Fourier Series Research Articles

An adaptive lattice Green's function method for external flows with two unbounded and one homogeneous directions

We solve the incompressible Navier-Stokes equations using a lattice Green's function (LGF) approach, including immersed boundaries (IB) and adaptive mesh refinement (AMR), for external flows with one homogeneous direction (e.g. infinite cylinders of arbitrary cross-section). We hybridize a Fourier collocation (pseudo-spectral) method for the homogeneous direction with a specially designed, staggered-grid finite-volume scheme on an AMR grid. The Fourier series is also truncated variably according to the refinement level in the other directions. We derive new algorithms to tabulate the LGF of the screened Poisson operator and viscous integrating factor. After adapting other algorithmic details from the fully inhomogeneous case [1], we validate and demonstrate the new method with transitional and turbulent flows over a circular cylinder at Re=300 and Re=12,000, respectively.

Acoustic properties and attenuation of coupled shaft-submarine hull system under various excitation transfer paths

Pump-jet propulsor excitation transfers to submarine hull along rotor-shaft and duct-stator paths simultaneously. The investigations on the effects of excitation transfer paths on structural vibration and acoustic radiation of submarine are limited. The present work aims to investigate vibro-acoustic characteristics of coupled shaft-submarine hull system utilizing a theoretical wavenumber analysis method and conduct acoustic design. The energy functional of the coupled structure-fluid system of the research object is first developed, and the displacement components of the jointed shell and the acoustic pressure are expanded by the Fourier series along circumferential direction. This allows for obtaining vibro-acoustic responses in the circumferential wavenumber-frequency domain, by which the predominant wavenumbers contributing to acoustic radiation are identified. The discussions reveal that the modes n = 0 and n = 1 respectively dominate the acoustic radiation under axial and vertical rotor loads. The acoustic radiations under duct-stator load are mainly contributed by mode n = 0, and the higher order modes n = 1 and n = 2 determine several acoustic peaks. Furthermore, two acoustic design schemes are proposed to suppress the wavenumbers with high radiation efficiency. It is proven that the design of the symmetric inner foundation and the application of new material are two efficient ways to improve acoustic performance of the submarine.

Buckling and vibro-acoustic behavior analysis of laminated immersed cylindrical shells with hydrostatic pressure based on Chebyshev-Fourier spectral approach

The buckling and vibroacoustic behaviour of laminated cylindrical shell structures is vital for the safety and operational efficiency of underwater vehicles. This paper introduces a Chebyshev-Fourier spectral approach for the comprehensive analysis of buckling and vibro-acoustic behaviour in water-immersed laminated cylindrical shells under external hydrostatic pressure. The approach combines Chebyshev polynomials and Fourier series to approximate the displacement of shell and acoustic pressure of the fluid-structure-interaction (FSI) surface in cylindrical coordinates. An approximate solution is used in the axial direction, while an analytical solution is used in the circumferential direction, allowing for the analysis of different circumferential wave numbers n. The influence of external fluid pressure is characterized through the Helmholtz boundary integral equation, while the effect of hydrostatic pressure is deduced using linear elastic theory. From a theoretical derivation standpoint, hydrostatic pressure leads to a reduction in equivalent stiffness, while the external fluid contributes to an increase in the effective mass matrix. Validated against established literature analysis, the study finds that hydrostatic pressure decreases the natural frequencies, though the impact is less significant for lower circumferential wave numbers. For these lower wave numbers, the overall radiated sound power remains largely independent of external hydrostatic pressure, regardless of being in air or water. However, at higher circumferential wave numbers, the effect is more pronounced. This study not only advances the theoretical framework for analysing submerged cylindrical shells but also provides crucial insights into designing more robust structures for underwater applications.

