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In this paper, the improved element-free Galerkin (IEFG) method is used for solving 3D advection-diffusion problems. The improved moving least-squares (IMLS) approximation is used to form the trial function, the penalty method is applied to introduce the essential boundary conditions, the Galerkin weak form and the difference method are used to obtain the final discretized equations, and then the formulae of the IEFG method for 3D advection-diffusion problems are presented. The error and the convergence are analyzed by numerical examples, and the numerical results show that the IEFG method not only has a higher computational speed but also can avoid singular matrix of the element-free Galerkin (EFG) method.

Currently, meshless method has been applied successfully to deal with various kinds of problems in the fields of science and engineering. When solving large deformation problems and dynamic propagation of cracks, meshless method can obtain greater precision than finite element method [

As an important meshless method, the EFG method has been applied to different kinds of engineering problems. In the EFG method, the moving least-squares (MLS) approximation is employed to construct the shape function. Because the MLS approximation is based on the least-squares method [

Cheng and Chen studied the IMLS approximation [

Based on the MLS approximation with a singular weight function, the interpolating MLS method was proposed by Lancaster et al. [

In order to overcome the difficulties caused by singular weight function in the interpolating MLS method, an improved interpolating least-squares (IIMLS) method based on nonsingular weight function was presented by Wang et al. Based on the IIMLS method, the improved interpolating EFG method is presented for potential [

By combining the dimension splitting method and meshless methods, the hybrid complex variable EFG method [

In this paper, combining the IMLS approximation and the Galerkin weak form, the IEFG method is used for solving 3D advection-diffusion problems. The IMLS approximation is used to form the trial function, the penalty method is applied to introduce the essential boundary conditions, the Galerkin weak form and the difference method are used to obtain the final discretized equations, and then the formulae of the IEFG method for 3D advection-diffusion problems are presented.

In the section of numerical examples, the weight functions, the scale parameter, the penalty factor, the node distribution, and the time step are discussed, respectively. The numerical results show that the IEFG method in this paper has a higher computational speed. And the advantage that the IEFG method can avoid the singular matrix is given.

The approximation of a function

In general,

The local approximation of equation (

Define

Equation (

From

Equation (

Then from equation (

Substituting equation (

This is the IMLS approximation, in which

For 3D advection-diffusion problems, the governing equation is

The equivalent functional of 3D advection-diffusion problem can be obtained as

Suppose that

From

In the cubic domain

From the IMLS approximation, at the time of

From equations (

Substituting equations (

Analyzing the integral terms in equation (

Substituting equations (

Then, using the finite difference method to separate the time, the relation of

Three numerical examples of 3D advection-diffusion problems are solved with the EFG and the IEFG method, and the computational accuracy and efficiency are compared, respectively.

Define the relative error as

In this section, the linear basis function is employed and the node distribution is regular. Moreover,

The first example we considered is a 3D advection-diffusion problem:

The problem domain is

The analytical solution is

In this example, we discuss the effects of the weight functions, the scale parameter, the penalty factor, and the node distribution on the solution of the IEFG method, respectively.

The time step

When the cubic spline function is used,

Thus, when we use the cubic spline function as the weight function, higher accuracy can be obtained.

The relative error of the IEFG method for different

The same time step, weight function, node distribution, and background integral grid are used, respectively;

The relative error of the IEFG method for different

The same weight function and time step are used, respectively;

The relative error of the IEFG method for different number of nodes.

The same weight function is used;

The relative error of the IEFG method for different time step.

The EFG method is selected to solve this example,

The comparison of the CPU time of the EFG and the IEFG methods under the same relative errors is shown in Table

Comparison of the CPU time of two methods under the same relative error.

