Lecture 1 -- Introduction to CEM

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11/25/2015

Lecture 1 Slide 1

EE 5320

Computational Electromagnetics (CEM)

Lecture #1

Introduction to CEM These notes may contain copyrighted material obtained under fair use rules. Distribution of these materials is strictly prohibited

InstructorDr. Raymond Rumpf (915) 747 ‐[email protected]

Outline

• What is CEM?• CEM wisdom• General concepts in CEM• Classification of methods• Overview of methods

Lecture 1 Slide 2



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Lecture 1 Slide 3

What is CEM?

Lecture 1 Slide 4

Computational Electromagnetics

DefinitionComputational electromagnetics (CEM) is the procedure we must follow to model and simulate the behavior of electromagnetic fields in devices or around structures.

Most often, CEM implies using numerical techniques to solve Maxwell’s equations instead of obtaining analytical solutions.

Why is this needed?

Very often, exact analytical solutions, or even good approximate solutions, are not available. Using a numerical technique offers the ability to solve virtually any electromagnetic problem of interest.

1 out

in

cosh2c

r Z

r

?c Z



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Lecture 1 Slide 5

Popular Numerical Techniques• Transfer matrix method• Scattering matrix method• Finite ‐difference frequency ‐domain• Finite ‐difference time ‐domain• Transmission line modeling method• Beam propagation method• Method of lines• Rigorous coupled ‐wave analysis• Plane wave expansion method• Slice absorption method• Finite element analysis• Method of moments• Boundary element method• Spectral domain method• Discontinuous Galerkin method

Lecture 1 Slide 6

CEM Wisdom



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Lecture 1 Slide 7

The Key to Computation is VisualizationIs there anything wrong? If so, what is it?

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Lecture 1 Slide 8

Golden Rule #1

(1.234567…) + (0.0123456…) = Lost two digits of accuracy!!

Why?

Solution: NORMALIZE EVERYTHING!!!

0

0

E E

0

0

H H

or

0 x k x

0 y k y

0 z k z

00 1 m

All numbers should equal 1.



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Lecture 1 Slide 9

Golden Rule #2

1. Golden Rule #1.2. Finite floating point precision introduces round ‐off errors.

Why?

Solution: MINIMIZE NUMBER OF COMPUTATIONS!!!

1. Take problems as far analytically as possible.2. Avoid unnecessary computations.

2 2

2exp

R x y R

g R

2 2

2

2exp

r x y

r g r

Never perform calculations.

Lecture 1 Slide 10

Golden Rule #3

1. It will run faster and more reliably.2. Easier to catch mistakes.3. Easier to troubleshoot.4. Easier to pick up again at a later date.5. Easier to modify.

Why?

Solution

1. Outline your code before writing it.2. Delete obsolete code.3. Comment every step.4. Use meaningful variable names.

Write clean code.• Well organized• Well commented

• Compact• No junk code



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Lecture 1 Slide 11

The CEM Process

There is a rhythm to computational electromagnetics

and it repeats itself constantly.

Starts with Maxwell’s equations and derives all the necessary equations to implement the algorithm in MATLAB.• Equations everywhere! Only a few are needed.• Implementation does not resemble the formulation.

Organizes the equations derived in the formulation and considers other numerical details.• Consider all numerical best practices.• Should end with a detailed block diagram.

Actually implements the algorithm in computer code.• Implementation should be simple and minimal.• The “art” of simulation begins here.• Practice, practice, practice!

Lecture 1 Slide 12

Don’t Be LazyA little extra time making your program more efficient or simulating a device in a more intelligent manner can save you lots of time, energy, and aggravation.



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Formulation Wisdom

• Derive equations as far and as simple as possible.• Build big/complicated matrices from small/simple

matrices. – This usually requires converting to matrix form early in

the formulation.• Make your formulation documents very detailed.• A good understanding of the formulation gives

you the ability to modify your algorithm or to

add/subtract features.

Lecture 1 Slide 13

Implementation Wisdom

• Make a detailed block diagram!• Include only the equations you will code.• Add all other steps

– Sources – Building devices – Extracting information – Post ‐processing data – Etc.

Lecture 1 Slide 14



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Coding Wisdom• Work hard to write clean, simple, and well

commented code. – Indent code inside loops, if statements, etc. – Let linear algebra do the work for you. – Use lots of comments.

• Match your code to your formulation. – Try to use the same variables.

