Lagrange Interpolation Algorithm Research Articles

An improved displacement adjustment method for overlarge deformation of valve at the cryogenic temperatures

Cryogenic valves are mostly designed at the room temperatures and served at the cryogenic temperatures. The wide temperature difference between design and service conditions causes the overlarge deformation of the valve stem structure, resulting in the cryogenic valve prone to leakage, stuck and other unsafety issues. In this study, an improved displacement adjustment (IDA) compensation method is proposed to solve the problem of size over-tolerance and shape deformation of the valve stem at the cryogenic temperatures. The shape of the valve stem is redesigned by combining IDA and the finite element method to meet its ideal service requirements under the cryogenic conditions. One of the features of the IDA method is to optimize the compensation direction by considering the angle change of normal vector on valve stem surface. Another is to improve the iteration efficiency by recognizing the compensation factor through the Lagrange interpolation algorithm. The simulation and experimental results show that the proposed compensation method can effectively reduce the size and shape deviation of the valve stem. Therefore, this study provides a new method to reduce the stem deviation at the cryogenic temperatures by the shape compensation, which has reference significance for the design of other low temperature components.

Comparative study of the implementation of the Lagrange interpolation algorithm on GPU and CPU using CUDA to compute the density of a material at different temperatures

Graphics Processing Units (GPUs) are microprocessors attached to graphics cards, which are dedicated to the operation of displaying and manipulating graphics data. Currently, such graphics cards (GPUs) occupy all modern graphics cards. In a few years, these microprocessors have become potent tools for massively parallel computing. Such processors are practical instruments that serve in developing several fields like image processing, video and audio encoding and decoding, the resolution of a physical system with one or more unknowns. Their advantages: faster processing and consumption of less energy than the power of the central processing unit (CPU). In this paper, we will define and implement the Lagrange polynomial interpolation method on GPU and CPU to calculate the sodium density at different temperatures Ti using the NVIDIA CUDA C parallel programming model. It can increase computational performance by harnessing the power of the GPU. The objective of this study is to compare the performance of the implementation of the Lagrange interpolation method on CPU and GPU processors and to deduce the efficiency of the use of GPUs for parallel computing.

Mapped Chebyshev pseudospectral methods for optimal trajectory planning of differentially flat hypersonic vehicle systems

This paper presents a novel trajectory planning algorithm for hypersonic glide vehicles based on differentially flatness theory and mapped Chebyshev pseudospectral method (MCPM). Firstly, the flatness property of a hypersonic glide vehicle dynamics system is extended explicitly by introducing a pseudo control input along the opposite direction of aerodynamics drag. In this way, all the states and controls of the system can be represented by flat outputs and their derivatives, and the trajectory planning problem is formulated into the flat output space with differential constraints entirely eliminated. Secondly, the optimization problem in the flat output space is converted into a nonlinear programming one via the mapped Chebyshev pseudospectral method. The MCPM employs the conformal mapping principle to discretize the flat outputs at the time interval, in terms of which, the closely clustered Chebyshev points near the boundary are stretched and the ill-conditions of the differential matrix is improved. The Barycentric Lagrange interpolation algorithm is then utilized to interpolate the flat output functions, which is seized of better approximation properties than polynomial interpolation. Finally, the transformed nonlinear programming problem are solved by sequence quadratic program technique. The states and controls are then generated by the expressions of the flatness outputs. At the end of this paper, numerical simulations are performed to evaluate the proposed approaches, and comparisons are made with traditional methods. Simulation results demonstrate that the proposed differentially flatness based mapped Chebyshev pseudospectral method possessed better computing efficiency than traditional ones without losing the merits of high precision.

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.

Highly accurate simplified lattice Boltzmann method

In this paper, we present a highly accurate simplified lattice Boltzmann method (HSLBM) which can achieve the third-order of accuracy in space. By introducing virtual streaming nodes and decoupling the streaming distance from the mesh spacing, HSLBM effectively combines the local second-order simplified and highly stable lattice Boltzmann method (SHSLBM) and the overall high order scheme. The correlation between the streaming distance and the mesh spacing is accomplished by a high-order Lagrange interpolation algorithm. Through a series of tests, it is found that using 5 interpolation points and setting streaming distance as 1/5 of the mesh spacing can give optimal results. In general, HSLBM improves the accuracy of SHSLBM from the second order to the third order, while maintaining its merits like low memory cost, convenient implementation of physical boundary conditions, and good numerical stability. To match the accuracy of the flow solver, a linear extrapolation scheme with the third-order of accuracy is also proposed to determine the boundary values of the non-equilibrium distribution functions. Various benchmark tests are performed to demonstrate the robustness of HSLBM in simulating two- and three-dimensional incompressible viscous flows as well as its flexibility in problems with curved boundaries and on body-fitted meshes.

Three-dimensional simplified and unconditionally stable lattice Boltzmann method for incompressible isothermal and thermal flows

In this paper, a three-dimensional simplified and unconditionally stable lattice Boltzmann method (3D-USLBM) is proposed for simulating incompressible isothermal/thermal flows. This method is developed by reconstructing solutions to the macroscopic governing equations recovered from the lattice Boltzmann equation and resolved in a predictor-corrector scheme. The final formulations of 3D-USLBM only involve the equilibrium and the non-equilibrium distribution functions. Among them, the former is calculated from the macroscopic variables and the latter is evaluated from the difference between two equilibrium distribution functions at different locations and time levels. Thus, 3D-USLBM directly tracks the evolution of macroscopic variables, which yields lower cost in virtual memory and facilitates the implementation of physical boundary conditions. A von Neumann stability analysis was performed on the present method to theoretically prove its unconditional stability. By imposing a regular Lagrange interpolation algorithm, this method can be flexibly extended to a non-uniform Cartesian mesh or body-fitted mesh with curved boundaries. Four numerical tests, that is, plane Poiseuille flow, 3D lid-driven cavity flow and 3D natural convection in a cubic cavity, and concentric annulus, were conducted to verify the stability, accuracy, and flexibility of the presented method.

