Numerical Green s Function Techniques

Transcription

1 Numerical Green s Function Techniques Andrew J. Christlieb University Department of This work supported by AFOSR and AFRL.

2 Co-Workers Senior Collaborators Iain Boyd (U Mich) Jean-Luc Cambier (AFRL: Edward s) Keith Cartwright (AFRL: Kirtland) Nick Hitchon (U Wisc) Robert Krasny (U Mich) (Co-Pi: AFOSR grant F012457) Georg Raithel (U Mich) John Verboncoeur (UC-Berkley) Post Doc Collaborators Lyudmyla Barannyk (U.Mich) (Supported AFRL/AFOSR-F013830) Graduate Student Collaborators co-advisor with Iain D. Boyd: Jerry Emhoff, PhD co-advisor with Georg Raithel: Spencer Olson, PhD Undergraduate Student Collaborators co-supervisor with Robert Krasny: Benjamin E. Sonday. (Undergraduate research supported by AFOSR grant F012457)

3 Workshop Mission Statement The key difficulty encountered when optical fields meet nanostructures is a fundamental mismatch in scales. ( ) This mismatch gives rise to phenomena (as described above) not encountered in conventional optics. ( ) challenges include the development of multiscale tools for analyzing the Maxwell equations in complex media; the study of scattering problems for highly-localized wave-fields; and the development of mathematical tools and associated inverse scattering problems for image reconstruction with subwavelength resolution.?plasma?

4 Outline Overview Goal: Develop grid-free simulation tools for complex multi-scale problems Current Grid-Free Work 1. Develop grid-free field solvers with fully Lagrangian fluid and kinetic plasma models 2. Develop grid-free Monte Carlo simulation method for modeling the effects of collisions Future Directions Electromagnetics Conclusions

5 Plasma Solid Liquid Gas E E E Plasma For an Argon Plasma T e ~3eV or Greater 1eV =10,000K

6 Examples of Plasma Sun Energy (Controlled Fusion) ~99% of the Universe in Plasma State Lightning Multi-Scale Electromagnetic Space ~ Time ~ 10 18

7 metal QuickTime?and a Helicon Plasma Sources Goal Dielectric Helicon Sources Plasmonics Types of Plasma Waves Surface Evanescent plasma Waves impart energy to plasma Surface - Stochastic Heating Evanescent - Ohmic Heating Efficiency Type Heating

8 Governing Equations Boltzmann t f i + v x f i + F i m i v f i = Lorenz F i = q i (E + v B) Maxwell H = ε t E + J, µ t H = E, εe = J = q i i q i i k f i dv, µh = 0, v f i dv δ{ f k f i } Key: f - Distribution Function (t,x,v) - 6D + time δ - Collision Operator m - Mass of Species i q - Charge on Species i F - Force on Species i E - Electric Field H - Magnetic Field J - Current Density B = µh

9 Observables Density Bulk Velocity Total Energy n i = u i = Q = f i dv v f i dv 1 2 m 2 i v 2 f i dv Chapman-Enskog f i (t,x,v) = n i (t,x)γ i (v) Sub. into Boltzmann Eq. Take First Three Moments Assume γ i (v)-gaussian t n i + (n i u i ) = S o (n k ) L o (n i ) n i ( t u i + u i u i )= q i n i (E + u i B) p + S 1 (n k ) L 1 (n i ) t e i + u i e i = ( q i n i (u i E) + (u i p) )+ S 2 (n k ) L 2 (n i ) Note: Must choose a closure relating p to the other variables.

