Tutorial: Computation for Advanced Accelerators
Thomas M. Antonsen Jr,
IREAP, University of Maryland
College Park MD, 20742
AAC 2012
Work supported by NSF, and DOE
Computation has become
integral to Accelerator Science
Why is Computation Important ?
- Understanding of basic physical processes
- Understanding and diagnosing particular experiments
- Designing improved experiments
- Optimizing designs
Areas of Progress
Improved Physics Models
Find better descriptions of what’s going on inside a an
accelerator
More Efficient Algorithms
Solve the governing equations with fewer operations
and/or less memory usage.
New Computing Architectures
Restructure your code to take advantage of new
hardware.
Examples of Areas where Computation
is Important
Particle trajectories in analytic fields*
Space charge fields*
Magnetic fields due to currents in coils
RF fields in cavities and waveguides
Radiation generated by energetic particles
Multipactor
Plasma-Based Acceleration
and more
*See R. Ryne http://uspas.fnal.gov/materials/09UNM/ComputationalMethods.pdf
Meshes and Grids
Calculations are carried out on digital computers
Time is discretized
Unstructured mesh
Space may also be discretized - meshes or grids
If field is a dynamical variable (Maxwell’s
Equations) a mesh is needed
If field describes pair-wise interaction forces
(Coulombs law) it can be calculated with or without a
grid.
Structured mesh
Tree Code Calculation of Coulomb interaction without grids
J. Barnes and P. Hut, “A hierarchical force-calculation algorithm,” Nature, vol. 324, pp. 446–
449, 1986
Direct application of Coulomb’s law to N particles requires
O(N2) operations. Not practical for large N.
Tree code: Divide domain into cells
Each cell has fewer than NL particles
Particles within same cell interact pairwise
Particles in different cells interact through
multipole moments
O(NlogN) operations possible
Mesh Types
Structured Mesh
Unstructured Mesh
Complex geometries may be represented
using a multi-block representation:
conformal meshes: boundary of
mesh conforms with physical
boundary
Existing
Discretization Techniques
u
Finite Integration Technique
Applicable to orthogonal Cartesian
mesh
Degrees of freedom are integrals of
fields over appropriate mesh elements
u
Finite Element Method
Applicable to curvilinear mesh
(structured or unstructured)
Field described by interpolating basis
functions in each cell
Basis & test functions
1
xˆ
-½
Suffers due to non-conformal
representation of geometry
0
½
Solves integral (weak) form of 2nd
order differential equations
Evaluates matrix elements using
overlap integrals of basis & test
functions
Large number of coupling terms
Fields and Structured Mesh for FDTD solution
of Maxwell’s Equations
Yee Mesh
(1966)
•
•
•
•
Widely used
Simple (explicit)
Second order accurate
Stable if Dt <cdx
Electric fields defined on edges
Magnetic fields defined on faces
Electric fields defined at integer time steps
Magnetic fields defined at half integer time steps
Leap Frog Explicit FDTD
Solutions to Maxwell’s Equation
¶H z æ ¶Ex
ö
m
=ç
- ....÷
¶t
è ¶y
ø
Ext (x +
time differencing
D
, y + D)
2
ö
H zt +dt /2 - H zt -dt /2 æ Ext (y + D) - Ext (y)
m
=ç
- ....÷
Dt
D
è
ø
D
D
H z (x + , y + )
2
2
Alternately apply Faraday’s law to faces
¶
- m ò H z dA = ò Eids
¶t
D
E (x + , y)
2
t
x
Explicit: knowing E at time t allows for
direct calculation of H at time t+Dt/2
Finite Differencing Plane Waves
