复合材料中带有接触热阻热传导问题的非拟合网格法研究

Numerical simulations of the heat conduction in composite media are often encountered in many important engineering applications, such as the control of the heat transfer process between components of spacecraft in extreme environments, the design of high-performance thermal energy storage, and numerous others. In practical applications, the composite materials often do not contact well and there exist gaps between the contacting materials. The non-perfect conduct leads to thermal contact resistance effect which results in a discontinuity of the temperature across the interface. Unfitted meshes has attracted a lot of attention because that the meshes is easy to generate, independent of the interface, especially convenient for moving interface problems. For the classical second-order elliptic interface problems, there are a lot of unfitted mesh methods in the literature. However, to our knowledge, there seem few unfitted mesh methods for the interface problem with thermal contact resistance. In the project, we intend to design and analyze an unfitted mesh method for the heat conduction in composite media with thermal contact resistance. Our goal is that the unfitted mesh method is convergent and stable independenty of the interface and can solve the moving interface problem successfully.

复合介质中带有接触热阻热传导问题的数值模拟在实际工程中有很广泛的应用，例如极为恶劣环境下航天器各组件之间传热过程的控制、高性能热能储存材料的制造等等。在实际问题中，不同材料之间不可能充分接触，接触面之间存在着间隙。这样，不同材料的非充分接触导致了接触热阻的存在。最直接的物理现象是，温度在跨过交界面上产生跳跃。对于这样的界面问题，非拟合网格由于其网格生成简单、不依赖于界面且更易于处理移动界面问题而受到广泛关注。目前，文献中存在的非拟合网格方法主要针对于经典的二阶椭圆界面问题，而对于带有接触热阻问题少有研究。本课题拟研究复合材料中带有接触热阻热传导问题的非拟合网格法。我们的目标是使得设计出的非拟合网格法具有界面无关的收敛性和稳定性，并且能够成功处理移动界面问题。

本项目研究了带有接触热阻跳跃条件的稳态和非稳态等一系列界面问题的非拟合网格方法及其理论分析。对非齐次跳跃界面问题，我们提出基于水平集函数的高阶非拟合网格方法并且严格证明了该方法的最优H1和L2误差估计。对带有间断系数和非齐次跳跃问题，我们提出了部分罚浸入界面有限元方法，并且理论证明了该方法在能量范数下具有最优收敛性。对变系数界面问题，我们提出了一种基于增广技巧的浸入界面有限元方法。该方法中的增广变量通过GMRES迭代求解。数值实验表明GMRES迭代次数与网格尺寸无关并且该方法具有最优收敛精度。