Introduction to Finite-Element-Methods In Electromagnetism

Nathan S. Froemming ([email protected])Northern Illinois University, PHYS 790DTue 08 Oct 2019

Introduction to Finite-Element-Methods In Electromagnetism

08 Oct 2019 Nathan S. Froemming | Introduction to Finite-Element Methods in Electromagnetism

Main Goal for Today….

• Be able to understand, setup, and solve basic problems in electrostatics and magnetostatics using finite-element methods, e.g. “Finite Element Method Magnetics” (FEMM), on your local Windows/Linux/Mac computer

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(quick real-time example)

http://www.femm.info/wiki/HomePage


Outline

• Overview LaPlace’s equation and solutions- Cauchy-Riemann equations; Connection to electromagnetism (EM)- Conformal mappings- Shortcomings• Finite-Element methods and codes (OPERA, COMSOL, FEMM, …)• Getting FEMM setup on your local machine• FEMM electric example- Problem setup: Electric dipole/quadrupole- How to extract the potential/E-field from FEMM• FEMM magnetic example (next time)- Problem setup: H-dipole, quadrupole (more challenging)- How to extract the potential/B-field from FEMM• Pro Tips And Tricks (next time)- Scripting in `lua` and `python`; Jupyter notebooks; WebPlotDigitizer; etc.• Conclusions

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• I am Prof. Syphers’ postdoc. I am stationed at Fermilab, and I work closely with Accelerator Division (AD) and the Muon g-2 Experiment (E989)

But first, A Few Quick Words About Me….

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Example Finite-Element Field


• [Accelerator] Physics has many applications elsewhere, e.g. medicine

• The ability to perform high-level data analysis is always sought after!

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I’ll Also Try To Talk About The Bigger Picture Along The Way

Date Presenter I Presentation Title �6

Outline



Laplace’s Equation

• In accelerator physics, most beamline elements (e.g. dipoles and quads) are static or slowly varying, so time-dependent terms in Maxwell’s equations are zero or negligible

• Additionally, we try not to pass the beam through other material (beam pipe, pole tip, electrode), so Maxwell’s equations simplify further:

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r ·B = 0, r⇥E+@B

@t= 0,

r ·E = ⇢/✏0, r⇥B� @E

@t= µ0j.

<latexit sha1_base64="lZatgnEh6MY1lExicLGHsTdBfFw=">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</latexit>

r ·B = 0, r⇥E = 0,

r ·E = 0, r⇥B = 0.<latexit sha1_base64="2+dpAhh0ZybsVKx2R2ykjADYBEg=">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</latexit>



• The curl equations imply the field may be derived from a potential,

• The divergence equations then imply Laplace’s equation,

• Also, don’t forget about these 3 copies of Laplace’s equation,

• Laplace’s equation is all over the place! We should spend some time trying to understand how to solve it….

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r · {E,B} = r · (�r{Ve, Vm}) = �r2{Ve, Vm} = 0.<latexit sha1_base64="vc3pMI/sd5TMQ+2oeGQUsaZqBuo=">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</latexit>

r⇥B = r⇥ (r⇥A) = r(r ·A)�r2A = 0 ) �r2A = 0<latexit sha1_base64="WJ3nuWYPdjvGvCdGK94hIDJneHQ=">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</latexit>

(Coulomb gauge)0

r⇥ {E,B} = 0 ) {E,B} = �r{Ve, Vm}.<latexit sha1_base64="MtHvbltQGwWhFPg4ONguVkh3pOo=">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</latexit>


• EM fields are derivable from potentials. The complex plane really helps with the analysis…

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f(z) = u(z) + iv(z), z ⌘ x+ iy 2 C.<latexit sha1_base64="CR26TOkeoX8PUsTI0iyAwntjX0I=">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</latexit>

f 0(z) ⌘ @f

@z= lim

�z!0

�f

�z= lim

�x!0�y!0

�u+ i�v

�x+ i�y<latexit sha1_base64="CfZUSiahoOkUQz9/A/WFxw7wMW4=">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</latexit>

z0*

ℂ

x

y

Suppose we have a particle here. We need to know the field (since F = q(E + v x B)), so we need to calculate a derivative of the potential

08 Oct 2019 Nathan S. Froemming | Introduction to Finite-Element Methods in Electromagnetism�10

• EM fields are derivable from potentials, and the complex plane really helps with the analysis:



z0*

ℂ

x

y

z

f 0(z) ⌘ @f

@z= lim

�z!0

�f

�z= lim

�x!0�y=0

�u+ i�v

�x+ i�y=

@u

@x+ i

@v

@x<latexit sha1_base64="gzBLkCB5unjWRULFbOXEw3ZCkr8=">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</latexit>

