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Forum for Electromagnetic Research Methods and Application Technologies (FERMAT)
Cheng-Yi Tian, Yan Shi, and Long Li School of Electronic Engineering, Xidian University, China
Abstract: 1. A low storage scheme based on the universal matrices is developed for the discontinuous Galerkin time-domain method (DGTD). The
proposed method can reduce the memory consumption dramatically with an acceptable raise in CPU time cost. 2. The weighted Laguerre polynomials (WLP) scheme is integrated into the DGTD, thus leading to an unconditionally stable computational
method. The resulted system uses the WLP as the temporal base and can be solved through many direct/iterative sparse linear system solvers. The WLP-DGTD method is suitable for the multi-scale simulations and eliminate the late-time instability of UPML in the DGTD method.
3. Boundary Integral Method is combined with the WLP-DGTD method, and the computational accuracy versus mesh size and penalty factor is studied.
Keyword: universal matrices, weighted Laguerre polynomial, unconditionally stable, boundary integral method, multi-scale problem
These slides are adopted from an oral presentation at ICCEM 2016, Guangzhou, Guangdong, China.
Unconditionally Stable Laguerre Polynomial-Based Discontinuous Galerkin Time-Domain Method
Cheng-Yi Tian received the B.E. degree in electrical engineering from the School of Electric Engineering, Xidian University, Xi’an, China, in 2013, where he is currently working towards the Ph.D. degree. His research interest is computational electromagnetics with an emphasis on DGTD and the unconditionally stable techniques in time domain. Yan Shi received the B.Eng. and Ph.D. degrees in electromagnetic fields and microwave technology from Xidian University, Xi’an, China, in 2001 and 2005, respectively. From 2007 to 2008, he worked at City University of Hong Kong, Hong Kong, China, as a Senior Research Associate. He was awarded a scholarship under the China Scholarship Council and was invited to visit the University of Illinois at Urbana-Champaign as a Visiting Postdoctoral Research Associate in 2009. He is currently Professor with School of Electronic Engineering, Xidian University. He is the author or coauthor of over 80 technical papers and a book Notes on catastrophe theory (Science Press, 2015). His research interests include computational electromagnetics, metamaterial, and antenna. Prof. Shi is a member of IEEE and senior member of The Chinese Institute of Electronics. He
received Program for New Century Excellent Talents in University from Ministry of Education of China and New Scientific and Technological Star of Shaanxi Province from Education Department of Shaanxi Provincial Government.
Long Li (Senior Member, IEEE) was born in Guizhou Province, China. He received the B.E. and Ph.D. degrees in electromagnetic fields and microwave technology from Xidian University, Xi’an, China, in 1998 and 2005, respectively. He joined the School of Electronic Engineering, Xidian University, in 2005 and was promoted to Associate Professor in 2006. He was a Senior Research Associate in the Wireless Communications Research Center, City University of Hong Kong, Hong Kong, in 2006. He received the Japan Society for Promotion of Science (JSPS) Postdoctoral Fellowship and visited Tohoku University, Sendai, Japan, as a JSPS Fellow from November 2006 to November 2008. He was a Senior Visiting
Scholar at the Pennsylvania State University, State College, PA, USA, from December 2013 to July 2014. He is currently a Professor in the School of Electronic Engineering, Xidian University. His research interests include metamaterials, computational electromagnetics, electromagnetic compatibility, novel antennas, and wireless power transfer technology. Dr. Li received the Nomination Award of the National Excellent Doctoral Dissertation of China in 2007. He won the Best Paper Award at the International Symposium on Antennas and Propagation in 2008. He received the Program for New Century Excellent Talents in University of the Ministry of Education of China in 2010; the First Prize of Awards for Scientific Research Results offered by Shaanxi Provincial Department of Education, China, in 2013; and the IEEE APS Raj Mittra Travel Grant Senior Researcher Award in 2014. Dr. Li is a Senior Member of the Chinese Institute of Electronics (CIE).
