Improved Plasma and High Power Electromagnetic … Plasma and High Power Electromagnetic Modeling...

Improved Plasma and High Power Electromagnetic Modeling within the

Improved Concurrent Electromagnetic Particle-In-Cell (ICEPIC) Code

5 August 2015

Supported financially by AFOSR and computationally by HPCMO

A.D. Greenwood, J. F. Hammond, and N. P. Lockwood

Directed Energy Directorate Air Force Research Laboratory

Approved for public release, distribution is unlimited.

Presenter

Presentation Notes

Title slide, self explanatory.

2 Approved for public release; distribution is unlimited.

Objectives/Motivations

• Continued development of advanced, accurate, and numerically efficient simulation algorithms for modeling electromagnetic devices.

• Specifically, improve plasma and high power electromagnetic modeling within the Improved Concurrent Electromagnetic Particle-In-Cell (ICEPIC) code by:

– Developing alogorithms for effective modeling of highly non-linear dielectrics and ferrites

– Couple new surface chemistry and outgassing models into the PIC method

– Integrate a Bolzmann transport model into the PIC method

– Improve code validation and verification


Potential Applications

• Generation of high power microwaves using compact non-linear transmission lines

• Improved accuracy of high power electromagnetic device simulations

• Accurate prediction of neutral outgassing and arc formation within a realistic HPM system.

• Modeling of high pressure discharges and air breakdown in HPM systems.


Non-linear Material Modeling

• Model generation of high power microwaves by non-linear transmission line (NLTL)

• Ferro-electric materials

– Fit polynomial to E vs D data

– Flux Corrected Transport (FCT) to remove spurious oscillations

• Magnetic Ferrites

– Couple Landau-Lifshitz-Gilbert (LLG) equation to Maxwell’s Equations

– Captures hysteresis of materials

– FCT sometimes needed to remove oscillations

Presenter

Presentation Notes

The non-linear modeling portion of this effort was largely complete in prior years. We have looked at both ferro-electric and magnetic ferrite materials. The application is non-linear transmission lines to generate high power microwaves. Much of the finite-difference time-domain literature that considers non-linear materials restricts itself to materials that are weekly non-linear. Our materials of interest generate spurious grid oscillations; we implemented a flux corrected transport algorithm to remove these. We also looked at magnetic ferrite materials. Sometimes flux corrected transport is needed for these materials as well.


Non-linear modeling results

• Ferro-electric • Magnetic Ferrite

Ferrite

Outer Conductor

Inner Conductor

Ferrite

Outer Conductor

Input Output

Presenter

Presentation Notes

Here are some results of non-linear material modeling. In the animation on the left, spurious grid oscillations form in the uncorrected FDTD at the top. At the bottom FCT version, the pulse steepens according to the physics, but the spurious oscillations are suppressed. ICEPIC also captures the correct rise time trend for an NLTL based on magnetic ferrite material.


Surface Chemistry/Outgassing Model

𝑑𝑑𝑑𝑑(𝑡𝑡)𝑑𝑑𝑡𝑡

= −𝑘𝑘𝑑𝑑𝑑𝑑(𝑡𝑡)𝑥𝑥 exp −𝐵𝐵𝐵𝐵𝑘𝑘𝑘𝑘

− 𝑑𝑑 𝑡𝑡 𝐽𝐽𝑒𝑒𝜎𝜎 +𝐶𝐶𝑃𝑃𝑔𝑔𝑀𝑀𝑘𝑘

(1 − 𝑑𝑑(𝑡𝑡)𝑑𝑑𝑎𝑎� )

• 1st Term -- Thermal Desorption • 2nd Term -- Electron Stimulated Desorption (ESD) • 3rd Term -- Adsorption

• Type of thermal desorption determines value of x

Thermal Desorption Langmuir-Hinshelwood x=2

Adsorption

• Purpose: Understand and mitigate neutral outgassing & ion formation in HPM sources through improved design

• Neutral outgassing/ion formation limits output power of HPM sources (vacuum breakdown)

* Dietrich Menzel and R. Gomer, “Desorption from Metal Surfaces by Low Energy Electrons”, J. Chem Phys. 41 3311 (1964)

