A method for compact modeling linear time-invariant structures of high frequency engineering on the level of electromagnetic fields

电磁场频域仿真的数值方法，如有限元（FE）方法，可以在低误差水平下分析高频工程的线性时不变系统，并在建模色散材料特性方面提供高度的灵活性。然而，它们导致了大规模的代数系统和高计算成本。模型阶约简（MOR）方法提供了一种将有限元系统转换为低维模型的系统方法，这种低维模型具有中等内存需求、较短的求解时间和可控的误差。在项目第一阶段开发的MOR方法在低频状态下表现出稳定的行为，并且能够处理由频率相关横向场模式的波导模式激发的系统。忽略波导馈电中的损失，该方法成功地构建了分布参数系统、其FE离散化以及由此产生的降阶系统（ROM），从而在整个建模链中可证明地保留了基本系统属性；这包括被动性、因果性、稳定性、互惠性和能量。这项更新建议在以下方面扩展了所建议的方法：由于许多应用程序需要时域模拟，因此第一个目标是对ROM进行转换，ROM最初位于频域。由于其保存特性，可以省略竞争性方法所需的事后钝化。此外，ROM的定义良好的结构使在非限制性假设下实现时域描述符或状态空间模型成为可能，因此，使用快速递归卷积方法。在波导馈电中包含损耗扩展了该方法的应用范围，并完善了其理论基础。与无损情况相比，一个额外的困难是波导模式不再是能量解耦的。本研究计划在端口-哈密顿系统的理论框架内推广模型，通过频率相关耗散材料模型扩展波导的数值方法，并更新时域变换。实际应用需要已知错误限制的rom。同样，对于给定的容错性，最好将ROM尺寸最小化。为此，需要一种自适应MOR方法，根据需要选择合适的扩展频率并调整ROM顺序。作为项目的一部分，非色散情况的现有方法应扩展到频率相关模式模式的情况。

Numerical methods of electromagnetic fields simulation in the frequency domain -- such as the finite-element (FE) method -- enable the analysis of linear time-invariant systems of high-frequency engineering at low levels of error and provide high flexibility in modeling dispersive material properties. However, they lead to algebraic systems of large scale and high computational cost. Methods of model-order reduction (MOR) provide a systematic approach to transforming FE systems into low-dimensional models that stand out for moderate memory requirements, short solution times, and controllable error.The MOR methods developed in the first phase of the project exhibit stable behavior in the low-frequency regime and are capable of handling systems that are excited by waveguide modes of frequency-dependent transversal field patterns. Neglecting losses in the waveguide feeds, the methods succeed in formulating the distributed-parameter system, its FE discretization, and the resulting reduced-order system (ROM) in such a way that essential system properties are provably preserved throughout the entire modeling chain; this includes passivity, causality, stability, reciprocity, and energy.This renewal proposal extends the suggested methodology in the following aspects:1. Since many applications require time-domain simulations, the first goal is to transform the ROM, which is originally residing in the frequency-domain. Thanks to its preservation properties, the a-posteriori passivation required by competing methods may be omitted. Moreover, the well-defined structure of the ROM enables the realization of a descriptor or state-space model in the time-domain under non-restrictive assumptions and, in consequence, the use of fast recursive methods of convolution.2. The inclusion of losses in the waveguide feeds extends the range of application of the methodology and completes its theoretical foundations. One extra difficulty compared to the lossless case is that the waveguide modes are no longer energetically decoupled. The plan is to generalize the modeling within the theoretical framework of port-Hamiltonian systems, to extend the numerical methods for waveguides by frequency-dependent dissipative material models, and to update the time-domain transformation.3. Real-world applications require ROMs of known error limits. Likewise, it is desirable to minimize the ROM dimension for a given error tolerance. For these purposes, a self-adaptive MOR method is required which chooses appropriate expansion frequencies and adjusts the ROM order according to the requirements. As part of the project, an existing method for the non-dispersive case shall be extended to the case of frequency-dependent mode patterns.