• Do not fix your code with incorrect equations. – Changing signs arbitrarily is a common way to make

things work, but you are hiding a problem and possibly creating more.

Lecture 1 Slide 15

Lecture 1 Slide 16

Simulation Wisdom #1

Simulate devices in multiple steps using models of increasing complexity.

Avoid the temptation to jump straight to the big, bad, and ugly 3D simulation in all of its glorious complexity.

Model your device with slowly increasing levels of complexity.

You will get to your final answer much faster this way!

R. C. Rumpf, “Engineering the dispersion and anisotropy in periodic electromagnetic structures,” Solid State Physics 66, 2015.



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Lecture 1 Slide 17


It must be standard practice to ensure your results are converged.

Grid Resolution

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f r a c t i v e

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supn

Lecture 1 Slide 18


Those who simulate the most, trust the simulations the least .

Never trust your code or your results. Benchmark. Benchmark. Benchmark.



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Lecture 1 Slide 19

Final Word of Wisdom

Do not EVER share your codes ! Only bad things will happen.

The best thing that can happenis that you become useless.

Instead, offer to simulate devices forthem and make yourself a collaborator.

Lecture 1 Slide 20

General Concepts in Computational EM



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Lecture 1 Slide 21

Physical Vs. Numerical Boundary Conditions

Physical Boundary ConditionsPhysical boundary conditions refer to the conditions that must be satisfied at the boundary between two materials. These are derived from the integral form of Maxwell’s equations.

Numerical Boundary ConditionsNumerical boundary conditions refer to the what is done at the edge of a grid or mesh and how fields outside the grid are estimated.

Tangential components are continuous

A

Lecture 1 Slide 22

Full Vs. Sparse Matrices

Full Matrices Sparse Matrices

A

Full matrices have all non ‐zero elements.

They tend to look banded with the largest numbers running down the main diagonal.

Sparse matrices have most of their elements equal to zero. They are often more than 99% sparse.

It is most memory efficient to store only the non ‐zero elements in memory.

They tend to “banded” matrices with the largest numbers running down the main diagonal.



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Lecture 1 Slide 23

Integral Vs. Differential Equations (1 of 2)

Integral Equations

Differential Equations

, f x x dx g x

df x

f x g xdx

Differential equations calculate a quantity at a specific point using only information from the local vicinity. They are usually written for points distributed throughout a volume and lead to formulations with sparse matrices. They require boundary conditions.

Integral equations calculate a quantity at a specific point using information from the entire domain. They are usually written around boundaries and lead to formulations with full matrices. They do not require boundary conditions.

Lecture 1 Slide 24

Frequency ‐Domain Vs. Time ‐Domain

This is what a frequency ‐domain code calculates.

This is what a time ‐domain code calculates.

Frequency ‐domain solutions are at a single frequency. Time ‐domain solutions look different because there are inherently a broad range of frequencies involved.



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Lecture 1 Slide 25

Definition of “Convergence”

Virtually all numerical methods have some sort of “resolution”

parameter that when taken to infinity solves Maxwell’s equations exactly. In practice, we cannot this arbitrarily far because a computer will run out of memory and simulations will take prohibitively long to run.

There are no equations to calculate what “resolution” is needed to obtain “accurate” results. Instead, the user must look for convergence. There are, however, some good rules of thumb to make an initial guess at resolution.

Convergence is the tendency of a calculated parameter to asymptotically approach some fixed value as the resolution of the model is increased. A converged solution does not imply an accurate solution !!!

Tips About Convergence

• Make checking for convergence a habit that you always perform.

• When checking a parameter for convergence, ensure that it is the only thing about the simulation that is changing.

• Simulations do not get more “accurate” as resolution is increased. They only get more

“converged.”

Lecture 1 Slide 26



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Lecture 1 Slide 27

How Do You Know if Your Model Works?

In many cases, you may not know.

1. BENCHMARK, BENCHMARK, BENCHMARK2. CONVERGENCE, CONVERGENCE, CONVERGENCE

Common Sense – Check your model for simple things like conservation of energy, magnitude of the numbers, etc.

Benchmark – You can verify your code is working by modeling a device with a known response. Does your model predict that response?

Convergence – Your models will have certain parameters that you can adjust to improve “accuracy” usually at the cost of computer memory and run time. Keep increasing “accuracy” until your answer does not change much any more.

When modeling a new device, benchmark your model using as similar of a device as you can find which has a known response. Compare your experimental results to the model. Do they agree? Reconcile any differences.

