Sinusoidal Function Research Articles

Detecting self-organising patterns in crowd motion: effect of optimisation algorithms

The escalating process of urbanization has raised concerns about incidents arising from overcrowding, necessitating a deep understanding of large human crowd behavior and the development of effective crowd management strategies. This study employs computational methods to analyze real-world crowd behaviors, emphasizing self-organizing patterns. Notably, the intersection of two streams of individuals triggers the spontaneous emergence of striped patterns, validated through both simulations and live human experiments. Addressing a gap in computational methods for studying these patterns, previous research utilized the pattern-matching technique, employing the Nelder-Mead Simplex algorithm for fitting a two-dimensional sinusoidal function to pedestrian coordinates. This paper advances the pattern-matching procedure by introducing Simulated Annealing as the optimization algorithm and employing a two-dimensional square wave for data fitting. The amalgamation of Simulated Annealing and the square wave significantly enhances pattern fitting quality, validated through statistical hypothesis tests. The study concludes by outlining potential applications of this method across diverse scenarios.

Compliant mechanism design of combined aircraft wing for stable separation

Abstract Stable separation is a crucial condition that must be met in order for combined aircraft to successfully engage in cooperative flight. In order to achieve the desired fast and controlled separation, this paper proposes a novel design for a torque-driven compliant separation mechanism. By taking into account the compliance characteristics of a sinusoidal acceleration function curve, a mechanical model for the separation mechanism is developed. By utilising the Coulomb friction law, an accurate determination of the aerodynamic load distribution under various conditions is achieved. Subsequently, the relationship between the unlocking moment and the aerodynamic load is derived based on these findings. Through the utilisation of the finite element method, a model of the separation mechanism is generated. To ensure the safety and reliability of the compliant separation mechanism, the mechanical properties of the structural materials are thoroughly analysed under the maximum aerodynamic load. Subsequently, the separation mechanism structure is constructed and subjected to testing in order to showcase the compliance characteristics. In addition, this paper conducts a simulation to analyse the impact of flight speed and angle-of-attack on the separation process. By doing so, the optimal conditions for separation are determined. The methods and findings presented in this study have the potential to contribute valuable insights to the design of combined aircraft.

Linearizing the vertical scale of an interferometric microscope and its effect on step-height measurement.

The vertical scale calibration of an interferometric microscope is important for establishing traceability of surface topography measurements to the International System of Units (SI) unit of length, the meter. Building on the calibration procedure for the amplification coefficient developed by de Groot and Beverage [Proc. SPIE 9526, 952610 (2015)], this paper describes a calibration procedure that yields the response curve for the entire vertical scan motion of a coherent scanning interferometric microscope. The method requires only a flat mirror as an artifact, a narrow band spectral filter, an aperture to reduce the effective numerical aperture, and the ability to raise and lower the microscope head so that the center of the interferogram can be varied within the scan range. The local frequency of the interferogram is determined by fitting sections of the interferogram to a sinusoidal function. The nonlinearity determined from the local frequency data can be used to estimate the uncertainty in uncorrected vertical height measurements. We describe how optical profile data can be corrected for nonlinearity due to dynamic effects in the scan motion and show that the correction improves the reproducibility of step height measurements by at least a factor of three and close to that of the repeatability.

Bloom dynamics under the effects of periodic driving forces

Phytoplankton bloom received considerable attention for many decades. Different approaches have been used to explain the bloom phenomena. In this paper, we study a Nutrient–Phytoplankton–Zooplankton (NPZ) model consisting of a periodic driving force in the growth rate of phytoplankton due to solar radiation and analyse the dynamics of the corresponding autonomous and non-autonomous systems in different parametric regions. Then we introduce a novel aspect to extend the model by incorporating another periodic driving force into the growth term of the phytoplankton due to sea surface temperature (SST), a key point of innovation. Temperature dependency of the maximum growth rate (μmax) of the phytoplankton is modelled by the well-known Q10 formulation: μmax=μ0∗(Q10)T/10, where μ0 is maximum growth at 0oC. Stability conditions for all three equilibrium points are expressed in terms of the new parameter ρ2, which appears due to the incorporation of periodic driving forces. System dynamics is explored through a detailed bifurcation analysis, both mathematically and numerically, with respect to the light and temperature dependent phytoplankton growth response. Bloom phenomenon is explained by the saddle point bloom mechanism even when the co-existing equilibrium point does not exist for some values of ρ2. Solar radiation and SST are modelled using sinusoidal functions constructed from satellite data. Our results of the proposed model describe the initiation of the phytoplankton bloom better than an existing model for the region 25–35° W, 40–45° N of the North Atlantic Ocean. An improvement of 14 days (approximately) is observed in the bloom initiation time. The rate of change method (ROC) is applied to predict the bloom initiation.

