Series Expansion Method Research Articles

Invariant analysis, exact solutions and conservation laws of (2+1)-dimensional time fractional Navier–Stokes equations

In this paper, we investigate the exact solutions and conservation laws of (2 + 1)-dimensional time fractional Navier–Stokes equations (TFNSE). Specifically, Lie symmetries and corresponding one-dimensional optimal system for TFNSE in Riemann–Liouville sense are obtained. Then, based on the admitted symmetries and optimal system, we reduce these equations to one-dimensional equations or (1 + 1)-dimensional fractional partial differential equations (PDEs) with the help of Erdélyi–Kober fractional differential operator and compound variable transformation. In addition, we solve the reduced PDEs applying power series expansion method and invariant subspace method, respectively. Furthermore, the conservation laws of TFNSE are derived using new Noether theorem.

Invariant Analysis, Analytical Solutions, and Conservation Laws for Two-Dimensional Time Fractional Fokker-Planck Equation

The main purpose of this paper is to apply the Lie symmetry analysis method for the two-dimensional time fractional Fokker-Planck (FP) equation in the sense of Riemann–Liouville fractional derivative. The Lie point symmetries are derived to obtain the similarity reductions and explicit solutions of the governing equation. By using the new conservation theorem, the new conserved vectors for the two-dimensional time fractional Fokker-Planck equation have been constructed with a detailed derivation. Finally, we obtain its explicit analytic solutions with the aid of the power series expansion method.

A Symplectic Conservative Perturbation Series Expansion Method for Linear Hamiltonian Systems with Perturbations and Its Applications

A Symplectic Conservative Perturbation Series Expansion Method for Linear Hamiltonian Systems with Perturbations and Its Applications

Heat transport in shear-driven flow with axial conduction

The classical Graetz problem of fluid entering a long channel possessing different temperature from that of fluid is extended here to the case where the upper wall of the channel is not stationary. Formulation of the current problem incorporates the simple Couette flow driven by the channel movement and taking into account the axial heat conduction, besides the usual convection. Analytical solutions in the thermally developing regime regarding the temperature distribution as well as the heat transfer rates from both the upper moving and the lower fixed channel walls are then obtained by means of the series expansion method making use of the proper orthogonal family of eigenfunctions. Simple approximate formulae for the rates of heat transfer at the channel entrance valid for large Peclet numbers and at the far downstream locations valid for all Peclet numbers are further derived. More simplistic outcomes are also provided when the axial pressure gradient is assumed to be constant. Results are then discussed from low to high Peclet numbers, which are in compliant with the well-known results available in the literature for the channel flow within stationary walls. On the other hand, the rate of heat transfer from the moving wall is shown to be of high importance when Peclet number increases, which requires special attention in technological applications.

Complex WGM frequencies of gyroelectric cylindrical resonators

Asymptotic closed-form expressions for calculating complex transverse electric (TE)/transverse magnetic (TM) whispering gallery mode (WGM) frequencies in homogeneous gyroelectric circular cylindrical resonators of infinite length are derived. In addition, a volume integral equation (VIE)-cylindrical Dini series expansion method is extended to support the prediction of complex WGMs for continuously varying highly inhomogeneous gyroelectric circular cylindrical resonators. To this end, the entire domain orthogonal vectorial basis of VIE is extended to support very large indices of the involved Dini-type cylindrical vector wave functions via asymptotic closed-form expressions. This way, the eigenbasis required to solve the VIE becomes free of numerical instabilities arising when very large orders of the involved Bessel functions are employed. The complex frequencies obtained by the asymptotic closed-form expressions for the case of the homogeneous gyroelectric resonator, as well as those obtained by the VIE when the multilayered gyroelectric resonator is reduced to one layer, are validated by comparisons with the complex roots extracted by numerically solving the TE/TM characteristic equations obtained from the separation of variables method, using complex root finding techniques. We demonstrate the calculation of very high-order WGM frequencies for cylindrical resonators composed of homogeneous and highly inhomogeneous permittivity profiles. This asymptotic theory constitutes a rigorous tool that may serve for verifying method of analytical regularisation–based numerical solutions for other non-circular inhomogeneous cylinders, and for interpreting experimental data for applications such as WGM lasing, refractometric sensing, and magneto-optic coupling.