Lecture 1 Slide 28

Classification of Methods



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Lecture 1 Slide 29

Classification by Size ScaleLow Frequency Methods High Frequency Methods

0 a 0 a Structural dimensions are on the order of the wavelength or smaller.

Polarization and the vector nature of the field is important.

Structural dimensions much larger than the wavelength.

Fields can be accurately treated as scalar quantities.

• Finite ‐difference time ‐domain• Finite ‐difference frequency ‐domain• Finite element analysis• Method of moments• Rigorous coupled ‐wave analysis•

Method of

lines• Beam propagation method

• Boundary element method• Spectral domain method• Plane wave expansion method

• Ray tracing• Geometric theory of diffraction• Physical optics• Physical theory of diffraction• Shooting and bouncing rays

Lecture 1 Slide 30

Classification by ApproximationsRigorous MethodsA method is rigorous if there exists a “resolution” parameter that when taken to infinity, finds an exact solution to Maxwell’s equations.• Finite ‐difference time ‐domain• Finite ‐difference frequency ‐domain• Finite element method• Rigorous coupled ‐wave analysis• Method of lines

Full Wave MethodsA method is full wave if it accounts for the vector nature of the electromagnetic field. A full wave method is not necessarily rigorous.• Method of moments• Boundary element method• Beam propagation method

Scalar MethodsA method is scalar if the vector nature of the field is not accounted for.• Ray tracing



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Lecture 1 Slide 31

Comparison of Method TypesFrequency ‐Domain Time ‐Domain

Semi ‐AnalyticalFully Numerical

Fourier ‐SpaceReal ‐Space

+ wideband simulations+ scales near linearly+ active & nonlinear devices+ easily locates resonances

‐ longitudinal periodicity

‐ sharp resonances‐ memory requirements‐ oblique incidence

+ resolves sharp resonances+ handles oblique incidence+ longitudinal periodicity+ can be very fast

‐ scales at best N logN

‐ can miss sharp resonances‐ active & nonlinear devices

+ better convergence+ scales better than SA+ complex device geometry

‐ memory requirements‐ long uniform sections

+ very fast & efficient+ layered devices+ less memory

‐ convergence issues‐ scales poorly‐ complex device geometry

+ high index contrast+ metals+ resolving fine details+ field visualization

‐ slow for low index contrast + moderate index contrast+ periodic problems+ very fast and efficient

‐ field visualization‐ formulation difficult‐ resolving fine details

Unstructured GridStructured Grid+ easy to implement+ rectangular structures

+ easy

for

divergence

free

‐ less efficient‐ curved surfaces

+ most efficient+ handles larger structures

+ conforms

to

curved

surfaces

‐ difficult to implement‐ spurious solutions

Integral BasedDifferential Based+ sparse matrices+ easier to formulate+ easier to implement

‐ volume mesh‐ spurious solutions

+ surface mesh+ Very efficient for many structures

‐ full matrices‐ more difficult to formulate‐ more difficult to implement

Lecture 1 Slide 32

Multiphysics Simulations

A multiphysics simulation is one that accounts for multiple simultaneous physical mechanisms at the same time.

• Electromagnetic• Thermal• Fluids• Motion• Chemical• Acoustic• Optical



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Lecture 1 Slide 33

Any Method Can Do Anything

Any method can be made to do anything.

The real questions are:• What devices and information is a

particular method

best

suited

for?• How much of a “force fit” is it for that

method?

Lecture 1 Slide 34

Overview of the Methods



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Lecture 1 Slide 35

Transfer Matrix Method (1 of 2)

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,ref

x

y

E

E

,trn

,trn

x

y

E

E

,1

,1

x

x

E

E

,2

,2

x

x

E

E

,ref ,11

,ref ,1

x x

y x

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E E

T

2T

3T

Transfer matrices are derived that

relate the

fields

present

at

the

interfaces between the layers.

Transmission through all the layers is described by multiplying all the individual transfer matrices.

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,2 ,12

,2 ,1

x x

x x

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E E

T

,trn ,2

3,trn ,2

x x

y x

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E E

T

global 3 2 1T T T T

,trn ,ref

global,trn ,ref

x x

y y

E E

E E

T

Transfer Matrix Method (2 of 2)

• Very fast and efficient• Rigorous• Near 100% accuracy• Unconditionally stable• Robust• Simple to implement• Thickness of layers can be anything• Able to exploit longitudinal periodicity• Easily incorporates material dispersion• Easily accounts for polarization and

angle of incidence• Excellent for anisotropic layered

materials

Lecture 1 Slide 36

• Limited number of geometries it can model.