Electron Stimulated Desorption

𝑑𝑑 𝑡𝑡 𝑘𝑘𝑑𝑑 𝐵𝐵𝐵𝐵

𝑘𝑘 𝜎𝜎 𝑑𝑑𝑎𝑎

- Adlayer - adatoms/area

- Thermal detachment rate

- Adatom binding energy - Adatom vacancies/area - ESD collision cross-section

- Surface temperature

𝐽𝐽𝑒𝑒 - Electron current density

Neutral Outgassing in a 0-D geometry

Presenter

Presentation Notes

Purpose: Understand and mitigate neutral outgassing & ion formation in HPM sources Neutral outgassing limits the output power of HPM sources due to the phenomenon known as vacuum breakdown. The neutral outgassing equation provides a 0-D model for the change in the adatom number density on a surface as a result of electron stimulated desorption, adsorption, and thermal desorption. The terms in the equation 1st Term -- Thermal Desorption of atoms occurs due to heating of the surface 2nd Term -- Electron Stimulated Desorption (ESD) occurs due to electrons impacting a surface liberating an molecule or atom (called an adatom) and potentially ionizing the molecule to form an ion 3rd Term – When a molecule/atom impacts the surface an adsorption event which adds it to the adlayer can occur or a scattering event which results in the molecule/atom remaining in gas phase Neutral outgassing model was developed in the work of Menzel and Gomer, Coneo, and others over fifty years


ESD & Basic Thermal Desorption Implemented in ICEPIC

Thermal Desorption

Experiment/simulation desorption angle distribution for H2-Cu(100) at 1100 K.

Desorption Angular Distribution

• Thermal Desorption Algorithm 1. Determine probability of surface emitting adsorbate in ∆𝑡𝑡

2. Determine angle/energy of particle desorption through conservation of energy/thermal energy of surface

(3) 𝑷𝑷𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅 𝑡𝑡 = 1 − exp − kdexp −𝐵𝐵𝐵𝐵𝑘𝑘𝑇𝑇

𝑑𝑑𝑥𝑥(𝑡𝑡)∆𝑡𝑡

• Dynamic time step for electrostatic ICEPIC – obeys particle Courant-Fredrick-Levy condition – Required for long term temporal evolution of slow gas

molecules relative to fast electrons and ions

• Electron Stimulated Desorption (ESD) Algorithm 1. Determine probability of desorption

2. Determine angle/energy of particle desorption through conservation of energy/thermal energy of surface

(4) 𝑷𝑷𝑒𝑒𝑒𝑒𝑑𝑑 𝑡𝑡 = 1 − exp −𝑑𝑑 𝑡𝑡 𝑣𝑣𝑒𝑒𝜎𝜎∆𝑡𝑡

kdexp −𝐵𝐵𝐵𝐵𝑘𝑘𝑇𝑇

𝑑𝑑𝑥𝑥(𝑡𝑡)∆𝑡𝑡<0.1 Where

Presenter

Presentation Notes

ESD & a basic thermal desorption was implemented in ICEPIC The thermal desorption algorithm first determines the probability of surface emitting adsorbate in a time step ∆𝑡. then the algorithm determines the angle/energy of particle desorption through conservation of energy with breaking the bond to the surface and the thermal energy of surface which is the energy given to the desorbed particle.


Adsorption Algorithm Implemented in ICEPIC

𝑆𝑆 𝜃𝜃 = 𝑆𝑆0 1 − 𝜃𝜃 + 𝑆𝑆0∗(1 − 𝜃𝜃)(𝑞𝑞𝑚𝑚𝜃𝜃

1 − 𝑞𝑞𝑚𝑚𝜃𝜃)

𝑑𝑑𝑑𝑑(𝑡𝑡)𝑑𝑑𝑡𝑡

= 𝑆𝑆(𝑑𝑑 𝑡𝑡 )𝑝𝑝𝑁𝑁𝐴𝐴2𝜋𝜋𝑚𝑚𝑘𝑘𝑇𝑇

𝐴𝐴𝑁𝑁𝑠𝑠

1 − 𝜃𝜃(𝑡𝑡) − 𝑟𝑟𝑑𝑑𝑖𝑖 𝜃𝜃(𝑡𝑡))

1. Determine if gas molecule will adsorb to surface using eqn. (1)

2. Does gas collide with an adatom or with the surface. If R ≤ θ, where 0 ≤ R ≤ 1 then the gas molecule collides with an adatom, else the surface.

3. The gas molecule scatters using a scattering method or kernel. Utilizes the Cercignani-Lapis-Lord scattering kernel.

4. If the post-collisional normal translational energy Etr,n, of the gas molecule is sufficient to escape the Potential Energy Surface (PES) (Etr,n,≥ 2εLJ), then gas molecule scatters.

