Fourier Series Research Articles

Discovering non-associated pressure-sensitive plasticity models with EUCLID

We extend (EUCLID Efficient Unsupervised Constitutive Law Identification and Discovery)—a data-driven framework for automated material model discovery—to pressure-sensitive plasticity models, encompassing arbitrarily shaped yield surfaces with convexity constraints and non-associated flow rules. The method only requires full-field displacement and boundary force data from one single experiment and delivers constitutive laws as interpretable mathematical expressions. We construct a material model library for pressure-sensitive plasticity models with non-associated flow rules in four steps: (1) a Fourier series describes an arbitrary yield surface shape in the deviatoric stress plane; (2) a pressure-sensitive term in the yield function defines the shape of the shear failure surface and determines plastic deformation under tension; (3) a compression cap term determines plastic deformation under compression; (4) a non-associated flow rule may be adopted to avoid the excessive dilatancy induced by plastic deformations. In contrast to traditional parameter identification methods, EUCLID is equipped with a sparsity promoting regularization to restrain the number of model parameters (and thus modeling features) to the minimum needed to accurately interpret the data, thus achieving a compromise between model simplicity and accuracy. The convexity of the learned yield surface is guaranteed by a set of constraints in the inverse optimization problem. We demonstrate the proposed approach in multiple numerical experiments with noisy data, and show the ability of EUCLID to accurately select a suitable material model from the starting library.

Stability of Tollmien-Schlichting modes in magnetohydrodynamic boundary layer flow in porous medium: energy budget analysis

Abstract Linear temporal and spatial stability analyses of the magnetohydrodynamic boundary layer flow over a wedge embedded in a porous medium have been carried out to analyze the effects of pressure gradient, Hartmann and Darcy numbers. Firstly, we determine the base state velocity profiles by imposing suitable similarity transformations on governing boundary layer equations and then find linear perturbed equations involving Reynolds number and disturbance wavenumber using Fourier modes. The Chebyshev spectral collocation method which provides insight into the complete structure of the eigenspectrum is used. The effect of Hartmann and Darcy numbers on the boundary layer is to stabilize the flow for all adverse pressure gradient parameters while only unstable modes are noticed in the absence of these effects. The noticed unstable modes are always part of the wall mode for which the phase speeds are found to approach zero. The eigenspectrum for all involved parameters has a balloon-like structure with the appearance of wall mode instability. The critical Reynolds number is found to be increasing for increasing pressure gradient, Hartmann and Darcy numbers. For an adverse pressure gradient, the energy budget shows that the energy production due to Reynolds stress dominates the viscous dissipation which results in destabilization of the flow, while the kinetic energy due to magnetic field and porous medium plays a role in stabilizing the flow. The physical dynamics behind these interesting modes are discussed.

SineKAN: Kolmogorov-Arnold Networks using sinusoidal activation functions

Recent work has established an alternative to traditional multi-layer perceptron neural networks in the form of Kolmogorov-Arnold Networks (KAN). The general KAN framework uses learnable activation functions on the edges of the computational graph followed by summation on nodes. The learnable edge activation functions in the original implementation are basis spline functions (B-Spline). Here, we present a model in which learnable grids of B-Spline activation functions are replaced by grids of re-weighted sine functions (SineKAN). We evaluate numerical performance of our model on a benchmark vision task. We show that our model can perform better than or comparable to B-Spline KAN models and an alternative KAN implementation based on periodic cosine and sine functions representing a Fourier Series. Further, we show that SineKAN has numerical accuracy that could scale comparably to dense neural networks (DNNs). Compared to the two baseline KAN models, SineKAN achieves a substantial speed increase at all hidden layer sizes, batch sizes, and depths. Current advantage of DNNs due to hardware and software optimizations are discussed along with theoretical scaling. Additionally, properties of SineKAN compared to other KAN implementations and current limitations are also discussed.

Methodology for Designing Vibration Devices with Asymmetric Oscillations and a Given Value of the Asymmetry of the Driving Force