Research on high-precision hole measurement based on robot vision method

A high-precision vision detection and measurement system using mobile robot is established for the industry field detection of motorcycle frame hole and its diameter measurement. The robot path planning method is researched, and the non-contact measurement method with high precision based on visual digital image edge extraction and hole spatial circle fitting is presented. The Canny operator is used to extract the edge of captured image, the Lagrange interpolation algorithm is utilized to determine the missing image edge points and calculate the centroid, and the least squares fitting method is adopted to fit the image edge points. Experimental results show that the system can be used for the high-precision real-time measurement of hole on motorcycle frame. The absolute standard deviation of the proposed method is 0.026 7 mm. The proposed method can not only improve the measurement speed and precision, but also reduce the measurement error.

Application of Lagrange Interpolation Algorithm in Beijing Carbon Emissions Prediction

Based on the depth analysis of the related literature and the present situation of energy consumption of Beijing’s carbon emissions, through the MATLAB programming on the Lagrange interpolation algorithm, the paper predicts the carbon emissions from energy consumption in Beijing’s economic growth. According to the relevant historical and predicted data, the paper examines the process of Carbon Emission Trend of energy consumption in Beijing. The results show that: Beijing carbon emissions showed "Y=X3" type growth. Combining the current situation, the paper analyzes the results, and finally provides decision support to the government, to promote the relationship between economic growth and energy consumption of carbon emissions in Beijing city into the inverted "U" trend.

A Hierarchical Access Control Scheme Based on Lagrange Interpolation and ELGamal Algorithm with Numerical Experiments

It is important to notice that the access control mechanism has been widely applied in various areas, such as on-line video systems, wireless network, and electronic documents. We propose an access control mechanism which is constructed based on two mathematical fundamentals: Lagrange interpolation and ElGamal algorithm. We conduct performance analysis to compare the efficiency of our proposed scheme with that of several related published schemes in both key generation phase and key derivation phase. Our new scheme is proven to be more efficient. It is shown, as expected, a more efficient scheme provides relatively less security and a more secure scheme is relatively less efficient for private keys of the same size.

Adaptive redundant lifting wavelet transform based on fitting for fault feature extraction of roller bearings.

A least square method based on data fitting is proposed to construct a new lifting wavelet, together with the nonlinear idea and redundant algorithm, the adaptive redundant lifting transform based on fitting is firstly stated in this paper. By variable combination selections of basis function, sample number and dimension of basis function, a total of nine wavelets with different characteristics are constructed, which are respectively adopted to perform redundant lifting wavelet transforms on low-frequency approximate signals at each layer. Then the normalized lP norms of the new node-signal obtained through decomposition are calculated to adaptively determine the optimal wavelet for the decomposed approximate signal. Next, the original signal is taken for subsection power spectrum analysis to choose the node-signal for single branch reconstruction and demodulation. Experiment signals and engineering signals are respectively used to verify the above method and the results show that bearing faults can be diagnosed more effectively by the method presented here than by both spectrum analysis and demodulation analysis. Meanwhile, compared with the symmetrical wavelets constructed with Lagrange interpolation algorithm, the asymmetrical wavelets constructed based on data fitting are more suitable in feature extraction of fault signal of roller bearings.

Map Matching Using De-Noise Interpolation Kohonen Self-Organizing Maps

In this work, we have proposed a de-noise interpolation Kohonen Self-Organizing Maps(DNIKSOM) -based method for the Map matching(MM). It has been seen that there are some problems in the MM, such as large error range of the original position information, low match accuracy and so on. Therefore, in MM problem to achieve high accuracy, it is necessary to consider the topography of roads and the requirement for match accuracy lying within the original position information in the matching process. In the present study, Kohonen Self-Organizing Maps(KSOM) in the field of pattern recognition possesses good performance. Now to get more valuable position information, A kind of de-noise algorithm for Kohonen neural network is proposed to meet the case that neural network may not be trained sufficiently with consideration for the topography of roads. And a kind of Lagrange interpolation algorithm is also proposed to meet the requirements for matching accuracy. These processes make the amended position information closer to the true value. In this application to a city’s MM, we investigate DNIKSOM’s ,KSOM’s and Centroid localization algorithm’s location performance on a original position data set. Finally, the comparison of experimental results shows that DNIKSOM has better location performance than others.

FFT applications to plane-polar near-field antenna measurements

The four-point bivariate Lagrange interpolation algorithm was applied to near-field antenna data measured in a plane-polar facility. The results were sufficiently accurate to permit the use of the FFT (fast Fourier transform) algorithm to calculate the far-field patterns of the antenna. Good agreement was obtained between the far-field patterns as calculated by the Jacobi-Bessel and the FFT algorithms. The significant advantage in using the FFT is in the calculation of the principal plane cuts, which may be made very quickly. Also, the application of the FFT algorithm directly to the near-field data was used to perform surface holographic diagnosis of a reflector antenna. The effects due to the focusing of the emergent beam from the reflector, as well as the effects of the information in the wide-angle regions, are shown. The use of the plane-polar near-field antenna test range has therefore been expanded to include these useful FFT applications.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>