Series Expansion Method Research Articles

Discrete singular convolution and Taylor series expansion method for free vibration analysis of beams and rectangular plates with free boundaries

Since the excellent performance of the discrete singular convolution (DSC) for high order frequency analysis has not been demonstrated for rectangular plates with free edges and corners thus far, this paper proposes a novel approach in applying the DSC for vibration analysis of beams and rectangular plates with arbitrary boundary conditions. The Taylor series expansion method is used to apply the boundary conditions. The existing difficulty encountered at plate corner points is overcome by the proposed method. The essential idea of the novel method is that two Taylor series expansions with different orders are used to eliminate the fictitious points outside the physical domain. Various beam and rectangular plate problems are solved by the proposed DSC and the results are compared with exact solutions, finite element data and results obtained by the differential quadrature method (DQM) using much higher number of grid points. It is demonstrated that the Taylor series expansion method to apply the boundary conditions is general and accurate. It is also found that the performance of the proposed DSC is excellent and independent of boundary conditions.

A new hybrid uncertainty optimization method for structures using orthogonal series expansion

This paper proposes a new hybrid uncertain design optimization method for structures which contain both random and interval variables simultaneously. The optimization model is formulated with the feasible robustness and the reliability of the worst scenario. The hybrid uncertainty is quantified by using the orthogonal series expansion method that integrates the Polynomial Chaos (PC) expansion method and the Chebyshev interval method within a uniform framework. The design sensitivity of objective and constraints will be developed to greatly facilitate the use of gradient-based optimization algorithms. The numerical results show that this method will be more possible to seek the feasible solution.

Radiated flow of chemically reacting nanoliquid with an induced magnetic field across a permeable vertical plate

Impact of induced magnetic field over a flat porous plate by utilizing incompressible water-copper nanoliquid is examined analytically. Flow is supposed to be laminar, steady and two-dimensional. The plate is subjected to a regular free stream velocity as well as suction velocity. Flow formulation is developed by considering Maxwell–Garnetts (MG) and Brinkman models of nanoliquid. Impacts of thermal radiation, viscous dissipation, temperature dependent heat source/sink and first order chemical reaction are also retained. The subjected non-linear problems are non-dimensionalized and analytic solutions are presented via series expansion method. The graphs are plotted to analyze the influence of pertinent parameters on flow, magnetism, heat and mass transfer fields as well as friction factor, current density, Nusselt and Sherwood numbers. It is found that friction factor at the plate is more for larger magnetic Prandtl number. Also the rate of heat transfer decayed with increasing nanoparticles volume fraction and the strength of magnetism.

An Efficient Series Solution for Nonlinear Multiterm Fractional Differential Equations

In this paper, we introduce an efficient series solution for a class of nonlinear multiterm fractional differential equations of Caputo type. The approach is a generalization to our recent work for single fractional differential equations. We extend the idea of the Taylor series expansion method to multiterm fractional differential equations, where we overcome the difficulty of computing iterated fractional derivatives, which are difficult to be computed in general. The terms of the series are obtained sequentially using a closed formula, where only integer derivatives have to be computed. Several examples are presented to illustrate the efficiency of the new approach and comparison with the Adomian decomposition method is performed.

Theoretical and Experimental Comparison of Off-Grid Sparse Bayesian Direction-of-Arrival Estimation Algorithms

Off-grid sparse Bayesian learning algorithms for estimating the directions-of-arrival (DOAs) of multiple signals using an array of sensors are attractive in practice due to three primary reasons. First, these algorithms are fully automatic Bayesian algorithms and hence tuning of regularization parameters (hyperparameters) is not necessary. Second, since these algorithms are based on sparsity, they can produce high accuracy DOA estimates by exploiting the spatial sparsity of acoustic signals even when the signals are coherent. Third, they can also estimate the offset in the DOAs for signals, whose DOAs are not exactly aligned with the steering vectors. Two previously proposed off-grid sparse Bayesian DOA estimation algorithms are considered. The first off-grid model is based on the Taylor series expansion method (OGSBL-T algorithm) and the second is based on the linear interpolation method (OGSBL-I algorithm). The Cramer–Rao lower bound (CRLB) of the off-grid bias parameters for both the algorithms is derived for multiple snapshots. It is shown that the CRLB of the off-grid bias parameters for the OGSBL-T algorithm is significantly less than that for the OGSBL-I algorithm. It is also shown that the CRLBs of the off-grid bias parameters for both the algorithms get worse when we move from the broadside to the endfire directions. A simulation study is also carried out to characterize the performances of both the algorithms in terms of the root-mean-squared error in the DOA estimates. It is shown that the OGSBL-T algorithm performs comparably to the OGSBL-I algorithm when the signals are relatively broadside and better than the OGSBL-I algorithm when the signals are relatively endfire. Finally, the application of the OGSBL-T algorithm for high resolution DOA estimation in an underwater communication system is demonstrated by analyzing passive sonar data from the SWellEx-96 ocean acoustic experiment.

