Fourier Series Research Articles

Shannon’s Sampling Theorem for Set-Valued Functions with an Application

In this study, we defined a kind of Fourier expansion of set-valued square-integrable functions. In fact, we have seen that the classical Fourier basis also constitutes a basis for the Hilbert quasilinear space L2(−π,π,Ω(C)) of Ω(C)-valued square-integrable functions, where Ω(C) is the class of all compact subsets of complex numbers. Furthermore, we defined the quasi-Paley–Wiener space, QPW, using the Fourier transform defined for set-valued functions and thus we showed that the sequence sinc.−kk∈Z form also a basis for QPW. We call this result Shannon’s sampling theorem for set-valued functions. Finally, we gave an application based on this theorem.

Transverse resonance technique for analysis of a symmetrical open stub in a microstrip transmission line

Open stubs in a strip (microstrip) transmission line are one of the most common elements of planar circuits used in numerous devices in the various types of wireless systems. Therefore, the urgent problem is to develop an analyzing method for discontinuities in the form of the open stub in a microstrip transmission line at frequencies at which the high-frequency effects must be considered. In the paper, a technique of scattering characteristics calculating on a symmetrical microstrip open stub by transverse resonance method is presented. Boundary value problems for a rectangular volume resonator based on a microstrip transmission line with a symmetric open stub are solved for the three options boundary conditions in the symmetry plane and on the longitudinal boundaries. The intersection of the spectral curves obtained by the numerical solution of the “electric” and “magnetic” boundary value problems determines the minima of a reflection or transmission coefficients of fundamental wave on discontinuities. To algebraize the boundary value problems for the eigen frequencies of volume resonator with discontinuity, the corresponding two-dimensional functions of the magnetic potential are constructed, through which the components of the current density on the strip are determined. The functions of magnetic potential were defined by decomposing them into expansion by Fourier series, which ensures stable convergence of the series and numerical calculation algorithm. The developed technique has been tested by calculating the eigenfrequency spectra of an open microstrip stub using the transverse resonance method on the example of an open stub in a microstrip transmission line with a resonant frequency of about 3.0 GHz. Also, a technique for numerical solutions of “electric” and “magnetic” boundary-value problems for resonators with two electrodynamically coupled symmetric open stubs in a microstrip transmission line is developed.

IDENTIFICATION OF AN ARBITRARY MOVING AXISYMMETRIC LOAD ACTING ON A CYLINDRICAL SHELL

Various types of external non-stationary loads can act on various structural elements and cylindrical shells of finite length in particular distributed and concentrated, stationary and moving load. When using various external load identification methods, the type of external load is usually known. In prac- tice this is not always the case. The purpose of the study is to develop a method for identifying an arbitrary axisymmetric load acting on an elastic cy- lindrical shell of finite length, which can be used to identify a moving load. To model the non-stationary load of a cylindrical shell, a system of differ- ential equations of the refined Timoshenko theory for medium thickness shells was used. The solution to this system of differential equations is ob- tained by expanding the unknown functions into Fourier series and applying the integral Laplace transform. The solution of the inverse problem was obtained using the theory of integral equations and the Tikhonov regularization method. As a result of the study, a solution of the inverse problem of solid mechanics to identify an arbitrary axisymmetric non-stationary load was obtained. A numerical experiment was carried out to use the developed method to identify a moving non-stationary load acting on a simply supported cylindrical shell of medium thickness. The simulation results indicate a fairly accurate identification of both the change in time and the distribution along the shell of a non-stationary axisymmetric moving load. A method has been developed for identifying an external non-stationary load randomly distributed along a cylindrical shell, which makes it possible to identify a moving load without preliminary information about the type of this load. The developed method allows us to reproduce moving non-stationary loads, which are often encountered in practice, and expand it to other types of structural elements.

