On the boundary conditions of magnetic field in OpenFOAM and a magnetic field solver for multi-region applications

This work begins with the discretization of the governing equations of magnetic field for different type of magnetic media over a two-dimensional rectangular domain using a Cartesian grid system. The basic rules that the coefficients of the discretization equation should obey, to ensure physical realism and overall balance, are discussed. In order to satisfy the rule, “consistency at control-volume faces,” the formulation of the permeability at the interface between two adjacent control volumes is derived. It reveals that the interface permeability is equal to the anti-linear interpolation of that on the cell centroids under the assumption of flux consistency at control-volume faces. For the purpose of effectively calculating magnetic fields using the FVM based OpenFOAM toolbox, the representations of boundary conditions on the interfaces between different type of media are for the first time derived. A C++ class named magPhiMixedFvPatchField describing the newly derived boundary conditions is defined based on OpenFOAM libraries. The new open-source solver labelled as magneticMultiRegionFoam, for the simulation of static magnetic field with multi-regions, based on FVM in the OpenFOAM framework is for the first time developed and validated. Comparison of the results of the predicted magnetic field for a cylindrical magnet in air with that obtained from Comsol software and the analytical method verifies the fidelity of the solver implementation. For the case of calculating the magnetic levitation forces exerted on a magnet immersed in ferrofluid, the results obtained from magneticMultiRegionFoam show better consistency with the experimental ones than Comsol software.本研究首先采用笛卡尔网格系统，在二维矩形计算域内对不同类型磁性介质的磁场控制方程开展离散化处理。为保障物理真实性与整体守恒性，本文探讨了离散化方程系数需遵循的基本准则。为满足「控制体面通量一致性」准则，本文推导了相邻两控制体界面处的磁导率表达式。研究表明，在控制体面通量一致性假设下，界面磁导率等于单元形心处磁导率的逆线性插值结果。针对基于有限体积法（Finite Volume Method, FVM）的OpenFOAM工具箱开展高效磁场计算的需求，本文首次推导了不同介质界面处的边界条件表达式。基于OpenFOAM库，本文定义了名为magPhiMixedFvPatchField的C++类，用于描述上述新推导的边界条件。本文首次基于OpenFOAM框架下的有限体积法，开发并验证了一款用于多区域静磁场仿真的开源求解器magneticMultiRegionFoam。将空气中圆柱形永磁体的磁场仿真结果与Comsol软件及解析法所得结果对比，验证了该求解器实现的保真度。针对浸没于铁磁流体中的永磁体所受磁悬浮力的计算场景，magneticMultiRegionFoam的计算结果相较Comsol软件，与实验数据的一致性更优。