Senior Design Student: Eric Smith
Supervisor: Dr. Branislav Notaros
Graduate Students: Ana Manic, Elene Chobanyan, Sanja Manic, Nada Sekeljic
What is Computational Electromagnetics? Solving Maxwell’s Equations Using Computational
Techniques
t
BE
t
DJH
ρ D
0 BMichael Faraday (1791-1867) James Clerk Maxwell (1831-1879)
James Clerk Maxwell, “A Treatise on Electricity and Magnetism,” 1873
Branislav Notaros, “Electromagnetics”
What is Geometry Processing? A structure’s volume or surface is automatically or
manually approximated by a set of interconnected elements
A.K.A Meshing
[1] [2]
Quadrilateral Element Description Locally defined, equidistant parametric coordinated
nodes mapped to Cartesian coordinate system through node placement and an interpolating polynomial function [3]
u
u
v
v
x
z
r()
u,v
yO
v = 1
v = 1
u = 1
u = 1
u = 1
v = 1
v = 1
u =
1
v
u
1,1 vu[4]
u v
vu
K
m
K
n
Kn
Kmmn vLuLvu
0 0
)()(),( rr
Quadrilateral Element Description (Cont.)
u00
10
01
11
v
u
v
00
01
02
03
0414
2434
441323
3343
1222
32
4211
21
31
4110 2030
40
Why Geometry Processing? Required for Computational Electromagnetics
Techniques
Computational Electromagnetics Tools Developed by B. Notaros and Students Method of moments (MoM) - Surface Integral Equation (SIE)
MoM - Volume Integral Equation (VIE)
Finite Element Method (FEM)
High Frequency Methods [Physical Optics (PO),…]
Hybrid Methods (FEM-MoM, MoM-PO, VIE-SIE, …)
Domain Decomposition (DD) – Diakoptics
Time Domain (TD) Techniques
Why Geometry Processing? (Cont.) Method of Moments
Numerical computation method used in several areas of science and engineering
f_a=discretized approximation of unknown function
a_i=unknown coefficient
f_i=defined known function (basis function)
L -> Linear Operator
f = Unknown Function (Current Sources)
g = Known function (Excitation)
gfL )(
N
iiia faf
1
gfLa i
N
ii
)(1
S S
iii
N
ii gdSfdSfLfa )(
1 [5]
1)
2)
3)
4)
Project Goal To create an automatic mesher which approximates as
general as possible structures by 1st to 4th quadrilaterals
Budget Senior Design Project Allowance $100
Considered Mesher
CATIA $1,560
Presentation Equipment for E-days (<$20)
Impact Improved Methods in Computational Electromagnetics
Improvements in Tools for Research, Design, and Development in Many Areas which Require Electromagnetic Solvers
Communication Systems
Antenna
Radar
High Frequency Electronics
Medical Imaging
Electromagnetic Compatibility and Interference
Acknowledgements Supervisor
Dr. Branislav Notaros
Website -- http://www.engr.colostate.edu/~notaros/
Electromagnetics Lab Graduate Students
Ana Manic
Elene Chobanyan
Sanja Manic
Nada Sekeljic
References [1] Yang. et. al. (2011, Dec.). “General Planar Quadrilateral Mesh Design Using Conjugate Direction Field.” ACM Transactions on Graphics. [Online]. (30)6. Available: http://research.microsoft.com/en-us/um/people/yangliu/publication/cdf.pdf [Dec. 3, 2012]. [2] D. Bommes, H. Zimmer, and L. Kobbelt. (2009, Aug.). “Mixed Integer Quandrangulation.” ACM Transactions on Graphics. [Online]. (28)3. Available: http://delivery.acm.org/10.1145/1540000/1531383/a77-bommes.pdf?ip=129.82.231.3&acc=ACTIVE%20SERVICE&CFID=152416474&CFTOKEN=93616921&__acm__=1354573047_b8c6b83b6c92941ab832b93705c0140a [Dec. 3, 2012].
References (Cont.) [3/4] M. Djordjevic and B. M. Notaros. (2004,Aug.). “Double Higher Order Method for Surface Integral Equation Modeling of Metallic and Dielectric Antennas and Scaterrers” IEEE Transactions on Antennas and Propogation. [Online]. (52)8. Available: http://www.engr.colostate.edu/~notaros/Papers/T-AP_Aug_2004.pdf [Dec. 3, 2012].
[5] B. Kolundzija and A. Djordjevic. “Method of Moments” in Electromagnetic Modeling of Composite Metallic and Dielectric Structures, 1st Ed., Vol. 1. Ed. Massachusetts: Artech House Inc, 2002, pp. 27-35.
[11] Internet: http://www.mathworks.com/, [Dec. 3, 2012].