Series Expansion Method Research Articles

Unsteady solute dispersion in blood rheology with reversible phase exchange at the artery wall

The effect of reversible phase exchange between the flowing fluid and wall tissues of arteries in the unsteady dispersion of solute in blood flow through a narrow artery is analysed mathematically, modelling the blood as Casson fluid. The resulting convective diffusion equation along with the initial and boundary conditions is solved analytically using the derivative series expansion method. The expressions for the negative asymptotic phase exchange, negative asymptotic convection, longitudinal diffusion coefficient and mean concentration are obtained. It is noted that when the solute disperses in blood flow through a narrow artery, the negative exchange coefficient, the negative convection coefficient increase and the longitudinal diffusion coefficient decreases with the increase of the Damköhler number and partition coefficient.

Virial series expansion and Monte Carlo studies of equation of state for hard spheres in narrow cylindrical pores.

In this paper, the virial series expansion and constant pressure Monte Carlo method are used to study the longitudinal pressure equation of state for hard spheres in narrow cylindrical pores. We invoke dimensional reduction and map the model into an effective one-dimensional fluid model with interacting internal degrees of freedom. The one-dimensional model is extensive. The Euler relation holds, and longitudinal pressure can be probed with the standard virial series expansion method. Virial coefficients B_{2} and B_{3} were obtained analytically, and numerical quadrature was used for B_{4}. A range of narrow pore widths (2R_{p}), R_{p}<(sqrt[3]+2)/4=0.9330... (in units of the hard sphere diameter) was used, corresponding to fluids in the important single-file formations. We have also computed the virial pressure series coefficients B_{2}^{'}, B_{3}^{'}, and B_{4}^{'} to compare a truncated virial pressure series equation of state with accurate constant pressure Monte Carlo data. We find very good agreement for a wide range of pressures for narrow pores. These results contribute toward increasing the rather limited understanding of virial coefficients and the equation of state of hard sphere fluids in narrow cylindrical pores.

An efficient technique for the indirect BEM for multi-frequency acoustic analysis using Green’s function approximation

We present an efficient technique for multi-frequency acoustic analysis by the variational indirect boundary element method. The system coefficient matrix of the boundary element method is frequency dependent and requires cumbersome computation in multifrequency acoustic analysis. To eliminate the frequency dependency of a system coefficient matrix of the boundary element method, the free space Green’s function is approximated using Taylor series. Unlike the existing Taylor series expansion method, the present technique deals with the approximated series of the Green’s function using matrix decomposition and multiplication. For multi-frequency acoustic analysis, the coefficient matrix of the frequency independent system is calculated only at the first frequency; from the remaining frequencies, it is calculated as the product of the wave number matrix and the frequency independent coefficient matrix at the first frequency. The efficiency of the present technique is demonstrated through numerical examples. The results are compared with those of the conventional indirect method and the existing series expansion method by Wang.

Numerical Simulation of Heat Flux in Unsteady Heat Transfer for Large Floating-Roof Tanks

Increasing demand for oil storage tanks required we develop new models for large tanks that are adapted to use under extreme conditions. In order to avoid solidification of the oil in the tank at low temperatures, we need to accurately represent the behavior of heat flux in the tank. In this paper, we propose a step-by-step algorithm for determining the heat transfer coefficient based on the law of conservation of energy, when is then applied in a discrete unsteady heat transfer equation derived using the Taylor series expansion method and solved numerically. We studied the effect of the liquid level and the average temperature in the tank on the heat flux density, and using this as the basis we were able to set up a theoretical model for optimizing the design and use of floating-roof tanks.

A Constrained Scheme for High Precision Downward Continuation of Potential Field Data

To further improve the accuracy of the downward continuation of potential field data, we present a novel constrained scheme in this paper combining the ideas of the truncated Taylor series expansion, the principal component analysis, the iterative continuation and the prior constraint. In the scheme, the initial downward continued field on the target plane is obtained from the original measured field using the truncated Taylor series expansion method. If the original field was with particularly low signal-to-noise ratio, the principal component analysis is utilized to suppress the noise influence. Then, the downward continued field is upward continued to the plane of the prior information. If the prior information was on the target plane, it should be upward continued over a short distance to get the updated prior information. Next, the difference between the calculated field and the updated prior information is calculated. The cosine attenuation function is adopted to get the scope of constraint and the corresponding modification item. Afterward, a correction is performed on the downward continued field on the target plane by adding the modification item. The correction process is iteratively repeated until the difference meets the convergence condition. The accuracy of the proposed constrained scheme is tested on synthetic data with and without noise. Numerous model tests demonstrate that downward continuation using the constrained strategy can yield more precise results compared to other downward continuation methods without constraints and is relatively insensitive to noise even for downward continuation over a large distance. Finally, the proposed scheme is applied to real magnetic data collected within the Dapai polymetallic deposit from the Fujian province in South China. This practical application also indicates the superiority of the presented scheme.

