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H. Li,M. Liu,Y. Wang,and Q. Cao,“ DGTD方法在电磁腔特性分析中的应用”,在2016 IEEE国际电磁学研讨会上,iWEM 2016-Proceeding,2016,编号。 5,第5-7页。
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The Application of the DGTD Method in the
Analysis of Electromagnetic Cavity Properties
Huiping Li
1,2
, Meilin Liu
3
, Yi Wang
1
, Qunsheng Cao
1
1
College of Electronic and Information Engineering, Nanjing University of Aeronautics& Astronautics, Nanjing 211106, China
2
School of Physics and Electronics, Henan University, Kaifeng 475004, China
3
Shanghai Institute of Satellite Engineering, Shanghai, China
Abstract- In this paper, the electromagnetic computation and
simulation of a cavity filled with vacuum and dielectric are
discussed. An efficient discontinuous Galerkin time domain
(DGTD) method is given in detail, it employs an orthonormal
basis (for spatial discretization) and Runge-Kutta method (for
temporal integration) to solve Maxwell’s equations. The
numerical experiment is carried out on different order of
approximation and number of elements in computational
domain. Compared with the exact analytical solves, DGTD
method could provides optimal convergence, slight error and
robust.
I
. INTRODUCTION
Electromagnetic cavity is a hollow closed composed by
conductors chamber wall, it is the effective tools produce
high-frequency oscillatory, than LC circuit is used more
widely oscillation component. The DGTD is a powerful and
intelligent computational electromagnetic method, which
combines the local high order basis functions of finite
element time domain (FETD) method and the high efficient
numerical fluxes of finite volume time domain (FVTD)
method [1]. The main properties of DGTD is the local basic
functions which not only allows for using explicit time
integration schemes but also provides domain
decomposition on the element level, offering the
opportunity of highly parallel implementations. Because the
mass matrix is local and thus can be inverted at very little
cost, it has more flexibility than the FETD. Compared with
the FVTD, it achieves high-order accuracy on general grids
through the local element-based basis [2,3]. Besides, the
DGTD is more accurate than the FDTD for complex
geometry and unstructured object [4].
II. T
HE DGTD THEORY
We define element by element a local smooth basis of
vector test function and enforce the residue of Maxwell’s
equations orthogonal to each basis function. Suppose
Ω
is
the physical domain and ∂
Ω
is the boundary. Let
u
xt
denote a function of position
x
∈
Ω
and time
t
∈(0,
T] which satisfies the one-dimensional Maxwell’s equations,
() ()
0,0 =
∂
∂
+
∂
∂
=
∂
∂
+
∂
∂
x
E
t
H
x
x
H
t
E
x
με
(1)
where E represents the electric fields, H represents the
magnetic fields. The material parameters
εx
and
μx
, reflect the electric permittivity and magnetic
permeability, respectively.
We assume to solve Maxwell’s equations in a fix domain
Ω
=[
l r
], it denotes the internal and external value for 1D
case. Moreover, the
Ω
is filled by K non-overlapping
elements, i.e.,
kk
r
k
l
Dxxx =∈ ],[
. On each of these elements
we introduce two complementary expressions for the local
solution.
()
()
()
()
()
()
()
()
x
txH
txE
x
tH
tE
txH
txE
k
i
N
n
k
i
k
h
k
i
k
h
n
N
n
k
h
k
h
k
h
k
h
pp
A
∑∑
==
⎥
⎦
⎤
⎢
⎣
⎡
=
⎥
⎦
⎤
⎢
⎣
⎡
=
⎥
⎦
⎤
⎢
⎣
⎡
11
,
,
ˆ
ˆ
,
,
ψ
(2)
here, in the first one, known as the modal form, we use a
local N-th order (N=N
p
-1) polynomial basis,
ψ
n
(x), in which
case the N
p
expansion coefficients are the unknowns. In the
other form, known as the nodal representation, the
polynomial through the interpolating Lagrange
polynomial,
()
x
k
i
A
, and
()()
[]
T
k
i
k
h
k
i
k
h
txHtxE ,,,
are the unknowns.
We want to find a piecewise N-th order polynomial
approximation (E
h
,H
h
) to the solution of the global solution
(E, H). Introducing the nodal form Eq.(2) into Eq.(1) and
requiring Maxwell’s equations to be satisfied locally on the
strong discontinuous Galerkin form yields the local semi-
discrete scheme,
()
()
[]
{}
()
()
[]
{}
k
r
k
l
k
r
k
l
x
x
k
h
kk
hr
kk
k
h
x
x
k
h
kk
hr
kk
k
h
EExE
Jdt
dH
HHxH
Jdt
dE
∗
∗
−+−=
−+−=
A
A
1-
1-
MD
MD
μ
ε
1
1
(3)
Here, J represents the local transformation Jacobian, D
r
is
the differentiation matrix and M is the local mass. E
*
and H
*
represent the electric and the magnetic numerical flux,
respectively. We choose the simplest Lax-Friedrichs-type
numerical flux to connect elements. Furthermore, this type
contains the central flux and the upwind flux (can refer to the
references in [3]).
For clarity of notation, let us further simplify Eq.(3).
Suppose u
h
is an unknown vector, which can be used to
represent E
h
or H
h
.
()
tu
dt
du
hh
h
,L=
(4)
Utilize the fourth-order Runge–Kutta formula to integrate
in the temporal dimension. The coefficients a
i
, b
i
and c
i
are
given in [5].
() ()
() () ()
()
() ( )
()
.
],5,,1[
,
,,
,
51
)(1
11
0
pu
i
kbpp
tctptkak
up
n
h
i
i
ii
i
ni
h
i
i
i
n
=
∈
+=
Δ+Δ+=
=
+
−
−−
"
L
(5)
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