Original articlesThree-dimensional transient heat conduction analysis by boundary knot method
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 and boundary . The governing equation can be stated as with Dirichlet/essential boundary condition Neumann/natural boundary condition and the initial condition where and denote the boundaries where Dirichlet and Neumann conditions are applied, respectively; 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 where and are the analytical and numerical solutions at , respectively. 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
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