RI-IGABEM based on generalized-[formula omitted] method in 2D and 3D elastodynamic problems

The isogeometric analysis boundary element method (IGABEM) has a broad application prospect due to its exact geometric representation and good approximation properties. In this paper, a novel radial integration IGABEM (RI-IGABEM) based on the generalized-α method is proposed to solve 2D and 3D elastodynamic problems of homogeneous and inhomogeneous materials. First of all, the elastostatics Kelvin fundamental solution is used as the fundamental solution of the problem. In order to preserve the advantage of IGABEM, i.e. only boundary is discretized, the radial integration method (RIM) is applied to transform the domain integral caused by the material heterogeneity and the inertia term into an equivalent boundary integral by means of applied points. In addition, using a simple transformation method, the rigid-body technique is applied to solve the strongly singular integrals, and the Telles scheme and the power series expansion method are used to solve the weakly singular integrals in RI-IGABEM respectively. Furthermore, the generalized-α method is adopted to solve the time domain problem, which can improve the stability of numerical results by effectively filtering out the false response of high frequency and minimizing the attenuation of low frequency response. A number of 2D and 3D examples, such as those with homogeneous materials, functionally gradient materials, and material defects and inclusions, are used to demonstrate the ability of the scheme to simulate the elastodynamic problems.

Probability analysis of steady-state voltage stability considering correlated stochastic variables

The penetration of renewable generation, such as photovoltaic and wind power in power networks will significantly impact the system’s stability in future, due to randomness and intermittency in the natural environment. Moreover, the correlations between renewable generation may enhance the impact on power system's stability. In this paper, cumulant-based maximum entropy method (CMEM) combined with Nataf Transform (NT) is proposed to analyze the steady-state voltage stability problem considering correlations and uncertainties of power injections and consumptions. The proposed methodology is tested on the IEEE 30-node and IEEE 57-node test systems. Taking the results of Monte Carlo method (MCM) as the experimental control group, this paper compares the proposed method with series expansion method (SEM), discusses and analyzes their calculation results and speed under different scenarios, and the results of the comparison prove the validity, accuracy and rapidity of the CMEM.

IG-DRBEM of three-dimensional transient heat conduction problems

In this paper, the isogeometric dual reciprocity boundary element method (IG-DRBEM) is proposed to solve three-dimensional transient heat conduction problems. It is well known that the error of traditional BEM mainly comes from element dispersion, and the introduction of isogeometric ideas makes BEM become a veritable high-precision numerical method. At present, most of the problems solved by isogeometric BEM (IGBEM) are time-independent. The reason is similar to the traditional BEM, which cannot avoid solving domain integrals when solving time-dependent problems. In this paper, based on the potential fundamental solution the boundary-domain integral equation is obtained by the weighted residual method, where the classic dual reciprocity method is adopted to transform domain integrals into boundary integrals. Meanwhile, a two-level time integration scheme is used to solve the discretized differential equations. In addition, the adaptive integration scheme, the radial integral transform method and the power series expansion method are adopted to solve the boundary regular, nearly singular and singular integrals. Several classical numerical examples show that the presented method has good numerical stability and high precision by considering different factors such as the approximation function, the time step, the number of interior points and so on.

Generalized WKB theory for electron tunneling in gapped α−T3 lattices

We generalize Wentzel-Kramers-Brillouin (WKB) semi-classical equations for pseudospin-1 $\alpha-\mathcal{T}_3$ materials with arbitrary hopping parameter $0 < \alpha < 1$, which includes the dice lattice and graphene as two limiting cases. In conjunction with a series-expansion method in powers of Planck constant $\hbar$, we acquired and solved a system of recurrent differential equations for semi-classical electron wave functions in $\alpha-\mathcal{T}_3$. Making use of these obtained wave functions, we analyzed the physics-related mechanism and quantified the transmission of pseudospin-1 Dirac electrons across non-rectangular potential barriers in $\alpha-\mathcal{T}_3$ materials with both zero and finite band gaps. Our studies reveal several unique features, including the way in which the electron transmission depends on the energy gap, the slope of the potential barrier profile and the transverse momentum of incoming electrons. Specifically, we have found a strong dependence of the obtained transmission amplitude on the geometry-phase $\phi = \tan^{-1} \alpha$ of $\alpha-\mathcal{T}_3$ lattices. We believe our current findings can be applied to Dirac cone-based tunneling transistors in ultrafast analog RF devices, as well as to tunneling-current control by a potential barrier through a one-dimensional array of scatters.

