Numerical analysis of a finite element method for the electromagnetic concentrator model
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Numerical analysis of a finite element method for the electromagnetic concentrator model Yunqing Huang1 · Jichun Li2 Received: 25 January 2020 / Accepted: 1 October 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we consider the electromagnetic concentrator model obtained through transformation optics. This model is formed by a system of coupled time-dependent Maxwell’s equations with three unknowns, which makes the analysis and simulation much more challenging compared to the standard Maxwell equations. In our previous work (W. Yang, J. Li, Y. Huang, and B. He, Commun. Comput. Phys., 25(1), pp. 135– 154, 2019), we proposed a finite element time-domain (FETD) method with edge elements for solving this model efficiently without any theoretical analysis. Here, we provide a rigorous analysis for the mathematical modelling equations and the FETD method proposed there. Keywords Maxwell’s equations · Finite element method · Edge elements · Metamaterial Mathematics Subject Classification (2010) 78M10 · 65N30 · 65F10 · 78-08
1 Introduction Since the year 2000, the electromagnetic (EM) metamaterial has been a very active research topic (cf. monographs [13, 15, 19, 24] and references therein) due to its many revolutionary potential applications. Some interesting applications include the Communicated by: Jan Hesthaven Jichun Li
[email protected] Yunqing Huang [email protected] 1
Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan, China
2
Department of Mathematical Sciences, University of Nevada Las Vegas, Las Vegas, NV 89154-4020, USA
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Adv Comput Math
(2020) 46:77
invisibility cloak (e.g., [14, 18, 28]), design of optical black hole (cf. [33] and references therein), EM concentrator, rotator and splitter (cf. [31, 35, 36] and references therein), etc. A good EM concentrator can focus incident waves coming from arbitrary directions and is reflectionless due to inherent impedance matching. Hence, the EM concentrator can play an important role in harnessing of light in solar cells or similar devices. In our recent work [35], we first derived the mathematical governing equations for the EM concentrator, rotator, and splitter, and developed a finiteelement time-domain (FETD) method to simulate the wave propagation phenomena in these devices. For example, we can obtain a cylindrical EM concentrator by compressing a circle with radius b = 0.3 m into a smaller circle with radius a = 0.1 m, while keeping the outer circle c = 0.4 m unchanged. The simulation of the EM concentrator is carried out by solving the two-dimensional (2D) time-dependent Maxwell equations plus some constitutive equations. For the simulation, we bury the EM concentrator in a physical domain = [−0.8, 0.8] m × [−0.8, 0.8] m surrounded by a perfectly matched layer. Some snapshots of the Ey field obtained from our simulation of wave propagation with time step 0.8 ps in the EM concentrator device are presented in Fig. 1, which s
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