Predicting Apple Firmness and Soluble Solids Content Based on Hyperspectral Scattering Imaging Using Fourier Series Expansion

Abstract. This article reports on using a Fourier series expansion method to extract features from hyperspectral scattering profiles for apple fruit firmness and soluble solids content (SSC) prediction. Hyperspectral scattering images of ‘Golden Delicious’ (GD), ‘Jonagold’ (JG), and ‘Delicious’ (RD) apples, harvested in 2009 and 2010, were acquired using an online hyperspectral imaging system over the wavelength region of 500 to 1000 nm. The moment method and Fourier series expansion method were used to analyze the scattering profiles of apples. The zeroth-first order moment (Z-FOM) spectra and Fourier coefficients were extracted from each apple, which were then used for developing fruit firmness and SSC prediction models using partial least squares (PLS) and least squares support vector machine (LSSVM). The PLS models based on the Fourier coefficients improved the standard errors of prediction (SEP) by 4.8% to 19.9% for firmness and by 2.4% to 13.5% for SSC, compared with the PLS models using the Z-FOM spectra. The LSSVM models for the prediction set of Fourier coefficients achieved better SEP results, with improvements of 4.4% to 11.3% for firmness and 2.8% to 16.5% for SSC over the LSSVM models for the Z-FOM spectra data and 3.7% to 12.6% for firmness and 5.4% to 8.6% for SSC over the PLS models for the Fourier coefficients. Experiments showed that Fourier series expansion provides a simple, fast, and effective means for improving Keywords: Apples, Firmness, Fourier series expansion, Hyperspectral scattering imaging, Least squares support vector machine, Partial least squares, Soluble solids content.

Paramagnetic susceptibility and correlation functions of a [formula omitted]-dimensional classical Heisenberg ferromagnet via the two-time Green function method

In the present paper we investigate the paramagnetic susceptibility and the short-range order correlation functions of a d-dimensional classical isotropic ferromagnetic Heisenberg model with short-range exchange interactions by employing the two-time Green function method in classical statistical mechanics. Here we use Tyablikov–Callen-like decouplings for higher order Green functions and a formula for magnetization recently obtained by extension of the well known Callen method developed many years ago for the quantum isotropic Heisenberg model. Although our analysis is true for any temperature and dimensionality, we focus on one-, and some two- and three-dimensional lattices of experimental interest and derive asymptotic expressions for susceptibility and correlation functions within the paramagnetic phase close to the phase boundary and in the high-temperature regime. Besides, we present a Fourier series expansion method for deriving the high-temperature behaviors of the correlation functions. Our predictions, as obtained from a genuine classical many-body framework, may constitute a good reference point for the quantum counterparts emerging in the classical limit at the same level of approximation. Of course, although the classical spin models have unrealistic properties at sufficiently low temperatures, our classical analysis provides, in a relatively simple way as compared to a quantum treatment, an experimentally interesting scenario at finite temperature and dimensionalities d≥1.

An Orthogonal Series Expansions Method to Hedge and Price European-Type Options

Closed-form pricing formulae and option Greeks are obtained for European-type options using an orthogonal polynomial series--complex Fourier series. We assume that risky assets are driven by exponential L\'evy processes and stochastic volatility models. We provide a succinct error analysis to demonstrate that we can achieve an exponential convergence rate in the pricing method in many cases as long as we choose the correct definite computational interval. As a novel pricing method, we also numerically demonstrate that complex Fourier series are more accurate and computationally effective when implemented to price in/at/out-of-the-money options or options with complicated contingent claims.