Analytical seismic model for a tunnel with segmental liner buried in the half-space soil

In this paper, by using the wave function expansion (WFE) and the expansion of the cylindrical wave into plane wave (ECPW) methods, an analytical method for the dynamic response of a circular tunnel with segmental liner buried in the half-space soil to harmonic elastic waves is developed. The liner of the tunnel is supposed to consist of several segments and joints. Both the segments and joints are treated as open cylindrical shells, and the segment and joint shells together thus form an equivalent continuous shell (ECS) liner, and the thin shell theory is utilized to describe its vibration. The scattered wave field due to the presence of the tunnel and boundary of the half-space soil is divided into two parts, namely, the direct scattered wave field and secondary scattered wave field. The expressions for the direct scattered cylindrical waves in the soil is determined via the WFE method, while the secondary scattered waves form the boundary of the half-space soil are obtained by the ECPW method together with the application of the boundary condition along the soil surface. Applying the cylindrical shell theory on the ECS liner and using the Fourier series expansions for the variables and parameters of the ECS liner along the azimuthal direction together with introducing the Fourier component constitutive relation for the ECS liner, a system of equations for the Fourier component ECS displacements are derived with the differential equations for the ECS liner displacements. By using the system of equations, the expressions for the free and scattered wave fields and the continuity conditions at the interface between the liner and soil, the system of equations for the ECS liner coupled with the soil are derived. By using the developed analytical method for the tunnel, some results of the ECS tunnel under different incident waves are given.

Solution of the Poisson equation by the boundary integral method

PurposeThe boundary integral method (BIM) is very attractive to practicing engineers as it reduces the dimensionality of the problem by one, thereby making the procedure computationally inexpensive compared to its peers. The principal feature of this technique is the limitation of all its computations to only the boundaries of the domain. Although the procedure is well developed for the Laplace equation, the Poisson equation offers some computational challenges. Nevertheless, the literature provides a couple of solution methods. This paper revisits an alternate approach that has not gained much traction within the community. The purpose of this paper is to address the main bottleneck of that approach in an effort to popularize it and critically evaluate the errors introduced into the solution by that method.Design/methodology/approachThe primary intent in the paper is to work on the particular solution of the Poisson equation by representing the source term through a Fourier series. The evaluation of the Fourier coefficients requires a rectangular domain even though the original domain can be of any arbitrary shape. The boundary conditions for the homogeneous solution gets modified by the projection of the particular solution on the original boundaries. The paper also develops a new Gauss quadrature procedure to compute the integrals appearing in the Fourier coefficients in case they cannot be analytically evaluated.FindingsThe current endeavor has developed two different representations of the source terms. A comprehensive set of benchmark exercises has successfully demonstrated the effectiveness of both the methods, especially the second one. A subsequent detailed analysis has identified the errors emanating from an inadequate number of boundary nodes and Fourier modes, a high difference in sizes between the particular solution and the original domains and the used Gauss quadrature integration procedures. Adequate mitigation procedures were successful in suppressing each of the above errors and in improving the solution accuracy to any desired level. A comparative study with the finite difference method revealed that the BIM was as accurate as the FDM but was computationally more efficient for problems of real-life scale. A later exercise minutely analyzed the heat transfer physics for a fin after validating the simulation results with the analytical solution that was separately derived. The final set of simulations demonstrated the applicability of the method to complicated geometries.Originality/valueFirst, the newly developed Gauss quadrature integration procedure can efficiently compute the integrals during evaluation of the Fourier coefficients; the current literature lacks such a tool, thereby deterring researchers to adopt this category of methods. Second, to the best of the author’s knowledge, such a comprehensive error analysis of the solution method within the BIM framework for the Poisson equation does not currently exist in the literature. This particular exercise should go a long way in increasing the confidence of the research community to venture into this category of methods for the solution of the Poisson equation.