Thermoelastic vibration and stability of temperature-dependent carbon nanotube-reinforced composite plates

The present article investigates the thermoelastic vibration and stability characteristics of carbon nanotube-reinforced composite (CNTRC) plates in thermal environment. The CNTRC plates are made up of four different types of uniaxially aligned reinforcements. The single-walled carbon nanotubes (SWCNTs) reinforcement is either uniformly distributed (UD) or functionally graded (FG) according to linear functions of the thickness direction. The material properties, of both matrix and CNTs, are temperature-dependent and the effective elastic coefficients are evaluated by using a micromechanical model. The governing equations (GEs) are derived in their weak-form by using Hamilton’s Principle in conjunction with the method of the power series expansion of the displacement components. The Ritz method, based on highly stable trigonometric trial functions, is used as solution technique. Convergence and stability of the proposed formulation have been thoroughly analyzed by assessing many higher-order plate models. Thermal and mechanical pre-stresses are taken into account. Moreover, the effect of significant parameters such as length-to-thickness ratio, volume fraction, aspect ratio, loading-type, CNTs distribution as well as boundary conditions is discussed.

A series expansion method aided design of CCII controller for a TITO system

Current mode circuits have attracted the attention of the design engineers due to their unique characteristics that are not found in their voltage mode counterparts. This article presents a series expansion method based design of current mode circuits for the control of a two input two output (TITO) system. The unique contribution of this work is the realization of the TITO controller using the second generation current conveyor circuits. The controller was practically implemented using AD844 integrated circuits. Simulation and the experimental results are presented for a pragmatic case study to demonstrate the efficacy of the proposed design methodology.

Bi-directional propagation leaky modes in a periodic chain of dielectric circular rods.

In this paper, a periodic chain composed of two-dimensional dielectric cylindrical inclusions was studied based on the Fourier series expansion method with perfectly matched layers. Phase and attenuation constants associated with guided modes, forward propagation leaky modes, and backward propagation leaky modes, were conceptually proposed and numerically examined. In particular, the relationships between the backward propagation mode, leaky mode, and propagation constant were explained in the second-order Bragg reflection region. This simple structure was investigated with the goal of realizing an efficient guiding device. Phase and attenuation constant results were compared with the results obtained using the Lattice Sums technique with the T-matrix approach and FDTD method; very good agreement was observed between these methods.

Diamond lattice Heisenberg antiferromagnet

We investigate ground-state and high-temperature properties of the nearest-neighbour Heisenberg antiferromagnet on the three-dimensional diamond lattice, using series expansion methods. The ground-state energy and magnetization, as well as the magnon spectrum, are calculated and found to be in good agreement with first-order spin-wave theory, with a quantum renormalization factor of about 1.13. High-temperature series are derived for the free energy, and physical and staggered susceptibilities for spin S = 1/2, 1 and 3/2, and analysed to obtain the corresponding Curie and Néel temperatures.

Time-dependent reliability assessment of fatigue crack growth modeling based on perturbation series expansions and interval mathematics

A novel method of the time-dependent reliability, which combines the perturbation series expansions with the interval mathematics, is presented in this study for life prediction of fatigue crack growth problems. Distinct from the treatment in statistical analysis, characteristic parameters that exist in structural crack propagation model are described as unknown-but-bounded (UBB) variables with perturbation terms owing to the fact of variability of geometric sizes, material properties, and loading conditions. Then, based on the perturbation principle and the Taylor extension approach, a new interval perturbation series expansion method (PSEM) to predict boundary rules of the fatigue crack length a(t) is derived. Meanwhile, the auto-correlation features between a(tk) and a(tk+1) are also confirmed by employing the interval process theory. Additionally, inspired by the classical out-crossing approach in random process issues, a non-probabilistic time-dependent reliability (NTR) index Rs(T), as a safety measure for in-service engineering structures with crack, is defined, and its solution algorithm is expounded in details. Numerical examples are eventually proposed to demonstrate the validity of the developed methodology of uncertainty quantification and reliability evaluation.

