Double Fourier Research Articles

Unsteady double-diffusive Brinkman–Bénard convection in cylindrical enclosure saturated with hybrid bi-viscous Bingham nanoliquid

This paper aims to present an analytical as well as a comparative study to investigate the effect of the Brinkman porous medium on the onset of regular and chaotic motion in cylindrical enclosures of different heights. The hybrid bi-viscous Bingham nanoliquid is considered as the working fluid. Modified Brinkman–Buongiorno and bi-viscous Bingham fluid models are incorporated to obtain the flow governing dynamics. The thermophysical properties of the hybrid nanoliquid are calculated using phenomenological laws and the mixture theory. The study is carried out for the axisymmetric mode, and the Bessel functions are taken as the eigenfunctions of the problem. Double Fourier–Bessel series expansions are used for weakly non-linear stability analysis. The limiting cases of the study are obtained, and the results on the onset of convection, heat, and mass transport are discussed graphically. The behavior of the dynamical system is analyzed using the maximum Lyapunov exponent plot, the bifurcation diagram, and phase plots. Outcomes suggest that convection sets in earlier in the water-based hybrid nanoliquid than in the bi-viscous Bingham hybrid nanoliquid. The use of Single-walled carbon nanotubes enhances the heat transfer rate by approximately 17%. Further, it is concluded that a tall cylindrical enclosure is the most favorable geometry for achieving a higher heat transfer rate among the others.

Delta function of the Fourier transform

In the analysis of the double window Fourier transform, it is found that both components of the direct transformation contain an equal amount of information sufficient to restore the original time signal. A double conversion diagram has been constructed. For any initial signal, there is a conjugate signal having the same frequency spectrum. Analysis of the harmonic signal transformation showed that at an arbitrary phase of the harmonic, the frequency representation contains two functions: conditionally unipolar with a maximum at the initial frequency and bipolar with a zero value at the specified frequency. For classical Fourier transformations with an infinite limit of integration, these functions turn into two delta functions with the same properties. The practical application of the found patterns is currently limited by the lack of an algorithm that allows the use of data from only one component.

Сильная аппроксимация функций двойными рядами Фурье в обобщённых гёльдеровых пространствах

We study some problems of strong approximation of the functions of double Fourier series in the generalized Holder spases. Order-accurate estimates for the strong summability functionals of double Fourier series are established, as wel as their approximation properties in the indicated spases.

Thermal Control of Quasi-2-Level Super-Switch by Power Routing

The requirement of power converters for Medium Voltage (MV) and High Voltage (HV) applications accelerates faster than the development of semiconductor switches capable of handling such voltage levels. For that reason, it is common to use the series connection of semiconductors with auxiliary snubbers. In contrast, an increased number of semiconductors is more likely to cause reliability problems due to increased redundancy, unequal thermal conditions, or different remaining lifetime. On the other hand, the quasi-2-level super-switch (Q2L SS) yields increased flexibility, enabling precise readjustments of the voltage and switching losses of individual devices of the converter due to the utilization of small capacitors. The power routing capability of the Q2L SS to redistribute the switching losses from the overheated semiconductors is presented in this paper. The influence of power routing on performance of Q2L SS is investigated with the Double Fourier analysis and experimental results.

A meshfree approach for vibro-acoustic analysis of combined composite laminated shells with variable thickness

The vibro-acoustic responses of combined composite laminated shell structures with variable thickness immersed in acoustic medium is investigated by using a meshfree method in this paper. The theoretical model of structural field is established by employing the energy principle in conjunction with the first-order shear deformation theory (FSDT) and the theoretical model of acoustic field are established by using Kirchhoff-Helmholtz integral equation. The displacement components of shell segments and the sound pressure variables of acoustic medium are described by using the meshfree Tchebychev point interpolation (TPIM) shape function and Double Fourier series in meridional and circumferential directions. The non-uniqueness problem of Kirchhoff-Helmholtz integral equation is overcome by adopting CHIEF method. The validation of the presented theoretical model is proved by comparing the results calculated by the proposed model with those results of existing literature and FEM/BEM. Based on the established theoretical model, vibro-acoustic responses of combined composite laminated shell structures with variable thickness immersed in acoustic medium are performed by analyzing the influences of thickness parameter, external load and boundary condition on the sound pressure level (SPL) and sound power level (SWL) of combined composite laminated shell structures. The above investigations can provide the theoretical thesis and technique guidance of preliminary design of combined composite laminated shell immersed in various acoustic medium.

