Lagrange Interpolation Method Research Articles

A new algorithm of dangerous points accurately identification on mechanical structure under dynamic loads

Identification of dangerous moment of mechanical structure is a key process for dynamic response optimization. The dangerous moment of identification method for solution space of dynamic response based spectral element was proposed according to the insufficient of traditional method, which was difficult to accurately determine structure dangerous moment. The solution space of structure dynamic response was obtained by using the modal superposition method. The matrices of time points and their corresponding solution space values of dynamic stress were constructed. The Lagrange interpolation method was applied in the solution space domain to getting dynamic response function of high precision. Finally, the most dangerous moment of structure under dynamic loads was found by executing the global optimizer, though constructing a mathematical model to identify dangerous moment. The theoretical research results were applied to engine piston with dynamic loads to verify, and results showed that dynamic stress and dynamic displacement was not simultaneously reach extreme moment of the dynamic load, due to the nonlinear relationship between stress-strain and strain-displacement of structural. The existence of a time gap between them leads to dynamic optimization design is inaccurate. Dynamic optimization design should also consider two dangerous moments in order to avoid structure inaccurate design. This paper is the foundation of dynamic response optimization of piston.

Design of Variable Fractional Delay Filter using FIR Filter Approximation

Beamforming plays an important role in the field of wireless communication. Beamforming means combination of a radio frequency (RF) signals from multiple antennas to form a single direction beam. This technique improves the quality of communication and reduces the interference of signal. In beam forming technique, different phases signals can be achieved with different signals and the received phase delay signals are converted into same phase, multiply with weight factor and combined this signals to form a beam in desired direction. The required phase delays are generated by using a Variable fractional delay filter. Variable fractional delay filter is design by using a direct form of a FIR filter structure. Variable fractional delay filter is calculated by two different phase signals from digital antennas and those two different phase signals are converted to in- phase and added together to form a beam forming. As the order of the filter increases, the delay also increases. The filter coefficients of the variable fractional delay filter are calculated my using a Lagrange interpolation method. The variable fractional delay filter is designed by using software Xilinx version 14.3.

Design of Variable Fractional Delay Filter using FIR Filter Approximation

Beamforming plays an important role in the field of wireless communication. Beamforming means combination of a radio frequency (RF) signals from multiple antennas to form a single direction beam. This technique improves the quality of communication and reduces the interference of signal. In beam forming technique, different phases signals can be achieved with different signals and the received phase delay signals are converted into same phase, multiply with weight factor and combined this signals to form a beam in desired direction. The required phase delays are generated by using a Variable fractional delay filter. Variable fractional delay filter is design by using a direct form of a FIR filter structure. Variable fractional delay filter is calculated by two different phase signals from digital antennas and those two different phase signals are converted to in- phase and added together to form a beam forming. As the order of the filter increases, the delay also increases. The filter coefficients of the variable fractional delay filter are calculated my using a Lagrange interpolation method. The variable fractional delay filter is designed by using software Xilinx version 14.3

Maximum power point tracking using lagrange interpolation method to optimized photovoltaic on dc microgrid system

Electrical energy is very important for everyday life. The need of electrical energy increasing every year. However, the solar panel source has a weakness. The output power of solar panel is still fluctuating depending on ambient temperature and solar irradiantions on the solar panel. Irradiation is the radiant of energy that comes from the sun. To overcome the problem the Maximum Power Point Tracking is needed so it can make solar panel work at its optimum point. This Maximum Power Point Method is looking for the largest current and voltage read by solar panel using the Lagrange Interpolation method. By using Maximum Power Point Tracking, the output power is expected to be optimal with constant voltage and current so that the battery’s lifetime is not damaged quickly. This method can track Maximum Power Point whenever irradiation changes occur. The converter used in this Maximum Power Point Tracking is the SEPIC Converter. This converter can convert DC to DC voltage in higher or lower voltage without changing its polarity. In this paper, Lagrange Interpolation method has average error in output power of 1,96%.