Thermodynamic analysis of Brownian coagulation based on moment method

In this article, the thermodynamic constraints of Brownian coagulation are proposed based on the binary perfectly inelastic collision theory and the principle of maximum entropy. The constraints can be expressed as inequalities with the Taylor series expansion method of moments. The inequalities establish the relationship between the physical parameters (such as temperature and specific surface energy) and the first three integral moments of particles. The inequalities are verified to determine the critical time for Brownian coagulation to reach the thermodynamic equilibrium. The critical time is proportional to the specific surface energy and inversely related to the temperature, which can be used to determine whether the particle size distribution reaches self-preserving form. Moreover, the critical states provide a new approach for detecting the particle specific surface energy with the moment method. The results further explain Brownian coagulation and offer opportunities to improve environmental quality from the viewpoint of thermodynamics.

Magnetoelectric effect in concentric quantum rings induced by shallow donor

We study the alteration of the magnetic and electric properties induced by the off-axis donor in a double InAs/GaAs concentric quantum ring. To this end we consider a model of an axially symmetrical ring-like nanostructure with double rim, in which the thickness of the InAs thin layer is varied smoothly in the radial direction. The energies and of contour plots of the density of charge for low-lying levels we find by using the adiabatic approximation and the double Fourier-Bessel series expansion method and the Kane model. Our results reveal a possibility of the formation of a giant dipole momentum induced by the in-plane electric field, which in addition can be altered by of the external magnetic field applied along the symmetry axis.

Series expansion method for determination of order of 3-dimensional bar-joint mechanism with arbitrarily inclined hinges

A general method for evaluating the order of infinitesimal mechanism is presented for 3-dimensional bar-joint mechanism, which is defined as assembly of rigid bars connected to nodes by hinges with arbitrarily inclined directions. Compatibility conditions of translations and rotations are derived at bar-ends with respect to the generalized displacements consisting of displacements and rotations of nodes and bars. Degree of kinematic indeterminacy is computed using singular value decomposition of the linearized compatibility matrix. The order of infinitesimal mechanism is determined by the existence condition of higher-order coefficients of the generalized displacements with respect to the path parameter of deformation. The coefficients are obtained in a similar manner as stability analysis of geometrically nonlinear structures, where symbolic computation software package is used for analytical derivation of equations. The detailed procedure of analysis is shown through the numerical examples of a two-bar and a four-bar linkages, and the results are verified using a finite element analysis software.

Lie symmetry analysis and explicit solutions for the time fractional generalized Burgers–Huxley equation

In this work, we study the time fractional generalized Burgers–Huxley equation with Riemann–Liouville derivative via Lie symmetry analysis and power series expansion method. We transform the governing equation to nonlinear ordinary differential equation of fractional order using its Lie point symmetries. In the reduced equation, the derivative is in Erdelyi–Kober sense. We apply power series technique to derive explicit solutions for the reduced equation. The convergence of the obtained power series solutions are also derived. Some interesting Figures for the obtained solutions are presented.

DIRECT METHOD OF LIE-ALGEBRAIC DISCRETE APPROXIMATIONS FOR ADVECTION EQUATION

We propose the direct method of Lie-algebraic discrete approximation for numerical solving the Cauchy problem for advection equation in this paper. Discretization of the equation is performed by means of the Lie-algebraic quasi representations for space variables of the equation and by means of Taylor series expansion and small parameter method for the time variable. Such combination of approaches leads to a factorial rate of convergence with respect to all variables in the equation if the quasi representations for differential operator are built by means of Lagrange interpolation. The approximation properties and error estimations for the proposed scheme are investigated. The factorial rate of convergence for the proposed numerical scheme has been proven. Key words: direct method of Lie-algebraic discrete approximations, advection equation, finite dimensional quasi representation, Lagrange polynomial, small parameter method, factorial convergence.

Semi-analytical solution of horizontally composite curved I-beam with partial slip

This paper presents a semi-analytical solution of simply supported horizontally composite curved I-beam by trigonometric series. The flexibility of the interlayer connectors between layers both in the tangential direction and in the radial direction is taken into account in the proposed formulation. The governing differential equations and the boundary conditions are established by applying the variational approach, which are solved by applying the Fourier series expansion method. The accuracy and efficiency of the proposed formulation are validated by comparing its results with both experimental results reported in the literature and FEM results.

