Forum for Electromagnetic Research Methods and Applications Technologies (FERMAT)

Unstructured-grid and conservative particle-in-cell algorithm: Application to micromachined

beam-plasma slow-wave structures

D. Y. Na1, Y. A. Omelchenko2 and F. L. Teixeira1

1ElectroScience Laboratory, The Ohio State University, Columbus Ohio USA 2Trinum Research, San Diego, California, USA

Abstract: We present an accurate and efficient electromagnetic particle-in-cell (EMPIC) algorithm on unstructured grids for the analysis and design of axisymmetric slow-wave structures. The use of unstructured grids allows for more fidelity in the modeling of micromachined geometries. The use of a reduced dimensionality algorithm decreases the computational costs significantly and enables its integration as a forward engine into a design loop. Special gather and scatter steps are employed to yield accurate beam plasma dynamics and exact charge conservation on unstructured grids. We provide numerical examples involving travelling-wave tube amplifiers designed to harness bunching effects arisen from Cherenkov radiation from plasma electron beams.

Keywords: Maxwell-Vlasov equations, particle-in-cell, plasma, vacuum electronics devices.

References:

[1] R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles, 2nd ed., New York: McGraw-Hill, 1981.

[2] D. L. Bruhwiler et al., “Particle-in-cell simulations of plasma accelerators and electron-neutral collisions,” Phys. Rev. ST Accelerators and Beams, vol. 4, p. 10132, 2001.

[3] R. Zheng, P. Ohlckers, X. Chen, ”Particle-in-Cell Simulation and Optimization for a 220-GHz Folded-Waveguide Traveling-Wave Tube,” IEEE Trans. Electron Devices , vol. 58, no. 7, pp. 2164-2171, 2011.

[4] Z.-H. Li, “Investigation of an Oversized Backward Wave Oscillator as a High Power Microwave Generator,” Appl. Phys. Lett., vol. 92, p. 054102, 2008.

[5] H. Moon, F. L. Teixeira, and Y. A. Omelchenko, “Exact charge-conserving scatter-gather algorithm for particle-in-cell simulations on unstructured grids: A geometric perspective,” Comput. Phys. Comm., vol. 194, pp. 43-53, 2015.

[6] F. L. Teixeira, “Lattice Maxwell’s equations,” PIER, vol. 148, pp.113-128, 2014.

[7] C. S. Meierbachtol, A. D. Greenwood, J. P. Verboncoeur, and B. Shanker, “Conformal electromagnetic particle in cell: A review,” IEEE Trans. Plasma Sci., vol. 43, pp. 3778-3793, 2015.

[8] B. He and F. L. Teixeira, “Differential forms, Galerkin duality, and sparse inverse approximations in finite element solutions of Maxwell equations,’ IEEE Trans. Antennas Propagat., vol. 55, pp. 1359-1368, 2007.

[9] B. He and F. L. Teixeira, “Geometric finite element discretization of Maxwell equations in primal and dual spaces,” Phys. Lett. A, vol. 349, pp. 1-14, 2006.

[10] B. He and F. L. Teixeira, “On the degrees of freedom of lattice electrodynamics,” Phys. Lett. A, vol. 336, no. 1, pp. 1–7, 2005.

[11] F. L. Teixeira and W. C. Chew, “Lattice electromagnetic theory from a topological viewpoint,” J. Math. Phys., vol. 40, no. 1, pp. 169-187, 1999.

[12] J.-L. Vay, “Simulation of beams or plasmas crossing at relativistic velocity,” Phys. Plasmas, vol. 15, 056701, 2008.

[13] B. Donderici and F. L. Teixeira, “Mixed finite-element time-domain method for transient Maxwell equations in doubly dispersive media,” IEEE Trans. Microwave Theory Tech., vol. 56, pp. 113-120, 2008.

[13] D.-Y. Na, H. Moon, Y. A. Omelchenko, and F. L. Teixeira, “Local, explicit, and charge-conserving electromagnetic particle-in-cell algorithm on unstructured grids,” IEEE Trans. Plasma Sci., vol. 44, pp. 1353-1362, 2016.