• Only handles linear, homogeneous and infinite slabs.

• Cannot account for diffraction effects

• Inefficient for transient analysis

This method is good for…1. Modeling transmission and reflection from layered devices.2. Modeling layers of anisotropic materials.

Benefits Drawbacks



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Lecture 1 Slide 37

Finite ‐Difference Frequency ‐Domain (1 of 2)

E E y z z y xx x

E E z x x z yy y

E E x y y x zz z

H H y z z y xx x

H H z x x z yy y

H H x y y x zz z

j

j

j

j

j

j

D e D e μ h

D e D e μ h

D e D e μ h

D h D h ε e

D h D H ε e

D h D h ε e

y z x

x z y

y x z

y z x

x z y

y x z

E E j H

y z

E E j H

z x E E

j H x y

H H j E y z

H H j E

z x H H

j E x y

1

Ax b

x A b

source

Space is converted to a grid and Maxwell’s

equations are

written

for

each

point

using

the finite ‐difference method.

This large set of equations is written in matrix form and solved to calculate the fields.

2 2, , , , y y z z z

E x y z E x y z E y y

x

y

z

e

x e

e

Finite ‐Difference Frequency ‐Domain (2 of 2)

• Accurate and robust• Highly versatile• Simple to implement• Easily incorporates dispersion• Excellent for field visualization• Error mechanisms are well

understood• Good method for metal devices• Excellent for volumetrically

complex devices• Good scaling compared to other

frequency ‐domain methods

Lecture 1 Slide 38

• Does not scale well to 3D• Difficult to incorporate

nonlinear materials• Structured grid is inefficient• Difficult to resolve curved

surfaces• Slow and memory innefficient

This method is good for…1. Modeling 2D devices with high volumetric complexity.2. Visualizing the fields.3. Fast and easily formulation of new numerical techniques.

Benefits Drawbacks



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Lecture 1 Slide 39

Finite ‐Difference Time ‐Domain (1 of 2)

Reflection PlaneTF/SF Planes

Unit cell of real device

Spacer Region

Spacer Region

Transmission Plane

Fields are evolved by iterating Maxwell’s

equations in small time steps.

Maxwell’s equations are enforced at each point at each time step.

Finite ‐Difference Time ‐Domain (2 of 2)

• Excellent for large ‐scale simulations. Easily parallelized.

• Excellent for transient analysis.• Accurate, robust, rigorous, and mature• Highly versatile• Intuitive to implement• Easily incorporates nonlinear behavior• Excellent for field visualization and learning

electromagnetics• Error mechanisms are well understood• Good method for metal devices• Excellent for volumetrically complex devices• Scales near linearly• Able to simulate broad frequency response in one

simulation• Great for resonance “hunting”

Lecture 1 Slide 40

• Tedious to incorporate dispersion• Typically has a structured grid

which is less efficient and doesn’t conform well to curved surfaces

• Difficult to resolve curved

surfaces• Slow for small devices• Very inefficient for highly

resonant devices

This method is good for…1. Modeling big, bad and ugly problems.2. Modeling devices with nonlinear material properties.3. Simulating the transient response of devices.

Benefits Drawbacks



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Lecture 1 Slide 41

Transmission ‐Line Modeling Method (1 of 2)

Space is interpreted as a giant 3D circuit.

Waves propagating through space are represented as current and voltage in extended circuits.

Also called transmission ‐line matrix method (TLM).

Transmission ‐Line Modeling Method (2 of 2)

• Essentially the same benefits at FDTD and FDFD.

• Excellent for large ‐scale simulations. Easily parallelized.

• Excellent for transient analysis.• No convergence criteria.• Inherently stable.• Time ‐ and frequency ‐domain

implementations exist.• Excellent fit with network theory

in microwave engineering.

Lecture 1 Slide 42

• Essentially the same drawbacks as FDTD and FDFD.

This method is good for…1. Modeling big, bad and ugly problems.2. Hybridizing models with microwave devices.3. Representing digital waveforms.

Benefits Drawbacks



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Lecture 1 Slide 43

Beam Propagation Method (1 of 2)

The beam propagation method (BPM) is a simple method to simulate

“forward” propagation through a device. It calculates the field one plane at a time so it does not need to solve the entire solution space at once.

