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A meshfree method namely, element free Gelerkin (EFG) method, is presented in this paper for the solution of governing equations of 2-D potential problems. The EFG method is a numerical method which uses nodal points in order to discretize the computational domain, but where the use of connectivity is absent. The unknowns in the problems are approximated by means of connectivity-free technique known as moving least squares (MLS) approximation. The effect of irregular distribution of nodal points on the accuracy of the EFG method is the main goal of this paper as a complement to the precedent researches investigated by proposing an irregularity index (II) in order to analyze some 2-D benchmark examples and the results of sensitivity analysis on the parameters of the method are presented.

Partial differential equations arise in connection with various physical and geometrical problems in which the functions involved depend on two or more independent variables, usually on time t and on one or several space variables [

Mesh based numerical methods, such as finite element method (FEM) and boundary element method (BEM), have been the primary numerical techniques in engineering computations. In spite of the positive points of the finite element method, it still suffers from high preprocessing time, low accuracy of stresses, difficulty in incorporating adaptivity and it is also not an ideal tool for certain classes of problems, e.g. large deformations, material damage, crack growth, and moving boundaries [

In the past few decades, a variety of new meshless methods have been developed, including the smoothed particle hydrodynamics (SPH) method [

Recently several meshless methods are proposed in order to solve potential problems. The improved EFG method [

The element free Galerkin (EFG) method that was developed by Belytschko et al. [

The element free Galerkin method is presented in this paper to solve potential problems, and the effect of irregularity distribution of nodal points by using a proposed irregularity index (II) that was not considered in the previous researches for the EFG method, is investigated. In what follows, the construction of MLS shape functions is first explained. EFG method for discretization of the governing differential equation is then explained. Several 2-D potential problems are solved using the proposed method; sensitivity analysis on the parameters of the proposed method is also carried out, and the results are presented.

MLS is a very important component of the element free Galerkin (EFG) method for the approximation of the field variables. The MLS approximation u^{h} of a scalar function u at point x is given as

where P(x) is a polynomial basis function of the spatial coordinates, m is the number of monomial terms in the basis function, and

The polynomial basis function P(x) is built from Pascal’s triangle and pyramid for 2- and 3-D problems, respectively. In 2-D problems, linear and quadratic basis functions are given as

The unknown coefficients in Equation (1) can be found by minimizing the following weighted least squares method.

where

Equation (5) using vector notation can be written as:

The minimum of J with respect to

This leads to the following system of linear equations

Here

And U is

Putting

where

where

and the index after the comma is a spatial derivative.

Weight function is an important part of the MLS approximation. There are no predefined rules to select the weight function for a particular application, but the weight function that could be used for meshless methods should have the following properties:

1) Its value should be maximized at the node and decrease with the distance from the node.

2) Smooth and non-negative.

3) It should have a compact support, i.e. non-zero over a small neighborhood of a node. This compact support is known as the influence domain of a node (nodal point).

Influence domain of a nodal point is a very important concept in meshless methods, as it determines the region in which it has influence. The size of influence domain for a node i is

where

Consider a Poisson’s partial differential equation in a two dimensional domain

where

where n is the outward normal vector to the boundary.

The MLS shape functions do not satisfy the Kronecker delta property, i.e.

In this paper, the penalty method is used to enforce the essential boundary condition. The use of penalty method produces system of equations of the same dimension that FEM produces for the same number of nodes, and the modified stiffness matrix is still positively defined; moreover, the symmetry and the bandedness of the system matrix are preserved [

In the EFG method, the essential boundary condition has the form

where

Consider the problem stated in Equation (18), a penalty factor is applied to penalize the difference between the potential of the MLS approximation and the prescribed potential on the essential boundary [

where

The final system of equation of the EFG formulation with penalty method is

where

The additional matrix

And the vector

To demonstrate the efficiency and accuracy of the EFG method in dealing with irregular distribution of nodal points, following irregularity index (II) is proposed in this paper

where

In this section, three 2-D numerical examples are solved to demonstrate the efficiency and accuracy of the proposed method. The effect of irregularity in distribution of nodal points is investigated by using of a proposed irregularity index (II) and the results are compared with the existing analytical solutions.

Consider the following 2-D Poisson’s equation

with the following Dirichlet and Neumann boundary conditions

the analytical solution of the aforementioned Poisson’s equation is

The above-mentioned problem is solved using two different sets of 81 distributed nodes. In all of these cases, the polynomial basis function is considered as

There are different parameters in the EFG method that affect the obtained results. In this paper a sensitivity analysis is carried out on these parameters. Number of nodal points, number of Gauss points, ratio of influence domain, number of monomial terms in the basis function, and the type of weight function, are the parameters that are analyzed. For the sensitivity analysis the following error norm has been used

where

The results of

The results of

This problem is solved here with different values of irregularity index to present the effect of irregularity distribution of nodal points. This analysis is done by using a proposed index that is shown in