Identifiability of mathematical models in medical biology

The analysis of biological data is a key topic in bioinformatics, computational genomics, molecular modeling, and systems biology. The methods covered in this article can reduce the cost of experiments aimed at obtaining biological data. The problem of the identifiability of mathematical models in physiology, pharmacokinetics, and epidemiology is considered. The processes considered are modeled using nonlinear systems of ordinary differential equations. The mathematical modeling of dynamic processes is based on the use of the mass conservation law. The problem of estimating the parameters characterizing the process under study raises the question of nonuniqueness. When the input and output data are known, it is useful to perform an a priori analysis of the relevance of these data. The definition of the identifiability of mathematical models is considered. Methods for the analysis of the identifiability of dynamic models are reviewed. In this review article, the following approaches are considered: the transfer function method applied to linear models (useful for the analysis of pharmacokinetic data, since a large class of drugs is characterized by linear kinetics); the Taylor series expansion method applied to nonlinear models; the differential algebra methods (the structure of this algorithm allows it to be run on a computer); and a method based on graph theory (this method allows for the analysis of the identifiability of the model and finding a proper reparametrization reducing the initial model to an identifiable one). The need to perform a priori identifiability analysis before estimating the parameters characterizing any process is demonstrated with several examples. The examples of identifiability analysis of mathematical models in medical biology are presented.

A Novel Positioning Algorithm Based on Self-adaptive Algorithm of RBF Network

Much attention has been paid to Taylor series expansion (TSE) method these years, which has been extensively used for solving nonlinear equations for its good robustness and accuracy of positioning. A Taylor-series expansion location algorithm based on the RBF neural network (RBF-TSE) is proposed before to the performance of TSE highly depends on the initial estimation. In order to have more accurate and lower cost,a new Taylor-series expansion location algorithm based on Self-adaptive RBF neural network (SA-RBF-TSE) is proposed to estimate the initial value. The proposed algorithm is analysed and simulated with several other algorithms in this paper.

Physical constraints on the non-dimensional absorption coefficients of compressional and shear waves for viscoelastic cylinders

BackgroundNormalized absorption coefficients for the longitudinal and shear waves in viscoelastic (polymer-type) materials, extracted from non-fictional experimental data showed anomalous effects, such as the generation of a negative radiation force (NRF) in plane progressive waves, negative energy absorption and extinction efficiencies and a scattering enhancement, not in agreement with energy conservation. ObjectiveThe objective of this work is directed towards analyzing those anomalies from the standpoint of energy conservation. Physical conditions which demonstrate that the ratio of the normalized absorption coefficients cannot be of arbitrary value but depends on the ratio of the square of the compressional and shear wave speeds, are established and discussed. MethodThe necessary physical condition for the validity of the linear viscoelastic (VE) model for any passive (i.e. that does not generate energy) polymeric cylinder with an ultrasonic absorption of hysteresis-type submerged in a non-viscous fluid requires that the absorption efficiency be positive (Qabs>0) since there are no active radiating sources inside the core material. This condition imposes restrictions on the values attributed to the normalized absorption coefficients for the compressional and shear-wave wavenumbers for each partial-wave mode n. The forbidden values produce anomalous/unphysical NRF, negative absorption and extinction efficiencies, as well as an enhancement of the scattering efficiency using plane progressive waves, not in agreement with energy conservation. ResultsBased on the partial wave series expansion method in cylindrical coordinates, numerical results for the radiation force, extinction, absorption and scattering energy efficiencies assuming plane progressive wave incidence are performed for three VE polymer cylinders immersed in a non-viscous host liquid (i.e. water) with particular emphasis on the shear-wave absorption coefficient, the dimensionless size parameter ka (where k is the wavenumber and a is the radius of the cylinder) and the partial-wave mode number n. Physical and mathematical conditions are established for the non-dimensional absorption coefficients of the longitudinal and shear waves for a cylinder (i.e. the 2D case) in terms of the sound velocities in the VE material. The physical condition for the spherical 3D case is also noted. ConclusionFor passive materials, the physical conditions must be always satisfied to allow accurate computations of the acoustic radiation force, torque, and energy absorption, extinction and scattering efficiencies for VE cylinders having a hysteresis type of absorption (such as polymers and plastics), and submerged in a non-viscous fluid. The physical conditions must be always satisfied regardless of the shape of the incident field. They also serve to validate and verify experimental data for VE materials and test the accuracy of related numerical computations.

Imaging of Broadband Noise from Rotating Sources in Uniform Axial Flow

A method is presented for imaging rotating broadband sources with a microphone array. The microphone array rotates virtually with the same angular frequency as the source such that the pressure data are mathematically transformed into a rotating reference frame and the influence of the moving source (Doppler shift) is compensated. In contrast with other works, this method works completely in the frequency domain by using the spherical harmonic series expansion method to calculate a modified free-space Green function in the rotating system. An analytical solution for the sound radiation from the rotating sound source, which considers motion compensation and a uniform subsonic axial flow, is deduced. Simulations on a rotating point source considering axial flow are performed. Measurements with a fan are examined for different frequencies with standard delay and sum beam-forming and high-resolution algorithms.