Analytical Solutions for a Fully Coupled Hydraulic‐Mechanical‐Chemical Model With Nonlinear Adsorption

ABSTRACTAdsorption characteristics play a crucial role in solute transport processes, serving as a fundamental factor for evaluating the performance of clay liners. Nonlinear adsorption isotherms are commonly found with metal ions and organic compounds, which introduce challenges in obtaining analytical solutions for solute transport models. In this study, analytical solutions are proposed for a fully coupled hydraulic‐mechanical‐chemical (HMC) model that accounts for both the Freundlich and Langmuir isotherms. To mitigate the difficulties arising from the variable coefficients, the system of second‐order partial differential equations involving three variables is linearized. The method of separation of variables, theory of integration, and Fourier series are utilized to derive analytical solutions. The analytical method presented can potentially be extended to a broad spectrum of nonlinear adsorption isotherms. The results reveal a 56.5% reduction in solute breakthrough time under the Freundlich isotherm and a remarkable 2.6‐fold extension under the Langmuir isotherm when compared to the linear isotherm. The adsorption constants of the Freundlich and Langmuir isotherms exhibit a positive correlation with breakthrough time, while the exponent of the Freundlich isotherm and the maximal adsorption capacity in the Langmuir isotherm demonstrate a negative association with breakthrough time. This study enhances the precision of solute transport prediction and provides a more scientific assessment of clay liner performance.

A meshfree formulation for size-dependent thermal buckling and post-buckling behaviour of porous microplates on elastic foundation subjected to localized heating

The semi-analytical framework is developed for in-plane thermal stresses within the microplates under localized heating. Localized heating is modelled using Fourier Series expansions over the complete domain of the plate. These stresses within the plate are estimated after solving the in-plane thermoelasticity problem using Airy's stress approach. Utilizing these stresses, present work also studies the buckling and post-buckling behaviour of a porous metal foam microplate resting on Winkler and Pasternak elastic foundations. The plate is modelled using Reddy's third-order shear deformation theory (TSDT) and von Kármán geometric nonlinearity, and the size-dependent effects are included using the modified strain gradient theory (MSGT). Galerkin's weighted residual method converts the partial differential equations (PDEs) into algebraic equations. The post-buckling equilibrium path is obtained using the modified Newton-Raphson method. The mode shapes of the microplate for the various configurations of the plate with different boundary conditions due to localized thermal loading are plotted. Also, these mode shapes are compared with the mode shapes of the microplate due to in-plane mechanical loading. The parametric effect of porosity, elastic foundation parameters, thickness of plate, size of plate, boundary conditions and loading concentrations on the buckling and post-buckling behaviour of the plate is studied.

Two-dimensional analysis of electrical behavior in composite semiconductor shell structures with magnetic field tuning

To precisely investigate the electric properties and reveal the intrinsic interaction mechanisms between multiple fields in the composite magneto-electric-semiconductor shell structure, two-dimensional analyses were conducted within the framework of the third-order shear deformation theory and the coupled field theory. The governing equations for the composite piezoelectric semiconductor (PS) shell structures were derived using the principle of virtual work. Field distributions of the composite PS shell structures were obtained by expanding basic field quantities into Fourier series along the width direction and using the differential quadrature method. Prior to analysis, the convergence and correctness of the adopted method were evaluated. Several numerical examples were presented to demonstrate how magnetic manipulation can tune electrical behavior. It was found that inhomogeneous shear strain induced by applied magnetic fields is the key factor affecting potential variation in the composite PS shells. The appearance of potential barriers and wells in the composite PS shell structure with an opposite c axis is explained thoroughly. Additionally, the effect of magnetic field non-uniformity on the composite PS shell structure was investigated using an equivalent body force model, offering valuable insights for designing thin-film devices.

Decoupling and predicting natural gas deviation factor using machine learning methods

