一种基于磁通和电磁能流的磁扩散问题求解格式

doi:10.19596/j.cnki.1001-246x.8430

计算物理 ›› 2022, Vol. 39 ›› Issue (4): 379-385.DOI: 10.19596/j.cnki.1001-246x.8430

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一种基于磁通和电磁能流的磁扩散问题求解格式

严春晖¹^,²(), 肖波²^,*(), 王刚华², 陆禹¹^,², 李平²

1. 中国科学技术大学近代力学系, 安徽合肥 230027
2. 中国工程物理研究院流体物理研究所, 四川绵阳 621900

收稿日期:2021-07-28 出版日期:2022-07-25 发布日期:2022-11-17
通讯作者: 肖波
作者简介:
严春晖, 男, 博士研究生, 研究方向为磁流体力学, E-mail: ychui@mail.ustc.edu.cn
基金资助:
国家自然科学基金(12075226)

A Finite Volume Scheme Based on Magnetic Flux and Electromagnetic Energy Flow for Magnetic Field Diffusion Problems

Chun-hui YAN¹^,²(), Bo XIAO²^,*(), Gang-hua WANG², Yu LU¹^,², Ping LI²

1. Department of Modern Mechanics, University of Science and Technology of ChinaHefei, Anhui 230027, China
2. Institute of Fluid Physics, China Academy of Engineering Physics, Mianyang, Sichuan 621900, China

Received:2021-07-28 Online:2022-07-25 Published:2022-11-17
Contact: Bo XIAO

摘要/Abstract

摘要：

针对一维磁场扩散问题设计一种显式的有限体积离散格式。格式的第一个特征是不仅将磁场的扩散表达为单元边界上的磁通流, 同时将能量方程中的欧姆加热也表达为单元边界上的电磁能量流, 该特征能够更好地保证磁场能与内能的总量守恒。格式的第二个特征是将单元边界上的磁通量和电磁能通量进行截断, 在极端电阻率存在的磁扩散问题求解过程中, 该特征能够一定程度上放宽稳定性条件对显式格式时间步长的限制。

关键词: 非线性磁扩散, 极端电阻率, 欧姆加热, 有限体积法

Abstract:

An explicit finite volume discrete scheme is designed for one-dimensional magnetic field diffusion problems. The first characteristic of the scheme is that, the diffusion process of magnetic field is described as magnetic flux and the ohmic heating in energy equation is presented as electromagnetic energy flow at the element boundary as well, which guarantees the conservation of the sum of electromagnetic energy and internal energy well. The second characteristic of the scheme is that it truncates magnetic flux and electromagnetic energy flux on boundary of the element. Combined with time step amplification, the truncation could break through the limitation of stability condition on explicit scheme time step to some extent in magnetic diffusion problems with extreme resistivity.

Key words: nonlinear magnetic field diffusion, extreme resistivity, ohmic heating, finite volume scheme

严春晖, 肖波, 王刚华, 陆禹, 李平. 一种基于磁通和电磁能流的磁扩散问题求解格式[J]. 计算物理, 2022, 39(4): 379-385.

Chun-hui YAN, Bo XIAO, Gang-hua WANG, Yu LU, Ping LI. A Finite Volume Scheme Based on Magnetic Flux and Electromagnetic Energy Flow for Magnetic Field Diffusion Problems[J]. Chinese Journal of Computational Physics, 2022, 39(4): 379-385.

图/表 7

图1 相邻单元格间物理量参数

Fig.1 Physical factors in adjacent grid cells

图2 两种离散格式得到的1 μs时刻强磁扩散分布

Fig.2 Magnetic diffusion distributions at 1 μs calculated with two discretization schemes

表1 两种离散格式在初始和1 μs时刻的总磁通量与总能量

Table 1 Total magnetic flux and total energy at initial time and 1 μs calculated with two discretization schemes

	初始磁通量/(T·m)	1 μs时刻总磁通量/(T·m)	初始总能量/J	1 μs时刻总能量/J
节点控制体格式	0.1	0.1	3.978 87 × 10⁸	4.373 69 × 10⁸
本文计算格式	0.1	0.1	3.978 87 × 10⁸	3.978 87 × 10⁸

图3 恒定电阻率金属平板模型中，不同时间步长取值计算得到的0.5 μs时刻磁扩散分布

Fig.3 Magnetic diffusion distributions at 0.5 μs calculated with different time steps in a metal plate model of constant resistivity

图4 台阶电阻率模型(简化Burgess模型，电阻率ηL是初始电阻率ηS的2~5个量级。)

Fig.4 A step-shaped resistivity model (In a simplified Burgess model, resistivity ηL is 2-5 orders of magnitude of the initial resistivity ηs.)

图5 不同的时间步长计算得到的0.5 μs时刻峭面磁扩散分布

Fig.5 The sharp-front magnetic diffusion distributions at 0.5 μs calculated with different time steps

图6 两种极端不同常数电阻率区域的金属平板模型，不同时间步长得到的0.5 μs时刻磁扩散分布

Fig.6 Magnetic diffusion distributions at 0.5 μs calculated with different time steps in a metal plate model with two extremely different resistivity regions

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一种基于磁通和电磁能流的磁扩散问题求解格式

A Finite Volume Scheme Based on Magnetic Flux and Electromagnetic Energy Flow for Magnetic Field Diffusion Problems

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