Polynomial Variation Research Articles

Implementation of the Gauss-Kronrod Quadrature Method (G7, K15) on 2D Gravity Anomaly Modeling in Basins with a Polynomial Variation of Density Distribution with Depth

Forward modeling of 2D gravity anomalies, considering density contrasts that vary polynomially with depth, was performed to examine basin structures. This process involved two main stages: deriving analytical formulas and executing numerical integration. The Gauss-Kronrod Quadrature Method, utilizing 7 Gauss points and 15 Kronrod points, was employed to precisely compute these integrals. Initial modeling applied to theoretical basement scenarios with fixed density contrasts showed gravity anomalies that accurately reflected the curvature of the basement. To validate the approach, it was then applied to real-world cases including the Sebastian Vizcaino Basin, San Jacinto Graben, and Sayula Basin. By incorporating suitable density contrasts, modeling lengths, and basement curvature shapes, the results revealed that both fixed-density and depth-variable density models produced gravity anomalies with patterns consistent with the actual basement curvature. These findings validate the modeling technique’s effectiveness in representing real geological features accurately. The study confirms that the Gauss-Kronrod Quadrature Method (G7, K15) is robust for analyzing 2D gravity anomalies, providing a reliable tool for understanding the influence of varying density contrasts on gravity responses.

Water resource optimization bi-level coupling model and carrying capacity of a typical plateau basin based on interval uncertainty stochastic programming

Abstract The bi-level programming coupling model of uncertainty constraints and interval parameter programming is developed to optimize the allocation of water resources and conduct a comprehensive analysis of water resource carrying capacity. The model uses an uncertainty credibility number set and interval value to deal with uncertain factors, and analyses the water resources allocation of Longchuan River in central Yunnan. The competition mechanism and polynomial variation improved algorithm are used to analyze the water consumption, economic benefits and satisfaction in different planning periods when λ = 0.7, 0.8, 0.9, 1.0. The results show that the uncertain bi-level coupling model can cause changes in water allocation, pollutant discharge, system efficiency, etc., and can also effectively balance the mutual constraints between economic benefits and environmental pollution discharge, ensuring a good development trend in the planning year. The water diversion from other basins such as the Central Yunnan Water Diversion Project was transferred to Longchuan River Basin to increase the water supply, and the carrying capacity was further improved, with an increase of water resources by 25.9%. The model research has certain practical and strategic significance for maintaining the sustainable development of the ecological environment in the Longchuan River Basin

Regional geomagnetic core field and secular variation model over the Iberian Peninsula from 2014 to 2020 based on the R-SCHA technique

The Earth’s magnetic field originated in the fluid core, the so-called core field, is the dominant contribution to the geomagnetic field. Since ancient times, the core geomagnetic field has been used primarily for geographical orientation and navigation by means of compasses. Nowadays, thanks to the large amount of geomagnetic data available, core field models can be developed on a global or regional scale. Global models resolve large-scale geomagnetic field features, while regional models can resolve greater detail over a particular region. The spherical harmonic cap analysis is a widely used technique for regional-scale modelling of the geomagnetic field. In this work we have developed a regional model of the core field and its secular variation between 2014.5 and 2020.5 over the Iberian Peninsula, based on data from Swarm satellites, geomagnetic observatories and repeat stations. Its performance has been validated by comparing the fit to the available geomagnetic data using the regional model and the global models IGRF and CHAOS over the whole spatio-temporal range studied. In order to optimise the model, a detailed study of its input parameters has been carried out, showing that not all parameters have an equal influence on the modelling. This new model reproduces the input data with a root mean square error of 2.9 nT, improving the outcome of global models on this region. The results of this work will allow the Spanish Instituto Geográfico Nacional to produce the magnetic cartography of Iberia and the Balearic Islands in 2020.0, which for the first time will be based on a regional core field model, replacing the polynomial variation method used in the past.Graphical abstract

An experimental study to investigate the effect of pitch angle and wave characteristics on the performance of a horizontal axis tidal turbine

