarity of the spatial distributions and the magnitudes between VTEC values and the corresponding estimated B-spline coefficients. This similarity allows for deducing physical interpretations from the coefficients. In this context, an empirical adaptive model to account for the dynamic model uncertainties, representing the temporal variations of VTEC errors, is developed in this work according to the structure of B-spline coefficients. For the validation, the differential slant total electron content (dSTEC) analysis and a comparison with Jason-2/3 altimetry data are performed. Assessments show that the quality of the VTEC products derived by the presented algorithm is in good agreement, or even more accurate, with the products provided by IGS ionosphere analysis centers within the selected periods in 2015 and 2017. Furthermore, we show that the presented approach can be applied to different ionosphere conditions ranging from very high to low solar activity without concerning time-variable model uncertainties, including measurement error and process noise of the KF because the associated covariance matrices are computed in a self-adaptive manner during run-time.

Spectral Analysis of Fractional Hyperbolic Diffusion Equations with Random Data

The paper studies the fundamental solutions to fractional in time hyperbolic diffusion equation or telegraph equations and their properties. Then it derives the exact solutions of the fractional hyperbolic diffusion equation with random data in terms of series expansions of isotropic in space spherical random fields on the unit sphere. Numerical illustration are presented to illustrate the theoretical results.

Dynamic behavior of thin-walled elements of aircraft made of composite materials, excited by heat shock

To improve the performance characteristics of modern aerospace systems, research is conducted and expensive programs are being carried out to provide for reducing the weight of the aircraft structure through the use of new, more promising materials, which include the so-called composite materials. Special attention is paid to the dynamic behaviour of composite structures under the influence of high-intensity heat fluxes of various physical nature. The paper considers the dynamic behaviour of composite structures of modern aerospace systems under the influence of high-intensity heat fluxes. As an example, the axisymmetric transverse vibrations of a composite circular plate connected to an elastic base, excited by thermal shock, are investigated. The plate material is modelled with a three-layer composite. To describe the kinematics of an asymmetric plate pack, the hypotheses of a broken normal are accepted. In thin bearing layers, Kirchhoff's hypotheses are valid. In a relatively thick lightweight core, the normal does not change its length, remains rectilinear, but rotates through some additional angle. The base reaction is described by the Winkler model. The statement of the initial-boundary value problem is given. The analytical solution is obtained as a series expansion in terms of eigenfunctions. Its numerical parametric analysis is carried out.

Describing Broadband Terahertz Response of Graphene FET Detectors by a Classical Model

Direct power detectors based on field-effect transistors are becoming widely used for terahertz applications. However, accurate characterization at terahertz frequencies of such detectors is a challenging task. The high-frequency response is dominated by parasitic coupling and loss associated with contacts and overall layout of the component. Moreover, the performance of such detectors is complicated to predict since many different physical models are used to explain the high sensitivity at terahertz frequencies. This makes it hard to draw important conclusions about the underlying device physics for these detectors. For the first time, we demonstrate accurate and comprehensive characterization of graphene field-effect transistors from1 GHz to 1.1 THz, simultaneously accessing the bias dependence, the scattering parameters, and the detector voltage responsivity. Within a frequency range of more than1 THz, and over a wide bias range, we have shown that the voltage responsivity can be accurately described using a combination of a small-signal equivalent circuit model, and the second-order series expansion terms of the nonlinear dc I-V characteristics. Without bias, the measured low-frequency responsivity was 0.3 kV/W with the input signal applied to the gate, and 2 kV/W with the input signal applied to the drain. The corresponding cutoff frequencies for the two cases were 140 and 50 GHz, respectively. With a300-GHz signal applied to the gate, a voltage responsivity of 1.8 kV/W was achieved at a drain-source current of 0.2 mA. The minimum noise equivalent power was below 30 pW/ √Hz in cold mode. Our results show that detection of terahertz signals in graphene field-effect transistors can be described over a wide frequency range by the nonlinear carrier transport characteristics obtained at static electrical fields. This finding is important for explaining the mechanism of detection and for further development of terahertz detectors.

Fracture analysis of rock specimens weakened by rounded-V and U-shaped notches under pure mode I loading

In this paper, the mode I fracture resistance of rock specimens weakened by rounded V (RV) and U-shaped notches is investigated experimentally and theoretically. For this purpose, several experiments were conducted on the Brazilian disks with central RV and U-shaped notches under pure mode I loading. Then, the fracture resistances of tested specimens were justified by a modified form of the border stress criterion. The border stress criterion states that the onset of fracture takes place when the tangential stress at a critical distance from notch border attains to a critical value. In the conventional border stress criterion, only the first term of stress series expansion around the notch border is used. The critical distance in the rock materials is relatively large and the first term cannot provide accurate description for stress field at the critical distance. Thus, the more accurate stress field is obtained from finite element analysis. It is shown that the modified border stress criterion can predict the fracture resistance of notched specimens in good agreement with experimental results.

