Hankel Transform Research Articles

Fast and accurate computation of projected two-point functions

We present the 2-point function from Fast and Accurate Spherical Bessel Transformation (2-FAST) algorithm for a fast and accurate computation of integrals involving one or two spherical Bessel functions. These types of integrals occur when projecting the galaxy power spectrum $P(k)$ onto the configuration space, $\xi_\ell^\nu(r)$, or spherical harmonic space, $C_\ell(\chi,\chi')$. First, we employ the FFTlog transformation of the power spectrum to divide the calculation into $P(k)$-dependent coefficients and $P(k)$-independent integrations of basis functions multiplied by spherical Bessel functions. We find analytical expressions for the latter integrals in terms of special functions, for which recursion provides a fast and accurate evaluation. The algorithm, therefore, circumvents direct integration of highly oscillating spherical Bessel functions.

Analytical and numerical solutions of two-dimensional general rate models for liquid chromatographic columns packed with core–shell particles

This work is concerned with the analytical and numerical solutions of linear and nonlinear two-dimensional general rate models (2D-GRMs) describing the transport of single-solute and multi-component mixtures through chromatographic columns of cylindrical geometry packed with core–shell particles. The finite Hankel and Laplace transformations are successively applied to derive analytical solutions for a single-solute model considering linear adsorption isotherms and two different sets of boundary conditions. Moreover, analytical temporal moments are derived from the Laplace domain solutions. The process is further analyzed by numerically approximating the nonlinear 2D-GRM for core–shell particles considering multi-component mixtures and nonlinear Langmuir isotherm. A high resolution finite volume scheme is extended to solve the considered 2D-model equations. Several case studies of single-solute and multi-component mixtures are considered. The derived analytical results are validated against the numerical solutions of a high resolution finite volume scheme. Typical performance criteria are utilized to analyze the performance of the chromatographic process. The results obtained are considered to be useful to support further development of liquid chromatography.

Analysis of Square and Circular Planar Spiral Coils in Wireless Power Transfer System for Electric Vehicles

Nowadays, wireless power transfer (WPT) has attracted great attention from researchers because it is a safe, convenient, and reliable way to recharge electric vehicles. Square and circular coils are most commonly used in WPT systems. However, there are few analyses of comparing these two geometry windings in detail. In this paper, two analytical models of both square and circular planar spiral coils between two semi-infinite substrates are developed based on Fourier–Bessel transformation and dual Fourier transformation. An analytical calculation of the self- and mutual inductance can be carried out with respect to the main parameters of the WPT systems such as the line spacing, the number of turns of the coils and the properties of the substrate. The analytical models can be used to investigate the different tendencies of self- and mutual inductance of square and circular coils, which is helpful for speeding up the design process. Finally, to compare the lateral misalignment tolerance between square and circular coils, a ${{\text{600}}\,\text{mm}\,\times \,{\text{600}}\,\text{mm}}$ with a nominal 200-mm-gap prototype has been built and tested.

Two-Stage Beamforming for Phased Array Imaging Using the Fast Hankel Transform.

An ultrasound scan generates a huge amount of data. To form an image, this data has to be transferred to the imaging system. This is an issue for applications where the data transfer capacity is limited such as hand-held systems, wireless probes, and miniaturized array probes. Two-stage beamforming methods can be used to significantly reduce the data transfer requirements. In the first stage, which is applied in-probe, the amount of data is reduced from channel to scanline data. In the imaging system, the data are then beamformed to obtain images, which are synthetically focused over the entire image. Currently, two approaches exist for the second stage. The first approach is a time-of-flight (TOF) approach called synthetic aperture sequential beamforming (SASB), which has been developed for both linear and phased arrays. SASB does, however, introduce artifacts in the image that can be reduced by tapering the first-stage scanlines at the cost of lateral resolution. The second approach is based on the wave equation, but a computationally efficient method for phased arrays that is producing sector scan data is lacking. Here, we propose an algorithm that uses the fast Hankel transform to obtain a fast algorithm. The imaging performance of this method is evaluated with simulations and experiments. Compared with PSASB, which is an adaption of SASB for phased arrays, our method requires a similar amount of operations to construct the entire image and there is no tradeoff between resolution and artifacts. These results show the advantage of using the wave equation instead of a TOF approach.

