A Novel Periodic Boundary Condition Treatment in Electrodynamics Wave Interaction With Small Structures

2007 ◽  
Author(s):  
Kang Fu ◽  
Pei-Feng Hsu
Keyword(s):  
Boundary Condition ◽  
Time Domain ◽  
Maxwell’S Equations ◽  
Periodic Boundary Condition ◽  
Maxwell's Equations ◽  
Periodic Boundary ◽  
Numerical Study ◽  
Large Angle ◽  
Radiative Properties ◽  
Angle Of Incidence

In the numerical study of the radiative properties of micro- and nano-structure devices, for example, the random roughness surfaces, grating surfaces, periodic photonic devices, the periodic boundary condition are frequently used to simulate device size much larger than the incident wavelength. Existing methods in handling the periodic boundary condition in the solution of the Maxwell’s equations are too limiting. A novel method is developed to efficiently treat such boundary conditions. The concept is not limited to any particular solution method of the Maxwell’s equations. The salient feature is to convert the phase difference between the corresponding boundaries from the time domain to frequency domain using a phasor diagram approach. The resulting electromagnetic field vector component equations at the boundaries are successfully tested in a finite-difference time-domain code at large angle of incidence, up to 80°, on a finite length, flat, and dielectric surface. The computed reflectivity is in good agreement with the analytical value calculated by Fresnel reflectivity.

Predictions of Radiative Properties of Patterned Silicon Wafers by Solving Maxwell’s Equations in the Time Domain

2003 ◽  
Author(s):  
J. Liu ◽  
S. J. Zhang ◽  
Y. S. Chen
Keyword(s):  
Time Domain ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Film Theory ◽  
Fdtd Method ◽  
Absorbing Boundary Conditions ◽  
Silicon Wafers ◽  
Radiative Properties ◽  
Computational Domain ◽  
Electromagnetic Model

A rigorous electromagnetic model is developed to predict the radiative properties of patterned silicon wafers. For nonplanar structures with characteristic length close to the wavelength of incident radiation, Maxwell’s equations must be used to describe the associated radiative interaction and they are solved by the finite difference time-domain (FDTD) method. In the die area, only one period of the structure is modeled due to its periodicity in geometry. To truncate a computational domain, both the Mur condition and perfectly matched layer (PML) technique are available to absorb outgoing waves. With the steady state time-harmonic electromagnetic field known, the Poynting vector is used to calculate the radiative properties. Due to its importance, the reflection error is checked at first for two absorbing boundary conditions. As expected, the PML technique yields much lower errors than the Mur condition and it is thus used in this study. To validate the present model, radiative interactions with a planar structure and a nonplanar structure are investigated, and predicted reflectivities are found to match available other solutions very well. To demonstrate the importance of the present study, a patterned wafer consisting of periphery and die area is also investigated. While the thin film theory is accurate for the wafer periphery, the rigorously electromagnetic model described in this study is found to be necessary to accurately predict the radiative properties in the die area.

scholarly journals Solutions for impulsive fractional pantograph differential equation via generalized anti-periodic boundary condition

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Idris Ahmed ◽  
Poom Kumam ◽  
Jamilu Abubakar ◽  
Piyachat Borisut ◽  
Kanokwan Sitthithakerngkiet
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Fixed Point ◽  
Fractional Differential Equation ◽  
Existence And Uniqueness ◽  
Periodic Boundary Condition ◽  
Periodic Boundary ◽  
Fixed Point Theorems ◽  
Uniqueness Of The Solution ◽  
Fractional Differential

Abstract This study investigates the solutions of an impulsive fractional differential equation incorporated with a pantograph. This work extends and improves some results of the impulsive fractional differential equation. A differential equation of an impulsive fractional pantograph with a more general anti-periodic boundary condition is proposed. By employing the well-known fixed point theorems of Banach and Krasnoselskii, the existence and uniqueness of the solution of the proposed problem are established. Furthermore, two examples are presented to support our theoretical analysis.

scholarly journals A new time-delayed periodic boundary condition for discrete element modelling of railway track under moving wheel loads

2021 ◽  
Vol 23 (4) ◽  
Author(s):  
Huiqi Li ◽  
Glenn McDowell ◽  
John de Bono
Keyword(s):  
Boundary Condition ◽  
Periodic Boundary Condition ◽  
Periodic Boundary ◽  
Discrete Element ◽  
Contact Forces ◽  
Railway Track ◽  
Discrete Element Modelling ◽  
Fixed Boundary ◽  
Element Modelling ◽  
New Time

Abstract A new time-delayed periodic boundary condition (PBC) has been proposed for discrete element modelling (DEM) of periodic structures subject to moving loads such as railway track based on a box test which is normally used as an element testing model. The new proposed time-delayed PBC is approached by predicting forces acting on ghost particles with the consideration of different loading phases for adjacent sleepers whereas a normal PBC simply gives the ghost particles the same contact forces as the original particles. By comparing the sleeper in a single sleeper test with a fixed boundary, a normal periodic boundary and the newly proposed time-delayed PBC (TDPBC), the new TDPBC was found to produce the closest settlement to that of the middle sleeper in a three-sleeper test which was assumed to be free of boundary effects. It appears that the new TDPBC can eliminate the boundary effect more effectively than either a fixed boundary or a normal periodic cell. Graphic abstract

An analytical solution of a two-dimensional temperature field in a hollow cylinder under a time periodic boundary condition using Fourier series

2009 ◽  
Vol 223 (8) ◽  
pp. 1889-1901 ◽  
Author(s):  
G Atefi ◽  
M A Abdous ◽  
A Ganjehkaviri ◽  
N Moalemi
Keyword(s):  
Boundary Condition ◽  
Temperature Field ◽  
Fourier Series ◽  
Temperature Distribution ◽  
Analytical Solution ◽  
Hollow Cylinder ◽  
Periodic Boundary Condition ◽  
Periodic Boundary ◽  
Separation Of Variables ◽  
Two Dimensional

The objective of this article is to derive an analytical solution for a two-dimensional temperature field in a hollow cylinder, which is subjected to a periodic boundary condition at the outer surface, while the inner surface is insulated. The material is assumed to be homogeneous and isotropic with time-independent thermal properties. Because of the time-dependent term in the boundary condition, Duhamel's theorem is used to solve the problem for a periodic boundary condition. The periodic boundary condition is decomposed using the Fourier series. This condition is simulated with harmonic oscillation; however, there are some differences with the real situation. To solve this problem, first of all the boundary condition is assumed to be steady. By applying the method of separation of variables, the temperature distribution in a hollow cylinder can be obtained. Then, the boundary condition is assumed to be transient. In both these cases, the solutions are separately calculated. By using Duhamel's theorem, the temperature distribution field in a hollow cylinder is obtained. The final result is plotted with respect to the Biot and Fourier numbers. There is good agreement between the results of the proposed method and those reported by others for this geometry under a simple harmonic boundary condition.

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