Unconditionally Stable Laguerre Polynomial-Based Discontinuous Galerkin Time-Domain
Method
Cheng-Yi Tian, Yan Shi, and Long Li
School of Electronic EngineeringXidian University
Xi’an, Shaanxi, China
ICCEM 2016Guangzhou, Guangdong, China
February 24, 2016
1
DGTD Fundamentals
Memory Efficient Scheme Based on Universal Matrices
DGTD Based on Marching-On-in-Degree (DGTD-MOD)
Hybrid Boundary Integral Method and DGTD-MOD
Conclusion
Outline
DGTD Fundamentals
0
st
t
EH J
HE
( )( ) ( ) (
*
)
( )
m m
m
mm m m
mi s mi
V V
mi ffk V
m
k
dV dVt
dS
EΦ J Φ
n H
H
Φ
( )( ) ( )
*( )
m m
m
mm m
mi mi
V V
mi
fk V
m
fk
dV dVt
dS
HΨ
nΨ E
Ψ E
Maxwell’s Equation
Weak Form
Numerical Flux
( ) ( )
0
, ,eN
m m
i i
i
t t
E r e r Φ r
( ) ( )
0
, ,hN
m m
i i
i
t t
H r h r Ψ r
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
*( ) ( )
*(( ( )) )( )
i i i i i
H Efk fk
i i i i i
E Hfk fk
i i
fk
i i
fk
n H
n
n H H n E
n E E n HE
Spatial Discretization
Galerkin Method
( ) ( ) ( )
( ) ( ) ( )
i j i
fk
i j i
fk
H H H
E E E
2
(Source: Google.com)(Source: [1])
(Source: [2])
DGTD Fundamentals
3
Numerical Flux
Numerical flux Centered Penalized Upwind
0.5
0.5
0
0
E
H
E
H
( )Y Y Y
( )Z Z Z
( )Y Y
( )Z Z
( )Y Y Y
( )Z Z Z
1 ( )Y Y
1 ( )Z Z
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
*( ) ( )
*(( ( )) )( )
i i i i i
H Efk fk
i i i i i
E Hfk fk
i i
fk
i i
fk
n H
n
n H H n E
n E E n HE
DGTD Fundamentals
( )( ) ( )
( )( ) ( )
ii i j i i i j i ii i
H E
ii i j i i i j i ii i
E H
d
dt
d
dt
eM S h P h F h P e F e
hM S e P e F e P h F h
Semi-discrete form
• Elementwise operation• Highly parallel computation structure• High-order accuracy• Unstructured mesh• …
• CFL constraint in explicit time discretization method
• Large number of DOFs• …
Advantage Disadvantage
4
Challenges In DGTD
Multiscale ProblemsAlternative Explicit Schemes
• Strong Stability Preserving RK (SSP-RK) Method
[3]
• Predictor-Corrector Time Integration Method [4]
• Tailored LSERK Method [5]
Implicit and Implicit-Explicit Time Schemes
• IMEX Crank-Nicolson Method [6]
• IMEX RK Method [7]
Local Time Stepping (LTS) Scheme [8,9]
Large Scale Problems
Parallel Computation
• Multi-core MPI-based Parallel Computation
• Multi-core OpenMP-based Parallel Computation
• Multi-GPU Acceleration Techniques (MPI-CUDA or MPI-OpenGL Hybrid Scheme) [10]
Hybrid Method
• DGTD-FDTD Method [11]
• DGTD-SETD Method [12]
• DGTD-FETD Method [12]
• DGTD–PSTD Method [12]
Multiphysics and Non-linear Problems
5
Microwave or optical materials and devices
• Graphene
• traveling wave tube (particle in cells)
• microwave plasma system
Electromagnetic-thermal-mechanical problem• Maxwell-heat equations
• Maxwell-acoustic/elastic wave equations
Electrodynamics-fluid problem
• Maxwell-Navier-Stokes equations
6
Reference[1]. S. Dosopoulos, J.F. Lee “Interconnect and lumped elements modeling in interior penalty discontinuous Galerkin time-domain
methods”, Journal of Computational Physics, vol. 229(12), pp. 8521-8536, 2010.