Dynamics of Carbon and Water Fluxes over Cropland and Agroforest Ecosystems on the Southern Chinese Loess Plateau

Studies on the spatiotemporal dynamics in ecosystem carbon and water exchanges are essential in predicting the effects of climate change on regional carbon and energy budgets. Using the eddy covariance technique, carbon and water fluxes were observed in a typical winter wheat ecosystem (WWE) and an agroforest ecosystem (AFE) in the southern Loess Plateau from 2004 to 2010. The seasonal and inter-annual variability in gross primary productivity (GPP), net ecosystem exchange (NEE), evapotranspiration (ET), and water use efficiency (WUE) were examined and the main influencing factors were identified using the Pearson correlation. The results indicate that the seasonal GPP and NEE showed a bimodal distribution in WWE, while this was unimodal in AFE. The sinusoidal function did well in the characterization of seasonal ET dynamics for both ecosystems, with the determination coefficients being 0.85 and 0.94, respectively. In WWE and AFE, the annual mean GPP were 724.33 and 723.08 g C m−2 a−1, respectively, and the corresponding ET were 392.22 and 410.02 mm a−1. However, the difference in NEE between the two ecosystems was obvious, NEE were −446.28 and −549.08 g C m−2 a−1, respectively, showing a stronger carbon sink in AFE. There were strong coupling relationships between the GPP and ET of both ecosystems; the overall slopes were 1.71 and 1.69, respectively. The seasonal trend of WUE was bimodal in WWE, with peak values of 3.94 and 3.65 g C kg−1 H2O, occurring in November and April, respectively. However, the monthly WUE in AFE had one single peak of 4.07 g C kg−1 H2O in January. Photosynthetically active radiation (PAR) and soil temperature (Ts) were most positively correlated with GPP, net radiation (Rn) and Ts were the major factors influencing ET, while vapor pressure deficit (VPD) and soil water content (SWC) were the major influencing factors for WUE. These results provide observational support for regional carbon neutrality simulations.

Decomposition via elastic-band transform

In this paper, we propose a novel decomposition method using elastic-band transform (EBT), which mimics eye scanning and is utilized for multiscale analysis of signals. The proposed EBT-based method can efficiently extract the features of various signals with the following three advantages. First, it is a data-driven approach that extracts several important modes based solely on data without using predetermined basis functions. Second, it does not assume that the signal consists of (locally) sinusoidal intrinsic mode functions, which is a common assumption in existing methods. Therefore, the proposed method can handle a wide range of signals. Finally, it is robust to noise. A practical algorithm for decomposition is presented, along with some theoretical properties. Simulation examples and real data analysis results show promising empirical properties of the proposed method.

A novel entanglement functions-based 4D fractional-order chaotic system and its bifurcation analysis

A novel 4D fractional-order chaotic entanglement system based on sinusoidal functions is established in this paper. We aim to reveal the relationship between the dynamical behavior of the new system and its entanglement coefficients. It is found that the equilibrium point of the system varies regularly with the successive change of the entanglement coefficient. The supercritical pitchfork bifurcation phenomenon of the new system is discussed based on the fractional-order stability theory. Furthermore, sufficient conditions and threshold for supercritical Hopf bifurcation caused by the entanglement coefficient are provided. Finally, the route to chaos of the new system is explored utilizing multiple numerical indicators, such as spectral entropy complexity, bifurcation diagrams, Lyapunov exponential spectrum, phase portraits, and 0–1 test curves. The results indicate that in addition to various chaotic attractors, there are phenomena such as period-doubling bifurcations, period windows, and coexisting symmetric attractors (periodic or chaotic).