This study investigates the performance characteristics of a scaled-down three-bladed tidal turbine in a towing tank. The effects of pitch angle, electrical load, wave characteristics, and current velocity on turbine performance are examined. The experimental model, a 1:6 scale horizontal-axis tidal turbine with a rotor diameter of 1.15 m, is tested in the Strait of Khoran in Iran, selected based on tidal potential evaluation. After calibration and uncertainty analysis, experimental tests are conducted using the Taguchi method to optimize parameters and explore turbine performance in different conditions: current-only and wave-current interactions. Results demonstrate the direct relationship between power, momentum, thrust, rotor speed, and current velocity. Notably, power and momentum coefficients exhibit nonlinear (polynomial) variations at different tip speed ratios (TSRs). The presence of waves consistently enhances turbine performance compared to the no-wave mode, albeit with a lesser impact than other variable parameters. A pitch angle of 18.77° in the presence of waves yields a significant increase of up to 31.1% in power coefficient and up to 23.6% in momentum coefficient, compared to the no-wave mode. The optimal blade angle generally falls within the range of 30–35 mm, with 35 mm (18.77°) and 30 mm (22.72°) as the respective optimum pitch angles for wave-current interaction and the no-wave state. Optimal electrical load is achieved by setting Rheostat equal to 20 Ω, resulting in a remarkable 11% to 40% increase in system efficiency. In the upstream flow, the maximum power and momentum coefficients are observed as 0.219 and 0.033, respectively, at a TSR of 6.62 in the presence of waves with a dimensionless wave number of 3.43. Similarly, in the no-wave mode, the optimal power and momentum coefficients are obtained as 0.205 and 0.0286, respectively, at a TSR of 7.18. These findings highlight the significance of waves with a dimensionless wave number of 3.43.

Fast calculation of gravitational effects using tesseroids with a polynomial density of arbitrary degree in depth

Fast and accurate calculation of gravitational effects on a regional or global scale with complex density environment is a critical issue in gravitational forward modelling. Most existing significant developments with tessroid-based modelling are limited to homogeneous density models or polynomial ones of a limited order. Moreover, the total gravitational effects of tesseroids are often calculated by pure summation in these methods, which makes the calculation extremely time-consuming. A new efficient and accurate method based on tesseroids with a polynomial density up to an arbitrary order in depth is developed for 3D large-scale gravitational forward modelling. The method divides the source region into a number of tesseroids, and the density in each tesseroid is assumed to be a polynomial function of arbitrary degree. To guarantee the computational accuracy and efficiency, two key points are involved: (1) the volume Newton’s integral is decomposed into a one-dimensional integral with a polynomial density in the radial direction, for which a simple analytical recursive formula is derived for efficient calculation, and a surface integral over the horizontal directions evaluated by the Gauss–Legendre quadrature (GLQ) combined with a 2D adaptive discretization strategy; (2) a fast and flexible discrete convolution algorithm based on 1D fast Fourier transform (FFT) and a general Toepritz form of weight coefficient matrices is adopted in the longitudinal dimension to speed up the computation of the cumulative contributions from all tesseroids. Numerical examples show that the gravitational fields predicted by the new method have a good agreement with the corresponding analytical solutions for spherical shell models with both polynomial and non-polynomial density variations in depth. Compared with the 3D GLQ methods, the new algorithm is computationally more accurate and efficient. The calculation time is significantly reduced by 3 orders of magnitude as compared with the traditional 3D GLQ methods. Application of the new algorithm in the global crustal CRUST1.0 model further verifies its reliability and practicability in real cases. The proposed method will provide a powerful numerical tool for large-scale gravity modelling and also an efficient forward engine for inversion and continuation problems.

Dynamic Economic Environmental Dispatch of Power System Using An Improved Whale Algorithm

With the global energy crisis and environmental problems becoming increasingly serious, the construction of a new power system containing multiple renewable energy sources for power generation has become a current research direction. In this study, a dynamic economic environmental dispatch model for power systems containing wind and thermal power generation is proposed to promote renewable energy consumption while ensuring system power balance. Firstly, a comprehensive objective function that integrates system operating costs and pollutant emission targets is established, and system active balance constraints and unit output constraints are considered. Secondly, an improved whale optimization algorithm is proposed to deal with the solution problem of the dynamic economic environmental dispatch model by introducing circle chaos mapping, dynamic sine cosine factor, and polynomial variation strategy. Finally, the optimization capability of the proposed model is verified by running a unit test system. The proposed model promotes the consumption and utilization of wind and photovoltaic resources and positively impacts achieving carbon peaking and carbon neutrality.