[2]. L. D. Angulo, J. Alvarez, M. F. Pantoja, S. G. Garcia, and A. R. Bretones, “Discontinuous Galerkin time domain methods in
computational electrodynamics: state of the art,” Forum for Electromangetic Research Methods and Application Technologies (FERMAT),
vol. 10, pp. 1-24, 2015.
[3]. D. Sarmany, M. A. Botchev, and J. J. Vegt, “Dispersion and dissipation error in high-order Runge-Kutta discontinuous Galerkin
discretisations of the Maxwell equations,” J. Sci. Comput., vol. 33, pp. 47–74, October 2007.
[4]. A. Glaser and V. Rokhlin, “A new class of highly accurate solvers for ordinary differential equations,” J. Sci. Comput., vol. 38, no. 3,
pp. 368–399, 2009.
[5]. J. Niegemann, R. Diehl, and K. Busch, “Efficient low-storage Runge–Kutta schemes with optimized stability regions,” J. Comput.
Phys., vol. 231, no. 2, pp. 364 – 372, 2012.
[6]. V. Dolean, H. Fahs, L. Fezoui, and S. Lanteri, “Locally implicit discontinuous Galerkin method for time domain electromagnetics,” J.
Comput. Phys., vol. 229, pp. 512–526, Jan. 2010.
[7]. A. Kanevsky, M. H. Carpenter, D. Gottlieb, and J. S. Hesthaven, “Application of implicit-explicit high order Runge-Kutta methods to
discontinuous-Galerkin schemes,” J. Comput. Phys., vol. 225, no. 2, pp. 1753 – 1781, 2007.
[8]. S. Dosopoulos, Interior Penalty Discontinuous Galerkin Finite Element Method for the Time-Domain Maxwell’s Equations. PhD
thesis, Ohio State University, 2012.
[9]. L. D. Angulo, J. Alvarez, F. Teixeira, M. Pantoja, and S. Garcia, “Causal-path local time-stepping in the discontinuous Galerkin
method for Maxwell’s equations,” J. Comput. Phys., vol. 256, pp. 678 – 695, 2014.
[10]. Stylianos Dosopoulos, Judith D. Gardiner, and Jin‐Fa Lee, “An MPI/GPU parallelization of an interior penalty discontinuous
Galerkin time domain method for Maxwell’s equations,” RADIO SCIENCE, VOL. 46, RS0M05, 2011.
[11]. Salvador G. Garcia, M. Fern´andez Pantoja, A. Rubio Bretones, R. G´omez Mart´ın, and Stephen D. Gedney, “A hybrid DGTD–
FDTD method for RCS calculations,” APS, pp. 3500-3503, 2007.
[12]. J.F Chen, Q.H. Liu, “Discontinuous Galerkin Time-Domain Methods for Multiscale Electromagnetic Simulations: A Review,”
Proceedings of the IEEE, Vol. 101, No. 2, pp. 242-254, 2013.
7
Outline
DGTD Fundamentals
Memory Efficient Scheme Based on Universal Matrices
DGTD Based on Marching-On-in-Degree (DGTD-MOD)
Hybrid Boundary Integral Method and DGTD-MOD
Conclusion
Low Storage Scheme Based on Universal Matrices
( )( ) ( ) ( )
( )( ) ( ) ( )
kk j k i k j k ik k k
H E
ik j k i k j k ik i k
E H
d
dt
d
dt
eM S h P h F h P e F e
hM S e P e F e P h F h
Semi-discrete form
8
Shared Matrices[1]
ˆ
ˆ
k
k
k
k
i j
V
k
i j
V
k
i j
V
dV
dS
dS
S Φ Φ
F Φ n Φ
P Φ n Φ
Independent
Matrices
• M, size N×N, full matrix
• Pν, size N×N, sparse matrices
• Fν, size N×N, sparse matrices
Duplicate unknowns on the same edges/faces
Memory cost
[1]. L. D. Angulo, J. Alvarez, M. F. Pantoja, S. G. Garcia, and A. R. Bretones, “Discontinuous Galerkin Time Domain Methods in
Computational Electrodynamics: State of the Art,” Forum for Electromagnetic Research Methods and Application Technologies (FERMAT) .
k
k
i j
V
dV M Φ Φ
z
yx
1
1
1
L1
L2
L3
, ,i i i i iL x y z a b x c y d z Barycentric coordinate
3
1 2 3
1
, , , ,i i im m
m
x y z L L L L
Φ ΦHierarchical Vector basis
3
1 1
6m
k mn
k m n
m n
V L L
M T
3 2 3
3 2 1
1 1 1
1 2 30 0 0
L L Lmn
ij mn im jn in jmL L L
T dL dL dL
Geometry Free!
