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High order FDTD methods for electromagnetic systems in dispersive inhomogeneous media.

机译：弥散非均匀介质中电磁系统的高阶FDTD方法。

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摘要

This dissertation presents matched interface and boundary time-domain (MIBTD) methods for solving both transverse magnetic (TM) and transverse electric (TE) Maxwell's equations in non-dispersive and dispersive media with complex interfaces and discontinuous wave solutions. In this thesis, five following problems will be discussed: (1) Dielectric interface problems; (2) Debye dispersive interface problems in TM mode; (3) Drude dispersive interface problems in TM mode; (4) Debye dispersive interface problems in TE mode; and (5) Perfectly matched layer (PML) boundary conditions for dispersive interface problems. It is well known in the electromagnetic interface problems that field components across the interfaces are often nonsmooth of even discontinuous. Consequently, the finite-difference time-domain (FDTD) algorithms without a proper interface treatment will cause a staircasing error when dealing with arbitrary interfaces; and only first-order of accuracy is achieved by those FDTD methods. Thus, to restore the accuracy reduction of the collocation FDTD approach near an interface, the physical jump conditions relating discontinuous wave solutions on both sides of the interface must be rigorously enforced. For this purpose, a novel matched interface and boundary (MIB) scheme is proposed to handle material interface problems, in which new jump conditions are derived so that the discontinuous and staggered features of electric and magnetic field components can be accommodated. That results in the staircasing error is totally eliminated in the dielectric interface problems. However, in the dispersive materials like Debye media, interface conditions are now time-dependent. Thus, interface auxiliary differential equations (IADEs) are utilized to describe the transient changes in the regularities of electromagnetic fields across a Debye dispersive interface. In addition, in TM mode, to assist the track of the jump condition information along the interface, a novel hybrid system, which couples the wave equation for the electric component with Maxwell's equations for the magnetic components, is constructed based on the auxiliary differential equation (ADE) approach. As a result, the staircasing error is also eradicated for the Debye interface problems. However, this MIBTD approach is only designed for Debye material equations formed by first-order ADE. Because of that, the MIBTD algorithm for the problem (2) cannot be directly extended to solve Drude dispersive interface problems having second-order ADE. To achieve high order accuracy for the problem (3), a novel hybrid Drude-Maxwell system and IADEs are also formulated to update the regularity change of the field components across interfaces so that the staircasing error is free in the numerical results. In the dispersive interface problems in TE mode, the jump conditions of the electric components become more complicated than in the TM mode case, and rigorously depend on the unknown flux density fields. Therefore, the standard Maxwell's equations are taken into consideration instead of the hybrid system. The leapfrog scheme is employed to simplify the complexities of the jump conditions' derivations in the TE mode, whereas the fourth-order Runge-Kutta method is exploited in the other cases. In any material interface problems, effective MIB treatments are proposed to rigorously impose the physical jump conditions which are not only time dependent, but also couple both Cartesian directions and different field components. Based on a staggered Yee lattice, the proposed MIB schemes can achieve up to sixth order-accuracy in dealing with the straight interfaces, while the uniform second-order accuracy is always maintained in solving irregular interfaces with constant curvatures, general curvatures, and nonsmooth corners. Based on the numerical verification, our MIBTD algorithms are conditionally stable and more cost-efficient than the classical FDTD methods. Finally, the Berenger's PML is successfully used as absorbing boundary condition (ABC) for the dispersive interface problems. The numerical results are provided to validate the efficiency of that PML ABC.
机译：本文提出了在界面复杂且界面不连续的非色散和色散介质中求解横向磁和麦克斯韦方程的界面和边界时域匹配方法。本文将讨论以下五个问题：（1）介电界面问题； （2）TM模式下的德拜色散接口问题； （3）TM模式下的Drude色散接口问题； （4）TE模式下的德拜色散接口问题； （5）色散界面问题的完全匹配层（PML）边界条件。在电磁接口问题中众所周知，跨接口的场分量通常不平滑甚至不连续。因此，在处理任意接口时，如果不进行适当的接口处理，则时差有限时域（FDTD）算法将引起阶梯错误。而那些FDTD方法仅能达到一级精度。因此，为了恢复界面附近的并置FDTD方法的精度降低，必须严格执行与界面两侧不连续波解相关的物理跳跃条件。为此，提出了一种新颖的匹配界面和边界（MIB）方案来处理材料界面问题，其中推导了新的跳跃条件，以便可以适应电场和磁场分量的不连续和交错特征。从而消除了介电界面问题中的阶梯误差。但是，在像Debye介质这样的分散材料中，界面条件现在与时间有关。因此，界面辅助微分方程（IADE）用于描述跨Debye色散界面的电磁场规则性的瞬态变化。此外，在TM模式下，为了辅助沿着界面跟踪跳跃条件信息，基于辅助微分方程构造了一个新颖的混合系统，该系统将电分量的波动方程与磁分量的麦克斯韦方程耦合在一起（ADE）方法。结果，对于Debye接口问题，也消除了楼梯错误。但是，此MIBTD方法仅设计用于由一阶ADE形成的Debye材料方程式。因此，无法将问题（2）的MIBTD算法直接扩展为解决具有二阶ADE的Drude色散接口问题。为了实现问题（3）的高阶精度，还制定了新型混合式Drude-Maxwell系统和IADE，以更新界面上场分量的规则性变化，从而在数值结果中消除了阶梯误差。在TE模式下的色散界面问题中，电子元件的跳变条件比TM模式下的情况更加复杂，并且严格取决于未知的磁通密度场。因此，考虑标准麦克斯韦方程而不是混合系统。采用跳越方案简化了TE模式下跳跃条件推导的复杂性，而在其他情况下则采用了四阶Runge-Kutta方法。在任何物质界面问题中，都提出了有效的MIB处理方法，以严格施加不仅取决于时间的物理跳变条件，而且还要结合笛卡尔方向和不同的场分量。基于交错的Yee格，提出的MIB方案在处理直线界面时可以达到六阶精度，而在求解具有恒定曲率，大曲率和不光滑拐角的不规则界面时，始终保持一致的二阶精度。 。基于数值验证，我们的MIBTD算法在条件上稳定，并且比传统FDTD方法更具成本效益。最后，将Berenger的PML成功用作吸收边界条件（ABC），以解决色散界面问题。提供了数值结果以验证该PML ABC的效率。

著录项

  • 作者

    Nguyen, Duc Duy.;

  • 作者单位

    The University of Alabama.;

  • 授予单位 The University of Alabama.;
  • 学科 Mathematics.;Electromagnetics.
  • 学位 Ph.D.
  • 年度 2015
  • 页码 212 p.
  • 总页数 212
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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