Mathematical Analysis and Simulation of Field Models in Accelerator Circuits
Mathematical Analysis and Simulation of Field Models in Accelerator Circuits
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加速器电路场模型的数学分析与仿真
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发表时间：
2021
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通讯作者：
I. C. Garcia
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作者：
I. C. Garcia
通常在电气工程中，采用网络建模方法来模拟设备及其周围电路，其中每个设备都被认为是电压-电流关系。然而，对于某些应用，这种简化不能产生所需的精度。在这些情况下，可以进行精细建模，其中将对所需物理量进行建模的空间分布偏微分方程耦合到经典网络方程。由此产生的耦合方程组往往表现出多尺度，多速率，甚至多物理的行为，处理与涉及的算法，以便有效地模拟它。因此，其结构分析是重要的，数值处理系统适当，并确保算法收敛。本论文主要研究这类系统的数学分析及其仿真。
从半离散精细模型电路得到的方程组是典型的微分代数方程。他们的数值和分析的困难进行了研究的背景下，他们的微分代数指数。为此，三个广义电路元件的定义，允许分类的细化模型。由此，整个耦合系统的指数可以通过电路的拓扑性质来指定。利用广义单元定义对麦克斯韦方程组的几种近似进行了分类，得到了场路耦合系统的折射率特性。
对于模拟两个算法进行了研究。本文首先分析了超导线圈上存在涡流效应的磁准静态场场路耦合系统的联合仿真波形松弛法。通过优化的施瓦茨方法加快了算法的收敛速度。这里，两个子系统之间的信息交换通过耦合条件的线性组合来改善。为了进一步加快模拟时间，并行的时间方法Parareal进行了分析。该算法研究的背景下，微分代数方程，通过研究其适用性的非线性更高的指标系统所产生的，例如，从电路仿真。最后，提出了两种方法结合Parareal和波形松弛。其中之一是专门设计的场路耦合系统，并产生一个微观宏类Parareal算法。然而，其背后的思想可以应用于其他类型的耦合系统。
场路耦合系统的数值试验，强调从数学理论得到的结果，以及测试所提出的算法的效率。
Typically in electrical engineering a network modelling approach for the simulation of devices and their surrounding circuitry is taken, where each device is considered by a voltage-to-current relation. For some applications, however, this simplification does not yield the required accuracy. In these cases, refined modelling can be performed, where a spatially distributed partial differential equation modelling the required physical quantity is coupled to the classic network equations. The resulting coupled system of equations often exhibits a multiscale, multirate and even multiphysical behaviour that is tackled with involved algorithms so as to efficiently simulate it. Its structural analysis is therefore important, to numerically treat the system appropriately and to ensure that the algorithms converge properly. This thesis deals with the mathematical analysis of these type of systems as well as their simulation.
The systems of equations obtained from circuits with semidiscrete refined models are typically differential algebraic equations. Their numerical and analytical difficulties is studied in the context of their differential algebraic index. For that, three generalised circuit element definitions are given, that allow the classification of the refined models. Hereby, the index of the entire coupled system can be specified by means of topological properties of the circuit. Several approximations to Maxwell’s equations are classified with the generalised element definitions to obtain the index properties of the field-circuit coupled systems.
For the simulation two algorithms are studied. First the co-simulation waveform relaxation method is analysed for field-circuit coupled systems arising from magnetoquasistatic fields with eddy current effects on superconducting coils. The convergence of the algorithm is sped up by means of optimised Schwarz methodologies. Here, the information exchange between both subsystems is improved by a linear combination of the coupling conditions. To further speed up simulation time, the parallel-in-time method Parareal is analysed. The algorithm is investigated in the context of differential algebraic equations by studying its applicability to nonlinear higher index systems arising e.g. from circuit simulation. Finally, two approaches are proposed for the combination of Parareal and waveform relaxation. One of them is specifically designed for field-circuit coupled systems and yields a micro-macro-like Parareal algorithm. However, the idea behind it can be applied to other type of coupled systems.
Numerical tests of field-circuit coupled systems are made to underline the results obtained from the mathematical theory as well as test the efficiency of the proposed algorithms.