9
0.5,
1,mn
m n
else
Low Storage Scheme Based on Universal Matrices
Only 6 matrices need to be calculated and stored
,
,
,
3 3,
1 1
ˆ ˆ
ˆ ˆ
k f
k f
k f
i j
V
f mn
V m n
m n
dS
S L L
P Φ n n Φ
n n P
,
,
,
3,
1 1
ˆ ˆ
ˆ ˆ
k f
k f
k f
i j
V
mf mn
V m n
m n
dS
S L L
F Φ n n Φ
n n F
,
,
k f
f mn
ij mn im jn in jm
V
dS
Fwhere
,
,
k f
f mn
ij mn im jn
V
dS
Pwhere
6×4×9×4 matrices
1
1
1
L1
L2
L3
1
23
4
Different NodeNumber Orders
Face Index for the adjacent nthelement
Each Pv is decomposedinto 9 matrices
Face Index for the mth element 10
Low Storage Scheme Based on Universal Matrices
Only 6×4 matrices need to be stored
11
Conventional Scheme Low Storage Scheme
M: NDOFs×NDOFs, Ne
Pν: NDOFs×NDOFs, 4×Ne
Fν: NDOFs×NDOFs, 4×Ne
Tmn: NDOFs×NDOFs, 6 Fmn: NDOFs×NDOFs, 24 Pmn: NDOFs×NDOFs, 864
S : NDOFs×NDOFs, 1 Pκ: NDOFs×NDOFs, 4 Fκ: NDOFs×NDOFs, 4
The comparison of memory cost and CPUtime between low storage scheme andconventional scheme
PEC sphere with r=3m
Low Storage Scheme Based on Universal Matrices
6 12 20 300
3
6
9
12
15
Rat
io
DOFs/element
Memory ratio of conventional method to proposed method
CPU time ratio of proposed method to conventional method
12
0
30
60
90
120
150
180
210
240
270
300
330
-30
-25
-20
-15
-10
-5
0
-30
-25
-20
-15
-10
-5
0
No
rmal
ized
Rad
itio
n P
atte
rn (
dB
)
Low Storage Scheme Based on Universal Matrices
7 slotted waveguide
Elements Conventional Proposed
338279 7356.9MB 1587.1MB
0
30
60
90
120
150
180
210
240
270
300
330
-30
-25
-20
-15
-10
-5
0
-30
-25
-20
-15
-10
-5
0
Norm
aliz
ed R
adia
tion P
atte
rn (
dB
)
LS-DGTD
FEM
1 slotted waveguide
7 slotted waveguide4.6 times reduction
1st order basis function
13
0 60 120 180 240 300 360-45
-40
-35
-30
-25
-20
-15
-10
-5
0
5
10
15
Bis
tati
c R
CS
(d
Bm
)
(deg.)
