In this study, a novel scaled coordinate transformation boundary element method (SCTBEM) is proposed to solve the transient heat transfer problem of three-dimensional (3D) functionally gradient materials. In order to compute the coefficient matrix only once when solving transient problems, the fundamental solution of Laplace operator is used to derive the boundary-domain integral equation. To maintain advantages of the boundary element method in dimensionality reduction, this study adopts the SCT technique proposed by Yu et al., to transform the domain integral into the boundary integral. With the aim of determining high precision heat flux, the dual interpolation technique is introduced for deriving integral equations only from the internal nodes of the surface element, which unifies the corner problem and achieves the coalescence of degrees of freedom. It is noteworthy that this study establishes the precise integration solution of the first order ordinary differential equation by means of Padé expansions without matrices inversion to improve the accuracy and efficiency of the solution. Numerical results show that both temperature and heat flux of 3D functionally gradient materials are highly accurate and stable, even for complex multi-connection models.

中文翻译：

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一种具有无反演 Padé 级数扩展的新型 SCTMEM，用于 FGM 中的 3D 瞬态传热分析


在本研究中，提出了一种新的缩放坐标变换边界元方法 （SCTBEM） 来解决三维 （3D） 功能梯度材料的瞬态传热问题。为了在求解瞬态问题时只计算一次系数矩阵，采用拉普拉斯算子的基本解推导边界域积分方程。为了保持边界元法在降维方面的优势，本研究采用 Yu 等人提出的 SCT 技术，将域积分转换为边界积分。为了确定高精度热通量，引入了对偶插值技术，仅从面单元的内部节点推导出积分方程，从而统一了角问题并实现了自由度的合并。值得注意的是，该文采用无矩阵反演的 Padé 展开方法建立了一阶常微分方程的精确积分解，以提高解的准确性和效率。数值结果表明，即使对于复杂的多连接模型，三维功能梯度材料的温度和热通量也具有很高的准确性和稳定性。