In mechanical engineering, the building industry, and many other branches of industry, vibration machines are widely used, in which circular and directed oscillations predominate in the form of movement of the working equipment. This article examines methods for generating asymmetric oscillations, which are estimated by a numerical parameter, namely by the coefficient of asymmetry of the magnitude of the driving force when changing the direction of action in a directed motion within each period of oscillations. It is shown that for generating asymmetric mechanical vibrations, vibration devices are used, consisting of vibrators of directed vibrations, called stages. These stages form the total asymmetric driving force. The behavior of the total driving force of asymmetric vibrations and the working equipment of the vibration machine are described by analytical equations, which represent certain laws of motion of the mechanical system. This article presents a numerical analysis of methods for obtaining laws of motion for a two-stage, three-stage, and four-stage vibration device with asymmetric oscillations. An analysis of the methodology for obtaining a generalized law of motion for a vibration device with asymmetric oscillations is performed based on the application of polyharmonic oscillation synthesis methods. It is shown that the method of forming the total driving force of a vibration device based on the coefficients of the terms of the Fourier series has limited capabilities. This article develops, substantiates, and presents a generalized method for calculating and designing a vibration device with asymmetric oscillations by the value of the total driving force and a given value of the asymmetry coefficient in a wide range of rational designs of vibration machines. The proposed method is accompanied by a numerical example for a vibration device with an asymmetry coefficient of the total driving force equal to 10.

CONSUMER PRICE INDEX MODELING USING A MIXED TRUNCATED SPLINE AND KERNEL SEMIPARAMETRIC REGRESSION APPROACH

Some semiparametric regression model approaches include spline, kernel, Fourier series, and wavelet. Semiparametric regression modelling can involve more than one independent variable (multivariable), a parametric approach is usually combined with one of the nonparametric approaches, such as combining a parametric approach with a nonparametric kernel. If a consumer price index model can be built based on the variables that influence it, predictions of consumer price percentages can be made, which it is hoped will help the government determine policies to control consumer price inflation, especially in NTB Province. The data used in this research includes the consumer price index and the factors that influence it according to districts/cities in NTB Province from 2022 to April 2024. The data source was obtained from secondary data at BPS NTB Province. This research design uses a mixed semiparametric approach of truncated spline and kernel regression. Based on calculations, the predicted results of the consumer price index in NTB Province show that the predicted data graph is very close to the actual data . Modelling the consumer price index in NTB Province is a model with 2 knot points, where the model efficiency has the smallest GCV value of 0.001507. The model goodness value is 0.99, meaning that the variables used can explain 99% of the model variability.

Quantum integration of decay rates at second order in perturbation theory

Abstract We present the first quantum computation of a total decay rate in high-energy physics at second order in
perturbative quantum field theory. This work underscores the confluence of two recent cutting-edge advances.
On the one hand, the quantum integration algorithm Quantum Fourier Iterative Amplitude Estimation (QFIAE),
which efficiently decomposes the target function into its Fourier series through a quantum neural network before quantumly integrating the corresponding Fourier components. On the other hand, causal unitary in the
loop-tree duality (LTD), which exploits the causal properties of vacuum amplitudes in LTD to coherently generate all contributions with different numbers of final-state particles to a scattering or decay process, leading to
singularity-free integrands that are well suited for Fourier decomposition. We test the performance of the quantum algorithm with benchmark decay rates in a quantum simulator and in quantum hardware, and find accurate
theoretical predictions in both settings.

Stability of stationary states for mean field models with multichromatic interaction potentials

Abstract We consider weakly interacting diffusions on the torus, for multichromatic interaction potentials. We consider interaction potentials that are not $H$-stable, leading to phase transitions in the mean field limit. We show that the mean field dynamics can exhibit multipeak stationary states, where the number of peaks is related to the number of nonzero Fourier modes of the interaction. We also consider the effect of a confining potential on the structure of non-uniform steady states. We approach the problem by means of analysis, perturbation theory and numerical simulations for the interacting particle systems and the PDEs.

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.

Axial Vibration of a Viscoelastic FG Nanobeam with Arbitrary Boundary Conditions

ObjectiveThis study investigates the axial vibration of a viscoelastic functionally graded (FG) nanobeam under deformable boundary conditions for the first time. The primary focus is on exploring the effects of damping and scale parameters on the dynamic behavior of the nanobeam.MethodsThe governing equation of the viscoelastic FG nanobeam is formulated by incorporating nonlocal elasticity theory and the Kelvin-Voigt viscoelastic model. The Fourier sine series is chosen as the axial displacement function, with higher-order derivatives obtained using Stokes transforms. The Fourier coefficient is determined through the governing equation and incorporated into the deformable boundary conditions. The resulting eigenvalue problem provides solutions for both rigid and constrained general boundary conditions.ConclusionsThe study presents solutions for various boundary conditions, comparing the results with existing literature. The analysis reveals significant findings, including the observation that damping has a greater influence on fundamental frequencies in higher modes, and that the impact of damping decreases as the nonlocal scale parameter increases. These findings are presented through tables and graphs to highlight the effects of damping and scale parameters on the dynamic behavior of the nanobeam.