Accurately predicting the deviation factor (Z-factor) of natural gas is crucial for the estimation of natural gas reserves, evaluation of gas reservoir recovery, and assessment of natural gas transport in pipelines. Traditional machine learning algorithms, such as Support Vector Machine (SVM), Extreme Gradient Boosting (XGBoost), Light Gradient Boosting Machine (LightGBM), Artificial Neural Network (ANN) and Bidirectional Long Short-Term Memory Neural Networks (BiLSTM), often lack accuracy and robustness in various situations due to their inability to generalize across different gas components and temperature-pressure conditions. To address this limitation, we propose a novel and efficient machine learning framework for predicting natural gas Z-factor. Our approach first utilizes a signal decomposition algorithm like Variational Mode Decomposition (VMD), Empirical Fourier Decomposition (EFD) and Ensemble Empirical Mode Decomposition (EEMD) to decouple the Z-factor into multiple components. Subsequently, traditional machine learning algorithms is employed to predict each decomposed Z-factor component, where combination of SVM and VMD achieved the best performance. Decoupling the Z-factors firstly and then predicting the decoupled components can significantly improve prediction accuracy of all traditional machine learning algorithms. We thoroughly evaluate the impact of the decoupling method and the number of decomposed components on the model’s performance. Compared to traditional machine learning models without decomposition, our framework achieves an average correlation coefficient exceeding 0.99 and an average mean absolute percentage error below 0.83% on 10 datasets with different natural gas components, high temperatures, and pressures. These results indicate that hybrid model effectively learns the patterns of Z-factor variations and can be applied to the prediction of natural gas Z-factors under various conditions. This study significantly advances methodologies for predicting natural gas properties, offering a unified and robust solution for precise estimations, thereby benefiting the natural gas industry in resource estimation and reservoir management.

Phase-field modeling of interdiffusion between dissimilar Fe-Cr-Ni alloys during non-isothermal hot isostatic pressing

Powder metallurgy hot isostatic pressing (PM-HIP) has emerged as a promising alternative to welding for joining dissimilar metals. During HIP, interfacial bonding is mediated by solid state diffusion. The interdiffusion zone across the interface depends on processing conditions, calling for the need for accurate numerical tools capable of simulating interdiffusion and possible phase transformation in order to optimize processing parameters. Here, a phase-field (PF) model based on CALPHAD-based free energy functionals is developed to simulate the interdiffusion and phase evolution between dissimilar Fe–Cr–Ni based steels undergoing HIP and is demonstrated using the interface between 316L and SA508 steels. To overcome the numerical challenges caused by the singular magnetic and entropy terms in the CALPHAD free energy models in the Fe–Cr-Ni system, polynomial functions are fitted with temperature dependent coefficients represented by Fourier series to accurately describe the phase stability of both fcc and bcc phases in the composition and temperature space. This enables simulations of non-isothermal HIP cycles. Diffusivity data from commercial software and literature are taken to parameterize the kinetic parameters. A discrete nucleation model is incorporated for possible phase transformation. The modified thermodynamic models are validated against previous experiments at 923 K and 1273 K. The interdiffusion kinetics are benchmarked against new HIP experiments joining powder and bulk 316L to bulk SA508 with three different HIP cycles. The good agreement between simulations and experiments on both phase stability and interdiffusion indicate that the model is suitable for simulating interdiffusion between Fe–Cr–Ni alloys during HIP cycles. It is also found that using powder and bulk 316L gives similar interdiffusion profiles at elevated temperature when a dense interface forms during HIP.

Soft Magnetoelastic Tactile Multi-Sensors with Energy-Absorbing Properties for Self-Powered Human-Machine Interfaces.

Tactile sensors play a key role in human-machine interfaces (HMIs) for augmented and virtual reality, point-of-care devices, and human-robot collaboration, which show the promise of revolutionizing our ways of life. Here, we present a sensor (EMTS) that utilizes the magnetoelastic effect in a soft metamaterial to convert mechanical pressure into electrical signals. With this unique mechanism, the proposed EMTS simultaneously possesses self-powering, waterproof, and compliant features. The soft metamaterial is essentially a porous magnetoelastomer structure designed based on the Fourier series expansion, which allows for programmable mechanical response and sensing performance of the EMTS. Fabricated by simple 3D-printed molds, the EMTS also holds potential for low-cost production. Particularly, the porous magnetoelastomer structure comes with selectable buckling instabilities that can significantly enhance biomechanical-to-electrical energy conversion. Also, with the embedded magnetic microparticles, the energy-absorbing performance of the sensor is greatly improved, which is highly beneficial to HMIs. To pursue practical applications, the EMTSs are further integrated with two systems as control and perception modules. It is demonstrated that the EMTS is able to identify different hand gestures to control a lighting system even in a high-humidity environment. Also, the EMTS stands out for its superior capability of simultaneous impact perception and energy absorption in drop tests. Overall, with its compelling array of features, the presented EMTS gives impetus to multi-sensing technology and practically enables a variety of HMI applications.

