This paper applies a semi-analytical boundary collocation method, the singular boundary method (SBM), in conjunction with the dual reciprocity method (DRM) and Laplace transformation technique to solve anomalous heat conduction problems under functionally graded materials (FGMs). In this study, transient heat conduction equation with Caputo time fractional derivative is considered to describe anomalous heat conduction phenomena. In the present numerical implementation, Laplace transformation and numerical inverse Laplace transformation are used to deal with the Caputo time fractional derivative, which avoid the effect of time step on the computational efficiency of the time fractional derivation approximation. The SBM in conjunction with the DRM is employed to obtain the high accurate results in the solution of Laplace-transformed time-independent nonhomogeneous problems. To demonstrate the effectiveness of the proposed method for anomalous heat conduction analysis under functionally graded materials, three numerical examples are considered and the present results are compared with known analytical solutions and COMSOL simulation.

本文应用半解析边界配置方法，奇异边界方法（SBM），对偶互易方法（DRM）和拉普拉斯变换技术来解决功能梯度材料（FGM）下的异常导热问题。在本研究中，考虑了具有Caputo时间分数导数的瞬态热传导方程来描述异常热传导现象。在本数值实现中，使用拉普拉斯变换和数值拉普拉斯逆变换来处理Caputo时间分数导数，避免了时间步长对时间分数导数近似的计算效率的影响。SBM与DRM结合使用，可在拉普拉斯变换的时间无关的非齐次问题的求解中获得高精度的结果。为了证明所提出的方法在功能梯度材料下进行异常导热分析的有效性，考虑了三个数值示例，并将这些结果与已知的解析解和COMSOL仿真进行了比较。