Assume all fields vary as
exp[i(k × x - w t)]
¶
e-iwDt /2 - eiwDt /2
sin(wDt / 2)
®
= -i
¶t
Dt
Dt / 2
¶
eikx D/2 - e-ikx D/2
sin(kx D / 2)
®
=i
¶x
D
D/2
Dispersion relation for Plane waves
w 2 = c2 ( kx2 + ky2 + kz2 )
Continuous
FDTD
2
2
2
é
sin(k
D
/
2)
æ
ö æ sin(kz D / 2) ö ù
æ sin(wDt / 2) ö
y
2 æ sin(kx D / 2) ö
+ç
çè
÷ø = c êçè
÷ø + ç
÷ø ú
÷
è
è
Dt / 2
D/2
D/2 ø
D/2
êë
úû
2
Stability - Courant Freidricks Lewy
Condition (CFL)
Dispersion Relation
2
2
2
é
ù
sin(k
D
/
2)
æ
ö
sin(k
D
/
2)
sin(k
D
/
2)
æ
ö
æ sin(wDt / 2) ö
æ
ö
y
2
z
x
+ç
çè
÷ø = c êçè
÷ø + ç
÷ø ú
÷
è
è
ø
Dt / 2
D
/
2
D
/
2
D
/
2
êë
úû
2
2
Left side < æç ö÷
è Dt ø
2
Maximum of Right Side
k x = k y = kz =
2
Stable if
æ 2ö
2æ 2ö
>
3c
çè ÷ø
çè ÷ø
Dt
D
CFL Condition
2
Dt <
D
3c
æ 2ö
3c ç ÷
è Dø
2
p
D
º
kg
2
2
Dispersion and Accuracy
Dispersion in discrete system
is periodic.
Departures from continuous
system are quadratic in Dt, D.
Waves have phase velocity <c
Numerical Cerenkov
Radiation - Requires filter
More complicated evaluation of curl
Conducting Boundaries on Structured Grids
When boundary does not
coincide with mesh FDTD must
be modified.
Boundary
Without modification errors of
order D are introduced.
S. Dey, R. Mittra, A locally conformal finite-difference time-domain (FDTD)
algorithm modeling modeling three-dimensional perfectly conducting objects,
IEEE Microwave Guided Wave Lett. 7 (1997) 273
Problem: If cells are small CFL condition is violated
Separate modification to compute surface losses.
ADI (Alternating Dimension Implicit)
Solutions to Maxwell’s Equation)
Scalar: Douglas, Peaceman, and Ratchford (1955)
ME: Namiki T (1999), Zheng, Chen, Zhang (2000)
¶H z æ ¶Ex
ö
m
=ç
- ....÷
¶t
è ¶y
ø
time differencing
•
ö
H zt + dt /2 - H zt æ ¶Ext + dt /2
•
m
=ç
- ....÷
•
dt
è ¶y
ø
•
t + dt /2
t
t + dt / 2
E
-E
æ ¶H z
ö
e
=ç
- .....÷
x
x
dt
è
¶y
ø
Split time steps
Recently developed
Complex (implicit)
Not subject to Courant
condition, Dt > cdx
ö
¶Ex æ ¶H z
e
=ç
- .....÷
¶t è ¶y
ø
Second half of time step has x, y, and z permuted
Leapfrog ADI-FDTD equations
S. Cooke et al (2009)
1
n- ö
ée
Dt ¶2 ù æ n+ 12
é ¶ n ¶ nù
2
E
E
=
x
ê Dt 4 m ¶y 2 ú ç x
ê ¶y H x - ¶z H y ú
÷
ø
ë
û
ë
ûè
“curl” terms
Tri-diagonal operators
1
1
n+
n+ ù
é
é m Dt ¶2 ù n+1
¶
¶
n
2
2
ê Dt - 4e ¶y 2 ú H x - H x = ê ¶z Ey - ¶y Ez ú
ë
û
ë
û
(
u
u
u
)
Electric and magnetic field terms are offset in time
Electric fields only appear on the half time steps
Magnetic fields only appear on the full time steps
Still need to solve six tri-diagonal equations per time step
3 for the E-components, 3 for the H-components
ADI-FDTD uses 6 for the E-components (3 components, 2 half steps)
No additional explicit equations are necessary
PIC Algorithm - From Plasma Physics via Computer
Simulation C. K. Birdsall and A. B. Langdon
FDTD ME in here
Integration of Equations of Motion
Basic Leap Frog:
pt - pt -Dt = Dt F t -Dt /2
x t +Dt /2 - x t -Dt /2 = Dt pt / m
Modified for
Lorenz Force:
“Boris Push”
p- = pt -Dt +
Dt
qE t -Dt /2
2
p+ - pp + p=q +
´ Bt -Dt /2
Dt
2mc
pt +Dt = p+ +
Dt
qE t -Dt /2
2
p and x known on
staggered time grid
Half step with E
Full step with B
Half step with E
Interpolation of Fields and Accumulation
of Currents and Charges
E and J are known on the same grid, but are staggered in time
Ex
Ey
Ey
Ex
Jy
Forces interpolated from grid
to location of particles
Jx
Jy
Particle contribution to current
density accumulated on grid.