Couple-stress thermoelasticity of three-dimensional multilayered composites

This work presents a comprehensive analysis of field solutions for three-dimensional multilayered composites within the framework of the couple-stress thermoelasticity. Using double Fourier series expansion solutions, the powerful and elegant mathematical Stroh formalism is extended to include the thermal coupling with the couple-stress elasticity theory to capture small-scale microstructural effects. The size-dependent layered approach leads to concise and general eigenrelations for each homogeneous orthotropic plate, while the unconditionally stable dual-variable technique is used to propagate the extended displacement/temperature fields and force-/tangential couple-tractions through the internal interfaces of the multilayered structures. Closed-form expressions are derived for the recursive layer-to-layer relations, as well as for the temperature, heat flux, displacement, stress and couple-stress field components, with respect to simply-supported structures subjected to combined thermal and mechanical loading. Application examples, including highly anisotropic fiber-reinforced composites, show significant deviations from the classical thermoelasticity, demonstrating the critical importance of considering the microstructural characteristics of interface-dominated materials. The results result come from a series of scenarios, covering mechanical and thermomechanical loading conditions, changes in aspect ratios and stacking sequences within multilayered systems, and the consequences of varying the number of sandwich plates.

Free vibration and sound radiation of ABH-type functionally graded sandwich plates under elastic boundary restraints

Based on the first-order shear deformation theory, the free vibration and sound radiation of ABH-type functionally graded sandwich plates (ABH-type FG sandwich plate) under elastic boundary restraints are studied. The displacement field of the plate is described as an improved form of double Fourier cosine series and several closed-form auxiliary polynomial functions. The dynamic equation is solved by using the Rayleigh-Ritz method. The accuracy of the present method is validated and the effects of thickness variation coefficients and elastic boundary restraints on the vibr-acoustic characteristics are discussed. The research can provide an idea for the design of sandwich structures.

Wave decomposition method for the dynamic analysis of a cable-stayed beam subjected to a moving load considering structural damping

Wave analysis emerges as a compelling methodology to elucidate the phenomenon of energy propagation back and forth between junctions. To develop an efficient model for comprehending the propagation of waves in cable-stayed bridges during operation, a wave decomposition method for the dynamic analysis of viscoelastic cable-stayed beams subjected to moving loads is proposed. Initially, the cables and beams are treated as distinct subparts. The Kelvin-Voigt damping model is introduced, and the influence of moving loads is represented using the Dirac delta function. The exact frequency-domain solution of the global system is analytically derived through both the double Fourier transform approach (DFA) and the method of the reverberation-ray matrix (MRRM). Subsequently, an explicit expression for multi-wave components (EMW) is inferred from the global exact solution. These components are characterized by frequency-dependent and dispersion-dependent traits, enabling the extraction of wave components and energy flow across various waveguides. To determine an appropriate damping coefficient for the K-V model, the complex model method is employed to establish a mathematical connection between the K-V damping coefficient and the measurable damping ratio. After validating the EMW through comparison with finite element method (FEM) outcomes and model testing, we investigate the influence of moving loads and structural parameters on the energy distribution across wave frequencies and waveguide paths. This inquiry reveals the tendency of waves to select propagation pathways based on both frequency and load factors. Additionally, a wave velocity analysis is conducted to explore the mechanism underlying the augmentation of structural dynamic stiffness. This research holds promise in the realms of vibration control and the evaluation of bridge transmission efficiency.