*This use of this work is restricted solely for academic purposes. The author of this work owns the copyright and no reproduction in any form is permitted without written permission by the author. *

Unstructured-Grid and Conservative Particle-in-Cell Algorithm: Application to Micromachined Beam-Plasma Slow-Wave Structures

Dong-Yeop Na1, Yuri A. Omelchenko2, and Fernando L. Teixeira1

2016 IEEE AP-S/USNC-URSI Symposium

1ElectroScience Laboratory, The Ohio State University, Columbus, OH2Trinum Research Inc. San Diego, CA

2

Introduction

Modeling of beam-field interactions in vacuum electronics devices (VED)

Numerical Examples (TWT amplifier and BWO)

Particle-In-Cell (PIC) algorithm for plasma simulations: Field solver, gather, relativistic particle pusher, and scatter

OUTLINE

Concluding Remarks

3

Electromagnetic Particle-in-Cell (EM-PIC) Algorithms

Newton's Law

of MotionLorentz Force

Maxwell’s

Equations

Vlasov equation 𝑑𝑓𝑝 𝑟 𝑡 , 𝑣 𝑡 , 𝑡 /𝑑𝑡 = 0 (collisionless)

𝜕𝑓𝑝

𝜕𝑡𝑓𝑝 + 𝑣 ∙ 𝛻 𝑟𝑓𝑝 +

𝑞𝑝

𝑚𝑝𝐸 + 𝑣 × 𝐵 ∙ 𝛻𝑣𝑓𝑝 = 0

Shape factor 𝑓𝑝 represents spatial distribution

for an ensemble of particles (number density of particles), described by a smaller set of “super-particles” (coarse graining of the phase-space).

Electromagnetic particle-in-cell (EM-PIC) algorithm Numerical approach to solve the MV system Four major steps at each time-update

Maxwell-Vlasov System

4 steps for each time-update

Field-Update

Gather

Particle-Update

Scatter

Applications based on EM-PIC simulations Plasma physics and astrophysics Vacuum electron devices (VED)

4

Vacuum Electronics Devices (VED) VED are essential for high-power microwave (HPM) sources in variety of applications

including pulsed radar systems, RF amplification, and others. Slow-wave structures (SWS) are often used to slow down the phase velocity of waves and

excite Cerenkov radiation. Most VEDs are cylindrically axisymmetric (invariant along 𝜙).

Micromachined Slow-Wave Structures (SWS) Micromachining desirable for high performance operation at microwaves and, especially,

terahertz frequencies. Either structured grids with conformal capabilities or unstructured grids (general solution)

should be used to correctly capture the geometry of micromachined SWSs. Long-standing challenge: Conventional EM-PIC algorithms on unstructured grids do not

attain charge-conservation.

Main contribution of this work We introduce a EM-PIC algorithm for circularly symmetric VED which is (1) based on

unstructured grids and (2) attains charge conservation from first principles.

Motivation of Our Work

5

Reduction of Problem Dimensionality

3D cylindrical axisymmetry VED

(1) Problem geometry, (2) EM fields, and (3) sources are cylindrically axisymmetric (𝜕

𝜕𝜙= 0 ).

𝐸𝜌, 𝐸𝑧, 𝐵𝜙 in Ω𝑧

from charged particles to charged rings

(invariant w.r.t. 𝜙)

1. Reduction of the exterior derivative 𝑑 : 𝑑 =𝜕

𝜕𝜌𝑑𝜌 +

𝜕

𝜕𝑧𝑑𝑧

2. Consideration of only TM eigenmodes with 𝑚 = 0 (𝑚 is an index for azimuthal modes)

Ω

𝜕Ω = 𝑅 𝑧 (PEC)

𝑧

𝜌

𝜙

vacuum

Two useful constraints to reduce the dimensionality of 3D cylindrically axisymmetric problems