Characteristics and evolution of ground vortex under heaving ground conditions: Insights from large eddy simulation (LES) analysis

When an aircraft engine operates near the ground, the ambient vortices generated by the freestream velocity boundary layer and engine suction will gather and lift, forming ground vortices which may result in intake distortion, consequently affecting the engine operation stability. To develop a shipborne test platform, this study conducted large eddy simulation calculations using the ANSYS Fluent and based on the assumption of incompressible flow. The 3-D unsteady Navier-Stokes (N-S) equations were solved using the WALE sub-grid model. The analysis focused on the characteristics and evolution of ground vortices under heaving ground conditions within 1 s. The results indicate that the ground vortex exhibits strong unsteady characteristics. During the descending phase of the platform, the quantity of vortex formed beneath the intake duct would decrease while the trailing vortex becomes more pronounced. However, the structure of the inlet vortex remains relatively unchanged at each time step. The intensity of the vortex and pressure distortion coefficient exhibit significant fluctuations over time, and their temporal variations approximately follow a sinusoidal function, which is also the heaving motion of the platform. Generally, when the velocity of the platform reaches its maximum downward vertical value, the average vortex intensity of the ground vortex tends to be minimal while the pressure distortion coefficient approaches its maximum value. Conversely, when the velocity of the platform reaches its maximum upward vertical value, the opposite trend can be observed.

Non-linear behavior of functionally graded elastoplastic beam under torsion

The torsional behavior of beams graded in one and two directions under large displacements and angular deformations was analyzed using the power law and sinusoidal functions. Functionally graded material is elastoplastic, consisting of ceramic and metal. A nonlinear finite element method with isoparametric hexahedral elements was used. The finite element formulation was made using the updated Lagrangian formulation based on the virtual displacement principle. An iterative solution using Newton-Raphson and updated Newton-Raphson methods was used to solve the nonlinear equation system. The propagation of the plastic zone was calculated based on the flow theory of plasticity. Elastoplastic behavior and effective material properties were determined according to the TTO model. Numerical investigations have shown that functionally graded beams behave quite differently from homogeneous beams under torsion. Yielding of the material starts at the outer boundaries of the section of the homogeneous beams, and the plastic zone propagates symmetrically. On the other hand, yielding and propagation of plastic zones tend to shift to regions with more ceramic volume with higher effective Young modulus in functionally graded beams. Unlike homogeneous beams, the largest shear stresses can occur within the section rather than at the outer boundaries of the section. In beams graded from ceramic to metal using the power law, the section moves along the transverse direction in addition to the rotation. This transverse displacement occurs in the grading direction, and its magnitude is about 3% of the thickness at 12.5° rotation angle. Also, the shear stresses are not zero in the section's midpoint. The effects of material distribution on displacements, stresses, and plastic zone propagation were examined, and essential points were reported.

Evaluation of thermal radiation and Lewis number effects on oscillatory thermal-diffusive instabilities of counterflow premixed flame fed with moisty biomass particles

In this study, a thermal-diffusive model using a sinusoidal function for the oscillation term is presented. The model is employed to evaluate the pulsating behavior in a premixed counterflow flame fed with biomass particle cloud considering Lewis number and thermal radiation effects. In order to scrutinize the pulsating characteristics of a flame, time-related governing equations are written for the assumed multi-zone system and analytically solved utilizing Matlab and Mathematica software. To enforce continuity, suitable matching conditions for the temperature and heat flux at the interfaces are derived. Time-dependent characteristics of burning velocity, flame temperature, flame front position, reactants temperature, and mass distributions are presented, taking into account effective Lewis number, thermal radiation, and strain rate influences. An acceptable comparison with numerical simulations affirms the reliability of the mathematical procedure. The obtained results indicated that the pulsating instability of biomass-fueled premixed flames in counterflow arrangements significantly relevant not only to the effective Lewis number but also to thermal radiation transferred from the reaction zone to unburnt reactants. Furthermore, it is found that in counterflow premixed flames, strain rate, in addition to Lewis number and thermal radiation, can be the key to controlling the individual flame oscillation modes.

