scholarly journals A Dimension Splitting-Interpolating Moving Least Squares (DS-IMLS) Method with Nonsingular Weight Functions

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2424
Author(s):  
Jufeng Wang ◽  
Fengxin Sun ◽  
Rongjun Cheng
Keyword(s):  
Least Squares ◽  
Splitting Method ◽  
Weak Form ◽  
Moving Least Squares ◽  
Weight Functions ◽  
High Approximation ◽  
Approximation Accuracy ◽  
Approximation Function ◽  
Discrete Equations ◽  
The Matrix

By introducing the dimension splitting method (DSM) into the improved interpolating moving least-squares (IMLS) method with nonsingular weight function, a dimension splitting–interpolating moving least squares (DS-IMLS) method is first proposed. Since the DSM can decompose the problem into a series of lower-dimensional problems, the DS-IMLS method can reduce the matrix dimension in calculating the shape function and reduce the computational complexity of the derivatives of the approximation function. The approximation function of the DS-IMLS method and its derivatives have high approximation accuracy. Then an improved interpolating element-free Galerkin (IEFG) method for the two-dimensional potential problems is established based on the DS-IMLS method. In the improved IEFG method, the DS-IMLS method and Galerkin weak form are used to obtain the discrete equations of the problem. Numerical examples show that the DS-IMLS and the improved IEFG methods have high accuracy.

scholarly journals An Improved Interpolating Element-Free Galerkin Method Based on Nonsingular Weight Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
F. X. Sun ◽  
C. Liu ◽  
Y. M. Cheng
Keyword(s):  
Least Squares ◽  
Shape Function ◽  
Weak Form ◽  
Moving Least Squares ◽  
Weight Functions ◽  
Element Free Galerkin Method ◽  
Element Free Galerkin ◽  
Node Distribution ◽  
Mls Approximation ◽  
Element Free

Based on the moving least-squares (MLS) approximation, an improved interpolating moving least-squares (IIMLS) method based on nonsingular weight functions is presented in this paper. Then combining the IIMLS method and the Galerkin weak form, an improved interpolating element-free Galerkin (IIEFG) method is presented for two-dimensional potential problems. In the IIMLS method, the shape function of the IIMLS method satisfies the property of Kroneckerδfunction, and there is no difficulty caused by singularity of the weight function. Then in the IIEFG method presented in this paper, the essential boundary conditions are applied naturally and directly. Moreover, the number of unknown coefficients in the trial function of the IIMLS method is less than that of the MLS approximation; then under the same node distribution, the IIEFG method has higher computational precision than element-free Galerkin (EFG) method and interpolating element-free Galerkin (IEFG) method. Four selected numerical examples are presented to show the advantages of the IIMLS and IIEFG methods.

An Improved Interpolating Element-Free Galerkin Method for Elastoplasticity via Nonsingular Weight Functions

2016 ◽  
Vol 08 (08) ◽  
pp. 1650096 ◽  
Author(s):  
Fengxin Sun ◽  
Jufeng Wang ◽  
Yumin Cheng
Keyword(s):  
Least Squares ◽  
Numerical Solutions ◽  
Weak Form ◽  
Moving Least Squares ◽  
Weight Functions ◽  
Shape Functions ◽  
Computational Accuracy ◽  
Displacement Boundary ◽  
Element Free Galerkin ◽  
Element Free

An improved interpolating element-free Galerkin (IIEFG) method for elastoplasticity is proposed in this paper. In the IIEFG method, the shape functions are constructed by the improved interpolating moving least-squares (IIMLS) method, and the final system equations are obtained by using the Galerkin weak form of elastoplasticity. Compared with the interpolating moving least-squares (IMLS) method, the weight functions are not singular in the IIMLS method, in which the shape functions have the interpolating property. The IIMLS method has fewer unknown coefficients to be solved in the trial functions than the moving least-squares (MLS) approximation. Hence, the IIEFG method is able to directly enforce the displacement boundary condition and obtain numerical solutions with high computational accuracy and efficiency. To show advantages of the IIEFG method, some selected elastoplastic examples are given.