Number of nodal points | 25 | 36 | 64 | 81 |
---|---|---|---|---|

0.250 | 0.200 | 0.143 | 0.125 | |

_{}_{ } | 0.3944 | 0.0730 | 0.0031 | 0.0029 |

CPU TIME (Sec) | 0.6708 | 0.7800 | 0.8580 | 0.9572 |

Number of Gauss points | 96 | 320 | 480 | 1152 |
---|---|---|---|---|

0.0153 | 0.0032 | 0.0029 | 0.0022 | |

CPU TIME (Sec) | 0.6084 | 0.7800 | 0.9672 | 1.6848 |

Number of Gauss points | 96 | 320 | 480 | 1152 |
---|---|---|---|---|

0.3273 | 0.2289 | 0.1323 | 0.0577 | |

CPU TIME (Sec) | 1.3193 | 2.2932 | 2.3868 | 4.5396 |

Irregularity Index (II) | 0.5 | 0.0727 | 0.0143 | 0.0012 |
---|---|---|---|---|

0.0002 | 0.0577 | 0.0991 | 0.1907 | |

CPU TIME (Sec) | 1.6848 | 4.5396 | 0.9984 | 1.6068 |

Ratio of influence domain | 1.12 | 2 | 3 | 4.8 | 6.4 |
---|---|---|---|---|---|

0.0112 | 0.0026 | 0.0007 | 0.1156 | 0.6371 | |

CPU TIME (Sec) | 0.3276 | 0.7176 | 1.5912 | 4.4928 | 7.5660 |

Ratio of influence domain | 1.12 | 2 | 3 | 4.8 | 6.4 |
---|---|---|---|---|---|

0.7114 | 0.1583 | 0.0577 | 0.0134 | 0.0025 | |

CPU TIME (Sec) | 1.2168 | 1.5756 | 4.5396 | 13.3536 | 22.3237 |

The number of monomial terms in the basis function | 0 | 1 | 2 |
---|---|---|---|

0.0029 | 0.0007 | 0.0006 | |

CPU time (Sec) | 1.2792 | 1.5912 | 2.3868 |

The number of monomial terms in the basis function | 0 | 1 | 2 |
---|---|---|---|

0.0267 | 0.0577 | 0.1645 | |

CPU time (Sec) | 3.7284 | 4.5396 | 6.2400 |

Type of weight functions | Cubic spline | Quartic spline | Exponential |
---|---|---|---|

0.0006 | 0.0016 | 0.0070 | |

CPU time (Sec) | 2.3868 | 2.4180 | 2.3556 |

Type of weight functions | Cubic spline | Quartic spline | Exponential |
---|---|---|---|

0.0577 | 0.1428 | 0.2116 | |

CPU time (Sec) | 4.5396 | 3.2136 | 4.1340 |

The problem is solved again on a mesh of 81 regularly and irregularly distributed of nodal points with different ratio of influence domain and 1152 Gauss points. The effect of this parameter is investigated in

The number of monomial terms in basis function is the other parameter that can affect the performance of the EFG method. In this case, the problem domain is discretized with 81 regular and irregular nodal points with 1152 Gauss points. The ratio of influence domain in

According to the results of

The other parameter that affects the solution’s accuracy of the EFG method is the type weight function. In order to investigate this effect, the problem domain is discretized again with 81 regular and irregular meshes of nodes with 1152 Gauss points and three types of weight functions that are considered. It is also notable that the ratio of influence domain in both cases is considered 3.

It can be concluded from

The second example is a 2-D Poisson’s equation with Dirichlet boundary conditions on the torus. The equation is

with the following boundary conditions

and the analyticalsolutionofthisproblemis

here,

In this section, flow over a circular cylinder is considered. Such a flow can be generated by adding a uniform flow, in the positive x direction to a doublet at the origin directed in the negative x direction. The geometry of the example is shown in

and the exact solution is

where

The above-mentioned problem is solved using three different sets of 241 distributions of nodal points with 962 Gauss points. It is notable that the ratio of influence domain in all cases is considered 3 and the distribution of nodal points with different values of irregular index is shown in

A meshless method namely element free Galerkin (EFG) method is presented in this paper. In order to investigate the performance and accuracy of the method, some 2-D potential problems on regular and irregular distribution

of nodal points by using a proposed irregularity index (II) are analyzed and compared with the exact solution. A sensitivity analysis on the parameters of the EFG method is also carried out. From above analysis, it can be inferred that the errors are dramatically reduced by increasing the number of nodal points and Gauss points while they get nearly constant when more of them are added. It is also notable that the appropriate ratio of influence domain has been found to be 2 - 3 for regular mesh of nodal points, and in irregular mesh of nodal points, the errors are converged by increasing this ratio. Increasing the number of monomial terms in basis function is another factor that can improve the accuracy of the EFG method in regular distribution of nodal points while this effect is contradictory in comparison with irregular distribution of nodal points. The effect of using different type of weight functions is another parameter considered and the results indicate better performance of the method in using cubic spline weight function. Finally, it can be concluded that EFG method can be used to solve problems on irregular mesh of nodes with admissible performance.