Rectangular-to-elliptical shape transition intakes with various leading-edge for performance enhancement

This study explores the efficiency of streamline-tracing intake in scramjets within the hypersonic domain, where the Mach number (Ma) exceeds five. Focusing on a rectangular-to-elliptical shape transition intake following Busemann streamlines with a contraction ratio of 4.67, tailored for Ma 7.1, this study innovates beyond traditional design method, employing numerical simulations to interpolate iso-Ma points and shape the leading edge of the intake to enhance the performance. Three distinct leading-edge designs—using a sinusoidal function to closely track all iso-Ma points, a convex second-order polynomial to bolster compression and mass flow, and a concave polynomial to widen the starting envelope—were conceptualized. The research included experiments with 0.06 scaled models for validation and performance assessments of full-scale intakes using three-dimensional viscous simulations. Key performance metrics, such as the pressure ratio, total pressure recovery, mass flow rate, and exit Ma, were evaluated to reveal the capability of all three intake designs to operate effectively over a broad Ma range (3–7). A key finding of this study is the ability to selectively enhance either the compression capability or the total pressure recovery by strategically designing the leading edge of the intake through interpolation methods. This study underscores the importance of using numerical simulation results in the notching process as an effective design strategy to handle a wide range of off-design conditions with fixed geometry and achieve performance improvements in scramjet intakes.

Mass transport at the bottom of propagating surface waves over a rippled bottom

The sea surface can be described by means of the superposition of many sinusoidal functions. However, quite often the amplitude of each component turns out to be much smaller than its wavelength, and any component evolves independently of the others. Hence, it is common to investigate the dynamics of a simple monochromatic surface wave. Hereinafter, the flow generated by a monochromatic surface wave within the bottom boundary layer over a rippled sea bed is determined by means of the numerical integration of vorticity and continuity equations. The forcing term that drives the fluid motion within the boundary layer is evaluated assuming that the steepness of the monochromatic surface wave is much smaller than one and considering the first term of the Stokes expansion. Even though the irrotational flow that forces the viscous rotational flow near the sea bottom is symmetric with respect to the ripple crests, Blondeaux and Vittori [“A route to chaos in an oscillatory flow: Feigenbaum scenario,” Phys. Fluids A 3(11), 2492–2495 (1991a)] showed that the symmetry of the flow field is broken when the Reynolds number becomes larger than a threshold value Rδ,t1 that depends on the geometrical characteristics of the ripples. The results of Blondeaux and Vittori [“A route to chaos in an oscillatory flow: Feigenbaum scenario,” Phys. Fluids A 3(11), 2492–2495 (1991a)] suggest that, when the Reynolds number is larger than Rδ,t1 but not too far from it, a steady current is also generated. Hereinafter, the steady velocity component is determined as a function of the ripple characteristics.

Analyzing the stochastic dynamics of COVID-19 waves in Turkey using real data and piecewise sinusoidal functions

Analyzing the stochastic dynamics of COVID-19 waves in Turkey using real data and piecewise sinusoidal functions

Complexity of Neural Network Training and ETR: Extensions with Effectively Continuous Functions

The training problem of neural networks (NNs) is known to be ER-complete with respect to ReLU and linear activation functions. We show that the training problem for NNs equipped with arbitrary activation functions is polynomial-time bireducible to the existential theory of the reals extended with the corresponding activation functions. For effectively continuous activation functions (e.g., the sigmoid function), we obtain an inclusion to low levels of the arithmetical hierarchy. Consequently, the sigmoid activation function leads to the existential theory of the reals with the exponential function, and hence the decidability of training NNs using the sigmoid activation function is equivalent to the decidability of the existential theory of the reals with the exponential function, a long-standing open problem. In contrast, we obtain that the training problem is undecidable if sinusoidal activation functions are considered.

