具有非理想界面扩散方程的有限体积格式

The interface problems of the diffusion equation are widely appeared in the multi medium heat transfer problem. When the contact between different media is imperfect, that is, thin film or thin interlayer with weak conductivity between different media, thin interlayer can be regarded as a curve and surface without thickness. On the interface, the heat flux is continuous and the temperature is discontinuous. The amount of temperature jump is unknown and is proportional to the heat flux across the interface. This condition has a clear physical meaning and strong nonlinear characteristics. It brings great challenges to design numerical schemes and solve the problems. As we know, the related numerical methods are relatively few. In view of the classical finite volume schemes for diffusion equation are mostly designed for the perfect interface problems, which can’t be applied to problems with imperfect interface. Therefore, this project first establishes a nine points scheme to solve the diffusion equation with imperfect interface, such that it can effectively capture the temperature jump on both sides of the interface and satisfy the interface connection condition. In addition, the monotone finite volume scheme will be further studied to ensure that the numerical solution is non-negative on the distorted mesh, and the non physical oscillation will be suppressed. Finally, the proposed scheme is applied to the numerical simulation of the heat transfer process of composite materials with imperfect interface problems. This research has important theoretical significance and application value in the field of material science, multiphase flow and other engineering fields.

扩散方程的界面问题广泛出现在多介质传热问题中。当不同介质间的接触为非理想时，即不同介质间存在弱传导性的薄膜或薄夹层，可以将薄层看作无厚度的曲线或曲面。在非理想界面上满足热通量连续，温度间断的特点，且温度的跳跃量未知并与穿过界面的热通量成比例。该条件具有明确的物理意义及强非线性特征，对数值格式的设计和求解带来不小的挑战，相关数值方法的研究报道也较为少见。鉴于扩散方程的经典有限体积格式大多针对理想界面问题，不能适用于非理想界面问题且不满足单调性。因此，本项目首先建立具有非理想界面扩散方程在扭曲网格上的九点格式，使其能够有效捕捉界面两侧温度的跳跃，且满足界面连接条件。此外，进一步研究单调有限体积格式，使其在扭曲网格上保证数值解非负，且抑制非物理振荡。最后，将所研究的格式应用于复合材料具有非理想界面问题的传热过程的数值模拟。本研究对材料科学，多相流等问题都具有重要的理论意义和实际应用价值。

扩散方程的界面问题广泛出现在多介质传热问题中。当不同介质间的接触为非理想时，即不同介质间存在弱传导性的薄膜或薄夹层，可以将薄层看作无厚度的曲线或曲面。在非理想界面上满足热通量连续，温度间断的特点，且温度的跳跃量未知并与穿过界面的热通量成线性比例关系。该条件具有明确的物理意义对数值格式的设计和求解带来不小的挑战，相关数值方法的研究报道较为少见。鉴于扩散方程的经典有限体积格式大多针对理想界面问题，不能适用于非理想界面问题且不满足单调性。因此，本项目研究单调有限体积格式，使其能够有效捕捉界面两侧温度的跳跃，在扭曲网格上保证数值解非负，且抑制非物理振荡。首先，本项目针对具有非理想线性界面条件的扩散方程，基于贴体网格上构造了单调有限体积格式，该格式能够有效捕捉界面两侧的物理量跳跃，具有二阶精度且具有局部守恒性和单调性。然而，当界面几何复杂或者界面存在运动时生成贴体网格需要花费更多的时间。本研究采用浸入界面方法构造有限差分格式求解具有非理想界面的扩散方程。浸入界面方法允许界面几何任意切割网格，能够适用于任意复杂界面几何问题。本项目分别针对一维和二维具有非理想界面的扩散方程，提出了能够有效处理隐式线性界面连接条件，捕捉界面间断且具有二阶精度的数值格式，并对格式的收敛性稳定性等给出了理论分析。本项目所研究的数值格式可以应用于复合材料具有非理想界面问题的传热过程的数值模拟。项目研究成果对材料科学，多相流等问题都具有重要的理论意义和实际应用价值。