This work presents a data-driven magnetostatic finite-element solver that is specifically well-suited to cope with strongly nonlinear material responses. The data-driven computing framework is essentially a multiobjective optimization procedure matching the material operation points as closely as possible to given material data while obeying Maxwell's equations. Here, the framework is extended with heterogeneous (local) weighting factors - one per finite element - equilibrating the goal function locally according to the material behavior. This modification allows the data-driven solver to capture sharp gradients in the constitutive law of strongly nonlinear materials, which constitute problematic cases for standard data-driven solvers with a homogeneous (global) weighting factor, hindering their efficiency and accuracy. The local weighting factors are embedded in the distance-minimizing data-driven algorithm used for noiseless data, likewise for the maximum entropy data-driven algorithm used for noisy data. Numerical experiments based on a quadrupole magnet model with a soft magnetic material show that the proposed modification results in major improvements in terms of solution accuracy and solver efficiency. For the case of noiseless data, local weighting factors improve the convergence of the data-driven solver by orders of magnitude. When noisy data are considered, the convergence rate of the data-driven solver is doubled.

中文翻译：

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用于强非线性材料响应的数据驱动求解器

这项工作提出了一种数据驱动的静磁有限元求解器，它特别适合处理强非线性材料响应。数据驱动的计算框架本质上是一个多目标优化程序，在遵守麦克斯韦方程的同时，尽可能将材料操作点与给定的材料数据相匹配。在这里，框架扩展了异质（局部）加权因子 - 每个有限元素一个 - 根据材料行为局部平衡目标函数。这种修改允许数据驱动求解器捕获强非线性材料本构定律中的陡峭梯度，这对于具有均匀（全局）加权因子的标准数据驱动求解器构成了问题，阻碍了它们的效率和准确性。局部加权因子嵌入在用于无噪声数据的距离最小化数据驱动算法中，同样用于用于噪声数据的最大熵数据驱动算法。基于具有软磁材料的四极磁铁模型的数值实验表明，所提出的修改在求解精度和求解器效率方面产生了重大改进。对于无噪声数据，局部加权因子将数据驱动求解器的收敛性提高了几个数量级。当考虑噪声数据时，数据驱动求解器的收敛速度加倍。基于具有软磁材料的四极磁铁模型的数值实验表明，所提出的修改在求解精度和求解器效率方面产生了重大改进。对于无噪声数据，局部加权因子将数据驱动求解器的收敛性提高了几个数量级。当考虑噪声数据时，数据驱动求解器的收敛速度加倍。基于具有软磁材料的四极磁铁模型的数值实验表明，所提出的修改在求解精度和求解器效率方面产生了重大改进。对于无噪声数据，局部加权因子将数据驱动求解器的收敛性提高了几个数量级。当考虑噪声数据时，数据驱动求解器的收敛速度加倍。