A data-driven technique for discovering the dynamical system with rigid impact characteristic

We propose a data-driven technique for discovering the equation of motion of the dynamical system with rigid impact. The method first discovers a dynamical system close to impact by the Fourier series with a high rate of convergence, known as the double-even extended series. Then, we use the system to construct an impact mapping, which maps the data close to impact to an estimated impact instant. By minimizing the error of impact mapping, we find the location of impact surface and energy lost during impact that generally satisfies the data close to impact. Finally, we discover the equation of motion without impact by the double-even extended series The analyzed data can be collected at equal time intervals with measurement error, and there is no need to deliberately collect data at the impact instant. The technique is able to capture the impact characteristic when there is a lack of knowledge about the critical changes of the dynamics at the impact instant and the non-linear dynamical behaviors without impact. We test the identification ability of the new technique using impact dynamical systems connected with cubic damping term and strong non-linear damping, respectively. The identified systems accurately capture impact dynamics such as the long-time prediction with impact, multistable dynamical phenomenon, and chattering dynamics.

Propagation Characteristics of Electrostatic Surface Plasma Waves at the Spin‐Polarized Quantum Plasma–Vacuum Interface

ABSTRACTThis investigation explores the characteristics of electrostatic surface plasma waves within the framework of a spin‐polarized quantum plasma. Utilizing the spin‐polarized quantum hydrodynamic model and incorporating essential elements like Fermi pressure and Bohm potential, we derive the dispersion relation governing surface plasma waves at a plasma–vacuum interface. Through Fourier decomposition of the hydrodynamic model, we establish the dispersion relation that outlines the behavior of surface plasmons under conditions of small amplitude. Quantum effects, encompassing degenerate pressure, and Bohm potential are considered with specific attention given to the spin polarization effect, treating spin up, and spin down electrons as distinct species. The resulting dispersion relation demonstrates that, regardless of the degree of spin matching, Bohm potential significantly alters the phase speed in the limit of a large wave vector. Increasing spin mismatch in the quantum plasma leads to a decrease in the phase speed of the surface mode for a fixed value of the plasmonic coupling parameter . Our findings bear relevance to graphene‐based plasmonic systems, aligning with some of the observations reported in Gao et al. (2013) and Guo et al. (2019).

Approximation by Subsequences of Matrix Transform Means of Some Two-Dimensional Rectangle Walsh–Fourier Series

In the present paper we discuss the rate of the approximation by the matrix transform of special partial sums of some two-dimensional rectangle (decreasing diagonal) Walsh-Fourier series in Lp(G2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p(G^{2})$$\\end{document} space (1≤p<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\le p <\\infty $$\\end{document}) and in C(G2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C(G^{2})$$\\end{document}. It implies in some special case norm convergence. We also show an application of our results for Lipschitz functions. At the end of the paper we show the most important result, the almost everywhere convergence theorem. We note that T summation is a common generalization of the following known summation methods Cesàro, Weierstrass, Riesz and Picar and Bessel methods.

The Effect of Unsteady Inlet Boundary Conditions on the Aero-Thermal Behavior of High-Pressure Turbine Vanes – A Numerical Study Using Scale-Resolving Simulations

Abstract This paper presents the application of a novel method to prescribe unsteady boundary conditions to transient, scale-resolving computational fluid dynamics simulations of the high-pressure turbine in modern jet engines. The methodology is based on the compression of the interface data at the combustor–turbine interface, using proper orthogonal decomposition and Fourier series (PODFS). Doing so can reduce the stored data at the interface drastically. The capability of the PODFS method to produce realistic inlet boundary conditions was demonstrated in previous work. Here, the method is applied to a turbine case. The outlet data of a combustor simulation is used to create the PODFS boundary conditions for a scale-resolving simulation of a simplified first nozzle guide vane of the high-pressure turbine. This simulation is compared with simulations with steady-state boundary conditions to show the effect of unsteadiness in the inlet boundary condition on the aerodynamic and thermal behaviors of the turbine. While the aerodynamics show minor sensitivity against the way of applying the inlet boundary conditions, the thermal behavior of the vanes is strongly affected by the modeling of combustor unsteadiness.

