非线性感应加热问题的全离散有限元方法

数学物理学报 ›› 2022, Vol. 42 ›› Issue (4): 1238-1255.

非线性感应加热问题的全离散有限元方法

黄媛¹,支越³,康彤^1,²,王然^1,^2,*(),张红⁴

¹ 中国传媒大学数据科学与智能媒体学院北京 100024
² 中国科学院大学计算地球动力学重点实验室北京 100049
³ 北京市育才学校通州分校北京 101101
⁴ 北京汇文中学朝阳垂杨柳分校北京 100021

收稿日期:2021-03-30 出版日期:2022-08-26 发布日期:2022-08-08
通讯作者: 王然 E-mail:wangr@ucas.ac.cn
基金资助:
国家重点研发计划项目(2020YFA0713401);国家自然科学基金项目(U2039207);国家自然科学基金项目(42074108);国家自然科学基金项目(41904067);中国传媒大学中央高校基本科研业务费专项资金

Fully Discrete Finite Element Scheme for a Nonlinear Induction Heating Problem

Yuan Huang¹,Yue Zhi³,Tong Kang^1,²,Ran Wang^1,^2,*(),Hong Zhang⁴

¹ School of Data Science and Intelligent Media, Communication University of China, Beijing 100024
² Key Laboratory of Computational Geodynamics, University of Chinese Academy of Sciences, Beijing 100049
³ Tongzhou Branch of Beijing Yucai School, Beijing 101101
⁴ Chaoyang Chuiyangliu Branch of Beijing Huiwen Middle School, Beijing 100021

Received:2021-03-30 Online:2022-08-26 Published:2022-08-08
Contact: Ran Wang E-mail:wangr@ucas.ac.cn
Supported by:
the National Key Research and Development Program of China(2020YFA0713401);the National Science Foundation of China(U2039207);the National Science Foundation of China(42074108);the National Science Foundation of China(41904067);the Fundamental Research Funds for the Central Universities

摘要/Abstract

摘要：

该文研究了一个感应加热模型, 该模型为Maxwell方程与热传导方程的耦合. 在感应加热模型中, 假设磁场与磁感应强度之间存在非线性关系, 电导率依赖于温度函数. 基于磁场H的形式, 给出时间和空间上的全离散格式, 讨论模型的可解性, 并证明全离散格式的解收敛于时间半离散的解, 而已有文献已经证明时间半离散的解收敛于连续变分问题的解. 最后, 使用数值实验去验证理论结果.

关键词: 感应加热, Maxwell方程, 非线性, 有限元, 收敛性

Abstract:

It studies an induction heating model described by Maxwell's equations coupled with a heat equation. In the induction heating model, it assumes a nonlinear relation between the magnetic field and the magnetic induction field, and the electric conductivity is temperature dependent. It presents a fully discrete H-based finite element scheme in time and space and discusses its solvability. Moreover, it proves the fully discrete solution converges to a weak solution of the continuous problem. Finally, theoretical results are supported by some numerical experiments.

Key words: Induction heating, Maxwell's equations, Nonlinear, Finite elements, Convergence

中图分类号:

O241.82

黄媛,支越,康彤,王然,张红. 非线性感应加热问题的全离散有限元方法[J]. 数学物理学报, 2022, 42(4): 1238-1255.

Yuan Huang,Yue Zhi,Tong Kang,Ran Wang,Hong Zhang. Fully Discrete Finite Element Scheme for a Nonlinear Induction Heating Problem[J]. Acta mathematica scientia,Series A, 2022, 42(4): 1238-1255.