LS-DGTD
Elements Conventional Proposed
242457 4734.1MB 1045.5MB
7×7 Vivaldi antenna array
Low Storage Scheme Based on Universal Matrices
14
Outline
DGTD Fundamentals
Memory Efficient Scheme Based on Universal Matrices
DGTD Based on Marching-On-in-Degree (DGTD-MOD)
Hybrid Boundary Integral Method and DGTD-MOD
Conclusion
0 5 10 15 20 25 30 35 40-0.50
-0.25
0.00
0.25
0.50
0.75
1.00
st
p=4
p=3
p=2
p=1
Mag
nit
ude
p=0
( ) ( )
0
, ,eN
m m
i i
i
t t
E r e r Φ r
( ) ( )
0
, ,hN
m m
i i
i
t t
H r h r Ψ r
Spatial Discretization
( ) ( )
,
0 0
( )
,
0 0
( ) ( )
,
0 0
( )
,
0 0
,
'
,
'
p e
p e
p h
p h
N Nm m
p p i
p i
N Nm
p p i i
p i
N Nm m
p p i
p i
N Nm
p p i i
p i
t st
st
t st
st
E r e r
e r Φ r
H r h r
h r Ψ r
Spatial and TimeDiscretization
/2st
p pst e L st
𝜙𝑝 → 0 as 𝑡 → ∞
Satisfy the casual property of unknown fields
temporal basis
function
DGTD Based on Marching-On-in-Degree (DGTD-MOD)
15
Original weak form
Previous order
fields
1( ) ( ) ( )
, , ,
0
( ) ( )
, ,
( ) ( )
, ,
1
2m m
m m
m m
pm m m
mi p j q j mi p j
qV V
m n
H mi p j mi p j
V V
m n
E mi p j mi p j
V V
s dV dV
dS dS
dS dS
Φ e r e r Φ h r
Φ n h r Φ n h r
Φ n n e r Φ n n e r
Scale factor
16
( )
( ) ( ) ( ) ( ) ( ) ( )
m m m
mm i i i i i
mi mi mi H Efk fkfkV V V
dV dV dSt
EΦ Φ H Φ n H H n E
DGTD Based on Marching-On-in-Degree (DGTD-MOD)
Analytical derivative form
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
m m n n m
ee E eh H ee E eh H e
m m n n m
he E hh H he E hh H h
M c M c P c P c F
M c M c P c P c F
discrete form of MOD Scheme
1
( ) ( )
,
0
2m m
m m
m
m
m
ee mi mj E mi mjij
V V
eh mi mj H mi mjij
V V
ee E mi njij
V
eh H i njij
V
pm m
e mi q jijqV
sdV dS
dV dS
dS
dS
s dV
M Φ Φ Φ n n Φ
M Φ Ψ Φ n Ψ
P Φ n n Φ
P Φ n Ψ
F Φ e r
1
( ) ( )
,
0
2
m m
m m
m
m
m
he mi mj E mi mjij
V V
hh mi mj H mi mjij
V V
he E mi njij
V
hh H mi njij
V
pm m
h mi q jijqV
dV dS
sdV dS
dS
dS
s dV
M Ψ Φ Ψ n Φ
M Ψ Ψ Ψ n n Ψ
P Ψ n Φ
P Ψ n n Ψ
F Ψ h r
Global but Free From CFL Condition
17
DGTD Based on Marching-On-in-Degree (DGTD-MOD)
p pAx b
(1) (1) (1) (1)
, 1 , , 1 ,
( ) ( ) ( ) ( )
, 1 , , 1 ,
[ , , , , , ,
, , , , , ]
p E p E pNe H p H pNh
M M M M T
E p E pNe H p H pNh
c c c c
c c c c
x
(1) (1) ( ) ( )[ , , , , ]M M T
p e h e hb F F F F
Global Sparse Matrix
Direct Solver
Intel MKL PARDISO
Iterative Solver
GMRES with ILU Preconditioner
18
DGTD Based on Marching-On-in-Degree (DGTD-MOD)
WR-75 waveguide with an iris
MethodDGTD-MOT
(LSERK4)DGTD-MOD
Number of Elements 8226 8226
Time Step (s) / Time scale factor
5.925e-14 s 8e10
Marching-on steps 25000(in time) 150(in degree)
CPU times (s) 10510 647
Simulation parameters
0.0 0.5 1.0 1.5-1.0
-0.5
0.0
0.5
1.0
Ey (
V/m
)
Time (ns)
DGTD-MOD
DGTD-MOT
Electric Field in the sample point
19
9.5
25
mm
19.05 mm
4.7
62
5 m
m
9.525 mm
0.1mm
Late-Time Instability of UPML in DGTD-MOD
0 5 10 15
-1.0
-0.5
0.0
0.5
1.0
Ey
(V/m
)
Time (ns)
Analytical solution
DGTD-MOD-UPML
DGTD-MOT-ABC
Method DGTD-MOT DGTD-MOD
Time Step (s) / Time scale factor 7.02e-13 s 8e10
Marching-on steps >600,000(in time) 150(in degree)
0 10 20 30 40-20
-10
0
10
20
Ey
(V/m
)
Time (ns)
DGTD-MOT-UPML
DGTD-MOD-UPML
DGTD-MOT-ABC
Eliminate late-time instability!
