电磁场问题的建模与计算  

Mathematical modeling and computations for electromagnetic problems

作　　者：江雪[1] 

机构地区：[1]北京邮电大学理学院,北京100876

出　　处：《中国科学：数学》2016年第3期247-264,共18页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11401040);中央高校基本科研业务费专项资金(批准号:24820152015RC17)资助项目

摘　　要：三维电磁学涡流问题在大型变压器构件的铁损计算和磁通量模拟中具有重要应用.变压器中的取向硅钢片是多尺度结构且大小尺度之比达到10~6,对大型电磁设备直接求解三维非线性Maxwell方程组非常困难且不现实.本文分别基于磁矢位和磁场,对拟稳态Maxwell方程组提出两个宏观均匀化模型;建立均匀化模型的适定性;并且证明当硅钢片厚度趋于零时,微观问题的解在H(curl,?)中弱收敛且在L^2(?)中强收敛到宏观问题的解.另外还提出一个模拟硅钢片内三维涡流分布的新数学模型,该模型去掉了硅钢片的外覆绝缘漆,将系统大小尺度之比减小了3个数量级,从而可以直接模拟硅钢片结构的三维涡流分布.针对时变非线性涡流问题,本文建立新数学模型解的适定性理论;并且证明,当绝缘漆膜厚度趋于零时,新模型的解强收敛于原问题的精确解.通过对国际计算电磁学会公开基准问题TEAM Workshop Problem 21~c-M1进行数值模拟,验证了均匀化模型和新涡流模型的正确性和有效性.Three-dimensional eddy current problems are widely used in the iron loss computations and the magnetic flux simulation of large power transformers. GO silicon steel laminations in large transformers have multiple scales and the ratio of the largest scale to the smallest scale can be up to 10~6. Direct solution of threedimensional nonlinear Maxwell＇s equations is very challenging and unrealistic for large electromagnetic devices.Based on the magnetic vector potential and the magnetic field respectively, we propose two macro-scale models for the quasi-static Maxwell＇s equations. We establish the wellposedness of the homogenized model, and prove that the micro-scale solutions converge to the solutions of the macro-scale models weakly in H（curl, Ω） and strongly in L^2（Ω） as the thickness of lamination tends to zero. We also propose a new eddy current model for the nonlinear Maxwell＇s equations with laminated conductors. The new model omits coating films and thus reduces the scale ratio by 3 orders of magnitude. We establish the well-posedness of the new model and prove the convergence of the solution of the original problem to the solution of the new model as the thickness of coating films tends to zero.Both the macro-scale model and the new model are validated by finite element computations of an engineering benchmark problem—TEAM Workshop Problem 21~c-M1.

关 键 词：MAXWELL方程组 非线性涡流问题 有限元方法 

分 类 号：O241.82[理学—计算数学] O441[理学—数学]