On the adequacy of the polynomial approximation to the exponential growth curve model

Exponential growth curves are usually approximated with a finite order polynomial curve in the study of trending curves. This is done because the most popular way of removing the trend component is by differencing. This paper shows that the trend curve cannot be removed by differencing when the trend curve is the exponential growth curve. Exponential curves are transcendental functions which can be reduced to a finite order polynomial by Maclaurin series expansion. The objective of this paper is to examine the adequacy of the polynomial approximation of the exponential growth curve with respect to its growth rate and sample size. The coefficients of the associated polynomial curve were obtained theoretically by the use of Maclaurin series expansion method. Next, exponential growth curves with varying growth rates and sample sizes were simulated. Adequate polynomials were fitted to the simulated exponential growth curves and the coefficients obtained were compared with the theoretical coefficients using absolute error and paired tests. Results obtained show that adequacy depend on both growth rate and sample size. For the purpose of statistical analysis, the highest sample size of 28 is not useful, especially in times series analysis where the demand of samples of 60 or more is made.

Active electromagnetic invisibility cloaking and radiation force cancellation

This investigation shows that an active emitting electromagnetic (EM) Dirichlet source (i.e., with axial polarization of the electric field) in a homogeneous non-dissipative/non-absorptive medium placed near a perfectly conducting boundary can render total invisibility (i.e. zero extinction cross-section or efficiency) in addition to a radiation force cancellation on its surface. Based upon the Poynting theorem, the mathematical expression for the extinction, radiation and amplification cross-sections (or efficiencies) are derived using the partial-wave series expansion method in cylindrical coordinates. Moreover, the analysis is extended to compute the self-induced EM radiation force on the active source, resulting from the waves reflected by the boundary. The numerical results predict the generation of a zero extinction efficiency, achieving total invisibility, in addition to a radiation force cancellation which depend on the source size, the distance from the boundary and the associated EM mode order of the active source. Furthermore, an attractive EM pushing force on the active source directed toward the boundary or a repulsive pulling one pointing away from it can arise accordingly. The numerical predictions and computational results find potential applications in the design and development of EM cloaking devices, invisibility and stealth technologies.

Monte Carlo determination of the low-energy constants for a two-dimensional spin-1 Heisenberg model with spatial anisotropy

The low-energy constants, namely the spin stiffness $\rho_s$, the staggered magnetization density ${\cal M}_s$ per area, and the spinwave velocity $c$ of the two-dimensional (2D) spin-1 Heisenberg model on the square and rectangular lattices are determined using the first principles Monte Carlo method. In particular, the studied models have antiferromagnetic couplings $J_1$ and $J_2$ in the spatial 1- and 2-directions, respectively. For each considered $J_2/J_1$, the aspect ratio of the corresponding linear box sizes $L_2/L_1$ used in the simulations is adjusted so that the squares of the two spatial winding numbers take the same values. In addition, the relevant finite-volume and -temperature predictions from magnon chiral perturbation theory are employed in extracting the numerical values of these low-energy constants. Our results of $\rho_{s1}$ are in quantitative agreement with those obtained by the series expansion method over a broad range of $J_2/J_1$. This in turn provides convincing numerical evidence for the quantitative correctness of our approach. The ${\cal M}_s$ and $c$ presented here for the spatially anisotropic models are new and can be used as benchmarks for future related studies.

Buckling of cylindrical shells with measured settlement under axial compression

Buckling behavior of cylindrical shells with measured settlement under axial compression has been researched and analyzed in this paper. Using the method of Fourier series expansion, the measured settlement data is transformed into differential settlement. Applying the finite element simulation method, the differential settlement is applied to bottom of the cylindrical shell in order to simulate behavior of the cylindrical shell under differential settlement. Based on the deformed cylindrical shell caused by settlement, the finite element simulation method is applied to research buckling behavior of the cylindrical shell under axial compression considering the geometric nonlinearity and large deformation. The meridional membrane stress distribution in the circumferential direction of the cylindrical shell under differential settlement is obtained and compared with the differential settlement distribution. Effects of liquid storage on buckling behavior of the cylindrical shell are researched by applying hydrostatic pressure to the inner surface. Parametric analysis is carried out to research the effects of diameter-thickness ratio and height-diameter ratio of the cylindrical shell on the axial buckling capability. Results show that the cylindrical shell will be subjected to deformation and meridional membrane stress because of the differential settlement and the critical axial buckling load will be decreased then. Liquid storage is helpful for the cylindrical shell to remain stable. The cylindrical shell with larger height-diameter ratio and smaller diameter-thickness ratio is more capable to resist buckling under axial compression following differential settlement.