An isogeometric boundary element method for three dimensional potential problems

Isogeometric analysis (IGA) coupled with boundary element method, i.e. IGABEM, received a lot of attention in recent years. In this paper, we extend the IGABEM to solve 3D potential problems. This method offers a number of key improvements compared with conventional piecewise polynomial formulations. Firstly, the models for analysis in the IGABEM are exact geometrical representation no matter how coarse the discretization of the studied bodies is, thus the IGABEM ensures that no geometrical errors are produced in the analysis process. Secondly, a meshing process is no longer required, which means redundant computations are eliminated to allow analysis to be carried out with greatly reduced pre-processing. To accurately evaluate the singular integrals appearing in our method, the power series expansion method is employed. The integration surface is on the real surface of the model, rather than the interpolation surface, i.e. no geometrical errors. Thus, the value of integral is more accurate than the traditional boundary element method, which can improve the computation accuracy of the IGABEM. Some numerical examples for 3D potential problems are used to validate the solutions of the present method with analytical and numerical solutions available.

Stokes flow through a microchannel with periodical protuberances on the wall

In this study, we analyze the Stokes flow through a microchannel with semicircular periodic protuberances on the walls. The radius and period of semicircular protuberances and the stagger ratio of the staggered protuberance arrangement on both walls are arbitrary. Considering the periodicity and symmetry of the flow, the Fourier series expansion and least squared error method are applied. The stream function, pressure distribution in the flow region, and shear stress distribution on the semicircular protuberances are determined, and the results for some cases of arrangement of protuberances are presented in this paper. In particular, the average pressure gradient along the microchannel is calculated as a function of the size, period, and the stagger ratio of the protuberances on the walls. The results are compared with those obtained by lubrication analysis when the protuberances are close to the opposite protuberances or the opposite wall. As the limiting case of large periodicity, the induced pressure drops attributed to a single pair of semicircular protuberances on both walls and attributed to a single semicircular protuberance on a wall are also considered.

Axial acoustic radiation force on rigid oblate and prolate spheroids in Bessel vortex beams of progressive, standing and quasi-standing waves

The analysis using the partial-wave series expansion (PWSE) method in spherical coordinates is extended to evaluate the acoustic radiation force experienced by rigid oblate and prolate spheroids centered on the axis of wave propagation of high-order Bessel vortex beams composed of progressive, standing and quasi-standing waves, respectively. A coupled system of linear equations is derived after applying the Neumann boundary condition for an immovable surface in a non-viscous fluid, and solved numerically by matrix inversion after performing a single numerical integration procedure. The system of linear equations depends on the partial-wave index n and the order of the Bessel vortex beam m using truncated but converging PWSEs in the least-squares sense. Numerical results for the radiation force function, which is the radiation force per unit energy density and unit cross-sectional surface, are computed with particular emphasis on the amplitude ratio describing the transition from the progressive to the pure standing waves cases, the aspect ratio (i.e., the ratio of the major axis over the minor axis of the spheroid), the half-cone angle and order of the Bessel vortex beam, as well as the dimensionless size parameter. A generalized expression for the radiation force function is derived for cases encompassing the progressive, standing and quasi-standing waves of Bessel vortex beams. This expression can be reduced to other types of beams/waves such as the zeroth-order Bessel non-vortex beam or the infinite plane wave case by appropriate selection of the beam parameters. The results for progressive waves reveal a tractor beam behavior, characterized by the emergence of an attractive pulling force acting in opposite direction of wave propagation. Moreover, the transition to the quasi-standing and pure standing wave cases shows the acoustical tweezers behavior in dual-beam Bessel vortex beams. Applications in acoustic levitation, particle manipulation and acousto-fluidics would benefit from the results of the present investigation.

Predicting fatigue crack growth evolution via perturbation series expansion method based on the generalized multinomial theorem

In this study, a novel numerical calculation method is proposed to investigate the fatigue crack growth evolution in aluminum alloy sheets accounting for the measurement error. Unlike the deterministic numerical method, the initial crack length is considered to be a modified parameter with a small correction term due to the measurement error; the solution to the crack growth equation is expressed in the form of a perturbation series after introducing said small parameter. By combining the proposed perturbation series expansion method (PSEM) with the deterministic crack growth equation, a series of modified equations for predicting the crack length history are derived based on the generalized multinomial theorem. Further, by substituting the initial condition under perturbation into the modified equations, variations in crack length versus the cycle number can be obtained. The proposed method is verified by comparing numerical results with experimental data, and the results demonstrate that the proposed model is indeed feasible and effective for predicting fatigue crack growth evolution.

