Computational Electromagnetics - Introduction · Why study electromagnetics (EM)? Reason #1: It’s cool! I Waves and ﬁelds ≈ magic! I Action at a distance. I What are ﬁelds?

Computational ElectromagneticsIntroduction

Dr. Wallace

Fall 2009

Dr. Wallace Computational Electromagnetics

Outline

Background

Lecture Description

Laboratory Description


Why study electromagnetics (EM)?

Reason #1: It’s cool!

I Waves and fields ≈ magic!

I Action at a distance.

I What are fields? How do they work?



























Reason #2: It’s fundamental.

I Electric, electrical, optical devices governed by EM

I Applications of EM: Satellite comm., wireless, GPS, radar,optical storage, fiber-optic networks, ...

I Often use a simpler description (e.g. digital chip)

I What happens when simplifying assumptions break down(V=0/1, prop. delay model, no cross-talk, etc.)?

I Knowing EM: Can develop new models.











































Reason #3: It fosters mathematical maturity.I Electromagnetics applies

Multivariate calculusVector algebra and calculusPartial differential equationsFourier transform theoryComplex analysisAsymptotic analysisLinear algebraStatistics

I A good way to become comfortable with these!



Reason #3: It fosters mathematical maturity.

I Electromagnetics applies











Multivariate calculus

Vector algebra and calculusPartial differential equationsFourier transform theoryComplex analysisAsymptotic analysisLinear algebraStatistics





Multivariate calculusVector algebra and calculus

Partial differential equationsFourier transform theoryComplex analysisAsymptotic analysisLinear algebraStatistics





Multivariate calculusVector algebra and calculusPartial differential equations

Fourier transform theoryComplex analysisAsymptotic analysisLinear algebraStatistics





Multivariate calculusVector algebra and calculusPartial differential equationsFourier transform theory

Complex analysisAsymptotic analysisLinear algebraStatistics





Multivariate calculusVector algebra and calculusPartial differential equationsFourier transform theoryComplex analysis

Asymptotic analysisLinear algebraStatistics





Multivariate calculusVector algebra and calculusPartial differential equationsFourier transform theoryComplex analysisAsymptotic analysis

Linear algebraStatistics





Multivariate calculusVector algebra and calculusPartial differential equationsFourier transform theoryComplex analysisAsymptotic analysisLinear algebra

Statistics













What is computational EM?

Maxwell’s Equations

∇ · D = ρv

∇ · B = 0

∇× E = −∂B

∂t

∇× H =∂D

∂t+ J

I Maxwell’s equations with BC’s give exact solution.

I Problem: Only solvable for very simple problems!

I Solution #1: Make approximations, simplifying assumptions

I Solution #2: Discretize, solve numerically (CEM)

I No best CEM method: Generality, accuracy, efficiency, ease ofimplementation




∇ · D = ρv

∇ · B = 0

∇× E = −∂B

∂t

∇× H =∂D

∂t+ J









∇ · D = ρv

∇ · B = 0

∇× E = −∂B

∂t

∇× H =∂D

∂t+ J









∇ · D = ρv

∇ · B = 0

∇× E = −∂B

∂t

∇× H =∂D

∂t+ J









∇ · D = ρv

∇ · B = 0

∇× E = −∂B

∂t

∇× H =∂D

∂t+ J









∇ · D = ρv

∇ · B = 0

∇× E = −∂B

∂t

∇× H =∂D

∂t+ J







Outline

Background

Lecture Description



Concepts covered: 1. Basic numerical techniques

f(x)

. . .

∆x

xxN = ba = x1

I Needed in other methods

I Numerical integration

I Random number generation

I Monte-Carlo methods



f(x)

. . .

∆x

xxN = ba = x1







f(x)

. . .

∆x

xxN = ba = x1







f(x)

. . .

∆x

xxN = ba = x1






Concepts covered: 2. Finite-difference (FD) methods

∂f (x)

∂x≈ f (x + ∆x)− f (x −∆x)

2∆x

I Discretize PDEs directly

I AdvantagesI Conceptually simpleI Easy to implement

I DisadvantagesI Derivatives amplify errors (accuracy)I Volumetric method (high model order)

I But, still used in practice!



∂f (x)

∂x≈ f (x + ∆x)− f (x −∆x)

2∆x

I Discretize PDEs directlyI Advantages

I Conceptually simpleI Easy to implement





∂f (x)

∂x≈ f (x + ∆x)− f (x −∆x)

2∆x







∂f (x)

∂x≈ f (x + ∆x)− f (x −∆x)

2∆x



I Disadvantages

I Derivatives amplify errors (accuracy)I Volumetric method (high model order)




∂f (x)

∂x≈ f (x + ∆x)− f (x −∆x)

2∆x







∂f (x)

∂x≈ f (x + ∆x)− f (x −∆x)

2∆x






Concepts covered: 2. FD applications

∂f (x)

∂x≈ f (x + ∆x)− f (x −∆x)

2∆x

I Quasi-static: capacitive structures, transmission lines

I Dynamic: wave equation

I Finite-difference time-domain (FDTD)



∂f (x)

∂x≈ f (x + ∆x)− f (x −∆x)

2∆x






∂f (x)

∂x≈ f (x + ∆x)− f (x −∆x)

2∆x






∂f (x)

∂x≈ f (x + ∆x)− f (x −∆x)

2∆x





Concepts covered: 3. Method of Moments (MoM)

I Cast PDEs into integral form analytically

I Green’s function analysis

I Only unknowns on surface need be modeled

I Radiating boundaries included automatically

I Very efficient and accurate!