Fractional Fourier Series on the Torus and Applications

Fractional Fourier Series on the Torus and Applications

An analytical model for daily‐periodic slope winds. Part 2: Solutions

AbstractThis article presents an analytical model for the diurnal cycle of slope‐normal profiles of potential temperature and wind speed characterizing thermally driven slope winds, generated by a daily‐periodic surface energy budget. The model extends the solution proposed by Zardi and Serafin, originally formulated for a pure sinusoidal surface forcing temperature. To account for the asymmetric features characterizing the daytime and nighttime phases, a full Fourier series expansion is derived, the coefficients and phases of which are prescribed from the surface energy budget driven by the daily‐periodic radiation model described in Part 1 of the present work. The model is applicable for any slope angle () and orientation, at any latitude and elevation (up to 2500 m), and for all seasons. Despite some inherent limitations, the most remarkable being the absence of moist processes and latent heat fluxes, the model captures most key features of daily‐periodic slope wind systems, in particular the asymmetry between daytime and nighttime phases. Moreover, it allows exploration of the sensitivity of these flows to the various factors concurring in their development, and offers a basis for more realistic analytical solutions for slope winds.

Evaluations and relations for finite trigonometric sums

Evaluations and relations for finite trigonometric sums

Analytical method for free vibration of steel pipe piles partially submerged in water and penetrated in layered soil

This paper presents a novel analytical method for dynamic characteristics of steel pipe piles submerged in water based on the thin shell theory by using the state space method. The state equation with respect to the length direction is first derived by idealizing the steel pipe pile in water as a thin cylindrical shell, where the soil around the pile is modelled as an elastic foundation with stiffnesses in three directions and the surrounding seawater is considered through added equivalent mass. The governing equations for the free vibration of steel pipe piles are derived after the expansion of Fourier series for the state vectors along the circumferential direction. The frequency equation for free vibration and its corresponding mode shapes are then obtained, which is used to analyze the dynamic characteristics of steel pipe piles in water. Due to the advantage of the state space method, arbitrary boundary conditions of the steel pipe pile are conveniently achieved as well as the effects of seawater, soil, the piling hammer and guiding frame. Finally, the present analytical method is verified and demonstrated in detail by several numerical examples, and results show its simplicity and efficiency.

Intermediate-point bracing for columns: New stiffness–buckling load formulae

According to the stiffness equation provided in the American Institute of Steel Construction (AISC) design specifications for intermediate-point braces, the required point brace stiffness increases with the number of point braces. This research aims to formulate a new stiffness design equation for intermediate-point braces that accounts for the decline in required stiffness with an increasing number of braces. The brace stiffness (β)–buckling load (Pcr) relationships for columns with point braces were investigated by performing a buckling analysis using the energy method with the Fourier series. A formula was derived, defining the β−Pcr relationships for each buckling mode of the columns, regardless of the number of point braces and whether the buckling is elastic or inelastic. The AISC stiffness equations for point braces were significantly conservative under most buckling loads. The proposed stiffness design equation requires lower brace stiffness with an increasing number of point braces, resulting in smaller brace members than the AISC design equations. Additionally, the proposed strength design equation can set relatively high strength requirements for the brace, reflecting that stiffness governs the design of the point brace. These new equations offer a more precise and efficient method for calculating required brace stiffness, providing valuable insights to improve structural design capabilities.

Statistical inference on nonparametric regression model with approximation of Fourier series function: Estimation and hypothesis testing

Nonparametric regression is an approximation method in regression analysis that is not constrained by the assumption of knowing the regression curve. One of the functions to approximate the curve is a Fourier series function. The nonparametric regression model with approximation of a Fourier series function has been widely discussed by several researchers. However, discussions on statistical inference, particularly in partial hypothesis testing, has not been carried out previously. Therefore, the purpose of this research is to discuss the statistical inference on nonparametric regression model with approximation of a Fourier series function. The discussion includes parameter and model estimations, simultaneous and partial hypotheses testing. In the application, we use life expectancy data from East Java Province during 2022. Based on data analysis, we obtain a model estimation with an R-square value of 96.24 %. At a 5 % significance level, the parameters simultaneously have a significant influence on the model. Partially, four parameters are not significant. However, overall, the predictor variables significantly influence the life expectancy data.•The Fourier series function used is a Fourier series function introduced by Bilodeau (1992).•The model estimation is obtained by selecting the optimal number of oscillation parameters.•The statistical test is obtained using the LRT method.