Diatomic Molecules and Mass Spectrum of Heavy Quarkonium System with Kratzer- Screened Coulomb Potential (KSCP) through the Solutions of the Schrödinger Equation

In this work, we obtain solutions of the Schrödinger equation with Kratzer-screened Coulomb potential (KSCP) model using the series expansion method. Explicitly, we compute the bound state energy eigenvalues for selected diatomic molecules of N2, CO, NO, and CH, respectively, for the various vibrational and rotational quantum states and the numerical energy eigenvalues agree with the existing literature. Three special cases were considered. The energy eigenvalues are applied to obtain the mass spectra of heavy quarkonium system such as charmonium and bottomonium. The results agree with the experimental data and other recent theoretical studies.

Estimating Ruin Probability in an Insurance Risk Model with Stochastic Premium Income Based on the CFS Method

This paper considers the estimation of ruin probability in an insurance risk model with stochastic premium income. We first show that the ruin probability can be approximated by the complex Fourier series (CFS) expansion method. Then, we construct a nonparametric estimator of the ruin probability and analyze its convergence. Numerical examples are also provided to show the efficiency of our method when the sample size is finite.

Stress intensity factors for a centrally cracked Brazilian disk under non-uniformly distributed pressure

To simulate the real distribution of pressure on the specimen in the Brazilian test, non-uniform distribution is assumed to be the form of pressure distribution on the disk. The analytic solutions of the stress field for an uncracked Brazilian disk under several typical non-uniformly distributed pressures are obtained by the power series expansion method and mathematical method. Then the analytic solutions of the mode I and mode II stress intensity factors for a centrally cracked Brazilian disk under the same pressures are obtained by the weight function method. The results show that the influence of distribution form on stress intensity factors cannot be ignored for the case of large relative crack length and load distribution angle. The applicable ranges of the formulae to calculate stress intensity factors under concentrated force, uniformly and non-uniformly distributed pressure are given according to the analysis results. In addition, the loading conditions for pure mode I and II fracture are discussed when a disk with a central crack is subjected to non-uniformly distributed pressure.

A Method to Combine Residual Stress Measurements from XRD and IHD using Series Expansion

It is common practice to use various residual stress measurement methods to complement each other and fully define a through-thickness stress distribution. Incremental hole-drilling (IHD) and X-ray diffraction (XRD) are two of the most widely used techniques. Although IHD readily provides stress data to some depth, the method is susceptible to large stress uncertainties near the surface. XRD, in contrast, is most suited to finding near-surface stresses. Constrain the residual stress distributions obtained through series expansion by using XRD measurements, thereby obtaining full depth stress measurements with reduced uncertainty near the surface. The proposed method enforces suitable relationships between the amplitude coefficients of the series expansion such that the resultant stress distributions match the XRD measurements. The method is demonstrated on an aluminium alloy 7075 specimen of 10 mm thickness that underwent laser shock peening treatment. Strong correlation in calculated residual stress distributions was found between the proposed method, standard series expansion and the regularized integral method. The proposed method has reduced stress uncertainty near the surface when compared to both standard series expansion and integral methods of IHD due to its incorporation of near-surface XRD data. The proposed method allows XRD measurements to be rigorously incorporated into IHD results. The effect of XRD uncertainty on the overall IHD stress distribution is localised to the near-surface measurements.

