The dual reciprocity boundary element formulation for nonlinear diffusion problems |
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Affiliation: | 1. College of Civil Engineering and Architecture, East China Jiaotong University, Nanchang, Jiangxi, 330013, PR China;2. Department of Computational Science and Statistics, Nantong University, Nantong, Jiangsu, 226019, PR China;3. School of Civil Engineering and Urban Construction, Jiujiang University, Jiujiang, Jiangxi, 332005, PR China;1. State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, China;2. School of Mathematics and Statistics, Central South University, Changsha 410083, China |
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Abstract: | This paper presents an extension of the dual reciprocity boundary element method (DRBEM) to deal with nonlinear diffusion problems in which thermal conductivity, specific heat, and density coefficients are all functions of temperature. The DRBEM, recently applied to the solution of problems governed by parabolic and hyperbolic equations, consists in the transformation of the differential equation into an integral equation involving boundary integrals only, the solution of which is achieved by employing a standard boundary element discretization coupled with a two-level finite difference time integration scheme. Contrary to previous formulations for the diffusion equation, the dual reciprocity BEM utilizes the well-known fundamental solution to Laplace's equation, which is space-dependent only. This avoids complex time integrations that normally appear in formulations employing time-dependent fundamental solutions, and permits accurate numerical solutions to be obtained in an efficient way. For nonlinear problems, the integral of conductivity is introduced as a new variable to obtain a linear diffusion equation in the Kirchhoff transform space. This equation involves a modified time variable which is itself a function of position. The problem is solved in an iterative way by using an efficient Newton-Raphson technique which is shown to be rapidly convergent. |
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