Realtime visualization and optimization of
vacuum surfaces - Boyd Blackwell, ANU
• Real time tracing code BLINE (Summer Scholar: Antony Searle, ANU)
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multi-thread/processor
mesh accuracy
speed
hierarchial system element/mesh structure
• Perturbation method for iota (Summer Scholar: Ben McMillan, ANU/UMelb)
• Real-time optimization by simulated annealing
– demonstration
Minimal Confinement Geometries
• Simplest possible geometries with closed surfaces that
resemble real geometries, for testing codes
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fast direct evaluation, exact
iota ~ 1
aspect ratio ~ 5-10
highly 3D
enclose no conductors
• “triator”
– 4 simple elements (finite filaments)
– iota ~ 0.6, bean shaped, (similar to Tom Todds?)
• “1 element” toroidal helix
– slow evaluation
Mesh Interpolation
• cubic tri-spline on regular rectangular meshes
• copy of mesh in neighbourood stored to better fit in CPU cache
– derivatives stored only in local mesh (4 point eval from main mesh)
H-1
• mesh hierarchy underneath the hierarchy of
magnetic macro-elements
TFC
– e.g. H-1 has 3 meshes for main field, but one coarse mesh
for VF coils
– allows quick configuration exploration by varying currents
(linear combination I1M1 + I2M2 + I3M3)
VF
3 ea.
32×128×32
• mesh filled on demand and/or in background
– (see also Gourdon code, Zacharov’s code (Hermite polynomials))
Mesh Convergence
Convergence of mesh tracing (NEW)
Mesh spacing (log(m))
– distance to nearest conductor
recorded in each cell,
automatically revert to
direct calculation if too
y = 6.02x + 5.07
close.
-2
-1.5
0
-1
-0.5
0
-1
-2
-3
-4
-5
log(Error Sum)
y = 5.23x + 2.65
-6
log(Max Error)
Linear (log(Error Sum))
Linear (log(Max Error))
5th order or better in x
-7
Error (log())
• Meshes of 10-50MByte are
adequate even near edge
Multi-processing
• windows threads (posix under linux) (MISD)
– needs semaphore system (e.g. no tracing while loading a new mesh)
• multi-threaded code runs fine on single processor
– some priority tuning useful on single processor
• initial scheme
– tracing thread, display thread and mesh-filling threads
– large caches on Intel machines favour each thread working in distant
memory locations
• multi-threading  object oriented coding
Perturbation Calculation of iota
• Find a nearby rational surface by iteration ~middle order
– say ~ 30 circuits
B0
• Store B and derivatives along this closed path
• For each variation in the perturbing winding, integrate
x  B/B0
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where B is the perturbing field
and B0 the original field
(Alternatively integrate cpt of B in surface, normalized to B0
and the puncture spacing at that point ~ Boozer )
B
Accuracy of / I
• Check / I by ultra high
accuracy (1e-7) direct
calculation of 
• correction for area change
can be significant
Change in transform vs Coil current Ratio
Change in transform
0.02
0.01
0
0.88
0.9
0.92
0.94
0.96
0.98
1
1.02
1.04
1.06
-0.01
-0.02
-0.03
y = 0.3039x - 0.3038
R2 = 1
-0.04
Perturbation result: 0.315 cf 0.304
Machine Optimization of iota
• Minimization by steepest descent (but multi-variate)
• Simulated annealing
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virtual temperature T
accept a new configuration even if slightly worse (up to T)
“heat” to explore new configurations
“cool” to home in on optimum
• Annealing more tolerant of occasional anomalies in goodness
function, e.g. local minima or discontinuities (resonances)
“Reinvent” helical conductor in flexible heliac
• Constrain conductor to lie inside a torus, N=3
– (actually end-point and middle point fixed)
• Seek maximum transform for length  current
• Result is very close to the flexible heliac
R>Rmin constraint  “sawtooth coil”
• Constrain conductor to lie on a cylinder, N=3
• Seek maximum transform near the axis of a heliac per unit
length  current
• Reproduces approximate “sawtooth coil”
Conclusions and Future Work
• Very useful for following particles out of machine (so far, not a drift calculation)
• Very quick (50k/sec) configuration evaluation for varying current ratios in
existing coil system (e.g. H-1 flexibility studies)
• Fast evaluation (10k/sec) of new winding (“simple”) in arbitrarily complex
existing configuration
• Iota perturbation calculation works, and is fast.
• Well calculation implemented, but not debugged
• Possibly extend to island width as in Rieman & Boozer 1983
• optimization principle demonstrated
• “standard results” recovered
• real time operation  possibility of human guidance during optimization
Develop/find “Meta-Language” for description of symmetries and constraints