Original articles
Three-dimensional transient heat conduction analysis by boundary knot method

https://doi.org/10.1016/j.matcom.2018.11.025Get rights and content

Abstract

This paper makes the first attempt to apply the boundary knot method (BKM), in conjunction with dual reciprocity technique, for the solution of three-dimensional transient heat conduction problems. The BKM is a meshless, integration-free, easy-to-program boundary-only numerical technique for high-dimensional problems. The first step of our strategy is to use the finite difference method for temporal derivative to convert the transient heat conduction equation into a nonhomogeneous modified Helmholtz equation. And then the corresponding nonhomogeneous problem is solved using the proposed BKM strategy in conjunction with dual reciprocity technique. Four benchmark numerical examples are investigated in detail, and the numerical results show that the present scheme has the merits of high accuracy, wide applicability, good stability, and rapid convergence and is appealing to solve 3D transient heat conduction problems.

Introduction

During machining processes, heat conduction strongly influences the tool performances such as the tool wear, life as well as process quality. Thus, it is very useful and important to have a clear understanding about the thermal behaviors of materials. For general transient heat conduction problems, it is difficult to find the corresponding analytical solutions. Therefore, the numerical simulation becomes an important and effective approach to obtain the approximation solution of such problems.

The conventional and popular numerical methods, such as the finite difference method (FDM) [2], [27], the finite element method (FEM) [3], [8], [29] and the boundary element method (BEM) [11], [26], [28], [33], can be applied to transient heat conduction problems. The first two are referred to domain discretization techniques and have long been dominant numerical techniques in the simulation of real-world engineering applications. However, these two methods require tedious domain meshing which is often computationally costly and sometimes mathematically troublesome, especially for high dimensional problems. As an alternative approach, the BEM has long been touted to avoid such drawbacks due to its boundary-only discretization and semi-analytical merit. The BEM, however, involves quite sophisticated mathematics and somewhat difficult numerical integration of singular kernel functions [1], [32]. Moreover, surface meshing in a 3D irregular domain remains a nontrivial task.

Over the past decade, some considerable effort was devoted to eliminating the need for meshing. This led to the development of meshless methods which require neither domain nor boundary mesh generation. The methods developed so far include the element-free Galerkin method [22], the reproducing kernel particle method [7], the meshless local boundary integral equation method [31], the meshless local Petrov–Galerkin method [19], [30], the radial basis function collocation method [4], [18], the generalized finite difference method [10], [20], the method of fundamental solutions (MFS) [35], [36], the singular boundary method [13], [21], [24], [34], the boundary particle method [14] and the collocation Trefftz method [23], just mention a few. Among these methods, the MFS has emerged as a boundary-only collocation method with the merit of easy-to-program, free-integration, high accuracy, and fast convergence.

In the traditional MFS [9], [25], a fictitious boundary outside the problem domain is required for placing the source points in order to avoid the singularity of fundamental solutions. Despite intense research, such a fictitious boundary of the source points remains largely based on experiences and therefore troublesome in engineering applications. To remedy this drawback, Chen and Tanaka [6] developed a modified MFS formulation, called boundary knot method (BKM). This method employs the nonsingular general solutions instead of the singular fundamental solutions to avoid the singularities, and therefore, circumvents the controversial fictitious boundary associated with the traditional MFS. In a more recent study, Fu et al. [12] introduced Laplace and Kirchhoff transformation into the BKM formulation for two-dimensional (2D) steady-state and transient heat conduction analysis of nonlinear functionally graded materials.

This paper applies the boundary knot method, in conjunction with dual reciprocity technique (DRM), to solve 3D transient heat conduction problems. A brief outline of the paper is as follows. The BKM in conjunction with dual reciprocity method for 3D transient heat conduction is described in Section 2, followed by Section 3 which illustrates the numerical efficiency and accuracy of the proposed approach through several benchmark examples. Finally, some conclusions are drawn on numerical observations in Section 4.

Section snippets

Transient heat conduction

Consider a 3D heat conduction equation which describes the unsteady temperature distribution in a solid with domain Ω and boundary Γ. The governing equation can be stated as k2T(x,t)+Q(x,t)=ρcT(x,t)t,x=x,y,zΩ,with Dirichlet/essential boundary condition T(x,t)=T¯(x,t),xΓD,Neumann/natural boundary condition kT(x,t)n=q¯(x,t),xΓN,and the initial condition T(x,0)=T0,xΩ,where ΓD and ΓN denote the boundaries where Dirichlet and Neumann conditions are applied, respectively; k is the thermal

Numerical examples

In this section, four benchmark numerical examples are examined to verify the applicability of the proposed strategy to 3D transient heat conduction problems. The measured errors are defined as Relativeerror=|TˆiTiTi|,Ti>103|TˆiTi|,Ti103, Rerr=1Ni=1N|TˆiTiTi|2,Ti>1031Ni=1N|TˆiTi|2,Ti103, Rerr2=1nj=1n|fˆjfjfj|2,fj>1031nj=1n|fˆjfj|2,fj103, where Ti and Tˆi are the analytical and numerical solutions at xi, respectively. N denotes the total number of test points in the domain

Conclusion

This paper documents the first attempt to apply the boundary knot method, in conjunction with dual reciprocity technique, for the solution of 3D transient heat conduction problems. As a meshless boundary collocation method, the proposed BKM is easy-to-implement, requires little data preparation. Unlike the BEM, it avoids the troublesome and costly singular integrations on the boundary. In a summary of overall performances, the present methodology agrees pretty well with the corresponding exact

Acknowledgments

The authors thank the anonymous reviewers of this article for their very helpful comments and suggestions to significantly improve the academic quality of this article. The work described in this paper was supported by the National Science Funds of China (Grant Nos. 11772119, 11572111), the Foundation for Open Project of State Key Laboratory of Structural Analysis for Industrial Equipment, China (Grant No. GZ1707), the 111 Project, China (Grant No. B12032), the Fundamental Research Funds for

References (37)

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