GenFoo: a general Fokker-Planck solver with applications in fusion - PowerPoint PPT Presentation

GenFoo: a general Fokker-Planck solver with applications in fusion plasma physics L. J. Höök and T. Johnson Fusion plasma physics, EE, KTH, Stockholm, Sweden June 6, 2012 L.J. Höök (KTH) GenFoo: a general Fokker-Planck solver 2012-06-06 1 / 25

Outline • Fusion energy • Integrated Tokamak Modeling and ITER • Kinetic diffusion: Coulomb collisions • GenFoo: a general Fokker-Planck solver • Summary L.J. Höök (KTH) GenFoo: a general Fokker-Planck solver 2012-06-06 2 / 25

Magnetic confinement fusion • D+T => He4 + n + 17.59 Mev • Temperature at around 200 million degrees (K) • Confined by magnetic fields • Huge temperature gradients => turbulent plasma L.J. Höök (KTH) GenFoo: a general Fokker-Planck solver 2012-06-06 3 / 25

The ITER project "To demonstrate that it is possible to produce commercial energy from fusion." • Output power 500 MW, input power 50 MW • Height: 73 metres • Weight: 23,000 tons • Cost: 13 billion euros L.J. Höök (KTH) GenFoo: a general Fokker-Planck solver 2012-06-06 4 / 25

Integrated Tokamak Modeling Task Force in EU "The main mission of ITM-TF is to build a validated suite of simulation codes for ITER plasmas and to provide a software infrastructure framework for EU integrated modeling activities." from efda-itm.eu • Aim to offer a full simulation environment. • Provide a validated set of European modeling tools for ITER exploitation. • Provide an API for coupled codes written in different languages. L.J. Höök (KTH) GenFoo: a general Fokker-Planck solver 2012-06-06 5 / 25

Simple workflow in Kepler • Kepler: a graph based workflow tool for coupling ITM codes, kepler-project.org READ FROM DATABASE Linear MHD spectrum WRITE TO DATABASE 7 L.J. Höök (KTH) GenFoo: a general Fokker-Planck solver 2012-06-06 6 / 25

Spatio-temporal scales in plasma kinetic models • The physics in a confined heated plasma occur on a broad range of temporal and spatial scales. • Spatial scales from the electron gyroradius, 10 − 6 m to the size of the confinement device 10 1 m. • Temporal scales range from the electron gyroperiod, 10 − 10 s to the plasma pulse length 10 5 s. • Requires the 1 largest computational facilities in the world. 1Currently the K-computer in Japan (figure), with 10 petaflops per second L.J. Höök (KTH) GenFoo: a general Fokker-Planck solver 2012-06-06 7 / 25

The kinetic description of charged particles ∂ t f a ( r , v , t ) + v · ∂ f a ∂ r + e a ( E + v × B ) · ∂ f a � ∂ f a � ∂ ∂ v = m a ∂ t c � ∂ f a � � = C ab ( f a , f b ) ∂ t c b Properties: • 6D+1 dimensional • l.h.s. describes evolution of particle distribution on a macroscopic scale while r.h.s. models coulomb collisions on a microscopic scale. L.J. Höök (KTH) GenFoo: a general Fokker-Planck solver 2012-06-06 8 / 25

Coulomb collisions • The Coulomb collision process describes collisions between charged particles: • Dominated by small angle, long range collision in contrast to large angle, binary collision in neutral gases. L.J. Höök (KTH) GenFoo: a general Fokker-Planck solver 2012-06-06 9 / 25

The collision operator • Decorrelation allow collisions to be modeled as a 3 + 1 dimensional nonlinear Fokker-Planck equation: � ∂ 3 3 � A i ( v , f a , f b ) f a + 1 ∂ ∂ B i , j ( v , f a , f b ) ∂ � � ∂ t f a = − f a ∂ v i 2 ∂ v i ∂ v j c i i , j where A = C a ∇ φ ( v ) , B = C s ∇∇ Ψ( v ) and φ ( v ) = − 1 � f b ( v ′ ) | v − v ′ | d 3 ( v ′ ) 4 π Ψ( v ) = − 1 � | v − v ′ | f b ( v ′ ) d 3 ( v ′ ) 8 π are the so called Rosenbluth potentials. • It describes the evolution of a probability density on a mesoscopic scale. L.J. Höök (KTH) GenFoo: a general Fokker-Planck solver 2012-06-06 10 / 25

Dimension reduction by symmetries • Orbit averaging over the periodic action angle variables Z (the angles in action-angle system); gyro angles α , R bounce angle β and toroidal φ angle φ . ∂ f a ∂ t = ∇ v · ( −� A � · f a + � B � · ∇ v f a ) θ where r 1 � � X � = Xd φ d β d α . ( 2 π ) 3 • Orbit averaging reduces the 6D problem to a 3D adiabatic invariant space with complex internal boundaries.