Algorithm should conserve
¶r
charge.
+ Ñ× J = 0
¶t
Jx
J.Villasenor
and O.Buneman. Computer Physics Communications, 69. 306-316, (1992)
21
IVEC 2012
FDTD-PIC Time-stepping loop
Cooke NRL
S.
Charge is not computed,
only current is used.
H E J
P X
Advance H
Space charge fields occur
only through integration of
Maxwell’s equations.
Assign external current sources
Add new particles
Advance P, then X
(accumulating J)
Remove dead particles
Advance E
Charge-conserving methods
do this exactly – no need for
divergence-cleaning.
Plasma Based Accelerators
• Direct Acceleration in Modulated Channels
Physical Processes
Important
• Relativistic Motion
• Plasma Wave Generation / Cavitation
• Laser Self Focusing / Scattering
• Ionization
Not Important
• Turbulence - most plasma particles interact for a short time, several
plasma periods, then leave
Our job is “relatively” easy.
Simulation has played an important role in the development of the field.
Guiding our understanding
Designing new experiments
Relevant Time Scales (LWFA)
• Laser Period
l =800 nm
• Driver Duration ~ Plasma Period
TL = 2.7 fs
TD = 50 fs
• Propagation Time (driver evolution, L= 1 cm) TP = 3.3 104 fs
(driver evolution, L= 1 m) TP = 3.3 106 fs
Propagation Time >> Driver Duration >> Laser Period
TP >> TD >> TL
• Disparity in time scales leads to both complications and simplifications
MODELS
APPROACHES / APPROXIMATIONS
• Laser
Full EM
- Laser Envelope
Particles
- Fluid
Full Lorenz force
- Ponderomotive
• Plasma
Dynamic response -
Quasi-static
Hierarchy of Descriptions
1. Full Format Particle in Cell (PIC)
- Relativistic equations of motion for macro-particles
- Maxwell’s Equations on a grid
- Most accurate but most computationally intensive
• Laser
Full EM
- Laser Envelope
• Plasma
Particles
- Fluid
Full Lorenz force - Ponderomotive
Dynamic response -
Quasi-static
Windows and Frames
Three versions:
- Lab Frame
- Moving Window
- Boosted Frame
t=L/c
Lab Frame - # Grids
NLF = (L/dz)2 ~ (L/l)2
grid must
resolve l
dt=dz/c
- LD
Pulse length z=0
Moving Window - # Grids
NMW = (LD/L) NLF
z=L Propagation distance
Boosted Frame
J-L Vay, PRL98, 130405 (2007)
Lab Frame
t=L/c
dt' = gdt
Boost - g
t' =L/gc
z' =-gLD
z=0
z'=0
dz' = gdz
z=L
Qualifications:
-no backscatter. In boosted frame
gives upshift. Requires smaller dz'
-transverse grid size sets limit on
dt' < dx' /c= dx /c
Boosted frame - # grids
NBF = g-2 NMW
Optimum based on 1D
makes simulation “square”
Using conventional PIC techniques, 2-3 orders of magnitude
speedup reported in 2D/3D by various groups
Osiris: trapped injection
Vorpal: external injection
w/ beam loading
Warp: external injection
wo/ beam loading
J-L Vay
Reported speedups limited by various factors:
• laser transverse size at injection,
• statistics (trapped injection),
• short wavelength instability (most severe).