20
UPML is more accurate!
WR-75-circular waveguide junction and dual mode filter
8 9 10 11 12 13 14 15-60
-50
-40
-30
-20
-10
0
|Sij| (
dB
)
Frequency (GHz)
S11
by DGTD-MoD
S21
by DGTD-MoD
S11
by FEM in [1]
S21
by FEM in [1]
S11
by measurement in [1]
S21
by measurement in [1]
𝑠 = 1𝑒11, 𝑁𝑝 = 500
21
[1] O. Tuncer, B. Shanker,
and L. C. Kempel,
“Discontinuous Galerkin
inspired framework for
vector generalized finite
element methods,”
Antennas Propag. IEEE
Trans. On, vol. 62, no. 3,
pp. 1339–1347, 2014.
Dual mode filter𝑠 = 8𝑒10, 𝑁𝑝 = 3000Circular waveguide junction
10 11 12 13 14 15
-30
-20
-10
0
The proposed method
Measurement in [2]
S2
1 (
dB
)
Frequency (GHz)
[2] J. R. Montejo-Garai and J. Zapata, “Full-wave
design and realization of multicoupled dual-mode
circular waveguide filters,” IEEE T-MTT, vol. 43, pp.
1290-1297, June 1995
Parallel-planar waveguide filled with ENZ material
0.0
99
m
0.1
m
0 50 100 150 200 250 300-80
-60
-40
-20
0
Frequency (MHz)
Sij (
dB
)
S11
for DGTD-MOD
S21
for DGTD-MOD
S11
for FEM
S21
for FEM
t = 19 ns
t = 5.7 ns t = 12.3 ns
|E| (V/m)
22
Δt<2.78×10-20 S
23
Outline
DGTD Fundamentals
Memory Efficient Scheme Based on Universal Matrices
DGTD Based on Marching-On-in-Degree (DGTD-MOD)
Hybrid Boundary Integral Method and DGTD-MOD
Conclusion
Boundary Integral Method Integrated with DGTD-MOD
0 0,
S
V
S
ˆSJ n H ˆ ˆ
SM E n
n̂
Region I
Region II
Region III
Conventional PML and its disadvantages [1]
Regular PML Evanescent Wave
CFS-PMLLow-Frequency Propagation
Wave𝑺′ TF/SF surface
𝑺 Huygens’ surface
Γ Truncated surface
Illustration of DGTD-BI system
24
[1]. David Correia and Jian-Ming Jin, “Performance of regular
PML, CFS-PML, and second-order PML for waveguide problems,”
Inc. Microwave Opt. Technol. Lett. Vol. 48, pp. 2121–2126, 2006
1( ) ( ) ( ) ( ) ( )
, , , , ,
0
( ) ( )
, ,
1
2m m m m
m m
pm m m m n
mi p j q j mi p j H mi p j mi p j
qV V V V
m n
E mi p j mi p j
V V
s dV dV dS dS
dS dS
Φ e r e r Φ h r Φ n h r Φ n h r
Φ n n e r Φ n n e r
Recall the semi-discrete form of DGTD-MOD
Modify the fields in the virtual elements which are adjacent to the physical boundary Γ
S
Γ ( )
,
n
p jh r
( )
,
n
p je r
delay
Rt
c
25
Numerical Flux Central Upwind Penalized
𝜅𝐸1
2
𝑌+
𝑌 + 𝑌+𝑌+
𝑌 + 𝑌+
𝜅𝐻1
2
𝑍+
𝑍 + 𝑍+𝑍+
𝑍 + 𝑍+
𝜈𝐸 01
𝑌 + 𝑌+𝜏
𝑌 + 𝑌+
𝜈𝐻 01
𝑍 + 𝑍+𝜏
𝑍 + 𝑍+
Penalty factor
Boundary Integral Method Integrated with DGTD-MOD
0 0,
S
V
S
ˆSJ n H ˆ ˆ
SM E n
n̂
Region I
Region II
Region III
2
0
1,s s s sE L K HJ M JK LM
0
30
0
2
0
1 ( , )
4
1( , )