5. If the gas molecule cannot escape the PES, i.e. if Etot ≥ Eads and Ncolls ≤ 10, then return to Step 1 using post-collisional values. Otherwise, the gas molecule is adsorbed.

6. Temporally evolve adlayer coverage using eq. 2 εLJ is the Lennard-Jones potential well energy for adatom

Probability of particle adsorbing to the surface

Temporal evolution of adlayer coverage using the state space

(1)

(2) Bentley, Brook I., Greendyke, R. B. “New method of calculating adsorption and scattering for Xe-Pt(111) using Direct Simulation Monte Carlo techniques”, Phys. Of Plasmas

Simulation/experiment comparison for adsorption for Xe-Pt(111) at 800 K

Lobular Particle scattering from the surface

𝜃𝜃 - Fraction of Adlayer Coverage

Presenter

Presentation Notes

The adsorption algorithm was added to ICEPIC to determine how many gas phase molecules would stick or adsorb to the surface. If the molecule did not stick to the surface then the algorithm determines the energy and angle of the molecule scattering off of the surface. The temporal evolution of the adlayer (layer of atoms stuck to the surface) and how its configuration space changes is also modeled in ICEPIC using equation 2.


• ICEPIC & analytic results compared to non-thermal ESD outgassing in experiment – Electron beam pulse width 10 ms & 2 keV electrons – Cu surface with H2 adlayer – H2 is outgassed quickly

ICEPIC Simulation and Comparison to Experiment

Cross-section for desorption 3.5x10-21 cm2 (Menzel and Gomer)

• Analytic, ICEPIC, & experiment show increase in H2 pressure in the cell to 7.0x10-9 Torr

Experiment

ESD generated H2

ICEPIC Sim of e- Beam Hitting Surface

e-

Velocity (m/s)

Velocity (m/s)

ICEPIC

• A = -𝐽𝐽𝑒𝑒𝜎𝜎 + 𝐶𝐶𝑃𝑃𝑔𝑔𝑀𝑀𝑇𝑇

• B=−𝑘𝑘𝑑𝑑exp −𝐵𝐵𝐵𝐵

𝑘𝑘𝑇𝑇− 𝐶𝐶𝑃𝑃𝑔𝑔

𝑛𝑛𝑎𝑎 𝑀𝑀𝑇𝑇 𝑑𝑑 𝑡𝑡 =

Aexp 𝐴𝐴 ∗ 𝑡𝑡𝐵𝐵 exp 𝐴𝐴 ∗ 𝑡𝑡 + 1

+ 𝑑𝑑(0)

Analytic

H2 P

ress

ure

(Tor

r)

Time (s) Thermal Desorption Maintains pressure

Mostly ESD

Presenter

Presentation Notes

ICEPIC was compared to non-thermal ESD outgassing in an experiment performed by AFRL/RXPS on the Anode Material Characterization System The Electron beam pulse width was 10 ms & 2 keV electrons Cu surface with H2 adlayer of 0.4 – H2 is outgassed from the surface due to the electron beam and then lost at slower rate to the surface Analytic, ICEPIC, & experiment all showed increase in H2 pressure to approximately 7.0x10-9 Torr in the vacuum chamber during the electron beam pulse. We utilized experimentally obtained ESD cross-sections and accommodation coefficients from Manzel and Gomer and Bentley and Greendyke papers for the Cu – Hydrogen system in the AFRL/RXPS copper anode ESD experiment. The peak pressure is predominata


60 keV e- gun RGA(3)

Simulated anode

RGA

Anode load/lock

Surface Analysis • AES • XPS • UPS • EELS

Anode Materials Characterization System • High energy e- gun (CW or pulsed) • Surface characterization of anodes

(Before & after e- bombardment) • Measure TOF distribution of desorbed species (H2, CO) (Translational temperature) • Photoelectron-photoion coincidence (PEPICO) spectrometer (Vibrational state distribution of desorbed species)

PEPICO spectrometer

PEPICO spectrometer

Anode Material Characterization System (AMCS)

• Collaborative AFOSR LRIR w/ RXP & RDH Experiment – Novel nanomaterials for High

Power Density, Nonequilibrium Environments of DE Weapons

– Verify ICEPIC simulations against experiment

– Design materials that mitigate neutral outgassing/vacuum breakdown in HPM sources