Deep learning approach for skin melanoma and benign classification using empirical wavelet decomposition.

Melanoma is a malignant skin cancer that causes high mortality. Early detection of melanoma can save patients' lives. The features of the skin lesion images can be extracted using computer techniques to differentiate early between melanoma and benign skin lesions. A new model of empirical wavelet decomposition (EWD) based on tan hyperbolic modulated filter banks (THMFBs) (EWD-THMFBs) was used to obtain the features of skin lesion images by MATLAB software. The EWD-THMFBs model was compared with the empirical short-time Fourier decomposition method based on THMFBs (ESTFD-THMFBs) and the empirical Fourier decomposition method based on THMFBs (EFD-THMFBs). The accuracy rates obtained for EWD-THMFBs, ESTFD-THMFBs, and EFD-THMFBs models were 100%, 98.89%, and 83.33%, respectively. The area under the curve (AUC) was 1, 0.97, and 0.91, respectively. The EWD-THMFBs model performed best in extracting features from skin lesion images. This model can be programmed on a mobile to detect skin lesions in rural areas by a nurse before consulting a dermatologist.

An Approach to Near-Optimal Continuous-Thrust Solution for Plane Constellation Deployment

Paving the way in satellite constellation deployment, this article presents the Comprehensive Program (GCP) for planning constellation deployment missions by introducing a near-optimal solution for continuous-thrust maneuvers. The GCP establishes time constraints, enabling appropriate time scheduling and eliminating reconfiguration phases. It considers various cases of satellite departure and arrival sequences, collision avoidance constraints, and time limitations. A near-optimal solution for the continuous-thrust trajectory of each satellite is introduced, relying on the Fourier series approximation of the thrust pointing angle. The Fourier series with constant coefficients replaces the thrust angle in the satellite’s orbital motion equations in polar coordinates. The primary advantage of this near-optimal approach lies in its simplicity in accommodating space perturbations, posing no challenges in problem-solving. Additionally, the approach is beneficial due to its straightforward implementation and simple mathematical computations. This approach does not require pre-determining the revolution number of the transfer trajectory, unlike shape-based methods. The proposed method’s flexibility allows it to adapt and optimize the trajectory without prior assumptions about the number of revolutions required for the mission. Analyses demonstrate the proposed algorithm’s versatility and precision across various scenarios involving different orbital eccentricities, thrust forces, and thrust pointing angles. The constant coefficients of the Fourier series are determined by defining a set of objective functions and minimizing them using the Multi-Objective Particle Swarm Optimization algorithm (MOPSO). The first and second objective functions ensure that the transfer orbit reaches the desired deployment point, while the third and fourth ones guarantee the tangential conditions between the transfer orbit and the final orbit. An additional objective function minimizes the trajectory time. As a perturbation of interest, the effect of Earth’s oblateness is considered.

Manifold Triangular Mesh Editing Based on Finite Differences and Fourier Series

Manifold Triangular Mesh Editing Based on Finite Differences and Fourier Series

Hierarchical Distributed Adaptive Fault-Tolerant Control of Nonlinear Fractional-Order Multiagent Systems With Faults and Periodic Disturbances Using Event-Triggered Communication.

This article presents a distributed fault-tolerant control (FTC) scheme for nonlinear fractional-order (FO) multiagent systems (MASs) with the order lying in (0, 1], such that the proposed control architecture can be directly applied to both FO and integer-order (IO) systems without any modifications. To handle the unexpected actuator faults encountered by the FO MASs, a hierarchical FTC mechanism is developed for each system by constructing an event-triggered distributed FO estimator at the upper layer to estimate the leader system's output via conditionally triggered neighboring information, and an FTC unit at the lower layer to counteract the loss-of-effectiveness faults via Nussbaum function with FO criteria. To further address the unknown nonlinear functions involving bias faults and periodic disturbances, the Fourier series expansion technique is used to construct the input variables of fuzzy neural networks (FNNs), such that the FNNs with dynamically adjusted weight matrices, centers, and widths can be developed for each FO system to act as the learning module. It is shown by FO Lyapunov stability analysis that all follower systems can track the leader system against faults and periodic disturbances. Simulation results on FO systems and hardware-in-the-loop experiment results on IO fixed-wing unmanned aerial vehicles show the extensive feasibility of the developed scheme.