I Price: extra analytical work










































Concepts covered: 4. Variational Methods

∇2Φ(x , y) = −f (x , y)

I =1

2

∫∫S[Φ2

x + Φ2y − 2f (x , y)Φ]dS

I Generalization of the moment method

I Convert the PDE into minimization of an integral equationI Concepts:

I Variational calculusI Converting PDEs to variational form (and back)I Rayleigh-Ritz methodI Weighted residual method



∇2Φ(x , y) = −f (x , y)

I =1

2

∫∫S[Φ2

x + Φ2y − 2f (x , y)Φ]dS






∇2Φ(x , y) = −f (x , y)

I =1

2

∫∫S[Φ2

x + Φ2y − 2f (x , y)Φ]dS


I Convert the PDE into minimization of an integral equation

I Concepts:I Variational calculusI Converting PDEs to variational form (and back)I Rayleigh-Ritz methodI Weighted residual method



∇2Φ(x , y) = −f (x , y)

I =1

2

∫∫S[Φ2

x + Φ2y − 2f (x , y)Φ]dS






∇2Φ(x , y) = −f (x , y)

I =1

2

∫∫S[Φ2

x + Φ2y − 2f (x , y)Φ]dS





Concepts covered: 5. Finite-element Method (FEM)

I Powerful technique for solving PDEs

I Used in many different disciplines

I Divide domain into subdomains (finite elements)

I Derive governing equation in each element (var. principle)

I Assemble elements

I Solve resulting minimization problem







I Assemble elements








I Assemble elements








I Assemble elements








I Assemble elements








I Assemble elements



Concepts covered: 6. Modern Developments

Diffracted

Ray

Transmission

Ray

Ray "tube" or "cone"

Source Specular

Reflection

Receive Point

Reception Sphere

Progress still being made in CEM!

I Ray-tracing and hybrid methods

I Finite-volume generalizations of FDTD (CST)

I Fast-multipole method for MoM

I Domain decomposition for repeated structures




Diffracted

Ray

Transmission

Ray


Source Specular

Reflection

Receive Point

Reception Sphere









Diffracted

Ray

Transmission

Ray


Source Specular

Reflection

Receive Point

Reception Sphere









Diffracted

Ray

Transmission

Ray


Source Specular

Reflection

Receive Point

Reception Sphere









Diffracted

Ray

Transmission

Ray


Source Specular

Reflection

Receive Point

Reception Sphere








Preliminary Schedule

Week Topic1 Intro

Basic Numerical Techniques: Integration, Monte Carlo Methods2 FD Methods: Laplace Equation

Example: Transmission Lines3 1D Wave Equation, Absorbing Boundaries

Stability of FD Solutions4 Waveguides; Modal Solutions

FDTD: Yee cells, discretization, algorithm5 Materials, Sources, scattering

Post processing, 2D FDTD6 Perfectly Matched Layer (PML)

7 Reading DayMidterm

8 Method of Moments: Green’s Function, Laplace Eq.Capacitance Computation

9 Scattering problems, Far-FieldsRadiation problems

10 Volumetric MoMVariational Methods: Variational Calculus

11 Rayleigh Ritz MethodEigenvalue problems, Weighted residuals

12 Finite-Element Method (FEM)Element Assembly

13 Modern Developments: Ray-tracing; Hybrid MethodsFast Multipole Method

14 Finite-Volume Time DomainProject Presentations


Prerequisite Material

I Programming (mainly for lab)

I Engineering math: Multivariate calculus, differentialequations, Fourier transforms

I Vector analysis: orthogonal coordinate systems, dot/crossproduct, divergence, curl

I Basic electromagnetics


























Textbook

I No good comprehensive source on CEM

I Sadiku book (comprehensive, but a little too sketchy)

I Class notes (main source of information)

I Course website: http://www.xwallace.com/courses/cem


Textbook






Textbook






Textbook






Academic integrity

I Expected to follow as student at Jacobs

I Do your own work on assignments, labs

I You can get help, but ultimately, you should write your owncodes and own solutions

I I will not tolerate obviously copied work!


Academic integrity






Academic integrity






Academic integrity






Academic integrity






Grading

Weight of gradeHomework assignments 20%Midterm 40%Final 40%


Outline

Background

Lecture Description



Laboratory Part

Purpose of Laboratory

I Practical experience implementing algorithms

I How do you learn CEM without the C?

I Visualize what the math means

Format

I About one assignment per week

I Designed to take less than 3 hours

I Use own computer

I Any programming environment fine (Matlab, Octave, etc.)

I Project


Laboratory Part





Format



I Use own computer


I Project


Laboratory Part





Format



I Use own computer


I Project


Laboratory Part





Format



I Use own computer


I Project


Laboratory Part





Format



I Use own computer


I Project


Laboratory Part





Format



I Use own computer


I Project


Laboratory Part





Format



I Use own computer


I Project


Laboratory Part





Format



I Use own computer


I Project


Laboratory Part





Format



I Use own computer


I Project


Laboratory Part





Format



I Use own computer


I Project


Lab Grading

Weight of gradeLab Assignments 70%Project Writeup 15%Project Present 15%


Thank you

See you next time!


Documents

Computational Electromagnetics - Introduction · Why study electromagnetics (EM)? Reason #1: It’s cool! I Waves and ﬁelds ≈ magic! I Action at a distance. I What are ﬁelds?