2. Full EM vs. Laser Envelope
Driver Duration >> Laser Period
TD >> TL
• Required Approximation for Laser envelope:
wlaser tpulse >> 1,
rspot >> l
wp /wlaser <<1
• Advantages of envelope model:
-Larger time steps
Full EM stability: Dt < Dx/c
Envelope accuracy: Dt < 2p Dx2/lc
- No unphysical Cherenkov radiation
- Further approximations
• Advantages of full EM: Includes Stimulated Raman back-scattering
Also direct acceleration
Laser Envelope Approximation
• Laser + Wake field:
• Vector Potential:
E = E laser + E wake
A laser = A 0(x ,x ,t) exp ik 0x
• Laser Frame Coordinate:
+ c.c.
x = ct – z
• Envelope equation:
2 ¶æ
¶öˆ
¶2 ˆ
4p ˆ
2 ˆ
j
ç ik - ÷ A - 2 2 A + Ñ^ A = c ¶t è 0 ¶x ø
c ¶t
c
Necessary for:
Raman Forward
Self phase modulation
vg< c
Drop
(eliminates Raman back-scatter)
Validity of Envelope Equation
2
Extended Paraxial
–
True:
Requires :
k c 2 + w 2p
v g(w ) = c / 1 +
2 w2
v g(w ) = c 1 –
k 2c 2 + w 2p
1/2
w2
2
k c 2 , w 2p < < w 2
Extended Paraxial approximation
- Correct treatment of forward and near forward scattered radiation
- Does not treat backscattered radiation
- If grid is dense enough can treat Dw ~ w
3. Full Lorenz Force vs. Ponderomotive
Description
• Full Lorenz:
d pi
v ´B
= q E+ i c
dt
• Separation of time scales
E = E laser + E wake
dx i
= vi
dt
g =
p2
1+ 2 2
mc
• Requires small excursion
x(t) × E laser < < Elaser
x(t) = x(t) + x(t)
• Ponderomotive Equations
2
q A laser
Fp = – mc 
mc 2
2g
v ´ Bwake
dp
= q E wake +
+ Fp
c
dt
g =
q A laser
p2
1+ 2 2 +
mc
mc 2
2
2
Ponderomotive Guiding Center PIC Code: TurboWAVE
D.F. Gordon, et al. , IEEE TRANSACTIONS ON PLASMA SCIENCE , V 28 , 1224-1232 ( 2000 )
Fields are separated into high and low frequency components. The low frequency
component is treated as in an ordinary PIC code. The high frequency component is
treated using a reduced description which averages over optical cycles.
Low Frequency Cycle
High Frequency Cycle
Deposit Sources
j and r
Deposit Source
nq2/<gm>
Advance Fields
(Maxwell’s Equations)
Advance Laser
(Envelope Equation)
Lorentz Push
dP
= q( E + v ´ B)
dt
+
Ponderomotive Push
dP
q2
a2
=Ñ
dt
gm
4
INF&RNO
(INtegrated Fluid & paRticle simulatioN cOde)
C. Benedetti at al. (LBNL)
2D cylindrical + envelope for the laser (ponderomotive approximation)
 full PIC/fluid description for plasma particle (quasi-static approx. is also available)
 switching between PIC/fluid modalities (hybrid PIC-fluid sim. are possible)
 dynamical particle resampling to reduce on-axis noise
nd order upwind/centered FD schemes + RK2/RK4 (& Implicit) for time integration
 2
 linear/quadratic shape functions for force interpolation/charge deposition
 high order low-pass compact filter for current/field smoothing
 “BELLA”-like runs (10GeV in ~ 1m) become feasible in a few days on small machines

FLUID
PIC
35
4. Quasi - Static vs. Dynamic Wake
P. Sprangle, E. Esarey and A. Ting, Phys. Rev. Lett. 64, 2011 (1990)
Laser Pulse
PlasmaWake
Electron transit time:
c
Plasma electron
c - vz
t pulse
1 – vz / c
Electron transit time << Pulse modification time
Trapped electron
Advantages:
te =
d =  + c – v  + v ×
z


dt
t
x
fewer particles, less noise (particles marched in ct-z)
Disadvantages: particles are not trapped
Quasi Static Simulation Code WAKE
P. Mora and T. M. Antonsen Jr. - Phys Plasma 4, 217 (1997)
x=ct-z
Two Time Scales
1. “fast time”
t ~ TD ~ wp-1
particle trajectories and
wake fields determined
2. “slow time”
laser pulse evolves
diffraction
self-focusing
depletion
r
Particle trajectories
QuickPIC: 3D quasi-static particle code
Description
• Massivelly Parallel, 3D Quasi-static particle-incell code
• Ponderomotive guiding center for laser driver
• 100-1000+ savings with high fidelity
• Field ionization and radiation reaction included
• Simplified version used for e-cloud modeling
• Developed by UCLA + UMaryland + IST
•
•
•
•
•
Examples of applications
Simulations for PWFA experiments,
E157/162/164/164X/167 (Including Feb. 2007
Nature)
Study of electron cloud effect in LHC.