Variability of SST through Koopman Modes

Abstract The majority of dynamical systems arising from applications show a chaotic character. This is especially true for climate and weather applications. We present here an application of Koopman operator theory to tropical and global sea surface temperature (SST) that yields an approximation to the continuous spectrum typical of these situations. We also show that the Koopman modes yield a decomposition of the datasets that can be used to categorize the variability. Most relevant modes emerge naturally, and they can be identified easily. A difference with other analysis methods such as empirical orthogonal function (EOF) or Fourier expansion is that the Koopman modes have a dynamical interpretation, thanks to their connection to the Koopman operator, and they are not constrained in their shape by special requirements such as orthogonality (as it is the case for EOF) or pure periodicity (as in the case of Fourier expansions). The pure periodic modes emerge naturally, and they form a subspace that can be interpreted as the limiting subspace for the variability. The stationary states therefore are the scaffolding around which the dynamics takes place. The modes can also be traced to the Niño variability and in the case of the global SST to the Pacific decadal oscillation (PDO). Significance Statement We compute the Koopman modes for the tropical and global SST, demonstrating that significant dynamical modes can be identified also in complex high-dimensional datasets with continuous spectra. The Koopman modes are then used to identify the stationary subspace, namely, the limit subspace for the invariant evolution of the system.

A novel method of vehicle height control utilizing semi-active actuator

Inspired by the physical phenomenon that the balance position of the vehicle body changes due to a suspension system with asymmetric damping, this paper presents a novel method to control the vehicle height utilizing semi-active actuators. The proposed vehicle height control method uses the asymmetric damping of the semi-active actuator to control the raising and lowering of the vehicle body, which essentially harnesses the energy from the vehicle’s driving vibrations and has the characteristics of low cost, energy saving and fast response. Aiming at the strong nonlinearity and discontinuity of the piecewise linear nonlinear system of the vehicle body vibration, the homotopy analysis method (HAM) is used to solve the high-order analytical solution, the non-smooth terms are processed through Fourier series expansion, and the iteration method is combined to treat the iteration result as the initial guess solution for the next iteration, improving the convergence accuracy of the series solution and expands the application scope of the HAM. The feasibility of the presented method is investigated from an analytical point of view and the influences of some critical parameters on the responses of the system are analyzed. Finally, the magnetorheological damper (MRD) produced by LORD company is utilized as the actuator, and the vehicle height control experiments system is established by using NI-Labview software and NI-PXI hardware platform, The experiments are performed to verify the feasibility of the proposed vehicle height control method and the system response laws. The proposed method realizes the functions of active actuators using semi-active actuators, and tries a new method and approach for vehicle height control under specific working conditions, such as short-term obstacle surmounting, high-speed steering, emergency avoidance, and rollover prevention.

Uniform Convergence of Fourier Series for Hölder Continuous Functions and Functions of Bounded Variation

This article discusses two significant results about the uniform convergence of Fourier series. One is the uniform convergence of the Fourier series for Hölder-α(α>0) continuous functions, while the other is the uniform convergence of the Fourier series for functions that can be expressed as the sum of finitely many continuous monotonic functions. The two results are particularly useful. Because Hölder continuity condition ensures a specific regularity that facilitates the analysis and provides a robust framework for proving uniform convergence and many practical functions can be decomposed into monotonic components, making the theorem broadly applicable. By providing detailed proofs and applications, this study demonstrates the significant implications of uniform convergence in both theoretical and practical contexts. Moreover, this work paves the way for future research in higher-dimensional Fourier analysis, non-periodic functions, and adaptive Fourier techniques, underscoring the ongoing relevance and versatility of Fourier series in modern mathematics. Several well-known results emerge as corollaries of these findings, highlighting the robustness and applicability of these theorems in Fourier analysis.