Challenges of the adiabatic invariant model Drift � A � and diffusion � B � coefficients must be calculated numerically with external codes. They contain line integrals over the particle orbit � � g � = ( 2 π ) − 1 g ( R ( β ) , Z ( β )) d β • The space contains a bifurcation, plane Left or right?? 3D space L.J. Höök (KTH) GenFoo: a general Fokker-Planck solver 2012-06-06 12 / 25

Model reduction • Homogenize in r and assume local isotropy in velocity space => 1D-2D Fokker-Planck models however with singular drift and diffusion coefficients • Linearize: • Assume collisions against a known distribution, C ( f a , f b ) where f b is known. • Rosenbluth potentials have analytical solutions if f b is a local Maxwellian. L.J. Höök (KTH) GenFoo: a general Fokker-Planck solver 2012-06-06 13 / 25

Computational complexity vs physics output Different Fokker-Planck models 6D, nonlinear y t i x 3D, nonlinear e l p 3D, quasi-linear in m adiabatic invariants o c l 3D, linearized against background distribution a n o 3D, linearized against background Maxwellian i t a t u p 2D, linearized against background Maxwellian m o C 1D, linearized against background Maxwellian Physics output L.J. Höök (KTH) GenFoo: a general Fokker-Planck solver 2012-06-06 14 / 25

Computational complexity vs physics output Different Fokker-Planck models 6D, nonlinear y t i x 3D, nonlinear e GenFoo target models l p 3D, quasi-linear in m adiabatic invariants o c l 3D, linearized against background distribution a n o 3D, linearized against background Maxwellian i t a t u p 2D, linearized against background Maxwellian m o C 1D, linearized against background Maxwellian Physics output L.J. Höök (KTH) GenFoo: a general Fokker-Planck solver 2012-06-06 14 / 25

GenFoo, GEneral FOkker-Planck sOlver • Separate numerics and physics with an API for the operator coefficients. • Numerics: FEM or ( δ f ) Monte-Carlo. • Physics: convection + diffusion + source + initial values. • XML in and XML out design => compatible with XProc, the XML pipeline language, w3.org/TR/xproc. • Monte Carlo particles are stored in the h5Part ( HDF5 4 particles, vis.lbl.gov/Research/H5Part ) format. L.J. Höök (KTH) GenFoo: a general Fokker-Planck solver 2012-06-06 15 / 25

GenFoo solves ... • the Fokker-Planck equation in n variables x = x 1 , . . . , x n , n n ∂ 2 ∂ f A i ( x ) f ( x , t ) + 1 ∂ � � ∂ t ( x , t ) = − B ij ( x ) f ( x , t ) + H ( x ) ∂ x i 2 ∂ x i ∂ x j i i , j assuming natural boundary conditions where A i and B ij are the drift vector and diffusion tensor respectively. • The characteristics are described by an Itô stochastic differential equation (SDE) q � dX i ( t ) = A i ( X ( t )) dt + σ ij ( X ( t )) dW j ( t ) j where σ ij is defined by B ij = � q k σ ik σ kj and dW is a q -dimensional Wiener process with zero mean and dt variance. • Many realizations of the SDE give a distribution with the same solution as obtained by the FPE. L.J. Höök (KTH) GenFoo: a general Fokker-Planck solver 2012-06-06 16 / 25

Internal structure of GenFoo Fokker-Planck model Input GenFoo JIT FEM Monte Carlo Delta-f Monte Carlo Output L.J. Höök (KTH) GenFoo: a general Fokker-Planck solver 2012-06-06 17 / 25

The JIT compiler XML to CPP Operator XML Schema XSLT (XSD) Pipe, fork (FEM) factory g++ dynamic loading CPP to .so (dlopen) L.J. Höök (KTH) GenFoo: a general Fokker-Planck solver 2012-06-06 18 / 25

Defining an operator: Heston model in finance <?xml version="1.0"?> <Operator> <Dimensions> 2 </Dimensions> <Drift> <component index="0"> <value> (r-d)*x[0] </value> </component> <component index="1"><value>kappa*(theta-x[1])</value></component> </Drift> <Diffusion> <component indexColumn="0" indexRow="0" > <value> sqrt(x[1])*x[0] </value> </component> <component indexColumn="1" indexRow="1"> <value> xi*sqrt(abs(x[1])) </value> </component> </Diffusion> <Source> <value>0.0</value></Source> <InitialCondition><value> ...</value></InitialCondition> </Operator> L.J. Höök (KTH) GenFoo: a general Fokker-Planck solver 2012-06-06 19 / 25

3D radio-frequency heating L.J. Höök (KTH) GenFoo: a general Fokker-Planck solver 2012-06-06 20 / 25

GenFoo blueprints Implement... • a common plasma physics class. • stability schemes e.g. SUPG, standard Galerkin on augmented grid [Brezzi, 2005], suggestions are welcome! • time-dependent goal-oriented adaptivity. • output kernels. L.J. Höök (KTH) GenFoo: a general Fokker-Planck solver 2012-06-06 21 / 25