Numerical study of functional parameters of compact heat exchanger with the corrugated wall in steady state

Compact heat exchangers have several benefits over other types of heat exchangers, including compactness, a high surface area to volume ratio, and a low-temperature differential. They are frequently employed in the heat integrating process. Herein, a compact heat exchanger is simulated in a steady state using the AnsysFluent 17.2 software pack. The flow regime is laminar and water enters the exchanger at a temperature of 298.15 K, assuming a constant wall temperature condition. This study examines the effects of wall wave formation on fluid flow and heat transfer characteristics. The two parameters, wave altitude, and wave angle, are verified and the resultant corrugation is fitted as a sinusoidal function. As the wave height factor increases, the results show that the temperature and velocity values in the solution field climb. More than 4.5% more has been added to the highest temperature value. It is also shown that the velocity values rise with the wave angle coefficient. The greatest rise is 50%. The Performance Ratio is also examined in this study. The findings show that the angle coefficient of 0.5π and π and the altitude coefficient of 0.2 provide values for the Performance Ratio that are higher than the unit.

Computational Pulsatile Flow and Efficiency Analysis of Biocompatible Microfluidic Artificial Lungs for Different Fiber Configurations.

Average-sized microfluidic artificial lungs consisting of rows and columns of fiber bundles with different column to row aspect ratios (AR) are numerically analyzed for flow characteristics, maximum gas transfer performance, minimum pressure drop, and proper wall shear stress (WSS) values in terms of biocompatibility. The flow is fully laminar and assumed to be incompressible and Newtonian. The transport analysis is performed using a combined convection-diffusion model, and the numerical simulations are carried out with the finite element method. The inlet volumetric flow is modeled as a sinusoidal wave function to simulate the cardiac cycle and its effect on the device performance. The model is first validated with experimental studies in steady-state condition and compared with existing correlations for transient conditions. Then, the validated model is used for a parametric study in both steady and pulsatile flow conditions. The results show that increasing the aspect ratio in fiber configuration leads to converging gas transfer, higher pressure drop, and higher WSS. While determining the optimum configuration, the acceptable shear stress levels play a decisive role to ensure biocompatibility. Also, it is observed that the steady analysis underestimates the gas transfer for higher aspect ratios.

Approximate closed-form solutions for free vibration of circular arches with discrete lateral braces

Discrete lateral braces are commonly employed to enhance the out-of-plane stability of arches. However, these braces generate coupling effects among vibration modes, which significantly impact the vibration characteristics of the arches. This paper presents an analytical solution for the free vibration analysis of two-hinged arches with equally-spaced lateral translational and/or rotational braces. A simplified method of evaluating the fundamental frequency for the out-of-plane vibration of arches with equally-spaced lateral braces is introduced. The method relies on the conventional approach of representing arch braces through elastic springs, but here the out-of-plane deflection of the braced arch is constructed by adding related coupled sinusoidal functions to that of the unbraced arch, satisfying both the boundary and kinematic conditions. Subsequently, the fundamental frequency and threshold bracing stiffness or stiffness requirement are derived using the Rayleigh–Ritz method. Finite element analysis (FEA) is applied to investigate the flexural-torsional vibration mode and fundamental frequency of arches with discrete braces, and the results from FEA closely match the theoretical predictions. The effects of number of lateral translational and/or rotational brace, out-of-plane slenderness, included angle, bracing stiffness and bracing position on the fundamental frequency are comprehensively analyzed. It is found that when the fundamental frequency of the braced arch increases due to changes in bracing stiffness, the mode shape displays more intricate behavior, rather than following a straightforward low-to-high mode growth pattern. Furthermore, it is observed that in some cases, the addition of rotational bracing stiffness can markedly enhance the maximum fundamental frequency of the arch with both translational and rotational braces.

SP2D: Simplified pseudo 2D battery model by piecewise sinusoidal/quadratic functions of potential curves

The pseudo 2D battery model is a widely used tool for analysing the electrochemical behaviour of lithium-ion batteries. However, the high computational expense of solving the fully coupled governing equations is a significant challenge. In this study, a simplified pseudo 2D battery model has been developed to address this bottleneck. The model gets the electrolyte potential with relative potential plus reference potential, which are described by piecewise quadratic or sinusoidal function and determined by boundary conditions. Two electrodes are defined with uniform potentials. This approach provides a simple solution of charge transfer equations and decouples them from the governing equations of the pseudo 2D model. It shows excellent agreement with the original pseudo 2D battery model for a wide range of C-rates, drive cycles and is applicable to electrolytes with different conductivities. The proposed approach provides a simpler and faster solution than the pseudo 2D battery model, with negligible accuracy sacrifice. Consequently, this work is crucial for applying physics-based models to real-time battery control and management systems.