6

Reduction of Problem Dimensionality

Differential forms expressions for dynamic variables with a non orthonormal basis set

of 𝑑𝜌, 𝑑𝜙, 𝑑𝑧

ℰ = 𝐸𝜌𝑑𝜌 + 𝐸𝑧𝑑𝑧

ℬ = 𝐵𝜙𝑑𝑧 ∧ 𝑑𝜌

ℋ = 𝐻𝜙𝑑𝜙

𝒟 = 𝐷𝜌𝑑𝜙 ∧ 𝑑𝑧 + 𝐷𝑧𝑑𝜌 ∧ 𝑑𝜙

𝒥 = 𝐽𝜌𝑑𝜙 ∧ 𝑑𝑧 + 𝐽𝑧𝑑𝜌 ∧ 𝑑𝜙

𝒬 = 𝜌𝑣𝑑𝜌 ∧ 𝑑𝜙 ∧ 𝑑𝑧

𝐿𝐻𝑆 = 𝐻𝜙𝑑𝜙

𝑅𝐻𝑆 =⋆𝜇−1 𝐵𝜙𝑑𝑧 ∧ 𝑑𝜌

= 𝜇0−1𝐵𝜙 ⋆ 𝑑𝑧 ∧ 𝑑𝜌 = 𝜇−1 𝜌 𝐵𝜙𝑑𝜙

ℋ =⋆𝜇−1 ℬ

𝐿𝐻𝑆 = 𝐷𝜌𝑑𝜙 ∧ 𝑑𝑧 + 𝐷𝑧𝑑𝜌 ∧ 𝑑𝜙

𝑅𝐻𝑆 =⋆𝜖 𝐸𝜌𝑑𝜌 + 𝐸𝑧𝑑𝑧 =

= 𝜖0 𝐸𝜌 𝜌𝑑𝜙 ∧ 𝑑𝑧 + 𝐸𝑧 𝑑𝜌 ∧ 𝜌𝑑𝜙

= 𝜖 𝜌 𝐸𝜌𝑑𝜙 ∧ 𝑑𝑧 + 𝐸𝑧𝑑𝜌 ∧ 𝑑𝜙

𝒟 =⋆𝜖 ℰ

We introduce an (1) artificial inhomogeneous medium with dependency on ρ and (2) rescaled sources.

𝒥 =⋆ 𝒥⋆

𝒬 =⋆ 𝒬⋆

𝒥⋆ = 𝐽⋆𝜌𝑑𝜌 + 𝐽⋆𝑧𝑑𝑧

𝒬⋆ = 𝜌𝑣⋆

𝐿𝐻𝑆 = 𝐽⋆𝜌𝑑𝜙 ∧ 𝑑𝑧 + 𝐽⋆𝑧𝑑𝜌 ∧ 𝑑𝜙

𝑅𝐻𝑆 =⋆ 𝐽⋆𝜌𝑑𝜌 + 𝐽⋆𝑧𝑑𝑧

= 𝜌𝐽⋆𝜌 𝑑𝜙 ∧ 𝑑𝑧 + 𝜌𝐽⋆𝑧 𝑑𝜌 ∧ 𝑑𝜙

𝐿𝐻𝑆 = 𝜌𝑣𝑑𝜌 ∧ 𝑑𝜙 ∧ 𝑑𝑧

𝑅𝐻𝑆 =⋆ 𝜌𝑣⋆=

= 𝜌𝜌𝑣⋆𝑑𝜌 ∧ 𝑑𝜙 ∧ 𝑑𝑧

Hodge star operator for fields and sources*

* Vey much akin to “transformation optics”: Teixeira and Chew, J. Math Phys. 40 (1), 169-187 (1999).

7

scaled sourcesartificial

inhomogeneous medium

fields are TE𝜙-polarized

3D axisymmetric problems w.r.t. geometry, fields, and sources can be modelled by 2D equivalent problems by 𝐓𝐄𝝓-polarized fields, artificial inhomogeneous medium, and scaled sources.

𝑧

𝜌

𝜖 𝜌 = 𝜖0𝜌

𝜇 𝜌 = 𝜇0/𝜌

𝐸𝜌

𝐸𝑧𝐵𝜙

at the meridian (𝜌𝑧) plane (arbitrary 𝜙)

𝐽⋆𝜌′ 𝜌𝑣⋆

′

𝐽⋆𝑧′

The continuous ring of particles along 𝝓-axis matches a particle at the meridian plane.

