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Analytical solutions for anisotropic time-dependent heat equations with Robin boundary condition for cubic-shaped solid-state laser crystals

机译：具有Robin边界条件的立方固态激光晶体各向异性时变热方程的解析解

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摘要

The problem of finding analytical solutions for time-dependent or time-independent heat equations, especially for solid-state laser media, has required a lot of work in the field of thermal effects. However, to calculate the temperature distributions analytically, researchers often have to make some approximations or employ complex methods. In this work, we present full analytical solutions for anisotropic time-dependent heat equations in the Cartesian coordinates with various source terms corresponding to various pumping schemes. Moreover, the most general boundary condition of Robin (or impedance boundary condition), corresponding to the convection cooling mechanism, was applied. This general condition can be easily switched to constant temperature and thermal insulation as special cases. To this end, we first proposed a general approach to solving time-dependent heat equations with an arbitrary heat source. We then applied our approach to explore the temperature distribution for three cases: steady-state pumping or long transient, single-shot pumping or short transient, and repetitively pulsed pumping. Our results show the possibility of an easier and more accurate approach to analytical calculations of the thermal dispersion, thermal stresses (strains), thermal bending, thermal phase shift, and other thermal effects.
机译：为与时间相关或与时间无关的热方程，特别是固态激光介质，寻找解析解的问题在热效应领域中需要大量工作。但是，为了分析性地计算温度分布，研究人员常常不得不做出一些近似或采用复杂的方法。在这项工作中，我们提供了笛卡尔坐标系中各向异性时变热方程的完整解析解，其中各种源项对应于各种泵送方案。此外，应用了最常规的罗宾边界条件（或阻抗边界条件），对应于对流冷却机制。在特殊情况下，可以很容易地将此​​常规条件切换为恒温和隔热。为此，我们首先提出了一种使用任意热源求解与时间相关的热方程的通用方法。然后，我们应用我们的方法来探索三种情况下的温度分布：稳态泵浦或长瞬态，单次泵浦或短瞬态以及重复脉冲泵浦。我们的结果表明，可以采用一种更简便，更准确的方法来进行热扩散，热应力（应变），热弯曲，热相移和其他热效应的解析计算。

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