Convolution with the linear canonical Hankel transformation

In this work, we introduce translation and convolution for linear canonical Hankel transformations and studied some inequalities. For the particular values of linear canonical Hankel transformation (i.e., for Hankel–Clifford transformation), we investigate linear time-invariant filters. Furthermore, some applications of linear canonical Hankel transformation to a generalized non-linear parabolic equation and a canonical convolution integral equation are given.

A pair of linear canonical Hankel transformations and associated pseudo-differential operators

In this work, we have introduced a pair of linear canonical Hankel transformations and investigated some of its properties on Zemanian-type spaces. Moreover two versions of pseudo-differential operator associated with canonical Hankel transformations are defined and discussed its integral representation.

Effects of thermal and electric boundary conditions on fracture of three-dimensional thermopiezoelectric media

An arbitrarily shaped planar crack under different thermal and electric boundary conditions on the crack surfaces is studied in three-dimensional transversely isotropic thermopiezoelectric media subjected to thermal–mechanical–electric coupling fields. Using Hankel transformations, Green functions are derived for unit point extended displacement discontinuities in three-dimensional transversely isotropic thermopiezoelectric media, where the extended displacement discontinuities include the conventional displacement discontinuities, electric potential discontinuity, as well as the temperature discontinuity. On the basis of these Green functions, the extended displacement discontinuity boundary integral equations for arbitrarily shaped planar cracks in the isotropic plane of three-dimensional transversely isotropic thermopiezoelectric media are established under different thermal and electric boundary conditions on the crack surfaces, namely, the thermally and electrically impermeable, permeable, and semi-permeable boundary conditions. The singularities of near-crack border fields are analyzed and the extended stress intensity factors are expressed in terms of the extended displacement discontinuities. The effect of different thermal and electric boundary conditions on the extended stress intensity factors is studied via the extended displacement discontinuity boundary element method. Subsequent numerical results of elliptical cracks subjected to combined thermal–mechanical–electric loadings are obtained.

Dynamic response of a pile embedded in elastic half space subjected to harmonic vertical loading

An analytical method is developed to investigate the dynamic response of a pile subjected to harmonic vertical loading. The pile is modeled as a one-dimensional (1D) elastic rod. The elastic soil is divided into a homogeneous half space underlying the base of pile and a series of infinitesimally thin layers along the vertical shaft of pile. The analytical solution for the soil-pile dynamic interaction problem is obtained by the method of Hankel transformation. The proposed solution is compared with the classical plane strain solution. Arithmetical examples are presented to demonstrate the sensitivity of the vertical impedance of the pile to relevant parameters.

Effect of Deformation on Semi–infinite Viscothermoelastic Cylinder Based on Five Theories of Generalized Thermoelasticity

We considera dynamical problem for semi-infinite viscothermoelastic semi infinite cylinder loaded mechanically and thermally and investigated the behaviour of variations of displacements, temperatures and stresses. The problem has been investigated with the help of five theories of the generalized viscothermoelasticity by using the Kelvin – Voigt model. Laplace transformations and Hankel transformations are applied to equations of constituent relations, equations of motion and heat conduction to obtain exact equations in transformed domain. Hankel transformed equations are inverted analytically and for the inversion of Laplace transformation we apply numerical technique to obtain field functions. In order to obtain field functions i.e. displacements, temperature and stresses numerically we apply MATLAB software tools. Numerically analyzed results for the temperature, displacements and stresses are shown graphically.

Precision calculations of the cosmic shear power spectrum projection

We compute the spherical-sky weak-lensing power spectrum of the shear and convergence. We discuss various approximations, such as flat-sky, and first- and second- order Limber equations for the projection. We find that the impact of adopting these approximations is negligible when constraining cosmological parameters from current weak lensing surveys. This is demonstrated using data from the Canada-France-Hawaii Telescope Lensing Survey (CFHTLenS). We find that the reported tension with Planck Cosmic Microwave Background (CMB) temperature anisotropy results cannot be alleviated. For future large-scale surveys with unprecedented precision, we show that the spherical second-order Limber approximation will provide sufficient accuracy. In this case, the cosmic-shear power spectrum is shown to be in agreement with the full projection at the sub-percent level for l > 3, with the corresponding errors an order of magnitude below cosmic variance for all l. When computing the two-point shear correlation function, we show that the flat-sky fast Hankel transformation results in errors below two percent compared to the full spherical transformation. In the spirit of reproducible research, our numerical implementation of all approximations and the full projection are publicly available within the package nicaea at http://www.cosmostat.org/software/nicaea.