Response of semicircular canyons and movable cylindrical cavities to SH waves in anisotropic half-space geology

The development of tunnels or the laying of underground pipelines are essential engineering projects in modern society, and in canyon tunnels and underground pipeline projects, the surface motion and cavity edge motion have been topics of concern in ground vibration problems. We investigate the wave scattering problem in an elastic half-space anisotropic medium containing a semicircular canyon and a subsurface movable cylindrical cavity by using the wave function expansion method, the complex function method, and the mirror method. By deriving the governing equation and transforming it into the standard form of the Helmholtz equation satisfying the zero-stress boundary condition, we solve the corresponding displacement functions. Introducing a position correction coefficient, the scattered wavefield in a half-space anisotropic medium is constructed by the mirror method, which improves the problem of scattered wave source singularity in anisotropic half-space media. Then, combining the free boundary conditions with a Fourier series expansion method, we solve the unknown coefficients in the equations. The correctness of the method is verified by degenerating it to a classical analytic solution. Finally, using frequency- and time-domain analysis, we investigate the effects of the relevant parameters on the surface motion [Formula: see text], the dynamic stress concentration factor, and the displacement amplitude [Formula: see text]. The results indicate that rock anisotropy and the presence of semicircular canyons have a significant effect on the dynamic response of subsurface structures. This not only provides a theoretical basis for practical unlined tunnels or pipeline projects but also can provide a basis for seismic design of underground structures.

Long-range near-side correlation in e+e− collisions at 183-209 GeV with ALEPH archived data

The first measurement of two-particle angular correlations for charged particles with LEP-II data is presented. The study is performed using archived hadronic e+e− data collected by ALEPH at center-of-mass energies up to 209 GeV, above the W+W− production threshold, which provide access to unprecedented charged-particle multiplicities and more complex color-string configurations if compared to previous measurements at LEP-I energies. An intriguing long-range near-side excess is observed in the correlation function measured with respect to the thrust axis in the highest multiplicity interval (Ntrk≥50). Such a structure is not predicted by the Monte-Carlo simulation. The harmonic anisotropy coefficients vn, which result from the Fourier expansion of the two-particle correlation functions, were also measured for the first time in e+e− data, and compared to pythia 6 predictions and to the results obtained in proton-proton collisions. The results presented in the Letter provide novel experimental constraints on the formation of collective phenomena in point-like e+e− collisions.

Phase-Resolved Functional Lung MRI for Pulmonary Ventilation and Perfusion (V/Q) Assessment.

Fourier decomposition is a contrast agent-free 1H MRI method for lung perfusion (Q) and ventilation (V) assessment. After image registration, the time series of each voxel is analyzed with regard to the cardiac and breathing frequency components. Using a standard 2D spoiled gradient-echo sequence with a temporal resolution of ~300 ms, an image-sorting algorithm was developed to produce phase-resolved functional lung imaging (PREFUL) with an increased temporal resolution. Thus, it is feasible to evaluate regional flow volume loops (FVL) during tidal volume breathing and depict the propagation of the pulse wave during the cardiac cycle. This method can be applied at 1.5T or 3T with standard MR hardware without the necessity for sequence programming, as the described protocol can be implemented with the default SPGRE sequence on most systems. PREFUL ventilation MRI has been validated using 129Xe and 19F gas imaging with good regional agreement. Perfusion-weighted PREFUL MRI has been validated using SPECT as well as dynamic contrast enhanced (DCE) MRI. PREFUL has been tested in a dual center dual vendor setting and is currently applied in several ongoing multicenter trials. Furthermore, it is feasible across a range of field strengths (0.55T-3T) and different age groups, including newborns. Quantitative V/Q PREFUL MRI has been used in patients with cystic fibrosis, chronic obstructive pulmonary disease, chronic thromboembolic pulmonary hypertension, and corona virus disease-2019 to quantify disease and monitor treatment change after therapy. Furthermore, PREFUL V/Q imaging has been shown to predict transplant loss due to chronic lung allograft dysfunction in patients after lung transplantation. In summary, PREFUL MRI is a validated technique for quantitative ventilation and pulmonary pulse wave/perfusion imaging for regional pulmonary disease detection, quantification, and treatment monitoring with potential added value to the current clinical routine.

