Post on 18-Feb-2018
transcript
Finite Element Method A Perfectly Matched Layer for
Computational Electromagnetics Team: Aaron Smull (Undergraduate, ECE)
Ana Manic (Graduate, ECE)
Sanja Manic (Graduate, ECE)
Advisor: Dr. Branislav Notaros
Overview
I. Finite Element Method and Computational Electromagnetics
II. Project Goals
III. Theoretical Modeling in FEM
IV. Anisotropic and Inhomogeneous Media
V. Boundary Conditions and the Perfectly Matched Layer
VI. Applications
VII. Budget
VIII. Future Plans
Finite Element Method (FEM) in Computational Electromagnetics
www.ansys.com
Computational Electromagnetism focuses on modeling the interaction of electromagnetic fields with physical objects and the environment.
FEM is a numerical method that solves a partial differential equation (wave equation)
FEM requires a Finite Domain – boundary conditions need to be applied at the surface of the modeled domain.
0r20
-1r =−×∇×∇ EE εµ k
www.emcos.com
www.wipl-d.com
Project Goals
Primary Goal: Implement an efficient perfectly matched layer (PML) scheme for existing finite element method code in order to simulate open-space.
Requires modeling of inhomogeneous, anisotropic materials.
Speed up and optimize the existing code, increase readability and make it more user-friendly.
www.feko.info
PML
FEM Geometrical Modeling Discretization into generalized curvilinear hexahedron
;
))...()()...()(())...()()...()((
)(
0
1110
1110
∏≠=
+−
+−
−
−=
−−−−−−−−−−
=
K
mjj jm
j
Kmmmmmmm
KmmKm
uuuu
uuuuuuuuuuuuuuuuuuuuuL
; ),,(ˆ),,(0001
lnmmnl
K
l
K
n
K
m
M
i
Kii wvuwvuLwvu
wvuuvw rrr ∑∑∑∑
====== 1,,1 ≤≤− wvu
Ridged circular-to-rectangular waveguide H-plane T-junction
large domain: 2λ small domain: λ/10
FEM field modeling, testing, integration
( ) ( ) ( ) rukj
iuijk wPvPuwvu af ,, =
Juwr
vaaa ×
=
Jwvr
uaaa ×
=
Jvur
waaa ×
=
uu d/ra ∂=
ww d/ra ∂=
vv d/ra ∂=
( ) wvuJ aaa ⋅×=
( ) ( ) ( ) rvk
iivijk wPvuPwvu af ,, =
( ) ( ) ( ) rw
kjiwijk wvPuPwvu af ,, =
wijkwijk
N
k
N
j
N
ivijkvijk
N
k
N
j
N
iuijkuijk
N
k
N
j
N
i
wvuwvuwvufffE
1
0000
1
0000
1
0α+α+α= ∑∑∑∑∑∑∑∑∑
−
====
−
====
−
=
Basis functions - curl-conforming hierarchical polynomials of arbitrary orders:
Testing - Galerkin method
0r20
-1r =−×∇×∇ EE εµ kCurl-curl electric-field vector wave equation:
Field expansion
0d )1( r20
r=ε−×∇
µ×∇⋅∫∫∫
Vuijk Vk EEf
0d )1( r20
r=ε−×∇
µ×∇⋅∫∫∫
Vvijk Vk EEf
0d )1( r20
r=ε−×∇
µ×∇⋅∫∫∫
Vwijk Vk EEf
w
v
u
NkNjNi
,...,0,...,0,...,0
===
)1)(1)(1(3 +++= wvu NNNN
≥−≥−=+=−
=
odd ,3,even ,2,1
1,10,1
)(
iuuiuiuiu
uP
i
ii
Anisotropic and Inhomogeneous Media
PML requires modeling of general anisotropic, inhomogeneous media.
Lagrange interpolation scheme for medium parameters:
Reimplementation and Optimization
Lagrange Polynomials are now pre-computed and stored in lookup tables for fast access during matrix filling.
∑∑∑= = =
=
=eu
ev
ew e
wev
eu
M
m
M
n
M
p
Mp
Mn
Mm
emnp
ezz
ezy
ezx
eyz
eyy
eyx
exz
exy
exx
e wLvLuLwvuwvuwvuwvuwvuwvuwvuwvuwvu
wvu0 0 0
r,
r,r,r,
r,r,r,
r,r,r,
r )()()(),,(),,(),,(),,(),,(),,(),,(),,(),,(
),,( εεεεεεεεεε
ε
∏≠= −
−=
eue
uK
ill li
lKi uu
uuuL0
)( 1,,1 ≤≤− wvu
Validation of New Code
0 30 60 90 120 150 180-50
-40
-30
-20
-10
0
10
Nor
mal
ized
mon
osta
tic φ
φ R
CS
[dB
]
φ [degrees]
Higher order FEM HFSS 3 layers HFSS 5 layers HFSS 7 layers
m 1=aMHz 300=f
0 15 30 45 60 75 90 105 120 135 150 165 180-10
-8
-6
-4
-2
0
2
4
6
8
10
12
14
Norm
alize
d bi
stat
ic φφ
RCS
[dB]
φ [degrees]
Higher order FEM HFSS Incidence θ = 90°, φ = 90° θ = 90°, φ = 0°
=ε
100040004
r
m 1=a
MHz 150=f
;11;1000100089
)(
2
r ≤≤−
−= u
uuε
FEM: 1 curved hexahedral element HFSS: 412,592 tetrahedral finite elements
Boundary Conditions
• FEM requires boundary condition to terminate the computational domain.