Kinematic classification of mandibular movements in patients with temporomandibular disorders based on PCA

This retrospective study aimed to kinematically classify mandibular movements collected during Temporomandibular Disorders (TMD) treatment, employing Fourier transformation (FT), Principal Component Analysis (PCA), and K-means clustering (k-means), and to investigate their correlation with symptoms of pain-related TMD. The study included five TMD participants diagnosed with myalgia (age: 39–86 years, with an SD of 18.96) and three healthy participants (age: 32–42 years, with an SD of 5.13) with no stomatognathic problems. TMD participants underwent tailored treatment for their symptoms, and their maximum unassisted mouth opening (MMO) was recorded randomly with a motion capture system (ARCUS digma II, Kavo, Biberach, Germany) at multiple time points. MMO for healthy participants served as a control. The dataset comprising 28 trials, was transferred to the AnyBody Modeling System (AnyBody Technology, Aalborg, Denmark) to extract joint angle time series, which were then transformed into Fourier series. Subsequently, PCA and k-means clustering were conducted. Two clusters were identified: Cluster 1, predominantly composed of symptomatic trials, and Cluster 2, mainly consisting of asymptomatic trials. Distinct transition pathways between the clusters were observed among participants, corresponding to the alleviation of pain-related symptoms during TMD treatment. These findings suggest that this approach has potential as an effective tool for diagnosing and assessing TMD by identifying symptomatic kinematic patterns and tracking temporal changes in mandibular movement. Despite the small dataset, these results suggest promise for a novel functional assessment method for TMD based on kinematic features.

Direct mathematical method for real-time ischemic episodes detection from electrocardiograms using the discrete Hermite transform

A real-time automated identification technique is developed for the detection of ischemic episodes in long-term electrocardiographic (ECG) signals using mathematical expansions involving the Discrete Dilated Hermite Transform. The Discrete Hermite functions could be viewed as a set of orthogonal vectors that resemble a finite Fourier series. They are generated easily as eigenvectors of a symmetric tridiagonal matrix that commutes with the centered Fourier matrix. The Discrete Hermite Transform (DHmT) values are computed from a simple dot product between an individual ECG complex extracted from the European Society of Cardiology (ESC) ST-T database and the corresponding discrete Hermite function. These values are found to contain information about the ECG shape, highlighting changes between ST segment and T wave alterations which are the features of ischemic episodes. This information from the discrete Hermite transform, based on an orthonormal set of n-dimensional digital Hermite functions that serve as shape-identification functions, can be used to identify ischemic episodes from the ECG. The performance measures resulting from applying this method to detect ischemic episodes were Sensitivity 87 %, Specificity 86 %, and positive predictive accuracy 81 %. The computer time to analyze one heartbeat for ischemia with this method is 0.031 s on a standard PC.

Acoustic Cavity Boundary Impedance Identification Based on Hybrid Neural Network and Boundary-Smoothed Fourier Series Method

Acoustic Cavity Boundary Impedance Identification Based on Hybrid Neural Network and Boundary-Smoothed Fourier Series Method

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.

NONLINEAR DOUBLE-DIFFUSIVE CONVECTION IN AN ANISOTROPIC POROUS LAYER UNDER TIME-DEPENDENT ROTATION WITH INTERNAL HEATING AND SORET EFFECT

This study investigates the double-diffusive convection onset in a nonuniformly rotating anisotropic porous fluid layer under the influence of Soret and internal heating effects. The linear stability approach is employed to investigate the system when subjected to infinitesimal perturbations. The nonlinear case is investigated using a minimum truncated double Fourier series, leading to the derivation of nonlinear Lorenz-type equations. As a novel characteristic of the article, the newly developed local linearization block hybrid method is utilized to solve the nonlinear Lorenz-type equations. We observed that the method achieves convergence and accurate results with a large number of collocation points. Heat and mass transfers have been expressed in terms of Nusselt number and Sherwood number, respectively. The study also investigates the influence of time-dependent rotation and internal heat generation on heat and mass transfer in anisotropic porous layers, including the Soret effect. Among other findings, we noticed that rotation modulation and mechanical anisotropy enhance the rate of heat and mass transfer, potentially advancing the onset of convection in the system. Further, the dual effect of internal heat generation is observed in the presence of the Soret effect.