Presenter

Presentation Notes

Collaborative AFOSR LRIR w/ RXP and RDH experiment to understand and mitigate neutral outgassing The name of the AFOSR LRIR is Novel nanomaterials for High Power Density, Nonequilibrium Environments of DE Weapons Results will be used to verify ICEPIC simulations against the AMCS experimental results Purpose of the AMCS experiment is to understand and design materials for HPM sources that mitigate neutral outgassing/vacuum breakdown in HPM sources Notable components of the A node Materials Characterization System include High energy e- gun used for ESD and thermal desorption experiments (CW or pulsed) Surface characterization of anodes before & after e- bombardment to determine adlayer coverage Measure Time of Flight (TOF) i.e. the translational temperature distribution of desorbed species (H2, CO, CO2) Photoelectron-photoion coincidence (PEPICO) spectrometer which measures the Vibrational state distribution of desorbed molecules to understand adatom desorption processes Intention is to model the 3D experiment utilizing ICEPIC to better understand neutral outgassing experimental results.


Hybrid Boltzmann/PIC

• Objective: Integrate a Boltzmann transport model into the particle in cell Monte Carlo collision (PIC/MCC) method

• Tasks Completed:

– Implemented a finite difference time domain solver with PIC/MCC

– Implemented a Boltzmann Equation solver:

• Assumed the solution could be expanded into a spherical harmonic series (Legendre Polynomials)

• Converted equation to a system of PDE’s and used finite-difference approximations and an ODE integrator to step forward in time

– Combined the separate solvers into one code.

• Remaining Tasks/Challenges:

– Establish weightings to allow for accurate combination of separate methods to simulate evolution of particle distribution functions.

– Use weightings to complete the hybrid Boltzmann/MCC code

Presenter

Presentation Notes

Self-explanatory


Hybrid Boltzmann/PIC Results

• Good agreement between Boltzmann Equation solver and PIC/MCC code.

Presenter

Presentation Notes

The plot on the left shows the electron energy distribution function (EEDF) from the MCC/PIC simulation (blue stars) and the Boltzmann solver(red line). The plot on the right shows the mean speed(MCC-blue star, and Boltzmann-red dashed) and mean energy(MCC-green star, and Boltzmann-black line) of the electrons as a function of time (log scale in time). These plots show a good agreement between the separate solution techniques. Simulation parameter Values: Collision Types: Elastic Inelastic Excitation Energy (11:5 eV loss) Ionization Energy (15:8 eV loss, plus 1 electron) Reduced Electric Field, E=N = 60 Td (1Td = 1*10^-21 V*m^2) Initial Distribution is Gaussian Distribution with mean of 5:5 eV Boltzmann Solver Parameters (Method 1): L = 32, Terms in Expansion, k = 500, discretizations in energy space, Umax = 50 eV, Max Energy MCC Solver (ICEPIC + CHIMP) (Method 2): Number of Particles = 5*10^5


Space Charge Limited Emission

• Want analytic results to test particle-in-cell emission

• Child-Langmuir law in 1-D, Langmuir-Blodgett in 2-D/3-D

• Original Child-Langmuir/Langmuir-Blodgett expressions not relativistically correct

• Find closed form relativistic expression in 1-D

• Transform relativistic equations in 2-D/3-D for application of an Ordinary Differential Equation (ODE) integrator

• Compare results to ICEPIC

• Comparison is favorable, but convergence is slow (as expected)

Presenter

Presentation Notes

Child-Langmuir and Langmuir-Blodgett were published nearly 100 years ago and are well known. Langmuir-Blodgett expressions require tabulation or numerical integration to obtain a geometry parameter that is a function of the ratio of the anode radius to the cathode radius. For the fully relativistic equations, the effects of geometry and voltage are not separable as in the Langmuir-Blodgett approach. There are also some difficulties with numerically integrating the equations. However, the equations can be transformed into a form that can be used with a packaged ordinary differential equation integrator.


