Time Domain Numerical Methods for Electromagnetic Wave Propagation Problems in Complex Dispersive Dielectrics

该项目旨在开发和分析时域中涉及电磁波在复杂电介质中传播和散射的正问题的有效数值方法，这些介质由于多种极化机制而显示物理色散。 色散的、随时间变化的电磁波系统的正演解通常用于获得关于人类暴露于高能电磁场的安全标准。 这些数值模拟也可以用于反问题的制定，例如，在癌症检测。 在用于医疗应用的成像中，人们试图通过亚红外频率的电磁场来研究对象（人体）的内部结构，以通过研究介电特性（诸如介电常数、电导率、弛豫时间等）的变化来检测和表征癌症或其他异常。组织。 电磁波在复杂色散介质中的传播受与时间相关的麦克斯韦方程组控制，该方程组耦合到描述诱导宏观极化的演化的方程组。 介质的宏观极化的行为可以包括用于建模极性材料（例如水和活组织）的德拜型的一阶弛豫机制，以及洛伦兹型的二阶机制。 这两种机制将被用作模型问题。 研究者和一名研究生将沿着电磁波在色散介质中传播的混合有限元方法，发展算子分裂技术。 此外，虚拟域方法将被应用到这些材料中的波散射问题。 有效的吸收层终止计算边界，包括色散的反射将被构造和分析。 将对新方法的适定性、稳定性和收敛性进行分析。 研究生将研究现有的模拟方法，并与新开发的计算技术进行比较。

This project aims to develop and analyze efficient numerical methods for forward problems in the time domain involving the propagation and scattering of electromagnetic waves in complex dielectric media that display physical dispersion due to multiple polarization mechanisms. The forward solution of dispersive, time-dependent electromagnetic wave systems is often used to obtain safety standards regarding exposure of humans to high-energy electromagnetic fields. These numerical simulations can also be utilized in an inverse-problem formulation, for example, in cancer detection. In imaging for medical applications, one seeks to investigate the internal structure of an object (the human body) by means of electromagnetic fields at sub-infrared frequencies to detect and characterize cancer or other anomalies by studying changes in the dielectric properties (such as permittivity, conductivity, relaxation times, etc.) of tissues. Electromagnetic wave propagation in complex dispersive media is governed by the time-dependent Maxwell's equations coupled to equations that describe the evolution of the induced macroscopic polarization. The behavior of the media's macroscopic polarization may include first-order relaxation mechanisms of Debye type, used in modeling polar materials such as water and living tissue, as well as second-order mechanisms of Lorentz type. These two mechanisms will be used as model problems. The investigator and a graduate student will develop operator-splitting techniques along with mixed finite element methods for electromagnetic wave propagation in dispersive dielectrics. Additionally, fictitious domain methods will be applied to wave-scattering problems in these materials. Efficient absorbing layers for terminating computational boundaries involving dispersive dielectrics will be constructed and analyzed. Analysis of well-posedness, stability, and convergence of the new methods will be conducted. The graduate student will study existing simulation methods and make comparisons with the newly developed computational techniques.