The influence of selectable parameters in the element-free Galerkin method: A one-dimensional beam-in-bending problem

2009 ◽  
Vol 223 (7) ◽  
pp. 1579-1590 ◽  
Author(s):  
O F Valencia ◽  
F J Gómez-Escalonilla ◽  
J López-Díez
Keyword(s):  
Galerkin Method ◽  
Least Squares ◽  
Least Squares Method ◽  
Moving Least Squares ◽  
Shape Functions ◽  
Element Free Galerkin Method ◽  
One Dimensional ◽  
Element Free Galerkin ◽  
Axially Loaded ◽  
Element Free

Continuing with the analysis performed for the one-dimensional axially loaded bar problem, a beam in bending is analysed to understand the influence of the characteristic parameters that have any influence in the solution of this problem using the element-free Galerkin method (EFGM), one of the most popular meshless methods. Both accuracy and time cost are considered as the evaluation functions to perform such an analysis. Both functions provide a reasonable idea to consider EFGM as an adequate method to solve the problem considered in this article. As in a one-dimensional axially loaded bar problem, the parameters to be considered will be those that affect the solution: number of nodes in which the domain is modelled, the nodes scatter, the order of the polynomial base to generate shape functions, the order of the quadrature to solve integrals, and the support radius. Besides, as in a one-dimensional axially loaded problem, some cases with different loading and stiffness conditions are considered. However, in this analysis a generalized moving least squares method is used to create shape functions instead of the moving least squares.

Bernstein–Galerkin approach in elastostatics

2013 ◽  
Vol 228 (3) ◽  
pp. 391-404 ◽  
Author(s):  
D Garijo ◽  
ÓF Valencia ◽  
FJ Gómez-Escalonilla ◽  
J López Díez
Keyword(s):  
Galerkin Method ◽  
Numerical Approximation ◽  
Bernstein Polynomials ◽  
Weak Form ◽  
Moving Least Squares ◽  
Shape Functions ◽  
Element Free Galerkin Method ◽  
Galerkin Approach ◽  
Element Free Galerkin ◽  
Element Free

Since 1994, two main meshless methods have been developed and widely used: these are the element free Galerkin method and the meshless local Petrov-Galerkin method. Both methods solve partial differential equations by posing a numerical approximation to the solution using the moving least squares technique. Using Bernstein polynomials as the shape functions of Galerkin weak form-based methods improves the numerical approximation achieved at boundaries without losing accuracy inside the domain.

scholarly journals Bernstein polynomials in element-free Galerkin method

2011 ◽  
Vol 225 (8) ◽  
pp. 1808-1815 ◽  
Author(s):  
O F Valencia ◽  
F J Gómez-Escalonilla ◽  
D Garijo ◽  
J L Díez
Keyword(s):  
Boundary Conditions ◽  
Galerkin Method ◽  
Bernstein Polynomials ◽  
Partition Of Unity ◽  
Particle Method ◽  
Moving Least Squares ◽  
Shape Functions ◽  
Element Free Galerkin Method ◽  
Element Free Galerkin ◽  
Element Free

In the recent decades, meshless methods (MMs), like the element-free Galerkin method (EFGM), have been widely studied and interesting results have been reached when solving partial differential equations. However, such solutions show a problem around boundary conditions, where the accuracy is not adequately achieved. This is caused by the use of moving least squares or residual kernel particle method methods to obtain the shape functions needed in MM, since such methods are good enough in the inner of the integration domains, but not so accurate in boundaries. This way, Bernstein curves, which are a partition of unity themselves, can solve this problem with the same accuracy in the inner area of the domain and at their boundaries.

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