Convergence improvement of the simultaneous relaxation method used in the finite element analysis of incompressible fluid flows

Purpose This paper aims to present an improved finite element method used for achieving faster convergence in simulations of incompressible fluid flows. For stable computations of incompressible fluid flows, it is important to ensure that the flow field satisfies the equation of continuity in each element of a generally distorted mesh. The study aims to develop a numerical approach that satisfies this requirement based on the highly simplified marker-and-cell (HSMAC) method and increases computational speed by introducing a new algorithm into the simultaneous relaxation of velocity and pressure. Design/methodology/approach First, the paper shows that the classical HSMAC method is equivalent to a Jacobi-type method in terms of the simultaneous relaxation of velocity and pressure. Then, a Gauss–Seidel or successive over-relaxation (SOR)-type method is introduced in the Newton–Raphson iterations to take into account all the derivative terms in the first-order Taylor series expansion of a nodal-averaged error explicitly. Here, the nine-node quadrilateral (Q2–Q1) elements are used. Findings The new finite element approach based on the improved HSMAC algorithm is tested on fluid flow problems including the lid-driven square cavity flow and the flow past a circular cylinder. The results show significant improvement of the convergence property with the accuracy of the numerical solutions kept unchanged even on a highly distorted mesh. Originality/value To the best of the author’s knowledge, the idea of using the Gauss–Seidel or SOR method in the simultaneous relaxation procedure of the HSMAC method has not been proposed elsewhere.

Thermal tides in rotating hot Jupiters

ABSTRACT We calculate tidal torque due to semidiurnal thermal tides in rotating hot Jupiters, taking account of the effects of radiative cooling in the envelope and of the planets rotation on the tidal responses. We use a simple Jovian model composed of a nearly isentropic convective core and a thin radiative envelope. To represent the tidal responses of rotating planets, we employ series expansions in terms of spherical harmonic functions $Y_l^m$ with different ls for a given m. For low-forcing frequency, there occurs frequency resonance between the forcing and the g- and r-modes in the envelope and inertial modes in the core. We find that the resonance enhances the tidal torque, and that the resonance with the g- and r-modes produces broad peaks and that with the inertial modes very sharp peaks, depending on the magnitude of the non-adiabatic effects associated with the oscillation modes. We also find that the behaviour of the tidal torque as a function of the forcing frequency (or period) is different between prograde and retrograde forcing, particularly for long forcing periods because the r-modes, which have long periods, exist only on the retrograde side.

Seidel aberration coefficients: an alternative computational method.

An alternative method to compute Seidel aberrations is presented that utilizes real pseudoparaxial skew rays traveling through a rotationally-symmetric lens system as a Taylor series expansion in terms of object height and ray-direction spherical coordinates. Expressions for defocus, lateral magnification, and Seidel primary ray aberration coefficients are obtained in terms of numerically-determined higher-order partial derivatives of rays which are proximate to the optical axis. In contrast to commonly used methods, the new form of the aberration coefficients is related to unit entrance pupil radius and unit object height. The proposed methodology can be extended to derive higher-order ray aberration coefficients.

Analytical solutions to renormalization-group equations of effective neutrino masses and mixing parameters in matter

Neutrino oscillations in matter can be fully described by six effective parameters, namely, three neutrino mixing angles left{{tilde{theta}}_{12},{tilde{theta}}_{13},{tilde{theta}}_{23}right} , one Dirac-type CP-violating phase tilde{delta} , and two neutrino mass-squared differences {tilde{Delta}}_{21}equiv {tilde{m}}_2^2-{tilde{m}}_1^2 and {tilde{Delta}}_{21}equiv {tilde{m}}_3^2-{tilde{m}}_1^2 . Recently, a complete set of differential equations for these effective parameters have been derived to characterize their evolution with respect to the ordinary matter term aequiv 2sqrt{2}{G}_{mathrm{F}}{N}_eE , in analogy with the renormalization-group equations (RGEs) for running parameters. Via series expansion in terms of the small ratio αc ≡ Δ21/Δc with Δc ≡ Δ31 cos2 θ12+Δ32 sin2θ12, we obtain approximate analytical solutions to the RGEs of the effective neutrino parameters and make several interesting observations. First, at the leading order, {tilde{theta}}_{12} and {tilde{theta}}_{13} are given by the simple formulas in the two-flavor mixing limit, while {tilde{theta}}_{23}approx {theta}_{23} and tilde{delta}approx delta are not changed by matter effects. Second, the ratio of the matter-corrected Jarlskog invariant tilde{mathcal{J}} to its counterpart in vacuum mathcal{J} approximates to tilde{mathcal{J}}/mathcal{J}approx 1/left({widehat{C}}_{12}{widehat{C}}_{13}right) , where {widehat{C}}_{12}equiv sqrt{1-2{A}_{ast } cos 2{theta}_{12}+{A}_{{}^{ast}}^2} with A∗ ≡ a/Δ21 and {widehat{C}}_{13}equiv sqrt{1-2{A}_c cos 2{theta}_{13}+{A}_c^2} with Ac ≡ a/Δc have been defined. Finally, after taking higher-order corrections into account, we find compact and simple expressions of all the effective parameters, which turn out to be in perfect agreement with the exact numerical results.

High Performance Distribution Network Power Flow Using Wirtinger Calculus

The power flow problem can be compactly formulated using complex variables; its solution is however commonly computed using the real variable Newton-Raphson algorithm, where the complex variables are replaced by their real valued rectangular or polar coordinates. A Newton-Raphson solution in complex variables is hindered by the fact that the power flow equations are non-analytic in their phasor voltages, i.e., they do not have a Taylor series expansion in terms of these complex variables alone. By using Wirtinger calculus, the power flow equations can be expanded in terms of the phasor voltages and their conjugates, thus allowing the implementation of the complex variable Newton-Raphson algorithm. This paper specializes the solution of the complex variable Newton-Raphson power flow to distribution networks, and includes the particularities of modeling line drop compensators, reactive power limited generation, and voltage dependent loads. Numerical results presented on distribution networks with up to 3139 nodes show that the complex implementation is better suited to modern processors that employ single instruction multiple data (SIMD) processor extensions; in particular SIMD instructions result in a complex variable formulation that is faster than the classical real variable formulation, yet it has the distinct advantage of a simple software implementation.