• Ex: MoM-FEM, ABCs, PEC
• First order absorbing boundary conditions (ABC) for scattering problems implemented.
• In general, the perfectly matched layer (PML) should provide better results.
PML
FEM-ABC Scattering Analysis of Dielectric Sphere
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
RCS
Abso
lute
Erro
r [dB
]
a/λ
N = 2 N = 3 N = 4 N = 5 N = 6 N = 7 N = 8
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0-35
-30
-25
-20
-15
-10
-5
0
5
Mie's Series
1st order ABC Polyn. order Unknowns
N = 8 11176
σ/(a
2 π) [d
B]
a/λ
25.2m,2.5bm,1 =ε== ra
Optimal choice:
N = 2,3 for a/λ < 0.4,
N = 4 for 0.4 < a/λ < 0.5,
N = 5 for 0.5 < a/λ < 0.6,
N = 6 for 0.6 < a/λ < 0.7,
N = 7 for 0.7 < a/λ < 0.8,
N = 8 for 0.8 < a/λ < 1, NGL = N + 4
Entire-domain model of second geometrical order
V
Scatterer
µε ,
rn ˆˆ =
SV
µε 00 ,
rincE
incn̂
scn̂
scE
ABC
Accuracy of the results is limited by the accuracy of first-order ABC!
Perfectly Matched Layer
Absorptive layer designed to exhibit zero reflection of outgoing scattered waves (simulate open space)
Extra degree of freedom – complex stretching factors
zszsxs zyxs ∂
∂+
∂∂
+∂∂
=∇111
0r20
-1r =−×∇×∇ EE εµ kss
0r20
-1r
-1 =Λ−×∇Λ×∇ EE εµ
k
In terms of “stretched” coordinates:
=Λ
z
yx
y
xz
x
zy
SSS
SSS
SSS
00
00
00
PML
Current Work: Reformulation of The Wave Equation
Existence of the PML requires theoretical reformulation of our usual method.
Attenuating media does not allow wave excitation from the outside.
Unknowns are now coefficients for the scattered field, rather than the total field
SVkVScS
Incidentkji
V
Incidentkji
V
Incidentkji ∫∫∫ ⋅×∇×−⋅−×∇⋅×∇ d d d )()( -1
rˆˆ̂rˆˆ̂20
-1rˆˆ̂ EfEfEf µεµ
=⋅×∇×−⋅−×∇⋅×∇ ∫∫∫ SVkVDS
Scatteredkji
V
Scatteredkji
V
Scatteredkji d d d )()( -1
rˆˆ̂rˆˆ̂20
-1rˆˆ̂ EfEfEf µεµ
IncidentIncidentScatteredScattered kk EEEE r20
-1rr
20
-1r εµεµ
−×∇×∇=−×∇×∇
[ ] [ ] [ ]( ){ } { } { } { }IncidentIncidentIncidentScattered SBkASBkA −−=−− 20
20 α
Possible Applications
Medical Use: Scattering from the Human Body
Antenna Design
Military Use: Radar signature of aircraft
Budget ECE Senior Design Budget: $100 Cost for second semester presentation <$10 for poster materials All other costs are covered; development tools are licensed by
CSU.
Continuation of PML Implementation
Extension of project to include conformal PML and possibly second order PML
Extensive testing of scattering structures to determine optimal PML parameters
Further Code Optimizations
Future Plans
References
[1] Ilić, M. “Higher order hexahedral finite elements for electromagnetic modeling”, University of Massachusetts Dartmouth, May 2003.
[2] M. M. Ilic and B. M. Notaros, “Higher order large-domain hierarchical FEM technique for electromagnetic modeling using Legendre basis functions on generalized hexahedra,” Electromagnetics, vol. 26, no. 7, pp. 517–529, Oct. 2006.
[3] Jin, J. M. and D. J. Riley, Finite Element Analysis of Antennas and Arrays, John Wiley & Sons, New York, 2008.
[4] Jin, J. M., Theory and Computation of Electromagnetic Fields, Wiley, 2010.
[5] Elene Chobanyan, PhD Dissertation Defense Presentation, December 1, 2014
[6] http://www-sop.inria.fr/nachos/