Abstract A hybrid numerical-analytical methodology based on integral transforms is employed for solving transient three-dimensional heat conduction problems in heterogeneous media comprising arbitrarily space variable thermophysical properties, including configurations with multiple subdomains of different materials and geometries. The Generalized Integral Transform Technique (GITT) is used to solve for both the temperature distribution and the corresponding eigenvalue problem accounting for the spatially variable coefficients. A previously introduced single domain reformulation strategy is adopted when multiple subdomains are involved, so as to rewrite the heat conduction equation in terms of coefficients with abrupt variations in the spatial coordinates. The hybrid numerical-analytical approach is demonstrated for three classes of problems in which the thermophysical properties undergo significant variations throughout the domain, such as: (i) FGMs (Functionally Graded Materials) with three-dimensional space variable thermophysical properties; (ii) composite medium with inclusions of different geometries and thermophysical properties; (iii) a multi-scale/multi-material heat conduction problem in a IGBT (Insulated Gate Bipolar Transistor) module. In each considered application, both the temperature eigenfunction expansions and the associated eigenvalue problem solutions are critically analyzed in terms of convergence rates and co-verified with purely numerical solutions.

中文翻译：

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异质介质中的瞬态三维热传导：积分变换和单域公式

摘要 采用基于积分变换的混合数值分析方法来解决非均质介质中的瞬态三维热传导问题，包括任意空间可变热物理特性，包括具有不同材料和几何形状的多个子域的配置。广义积分变换技术 (GITT) 用于解决温度分布和相应的特征值问题，以解决空间可变系数。当涉及多个子域时，采用先前引入的单域重构策略，以便根据空间坐标突变的系数重写热传导方程。混合数值分析方法针对热物理特性在整个域中发生显着变化的三类问题进行了演示，例如：（i）具有三维空间可变热物理特性的 FGM（功能梯度材料）；(ii) 具有不同几何形状和热物理性质的夹杂物的复合介质；(iii) IGBT（绝缘栅双极晶体管）模块中的多尺度/多材料热传导问题。在每个考虑的应用中，温度特征函数展开和相关的特征值问题的解决方案都在收敛速度方面进行了批判性分析，并与纯数值解决方案共同验证。(i) 具有三维空间可变热物理特性的 FGM（功能梯度材料）；(ii) 具有不同几何形状和热物理性质的夹杂物的复合介质；(iii) IGBT（绝缘栅双极晶体管）模块中的多尺度/多材料热传导问题。在每个考虑的应用中，温度特征函数展开和相关的特征值问题的解决方案都在收敛速度方面进行了批判性分析，并与纯数值解决方案共同验证。(i) 具有三维空间可变热物理特性的 FGM（功能梯度材料）；(ii) 具有不同几何形状和热物理性质的夹杂物的复合介质；(iii) IGBT（绝缘栅双极晶体管）模块中的多尺度/多材料热传导问题。在每个考虑的应用中，温度特征函数展开和相关的特征值问题的解决方案都在收敛速度方面进行了批判性分析，并与纯数值解决方案共同验证。