Pinning of reaction–diffusion travelling waves in one-dimensional annular geometry

We analyse the complex dynamics arising in a one-dimensional bistable advection–reaction–diffusion equation on a bounded annular domain, along with its simplest nontrivial approximation in terms of a Fourier expansion. We investigate the pinning of travelling waves produced by spatial anisotropies and suggest possible biological implications in the context of cardiac action potential propagation failure due to acute ischemia.

Synchronized Approach Based on Empirical Fourier Decomposition for Accurate Assessment of Harmonics and Specific Supraharmonics

Synchronized Approach Based on Empirical Fourier Decomposition for Accurate Assessment of Harmonics and Specific Supraharmonics

Hydrodynamic modes in nano-channels

This paper investigates the local hydrodynamics of a dense fluid confined in nanoscale slit-pores with different heights. Using non-equilibrium molecular dynamics simulations of the fluid system, we induce a steady-state sinusoidal velocity profile across the channel having a characteristic wavelength, thus, probing the fluid response to a specific Fourier mode. As expected, for sufficiently large channel heights and wavelengths there is an excellent agreement between the hydrodynamic predictions and simulation data. As the wavelength decreases to around 5 molecular diameters, the classical hydrodynamics fails to predict the steady-state velocity profile; we attribute this to the non-local nature of the fluid response and the presence of density gradients in the wall–fluid interfacial region. Using generalized hydrodynamics and the Fourier spectrum of the density profile, we derive the strain rate amplitude and shear pressure corrections due to these two effects. The local relaxation from the steady-state to the zero flow situation is tracked for different channel heights and wavelengths. The relaxation is in general visco-elastic in the wall–fluid region, and we argue that this phenomenon is the mechanism behind the “enhanced viscosity” used in the literature. We also report a surprising dynamics for the fluid located between the wall–fluid region and bulk region, which cannot be explained by classical hydrodynamics; here, an initial exponential relaxation abruptly transitions into a linear relaxation. The work highlights the many different physical mechanisms present in nano-confined fluids, and that the fluid response is in general position and wavelength dependent.

Applications of Fourier Series on Signal processing and Heat conduction

This paper explores the application of the Fourier series in mathematical analysis and its significant impact on scientific fields such as signal processing and heat conduction. The study emphasizes the utility of Fourier series in decomposing complex functions into simpler trigonometric components, which facilitates the analysis of periodic functions. A primary focus is on the series’ role in approximating square waves in signal processing, demonstrating its effectiveness despite challenges like the Gibbs phenomenon. Additionally, the Fourier series is applied to solving the heat equation, where it models the evolution of temperature distribution over time in a medium. Techniques to improve convergence and mitigate oscillations, such as corrected Fourier series and summation methods like Cesàro and Fejér summation, are discussed to enhance the accuracy of approximations in cases with discontinuities. The results underline the Fourier series’ versatility in representing and analyzing both smooth and discontinuous functions, showcasing its importance in theoretical and applied mathematics.

Modeling of Electric Field and Dielectrophoretic Force in a Parallel-Plate Cell Separation Device with an Electrode Lid and Analytical Formulation Using Fourier Series.

Dielectrophoresis (DEP) cell separation technology is an effective means of separating target cells which are only marginally present in a wide variety of cells. To develop highly efficient cell separation devices, detailed analysis of the nonuniform electric field's intensity distribution within the device is needed, as it affects separation performance. Here we analytically expressed the distributions of the electric field and DEP force in a parallel-plate cell separation DEP device by employing electrostatic analysis through the Fourier series method. The solution was approximated by extrapolating a novel approximate equation as a boundary condition for the potential between adjacent fingers of interdigitated electrodes and changing the underlying differential equation into a solvable form. The distributions of the potential and electric fields obtained by the analytical solution were compared with those from numerical simulations using finite element method software to verify their accuracy. As a result, it was found that the two agreed well, and the analytical solution was obtained with good accuracy. Three-dimensional fluorescence imaging analysis was performed using live non-tumorigenic human mammary (MCF10A) cells. The distribution of cell clusters adsorbed on the interdigitated electrodes was compared with the analytically obtained distribution of the DEP force, and the mechanism underlying cell adsorption on the electrode surface was discussed. Furthermore, parametric analysis using the width and spacing of these electrodes as variables revealed that spacing is crucial for determining DEP force. The results suggested that for cell separation devices using interdigitated electrodes, optimization by adjusting electrode spacing could significantly enhance device performance.

Specific Signal Analysis of the Nikkei and Dow Jones Industrial Indices Using Fourier Series Post-1985 Plaza Agreement

Specific Signal Analysis of the Nikkei and Dow Jones Industrial Indices Using Fourier Series Post-1985 Plaza Agreement