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Homotopy method of fundamental solutions for solving nonlinear heat conduction problems

机译：解决非线性热传导问题的基本解的同伦方法

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In this study, we propose a meshless and boundary-type numerical method, namely the homotopy method of fundamental solutions (HMFS), to solve the steady-state nonlinear heat conduction problems in two dimensions. The HMFS is composed by the homotopy analysis method (HAM) and the method of fundamental solutions (MFS). In the solution procedure, the Kirchhoff transformation is employed to transform the nonlinear governing partial differential equation into the Laplace equation with nonlinear boundary conditions. Sequentially, the HAM is applied to convert the Laplace equation with nonlinear boundary conditions into a sequence of the Laplace equation with linear boundary conditions, which can be solved by the MFS. In order to solve strongly nonlinear problems, a convergence control parameter is introduced to ensure the solution convergence of the prescribed sequence of problems. Several numerical experiments were carried out to validate the proposed method. In addition, a multiple-precision computing is performed to demonstrate the exponential convergence of the HMFS in both the spatial and homotopy coordinates for solving nonlinear heat conduction problems. Finally, bi-material and irregular-domain problems are also considered.
机译：在这项研究中，我们提出了一种无网格边界类型的数值方法，即基本解的同伦方法（HMFS），以解决二维的稳态非线性导热问题。 HMFS由同构分析方法（HAM）和基本解方法（MFS）组成。在求解过程中，采用Kirchhoff变换将非线性控制偏微分方程转换为具有非线性边界条件的Laplace方程。顺序地，使用HAM将具有非线性边界条件的拉普拉斯方程转换为具有线性边界条件的拉普拉斯方程的序列，这可以通过MFS求解。为了解决强非线性问题，引入了收敛控制参数以确保规定问题序列的解收敛。进行了几次数值实验以验证所提出的方法。此外，进行了多精度计算以证明HMFS在空间和同伦坐标中的指数收敛性，从而解决了非线性导热问题。最后，还考虑了双材料和不规则域问题。

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