1 2, , , , eff

1

11

eff eff 4 4

H E i xx i x zz i x xx i yy i

i ii i y

n

j z j z n n

A μ D μ D μ ε I

e I A I A e

Beam Propagation Method (2 of 2)

• Simple to formulate and implement (FFT‐BPM is easiest)

• Numerically efficient for faster simulations

• Well established for nonlinear materials (unique for frequency ‐domain method).

•

Easily incorporates dispersion• Excellent for field visualization• Error mechanisms are well

understood• Well suited for waveguide circuit

simulation

Lecture 1 Slide 44

• Not a rigorous method• Limited in the physics it can

handle• Typically uses paraxial

approximation•

Typically neglects backward reflections• FFT‐BPM is slower, less stable,

and less versatile than FDM‐BPM

This method is good for…1. Nonlinear optical devices.2. Devices where reflections and abrupt changes in the

field are negligible (i.e. forward only devices)

Benefits Drawbacks



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Lecture 1 Slide 45

Method of Lines (1 of 2)

x

y

z

BCs

BCs

BCs

BCs

s o u r c e r

e f l e c t e d

t r a n s m i t t e d

The method of lines is a semi ‐analytical method.

Modes are computed in the transverse plane for each layer and propagated analytically in the z-direction.

Boundary conditions are used to matched the fields at the interfaces between layers.

Transmission through the entire stack of layers is then known and transmitted and reflected fields can be computed.

Method of Lines (2 of 2)

• Excellent for longitudinally periodic devices

• Rigorous method• Excellent for devices with high

index contrast and metals•

Good for resonant structures• Less numerical dispersion than

fully numerical methods• Easier field visualization than

RCWA

Lecture 1 Slide 46

• Scales very poorly in the transverse direction

• Cumbersome method for field visualization

• Less efficient than RCWA for dielectric structures.

• Rarely used in 3D analysis, but this may change with more modern computers

This method is good for…1. Long devices.2. Long devices with metals.

Benefits Drawbacks



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Lecture 1 Slide 47

Rigorous Coupled ‐Wave Analysis (1 of 2)

Field in each layer is represented as a set of plane waves at different angles.

Plane waves describe propagation through each layer.

Layers are connected by the boundary conditions.

Rigorous Coupled ‐Wave Analysis (2 of 2)

• Excellent for modeling diffraction from periodic dielectric structures.

• Extremely fast and efficient for all‐dielectric structures with low to moderate index contrast

• Accurate and robust• Unconditionally stable• Thickness of layers can be anything without

numerical cost• Excellent for longitudinally periodic

structures.• Excellent for structures large in the

longitudinal direction.• Easily incorporates polarization and angle of

incidence.

Lecture 1 Slide 48

• Scales poorly in transverse dimensions.

• Less efficient for high dielectric contrast and metals due to Gibb’s phenomenon.

•

Poor method

for

finite

structures.

• Slow convergence if fast Fourier factorization is not used.

This method is good for…1. Modeling diffraction from periodic dielectric structures2. Periodic devices with longitudinal periodicity

Benefits Drawbacks



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Lecture 1 Slide 49

Plane Wave Expansion Method (1 of 2)

The plane wave expansion method (PWEM) calculates modes that exist in an infinitely periodic lattice. It represents the field in Fourier ‐space as the sum of a large set of plane waves at different angles.

Plane Wave Expansion Method (2 of 2)

• Excellent for all‐dielectric unit cells

• Fast even for 3D• Accurate and robust• Rigorous method

Lecture 1 Slide 50

• Scales poorly.• Weak method for high

dielectric contrast and metals.• Limited to modal analysis.• Cannot model scattering.• Cannot incorporate dispersion.

This method is good for…1. Analyzing unit cells2. Calculating photonic band diagrams and effective material properties.

Benefits Drawbacks



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Lecture 1 Slide 51

Slice Absorption Method (1 of 2)

Virtually any method that converts Maxwell’s equations to a matrix equation can order the matrix to give it the following block tridiagonal form.

This allows the problem to be solved one slice at a time.

Slice Absorption Method (2 of 2)

• Excellent for modeling devices with high volumetric complexity

• Easily incorporates dispersion• Easily incorporate polarization

and oblique incidence• Potential for transverse devices•

Excellent for finite size devices• Excellent framework to hybridize

different methods.• Transverse sources• Stacking in three dimensions.

Lecture 1 Slide 52

• New method and not well understood.