Plasma afterburner design up to TeV
Efficient simulation of externally injected LWFA
Beam loading studies using laser/beam drivers
Chengkun Huang:
huangck@ee.ucla.edu
http://exodus.physics.ucla.
edu/
http://cfp.ist.utl.pt/golp/epp
New Features
• Particle tracking
• Pipelining
• Parallel scaling to 1,000+
processors
• Beta version of enhanced
pipelining algorithm: enables
scaling to 10,000+ processors
and unprecedented simulation
QUASI-STATIC CODE STRUCTURE
Laser
Fp
“slow time - t”
2
w
2 ¶æ
¶ö ˆ
p
2 ˆ
ˆ
ik
A
+
Ñ
A
=
A
0
^
c ¶t çè
¶x ÷ø
c 2g
“fast time x=ct-z”
Particles and Wake
dp^
1 é æ
v ´ Bö
ù
=
q
E
+
+
F
pú
dx c - v z êë è
c ø
û^
w 2p
c 2g
dr^
v
= ^
dx c - v z
Ñ´B=
4pq n
p
c g
+
particles
¶E
¶x
PARTICLES CONTINUED
constant of motion
:
• Hamiltonian:
H = H(x = ct - z,r^, p) = mc g + qf
2
H - cPz = const.
• Weak dependence on “t” in the laser frame
Transverse Dynamics
dp^
1 é æ
v ´ Bö
ù
=
q
E
+
+
F
pú
dx c - v z êë è
c ø
û^
dr^
v
= ^
dx c - v z
• Introduce potentials
B= Ñ´ A
E = -Ñf -
¶A
¶x
• Algebraic equation:
pz = pz (p ^,y = f - Az, a )
2
g = g (p ^,y = f - Az , a )
2
LOCAL FREQUENCY MODIFICATION
M. Tsoufras, PhD Thesis, UCLA
Intensity ne
Frequency shift is proportional
to propagation distance
2
z dw p
=w
2cw 2 dt
Dw
dne
< 0,
dt
dw
<0
dt
dne
> 0,
dt
dw
>0
dt
Assume complete cavitation
Relative frequency shift is
unity at the dephasing distance
zD = TD
cw 2
w 2p
Quasi Static Modeling of Pulse Depletion
W. Zhu Wednesday WG1
Comparison of Spectra; Modified Paraxial,
Full-wave, and FDTD-PIC
Conclusions
• Numerical simulation of Laser-Plasma interactions is a powerful tool
• A variety of models and algorithms exist
first principles
reduced modles
• Field is still advancing with new developments
Boosted Frame Calculations speed-up ~ g2
3D Parallel Quasi-static
speed-up ~ [w0/wp]2
GPUs
WAKE - Cavitation
P. Mora and T. Antonsen PHYSICAL REVIEW E, Volume: 53 R2068 (1996)
Intensity
Density
Trajectories
5
4
3
r
2
1
0
0
Complete cavitation
Suppression of Raman instability
Stable propagation for 30 Rayleigh lengths
10
20
x
30
40
Wake simulation of pulse
propagation in corrugated plasma channels
See WG #1 B. Layer Wed. PM, J. Palastro Thu. AM
l=800 nm
wch=15 m
ao=.35
t=1 ps
no=7x1018 cm-3
=.9
lm=.035 cm
3 cm ~ 90 corrugations ~ 40 TR
150 m
density
intensity
t=0 TR
Upper: corrugated
Lower: smooth
t=15 TR
t=30 TR
Suppression of Raman Instability
QUICKPIC
Quasi-Static Plasma Particles
Wake Fields
Dynamic Beam Particles
Laser Field