Numerical solution to Volterra Integro-Differential Equations of the second kind by hybrid one-step block method

The hybrid block one-step method of order four is presented and implemented to solve first order Volterra Integro-Differential Equations (VIDEs). The technique is developed using Lagrange interpolation method. The numerical solutions of VIDEs will be solved at two-point concurrently using the proposed numerical method. Properties of the method such as consistency, order and region of absolute stability are explored. After that, the linear and nonlinear of VIDEs has been solved and the numerical results show that the accuracy and efficiency of the hybrid block method is better than the other methods which is RK4 and VIM.

Research on spur face gear by electrochemical machining based on the complex surface mesh

The tooth surface of the spur face gear is a kind of complex space curved surface, in order to solve the technical defects in traditional NC milling manufacturing. This paper utilizes the complex tooth surface grid planning principle to establish the cathode surface firstly; Secondly, the orthogonal test, linear regression analysis and weighted comprehensive scoring system is adopted to obtain the optimal processing parameters; And finally, the Lagrange interpolation method is used to realize the modification of the cathode surface. The verification test shows that this method compared with the traditional NC milling, reduces the cost by 70% and the cycle by 75%, can realizes the rapid high-precision batch manufacturing of complex structure products.

A Compound Wind Power Forecasting Strategy Based on Clustering, Two-Stage Decomposition, Parameter Optimization, and Optimal Combination of Multiple Machine Learning Approaches

Given the large-scale exploitation and utilization of wind power, the problems caused by the high stochastic and random characteristics of wind speed make researchers develop more reliable and precise wind power forecasting (WPF) models. To obtain better predicting accuracy, this study proposes a novel compound WPF strategy by optimal integration of four base forecasting engines. In the forecasting process, density-based spatial clustering of applications with noise (DBSCAN) is firstly employed to identify meaningful information and discard the abnormal wind power data. To eliminate the adverse influence of the missing data on the forecasting accuracy, Lagrange interpolation method is developed to get the corrected values of the missing points. Then, the two-stage decomposition (TSD) method including ensemble empirical mode decomposition (EEMD) and wavelet transform (WT) is utilized to preprocess the wind power data. In the decomposition process, the empirical wind power data are disassembled into different intrinsic mode functions (IMFs) and one residual (Res) by EEMD, and the highest frequent time series IMF1 is further broken into different components by WT. After determination of the input matrix by a partial autocorrelation function (PACF) and normalization into [0, 1], these decomposed components are used as the input variables of all the base forecasting engines, including least square support vector machine (LSSVM), wavelet neural networks (WNN), extreme learning machine (ELM) and autoregressive integrated moving average (ARIMA), to make the multistep WPF. To avoid local optima and improve the forecasting performance, the parameters in LSSVM, ELM, and WNN are tuned by backtracking search algorithm (BSA). On this basis, BSA algorithm is also employed to optimize the weighted coefficients of the individual forecasting results that produced by the four base forecasting engines to generate an ensemble of the forecasts. In the end, case studies for a certain wind farm in China are carried out to assess the proposed forecasting strategy.

Designing on scaled absorbing materials at THz band using Ni@mica composite

The scaled measurement on the target model coated with the frequency dispersion absorber with a large-scaled factor was still an unsolved problem. In this work, the scaled absorbing material with the scale factor 100 was designed at Terahertz (THz) band using the hybrid Ni@mica particle. The designing method on the scaled absorber was composed with the particle distribution designing, permittivity interpolation method, and multilayer structure optimization. The complex permittivity and permeability were measured first on the THz time-domain spectroscopy (TDS) measurement system in frequency 0.8–1.8 THz as the Ni@mica particle-filling ratio of the absorber was 30%, and then, the reflection loss could be calculated. As the different holes were set in the absorbing squared body element, the transmission ratio and the reflection ratio of the absorbing composite were simulated in the computer simulation technology (CST) software, and the effective permittivity and permeability could be calculated. Then, the permittivity data base with different holes was built by the Lagrange interpolation method. Finally, the scaled absorber materials could be optimized for the magnetic coating plate at 1 THz and 0.8 THz, and the designed scaled absorber included two layers: the bottom layer thickness was 0.12 mm with volume content 30% and the upper layer thickness was 0.17 mm with volume content 27%. The calculation result showed that the radar cross section (RCS) deviation between the two similar models was − 40.02dB@8GHz and − 40.21dB@10GHz, which were very close to the theoretical value − 40 dB. It indicated that the Ni@mica was one of the effective candidates to construct the scaled absorbing material.