Electromagnetohydrodynamic flow and thermal performance in a rotating rough surface microchannel

Rough surfaces in microchannels effectively enhance liquid mixing, thermal performance, and chemical reactions in electrically actuated microfluidic devices. Rotation of the microchannel with surface roughness intensifies this enhancement. We investigate the combined effects of electromagnetohydrodynamics and surface roughness on transient rotating flow in microchannels. We present a mathematical model considering the variable zeta potential, heat transfer characteristics, and entropy generation within the microchannel. We obtain analytical solutions using the separation of variables method and Fourier series expansion. The surface roughness of the microchannel, when combined with rotation, impacts the temperature enhancement. Higher rotation rates result in the formation of multiple vortices. The secondary flow pushes the primary velocity toward the boundary layer, which affects the flow pattern. Surface roughness and electroosmotic flow significantly affect secondary flow, resulting in complex flow patterns and reversals. The interaction between centrifugal and viscous forces results in maximum velocities at the boundary layers. Higher roughness and electromagnetic effects enhance temperature by intensifying fluid-solid friction and joule heating. Surface roughness causes an increase in wall shear stress and friction factor, resulting in a higher Poiseuille number. Moreover, surface roughness increases entropy production by enhancing fluid mixing and internal friction despite improved heat transfer. Higher rotation also elevates entropy generation due to additional vortices induced by secondary flow.

An analytical method for broadband acoustic analysis of 2D cavities containing or bounded by porous materials

Real-life vibro-acoustic systems often involve porous treatments, resulting in complex-valued and frequency-dependent models that are challenging to solve. Traditional prediction techniques like the finite element (FE) method requires huge computational cost, especially in the mid to high frequency ranges. This paper develops a novel spectral dynamic stiffness (SDS) formulation, using very few number of degrees of freedom but describing the broadband acoustic behaviour of acoustic cavities with porous materials highly accurately. The method employs frequency-dependent shape function that satisfies exactly the (damped) Helmholtz equation to describe the (equivalent) acoustic pressure field, and also features an innovative approach to use the fast-convergent Modified Fourier series to describe any arbitrary acoustic BCs. Finally, the SDS matrices for cavities containing or bounded by porous materials are formulated in an analytical manner. It is demonstrated that the method exhibits a much higher computational efficiency over the FE package COMSOL, at least 6 times faster than the COMSOL with a similar level of accuracy, and benchmark solutions are provided. This promising method can provide a powerful tool for systems with porous materials that have frequency-dependent characteristics, paving the way for efficient and accurate acoustic analysis in complex engineering applications.

Elastodynamic behaviors of steady moving straight dislocation within thin nano film

The elastodynamic dislocation behaviors are of great interest for understanding the performances of structural alloys under intense dynamic loading conditions. The formation, propagations, and interactions of dislocations (such as injected dislocation, accelerating dislocation, steady moving dislocation at high constant speed) are quite different from static dislocations. For steady-moving dislocation within the isotropic infinite medium, the effects of surface and interface on steady-moving dislocations within limited space are still known. In this paper, we investigate the elastodynamic image stress simulation of steady moving dislocation within film of limited thickness at constant speed using Eigenstrain theory, Lorentz transformation, and steady dynamic equilibrium equations. We propose an efficient solution method that involves complex Fourier series, transforming partial differential equations into ordinary differential equations, and ultimately into a set of algebraic equations in spectral space. The effects of dislocation speed and position near the free surface on the image stress of steady-moving climbing and gliding dislocations within the thin film are examined. The results show that relativistic effects are significant for certain dislocation configurations and stress components, whereas other stress components are less sensitive to relativistic effects near the transonic speed region.