The use of the classical finite element method (FEM) to solve problems with magnetic composites leads to huge linear systems that are impossible to solve. Instead, homogenization and multiscale methods are often used with the composite material replaced by a homogeneous material with the homogenized constitutive law obtained by solving cell-problems representing the mesoscale material structure. For non-linear time-dependent problems, FEM is often used with a time-transient method (TTM) and the solution is obtained one time-step at a time. However, in cases where a steady-state solution is of interest, the multiharmonic method can be faster and more cost effective for the same accuracy of the time discretization. In addition, when solving magnetoquasistatic multiscale problems with TTM, the dynamic hysteresis in the homogenized fields can slow down or even impede the convergence of the macro-scale problem due to the possibly non-continuously differentiable homogenized material laws. This work presents a novel robust modelling approach for non-linear magnetoquasistatic problems combining multiharmonic method with the multiscale method.

中文翻译：

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3D 非线性磁准静态中的多谐波多尺度建模：由绝缘颗粒制成的复合材料

使用经典的有限元方法（FEM）来解决磁性复合材料的问题会导致无法解决的巨大线性系统。相反，通常使用均质化和多尺度方法，用均质材料代替复合材料，并通过解决代表介观材料结构的细胞问题获得均质化本构定律。对于非线性时间相关问题，FEM 通常与瞬态方法 (TTM) 一起使用，并且一次一个时间步长地获得解。然而，在需要稳态解的情况下，对于相同的时间离散精度，多谐波方法可以更快且更具成本效益。此外，当用TTM解决磁准静态多尺度问题时，由于均匀化材料定律可能是非连续可微的，均匀化场中的动态磁滞会减慢甚至阻碍宏观尺度问题的收敛。这项工作提出了一种将多谐波方法与多尺度方法相结合的非线性磁准静态问题的新颖鲁棒建模方法。