Displacement response analysis of a floating ice plate under a triangular pulse load

The dynamic equation of a viscoelastic floating ice plate under a triangular pulse load is solved analytically through the Hankel transformation and the Laplace transformation. The effects of physical and geometrical parameters of the problem on the displacement response as a function of time and spatial coordinate are discussed.

Generalized functions and Laguerre expansions

In this paper we study and characterize expansions of distributions in the Zemanian spaces, \({\mathcal {H}}_{\mu }\) and its dual \({\mathcal {H}}_{\mu }^{\prime }\) (\(\mu \ge -\frac{1}{2}\)) with respect to Laguerre functions. We obtain as applications of this result, the kernel Theorem and a structure Theorem for \({\mathcal {H}}_{\mu }^{\prime }\). We also introduce a new algebra of generalized functions in the sense of J. F. Colombeau such that it satisfies interesting properties involving the Hankel transformation and Hankel convolution.

Analysis of linear two-dimensional general rate model for chromatographic columns of cylindrical geometry

This work is concerned with the analytical solutions and moment analysis of a linear two-dimensional general rate model (2D-GRM) describing the transport of a solute through a chromatographic column of cylindrical geometry. Analytical solutions are derived through successive implementation of finite Hankel and Laplace transformations for two different sets of boundary conditions. The process is further analyzed by deriving analytical temporal moments from the Laplace domain solutions. Radial gradients are typically neglected in liquid chromatography studies which are particularly important in the case of non-perfect injections. Several test problems of single-solute transport are considered. The derived analytical results are validated against the numerical solutions of a high resolution finite volume scheme. The derived analytical results can play an important role in further development of liquid chromatography.

The relation between Bessel wavelet convolution product and Hankel convolution product involving Hankel transform

In this paper, the relation between Bessel wavelet convolution product and Hankel convolution product is obtained by using the Bessel wavelet transform and the Hankel transform. Approximation results of the Bessel wavelet convolution product are investigated by exploiting the Hankel transformation tool. Motivated from the results of Pinsky, heuristic treatment of the Bessel wavelet transform is introduced and other properties of the Bessel wavelet transform are studied.

Numerical modelling of flexible pavement incorporating cross-anisotropic material properties. Part I: Surface circular loading

Accurate numerical modelling of the behaviour of road pavement layers is an important requirement for the design and evaluation of road pavements. This modelling includes the prediction of pavement performance under the action of traffic loading and environmental factors. Depending on the complexity of the models, properties of pavement layers that may be considered are wide-ranging - from linear or nonlinear elastic to cross-anisotropic through to linear visco-elasto-plastic. Some properties, such as cross-anisotropic, are not only related to placement and compaction of the pavement layers, but are also inherent to the materials used. Other properties, such as linear visco-elasto-plastic, are specific to asphalt concrete and depend on the speed and magnitude of traffic loading, as well as the environment (temperature) in which the road is located. This paper presents basic theoretical derivation of numerical modelling of a flexible pavement considering cross-anisotropic material properties (with isotropic properties as a special case). The solutions derived in this paper are based on Hankel transformation of Navier's equations. The accuracy and validity of the solutions are verified through comparisons with a proprietary finite element method (FEM) package. For this purpose, a pavement structure composed of five main layers constituted by isotropic and cross-anisotropic (also known as transversely isotropic) material properties is analysed. In order to vary some of the layer properties with depth, the main layers were sub-layered, resulting in a 17-layer pavement system.

New results for the Hankel transform via the Hankel potential and Hankel resolvent

ABSTRACTIn this paper, we characterize the support of the Hankel transform under the behaviour of -norms of iterated Hankel resolvent. Next, we extend the result under the behaviour of -norms of iterated Hankel potentials.