𝑧

Reduction of Problem Dimensionality (cont’)

𝑧

𝜌

“ring” of particles

a particle at the meridian plane

8

𝑑ℰ = −𝜕ℬ

𝜕𝑡, 𝑑ℋ =

𝜕𝒟

𝜕𝑡+ 𝒥, 𝑑𝒟 = 𝒬, 𝑑ℬ = 0

⋆𝜖 𝑖𝑗 = 𝐾 𝜖 𝜌 𝓌𝑖1 ∧⋆ 𝓌𝑗

1

⋆𝜇−1𝑖𝑗

= 𝐾𝜇−1 𝜌 𝓌𝑖

2 ∧⋆ 𝓌𝑗2

ℰ = 𝑖=1𝑁1 𝔼𝑖

𝑛𝓌𝑖1 (1 − form) ℋ = 𝑖=1

𝑁0 ℍ𝑖

𝑛−1

2 𝓌𝑖0 (0 − form)

ℬ = 𝑖=1𝑁2 𝔹

𝑖

𝑛−1

2𝓌𝑖2 (2 − form) 𝒟 = 𝑖=1

𝑁1 𝔻𝑖𝑛 𝓌𝑖

1 (1 − form)

𝒬⋆ = 𝑖=1𝑁0 ℚ⋆𝑖

𝑛𝓌𝑖0 (0 − form)

𝒥⋆ = 𝑖=1𝑁1 𝕁⋆𝑖

𝑛+1

2𝓌𝑖1 (1 − form)

spatial discretization by expanding variables as

𝔹 𝑛+12 = 𝔹 𝑛−

12 + ∆𝑡 𝒟curl ∙ 𝔼 𝑛

⋆𝜖 ∙ 𝔼 𝑛+1 = ⋆𝜖 ∙ 𝔼 𝑛 + ∆𝑡 𝒟curl ∙ ⋆𝜇−1 ∙ 𝔹 𝑛+12 − 𝐼 ∙ 𝕁⋆

𝑛+12

Paring, generalized Stokes Theorem, and leapfrog time scheme

Discrete Hodge matricesMixed 𝓔 − 𝓑 FETD Scheme: Kim and Teixeira, IEEE T-AP, vol. 59, p. 2350 (2011)

Mixed 𝓔 − 𝓑 FETD Scheme for (2+1) Theory

𝒬 = 𝑖=1

𝑁2 ℚ𝑖𝑛 𝓌𝑖

0 (0 − form)

𝒥 = 𝑖=1

𝑁1 𝕁𝑖

𝑛+1

2 𝓌𝑖1 (1 − form)

𝐾 : primal mesh

𝐾 : dual mesh

𝑧

𝜌

ℰ

𝒥

ℬℋ

𝒟

⋆𝒬

𝒥⋆

𝒬

In primal mesh In dual mesh

9

Gather

EM field values are interpolated at the instant positions of particles by using interpolants of Whitney forms

𝐸𝑝𝑛 = 𝑖=1

𝑁1 𝔼𝑖𝑛𝑊𝑖

1 𝑟𝑝𝑛

𝐵𝑝𝑛 =

𝐵𝑝

𝑛+12+𝐵𝑝

𝑛−12

2= 𝑖=1

𝑁2𝔹

𝑖

𝑛+12+𝔹

𝑖

𝑛−12

2𝑊𝑖

2 𝑟𝑝𝑛

Relativistic particle pusher

Lorentz force and Newton’s law of motion Updates of particles’ position and velocity

Relativistic factor, 𝛾 = 1/ 1 − 𝑣𝑝 /𝑐2

Boris or Vay algorithms Beam focusing system (BFS)

𝑣𝑝

𝑛+12−𝑣𝑝

𝑛−12

Δ𝑡=

𝑞𝑝

𝛾𝑝𝑛𝑚𝑝0

𝐸𝑝𝑛 + 𝑣𝑝

𝑛 × 𝐵𝑝𝑛

𝑟𝑝𝑛+1− 𝑟𝑝

𝑛−1

Δ𝑡= 𝑣𝑝

𝑛+1

2

𝐸𝑝𝑛 𝐵𝑝

𝑛

𝔼2𝑛 𝔼1

𝑛

𝔼3𝑛

𝔹𝑖𝑛

𝑟𝑝𝑛

𝑟𝑝𝑛

Gather and Particle Pusher

𝔼1𝑛

𝔼2𝑛

𝔼3𝑛

face-wise constant

10

Beam Focusing System (BFS)