Geometry

• Separation of variables applies

– Parallel plates in 1-D

– Coaxial cylinders in 2-D

– Concentric spheres in 3-D

• Cathode at 𝒅𝒅𝒄𝒄 = 𝒅𝒅𝟏𝟏 or 𝒅𝒅𝟐𝟐 and potential 𝑽𝑽 = 𝟎𝟎

• Anode at 𝒅𝒅𝒂𝒂 = 𝒅𝒅𝟐𝟐 or 𝒅𝒅𝟏𝟏(opposite cathode) and potential 𝑽𝑽 = 𝑽𝑽𝟎𝟎 > 𝟎𝟎

• Space charge limit implies derivative of 𝑽𝑽 is zero at the cathode

𝒛𝒛 = 𝒅𝒅𝟏𝟏 𝒛𝒛 = 𝒅𝒅𝟐𝟐

𝒛𝒛 𝒅𝒅

𝒅𝒅𝟏𝟏 𝒅𝒅𝟐𝟐

Presenter

Presentation Notes

The geometry is simply a gap across which a voltage is applied. In the cylinder or sphere case, the cathode can be inside or outside the anode. Space charge limited emission implies that the emission neutralizes the electric field at the cathode, which means that the derivative of voltage with respect to radius is zero.


Parallel Plates

• R.J. Umstattd, C.G.Carr, C.L. Frezen, J.W.Luginsland, and Y.Y.Lau, “A Simple Derivation of Child-Langmuir Space Charge Limited Emission Using Vacuum Capacitance,” Am J Phys, Vol 73, No 2, Feb 2005.

• 𝒅𝒅𝟐𝟐𝑽𝑽𝒅𝒅𝒛𝒛𝟐𝟐

= 𝑱𝑱𝟏𝟏𝟏𝟏 𝑽𝑽𝒅𝒅+𝒎𝒎𝒄𝒄𝟐𝟐

𝝐𝝐𝒄𝒄 𝑽𝑽𝒅𝒅 𝑽𝑽𝒅𝒅+𝟐𝟐𝒎𝒎𝒄𝒄𝟐𝟐

• 𝑱𝑱𝟏𝟏𝟏𝟏 = 𝟒𝟒𝟗𝟗𝝐𝝐 𝟐𝟐𝒅𝒅

𝒎𝒎𝑽𝑽𝟎𝟎

𝟑𝟑𝟐𝟐

𝒅𝒅𝟐𝟐 𝟐𝟐𝑭𝑭𝟏𝟏𝟏𝟏𝟒𝟒

, 𝟑𝟑𝟒𝟒

; 𝟕𝟕𝟒𝟒

;− 𝑽𝑽𝟎𝟎𝒅𝒅𝟐𝟐𝒎𝒎𝒄𝒄𝟐𝟐

𝟐𝟐

• Expression is identical to classical Child-Langmuir with addition of the hypergeometric function

Presenter

Presentation Notes

For the parallel plate (1-D) case, the fully relativistic equation can be analytically integrated following the procedure applied to the classical equation in the referenced paper. The differential equation if found by substituting the current divided by the velocity for the charge density term in Poisson’s equation. The relativistic expression for electron velocity as a function of potential (voltage) is then substituted. Multiplying both sides of the equation by dV/dz allows one integration. The constant of integration can be shown to be zero, the square root of the equation taken, and all dependence on the potential (voltage) grouped into a single term. It is then possible to integrate again and solve for the current density. The hypergeometric function is the result of the second integration and provides the relativistic correction.


• Cylinders: 𝒅𝒅 𝒅𝒅𝟐𝟐𝑽𝑽𝒅𝒅𝒅𝒅𝟐𝟐

+ 𝒅𝒅𝑽𝑽𝒅𝒅𝒅𝒅

= 𝑱𝑱𝟐𝟐𝟏𝟏𝟐𝟐𝝅𝝅𝝐𝝐𝒄𝒄

𝑽𝑽𝒅𝒅+𝒎𝒎𝒄𝒄𝟐𝟐

𝑽𝑽𝒅𝒅(𝑽𝑽𝒅𝒅+𝟐𝟐𝒎𝒎𝒄𝒄𝟐𝟐)

• Spheres: 𝒅𝒅𝟐𝟐 𝒅𝒅𝟐𝟐𝑽𝑽𝒅𝒅𝒅𝒅𝟐𝟐

+ 𝟐𝟐𝒅𝒅 𝒅𝒅𝑽𝑽𝒅𝒅𝒅𝒅

= 𝑰𝑰𝟑𝟑𝟏𝟏𝟒𝟒𝝅𝝅𝝐𝝐𝒄𝒄

𝑽𝑽𝒅𝒅+𝒎𝒎𝒄𝒄𝟐𝟐

𝑽𝑽𝒅𝒅(𝑽𝑽𝒅𝒅+𝟐𝟐𝒎𝒎𝒄𝒄𝟐𝟐)