Enhanced DFT-Based Controller for Selective Harmonic Compensation in Active Power Filters

Among the various selective harmonic current controllers, the traditional repetitive-based discrete Fourier transform controller (DFTRC) is increasingly favored thanks to its unique merits of excellent selectivity and simplicity. However, the traditional approach suffers from disadvantages like slow dynamic responses, defective control structure, and sensitivity to frequency fluctuation. In order to alleviate aforementioned shortcomings, an enhanced discrete Fourier transform-based controller is proposed. The proposed controller is based on the generalized discrete Fourier transform to improve the overall control performances of dynamic responses, structure flexibility, and control robustness. The proposed controller provides individual positive-feedback path and gain coefficient for each harmonic component. Therefore, it not only maintains the advantage of excellent selectivity but also realizes the individual control parameters tuning for each harmonic frequency, which is impossible in traditional DFTRC owing to the congenital defects of control structure. In addition, a correction function is embedded in the forward path to provide the overall phase compensation and correct the plant for better characteristic. The Lagrange interpolation method is adopted to realize the fractional-order delay and its adaption to frequency variations with fixed sampling frequency. Detailed mathematical model along with optimized design principle is given to make full use of the advantages of the enhanced control structure. Extensive experimental results validate the effectiveness of the theoretical analysis and the improvement achieved by the proposed controller.

Paczynski yıldız modellemesinde kullanılmak için opaklık tabloları ve onların yıldız evrimlerine etkileri

We have constructed opacity tables for using in stellar modelling in a format particularly applicable to the Paczynski freeware program GOB. Our tables are derived from those of Allard and Weiss for lower metal content envelopes, using the Lagrange interpolation method. This allows accurate opacities to be given as numerical functions of density and temperature with given proportions of hydrogen (X), helium (Y), and ‘metals’ (Z). Inlek et al. showed that it is possible independently to check opacity effects from modern high-quality observational data. These tables will be useful for such purposes.

A piecewise spectral-collocation method for solving fractional Riccati differential equation in large domains

This paper addresses the approximate solution of the fractional Riccati differential equation (FRDE) in large domains. First, the solution interval is divided into a finite number of subintervals. Then, the Legendre–Gauss–Radau points along with the Lagrange interpolation method are employed to approximate the FRDE solution in each subinterval. The method has the advantage of providing the approximate solutions in large intervals. Additionally, the convergence analysis of the numerical algorithm is also provided. Three illustrative examples are given to illustrate the efficiency and applicability of the proposed method.

Application of Lagrange Polynomial for Interpolating Income Generation from Certain Students’ Fee Structure

This paper presents the findings of the study that aimed to examine the changing aspects of income generation from certain students’ fee structure. It used an empirical example of Higher learning institution from Tanzania to determine whether Lagrange interpolation method would emerge a useful tool for interpolating and extrapolating enrolments of undergraduate students. The functional values evaluated at all nodes on the given data set accurately match the enrolments records of each respective academic year, though with nonconformities at the additional interior nodes. The function obtained from polynomial curve fitting using Excel spreadsheet at replaced the Lagrange polynomial to compute the rate of income generated from undergraduate students’ fees for academic years 2012/2013 to 2018/2019. The relative errors committed by range from the number at the node to the number at the node . Enrolments trend however is not stable, as it fluctuates basing on the particular academic year that causes the rate of income generated from undergraduate students’ fees to act wildly. Being the cause therefore oscillatory behaviour perhaps lax if enrolments grew continuously in the successive academic years, however, this would mark progressive manner if there will be reciprocal efforts to improve physical and electronic resources imbued by human resources capacity. Higher learning institutions could diversify their income generation strategies from multiplicity sources of incomes by using various techniques, including rising students’ fees and downhill expenses without compromising quality of services and products, reducing dependency on Government subsidies, and strengthening the rate of students’ retention in higher learning. They may also utilize to the maximum the unobserved multiplicity sources of incomes for instance; consultation fees, project funds, and new competitive learning programs to support realization of their strategic objectives in coherence with their missions and visions. Keywords: Enrolment, Generation, Income, Interpolation, Lagrange, Polynomial DOI : 10.7176/MTM/9-5-05 Publication date :May 31 st 2019