Dynamic analysis of GPLs reinforced microcapsules subjected to moving micro/nanoparticles using mathematical modeling and deep-neural networks

In the realm of engineering, the use of cylindrical microcapsules containing moving micro/nanoparticles is crucial. Therefore, biomechanical engineers must focus on increasing its stability so that it may be used in a variety of scenarios. Thus far, graphene nanoplatelets, or GPLs, have shown their efficacy as an epoxy reinforcement in enhancing the material's characteristics in comparison to plain epoxy. This indicates that GPL-reinforced epoxy has the potential for use as the fiber composite's matrix phase. Thus, bending data of the thick cylindrical microcapsule with GPL reinforcement under a moving load is reported in this study. A modified couple stress theory or MCST with a single length scale parameter is proposed for simulating the micro-size structure. The high-order shear deformation theory are used to represent the displacement fields while taking the shear-strain function into account. The space-dependent equations are solved using the Laplace transform approach and the double Fourier series, respectively, using forced vibration analysis. The results are compared and validated with another published study in the literature after the modeling and solution of the governing equations of the GPLs reinforced microcapsules exposed to moving micro/nano particles. With the help of the findings from the mathematical modeling section, deep neural networks (DNN) with output, input, and hidden layers are used to mimic the existing microcapsule in order to provide further verification and introduce machine learning methodologies for addressing engineering challenges. A cylindrical microcapsule under motion under a micro/nanoparticle may be modeled in the area of biomechanical engineering using the current outputs obtained from mathematical modeling and DNN sections.

Optimization of time-dependent displacements of the axially graphene nanoplatelets reinforced micro-curved panels via swarm and genetic algorithm approaches

Micro-curved panels with moving micro/nanoparticles included are essential microstructures in the engineering field. So, improving its stability for use in various situations is an essential key for biomechanical engineers. There is potential for using graphene nanoplatelets (GPLs)-reinforced epoxy as the matrix phase in a fiber composite as it has been shown that GPLs successfully improve the properties of epoxy materials when compared to clean epoxy. So, in the current work, dynamic deflection of the axially reinforced thick micro-curved panel under a moving load is presented. Optimizing the geometry of composite materials is essential in structural design if you want to achieve the highest possible levels of strength or stability. In the present study, we use the genetic algorithm and particle swarm optimization to find the optimal radius ratio for lowering the displacement fields. For modeling the micro-size structure, modified couple stress theory with one length scale parameter is presented. The displacement fields are modeled using Taylor’s series higher-order terms considering strain in the thickness direction. Using forced vibration analysis, the space and time-dependent equations are solved via double Fourier series and state-space method, respectively. It is shown that reinforcing a microstructure with GPLs in the axial direction has lower deflection than in other directions. The results show that velocity and external loading of moving micro/nanoparticle, geometry properties, and length scale parameters have an important role in the time-dependent deflections of the axially reinforced thick micro-curved panel under moving micro/nanoparticles. The current outputs can be used in the biomechanical engineering field for modeling a micro-curved panel under moving micro/nanoparticles.

Dynamic response of fractional poroviscoelastic layered media subjected to moving loads

This study investigates the dynamic response of fractional poroviscoelastic layered media subjected to a moving harmonic load. The material viscosity includes two parts: the flow-dependent viscosity between the skeleton and pore water and the skeleton viscosity. Firstly, the governing equations of poroelastic media are derived based on Biot's theory and then transformed into ordinary differential equations by using the double Fourier integral transform. By incorporating fractional calculus theory and the dynamic correspondence principle, the applicability of these equations is extended to fractional poroviscoelastic media. The extended precise integration method (PIM) is employed to obtain solutions. Subsequently, the fractional Zener model and the presented method are validated. Finally, parametric studies are conducted to analyze the effects of fractional order, moving velocity, load frequency, and stratification on the dynamic behaviors of the media.