Approach1: Let Δy = 0, and approach z0 purely along the x-direction….





z0*

ℂ

x

yz

Approach2: Now let Δx = 0, and approach z0 purely along the y-direction

f 0(z) ⌘ @f

@z= lim

�z!0

�f

�z= lim

�x=0�y!0

�u+ i�v

�x+ i�y=

@v

@y� i

@u

@y<latexit sha1_base64="Y+rgA4nQtnrtfPlc1bk1zhEyZgs=">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</latexit>





z0*

ℂ

x

yz

If the derivative of f(z) exists, the two approaches outlined above must give the same result! Equating the real and the imaginary parts of the equations on the last two slides, we get the celebrated Cauchy-Riemann equations! 😊z

@u

@x= +

@v

@y,

@u

@y= �@v

@x.

<latexit sha1_base64="2owLUGfQJOwUEvUHNQ0D4XofXZ0=">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</latexit>

f 0(z) ⌘ @f

@z= lim

�z!0

�f

�z= lim

�x=0�y!0

�u+ i�v

�x+ i�y=

@v

@y� i

@u

@y<latexit sha1_base64="Y+rgA4nQtnrtfPlc1bk1zhEyZgs=">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</latexit>


• The Cauchy-Riemann equations are deeply connected to Laplace’s equation

• The Cauchy-Riemann (CR) equations imply two separate copies of Laplace’s equations, one for both the real and the imaginary parts of f(z)!

• On Thursday, we’ll talk more about multipole expansions, f(z) = Σ cn * zn



@u

@x= +

@v

@y,

@u

@y= �@v

@x.

<latexit sha1_base64="2owLUGfQJOwUEvUHNQ0D4XofXZ0=">AAACjnicjVFdS8MwFE3r16xfVR99CQ5BUEc7RYcyHPiyxwlOhbWMNE1dWPphkspK6c/xD/nmvzHdCuoU8UDgcO+5J8m5XsKokJb1rukLi0vLK7VVY219Y3PL3N65F3HKMenjmMX80UOCMBqRvqSSkceEExR6jDx445uy//BCuKBxdCezhLgheopoQDGSqjQ0Xx0vnhA/N2AFJ+AI506CuKSIwbT45JMCwjaER3OSly+SrDhWFs/PKfL/45jNHE/+cJwUDaMYmnWrYU0BfxK7InVQoTc03xw/xmlIIokZEmJgW4l089ITM1IYTipIgvAYPZGBohEKiXDzaZwFPFAVHwYxVyeScFr9OpGjUIgs9JQyRHIk5ntl8bfeIJVBy81plKSSRHh2UZAyKGNY7gb6lBMsWaYIwpyqt0I8QioaqTZoqBDs+S//JPfNhn3aaN6e1TutKo4a2AP74BDY4AJ0QBf0QB9gbU2ztUvtSjf1c72tX8+kulbN7IJv0Lsf51jDKA==</latexit>

@2xu = @x(@xu) = @x(@yv) = @y(@xv) = �@2

yu ) �(@2x + @2

y)u = 0

@2yv = @y(@yv) = @y(@xu) = @x(@yu) = �@2

xv ) �(@2x + @2

y)v = 0 .<latexit sha1_base64="JXuxb9IdPOTQ98oIYZvLUnUyYsc=">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</latexit>

Cauchy-Riemann (CR)


Conformal Transformations

• Conformal transformations of the complex plane are basically functions that preserve orientation and angles locally (“conformal”)

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Example from Mathematica

z

f(z)

ℂ

ℂExample: ElectricPotential / Field

http://demonstrations.wolfram.com/ElectrostaticFieldsUsingConformalMapping/


• Mathematical expressions quickly get out of control. Limited to all but this simplest cases. Nowadays, with computers, we can do much more….

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Conformal transformations (i.e. math equations on paper) were basically superseded by numerical methods on computers c. 1960-1970

Conformal Transformations => Finite-Element Methods


Finite-Element Methods (FEM)?

• Essentially, (FEM) “puts differential equations on a grid/mesh” and solves numerically with computers (fluid flow, heat flow, stress/strain, electromagnetism, etc.)

�16

“Finite Elements”(a.k.a. “mesh”)


• A simple example of solving Laplace’s equation via “grid relaxation”

�17

Finite-Element Methods (FEM): Laplace’s equation

Problem domain (“mesh”) Laplace’s equation on the grid

(quick derivation on the board)


• We can rewrite the last slide using “index notation”

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Problem domain (“mesh”) Laplace’s equation on the grid, u(i, j)

(this is the same equation as on the last slide)


• Let Δx = Δy = 1 for simplicity. Then the solution at u(i, j) is just the average value of its neighbors!