This method is good for…1. Modeling structures with high volumetric complexity2. Modeling finite size structures (i.e. not infinitely

periodic)

Benefits Drawbacks



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Lecture 1 Slide 53

Finite Element Method (1 of 2)Step 1: Describe Structure Step 2: Mesh Structure

Step 3: Build Global Matrix Step 4: Solve Matrix Equation

This is a VERY important and involved step.

1 1.0

2 2.5 01.5r

Iterate through each element to populate the global matrix.

Ax 0

Incorporate a source.

Ax b

Calculate field.

1x A b

Finite Element Method (2 of 2)

• Very mature method• Excellent representation of

curved surfaces• Unstructured grid is highly

efficient• Unconditionally stable• Scaling improved with domain

decomposition

Lecture 1 Slide 54

• Tedious to implement• Requires a meshing step

This method is good for…1. Modeling volumetrically complex structures in the frequency ‐

domain.

Benefits Drawbacks



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Lecture 1 Slide 55

Method of Moments (1 of 2)

Lf g

, ,n m n mn

a v Lv v g

1 11 1 1 2

1 22 1 2 2

,, ,

,, ,

, N N

a

a

a

v Lgv Lv v Lv

v Lgv Lv v Lv

v Lg

n nn

a Lv gn nn

a f v Galerkin Method Integral Equation• Usually uses PEC approximation• Usually based on current

2

22inc 2

2 4 L

jkr L

z z

j e E I z k dz

z r

• Converts a linear equation to a matrix equation

The Method of Moments

11 12 13 14 15 16 17 1

21 22 23 24 25 26 27 2

31 32 33 34 35 36 37 3

41 42 43 44 45 46 47 4

51 52 53 54 55 56 57 5

61 62 63 64 65 66 67 6

71 72 73 74 75 76 77 7

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Method of Moments (2 of 2)

• Extremely efficient analysis of metallic devices

• Full wave• Very fast• Excellent scaling using the fast

multipole method• No boundary conditions• Simple implementation• Mature method with lots of

literature• Can by hybridized with FEM

Lecture 1 Slide 56

• Not a rigorous method• Poor method for incorporating

dispersion and dielectrics• Long a tedious formulation• Inefficient for volumetrically

complex structures

This method is good for…1. Modeling metallic devices at radio frequencies2. Modeling large ‐scale metallic structures at radio frequencies

Benefits Drawbacks



11/25/2015

Lecture 1 Slide 57

Boundary Element Method (1 of 2)

Governing equation exists only at the boundary of a device so many fewer elements are needed.

400 elements5000 elements

The boundary element method (BEM) is also called the Method of

Moments, but is applied to 2D elements. The most famous element is the Rao‐Wilton ‐Glisson (RWG) edge element.

S. M. Rao, D. R. Wilton, A. W. Glisson, “Electromagnetic Scattering by Surfaces of Arbitrary Shape,” IEEE Trans. Antennas and Propagation, vol. AP‐30, no. 3, pp. 409 ‐418, 1982.

Boundary Element Method (2 of 2)

• Highly efficient when surface to volume ratio is low

• Excellent representation of curved surfaces

• Unstructured grid is highly

efficient• Unconditionally stable• Can be hybridized with FEM• Domain can extend to infinity• Simpler meshing than FEM

Lecture 1 Slide 58

• Tedious to implement• Requires a meshing step• Not usually a rigorous method• Inefficient for volumetrically

complex geometries

This method is good for…1. Modeling large devices with simple geometries.2. Modeling scattering from homogeneous blobs.

Benefits Drawbacks



11/25/2015

Lecture 1 Slide 59

Discontinuous Galerkin Method (1 of 2)

The discontinuous Galerkin method (DGM) combines features of the

finite element and finite ‐volume framework to solve differential equations.

Discontinuous Galerkin Method (2 of 2)

• Mesh elements can have any arbitrary shape.

• Fields may be collocated instead of staggered.

• Inherently a parallel method.•

Easily extended to higher ‐order of accuracy.• Allows explicit time ‐stepping• Low memory consumption (no

large matrices)

Lecture 1 Slide 60

• Tedious to implement• Requires a meshing step• Not usually a rigorous method• Inefficient for volumetrically

complex geometries

This method is good for…1. Solving very complex equations.2. Modeling very electrically large structures.3. Time ‐domain finite ‐element method.

Benefits Drawbacks

Documents

Lecture 1 -- Introduction to CEM