A Hermitian Cn finite cylindrical layer method for 3D size‐dependent buckling and free vibration analyses of simply supported FG piezoelectric cylindrical sandwich microshells subjected to axial compression and electric voltages

AbstractWithin the framework of the consistent couple stress theory (CCST), we develop a Hermitian Cn (n = 1, 2) finite cylindrical layer method (FCLM) for carrying out the three‐dimensional (3D) analysis of the size‐dependent buckling and free vibration behaviors of simply supported, functionally graded (FG) piezoelectric cylindrical sandwich microshells. The microshells of interest are placed under closed‐circuit surface conditions and subjected to axial compression and electric voltages. We derive a 3D weak formulation based on Hamilton's principle for this study. In the resulting formulation, the microshell is artificially divided into nl microlayers, with the elastic displacement components and the electric potential selected as the primary variables. By incorporating a layer‐wise kinematic model into our weak formulation, we develop a Hermitian Cn FCLM, which can be used for analyzing FG piezoelectric cylindrical sandwich microshells. Each primary variable is expanded as a double Fourier series in the in‐surface domain and is interpolated in the thickness direction using Hermitian Cn polynomials. The accuracy and the convergence rate of our Hermitian Cn FCLMs are validated by comparing the solutions they produce for FG piezoelectric cylindrical macroshells and FG elastic cylindrical microshells with the relevant exact and approximate 3D solutions which have been reported in the literature. The material length scale parameter of our FCLMs is set at zero in the comparison made with the FG piezoelectric macroshells. In contrast, the piezoelectric and flexoelectric effects are ignored in the comparison made with the FG elastic microshells. The impact of some essential factors on the critical load, critical voltage, and natural frequency of simply supported FG piezoelectric cylindrical sandwich microshells is assessed. The important factors are identified as piezoelectricity, flexoelectricity, the material length scale parameter, the inhomogeneity index, the radius‐to‐thickness ratio, the length‐to‐radius ratio, and the magnitude of the applied voltage and the applied load.

A Novel Modulation Strategy for Controllable Current Source Converter

Controllable current source converter (CSC) technology is an effective solution in addressing commutation failure issues. Modulation strategy is the fundamental approach for controlling the valve arm of the converter. Higher switching frequencies result in lower AC harmonics but increased losses in the commutation valve. Lower switching frequencies lead to reduced losses but higher harmonics. The major challenge in engineering currently lies in eliminating the direct conduction state of the valve arms in the current source converter and effectively suppressing harmonics. This paper first analyzes the valve arm signal allocation method in the CSC and examines the mechanism and impact of direct conduction. Then, based on double Fourier signal analysis theory, a SPWM modulation method utilizing the odd multiples of "150Hz" is proposed. Modulation range and signal synchronization methods are analyzed. The effectiveness of the method in terms of duty cycle, transmission characteristics, and saturation characteristics is analyzed. Finally, a comparison is made between this method and traditional linear modulation and asymmetric modulation methods.

On the Buckling of Simply Supported Anisotropic Sandwich Plates

Here, the elastic stability behavior of simply supported anisotropic sandwich flat panels subjected to mechanical in-plane loads is investigated using analytical approach. The formulation is based on first-order shear deformation theory and the shear correction factors employed are based on energy consideration that depends on the lay-up as well as material properties. The governing equations are obtained using the Raleigh-Ritz method assuming a combination of sine and cosine functions in the form of double Fourier series for the displacement fields. The effectiveness of the integrated formulation is tested for global characteristics considering examples related to multi-layered laminates and sandwich panels for which solutions are available.