Comparison | |||||
---|---|---|---|---|---|

0.1 s | 0.3 s | 0.5 s | 0.7 s | 0.9 s | |

Relative error (%) | 1.2654 | 2.5893 | 2.7382 | 2.7478 | 2.7483 |

CPU time of EFG (s) | 101.6 | 176 | 250.5 | 327.8 | 395.8 |

CPU time of IEFG (s) | 97.2 | 173.4 | 240.5 | 314.9 | 386.3 |

The numerical results and the analytical solutions along

The numerical results and the analytical solutions along

The numerical results and the analytical solutions along

Moreover, the IEFG method which can avoid singular matrix is another advantage when constructing the shape functions. If

The numerical results and the analytical solutions along

In the second example, a 3D advection-diffusion problem with source term is given as

The problem domain is

The analytical solution is

The EFG method is used to solve this example,

The comparison of the CPU time of the EFG and the IEFG methods under the same relative errors is shown in Table

Comparison of the CPU time of two methods under the same relative error.

Comparison | |||||
---|---|---|---|---|---|

0.1 s | 0.3 s | 0.5 s | 0.7 s | 0.9 s | |

Relative error (%) | 0.6156 | 0.8093 | 0.8109 | 0.8109 | 0.8109 |

CPU time of EFG (s) | 313.2 | 816.7 | 1338.4 | 1822.8 | 2325.8 |

CPU time of IEFG (s) | 303.9 | 785.3 | 1269.3 | 1763.8 | 2251.9 |

The numerical results and the analytical solutions along

The numerical results and the analytical solutions along

The numerical results and the analytical solutions along

Similarly, if

The numerical results and the analytical solutions along

In the third example, a 3D advection-diffusion equation is given as

The problem domain is

The analytical solution is

We set

The comparison of the CPU time of the EFG and the IEFG methods under the same relative errors is shown in Table

Comparison of the CPU time of two methods under the same relative error.

Comparison | |||||
---|---|---|---|---|---|

0.1 s | 0.3 s | 0.5 s | 0.7 s | 0.9 s | |

Relative error (%) | 0.1220 | 0.1402 | 0.1402 | 0.1402 | 0.1402 |

CPU time of EFG (s) | 366.2 | 588.2 | 781.4 | 987.1 | 1184.2 |

CPU time of IEFG (s) | 336.1 | 517.3 | 702.4 | 888.6 | 1064 |

The numerical results and the analytical solutions along

The numerical results and the analytical solutions along

The numerical results and the analytical solutions along

Similarly, if

The numerical results and the analytical solutions along

For this example, we can select different parameters,

The comparison of the CPU time of the EFG and the IEFG methods under the same relative errors is shown in Table

Comparison of the CPU time of two methods under the same relative error.

Comparison | |||||
---|---|---|---|---|---|

0.1 s | 0.3 s | 0.5 s | 0.7 s | 0.9 s | |

Relative error (%) | 0.5886 | 0.6330 | 0.6330 | 0.6330 | 0.6330 |

CPU time of EFG (s) | 100.4 | 175.1 | 249.1 | 324.7 | 398 |

CPU time of IEFG (s) | 98 | 170 | 246.6 | 316 | 393 |

The numerical results and the analytical solutions along

The numerical results and the analytical solutions along

The numerical results and the analytical solutions along

Similarly, if

The numerical results and the analytical solutions along

From this example, we can see that both diffusion dominated case and advection dominated one can be solved by using the EFG and the IEFG methods. Although two methods can obtain the solutions with similar computational accuracy, the computational speed of the IEFG is faster than the EFG method.

On the basis of the IMLS approximation, the IEFG method for 3D advection-diffusion problems is proposed in this paper.

In Section

From three examples, we can conclude that, compared with the EFG method for 3D advection-diffusion problems, the IEFG method has a higher computational efficiency.

The numerical solutions show that the IEFG method can avoid singular matrix when constructing the shape functions; then the deficiency of the EFG method is overcome.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (no. 11571223) and the Science and Technology Innovation Project of Shanxi Colleges and Universities (Grant no. 2020L0344).