Integration of Moving Least Squares Method and Multilevel B-Spline Method for Scattered Data Approximation

2011 ◽  
Vol 291-294 ◽  
pp. 2245-2249
Author(s):  
Shi Ju Yan ◽  
Bin Ge
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Scattered Data ◽  
Moving Least Squares ◽  
Scattered Data Approximation ◽  
Approximation Accuracy ◽  
Data Approximation ◽  
B Spline ◽  
Moving Least Squares Method ◽  
Method Accuracy

For scattered data approximation with multilevel B-spline(MBS) method, accuracy could be enhanced by densifying control lattice. Nevertheless, when control lattice density reaches to some extent, approximation accuracy could not be enhanced further. A strategy based on integration of moving least squares(MLS) and multilevel B-spline(MBS) is presented. Experimental results demonstrate that the presented strategy has higher approximation accuracy.

scholarly journals A Meshless Local Petrov-Galerkin Shepard and Least-Squares Method Based on Duo Nodal Supports

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Xiaoying Zhuang ◽  
Yongchang Cai
Keyword(s):  
Least Squares ◽  
Present Method ◽  
Least Squares Method ◽  
Partition Of Unity ◽  
Weak Form ◽  
Weight Functions ◽  
Kronecker Delta ◽  
Background Mesh ◽  
Support Domain ◽  
New Formulation

The meshless Shepard and least-squares (MSLS) interpolation is a newly developed partition of unity- (PU-) based method which removes the difficulties with many other meshless methods such as the lack of the Kronecker delta property. The MSLS interpolation is efficient to compute and retain compatibility for any basis function used. In this paper, we extend the MSLS interpolation to the local Petrov-Galerkin weak form and adopt the duo nodal support domain. In the new formulation, there is no need for employing singular weight functions as is required in the original MSLS and also no need for background mesh for integration. Numerical examples demonstrate the effectiveness and robustness of the present method.

THE INTERPOLATING ELEMENT-FREE GALERKIN (IEFG) METHOD FOR TWO-DIMENSIONAL ELASTICITY PROBLEMS

2011 ◽  
Vol 03 (04) ◽  
pp. 735-758 ◽  
Author(s):  
HONGPING REN ◽  
YUMIN CHENG
Keyword(s):  
Least Squares ◽  
Shape Function ◽  
Weak Form ◽  
New Method ◽  
Moving Least Squares ◽  
Two Dimensional ◽  
Element Free Galerkin ◽  
Elasticity Problems ◽  
Mls Approximation ◽  
Element Free

In this paper, a new method for deriving the moving least-squares (MLS) approximation is presented, and the interpolating moving least-squares (IMLS) method proposed by Lancaster is improved. Compared with the IMLS method proposed by Lancaster, a simpler formula of the shape function is given in the improved IMLS method in this paper so that the new method has higher computing efficiency. Combining the shape function constructed by the improved IMLS method with Galerkin weak form of the elasticity problems, the interpolating element-free Galerkin (IEFG) method for the two-dimensional elasticity problems is presented, and the corresponding formulae are obtained. In the IEFG method, the boundary conditions can be applied directly which makes the computing efficiency higher than the conventional EFG method. Some numerical examples are presented to demonstrate the validity of the method.

A NEW ELEMENT-FREE GALERKIN METHOD BASED ON IMPROVED COMPLEX VARIABLE MOVING LEAST-SQUARES APPROXIMATION FOR ELASTICITY

2012 ◽  
Vol 01 (01) ◽  
pp. 1250011 ◽  
Author(s):  
HONGPING REN ◽  
YUMIN CHENG
Keyword(s):  
Least Squares ◽  
Complex Variable ◽  
Weak Form ◽  
Moving Least Squares ◽  
Two Dimensional ◽  
Element Free Galerkin Method ◽  
Element Free Galerkin ◽  
Variable Element ◽  
Elasticity Problems ◽  
Element Free

In this paper, by constructing a new functional, an improved complex variable moving least-squares (ICVMLS) approximation is presented. Based on element-free Galerkin (EFG) method and the ICVMLS approximation, a new complex variable element-free Galerkin (CVEFG) method for two-dimensional elasticity problems is presented. Galerkin weak form is used to obtain the discretized equations and the essential boundary conditions are applied with Lagrange multiplier. Then the formulae of the new CVEFG method for two-dimensional elasticity problems are obtained. Compared with the conventional EFG method, the new CVEFG method has greater computational precision and efficiency. For the purposes of demonstration, some selected numerical examples are solved using the ICVEFG method.