Topological Design of Freely Vibrating Bi-Material Structures to Achieve the Maximum Band Gap Centering at a Specified Frequency

AbstractThe topological design of structures to avoid vibration resonance for a certain external excitation frequency is often desired. This paper considers the topology optimization of freely vibrating bi-material structures with fixed/varying attached mass positions, targeting at maximizing the frequency band gap centering at a specified frequency. A band gap measure index is proposed to measure the size of the band gap with a specified center frequency. Aiming at maximizing this measure index, the topology optimization problem is formulated on the basis of the material-field series-expansion (MFSE) method, which greatly reduces the number of design variables and at the same time keeps the capability to describe relatively complex structural topologies with clear boundaries. As the considered optimization problem is highly non-linear and may yield multiple local minima, a sequential Kriging-based optimization solution strategy is employed to effectively solve the optimization problem. This solution strategy exhibits a relatively strong global search capability and requires no sensitivity information. With the present topology optimization model and the gradient-free algorithm, relative large band gaps with specified center frequencies have been obtained for two-dimensional (2D) beams and three-dimensional (3D) plates, without specifying the frequency orders between which the desired band gap occurs in prior.

Longitudinal and lateral interparticle optical binding and extrinsic radiation force and torque on a pair of lossless dielectric cylinders of arbitrary sizes and the acoustical analogue

The purpose of this work is to develop an exact formalism and analytical equations for the interparticle electromagnetic/optical (longitudinal and lateral) binding forces arising between two lossless dielectric cylinders with different permittivities in a TE-polarized plane wave-field with an arbitrary incidence angle (in the polar plane). Additionally, the extrinsic/external optical radiation force and torque acting on the aggregate (i.e. the cluster of particle pair) are determined, and their exact mathematical expressions are derived. Based on the modal series expansion method in cylindrical coordinates in conjunction with the translational addition theorem, numerical simulations and computations for the longitudinal and lateral components of the interparticle and extrinsic forces as well as the axial component of the extrinsic torque show that attraction/repulsion, rotation clockwise or counter-clockwise, or neutrality, arise depending on the interparticle distance, the incidence angle, particle sizes and their dielectric permittivities. This investigation offers a complete analytical formalism without any approximations related to particle sizes. The numerical results computed based on exact mathematical expressions for interparticle and extrinsic physical observables can be used in the design and optimization of optical tweezers and related applications in optically-bound matter, particle manipulation, cloaking, and optically-assembled metamaterials with adjustable periodicities to name some examples. Also, the relevant analogy with the acoustic counterpart is noted.

Conformable Fractional Models of the Stellar Helium Burning via Artificial Neural Networks

The helium burning phase represents the second stage that the star used to consume nuclear fuel in its interior. In this stage, the three elements, carbon, oxygen, and neon, are synthesized. The present paper is twofold: firstly, it develops an analytical solution to the system of the conformable fractional differential equations of the helium burning network, where we used, for this purpose, the series expansion method and obtained recurrence relations for the product abundances, that is, helium, carbon, oxygen, and neon. Using four different initial abundances, we calculated 44 gas models covering the range of the fractional parameterα=0.5−1with stepΔα=0.05. We found that the effects of the fractional parameter on the product abundances are small which coincides with the results obtained by a previous study. Secondly, we introduced the mathematical model of the neural network (NN) and developed a neural network algorithm to simulate the helium burning network using a feed-forward process. A comparison between the NN and the analytical models revealed very good agreement for all gas models. We found that NN could be considered as a powerful tool to solve and model nuclear burning networks and could be applied to the other nuclear stellar burning networks.

Equivalent SDOF Load Transformation Factors for One-Way Members Subjected to Air Blast Loading