• Problem: 𝒅𝒅𝟐𝟐𝑽𝑽𝒅𝒅𝒅𝒅𝟐𝟐

→ ∞ as 𝒅𝒅 → 𝒅𝒅𝒄𝒄

• Solution: transform equations with 𝚿𝚿 = 𝑽𝑽𝟑𝟑 𝟐𝟐⁄

• Cylinders: 𝒅𝒅𝟐𝟐𝚿𝚿𝒅𝒅𝒅𝒅𝟐𝟐

= 𝟑𝟑𝑱𝑱𝟐𝟐𝟏𝟏𝟒𝟒𝝅𝝅𝝐𝝐𝒄𝒄𝒅𝒅

𝚿𝚿𝟐𝟐 𝟑𝟑⁄ 𝒅𝒅+𝒎𝒎𝒄𝒄𝟐𝟐

𝒅𝒅 𝚿𝚿𝟐𝟐 𝟑𝟑⁄ 𝒅𝒅+𝟐𝟐𝒎𝒎𝒄𝒄𝟐𝟐− 𝟏𝟏

𝒅𝒅𝒅𝒅𝚿𝚿𝒅𝒅𝒅𝒅

+ 𝟏𝟏𝟑𝟑𝚿𝚿

𝒅𝒅𝚿𝚿𝒅𝒅𝒅𝒅

𝟐𝟐

• Spheres: 𝒅𝒅𝟐𝟐𝚿𝚿𝒅𝒅𝒅𝒅𝟐𝟐

= 𝟑𝟑𝑰𝑰𝟑𝟑𝟏𝟏𝟖𝟖𝝅𝝅𝝐𝝐𝒄𝒄𝒅𝒅𝟐𝟐

𝚿𝚿𝟐𝟐 𝟑𝟑⁄ 𝒅𝒅+𝒎𝒎𝒄𝒄𝟐𝟐

𝒅𝒅 𝚿𝚿𝟐𝟐 𝟑𝟑⁄ 𝒅𝒅+𝟐𝟐𝒎𝒎𝒄𝒄𝟐𝟐− 𝟐𝟐

𝒅𝒅𝒅𝒅𝚿𝚿𝒅𝒅𝒅𝒅

+ 𝟏𝟏𝟑𝟑𝚿𝚿

𝒅𝒅𝚿𝚿𝒅𝒅𝒅𝒅

𝟐𝟐

Coaxial Cylinders/Concentric Spheres

Presenter

Presentation Notes

In the cylinder/sphere case (2D/3D), the equations can no longer be analytically integrated. Close examination also reveals that the second derivative of the potential (voltage) grows without bound as the radius approaches the cathode. This makes numerical integration difficult. However, the equations can be transformed using psi=V^(3/2). In the resulting equations, psi and its first derivative are zero at the cathode, and the second derivative of psi reaches a finite limit. Thus, psi can be approximated by a Taylor series near the cathode, which allows the numerical integration to start a short distance from the cathode and avoids the difficulties caused by the zero initial conditions.


Results

• Results of the relativistic calculations compared to classical and ultra-relativistic approximations

• Largest error at voltage where the approximations cross

Presenter

Presentation Notes

Another issue stems from the fact that the known anode voltages is an additional boundary condition, and the unknown current (density) appears as a coefficient in the equation. However, we can use either the classical Langmuir-Blodgett result or an ultra-relativistic approximation as an initial guess and use a rapidly converging bisection to find the current. Note that the initial guess always over-estimates the current, as shown in the plots.


Results

• ICEPIC grid convergence for inward directed current

• Emission algorithm converges slowly

Presenter

Presentation Notes

Here are some icepic results as a function of the number of cells across the gap. For these plots, the cathode radius is 3.3 cm and the anode radius is 1.4 cm, and the anode is at 250kV. Convergence is close to 1st order, which is not surprising considering the staircase approximation to the cathode geometry. Using this analytic result, we now have a good way to evaluate the convergence of new algorithms that claim to improve the situation. We are also looking at closed form approximations to the current expressions that are valid for over most of the radii and voltages of interest and plan a publication soon.


Conclusion

• Work on non-linear material modeling largely complete

• Initial implementation of surface chemistry/outgassing model is now in ICEPIC

• Hybrid Boltzmann/MCC code is in progress

• Work with space charge limited emission allows analytic validation of ICEPIC emission models

Presenter

Presentation Notes

Self-explanatory

Improved Plasma and High Power Electromagnetic … Plasma and High Power Electromagnetic Modeling...

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