Particle Acceleration

Particle Push

𝜌𝑝𝑛, 𝑧𝑝

𝑛Particle Acceleration

𝐵𝑧BFS 𝐸𝑝

𝑛, 𝐵𝑝𝑛

𝑣𝑝,𝜌

𝑛−12

𝑣𝑝,𝜙

𝑛−12

𝑣𝑝,𝑧

𝑛−12

𝑣𝑝,𝜌

𝑛+12

𝑣𝑝,𝜙

𝑛+12

𝑣𝑝,𝑧

𝑛+12

𝜌𝑝𝑛+1

𝑧𝑝𝑛+1

𝑧

helical trajectory of a particle

surface trajectory for continuous ring of particles along 𝜙

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

z [m]

[

m]

the particle trajectory at the meridian plane

BFS prevents particles to radially deviate from electron beams due to the repulsive forces

neglecting the motion of 𝜙 due to continuous ring of particles

It computes effects by self-fields and BFS and neglects 𝜙 motions due to continuous rings of particles

(3D particles’ surface trajectory under BFS)(Projection of the 3D particles’

trajectory into the meridian plane)

𝐵𝑧BFS

11

Exact Charge-Conserving Scatter

𝐼 ∙ ℚ⋆𝑛+1 − 𝐼 ∙ ℚ⋆

𝑛

Δ𝑡= − 𝒟div ∙ 𝐼 ∙ 𝕁⋆

𝑛+12

ℚ⋆,𝑛𝑖≡ 𝑞 𝑥, 𝜌𝑤𝑛𝑖

0 𝕁⋆,𝑒𝑖≡

𝑞

Δ𝑡𝑙, 𝜌𝑤𝑒𝑖

1

ℚ⋆,𝑛1𝑛+1 − ℚ⋆,𝑛1

𝑛

Δ𝑡=

𝑞

Δ𝑡

𝜌𝑓 + 𝜌2 + 𝜌3

3𝜆1𝑓−

𝜌𝑠 + 𝜌2 + 𝜌3

3𝜆1𝑠

=𝑞

Δ𝑡𝜌𝑓𝜆1

𝑓− 𝜌𝑠𝜆1

𝑠 + 𝜌2 + 𝜌3 𝜆1𝑓− 𝜆1

𝑠

− 𝕁⋆,𝑒1𝑛+

12 + 𝕁⋆,𝑒2

𝑛+12 = −

𝑞

Δ𝑡

𝜌𝑓 + 𝜌𝑠 + 𝜌3

3𝜆1𝑠𝜆2

𝑓− 𝜆1

𝑓𝜆2𝑠 −

𝜌𝑓 + 𝜌𝑠 + 𝜌2

3𝜆1𝑠𝜆3

𝑓− 𝜆1

𝑓𝜆3𝑠

= −𝑞

Δ𝑡𝜌𝑓𝜆1

𝑓− 𝜌𝑠𝜆1

𝑠 + 𝜌2 + 𝜌3 𝜆1𝑓− 𝜆1

𝑠

Discrete continuity equation

discrete currentdiscrete charge

ℚ⋆,𝑛1𝑛+1

ℚ⋆,𝑛1𝑛+1 − ℚ⋆,𝑛1

𝑛

=

𝕁⋆,𝑒1

𝑛+12

𝕁⋆,𝑒2

𝑛+12

𝕁⋆,𝑒1

𝑛+12 + 𝕁⋆,𝑒2

𝑛+12

𝑧

𝜌

𝑛1

𝑛2

𝑛3

𝑒1𝑒2

𝑒3

𝑟𝑠

𝑟𝑓

→ →

𝑟𝑠

𝑟𝑓difference addition

Net currents flowing outward from 𝑛1Rate of charge variation at 𝑛1

ℚ⋆,𝑛1𝑛

“areas weighted by 𝜌”

12

CC & SPAI 𝒌 = 𝟐 CC & LU dcmp. NCC & LU dcmp.

Par

ticl

e d

ist.