Degree of approximation of functions conjugate to the functions belonging to Lip((𝜉1,𝜉2)𝑟)-class through double Nörlund means

Abstract In this paper, we determine the error of approximation of functions, conjugate to the functions of two variables ( 2 ⁢ π 2\pi -periodic in both variables) belonging to the Lipschitz class Lip ( ( ξ 1 , ξ 2 ) ; r \operatorname{Lip}((\xi_{1},\xi_{2});r ) ( r ≥ 1 r\geq 1 ), through the double Nörlund means of its conjugate double Fourier series. Additionally, in the form of corollaries, we present some particular cases of the theorem proved here.

Semi-analytical vibration modeling of complex axisymmetric shells using shifted Legendre series

This paper presents a semi-analytical approach based on the shifted Legendre series for vibration modeling of complex axisymmetric shells. This method divides the structure along its generator into complex axisymmetric shells comprised solely of conical and spherical shell segments. A global cylindrical coordinate system is established to unify the coordinate system of complex axisymmetric shells. Coupling between segments and the simulation of arbitrary boundary conditions are achieved using artificial spring technology. Based on the first-order shear deformation theory (FSDT), the energy functional of each segment is derived. The circumferential displacement is expanded using Fourier series, while the meridional displacement is expanded using the shifted Legendre series at shifted Gauss-Lobatto nodes. Utilizing the inner product matrix and differential matrix of the shifted Legendre series, the integration and differentiation processes in the vibration characteristic equation are simplified into a product form, significantly enhancing computational efficiency. Through convergence analysis and comparison of the computational results of the present method with numerical simulation results, experimental results and literature results demonstrate the advantages of high convergence, high accuracy and fast solution speed of this solution method. On this basis, the vibration characteristics of ring-stiffened double-layer hemispherical shells are investigated. The results demonstrate the great applicability of this method for vibration modeling of complex axisymmetric shells in engineering.

A method for evaluating the design scheme of continuous measuring bridge of the instrumented wheelset

This paper proposes a method for evaluating the design scheme of continuous measuring bridge of the instrumented wheelset, based on the theory of continuous measurement of wheel-rail forces, Finite Element Analysis, and Levenberg-Marquardt algorithm. The elastic strain at each placement point of the strain gauge on the web surface of the wheel is expanded into Fourier series, and the general mathematical expressions of the output signals for each symmetric component, each bridge, and the synthesized bridge are provided. Nine typical measuring bridge schemes of strain gauge arrangement on the wheel web surface and seven typical measuring bridge schemes of strain gauge arrangement inside the measuring holes are given. The finite element model of the wheel is established using software ABAQUS, and the moving force is applied to the rolling surface of the wheel for simulation calculations. The output signals of 16 typical measuring bridges are then obtained. By utilizing the mathematical formulas of the bridge outputs and the simulation results, a matrix expression describing the relationship between amplitudes of all harmonics and the bridge output is given, and the amplitudes of all the former 20 orders harmonics of the bridge signal are obtained by applying Levenberg-Marquardt algorithm. Based on the analysis results of the harmonic amplitudes, a method is proposed to quantitatively evaluate the output signals of a single bridge (which can be either harmonic or triangular wave) and the synthesized bridge. The mathematical expressions of the evaluation indicators are provided within the range of 0 to 1. Finally, the analysis results of the evaluation indicators for 16 typical measuring bridges are presented and compared. The proposed method is not only applicable to the bridge design of the instrumented wheelset but also to the evaluation of signals from channels of the data collector when the instrumented wheelset is being calibrated on the test rig.