The equivalent single-degree-of-freedom (SDOF) load transformation factor expressions of the simply supported, fixed end and simply-fixed one-way members subjected to air blast loading are derived by finite series-expansion method or finite point-fitting method. Moreover, figures are used to show the relative relationship between the load transformation factors KL and the ratio of stand-off distance to structural span (γ). The typical calculation conditions show that the load transformation factors are sensitive to the piecewise function of peak overpressure when γ is less than 0.5. The load transformation factor will be a smooth curve for peak overpressure adopting a single smooth continuous function. For the blast-resistant structural design, if the error of 3% in the elastic stage and 5% in the plastic stage can be allowed, γ = 1.0 is the demarcation point between near field blast and far field blast. In addition, if the error of 0.5% can be allowed, γ ≥ 2.0 can be treated as the far field blast. The calculation results of far field blast show that the values of KL of simply supported, simply-fixed, and fixed-end members are 0.64, 0.58, and 0.54, respectively, which means that the load transformation factor is insensitive to the selection of shape function in vibration mode function or static deflection curve function. The calculation accuracy of KL expression for simply supported beam structure in this paper has been verified through near field blast experiments. This paper provides a clear numerical definition for the distinction between near field and far field explosions. It provides qualification criteria for analyzing explosion categories in blast-resistant structural design and a reference for the selection of theoretical calculation formulas.

A complex-variable cohesive finite element subroutine to enable efficient determination of interfacial cohesive material parameters

A new complex-variable version of a cohesive element is presented that provides highly accurate first order derivatives of the nodal displacements with respect to the cohesive fracture parameters. These sensitivities are provided as a byproduct of the analysis using the complex Taylor series expansion method. This information is useful for inversely determining the cohesive fracture parameters from experimental or synthetic data using a finite element-based approach. In particular, the PPR cohesive element (Park et al., 2009), was extended using complex variables as a user element for the well-known commercial finite element program, Abaqus. The source code for the element is provided as an educational resource. The advantage of having accurate first order derivatives on both accuracy and efficiency is demonstrated through numerical examples.

Longitudinal and transverse optical scattering asymmetry parameters for a dielectric cylinder in light-sheets of arbitrary wavefronts and polarization.

The asymmetry parameter is an important quantity used in radiative transfer modeling and scattering. This parameter specifies the amount of energy scattered by the particle along the direction of the incident illuminating field. However, a rigorous and complete analysis of the energetic scattering requires determining the energy scattered in the lateral direction as well. As such, the present work introduces generalized expressions for the scattering asymmetry parameters for a dielectric cylinder in arbitrary-shaped light-sheets, both along and perpendicular to the direction of the incident radiation. Both longitudinal and transverse scattering asymmetry parameters are defined, and their generalized expressions are obtained based on the (spatial) average cosine and sine of the scattering angle θ and the expression of the scattering cross section (or energy efficiency). The partial-wave series expansion method in cylindrical coordinates is used, and the resulting mathematical expressions depend on the beam-shaped coefficients and the scattering coefficients of the dielectric cylinder. Numerical results for arbitrary-shaped light-sheets illuminating a dielectric cylinder cross section located arbitrarily in space are presented and discussed. The longitudinal and transverse scattering asymmetry parameters defined here offer additional quantitative (quadratic) observables for the analysis of the energetic scattering in applications in electromagnetic scattering, optical light-sheet tweezers, radiative transfer computations, and remote sensing, to name a few examples.

Reliability analysis of horizontally loaded pile considering spatial variability of soil parameters

In the design and analysis of pile foundations, the horizontal load is an important factor that needs to be considered. The reliability analysis can take the uncertainty of various parameters into consideration and thus reduce the design deviations, especially the uncertainty of soil parameters. This paper proposes a reliability analysis method considering the spatial variability of soil for the horizontally loaded pile. Firstly, a parameter sensitivity analysis was performed by the Sobol sensitivity method, and the critical variables, including the undrained shear strength Cu, the soil strain ε50, which represents the soil strain when principal stress difference reaches half (50%) of the maximum value in triaxial test, and the horizontal load and applied moment, are identified and taken as the random variables. Moreover, the spatial variability of soil parameters (Cu and ε50) are simulated as one-dimensional random field through the orthogonal series expansion method. The failure probabilities of pile foundation under the failure modes of deflection and strength are evaluated by Monte Carlo simulation. Finally, the proposed method is verified through case study in which the influences of the spatial variability of soil parameters, the uncertainty of the horizontal load on the failure probability were investigated. The results show that the uncertainty of horizontal loads and Cu make significant effects on the reliability of pile foundations. The increase of correlation length of Cu and ε50 will lead to the increase of the mean value of pile response, which has different effects on the failure probability under different failure modes.