Self

-fie

lds

Proposed charge-conserving (CC) scheme vs. conventional non charge-conserving (NCC) scheme

+-

non-physical bunching

Cathode Anode

electron beam

spurious fields

Example 1: Vacuum Diode

13

Electron beam, input RF signal, SWS Sinusoidal Corrugated Circular Waveguide (SCCW)

𝑅0 = 0.04, 𝜖𝑅 = 0.025, 𝑝𝑐 = 0.02 Electron bunching due to forward Cerenkov radiation Confinement of the electron beam by BFS

Example 2: Traveling Wave Tube (TWT) Amp.

Dispersion relations for SCCW Distribution of the electron beam

𝑅0

2𝜖𝑅

𝑝𝑐

coupling point

14

200 400 600 800 1000

10-15

10-10

10-5

100

Nodal index, i

|NR

DG

L i

n|

Residual analysis for discrete Gauss law (DGL)

Example 2: Traveling Wave Tube (TWT) Amp. (cont’)

𝑁𝑅DGL𝑖

𝑛 = 1 −ℚ⋆ 𝑖

𝑛+1

𝑗=1𝑁0 𝒟div 𝑖,𝑗

𝑘=1𝑁1 ⋆𝜖 𝑗,𝑘 𝔼 𝑘

𝑛

𝒟div ∙ ⋆𝜖 ∙𝔼 𝑛+1− 𝔼 𝑛

∆𝑡= − 𝒟div ∙ 𝕁⋆

𝑛+1

2.

𝒟div ∙ ⋆𝜖 ∙ 𝔼 𝑛+1 − ℚ⋆𝑛+1 = 𝒟div ∙ ⋆𝜖 ∙ 𝔼 𝑛 − ℚ⋆

𝑛.

𝒟div ∙ ⋆𝜖 ∙ 𝔼 𝑛 − ℚ⋆𝑛 = 𝒟div ∙ ⋆𝜖 ∙ 𝔼 0 − ℚ⋆

0

res0res𝑛

DGL : 𝒟div ∙ ⋆𝜖 ∙ 𝔼 𝑛 = ℚ⋆𝑛

The residuals for DGL at any two successive time-stepsremain the same, in other words, if initial conditions have 𝒟div ∙ ⋆𝜖 ∙ 𝔼 0 = ℚ⋆

0, DGL for all time-steps is verified.

Normalized residuals for DGL

𝑡 = 400,000∆𝑡

double precision floor

residual level

Premultiplying 𝒟div ∙ ⋆𝜖 into DAL, using 𝒟div ∙ 𝒟curl = 0, and

rearranging the equation yield

Substituting DCE into above equation and rearranging it give

By induction,

15

Small perturbation by the electron beam generates RF signals in the positive feedback system (BWO). Rectangular corrugated cylindrical waveguide : 𝑅𝑖𝑛 = 0.04 [m], 𝑅𝑜𝑢𝑡 = 0.045 [m], 𝑝𝑐 = 0.045 [m]

𝑧 [𝑚]

Example 3: Backward Wave Oscillator (BWO)

𝑅𝑖𝑛

𝑝𝑐

𝑅𝑜𝑢𝑡

𝑉𝑎 = 500 [kV] yields relativistic electron beams with 𝑣𝑝 = 0.83𝑐.

Electron bunching due to backward Cerenkov radiation

16

New electromagnetic particle-in-cell (EM-PIC) algorithm developed to solveMaxwell-Vlasov system on unstructured grids.

Exterior calculus of differential forms1 was used to derive an exact charge-conserving gather and scatter schemes on unstructured grids2.

SPAI algorithm was used to accelerate the field update3. Relativistic particle pusher has been implemented by using Boris algorithm. Successfully applied to simulate plasma-beam dynamics in axisymmetric

VED devices, including travelling wave tube (TWT) amplifiers and backward wave oscillators (BWO).

Concluding Remarks

1F. L. Teixeira, PIER 148, 113-128 (2014). 2H. Moon, F. L. Teixeira, Y. A. Omelchenko, Comput. Phys. Comm. 194, 43-53 (2015).3D. Y. Na, H. Moon, Y. A. Omelchenko, F. L. Teixeira, IEEE Trans. Plasma Sci. 44, 1353-1362 (2016).

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This work has been supported by NSF ECCS-1305838, OSC PAS-0661 and

PAS-0110, and by the OSU Presidential Fellowship program.

Thank You!

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