A Transverse Dynamic Deflection Model for Thin Plate Made of Saturated Porous Materials

AbstractIn this article, a transverse dynamic deflection model is established for thin plate made of saturated porous materials. Based on the Biot’s model for fluid-saturated porous media, using the Love–Kirchhoff hypothesis, the governing equations of transverse vibrations of fluid-saturated poroelastic plates are derived in detail, which take the inertial, fluid viscous, mechanical couplings, compressibility of solid, and fluid into account. The free vibration and forced vibration response of a simply supported poroelastic rectangular plate is obtained by Fourier series expansion method. Through numerical examples, the effect of porosity and permeability on the dynamic response, including the natural frequency, amplitude response, and the resonance areas is assessed.

Airy acoustical–sheet spinner tweezers

The Airy acoustical beam exhibits parabolic propagation and spatial acceleration, meaning that the propagation bending angle continuously increases before the beam trajectory reaches a critical angle where it decays after a propagation distance, without applying any external bending force. As such, it is of particular importance to investigate its properties from the standpoint of acoustical radiation force, spin torque, and particle dynamics theories, in the development of novel particle sorting techniques and acoustically mediated clearing systems. This work investigates these effects on a two-dimensional (2D) circular absorptive structure placed in the field of a nonparaxial Airy “acoustical-sheet” (i.e., finite beam in 2D), for potential applications in surface acoustic waves and acousto-fluidics. Based on the characteristics of the acoustic field, the beam is capable of manipulating the circular cylindrical fluid cross-section and guides it along a transverse or parabolic trajectory. This feature of Airy acoustical beams could lead to a unique characteristic in single-beam acoustical tweezers related to acoustical sieving, filtering, and removal of particles and cells from a section of a small channel. The analysis developed here is based on the description of the nonparaxial Airy beam using the angular spectrum decomposition of plane waves in close association with the partial-wave series expansion method in cylindrical coordinates. The numerical results demonstrate the ability of the nonparaxial Airy acoustical-sheet beam to pull, propel, or accelerate a particle along a parabolic trajectory, in addition to particle confinement in the transverse direction of wave propagation. Negative or positive radiation force and spin torque causing rotation in the clockwise or the anticlockwise direction can occur depending on the nondimensional parameter ka (where k is the wavenumber and a is the radius) and the location of the cylinder in the beam. Applications in acoustic levitation, long-distance particle transport and manipulation, as well as acousto-fluidics directly benefit from the results of this analysis.

Unsteady mixed convection dusty fluid flow past a vertical wedge due to small fluctuation in free stream and surface temperature

We examine the unsteady characteristics of the mixed convection boundary layer flow of dusty fluid past a vertical wedge. The free stream and surface temperature are assumed to be fluctuating with small amplitude in time about a steady non-zero mean surface temperature and free stream velocity. The set of governing equations has been solved by two distinct methods, namely, the straightforward finite difference method for the entire frequency range, and the extended series solution for low frequency range and the asymptotic series expansion method for high frequency range. The effects of varying the ratio of the particle density to the gas density, κ, and Richardson’s number, Ri, are discussed in terms of the amplitudes and phase angles of the skin friction and heat transfer and the transient skin friction and heat transfer. The unsteady behaviors of streamlines and isolines of temperature are also observed with the change of these physical parameters as well as of the amplitude of oscillation, ϵ. The results show that the presence of particle into the fluid, buoyancy caused mixed convection and fluctuations of free stream and surface temperature significantly enhance the time-dependent skin friction and heat transfer.

Uncertain dynamic analysis for rigid-flexible mechanisms with random geometry and material properties

This paper proposes an uncertain modelling and computational method to analyze dynamic responses of rigid-flexible multibody systems (or mechanisms) with random geometry and material properties. Firstly, the deterministic model for the rigid-flexible multibody system is built with the absolute node coordinate formula (ANCF), in which the flexible parts are modeled by using ANCF elements, while the rigid parts are described by ANCF reference nodes (ANCF-RNs). Secondly, uncertainty for the geometry of rigid parts is expressed as uniform random variables, while the uncertainty for the material properties of flexible parts is modeled as a continuous random field, which is further discretized to Gaussian random variables using a series expansion method. Finally, a non-intrusive numerical method is developed to solve the dynamic equations of systems involving both types of random variables, which systematically integrates the deterministic generalized-α solver with Latin Hypercube sampling (LHS) and Polynomial Chaos (PC) expansion. The benchmark slider-crank mechanism is used as a numerical example to demonstrate the characteristics of the proposed method.