Senior Design Student: Eric Smith

Supervisor: Dr. Branislav Notaros

Graduate Students: Ana Manic, Elene Chobanyan, Sanja Manic, Nada Sekeljic

Presentation Overview Introduction

Project Description

The Solution

Present/Future Work

Wrap up

Background Theory

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)

Goal, Budget, and Impact

Project Goal To create an automatic mesher which approximates as

general as possible structures by 1st to 4th quadrilaterals

Large Domain vs. Small Domain

26

6 9

20 23

12

15 18

17

2124

27

v

u

w

Specifications One Element Edge to One Element Edge

Specifications (Cont.) Convex Quadrilaterals

Specifications (Cont.) Maximum Edge Length Two Times the Operating

Wavelength

Element Quality

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

Gmsh, Simplification, and Extension to Higher Order

Approach Development (First Semester’s Work) Please contact for more information

Solution Overview Please contact for more information

End Result

Implementation and Test

Current Work Simplification Algorithm Implementation in MATLAB

[11]

Plans for Next Semester Please contact for more information

Acknowledgements and Conclusion

Acknowledgements Supervisor

Dr. Branislav Notaros

Website -- http://www.engr.colostate.edu/~notaros/

Electromagnetics Lab Graduate Students

Ana Manic

Elene Chobanyan

Sanja Manic

Nada Sekeljic

Conclusion Solution Found

Implementation Next

Test After

Comments and Questions?

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].