Exact general solutions for the mode shapes of longitudinally vibrating non-uniform rods via Lie symmetries

A Lie symmetry method-based approach is proposed for systematically computing general solutions in closed-form for the mode shape equation of non-uniform and unconventional vibrating rods. The mode shape equation is modeled by the elementary rod theory, addressing polynomial, exponential, trigonometric, and hyperbolic cross-section variations. The method provides algorithmic order-reduction steps for solving the investigated mode shape equation, producing a first-order Riccati equation whose integration reveals the aimed solutions for the problem. Illustrative examples are presented, including original solutions in closed-form as well as solutions previously obtained in the literature by other approaches. Mode shapes from general solutions with appropriate rod boundary conditions are also considered for different examples. • Original general solutions establish new exact mode shapes of non-uniform rods. • The Lie symmetry method allows addressing new cross-section variations systematically. • Mode shapes derive from the general solutions with rod boundary conditions. • Lie symmetry applications have been highlighted for further problems of interest.

Greenhouses Detection in Guanzhong Plain, Shaanxi, China: Evaluation of Four Classification Methods in Google Earth Engine

Greenhouses used for agricultural production have been expanding around the world because it significantly increases crop yield. Meanwhile, it brings a series of environmental problems that should be considered in agricultural planning and management. The advent of the Google Earth Engine (GEE) cloud platform makes remote sensing image processing more convenient and efficient. It has been widely applied in multiple disciplines, but few studies have investigated the detection of greenhouses. In this research, four different classification methods were applied for comparing their performance in monitoring greenhouses in the Guanzhong Plain, Shaanxi, China using GEE: the Minimum Distance Classifier (MDC), the Support Vector Machine with three kernel functions (linear, SVM-L, polynomial, SVM-P, and radial basis function variations, SVM-R), the Classification and Regression Trees (CART), and the Random Forest (RF). Our results illustrate that these classification techniques’ overall accuracy is >84%. The most accurate classification results were obtained by the SVM-R classifier, with an overall accuracy of 94%, followed by the RF and CART classifier, while the MDC performed worst among these four classifiers. These results would be useful for greenhouse extraction in long time series and large-scale areas, which provides solid information for decision-makers and practitioners for agriculture planning and management.

Some Generalized Properties of Poly-Daehee Numbers and Polynomials Based on Apostol–Genocchi Polynomials

Numerous polynomial variations and their extensions have been explored extensively and found applications in a variety of research fields. The purpose of this research is to establish a unified class of Apostol–Genocchi polynomials based on poly-Daehee polynomials and to explore some of their features and identities. We investigate these polynomials via generating functions and deduce various identities, summation formulae, differential and integral formulas, implicit summation formulae, and several characterized generating functions for new numbers and polynomials. Finally, by using an operational version of Apostol–Genocchi polynomials, we derive some results in terms of new special polynomials. Due to the generic nature of the findings described here, they are used to reduce and generate certain known or novel formulae and identities for relatively simple polynomials and numbers.

Lightweight design of variable-angle filament-wound cylinders combining Kriging-based metamodels with particle swarm optimization

Variable-angle filament-wound (VAFW) cylinders are herein optimized for minimum mass under manufacturing constraints, and for various design loads. A design parameterization based on a second-order polynomial variation of the tow winding angle along the axial direction of the cylinders is utilized to explore the nonlinear steering-thickness dependency in VAFW structures, whereby the thickness becomes a function of the filament steering angle. Particle swarm optimization coupled with three Kriging-based metamodels is used to find the optimum designs. A single-curvature Bogner–Fox–Schmit–Castro finite element is formulated to accurately and efficiently represent the variable stiffness properties of the shells, and verifications are performed using a general purpose plate element. Alongside the main optimization studies, a vast analysis of the design space is performed using the metamodels, showing a gap in the design space for the buckling strength that is confirmed by genetic algorithm optimizations. Extreme lightweight while buckling-resistant designs are reached, along with non-conventional optimum layouts thanks to the high degree of thickness build-up tailoring.

A Guided Mutation Grey Wolf Optimizer Algorithm Integrating Flower Pollination Mechanism

<p>The basic Grey Wolf Optimizer (GWO) has some shortcomings, for example, the convergence speed is slow, it is easy to fall into local extremum, and high-dimensional optimization ability is poor and so on. In response to these shortcomings, an improved grey wolf algorithm which combines flower pollination mechanism, teaching mechanism and polynomial variation is proposed in this study. The flower pollination mechanism is integrated with GWO algorithm, Levy distribution is introduced into the global search of grey wolf population. And the double random mechanism is added in the local search, for these improvements, this algorithm’s overall optimization performance is improved. The teaching mechanism is added to wolf to improve the algorithm’s convergence speed. Polynomial mutation is applied to the individuals with poor optimization effect to improve the algorithm’s accuracy and its ability to jump out of local extremum. Theoretical analysis shows that the time complexity of the improved algorithm is the same as that of the basic algorithm. The test results of five representative comparison algorithms on multiple different characteristics and different dimensions of CEC2017 benchmark functions and two classical engineering problems show that FMGWO algorithm has high optimization accuracy, convergence speed and solution stability. Therefore, it has obvious advantages in global optimization.</p> <p> </p>

