High-order Lagrange Interpolation Research Articles

A novel modified conformable fractional grey time-delay model for power generation prediction

Many fractional-order grey models are proposed and discussed for electric power generation data analysis, which are helpful in enterprise production and policy scheduling. Considering that time-delay is a universal phenomenon in real life and engineering application, a new and comprehensive conformable fractional grey time-delay model is established by extending classical grey models with the forms of conformable fractional derivative, conformable fractional accumulation and delay parameters. Considering delay data is always unknown, Lagrange interpolation is used to estimate the time-delay data. Compared with linear estimation, high order Lagrange interpolation will provide more detail information in the fitting stage. Furthermore, the optimal and modified models are also disused for predicting the future power generation by the tested data in this paper. The errors between the simulated and real data were analyzed and predicted by autoregression model, which is good at revealing the inner trend for historical residuals. The accuracies of modeling and forecasting can be improved by the optimization algorithm and autoregression error estimation in this paper. The results show the optimal and modified models could be widely used in forecasting electrical time series data, which has high effectiveness and flexibility. The novel model could enrich the connotation of parameters and the physical significance of the traditional fractional grey model.

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.

High-order full-discretization methods for milling stability prediction by interpolating the delay term of time-delayed differential equations

The full-discretization method (FDM) has been proven effective for prediction on the regenerative chatter in many papers. However, the previous studies towards FDM just focused on high-order Lagrange interpolation for state term of time-delayed differential equations (DDEs), which formulates the dynamics model in milling process. It is well known that the discretization error caused by the delay term of DDEs would transmit to the state term inevitably; higher-order Lagrange interpolation for delay term is thus vital. In this paper, second-order, third-order, and fourth-order full-discretization methods using Lagrange interpolation for the delay term of DDEs (DFDMs) were firstly proposed. Then, influence on the accuracy, computational efficiency, and convergence rate of the proposed DFDMs was discussed in detail as the change of interpolation order. It was found that rise in accuracy and convergence rate of the proposed DFDM nearly stopped when the interpolation order for delay term was up to fourth order. Next, some researches on 2-degree-of-freedom (2-DOF) of dynamic system was studied and the results show that the proposed method using fourth-order Lagrange interpolation for the delay term of DDEs (4th DFDM) was effective. Finally, this paper verified the 4th DFDM by experiment and analyzed the prediction error of 4th DFDM, which may be caused by the modeling process of cutting force. The proposed DFDMs are developed to find a better method, which can update the existing FDM and make regenerative chatter’s prediction more efficient and precise.

ブロック構造型直交格子とIB法を用いたLEEコードの構築

Recently, acoustic analysis using the Linearized Euler Equation (LEE) on multi-block structured or unstructured mesh is focused. However, mesh generation around complicated geometries takes time on structured mesh and cost of high order calculation gets larger on unstructured mesh. In this research, a LEE code for aeroacoustic analysis is developed on block structured Cartesian mesh of Building-Cube Method. To make an accurate calculation, the Immersed Boundary Method is implemented for wall boundary treatment and high order Lagrange interpolation is implemented at the Cube boundary for data exchange. This code is validated through acoustics scattering problems around cylinders and the calculational errors are compared. The results show good agreement with analytical solutions and the usability of these methods.

Research on the Local Upwind DQ-Lagrange Method for Solving the Flow Field in Labyrinth Seal

The paper sets up an upwind local differential quadrature-Lagrange interpolation (DQ-Lagrange) method for solving the flow field in the interlocking labyrinth seal. The implementation of the Dirichlet boundary condition and the Neumann boundary condition is improved. The paper analyzes the influence of support domain size, the implementation of boundary condition and the upwind scheme on the accuracy of the calculation. Numerical simulation result shows that the high order Lagrange interpolation may cause numerical oscillation and the local differential quadrature method is recommended. The upwind support domain can improve the accuracy of the calculation. Solution accuracy may be better in case of that the velocity support domain is larger than the pressure support domain.

An explicit construction of interpolation nodes on the simplex

An open question concerns the spatial distribution of nodes that are suitable for high-order Lagrange interpolation on the triangle and tetrahedron. Several current methods used to produce nodal sets with small Lebesgue constants are recalled. A new approach is presented for building nodal distributions of arbitrary order, that is based on curvilinear finite-element techniques. Numerical results are shown which demonstrate that, despite the explicit nature of this construction, the resulting node sets are well suited for interpolation and competitive with existing sets for up to tenth-order polynomial interpolation. Matlab scripts which evaluate the node distributions on the equilateral triangle are included.

An element by element spectral element method for elastic wave modeling*

Abstract The spectral element method which combines the advantages of spectral method with those of finite element method, provides an efficient tool in simulating elastic waves equation in complex medium. Based on weak from the elastodynamic equations, mathematical formulations for Legendre spectral element method are presented. The wave field on an element is discretized using high-order Lagrange interpolation, and integration over the element is accomplished based upon the Gauss-Lobotto-Legendre integration rule. This results in a diagonal mass matrix which leads to a greatly simplified algorithm. In addition, the element by element examples are resented to in our method to reduce the memory sizes and improve the computation efficiency. Finally, some numerical examples are resented to demonstrate the spectral accuracy and the efficiency. Because of combinations of the finite element scheme and a spectral algorithms, the method can be used for complex models, including free surface boundaries and strong...