4
1( , )
4
S
t R c
S
S
F r t R cLF ds
R t
RF r d ds
R
RF r t R c ds
c R
2
3
1 ( , )
4
1( , )
4
S
S
F r t R c RKF ds
c t R
RF r t R c ds
R
1( ) ( )0
1 1 1
1( ) ( )
31 00
( )1
ˆ ˆ( ) ( ) ( )4 2
( )1
ˆ ˆ ( ) 2 ( 1) ( )2
n
N l llln n n
l l mS
n l m
l llln m l n
l m
l m
sRs ce r n h r n h r ds
R
sRR
c n h r n h rs R
1
1( ) ( ) ( )
2 21 1 1 1 10
( )1 1
ˆ ˆ ˆ ( ) ( ) ( ) ( )4 4 2
1
n
n n
N
Sn
N l N l llln n n
l l m llS S
n l n l m
ds
sRR
s R sRc n h r ds e r n e r n dsc c cR R
( )
31 1
ˆ( ) ( )4 n
N ln
l llS
n l
R sRe r n ds
cR
0
1
( ) ( ) ( ) ( )
( ) ( )
0
ll l l
l l l l
sR sRst st d st
c c
sR sRl l
c c
l l
with
26
Boundary Integral Method Integrated with DGTD-MOD
1 2 3 4 5 10 15 20-60
-55
-50
-45
-40
-35
-30
-25
Rel
ativ
e E
rror
(dB
)
min/h
mixed 1st order
mixed 2nd order
mixed 3rd order
Accuracy Benchmark
Electric Field Relative Error in a 0.5m×0.5m×0.25m parallel plate waveguide (s=1e10, Np=100).
Best accuracy
Bistatic RCS RMS error versus penalty factor. 1st order basis are used (s=1e10, Np=100).
0.0 0.2 0.4 0.6 0.8 1.0-50
-40
-30
-20
-10
0
RM
S E
rror
(dB
)
27
20 cm
1.5 cm
20
cm
1 cm
22 c
m
22 cm
TF/SF PlaneHuygens's Plane
Huygens's Plane
S-Parameter of SRR Ring
0 50 100 150 200 250 300-60
-50
-40
-30
-20
-10
0
Sij (
dB
m)
Frequency (MHz)
S11
by DGTD-TDBI-MOD
S21
by DGTD-TDBI-MOD
S11
by FEM
S21
by FEM
Split-ring-resonator structure: (a) geometry; (b) scattering
parameters.
(a)
(b)
𝑠 = 1𝑒10, 𝑁𝑝 = 100
28
RCS of Conical Antenna
0 30 60 90 120 150 180-75
-50
-25
0
25
Bis
tati
c R
CS
(d
Bsm
)
(degree)
DGTD-TDBI-MOD for 150 MHz
FEM for 150 MHz
DGTD-TDBI-MOD for 200 MHz
FEM for 200 MHz
DGTD-TDBI-MOD for 300 MHz
FEM for 300 MHz
t=15 ns t=20 ns
t=25 ns
|E| V/m
Conical antenna structure: (a) geometry; (b) bistatic RCS;
(c) electric field distribution.
(a)
(b)
(c) 29
Outline
DGTD Fundamentals
Memory Efficient Scheme Based on Universal Matrices
DGTD Based on Marching-On-in-Degree (DGTD-MOD)
Hybrid Boundary Integral Method and DGTD-MOD
Conclusion
30
Conclusion
• We proposed a memory efficient scheme for DGTD based on universal matrix, and the memory usage is significantly reduced with slightly higher CPU time.
• We integrated DGTD with Marching-On-in-Degree method. With this method, we eliminate the late-time instability of UPML in DGTD and overcome the CFL limitation imposed by the hmin.
• Boundary Integral Method is utilized in the DGTD-MOD method, numerical examples demonstrate the accuracy versus mesh size and penalty factor.
31
Thank You!
32