Static analysis of wind turbine blade using 1D FE model

The study presents a unique technique to determine the static response of wind turbine (WT) blade. A 1D Finite element (FE) beam model of WT blade is developed using thin-walled beam theory coupled with PreComp tool used to compute crosssectional stiffness with elastic coupling effects. A realistic 9.2 m long, WT blade is developed using different aerofoils with fourth order polynomial variation for twisting angle of blade span. Three different aerfoil sections NREL S818, NREL S825, and NACA 2412 are employed in the current study. For validation, the results of 1D model developed using MATLAB are compared with that of 3D WT blade model which is analyzed in ANSYS using NuMAD..

Analytical Solutions for Side-branch Impedance Producing Spatial Localization of Acoustic Waves in Ducts with Varying Cross Sections

Analytical mathematical models and solutions for spatial localization of acoustic waves through an impedance discontinuity produced by an intermediate damped side branch are studied in stationary media in ducts with varying cross sections. Three specific geometries, namely, with polynomial, sinusoidal, and exponential longitudinal variations, are investigated. The sound fields inside the ducts are modeled by Webster's horn equation. Traveling-wave solutions are obtained by appropriate transformations. The side-branch impedances required for spatial localization (confinement) of traveling and standing waves are found analytically and verified numerically using three-dimensional finite element analysis. The impact of the longitudinal variation of the duct's cross-sectional area (CSA) on the side-branch impedance is examined. It was found that the required side-branch resistance changes more than the reactance with the variation of the duct CSA. A measure of a traveling wave is defined to quantitatively examine the spatial localization of acoustic waves. It was found that the CSA corrections on the side-branch impedances are important. The results of this study reveal the quantitative relationships between the side-branch impedance and the CSA variations for zero reflection from the impedance discontinuity. The mathematical approach presented is potentially helpful for a design of a full anechoic termination and energy localization in duct systems.

A generalization of Edelman–Greene insertion for Schubert polynomials

Edelman and Greene generalized the Robinson--Schensted--Knuth correspondence to reduced words in order to give a bijective proof of the Schur positivity of Stanley symmetric functions. Stanley symmetric functions may be regarded as the stable limits of Schubert polynomials, and similarly Schur functions may be regarded as the stable limits of Demazure characters for the general linear group. We modify the Edelman--Greene correspondence to give an analogous, explicit formula for the Demazure character expansion of Schubert polynomials. Our techniques utilize dual equivalence and its polynomial variation, but here we demonstrate how to extract explicit formulas from that machinery which may be applied to other positivity problems as well.

The lengthening pendulum: Adiabatic invariance and bursting solutions

The adiabatic invariance of the action variable of a length varying pendulum is investigated in terms of the two different time scales that are associated with the problem. A length having a general polynomial variation in time is studied; an analytical solution for a pendulum with length which varies quadratically in time is obtained in the small angle approximation. We find that for length with quadratic time variation, the action neither converges (as it does for linear time variation), nor diverges (as it does for exponential time variation), but rather shows oscillatory behaviour with constant amplitude. It is then shown that for a pendulum length which has a combination of periodic and linear time variations, the action is no longer an adiabatic invariant and shows jumps with time. In the case in which the length varies sinusoidally in time, we demonstrate that the full nonlinear system exhibits bursting oscillations.

Atomistic approach for the evaluation of direct flexoelectric coefficients in gradient theory

A simplified two length-scale model is applied in direct flexoelectricity for two-dimensional problems. Numerical experiments are performed to obtain the two unknown flexoelectric parameters that appear in gradient theory in elasticity. They are determined by fitting the results from an atomistic model with an analytical solution for a simple problem employing gradient theory, namely, a square domain with a simple polynomial variation of displacements. This is demonstrated by a numerical example for alpha alumina α-Al2O3.

