A Stable Discontinuous Galerkin Time-Domain Method for Simulation of Thin Conductive Layers

Method for Simulation of Thin Conductive Layers

Item Type Conference Paper

Authors Özakın, M. Burak;Chen, Liang;Ahmed, Shehab;Bagci, Hakan Citation Ozakin, M. B., Chen, L., Ahmed, S., & Bagci, H. (2022). A Stable

Discontinuous Galerkin Time-Domain Method for Simulation of Thin Conductive Layers. 2022 IEEE International Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting (AP-S/URSI). https://doi.org/10.1109/ap-s/usnc- ursi47032.2022.9886126

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DOI 10.1109/AP-S/USNC-URSI47032.2022.9886126

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Link to Item http://hdl.handle.net/10754/681682

A Stable Discontinuous Galerkin Time-Domain Method for Simulation of Thin Conductive

Layers

M. Burak ¨ Ozakın

⁽¹⁾

, Liang Chen

⁽¹⁾

, Shehab Ahmed

^(1),(2)

, and Hakan Bagci

⁽¹⁾

(1)

Electrical and Computer Engineering Program

Computer, Electrical and Mathematical Science and Engineering Division

King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Saudi Arabia

{mehmet.ozakin, liang.chen, shehab.ahmed, hakan.bagci}@kaust.edu.sa

(2)

Ali I. Al-Naimi Petroleum Engineering Research Center

King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Saudi Arabia

Abstract

A conductive layer that is much thinner than the skin depth is often incorporated into electromagnetic solvers as a resistive boundary condition (RBC) to avoid fine meshes (and consequently small time steps for time-domain methods). When the discontinuous Galerkin time-domain (DGTD) method, which relies on an explicit Runge-Kutta (RK) scheme for time integration, is used in a simulation involving an RBC surface with high conductivity, the solution becomes unstable. In this work, to circumvent this bottleneck, a locally-implicit time marching scheme is developed. The proposed method uses implicit Euler forward and explicit Euler backward schemes to discretize the time derivatives in local DG equations associated with the elements that “touch” and do not “touch” the RBC surface, respectively. This yields a locally-implicit Euler (LIME) time marching scheme that maintains its stability in the presence of a highly conductive layer. Indeed, numerical results demonstrate that LIME-DGTD is more stable than RK-DGTD for a transmission problem involving a thin metal sheet.

I. INTRODUCTION

Many electromagnetic devices and systems involve thin but highly conductive layers. Especially in recent years, with the increasing interest in so-called two-dimensional (2D) materials (such as graphene, perovskite, etc.) [1], efficient modeling of thin and highly conductive layers in electromagnetic simulations has regained traction from computational electromagnetics (CEM) practitioners. In this context, if a conductive layer is much thinner than the skin depth, it can be incorporated into a electromagnetic solver as a resistive boundary condition (RBC) without loss of accuracy [2]–[8]. Since this approach avoids fine meshes (and consequently small time steps for time-domain methods), it significantly reduces the computational requirements of the electromagnetic simulations.

In recent years, the discontinuous Galerkin time-domain (DGTD) method has attracted significant attention in the CEM community since it allows for non-conformal meshes and easier implementations ofh-and/orp-adaptive mesh

refinement techniques [9]. In addition, if an explicit time integration method, such as Runge-Kutta (RK) schemes, is used, the resulting DGTD solver is very efficient and has a very small memory footprint. Having said that, when RK-DGTD is used in a simulation involving an RBC surface with high conductivity, the solution becomes unstable.

In this work, to circumvent this bottleneck, a locally-implicit time-integration scheme is developed. The proposed method uses implicit Euler forward and explicit Euler backward schemes to discretize the time derivatives in local DG equations associated with the elements that “touch” and do not “touch” the RBC surface, respectively. This yields a locally-implicit Euler (LIME) time marching scheme that maintains its stability in the presence of a highly conductive layer. Indeed, numerical results demonstrate that LIME-DGTD is more stable than RK-DGTD for a transmission problem involving a thin metal sheet.

II. FORMULATION

A. Resistive Boundary Condition (RBC)

RBCs can be incorporated into DGTD using the Rankine–Hugoniot jump relations between the fields of the virtual nodes (denoted with superscripts^∗ and^∗∗) on two sides of the thin metal sheet [8]:

ˆ

n×(H^∗−H^∗∗) =σ_s{{ˆn׈n×E}}

ˆ

n×(E^∗−E^∗∗) = 0

(1)

whereσ_s is the surface conductivity of the metal sheet, the average operator is denoted as{{.}}, andˆnis the unit surface normal vector. Then, the numerical flux for the DG discretization element surfaces that “touch” the thin metal sheet can be derived using (1) [8].