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参考文献 21

1	Akrivis G , Larsson S . Linearly implicit finite element methods for the time dependent Joule heating problem. BIT Numerical Mathematics, 2005, 45: 429- 442 doi: 10.1007/s10543-005-0008-1
2	Barglik J , Doležel I , Karban P , et al. Modelling of continual induction hardening in quasi-coupled formulation. Compel-the International Journal for Computation and Mathematics in Electrical and Electronic Engineering, 2005, 24 (1): 251- 260 doi: 10.1108/03321640510571273
3	Bermúdez A , Gómez D , Muniz M C , et al. Transient numerical simulation of a thermoelectrical problem in cylindrical induction heating furnaces. Advances in Computational Mathematics, 2007, 26: 39- 62 doi: 10.1007/s10444-005-7470-9
4	Bień M . Global solutions of the non-linear problem describing Joule's heating in three space dimensions. Mathematical Methods in the Applied Sciences, 2005, 28 (9): 1007- 1030 doi: 10.1002/mma.559
5	Bossavit A , Emson C , Mayergoyz I D . Méthodes Numériques en Électromagnétisme. Paris: Eyrolles, 1991
6	Chovan J , Geuzaine C , Slodička M . $\bm{A}$-$\phi$ formulation of a mathematical model for the induction hardening process with a nonlinear law for the magnetic field. Computer Methods in Applied Mechanics and Engineering, 2017, 321: 294- 315 doi: 10.1016/j.cma.2017.03.045
7	Elliott C M , Larsson S . A finite element model for the time-dependent Joule heating problem. Mathematics of Computation, 1995, 62 (212): 1433- 1453
8	Evans L C. Graduate Studies in Mathematics: Partial Differential Equations. Second ed. New York: American Mathematical Society, 2010
9	Hömberg D . A mathematical model for induction hardening including mechanical effects. Nonlinear Analysis: Real World Applications, 2004, 5 (1): 55- 90 doi: 10.1016/S1468-1218(03)00017-8
10	Hömberg D , Petzold T , Rocca E . Analysis and simulations of multifrequency induction hardening. Nonlinear Analysis: Real World Applications, 2005, 22: 84- 97
11	Kang T , Wang R , Zhang H . Potential field formulation based on decomposition of the electric field for a nonlinear induction hardening model. Communications in Applied Mathematics and Computational Science, 2019, 14 (2): 175- 205 doi: 10.2140/camcos.2019.14.175
12	Kang T , Wang R , Zhang H . Fully discrete $\bm{T}$-$\psi$ finite element method to solve a nonlinear induction hardening problem. Numerical Methods for Partial Differential Equations, 2021, 37: 546- 582 doi: 10.1002/num.22540
13	Nečas J . Introduction to the Theory of Nonlinear Elliptic Equations. Chichester: John Wiley & Sons Ltd, 1986
14	Nédélec J C . Mixed finite elements in $R^3$. Numerische Mathematick, 1980, 35: 315- 341 doi: 10.1007/BF01396415
15	Roubíček T . Nonlinear Partial Differential Equations with Applications. Berlin: Birkhöuser, 2005
16	Slodička M , Chovan J . Solvability for induction hardening including nonlinear magnetic field and controlled Joule heating. Applicable Analysis, 2017, 96: 2780- 2799 doi: 10.1080/00036811.2016.1243661
17	Sun D , Manoranjan V S , Yin H M . Numerical solutions for a coupled parabolic equations arising induction heating processes. Conference Publications, 2007, 2007 (Special): 956- 964
18	Vajnverg M . Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations. New York: John Wiley, 1973
19	Yin H M . On a nonlinear Maxwell's system in quasi-stationary electromagnetic fields. Mathematical Models and Methods in Applied Sciences, 2004, 14 (10): 1521- 1539 doi: 10.1142/S0218202504003787
20	Yin H M . Regularity of weak solution to Maxwell's equations and applications to microwave heating. Journal of Differential Equations, 2004, 200 (1): 137- 161 doi: 10.1016/j.jde.2004.01.010
21	Yin H M , Wei W . Regularity of weak solution for a coupled system arising from a microwave heating model. European Journal of Applied Mathematics, 2014, 25: 117- 131 doi: 10.1017/S0956792513000326

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