Analysis of an interface crack of arbitrary shape in a three-dimensional transversely isotropic magnetoelectrothermoelastic bimaterial—part 1: Theoretical solution

ABSTRACTThe extended displacement discontinuity boundary integral–differential equation method is developed for the analysis of an interface crack of arbitrary shape in a three-dimensional (3D), transversely isotropic magnetoelectrothermoelastic bimaterial. The extended displacement discontinuities (EDDs) include conventional displacement discontinuities, electric potential discontinuity, magnetic potential discontinuity as well as temperature discontinuity across the interface crack faces, correspondingly, while the extended stresses represent conventional mechanical stresses, electric displacement, magnetic induction, heat flux, etc. By virtue of the potential functions and Hankel transformation technique, the fundamental solutions for unit-point EDDs on the interface in a 3D transversely isotropic magnetoelectrothermoelastic bimaterial are derived, then the extended displacements and stresses are all obtained in terms of EDDs. An analysis method is proposed based on the analogy with the solution in an isotropic thermoelastic bimaterial. The singular indices and the singular behaviors of the near crack-tip fields are studied. The combined extended stress intensity factors for three new fracture modes are derived in terms of the EDDs and are compared with those in magnetoelectroelastic bimaterials.

Matlab Code for the Discrete Hankel Transform

Previous definitions of a Discrete Hankel Transform (DHT) have focused on methods to approximate the continuous Hankel integral transform without regard for the properties of the DHT itself. Recently, the theory of a Discrete Hankel Transform was proposed that follows the same path as the Discrete Fourier/Continuous Fourier transform. This DHT possesses orthogonality properties which lead to invertibility and also possesses the standard set of discrete shift, modulation, multiplication and convolution rules. The proposed DHT can be used to approximate the continuous forward and inverse Hankel transform. This paper describes the matlab code developed for the numerical calculation of this DHT.

Sinusoidal Frequency Modulation Fourier-Bessel Series for Multicomponent SFM Signal Estimation and Separation

Multicomponent sinusoidal frequency modulated (SFM) signals are widely used in radar, acoustics, and biomedicine. The instantaneous frequency (IF) characterizes important physical parameters of the real applications. In this paper, a sinusoidal frequency modulation Fourier-Bessel (SFMFB) series is defined for IF estimation. It provides the signal decomposition on the Bessel function basis with a finer resolution, which proposes an extension of the performance and the applicability of the classic Fourier-Bessel transform (FBT). Based on the property analysis of the SFMFB series, an algorithm of IF estimation and signal separation is introduced. Unlike the existing estimation methods which apply sliding windows to make an instantaneous approximation, the proposed method uses the global data, which provides a longer period gain, therefore achieving a better estimation performance. Moreover, considering that most estimation methods are invalid in multicomponent separation, the individual signals are well separated by the proposed algorithm, which facilitates the further monocomponent analysis. A performance comparison between the proposed method, the FBT, and another recently proposed sinusoidal frequency modulation Fourier transform (SFMFT) is also provided. Simulation results indicate that the proposed method outperforms the existing methods in estimation precision and computation load, and it is free of interference which exists in SFMFT.

ДИНАМИЧЕСКАЯ ОСЕСИММЕТРИЧНАЯ ЗАДАЧА ПРЯМОГО ПЬЕЗОЭФФЕКТА ДЛЯ КРУГЛОЙ БИМОРФНОЙ ПЛАСТИНЫ

The dynamic axisymmetric problem for a round bimorph structure consisting of the metal substrate and the axially polarized piezoceramic plate is considered. Its bending modes are caused by the action of the mechanical loading (normal tension) on the side surface; the loading is a function of the radial coordinate and time. The rigid and hinge supports of the plate are considered. The initial design relations are formulated for the piezoceramic material with the hexagonal crystal lattice of the 6 mm class. In order to solve the problem of the theory of electrodynamics in a three-dimensional formulation, the finite integral Hankel transformations along the axial coordinate and a generalized transformation (FIT) over the radial variable is used. At each stage, the standardization is carried out which allows implementing an appropriate transformation algorithm. In the first case the boundary conditions are presented in a mixed form; and in the second case they are homogeneous by introducing auxiliary functions. This approach allows gaining precise (within the used models) calculated ratios in a most general form. The built closed solution allows defining the frequency range of the axisymmetric oscillations, the stress-strain state and the nature of the changing induced electric field of the bimorph plate. This makes it possible to establish the conventional solutions of the designed devices, determine a way of fixing the electrical signal, pick up all the geometrical and physical characteristics of the typical elements of the piezoceramic transducers Also, the developed solution allows solving the problems of the elasticity and electroelasticity theory for circular thick and thin plates with an arbitrary number of layers under most general loading conditions without the use of kinematic hypotheses.