Fourier series method for the stability solution of simply supported thin beams on two-parameter elastic foundations of the Pasternak, Filonenko-Borodich, Hetenyi or Vlasov models

The analysis of stability problems of beams on two-parameter foundations (Bo2PFs) is an important part of their design for compressive loads. This work presents novel first principles derivation of the governing differential equations of elastic stability (GDES) of thin beams resting on two-parameter elastic foundations of the Pasternak, Filonenko-Borodich, Hetenyi or Vlasov models. The requirements of translational and rotational equilibrium of all the applied, reactive and internal forces on an infinitesimal segment of the Bo2PF and the laws of infinitesimal calculus were used to formulate the GDES as a fourth order ordinary differential equation (ODE) in terms of the transverse displacement function ux. The GDES is non-homogeneous in the presence of applied transverse load qx but homogeneous when qx vanishes. This study presents the Fourier series method (FSM) for solving the governing differential equation of stability (GDES) for the case of Dirichlet boundary conditions. The FSM has the advantage of amenability to differentiation, and integration due to the orthogonality properties of the sinusoidal functions. Implementation of the FSM by assuming the unknown function in the GDES as a Fourier series of infinite terms and the exploitation of orthogonalization simplifies the problem to an algebraic eigenvalue problem which is the characteristic buckling equation. The exact eigenvalues are found by algebraic solution of the buckling equation. The exact eigenvalues were used to find the exact buckling loads and the exact buckling load coefficients. The critical buckling load was found to correspond to the first buckling mode (n= 1), and is identical with previous solutions in the literature. Numerical calculations for the critical buckling load parameters Kcr were presented for the Bo2PF problem for values of the dimensionless foundation parameters k-1= 0, k-2= 0; k-1= 100, k-2= 0; k-1= 0, k-2= 1; k-1= 100, k-2= 100; k-1= 0, k-2= 2.5; k-1= 100, k-2= 2.5. The present solutions were compared with previous solutions for Kcr in the literature. The comparison shows that the present FSM results are identical with previous results obtained using various other methods such as Recursive Differentiation Method, Finite Element Method, Generalized Integral Transform Method (GITM) and Stodola-Vianello Iteration Method. The study has illustrated the effectiveness of the FSM for solving Bo2PFs.

Mathematical Prediction of Electrical Solar Energy Based on Solar Data for Two Main Cities of Chad: Mongo in the Centre and Pala in the South of Chad

The comparative study of the solar powers between two main cities of Chad is performed in the present work, the city of Mongo in the Centre and that of Pala in the South, with an aim of knowing which one of the two cities is more adequate for an installation of the solar power station, taking into account the regional climatic and environmental conditions of both cities. To do this, the graphical statistical analysis of long-term solar irradiance data and temperature is performed. The data used is that of the decade (2010-2020), based on solar radiation data handed by the National Aeronautics and Space Administration (NASA) and Photovoltaic Geographical Information System (PGIS) for Mongo in the centre and Pala in the south of Chad. The shape of the mean monthly irradiation has been plotted and has been approximated using the sinusoidal function through the mean square analysis. The temperature data has been also obtained by the same process and plotted versus irradiance in order to find the adequate mathematical relationship between them. For the statistical analysis, the maximum entropy principle has been used. As results, it is found that the maximum irradiance is obtained in March, which are 226.26kWh/m<sup>2</sup> for Pala and 219.355kWh/m<sup>2</sup> for Mongo, while the minimum irradiances are obtained in August, which are 151.67kWh/m<sup>2</sup> for Pala and 158.9kWh/m<sup>2</sup> for Mongo. The temperature data is also obtained and the mean monthly data plotted, showing that apart for the months of March and April, the the shapes of irradiation and temperatures are similar for both sites. Then it is found that the frequency and probability density distributions reach their maximum at the same dates.