An Extended Generalized Average Modeling Framework for Power Converters

The Generalized Averaged Modeling (GAM) technique is traditionally employed to capture the dynamic performance of power electronic converters. This paper proposes an improved version of it, named the Extended-GAM (EGAM) technique, which supports the multiplication of two Double Fourier Series (DFS) signals in the time domain. Multiplication of DFS signals in the time domain translates to the 2D-convolution of coefficients of the DFS terms of their equivalent Discrete Fourier Image (DFI) representations. Thus, the proposed EGAM technique, capable of capturing many harmonics present in the output of a power converter, effectively captures the dynamic behavior of power converters excited by two distinct frequencies. The proposed technique is then converted into an algorithm suitable for numerical platforms, which typically use Ordinary Differential Equation (ODE) solvers. The proposed algorithm is validated based on the observations of the effects of harmonic truncation. The efficacy of the proposed technique is assessed through a case study, wherein a single-phase inverter employs LC filters on both the dc-link and the ac-side. Finally, it is shown that the results obtained with the proposed method show an excellent congruence between simulation and hardware experimental models. Additionally, the proposed algorithm is packaged into a MATLAB toolbox and shared for future implementations.

VSG Control for Cascaded Three-Phase Bridge Based Battery Inverter

With the increasing number of new energy sources connected to the grid, the unbalanced output of three-phase grid-connected inverters and the lack of no inertia and damping characteristics in the traditional microgrid control system will seriously affect the stability of voltage, frequency, and power angle for microgrids. This paper proposes a novel cascaded three-phase bridge inverter topology for the battery system used for the electric vehicle. Compared with traditional cascaded H-bridge inverters, the proposed multilevel inverter can achieve self-adaptive balance for three phases. The mathematical model of a cascaded three-phase bridge inverter is established in this paper. Based on the voltage and current equations of a multilevel inverter, a new modulation strategy named carrier phase-shifted-distributed pulse width modulation (CPSD-PWM) was developed, which is more suitable for cascaded three-phase bridge inverters. The harmonic analytic equations of carrier phase-shifted pulse width modulation (CPS-PWM) and CPSD-PWM are constructed by the double Fourier analysis method. Compared with the traditional PWM modulation strategy, the CPSD-PWM can reduce the output harmonics and improve the balance of the three-phase output, which can realize the three-phase adaptive balance in the cascaded three-phase bridge inverter. This paper develops a cascaded three-phase bridge multilevel power converter system based on the virtual synchronous generator (VSG) control strategy. The voltage and frequency of inverter output can be accurately controlled in both island mode and grid-connected mode through active power-frequency regulation and reactive power–voltage regulation, and the stability of primary frequency regulation for the multilevel microgrid inverter can be improved by collaborative optimization of virtual inertia and virtual damping. The CPSD-PWM modulation strategy and VSG control strategy are verified by the simulation results and experimental data for the cascaded three-phase bridge inverter.

A double Fourier sphere method for d-dimensional manifolds

The double Fourier sphere (DFS) method uses a clever trick to transform a function defined on the unit sphere to the torus and subsequently approximate it by a Fourier series, which can be evaluated efficiently via fast Fourier transforms. Similar approaches have emerged for approximation problems on the disk, the ball, and the cylinder. In this paper, we introduce a generalized DFS method applicable to various manifolds, including all the above-mentioned cases and many more, such as the rotation group. This approach consists in transforming a function defined on a manifold to the torus of the same dimension. We show that the Fourier series of the transformed function can be transferred back to the manifold, where it converges uniformly to the original function. In particular, we obtain analytic convergence rates in case of Hölder-continuous functions on the manifold.

Combined effect of helical force and rotation on the double convection of a binary viscoelastic fluid in a porous medium: Linear and weakly nonlinear analysis

