Time-Domain Finite Element Methods for Maxwell's Equations in Metamaterials

verfasst von: Jichun Li, Yunqing Huang

Verlag: Springer Berlin Heidelberg

Buchreihe : Springer Series in Computational Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

The purpose of this book is to provide an up-to-date introduction to the time-domain finite element methods for Maxwell’s equations involving metamaterials. Since the first successful construction of a metamaterial with both negative permittivity and permeability in 2000, the study of metamaterials has attracted significant attention from researchers across many disciplines. Thanks to enormous efforts on the part of engineers and physicists, metamaterials present great potential applications in antenna and radar design, sub-wavelength imaging, and invisibility cloak design. Hence the efficient simulation of electromagnetic phenomena in metamaterials has become a very important issue and is the subject of this book, in which various metamaterial modeling equations are introduced and justified mathematically. The development and practical implementation of edge finite element methods for metamaterial Maxwell’s equations are the main focus of the book. The book finishes with some interesting simulations such as backward wave propagation and time-domain cloaking with metamaterials.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction to Metamaterials

Abstract

In this chapter, we start with a brief discussion on the origins of metamaterials, and their basic electromagnetic and optical properties. We then present some metamaterial structures and potential applications in areas such as sub-wavelength imaging, antenna design, invisibility cloak, and biosensing. After all these, we then move to the related mathematical problems by introducing the governing equations used to model the wave propagation in metamaterials. Finally, a brief overview of some popular computational methods for solving Maxwell’s equations is provided.

Jichun Li, Yunqing Huang

Chapter 2. Introduction to Finite Element Methods

Abstract

The finite element method (FEM) is arguably one of the most robust and popular numerical methods used for solving various partial differential equations (PDEs). Due to the diligent work of many researchers over the past several decades, the fundamental theory and implementation of FEM have been well established as evidenced by many excellent books published in this area (e.g., [4, 20, 21, 39, 51, 54, 65, 78, 158, 163, 243]).

Jichun Li, Yunqing Huang

Chapter 3. Time-Domain Finite Element Methods for Metamaterials

Abstract

In this chapter, we present several fully discrete mixed finite element methods for solving Maxwell’s equations in metamaterials described by the Drude model and the Lorentz model. In Sects. 3.1 and 3.2, we respectively discuss the constructions of divergence and curl conforming finite elements, and the corresponding interpolation error estimates. These two sections are quite important, since we will use both the divergence and curl conforming finite elements for solving Maxwell’s equations in the rest of the book. The material for Sects. 3.1 and 3.2 is quite classic, and we mainly follow the book by Monk (Finite element methods for Maxwell’s equations. Oxford Science Publications, New York, 2003). After introducing the basic theory of divergence and curl conforming finite elements, we focus our discussion on developing some finite element methods for solving the time-dependent Maxwell’s equations when metamaterials are involved. More specifically, in Sect. 3.3, we discuss the well posedness of the Drude model. Then in Sects. 3.4 and 3.5, we present detailed stability and error analysis for the Crank-Nicolson scheme and the leap-frog scheme, respectively. Finally, we extend our discussion on the well posedness, scheme development and analysis to the Lorentz model and the Drude-Lorentz model in Sects. 3.6 and 3.7, respectively.

Jichun Li, Yunqing Huang

Chapter 4. Discontinuous Galerkin Methods for Metamaterials

Abstract

In this chapter, we introduce several discontinuous Galerkin (DG) methods for solving time-dependent Maxwell’s equations in dispersive media and metamaterials. We first present a succint review of DG methods in Sect. 4.1. Then we present some DG methods for the cold plasma model in Sect. 4.2. Here the DG methods are developed for a second-order integro-differential vector wave equation. We then consider DG methods for the Drude model written in a system of first-order differential equations in Sect. 4.3. Finally, we extend the nodal DG methods developed by Hesthaven and Warburton [141] to metamaterial Maxwell’s equations in Sect. 4.4.

Jichun Li, Yunqing Huang

Chapter 5. Superconvergence Analysis for Metamaterials

Abstract

In this chapter, we first give a quick review of superconvergence analysis in Sect. 5.1. Then we carry out the superclose analysis for 3-D metamaterial Maxwell’s equations represented by the Drude model. The analysis for a semi-discrete scheme is presented in Sect. 5.2, which is followed by the analysis for two fully-discrete schemes in Sect. 5.3. In Sect. 5.4, a superconvergence result in the discrete l ₂ norm is proved. Finally, the superconvergence analysis is extended to the 2-D case in Sect. 5.5.

Jichun Li, Yunqing Huang

Chapter 6. A Posteriori Error Estimation

Abstract

In this chapter, we present some basic techniques for developing a posteriori error estimation for solving Maxwell’s equations. It is known that the a posteriori error estimation plays a very important role in adaptive finite element method. In Sect. 6.1, we provide a brief overview of a posteriori error estimation. Then in Sect. 6.2, through time-harmonic Maxwell’s equations, we demonstrate the fundamental ideas on how to obtain the upper and lower posteriori error estimates. In Sect. 6.3, we present a posteriori error estimator obtained for a cold plasma model described by integro-differential Maxwell’s equations.

Jichun Li, Yunqing Huang

Chapter 7. A Matlab Edge Element Code for Metamaterials

Abstract

In this chapter, we demonstrate the practical implementation of a mixed finite element method (FEM) for a 2-D Drude metamaterial model (5.1)–(5.4).

Jichun Li, Yunqing Huang

Chapter 8. Perfectly Matched Layers

Abstract

One common problem in computational electromagnetics is how to simulate wave propagation on an unbounded domain accurately and efficiently. One typical technique is to use the absorbing boundary conditions (ABCs) to truncate the unbounded domain to a bounded domain. The solution computed with an ABC on a bounded domain should be a good approximation to the solution originally given on the unbounded domain. Hence constructing a good ABC is quite challenging.

Jichun Li, Yunqing Huang

Chapter 9. Simulations of Wave Propagation in Metamaterials

Abstract

In this chapter, we present some interesting simulations of wave propagation in metamaterials. We start in Sect. 9.1 with a perfectly matched layer model, which allows us to reduce the simulation on an infinite domain to be realized on a bounded domain. Here we present a simulation demonstrating the negative refraction index phenomenon for metamaterials. In Sects. 9.2 and 9.3, we present invisibility cloak simulations using metamaterials in frequency domain and time domain, respectively. In Sect. 9.4, we present an interesting application of metamaterials for solar cell design. In Sect. 9.5, we end this chapter by presenting some open mathematical problems related to metamaterials.

Jichun Li, Yunqing Huang

Backmatter

Titel: Time-Domain Finite Element Methods for Maxwell's Equations in Metamaterials
verfasst von: Jichun Li
Yunqing Huang
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-33789-5
Print ISBN: 978-3-642-33788-8
DOI: https://doi.org/10.1007/978-3-642-33789-5