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.

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 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.

Solution of two-dimensional vector wave equations via a mixed least squares method

2015 ◽  
Vol 32 (7) ◽  
pp. 1893-1907
Author(s):  
Maria Tchonkova
Keyword(s):  
Least Squares ◽  
Wave Equations ◽  
Least Squares Method ◽  
Equations Of Motion ◽  
Time Discretization ◽  
Weak Form ◽  
Time Step ◽  
Content Type ◽  
Vector Wave ◽  
Backward Euler

Purpose – The purpose of this paper is to present an original mixed least squares method for the numerical solution of vector wave equations. Design/methodology/approach – The proposed approach involves two different types of unknowns: velocities and stresses. The approximate solution to the dynamic elasticity equations is obtained via a minimization of a least squares functional, consisting of two terms: a term, which includes the squared residual of a weak form of the time rate of the constitutive relationships, expressed in terms of velocities and stresses, and a term, which depends on the squared residual of the equations of motion. At each time step the functional is minimized with respect to the velocities and stresses, which for the purpose of this study, are approximated by equal order piece-wise linear polynomial functions. The time discretization is based upon the backward Euler scheme. The displacements are computed from the obtained velocities in terms of a finite difference interpolation. Findings – To test the performance of the method, it has been implemented in original computer codes, using object-oriented logic. One model problem has been solved: propagation of Rayleigh waves. The performed convergence study suggests that the method is convergent for both: velocities and stresses. The obtained results show excellent agreement between the exact and analytical solutions for displacement modes, velocities and stresses. It is observed that this method appears to be stable for the different mesh sizes and time step values. Originality/value – The mixed least squares formulation, described in this paper, serves as a basis for interesting future developments and applications.

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.

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.

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.

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.

scholarly journals Numerical Solutions of Mixed Integro-Differential Equations by Least-Squares Method and Laguerre Polynomial

2021 ◽  
pp. 309-323
Author(s):  
Hameeda Oda Al-Humedi ◽  
Ahsan Fayez Shoushan
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Least Squares ◽  
Present Method ◽  
Numerical Solutions ◽  
Least Squares Method ◽  
Laguerre Polynomial ◽  
New Technique ◽  
Order Linear ◽  
A New Technique

The main objective of this article is to present a new technique for solving integro-differential equations (IDEs) subject to mixed conditions, based on the least-squares method (LSM) and Laguerre polynomial. To explain the effect of the proposed procedure will be discussed three examples of the first, second and three-order linear mixed IDEs. The numerical results used to demonstrate the validity and applicability of comparisons of this method with the exact solution shown that the competence and accuracy of the present method.

‘Least squares’ method—theorem or law?

1980 ◽  
Vol 59 (9) ◽  
pp. 8
Author(s):  
D.E. Turnbull
Keyword(s):  
Least Squares ◽  
Least Squares Method
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