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 is a vector of coefficients given by

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 is the weight function of node at a point x which for simplicity it will be stated as.

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

The minimum of J with respect to is found by

This leads to the following system of linear equations

Here and are and matrices, respectively, and are given as

And U is vector and is given as

can be found using Equation (8);

Putting from Equation (12) into Equation (1) leads to

where is a vector of shape functions. The first derivative of the shape functions with respect to the spatial coordinates is also required for the numerical implementation and is given as

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 is a scaling parameter and is determined by searching for enough neighbor nodes such that matrix A in Equation (8) is invertible. In regular distribution of nodal points can be chosen as the distance between two neighboring nodes. In this paper, the cubic spline weight function is used;

where is the distance between node and point of interest. Weight function derivatives with respect to the spatial coordinates are also required for the shape function derivatives as given in Equation (14) and are given as follows [2] :

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

where is a source term. On one part of the boundary, is the Dirichlet boundary condition, and on the other part, is the Neumann boundary condition.

where n is the outward normal vector to the boundary.

The MLS shape functions do not satisfy the Kronecker delta property, i.e., and are termed as approximants instead of interpolants. The values obtained from the MLS approximation are therefore, not the same as the nodal values, i.e., and are known as nodal parameters. This leads to some difficulties in imposition essential boundary condition in contrast to conventional FEM [2] .

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 [2] .

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

where is the prescribed potential on the boundary.

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 [2] . The constrained Galerkin weak form uses the penalty method and with substituting the expression of MLS approximation of Equation (13) can then be posed as

where is a diagonal matrix of the penalty factor that for 2-D case. The penalty factor can be a function of the coordinates, and it can be different from one another. Although in practice the identical constant of a large positive number is assigned for penalty factor, which can be chosen by following method [2]

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

where

The additional matrix is the global penalty matrix assembled using the nodal matrix defined by

And the vector is caused by the essential boundary condition that its nodal vector has the form

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 and the ratio of influence domain is considered 3. The regular and irregular distribution of 81 nodal points for this problem is shown in Figure 1 and Figure 2. The analytical and EFG solution on a mesh of 81 nodal points with 96 and 1152 Gauss points along x axis are shown in Figure 3 and Figure 4, respectively, to assess the effect of number of Gauss points on the solution accuracy.

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 and is the quantity of analytical solution and numerical solution, respectively. For the sensitivity analysis, one of the parameters is changed while the others are constant. The result of this analysis is shown in Tables 1-10, and the computational time is presented.

The results of Table 1 indicate that the errors are dramatically reduced with increasing the number of nodal points while they get nearly constant when more nodal points are added. These results are also used to evaluate the convergence rate of the method with respect to nodal points and the results are shown in Figure 5.

The results of Table 2 and Table 3 quantitatively emphasize the rule of Gauss points on the accuracy of the EFG method and demonstrate high accuracy and low sensitivity of the proposed method in dealing with irregular distribution of nodal points.

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 Table 4 and a convergence rate is also demonstrates the obtained results in Figure 6. These results indicate the convergent behavior of the method as expected.

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 Table 5 and Table 6. The values of this ratio in Table 6 vary in the same way as Table 5 to have a better comparison between them. The results of Table 5 demonstrate that the appropriate interval of ratio of influence domain in regular distribution of nodal points is 2 - 3, while it is obvious from Table 6 that the errors are decreased by increasing this ratio.

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 Table 7 and Table 8 is considered 3 to have a better comparison between them.

According to the results of Table 7 and Table 8, the errors are diminished by increasing the number of monomial terms in basis function in regular distribution of nodal points, however, this effect is opposite in irregular distribution of nodal points because the higher number of monomial terms, the more nodal points are acquired in a favorable influence domain.

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 Table 9 and Table 10 in both cases, the solution’s accuracy obtained by cubic spline is more desirable than the other weigh functions.

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, and are assumed. The regular distribution of nodal points is shown in Figure 7. The above- mentioned problem is solved using two different sets of 460 distributed nodes that are shown in Figure 7 and Figure 8. The analytical and numerical solutions along r direction at any angle with 460 nodal points are plotted in Figure 9 and the ratio of influence domain is considered 3 for this problem again.

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 Figure 10 and the governing equation of that is as follows:

and the exact solution is

where is the fluid’s velocity. Due to the symmetry, only the one-quarter of the problem domain is considered. This domain with its boundary condition is shown in Figure 11.

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 Figure 12-14. The analytical and the EFG solutions along y axis are shown in Figure 15.