Polynomial-based modifiable blockchain structure for removing fraud transactions

Abstract Blockchain has become one of the most significant technology in financial fields. A hash-based blockchain holds the feature of strong tamper resistance. However, it is almost impossible to change fraud transactions in time, since starting a fork requires a lot of time and resources. Failing to remove fraudulent transactions in time is harmful to the entire economic environment. In order to deal with the modification problem, in this work, we propose a novel polynomial-based blockchain structure. Data segments are organized by a Lagrange interpolation method in each block. Polynomial functions are used to keep the order of blocks. The polynomial-based blockchain structure not only achieves the aim of modification but also supports differential control strategy on modification difficulty. Experimental results demonstrate that the polynomial-based blockchain structure is efficient and practical. Detailed theoretical and practical analysis showed that the polynomial-based modifiable blockchain structure has a wide range of application scenarios with the help of other techniques on cryptography and privacy preservation.

Necessary conditions of exponential stability for a class of linear uncertain systems with a single constant delay

In this paper, we will investigate the necessary conditions, described by the Lyapunov matrix, for the robust exponential stability for a class of linear uncertain systems with a single constant delay and time-invariant parametric uncertainties, which are some generalizations of the existing results on uncertain linear time-delay systems. As a medium step, several pivotal properties of parameter-dependent Lyapunov matrix are proposed, which set up the relationships between fundamental matrix and Lyapunov matrix for the considered system. In addition, to calculate the parameter-dependent Lyapunov matrix, we introduce the differential equation method and the Lagrange interpolation method, respectively. Furthermore, it is noted that the proposed necessary conditions can be used to estimate the range of time delay, when the linear uncertain time-delay system is robust exponential stability. Finally, the validity of the obtained theoretical results is illustrated via numerical examples.

A steady barycentric Lagrange interpolation method for the 2D higher‐order time‐fractional telegraph equation with nonlocal boundary condition with error analysis

In this paper, we consider the numerical solution of the time‐fractional telegraph equation with a nonlocal boundary condition. A novel barycentric Lagrange interpolation collocation method is developed to solve this equation. Two difficulties have been sorted: the singularity of the integration and the higher accuracy. At the same, we put forward a steady barycentric Lagrange interpolation technique to overcome the new “Runge” phenomenon in computation. Error estimates of the barycentric Lagrange interpolation and the time‐fractional telegraph system for the present method are presented in Sobolev spaces. High convergence rates of the proposed method are obtained and are consisted with the numerical values. Especially in the time dimension, we get the error bound, for h‐refinement and for nt‐density in the L2 norms. The numerical results obtained show that the proposed numerical algorithm is accurate and computationally efficient for solving time‐fractional telegraph equation. Experiments demonstrate the high convergence rates of the proposed method are consisted with the theoretical values.

Fully nonlinear simulation of wave–current interaction with an oscillating wave surge converter

On the basis of a time-domain HOBEM (higher-order boundary element method), a two-dimensional (2D) fully nonlinear numerical model is developed to investigate the wave energy capturing efficiency of an OWSC (oscillating wave surge converter) in the presence of uniform currents. In the model, the wave energy is extracted through a linear external damping coefficient, and the natural frequency of the OWSC is tuned to the wave characteristics by introducing an elastic and inertia coefficient. On the free surface, a five-point fourth order Lagrangian interpolation method is applied to determine the redistribution of meshes, and a rotation scheme of spatial coordinate system is employed to tackle the discontinuity of flux at the intersection points. The wave loads on the wetted body surface are then evaluated using two auxiliary functions through which the mutual dependence of body motion and fluid flow is decoupled. The present model is validated against the published experimental and numerical results for an OWSC in large motion excited by nonlinear waves. Simulations are performed to capture the hydrodynamic behavior of the OWSC with different currents. Numerical results are also provided to show the behavior dependence on the design parameters, such as current velocities, wave conditions, geometric characteristics and mechanical properties.