A Simplified Interpolating Moving Least-Squares Method and its Error Estimates

2011 ◽  
Vol 101-102 ◽  
pp. 271-274
Author(s):  
Ju Feng Wang
Keyword(s):  
Error Estimate ◽  
Least Squares ◽  
Shape Function ◽  
Least Squares Method ◽  
Delta Function ◽  
Moving Least Squares ◽  
Approximation Function ◽  
Kronecker Delta ◽  
Moving Least Squares Method ◽  
Mls Approximation

A disadvantage of the MLS approximation is that the shape function of this method does not satisfy the property of Kronecker Delta function. Thus developing an interpolating MLS approximation is very important. In this paper, the interpolating moving least-squares (IMLS) method presented by Lancaster and Salkauskas is discussed in detail and a simplified expression of the approximation function of the IMLS method is given. The simpler expression makes it more convenient to use this method. The error estimate of the approximation function also is discussed. And a numerical example is given to confirm the results.

scholarly journals Extended Peridynamics and Parameter Optimization Study Based on Moving Least Squares Method

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Jingwei Xu ◽  
Wei Hou ◽  
Shoucheng Luan ◽  
Shuting Mao ◽  
Guowei Liu ◽  
...  
Keyword(s):  
Weight Function ◽  
Least Squares ◽  
Least Squares Method ◽  
Moving Least Squares ◽  
Weight Functions ◽  
Optimization Study ◽  
Fitting Method ◽  
Moving Least Squares Method ◽  
Physical Information ◽  
The One

Based on the theory of peridynamics, the least squares and the moving least squares method are proposed to fit the physical information at nondiscrete points. It makes up for the shortcomings of the peridynamic method that only solves the discrete nodes and cannot obtain the physical information of other blank areas. The extended method is used to fit the one-way vibration problem of the rod, and the curve of the displacement of a nondiscrete node in the rod is extracted with time. The fitted displacement results are compared with the theoretical results to verify the feasibility of the fitting method. At the same time, the parameters in the fitting of the moving least squares method are optimized, and the effects of different tight weight functions and influence ranges on the results are analyzed. The results show that when the weight function is a power exponential function, the fitting effect increases with the decrease in the coefficient. When the weight function is a cubic spline weight function, a better fitting effect is obtained. And in the case of ensuring the fitting result, the affected area should be reduced as much as possible, and the calculation efficiency and precision can be improved.

Analyzing nonlinear large deformation with an improved element-free Galerkin method via the interpolating moving least-squares method

2016 ◽  
Vol 05 (04) ◽  
pp. 1650023 ◽  
Author(s):  
Yumin Cheng ◽  
Funong Bai ◽  
Chao Liu ◽  
Miaojuan Peng
Keyword(s):  
Least Squares ◽  
Large Deformation ◽  
Least Squares Method ◽  
Weak Form ◽  
Moving Least Squares ◽  
Displacement Boundary ◽  
System Equation ◽  
Element Free Galerkin ◽  
Element Free ◽  
Large Deformation Problems

Using the interpolating moving least-squares (IMLS) method to form the shape function, a novel improved element-free Galerkin (IEFG) method is presented for solving nonlinear elastic large deformation problems. To obtain the formulae of the IEFG method for elastic large deformation problems, we use the Galerkin weak form to obtain the discretized system equation, and use the penalty method to apply the displacement boundary conditions. Some selected numerical examples of two-dimensional elastic large deformation problems are given, and the numerical results are analyzed. From the examples, it is shown that the IEFG method in this paper has higher computational precision than the element-free Galerkin (EFG) method presented before.

scholarly journals Improvement of the Approximation Accuracy of LED Radiation Patterns

Electronics ◽  
2019 ◽  
Vol 8 (3) ◽  
pp. 337 ◽  
Author(s):  
Ivan Rachev ◽  
Todor Djamiykov ◽  
Marin Marinov ◽  
Nikolay Hinov
Keyword(s):  
Least Squares ◽  
Mathematical Analysis ◽  
Least Squares Method ◽  
Weighting Function ◽  
Radiation Patterns ◽  
Approximation Accuracy ◽  
Light Emitting ◽  
Approximation Function ◽  
Wide Range ◽  
Optoelectronic Systems

For the great variety of light-emitting diodes (LEDs), there exists a wide range of LED radiation patterns. An approach for constructing patterns of higher accuracy is here considered. The latter is required when the design of optoelectronic systems or their optimization is carried out analytically. A weighting function is introduced that allows increasing the gradient of the diagram of different widths. It has been selected through mathematical analysis of the emission diagrams of different LEDs used in optoelectronic systems. Based on the least squares method an algorithm is created, and programs are developed in MATLAB environment to estimate the parameters of the approximation function. Its accuracy is evaluated by comparison with the approximation with Lambert source of order n. The results show higher accuracy of the proposed approximation function compared to those obtained by conventional methods. Recommendations on the application of the proposed approach are given.

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