B. Locally-Implicit Euler (LIME) Time Integration

The semi-discritized form of the Maxwell equations obtained using the spatial DG discretization can be written as:

∂

∂t



 µH

εE



=





−CE+F^H(E,H) +F_RBC^H (E) CH+F^E(E,H) +F_RBC^E (E)



 (2)

where ε and µ are the permittivity and the permeability, C and F^H and F^E denote the discretized operators associated with curl and the “regular” numerical flux [9], respectively. F_RBC^H and F_RBC^E are the additional flux terms introduced only on the discretization element surfaces that touch the thin metal sheet.

To maintain the stability of the solution, the time derivative in (2) is discretized using the Euler forward method for the elements that touch the thin metal sheet (RBC elements) and the Euler backward method for the elements

Fig. 1. Computation domain andExdistribution in the whole domain at time step640of the simulation carried out using LIME-DGTD.

that do not touch the thin metal sheet (non-RBC elements). This yields the LIME time integration scheme as:

Hⁿ⁺¹_k =Hⁿ_k +∆t µ^k

−CkEⁿ_k +F_k^H(Eⁿ_k,Hⁿ_k,Eⁿ_k′,Hⁿ_k′)

(3)

Eⁿ⁺¹_k =Eⁿ_k+∆t ε^k

CkHⁿ_k+F_k^E(Eⁿ_k,Hⁿ_k,Eⁿ_k′,Hⁿ_k′)

(4)

Hⁿ⁺¹_r =Hⁿ_r+∆t µr

[−CrEⁿ⁺¹_r +F_RBC,r^H (Eⁿ⁺¹_r ,Eⁿ⁺¹_r′ ) (5)

+F_r^H(Eⁿ⁺¹_r ,Hⁿ⁺¹_r ,Eⁿ⁺¹_r′ ,Hⁿ⁺¹_r′ ,Eⁿ⁺¹_k′ ,Hⁿ⁺¹_k′ )]

Eⁿ⁺¹_r =Eⁿ_r +∆t εr

[CrHⁿ⁺¹_r +F_RBC,r^E (Eⁿ⁺¹_r ,Eⁿ⁺¹_r′ ) (6)

+F_r^E(Eⁿ⁺¹_r ,Hⁿ⁺¹_r ,Eⁿ⁺¹_r′ ,Hⁿ⁺¹_r′ ,Eⁿ⁺¹_k′ ,Hⁿ⁺¹_k′ )]

where subscriptkis the index of a non-RBC element,randr^′are the indices of an RBC element and its neighboring RBC element, respectively, k^′ is the index of a neighboring non-RBC element, which is touching either element k or elementr. The resulting DGTD scheme first updates the fields of all non-RBC elements explicitly using (3) and (4). Thus, Hⁿ⁺¹_k′ andEⁿ⁺¹_k′ are available before updating (5) and (6). Then, the fields of element randr^′ at time step(n+ 1) are updated together by solving the relevant local matrix system.

III. NUMERICALRESULTS

In this section, transient wave transmission through a thin metal sheet is studied to demonstrate the accuracy and the stability of the proposed method. The conductivity (σ)and the thickness of the metal sheet (d) are 10⁶S/m and 1µm, respectively, yielding σs = σd = 1 S. A monochromatic planewave generated by an aperture source with frequency 100 MHz is normally incident on the thin sheet as shown in Fig. 1. The excitation is applied for a duration of four periods. Note that the skin depth associated with the thin metal sheet at 100 MHz is 15.9µm, which is much larger thand. Also, note that periodic and absorbing boundary conditions are used on side and end surfaces of the computation domain.

The fields transmitted through the thin metal sheet are computed using LIME-DGTD and RK-DGTD. Fig. 2 compares E_x (ˆx-component of the electric field) recorded during these simulations at point (0.1 m,0.1 m,2.2 m)

1 1.5 2 2.5 3 3.5

Time (s) ₁₀^-8

-6 -4 -2 0 2 4 6

E x

10^-3

(a)

1 1.5 2 2.5 3 3.5

Time (s) ₁₀^-8

-1 0 1 2

E x

10¹⁹⁴

(b)

Fig. 2. Excomputed at point(0.1 m,0.1 m,2.2 m)using analytical solution, (a) LIME-DGTD, and (b) RK-DGTD.

(marked as “observation point” in Fig. 1) to the analytical solution. The figure clearly shows that LIME-DGTD is more stable than RK-DGTD.

ACKNOWLEDGMENTS

The authors thank the KAUST Supercomputing Laboratory (KSL) for providing the required computational resources.

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