A novel method for preform die design in forging process of an airfoil blade based on Lagrange interpolation and meta-heuristic algorithm

In this paper, a new method is introduced for optimization of a two-stage forging process of an airfoil blade. Geometrical dimensions of preform shape and cavity of the intermediate die were selected as input variables and uniformity of strain distribution, flash volume, and lateral forces were considered as objective functions. Design parameters of the intermediate die cavity were defined by Lagrange interpolation method. Then, using the design of experiments and implementation of proposed tests by the finite element method, the objective function values were obtained and afterwards artificial neural network modeling were applied for extracting the effectiveness of the input variables on the objectives. Finally, optimal amount of preform and intermediate die parameters was achieved using a multi-objective genetic algorithm based on a fuzzy method. Also, by using physical modeling of the optimal conditions, experimental tests of the forging process were executed. Results show that the trained artificial neural network model estimates the process with good accuracy and also, comparison of optimized and non-optimized two-stage forging reveals that in the former condition, flash volume and strain non-uniformity reduced more by 51% and 42%, respectively.

Numerical solution of potential problems using radial basis reproducing kernel particle method

Abstract The paper presents the radial basis reproducing kernel particle method (RRKPM) for potential problems. The proposed RRKPM can eliminate the negative effect of different reproducing kernel functions (RKF) on computational stability and accuracy. The integral weak form is used to derive a discretized system equation for potential problem, and the essential boundary condition is imposed by the Lagrange interpolation method, the corresponding governing equations of RRKPM are investigated and obtained. Compared with the conventional RKPM, the RRKPM has higher computational stability and accuracy. Finally, the RRKPM is applied to numerical simulation of potential problems, and the accuracy and stability of the RRKPM can be illustrated by the two numerical results.

Improved distortion correction method and applications for large aperture infrared tracking cameras

Accurate distortion correction is especially crucial for applications involving quantitative measurements such as on-orbit geometric positioning of remoting sensing cameras, depth measurement from stereoscopy, and target tracking and recognition from images. However, the conventional complicated correction methods and the unsatisfied results greatly limit their application prospects, especially in the aerospace field. In this paper, we propose an improved distortion correction method (IDCM) for large aperture infrared tracking cameras, addressing the distortion model and the principle point and distance with high-precision measurement points in the focal plane. Experimental results showed that the distortion errors corrected by the proposed method were 0.26 pixels in a near-infrared image and 0.19 pixels in a short-wavelength infrared image. These corrected distortion errors are better than that corrected by the two-dimensional Lagrange interpolation method (TDLIM) of 0.36 pixels in a near-infrared image and 0.29 pixels in a short-wavelength infrared image, and the conventional distortion model (CDM) of 0.54 pixels in a near-infrared image and 0.64 pixels in a short-wavelength infrared image, respectively.

ECG Signal Characterization Using Lagrange-Chebyshev Polynomials

An ECG (electrocardiogram) is a signal representing a combination of potentials that reflect the electrical activity of the heart. These signals are often corrupted by artifacts of high magnitude and frequency during acquisition, storage and transmission. Addition of these artifacts may change the morphology of the ECG signals and can hinder accurate interpretations; hence proper characterization is required for better clinical evaluation. In this paper ECG signals from MIT-BIH database are characterized using polynomial interpolation method. In this method, initially Chebyshev nodes are suitably chosen in the desired ECG interval and then these nodes are utilized by Lagrange interpolation method for restoration of ECG signals. The results of the interpolation are further improved by segmenting the ECG signals into suitable number of segments using Bottom-Up time series segmentation approach. The performance of the proposed method is analyzed in terms of standard ECG performance parameters: mean absolute deviation, root mean square deviation, percentage root mean square difference error, signal to noise ratio and cross correlation. Results obtained are compared with existing methods and are found to be superior and diagnostically acceptable. Practical implementation areas of the proposed method are also explored.