This paper first attempts to solve the transient heat conduction problem by combining the recently proposed local knot method (LKM) with the dual reciprocity method (DRM). Firstly, the temporal derivative is discretized by a finite difference scheme, and thus the governing equation of transient heat transfer is transformed into a non-homogeneous modified Helmholtz equation. Secondly, the solution of the non-homogeneous modified Helmholtz equation is decomposed into a particular solution and a homogeneous solution. And then, the DRM and LKM are used to solve the particular solution of the non-homogeneous equation and the homogeneous solution of the modified Helmholtz equation, respectively. The LKM is a recently proposed local radial basis function collocation method with the merits of being simple, accurate, and free of mesh and integration. Compared with the traditional domain-type and boundary-type schemes, the present coupling algorithm could be treated as a really good alternative for the analysis of transient heat conduction on high-dimensional and complicated domains. Numerical experiments, including two- and three-dimensional heat transfer models, demonstrated the effectiveness and accuracy of the new methodology.

Transient heat conduction exists widely in the thermal structure design of industrial equipment such as energy and chemical industry, as well as in the casting and heat treatment process of many new materials [

There are many kinds of conventional numerical methods applied to transient heat conduction problems, for instance, the finite element method (FEM) [

In order to get rid of the complexity of mesh generation and reduce the time of preprocessing, various meshless methods have devoted considerable attention. These approaches include the element-free Galerkin method [

For the above reasons, a new class of meshless collocation techniques, called the localized semi-analytical meshless methods, has been proposed to solve various mathematical and mechanical problems. Such methods mainly include the localized method of fundamental solution (LMFS) [

Very recently, an improved LBKM called the local knot method (LKM) [

Firstly, the temporal derivative is discretized by using the finite difference scheme. The governing equation is transformed into a non-homogeneous modified Helmholtz equation. Secondly, the solution of transformed equation is decomposed into a particular solution and a homogeneous solution of the corresponding homogeneous equation. Finally, the particular solution and the homogeneous solution are obtained by the dual reciprocity method (DRM) [

The outline of this article is as follows.

A bounded domain

The transient heat conduction problem is transformed into the modified Helmholtz equation by using the finite difference scheme to deal with the time variable. For the time span

Substituting

It should be pointed out that the numerical solution

For convenience,

Noted that

The particular solution only satisfies the non-homogeneous equation, rather than boundary condition. However, the homogeneous solution satisfies both homogeneous equation and corresponding boundary conditions. The next two subsections will introduce the dual reciprocity method and the LKM for approximating particular solution and homogeneous solution, respectively.

Obviously, the particular solution

This study uses the dual reciprocity method to approximate the particular solution

Accordingly, the particular solution

As we all know, there are many kinds of radial basis functions. The present study adopts the following function [

Substituting

In this subsection, the LKM is used to approximate the homogeneous solution

In the LKM,

Using the moving least squares theory, we define the following residual function:

The unknown coefficients

Then, a matrix equation can be obtained,

Further, the vector

In terms of

Let

In addition, the normal derivative can be calculated by

At internal nodes and boundary nodes, the following equations should be satisfied:

By combining

Three numerical examples are investigated to test the accuracy and effectiveness of the proposed scheme. To measure numerical accuracy, the following errors are adopted:

In the first example, we consider a transient heat transfer problem in a rectangular plate with a hole [

In the calculation, we set

Then, we consider the numbers of supporting points as

The LKM is applied to analyze the temperature field of the cubic domain in the second example. The shape parameters and initial temperature of the cube are shown in

The analytical solution is given by

In this example, we consider two cases: (a) All the boundaries satisfy the Dirichlet boundary conditions; (b) The boundary

In this example, the number of the supporting nodes is 60, and the time interval is

Additionally,

By comparing the results of the LKM and the BKM in different cases, the accuracy and convergence of this method are carefully validated. With the increase of the total number of nodes, the variation of the errors and the condition numbers of the two methods are listed in

N | Case 1 | Case 2 | ||||
---|---|---|---|---|---|---|

Max error | Global error | Cond (A) | Max error | Global error | Cond (A) | |

896 | 4.769 ^{−2} |
1.721 ^{−2} |
1.756E + 03 | 4.070 ^{−1} |
7.703 ^{−2} |
1.647E + 06 |

1215 | 3.417 ^{−2} |
1.112 ^{−2} |
3.076E + 03 | 3.295 ^{−1} |
5.454 ^{−2} |
8.672E + 04 |

3211 | 1.525 ^{−2} |
3.919 ^{−3} |
6.361E + 03 | 1.522 ^{−1} |
2.155 ^{−2} |
2.689E + 05 |

4725 | 1.475 ^{−2} |
3.614 ^{−3} |
2.768E + 04 | 1.045 ^{−1} |
1.395 ^{−2} |
5.691E + 06 |

6647 | 1.387 ^{−2} |
3.513 ^{−3} |
3.170E + 04 | 7.214 ^{−2} |
9.326 ^{−3} |
4.340E + 06 |

N | Case 1 | Case 2 | ||||
---|---|---|---|---|---|---|

Max error | Global error | Cond (A) | Max error | Global error | Cond (A) | |

896 | 4.578 ^{−2} |
1.598 ^{−2} |
1.426E + 18 | 2.475 ^{−1} |
4.402 ^{−2} |
1.245E + 20 |

1215 | 3.369 ^{−2} |
1.058 ^{−2} |
1.391E + 18 | 2.196 ^{−1} |
3.488 ^{−2} |
6.790E + 19 |

3211 | 1.713 ^{−2} |
4.014 ^{−3} |
8.996E + 17 | 1.711 ^{−1} |
1.616 ^{−2} |
5.385E + 20 |

4725 | 2.037 ^{−2} |
4.179 ^{−3} |
3.540E + 18 | 2.561 ^{−1} |
1.682 ^{−2} |
5.342E + 21 |

6647 | 1.710 ^{−2} |
4.477 ^{−3} |
2.048E + 19 | 1.390 ^{−1} |
1.164 ^{−2} |
1.849E + 21 |

Finally, we can observe from ^{−2}), and gradually decreases from the center to the boundaries. In case 2, we can find that the absolute error on the boundary ^{−2}), and decreases sharply from it to other boundaries. The above phenomenon is mainly caused by different boundary conditions.

In order to verify the proposed LKM for simulating 3D transient heat conduction in a multi-connected domain, the third example considers is a hollow cylinder.

In this case, the number of supporting nodes is set to be 60. The length of each time step is

This paper presented a novel coupling algorithm of the LKM and the dual reciprocity method to simulate the transient heat transfer numerically. The LKM is a semi-analytical and local meshless method which uses the non-singular general solution as the basis function of interpolation. Unlike the existing conventional methods, the coupled approach avoids mesh division, reduces condition number and expands the simulation scale.

Numerical results of the three benchmark examples involving simply-connected and multiply-connected domains have been investigated in detail. In the examples without exact solutions, our numerical results are compared with the FEM results from COMSOL software. Numerical experiments indicate that the proposed scheme is accurate and effective, and has small condition numbers.

This paper focuses on the application of the LKM in transient heat transfer problems, and shows its superior performance in solving such problems. However, there are still some issues that need to be addressed. The present study uses the Euler formula to discretize the time derivative, which has a certain impact on the accuracy and efficiency of the present method. Some more complex and accurate methods addressing the time derivative are interesting and will be discussed in more detail subsequently. Furthermore, the proposed methodology can be further optimized and improved to accurately solve the high-dimensional nonlinear heat transfer as well as the coupling heat transfer in complicated geometry.