Coupled state-space and Gauss integration procedures for static analysis of general functionally graded magnetoelectroelastic laminated plates

In this article, a 3D semi-analytical procedure is elaborated for static behavior of simply supported functionally graded magnetoelectroelastic laminated plates subjected to various types of loads. An arbitrary graded formulation is considered in the thickness direction. The laminated plates have been divided into some layers in which the materials properties vary through the thickness. In each layer, the solution has been established using the state-space approach coupled with the Gauss-Tchybechev integration. The predicted solution has been propagated from the bottom to the top layers of the plate using the propagator matrix method. The presented methodological procedure is general and allow considering various types of graded functionalities in the thickness direction. For the computation process, the piezoelectric material and the piezomagnetic material are used. Two kinds of functionally graded materials have been hindered. First, the polynomial variation of the materials properties through the thickness direction of the plate is considered. The effective properties of functionally graded materials are assumed to follow the law of mixture. Second, the exponential variation of the materials properties through the thickness of the plate was used. The obtained numerical results have been well compared with the available ones obtained by the 3D asymptotic approach, the modified Pagano method, the pseudo-Stroh formalism, and the finite elements method, respectively. The convergence study showed the accuracy, the efficiency, and the reliability of the presented methodological procedure.

Effect of rigid boundary on Rayleigh wave in an incompressible heterogeneous medium over an incompressible half-space

In the present problem, an attempt has been made to study the propagation of Rayleigh waves in an incompressible medium with polynomial variation (m) of rigidity over an incompressible half-space under rigid layer. Instead of using the Whittaker function, the expansion formula proposed by Newlands has been used for a better result in shallow depth. The velocity equation has been calculated and the results are shown in figures. The study in the assumed medium, the authors obtained that the phase velocity of Rayleigh waves increases except for the polynomial variation of rigidity m = 1, 2 and 3.

Effect of solvent polarity on extraction yield and antioxidant properties of phytochemicals from bean (Phaseolus vulgaris) seeds

The effect of solvent polarity on extraction yield and antioxidant properties of phytochemical compounds in bean seeds was studied. Seed flour of three varieties of bean was extracted in a series of organic solvents with increasing polarity (n-hexane, petroleum ether, chloroform, ethyl acetate, ethanol, acetone and water). Preliminary screening of phytochemicals showed the presence of tannins, flavonoids, cardiac glycosides, anthocyanins, terpenoids, carotenoids, ascorbic acid and reducing compounds in all extracts. One way analysis of variance (ANOVA) of results showed that extraction yield, phytochemical content and antioxidant properties were significantly influenced (p<0.05) by the polarity of extracting solvents. The regression analysis of data showed polarity-dependent second order polynomial variations in the extraction yield, phytochemical contents, antioxidant activity, reducing properties and free radical scavenging activity of each variety. Extraction in highly polar solvents resulted in high extract yield but low phenolic and flavonoid content as compared to non-polar ones. The polarity-dependent increase in total antioxidant activity and reducing properties indicates the extraction of strong antioxidant compounds in polar solvents. The study suggests the use of a combination of polar and nonpolar solvents to increase the extraction efficiency of phytochemicals with good antioxidant quality from the bean and other legume seeds.

A feature selection and multi-model fusion-based approach of predicting air quality

With the rapid development of China’s industrialization, the air pollution is becoming more and more serious. It is vital for us to predict the air quality for determining the further prevention measures of avoiding the brought disasters. In this paper, we are going to propose an approach of predicting the air quality based on the multiple data features through fusing the multiple machine learning models. The approach takes the meteorological data and air quality data for the past six days as one batch of input (the whole data set is for 46 days) and employs a multi-model fusion to provide an improved 24-hour prediction of PM2.5 pollutant concentration all over Beijing. During the above process, two focal feature groups are composed. The first focal feature group contains the historical meteorological data, while the second group includes the statistical information, the date information and the polynomial variations. Besides the two groups, we complement one million more data items by employing the time sliding means. Among the supplementary data, we select the most critical 500 features with Light Gradient Boosting Machine (LightGBM) model and send the features as the input to Gradient Boosting Decision Tree (GBDT) and LightGBM models. Meanwhile, we screen the most critical 300 features with eXtreme Gradient Boosting (XGBoost) model and send them as the input to the three prediction models. Referring to each of the models, we respectively gain the optimal parameters through grid search methods and then fuse the models’ contribution with the linear weighting. The experiments indicate that the proposed approach based on the weighting fusion is better than that provided by a single modeling scheme, and the loss value is 0.4158 under the SMAPE index.