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7/25/2019 Hammett Presentation
http://slidepdf.com/reader/full/hammett-presentation 1/14
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Supported by the MPPC and a PPPL LDRD project.
7/25/2019 Hammett Presentation
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R e l a t i v e C a p i t a l C
o s t
H98
n∝ nGreenwald
2
Improving Confinement Can Significantly
! Size & Construction Cost of Fusion Reactor
Well known that improving confinement & ! can lower Cost of Electricity / kWh, at fixed power output.
Even stronger effect if consider smaller power:
better confinement allows significantly smallersize/cost at same fusion gain Q (nT " E ).
Standard H-mode empirical scaling:" E ~ H I p
0.93 P -0.69 B0.15 R1.97 …
( P = 3VnT/ " E & assume fixed nT " E, q95 , ! N , n/nGreenwald ):
$ ~ R2 ~ 1 / ( H 4.8 B3.4 )
ITER std H=1, steady-state H~1.5
ARIES-AT H~1.5
MIT ARC H 89 /2 ~ 1.4
n ~ const.
R e l a t i v e C o n s t r u c t i o n C o s t
(Plots assumes cost∝ R2 roughly. Includes constraint on B @ magnet with ARIES-AT1.16 m blanket/shield, a/R=0.25, i.e. B = Bma (R-a-a BS )/R. Neglects current drive issues.)
Need comprehensive simulations to make casefor extrapolating improved H to reactor scales.
7/25/2019 Hammett Presentation
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7/25/2019 Hammett Presentation
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4
• Several advanced algorithms (some in planning) to significantly improve efficiency:-
-
-
-
•
DG: Efficient Gaussian integration --> ~ twice the accuracy / interpolation point:
• Standard interpolation: p uniformly-spaced points to get p order accuracy•
DG interpolates p optimally-located points to get 2p-1 order accuracy
•
Kinetic turbulence very challenging, benefits from all tricks we can find. Potentiallybig win: Factor of 2 reduction in resolution --> 64x speedup in 5D gyrokinetics
Goal: a robust code applicable for a wider range of fusion and non-fusion problems,capable of relatively fast simulations at low velocity resolution, but with qualitatively-
good gyro-fluid-like results, or fully converged kinetic results at high velocity resolution
w/ massive computing.
General goal: robust (gyro)kinetic codeincorporating several advanced algorithms
7/25/2019 Hammett Presentation
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Discontinuous Galerkin (DG) Combines Attractive
Features of Finite-Volume & Finite Element Methods
Standard finite-volume (FV) methods evolve just average value in each cell (piecewise
constant), combined with interpolationsDG evolves higher-order moments in each cell. I.e. uses higher-order basis functions,like finite-element methods, but, allows discontinuities at boundary like shock-capturing
finite-volume methods --> (1) easier flux limiters like shock-capturing finite-volume
methods (preserve positivity) (2) calculations local so easier to parallelize.
Hot topic in CFD & Applied Math: >1000 citations to Cockburn & Shu JCP/SIAM 1998.
5
7/25/2019 Hammett Presentation
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Discontinuous Galerkin (DG) Combines Attractive
Features of Finite-Volume & Finite Element Methods
Don’t get hung up on the word “discontinuous”. Simplest DG is piecewise constant:
equivalent to standard finite volume methods that evolve just cell averaged quantities.Can reconstruct smooth interpolations between adjacent cells when needed.
Need at least piecewise linear for energy conservation (even with upwinding).
DG has ~ twice the accuracy per point of FV, by optimal spacing of points within cell.
6
7/25/2019 Hammett Presentation
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7
7/25/2019 Hammett Presentation
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7/25/2019 Hammett Presentation
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Challenge for magnetic fluctuations in DG
9
This electrostatic field drives a current that is a square wave, and wants to
make a square wave A||(z). But projection of this square wave A|| onto a
continuous subspace gives A|| =0, as if ! =0. This gives very high frequency
mode at grid scale, requiring a very small time step $t < k ||,max vte / (k ! ,min % s ).
(x in these figures
should be z)
7/25/2019 Hammett Presentation
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Fix for magnetic fluctuations for DG
1
In order to conserveenergy, the projection
operator must be self-adjoint. We have
found a local filtering/projection operator
that is self-adjoint.
7/25/2019 Hammett Presentation
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!,#B55 <=#= (9R#"3 /9R# '"/1-8#/8<"#
11
•
Gkeyll is written in C++ and is inspired by framework efforts like Facets, VORPAL
(Tech-X Corporation) and WarpX (U. Washington). Uses structured grids with arbitrarydimension/order nodal basis functions.
•
Linear solvers from Petsc1 are used for inverting stiffness matrices.
•
Programming language Lua2, used as embedded scripting language to drivesimulations. (Lua in widely played games like World of Warcraft, some iPhone apps, ...)
•
MPI is used for parallelization via the txbase library developed at Tech-X
•
Package management and builds are automated via scimake and bilder, bothdeveloped at Tech-X.
•
(I am beginning to explore Julia / iJulia for postprocessing: http://julialang.org. Newhigh-level language oriented to scientific programming being developed at MIT. Fast,
parallelization,!)
7/25/2019 Hammett Presentation
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!"
Test Problem Geometry
ELM crash simulated as a sourceof plasma at the midplane
Target plates at edges of symmetric computationaldomain, midplane in the center
Evolve plasma and calculate heatflux vs. time at target plates
Eric Shi 1D ELM Divertor Heat Pulse Test Problem Graduate Seminar Talk 11 / 27
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7/25/2019 Hammett Presentation
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Axisymmetric SOL (Side View)
∂ζ = 0 Bζ Bξ
∂ =
Bξ
B ∂ξ ≈
Bξ
B ∂⊥ ∂⊥ =
Bζ
Bξ∂ ≈ ∂ξ
Eric Shi 1D ELM Divertor Heat Pulse Test Problem MPPC Workshop 6 / 22
−∂⊥
nimi
B2 ∂⊥φ
= e(ni − ne)
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7/25/2019 Hammett Presentation
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Gkeyll can now Model ELM Heat Pulse in 1D SOL
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(small differences because
initial conditions notprecisely specified.)
Gkeyll:
Full PIC: 1D Vlasov:
Full PIC: 1D Vlasov: