unconditionally stable laguerre polynomial-based ... · pdf filethe weighted laguerre...

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

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

( )( ) ( ) (

mm m m

mi s mi

mi ffk V

dV dVt

EΦ J Φ

( )( ) ( )

dV dVt

Maxwell’s Equation

Weak Form

Numerical Flux

( ) ( )

E r e r Φ r

( ) ( )

H r h r Ψ r

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

*( ) ( )

*(( ( )) )( )

i i i i i

H Efk fk

i i i i i

E Hfk fk

n H H n E

n E E n HE

Spatial Discretization

Galerkin Method

( ) ( ) ( )

(Source: Google.com)(Source: [1])

(Source: [2])

DGTD Fundamentals

Numerical Flux

Numerical flux Centered Penalized Upwind

( )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

n H H n E

n E E n HE

DGTD Fundamentals

( )( ) ( )

ii i j i i i j i ii i

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

Challenges In DGTD

Multiscale ProblemsAlternative Explicit Schemes

• Strong Stability Preserving RK (SSP-RK) Method

• 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

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

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.

Outline

DGTD Fundamentals

Conclusion

Low Storage Scheme Based on Universal Matrices

( )( ) ( ) ( )

kk j k i k j k ik k k

ik j k i k j k ik i k

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

Shared Matrices[1]

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) .

dV M Φ Φ

, ,i i i i iL x y z a b x c y d z Barycentric coordinate

, , , ,i i im m

x y z L L L L

Φ ΦHierarchical Vector basis

1 2 30 0 0

L L Lmn

ij mn im jn in jmL L L

T dL dL dL

Geometry Free!

Only 6 matrices need to be calculated and stored

P Φ n n Φ

F Φ n n Φ

ij mn im jn in jm

Fwhere

ij mn im jn

Pwhere

6×4×9×4 matrices

Different NodeNumber Orders

Face Index for the adjacent nthelement

Each Pv is decomposedinto 9 matrices

Face Index for the mth element 10

Only 6×4 matrices need to be stored

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

6 12 20 300

DOFs/element

Memory ratio of conventional method to proposed method

CPU time ratio of proposed method to conventional method

7 slotted waveguide

Elements Conventional Proposed

338279 7356.9MB 1587.1MB

tion P

LS-DGTD

1 slotted waveguide

7 slotted waveguide4.6 times reduction

1st order basis function

0 60 120 180 240 300 360-45

(deg.)

LS-DGTD

Elements Conventional Proposed

242457 4734.1MB 1045.5MB

7×7 Vivaldi antenna array

Outline

DGTD Fundamentals

Conclusion

0 5 10 15 20 25 30 35 40-0.50

( ) ( )

E r e r Φ r

( ) ( )

H r h r Ψ r

Spatial Discretization

( ) ( )

N Nm m

p p i i

N Nm m

p p i i

E r e r

e r Φ r

H r h r

h r Ψ r

Spatial and TimeDiscretization

p pst e L st

𝜙𝑝 → 0 as 𝑡 → ∞

Satisfy the casual property of unknown fields

temporal basis

function

Original weak form

Previous order

fields

1( ) ( ) ( )

( ) ( )

pm m m

mi p j q j mi p j

H mi p j mi p j

E mi p j mi p j

s dV dV

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

( ) ( ) ( ) ( ) ( ) ( )

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

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

discrete form of MOD Scheme

( ) ( )

ee mi mj E mi mjij

eh mi mj H mi mjij

ee E mi njij

eh H i njij

e mi q jijqV

sdV dS

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

( ) ( )

he mi mj E mi mjij

hh mi mj H mi mjij

he E mi njij

hh H mi njij

h mi q jijqV

sdV dS

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

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

(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

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

Time (ns)

DGTD-MOD

DGTD-MOT

Electric Field in the sample point

19.05 mm

9.525 mm

Late-Time Instability of UPML in DGTD-MOD

0 5 10 15

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

Time (ns)

DGTD-MOT-UPML

DGTD-MOD-UPML

DGTD-MOT-ABC

Eliminate late-time instability!

UPML is more accurate!

WR-75-circular waveguide junction and dual mode filter

8 9 10 11 12 13 14 15-60

|Sij| (

Frequency (GHz)

by DGTD-MoD

by FEM in [1]

by measurement in [1]

𝑠 = 1𝑒11, 𝑁𝑝 = 500

[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

The proposed method

Measurement in [2]

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 50 100 150 200 250 300-80

Frequency (MHz)

for DGTD-MOD

for FEM

t = 19 ns

t = 5.7 ns t = 12.3 ns

|E| (V/m)

Δt<2.78×10-20 S

Outline

DGTD Fundamentals

Conclusion

Boundary Integral Method Integrated with DGTD-MOD

ˆSJ n H ˆ ˆ

SM E n

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

[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( ) ( ) ( ) ( ) ( )

, , , , ,

( ) ( )

2m 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

E mi p j mi p j

s dV dV 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

Γ ( )

p jh r

p je r

Numerical Flux Central Upwind Penalized

𝜅𝐸1

𝑌 + 𝑌+𝑌+

𝑌 + 𝑌+

𝜅𝐻1

𝑍 + 𝑍+𝑍+

𝑍 + 𝑍+

𝜈𝐸 01

𝑌 + 𝑌+𝜏

𝑌 + 𝑌+

𝜈𝐻 01

𝑍 + 𝑍+𝜏

𝑍 + 𝑍+

Penalty factor

ˆSJ n H ˆ ˆ

SM E n

Region I

Region II

Region III

1,s s s sE L K HJ M JK LM

1 ( , )

1( , )

F r t R cLF ds

RF r d ds

RF r t R c ds

1 ( , )

1( , )

F r t R c RKF ds

RF r t R c ds

1( ) ( )0

1( ) ( )

ˆ ˆ( ) ( ) ( )4 2

ˆ ˆ ( ) 2 ( 1) ( )2

N l llln n n

l l mS

l llln m l n

sRs ce r n h r n h r ds

c n h r n h rs R

1( ) ( ) ( )

2 21 1 1 1 10

( )1 1

ˆ ˆ ˆ ( ) ( ) ( ) ( )4 4 2

N l N l llln n n

l l m llS S

n l n l m

s R sRc n h r ds e r n e r n dsc c cR R

ˆ( ) ( )4 n

R sRe r n ds

( ) ( ) ( ) ( )

( ) ( )

ll l l

l l l l

sR sRst st d st

sR sRl l

1 2 3 4 5 10 15 20-60

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

1.5 cm

TF/SF PlaneHuygens's Plane

Huygens's Plane

S-Parameter of SRR Ring

0 50 100 150 200 250 300-60

Frequency (MHz)

by DGTD-TDBI-MOD

by FEM

Split-ring-resonator structure: (a) geometry; (b) scattering

parameters.

𝑠 = 1𝑒10, 𝑁𝑝 = 100

RCS of Conical Antenna

0 30 60 90 120 150 180-75

(degree)

DGTD-TDBI-MOD for 150 MHz

FEM for 150 MHz

FEM for 200 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.

(c) 29

Outline

DGTD Fundamentals

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.

Thank You!

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