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Ask to see Prof.Syphers’ notebook :)


Outline



Getting FEMM Setup On Your Local Windows/Linux/Mac

• Instructions at the PHYS 790D course website (here)• Windows installation is very easy• Linux/Mac installation requires one additional step: Wine

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FEMM (a Windows application) WINE allows you to run Windows Apps on your Linux/Mac!

http://nicadd.niu.edu/~syphers/tutorials/FEMMinstall.html

https://www.winehq.org


• Electric problems are easier to setup in FEMM, so that’s where we’ll start• Two simple examples: (1) electric dipole, (2) electric quadrupole

�22

Electric storage ring: EDM measurements Example electric dipole FEMM simulation

E-fieldWe’ll look at

straight/curved longitudinal axes

FEMM Tutorial: Electrostatics



• Here we go! Start FEMM and create a new “Electrostatics Problem”

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• Can define global properties of the FEMM simulation here

�24



• Always save your work! (I chose filename = “femm-electric-dipole”)

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• Build a “vacuum chamber” shape. Start by defining corners & edges

�26

Add 4 corner points at {±10, ±10} by pressing <TAB> and entering

(x, y) coordinates



Now add edges by first selecting “segment” tool, then double-clicking any

two points

Selecting these two points will draw a segment

between them



• The vacuum chamber is at 0V — add a boundary condition

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• The vacuum chamber is at 0V — add a boundary condition

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Select line segments, press <SPACE BAR>, add boundary condition



• Add material, “Vacuum”

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Choose a name for your material, and

choose εx/ε0 = εy/ε0 = 1 (vacuum)



• Add material, “Vacuum”

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1. Select block-label tool. Press <TAB> and choose coordinates (x, y) = (0, 0)

2. Select your newly created block label at (x, y) = (0, 0)

3. Press <SPACE BAR> and apply your vacuum material definition from earlier

1 2

3



• Create the “outer electrode” by first drawing a line segment

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I chose electrode vertices (x, y) = (5, ±5) cm



• Define the potential for the outer electrode, e.g. +1kV

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• Apply potential to outer electrode, e.g. +1kV

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• Now create the “inner electrode” by drawing a line, as before

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Same steps as before. I chose (x, y) = (-5, ±5) cm

for vertices



• Define the potential for the inner electrode, e.g. -1kV

�36



• Apply potential to inner electrode, e.g. -1kV

�37




• Mesh, analyze, and see the results!

�38

1. Run mesh generator (mesh icon, yellow)2. Run analysis (gear icon)3. View results (glasses icon)

1, 2, 3



• This is what you should see. If not, I can help!

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• Suppose we want the E-field in the horizontal midplane….

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Same steps as before: Define a line via 2 points

(press <TAB>). Can hover mouse to see coordinates

listed at lower left.


• Now make a plot by clicking the “plot” icon. Can export to text file….

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• Our first results! (Does the plot make sense?)

�42

E = − ∇V


08 Oct ‘19 Nathan S. Froemming | Introduction to Finite Element Method Magnetics (FEMM)

• Now we’ll modify the dipole to make a quadrupole. Always save your work!

�43



• Add another couple of electrodes to make quadrupole

�44


Add 2 new electrodes. I chose to place the 2 new

electrodes at y = ±6 cm so they don’t touch the old

electrodes



Move old electrodes slightly for symmetry using the “move” tool


• Set electrode voltages for vertical focusing of positively-charged particles

�46


Apply the correct voltages for vertical focusing of

positively charged particles


• This is what your solution should look like. If not, I can help!

�47



• Plot the potential in the horizontal midplane. (Does it make sense?)

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• Plot the potential in the horizontal midplane. (Does it make sense?)

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• What about a curved longitudinal axis, i.e. cylindrical symmetry?

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• What about a curved longitudinal axis, i.e. cylindrical symmetry?

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• Does the new solution (with cylindrical symmetry) look any different?

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• Plot the potential in the horizontal midplane

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• Plot the potential in the horizontal midplane

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• Compare the two cases, i.e. Cartesian vs. Cylindrical coordinates

�55

E = − ∇VCurved

longitudinal axis

Straight longitudinal

axis

E = − ∇V



That’s All For Today — See You On Thursday!


Introduction to Finite-Element-Methods In Electromagnetism

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Transcript of Introduction to Finite-Element-Methods In Electromagnetism