Hammett Presentation

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.



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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.



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• Several advanced algorithms (some in planning) to significantly improve efficiency:-

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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



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



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



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Challenge for magnetic fluctuations in DG

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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)



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.



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•

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.

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Linear solvers from Petsc1 are used for inverting stiffness matrices.

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Programming language Lua2, used as embedded scripting language to drivesimulations. (Lua in widely played games like World of Warcraft, some iPhone apps, ...)

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MPI is used for parallelization via the txbase library developed at Tech-X

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Package management and builds are automated via scimake and bilder, bothdeveloped at Tech-X.

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(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,!)



!"

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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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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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:

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Hammett Presentation