Second Order Accurate Split-Step Scheme
time-t
Laser
Advance
t
Axial (dt)
n
gn 0
Vp
Plasma
Particle
Advance
t+dt
Axial (dt)
Transverse (dt)
s
Axial (dt/2)
n
gn 0
Transverse (dt)
s+ds
Vp
push particles
push particles
time-s
t
t+dt
Verification : Full PIC vs. Quasi-static PIC
Benchmark for different drivers:
QuickPIC vs. Full PIC
Excellent agreement
with full PIC.
100 to 10000 times
savings in CPU needs
e- driver
e- driver with
ionization
e+ driver
No noise issues and no
unphysical Cerenkov
radiation
laser driver
100 to 10000 CPU savings with “no” loss in accuracy
Iteration of Electromagnetic Field
Parallel electric field
Transverse electric field
Ez =
¶
¶
(f - Az ) = - y
¶x
¶x
Equation of motion
E˜ ^
æ ¶
ö
¶
E^ = -Ñ^f - A^ = -Ñ ^y - ç A^ + Ñ^ Az÷
¶x
è ¶x
ø
Iterate to find
Ampere’s law
Electromagnetic portion
E˜ ^
4p æ
¶ ö
j ^ ÷ = -Ñ2^ E˜ ^
ç Ñ^ j z +
c è
¶x ø
dp^
1 é æ
v ´ Bö
ù
=
q E+
+ Fp ú
ê
è
ø
dx c - v z ë
c
û^
Quasi-Static Field Representation
Lorenz
Transverse Coulomb
Ñ^ × A^ = 0
4p
-Ñ A =
j
c
2
^
-Ñ f = 4pr
2
^
Pro:
Simple structure
Compatible with 2D PIC
Con:
A carries “electrostatic” field
-Ñ 2^ A =
é ¶(f - Az ) ù
4p
j - Ñê
ú
c
¶
x
êë
úû
Pro:
A = 0 in electrostatic limit
Con:
non-standard field equations
Particle Promotion in WAKE
S. Morshed PoP, to be published
Osiris
Wake
“Plasma particles” for which
quasi-static violated are promoted
to “beam particles”
New “Leapfrog” ADI-FDTD Formulation
2
2
éu Combine
ù
D
t
D
t
a
half
step
forward
with
a
half
step
back:
æ Dt ö
æ
ö
-1
n +1 / 2
= e x E xn +
d y H zn - d z H yn - ç ÷ d y m z-1d x E yn
êe x - ç ÷ d y m z d y ú Ex
2
è 2 ø
è 2 ø
êë
úû
[
]
2
2
é
ù n -1/ 2
Dt
æ Dt ö
æ Dt ö
-1
n
n
n
= e x Ex d y H z - d z H y - ç ÷ d y m z-1d x E yn
êe x - ç ÷ d y m z d y ú E x
2
è 2 ø
è 2 ø
ëê
ûú
[
]
u Subtracting equations gives a new scheme:
2
é
ù n +1/ 2
æ Dt ö
-1
- E xn -1/ 2 ) = Dt [d y H zn - d z H yn ] Simple source
êe x - ç ÷ d y m z d y ú (E x
terms
è 2 ø
êë
úû
Tri-diagonal operator
“curl H” term
(
)
[
n +1/ 2
- Exn-1/ 2 = Dt d y H zn - d z H yn
Compare to FDTD: e x Ex
]
S.J.Cooke, M.Botton, T.M.Antonsen, Jr., B.Levush, “A leapfrog formulation of the 3-D ADI-FDTD algorithm”
Int. J. Numer. Model. 2009; 22:187–200
54
NRL
Plasma
Physics
Division
TurboWAVE Framework
PIC
EM pusher, ponderomotive
pusher, gather/scatter
Nonlinear
Optics
“SPARC”
Multi-species hydrodynamics
Anharmonic Lorentz model,
Quasistatic model
Field Solver Modules
Chemistry
Explicit, envelope, direct fields, coulomb gauge, electrostatic.
Lindman boundaries, simple conducting regions, PML media.
Arbitrary species,
reactions
Numerical Framework
Linear algebra, elliptical solvers, fluid advection…
TurboWAVE Base
Grid geometry, regions, domain decomposition, diagnostics, input/output…
Images from TurboWAVE
Gridded Gun Modeling
Contours are equipotential lines.
Colors represent current density.
Black circles represent grid wires.
NRL
Plasma
Physics
Division
Blowout Wakefield
Red = ion rich
Blue = electron rich
ve
k0
Utilizes 3D electromagnetic PIC
Utilizes 2D electrostatic PIC (slab or
cylindrical) with conducting regions
Envelope model in VORPAL enables full 3D
simulations of meter-scale LPA stages
Speedup of 50,000 shown
for meter-scale parameters
u Rigorously tested,
showing second-order
convergence and correct
laser group velocity
u Benchmarks against scaled
Ben Cowan
simulations show excellent
[1] B. Cowan et al., Proc. AAC 2008
agreement
[2] P. Messmer and D. Bruhwiler, PRST-AB
u
9, 031302 (2006)
[3] D. Gordon, IEEE Trans. Plasma Sci. 35,
1486 (2007)
[4] P. Mora and T. M. Antonsen, Phys.
Plasmas 4, 217 (1997)
Charge Conserving Algorithm
I1,j+1/2,k+1
k+1
I1,j+3/2,k+1
dx1
I2,j,k+1/2
k
j
dx2
I2,j+1,k+1/2
I1,j+1/2,k
j+1
I
2,j+2,k+1/2
I1,j+3/2,k
j+2
Full Format: PIC Algorithm
VORPAL
OSIRIS 2.0
osiris framework
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Massivelly Parallel, Fully Relativistic
Particle-in-Cell (PIC) Code
Visualization and Data Analysis
Infrastructure
Developed by the osiris.consortium: UCLA
+ IST
Widely used: UCLA, SLAC, USC, Michigan,
Rochester, IST, Imperial College, Max Planck
Inst.
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Examples of applications
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Mangles et al., Nature 431 529 (2004).
Tsung et al., Phys. Rev. Lett., 94 185002
(2004)
Mangles et al., Phys. Rev. Lett., 96, 215001
(2006)
Lu et al., Phys. Rev. ST: AB, 10, 061301
(2007)
I.A. Blumenfeld et al., Nature 445, 241
(2007)
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Frank Tsung: tsung@physics.ucla.edu
Ricardo Fonseca: ricardo.fonseca@ist.utl.pt
http://exodus.physics.ucla.edu
http://cfp.ist.utl.pt/golp/epp
New Features in v2.0
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Bessel Beams
Binary Collision Module
Tunnel (ADK) and Impact
Ionization
Dynamic Load Balancing
PML absorbing BC
Optimized Higher Order Splines
Parallel I/O (HDF5)
Boosted Frame in 1/2/3D
+10GeV self-injection in nonlinear regime
Controlled self-guided a0=5.8
Smooth
accelerating field
Injected
electrons
Laser
pulse
Boosted frame
7000x256x256 cells
~109 particles
3x104 timesteps
Υ=10
Samuel F. Martins et al.
Nature Physics V6, April 2010
~300x faster
than lab
7-12 GeV
1-2 nC