Time Series Analysis Techniques for Transient Electro- and Magneto-Quasistatic Field Simulations

该研究项目的目的是发展基于时间序列分析技术(信息论、非线性动力学、统计方法)的方法，以建立空间离散的非线性暂态电-拟静电场和磁准静电场问题的改进降阶模型(ROM)。基于空间离散方法(如有限元方法)的暂态电场准静电场模拟被用于电力传输系统的设计，特别是在具有非线性电导特性的电场应力分级材料的情况下。在电机、磁致动器、感应功率传输和加热装置等机电或电热能量转换系统的设计过程中，通常使用有限元模拟来计算瞬变磁准静电场问题，也称为涡流问题。在这里，铁磁材料的行为通常会导致所涉及的模型的非线性。由于通常情况下，随时间变化并受非线性规律支配的现象可能表现出复杂的、非系统的时间演化，因此，由于一些现场信号的弱可分离性，使用离散经验插值法(DEIM)内的奇异值分解获得的通常使用的简化基可能会失效，即用于非线性问题的模型降阶的固有正交分解(POD)方法的变体。因此，需要开发新的模型降阶方法，这些方法基于信息几何和统计的信息熵和散度概念，用作(相对)信息量的度量。这样的方法将涉及开发非最优贪婪内插节点选择的替代方案，该非最优贪婪内插节点选择通常用作诸如DEIM的非线性模型降阶方法的一部分，并且在许多情况下产生不符合标准的、即不可靠的降阶模型。此外，这项研究的目的是在有限元模拟的时间离散电磁场和磁场准静态场解的暂态信号分析中，探索新引入的策略的可能性，这些策略源于非线性动态系统理论。这项研究的重点是有效的快照选择，将空间离散化的场问题自动区域分解为仅具有线性或弱或强非线性解行为的子区域，以及可能的自适应空间离散化方案的网格加密技术。

The goal of this research project is to develop methods based on time series analysis techniques (information theory, nonlinear dynamics, statistical methods) in order to manufacture improved reduced order models (ROM) for spatially discretized nonlinear transient electro-quasistatic and magneto-quasistatic field problems. Transient electro-quasistatic field simulations based on spatial discretization schemes as e.g. the finite element method (FEM) are used for the design of electric power transmission systems, especially when featuring electric field stress grading materials with nonlinear electrical conductivity characteristics. Transient magneto-quasistatic field problems, also dubbed as eddy current problems, are often numerically calculated using FEM simulations within the design process of electro-mechanical or electro-thermal energy conversion systems such as e.g. electric machines, magnetic actuators, inductive power transfer and heating devices. Here, ferromagnetic material behavior typically contributes to the nonlinearity of the models involved. Since, in general, phenomena that vary with time and are governed by nonlinear laws may exhibit a complex, unsystematic time evolution, the commonly employed reduced basis that is obtained using a singular value decomposition within the discrete empirical interpolation method (DEIM), i.e., a variant of the proper orthogonal decomposition (POD) method used for the model order reduction of nonlinear problems, may fail due to the the weak separability of some of the field signals. Hence, novel model order reduction methods need to be developed that are based on information geometrical and statistical notions of entropy and divergence, to be used as measures of (relative) information content. Such an approach will involve developing alternatives to the non-optimal greedy interpolation node selection that is commonly used as part of nonlinear model reduction methods such as DEIM and in many cases generates substandard, i.e., non-reliable reduced order models. Furthermore, this research project aims at exploiting the possibilities of the newly introduced strategies with their origin in nonlinear dynamic system theory within the transient signal analysis of time-discrete electro- and magneto-quasistatic field solutions of FEM simulations. The focus of this research aims at an effective snapshot selection, an automated domain decomposition of the spatially discretized field problem into subregions with only linear or weak or strongly non-linear solution behavior as well as possible mesh refinement techniques for adaptive spatial discretization schemes.