In this paper, we considered the improved element-free Galerkin (IEFG) method for solving 2D anisotropic steady-state heat conduction problems. The improved moving least-squares (IMLS) approximation is used to establish the trial function, and the penalty method is applied to enforce the boundary conditions, thus the final discretized equations of the IEFG method for anisotropic steady-state heat conduction problems can be obtained by combining with the corresponding Galerkin weak form. The influences of node distribution, weight functions, scale parameters and penalty factors on the computational accuracy of the IEFG method are analyzed respectively, and these numerical solutions show that less computational resources are spent when using the IEFG method.

Heat conduction in anisotropic material has been widely applied in various branches of science and engineering. Unlike those of isotropic materials, the thermal conductivity of anisotropic materials varies with direction. Due to the complexity of such problems, analytical solutions are limited to only a few idealized cases. Therefore, how to obtain the internal temperature distribution of anisotropic materials effectively and accurately is one of the significant directions in scientific research.

Currently, lots of meshless methods have been applied for researching heat conduction in anisotropic materials, such as local meshless method [

As an important meshless method, the EFG method [

In order to eliminate the singular matrices, Cheng et al. studied the IMLS approximation [

By introducing the singular weight function into the MLS approximation, Lancaster et al. studied the interpolating MLS method [

Using the nonsingular weight function, Wang et al. developed the improved interpolating MLS method, using this method to construct the trial function, thus the improved interpolating EFG method are presented [

By combining the traditional finite difference method with various kinds of meshless methods, thus the hybrid EFG method [

In this study, the IEFG method is used for solving anisotropic steady-state heat conduction problem. The shape functions are established by using the IMLS approximation, using the penalty method to enforce the boundary condition, thus the final formulae of discretized equations of the IEFG method for anisotropic steady-state heat conduction problem can be derived by combining with the corresponding Galerkin weak form.

The influences of nodes number, weight functions, scale parameters and penalty factors on computational accuracy of the IEFG method are discussed by given examples, and numerical solutions show that the IEFG method for anisotropic steady-state heat conduction problems is convergent, compared with the traditional EFG method, less computational resources are spent when using the IEFG method.

For an arbitrary function

In general,

The local approximation is

Define

From

Then from

Substituting

This is the IMLS approximation [

The governing equation is

The equivalent functional of anisotropic steady-state heat conduction problems is

By applying the penalty method to enforce the boundary conditions, we can obtain

Let

In the problem domain, we select

From the IMLS approximation, we can obtain

From

Substituting

Analyzing all terms in

Let

Substituting

This is the IEFG method for 2D anisotropic steady-state heat conduction problem.

The formula of the relative error is

For simplicity, we select linear basis function in this section, and 4 × 4 Gaussian points are selected in each integral grid. Fourth numerical examples are presented, and the IEFG and the EFG methods are used to solve them, respectively.

The first example is

The problem domain is

In order to illustrate the advantages of the IEFG method for 2D anisotropic steady-state heat conduction problem, we should study the convergence of this method.

Using the IEFG method to solve it, the cubic spline weight function is selected, _{max }=_{ }1.19, ^{5},

The influences of weight function, scale parameter and penalty factor on solution of the IEFG method will be discussed, respectively.

1) Weighting function

If we select the cubic spline function, 17 × 15 regular nodes and 16 × 14 integral grids are selected respectively, _{ }=_{ }6.0 × 10^{5}, _{max }=_{ }1.19, thus the smaller relative error is 0.3292%. When the quartic spline function is used, same nodes and integral grids are used respectively, _{ }=_{ }4.0 × 10^{5}, _{max }=_{ }1.15, thus the smaller relative error is 0.3320%. It is shown that the relative error is slightly bigger when using the quartic spline function. In this section, we select the cubic spline function.

2) Scale parameter _{max}

The same nodes and integral grids are used respectively, _{ }=_{ }4.0 × 10^{5}, and the cubic spline function is used. _{max} and relative error. It is shown that, when _{max }=_{ }1.19, the smaller relative error is obtained.

3) Penalty factor

The same nodes, integral grids and the weight function are used respectively, _{max }=_{ }1.19, ^{5}∼1.0 × 10^{6}, the smaller relative error is obtained.

When the IEFG method is used to solve it, 17 × 15 regular nodes and 16 × 14 integral grids are selected, _{max }=_{ }1.19, ^{5}, and the cubic spline weight function is used; when the EFG method is used, keep all parameters consistent with the IEFG method, thus the same computational accuracy can be obtained, and the relative errors of two methods are equal to 0.3292%.

The second example we considered is an orthotropic medium, and the problem domain is a semi-circular ring, the outer and inner radii are 2 and 1, respectively. The governing equation is

The problem domain is

Using the IEFG method to solve it, 21 × 11 nodes (see _{max }=_{ }1.7, ^{3}, and the cubic weight function is used, thus the smaller relative error is 0.0735%; when the EFG method is used to solve it, keep all parameters consistent with the IEFG method, thus the same calculation accuracy is obtained. The CPU times of the EFG and the IEFG methods are 1.5 and 1.1 s, respectively.

The third example is a heat conduction problem in orthotropic material with internal heat source

The problem domain is

Using the IEFG method to solve it, 11 × 11 regular nodes and 10 × 10 integral grids are selected respectively, _{max }=_{ }1.2, ^{3}, the cubic spline weight function is used; when the EFG method is used, keep all parameters consistent with the IEFG method, thus the relative errors of both methods are equal to 0.0019%.

The fourth example is

The problem domain is

When the IEFG method is used to solve it, 15 × 15 regular nodes and 14 × 14 integral grids are selected, respectively, the cubic spline weight function is used, _{max }=_{ }1.3, ^{3}, thus the relative error is 0.0517%; when the EFG method is used, keep all parameters consistent with the IEFG method, thus the same calculation accuracy is obtained.

In order to compare the accuracy and efficiency of the IEFG and the EFG methods under different node distributions, we only change the parameters of the EFG method. When the EFG method is used, 8 × 8 regular nodes and 7 × 7 integral grids are selected, respectively, _{max }=_{ }1.27, ^{3}, thus the smaller relative error is 0.2274%, and the CPU time is 0.23 s.

In this paper, we considered the IEFG method for solving 2D anisotropic steady-state heat conduction problems.

From

Therefore, the study in this paper can broaden the scope of application of the IEFG method.