AbstractWe used linear stability theory based on the normal mode decomposition technique and nonlinear stability theory based on the minimum representation of double Fourier series to study the criterion of appearance of the stationary convection and the oscillatory convection; and the rate of heat and mass transfer in a binary viscoelastic fluid mixture in a rotating porous medium under the effect of helical force. We have determined the analytical expression of the Rayleigh number of the system as a function of the dimensionless parameters. Expressions for heat and mass transfer rates are determined as a function of Nusselt and Sherwood number, respectively. The transient behaviors of the Nusselt number and the Sherwood number are studied by solving the finite amplitude equations using the Runge‐Kutta method. Then, the effect of each dimensionless parameter on the system is studied pointed out interesting results. According to the analysis of the different results obtained, it appears that in a porous medium saturated by a binary mixture of viscoelastic fluid in rotation, the Taylor‐Darcy number , the delay time parameter λ2 and the mass Rayleigh‐Darcy number delay the onset of stationary and oscillatory convection. On the other hand, the helical force , the relaxation time parameter λ1, the ratio of diffusivities τ and the number of Vadasz accelerate the onset of stationary and oscillatory convection. In the case of the nonlinear stability study, at steady state, the helical force accelerates the rate of heat and mass transfer. On the other hand, the Taylor‐Darcy number , the diffusivity ratio τ and the mass Rayleigh‐Darcy number retard the rate of heat and mass transfer. In the unsteady state, the helical force , the relaxation time parameter λ1 and the mass Rayleigh‐Darcy number accelerate the rate of heat and mass transfer. This is not the case for the Taylor‐Darcy number and the delay time parameter λ2.

Sphere2Vec: A general-purpose location representation learning over a spherical surface for large-scale geospatial predictions

Generating learning-friendly representations for points in space is a fundamental and long-standing problem in machine learning. Recently, multi-scale encoding schemes (such as Space2Vec and NeRF) were proposed to directly encode any point in 2D or 3D Euclidean space as a high-dimensional vector, and has been successfully applied to various (geo)spatial prediction and generative tasks. However, all current 2D and 3D location encoders are designed to model point distances in Euclidean space. So when applied to large-scale real-world GPS coordinate datasets (e.g., species or satellite images taken all over the world), which require distance metric learning on the spherical surface, both types of models can fail due to the map projection distortion problem (2D) and the spherical-to-Euclidean distance approximation error (3D). To solve these problems, we propose a multi-scale location encoder called Sphere2Vec which can preserve spherical distances when encoding point coordinates on a spherical surface. We developed a unified view of distance-reserving encoding on spheres based on the Double Fourier Sphere (DFS). We also provide theoretical proof that the Sphere2Vec encoding preserves the spherical surface distance between any two points, while existing encoding schemes such as Space2Vec and NeRF do not. Experiments on 20 synthetic datasets show that Sphere2Vec can outperform all baseline models including the state-of-the-art (SOTA) 2D location encoder (i.e., Space2Vec) and 3D encoder NeRF on all these datasets with up to 30.8% error rate reduction. We then apply Sphere2Vec to three geo-aware image classification tasks - fine-grained species recognition, Flickr image recognition, and remote sensing image classification. Results on 7 real-world datasets show the superiority of Sphere2Vec over multiple 2D and 3D location encoders on all three tasks. Further analysis shows that Sphere2Vec outperforms other location encoder models, especially in the polar regions and data-sparse areas because of its nature for spherical surface distance preservation. Code and data of this work are available at https://gengchenmai.github.io/sphere2vec-website/.

Generalized Accurate Harmonic Calculation Method Based on Discretized Double Fourier Series to Solve Double-Pulse Phenomenon

Harmonic issues have always been paid attention to in electrical systems, and accurate calculation of the amplitude and phase of each harmonic has become one of the research focuses of scholars. This article develops a generalized harmonic closed-form analytical solution for a single-phase voltage source inverter (VSI) under digital sinusoidal pulse width modulation (DSPWM) and dead-time effects. This article is the first to point out the double-pulse phenomenon that can occur when using a double Fourier series to solve the harmonic spectrum due to the discontinuity of the dead-time effects and half-cycle integrals. Analysis and data comparison shows that this phenomenon can affect the accuracy of harmonic calculations. An discretized double Fourier series method is proposed to solve this phenomenon. Moreover, the problems of missing pulse caused by the modulation under a specific current phase and current zero-crossing are also solved, improving the harmonic calculation's accuracy and generality. Simulation analysis and the results of inverse fast Fourier transform (IFFT) proved the correctness of the optimization method. And the experimental results are obtained on the DSP and SIC platforms, which are in good agreement with the expression calculation results