10 Simulation Eulerean Semi-Lagrangian Lagrangian 1) CFL 2) Numerical Diffusion 3) Memory (100 3 X30 3 )~100GB v 1) Numerical Diffusion 1) Point Separation 2) Memory (100 3 X30 3 )~100GB dx dt = v v dv dt = F Fixed Mesh x Fixed Mesh x F i = 1 ε j {1,...,N}/i q j y G(x i y) y= x

11 Current Work Numerical Green s Function s lasov-poisson: t f i + v x f i + F i v f i = 0, m i F i = q i Φ i {+, } 2 Φ= 1 ( ε q + f + dv + q f dv)= ρ ε αφ + β Φ n = g(x,y) (x,y) Ω reen s Formula : Let : Free Space reen s Function ( u 2 v v 2 u) dω= ( v u u v) n ds Ω Ω u = G(x, y x 0, y 0 ) and v = Φ 2 G = δ(x x 0 )δ(y y 0 ) Φ(x 0, y 0 ) = ρ ε G dω ( Φ G G Φ ) n ds Ω Ω (jump conditions)

12 Volume Integral Lagrangian Points Φ p = q j Change of Variables to IC: f j j (t,y,v)g(y x) dvdy ε Γ t f j (t,y,v) dvdy = f j (t,y(t,y o,v o ),v(t,y o,v o )) J o (y,v)dv o dy o Γ t Lemma: Γ o f j (t,y(t,x o,v o ),v(t,x o,v o ))J o (y,v)= f j (0,y o,v o ) Potential: q Φ p = j f j j (0,y o,v o )G( y(t,y o,v o ) x) dv o dy o ε Γ o Trapezoidal Rule: M M G( y(t,y o,v o ) x)f j (0,y o,v o )dy o dv o = v x G( y(t,y l o,v k o ) x)q l,k Γ o k=1 l=1 + E j Q l,k = q j f j (0,y o l,v o k )w l w k

13 Boundary Integral Method Consistency: Limit to Boundary lim αφ + β Φ n v = g(l), l Ω r Ω Homogeneous Solution (PDE s-folland) Single Layer Potential Φ H (x o, y o ) = ( σ(y)g(x j,y j x 0, y 0 ))ds(y) Ω1 Double Layer Potential Φ H (x o, y o ) = ( γ(y) G(x j,y j x 0, y 0 ) n )ds(y) Ω2 Panel Method: σ i, γ i = constant on panel j, j = 1 to M Linear System: Aσ = b GMRES: A i,j ~ G(x i,y i x j,y j ) Boundary is equivalent to particles F i = 1 ε q j y G(x i y) y= x j O(N 2 )? j {1,...,N}/i

14 Fast Summation O(N log N) Compute E-Field at Red Particle N j=1 j red E red = Q j G(x red,x j ) Barnes-Hut treecode Particle Cluster Method (Monopole) Cluster - Bisection Algorithm E-Field - Divide and Conquer Red Particle High Order Approximations - Multipole K. Lindsay and R. Krasny, A particle method and adaptive treecode for vortex sheet motion in three-dimensional flow, J. Comput. Phys., 172, pp.~ , 2001.

15 Advanced Methodology Smoothed Kernel 2D : G(r) = ln( r 2 + δ 2 ) 2π For 2 and 3 dimensions G (k) recursion. 3D : G(r) = 1 4π r 2 + δ 2 Order (O p ) needed in Taylor expansion computed on fly Adaptive Point Insertion to Maintain Resolution A. Christlieb et.al. Grid-Free Plasma Simulation Techniques", IEEE TPS, 34 (2): Part 1 APR 2006

16 1D - Two Stream l -l ϖ f (0,ζ o,ϖ o ) f + (τ,ζ,ϖ) ζ = 2π Periodic Boundary Conditions f + = Cδ(ϖ) f = δϖ+ 3 a nc + εsin(ζ) + δϖ Electron Response: PIC Vs. BIT 3 a nc + εsin(ζ PIC-TechX BIT BIT-PI (400) QuickTime?and a TIFF (U are n

17 Exact Solution: Direct Sum Timing Results PIC Vs. BIT N-Particles: Randomly Placed MxM-Mesh: PIC Multi-Grid Treecode: Tuned to match error Time (Sec.) Rel. Error 300, , # Particles DS MG-256 MG-512 MG-2048 TC-1 TC , , , , % 0% 0% % 13% 14% % 8% 10% % 1% 1% 8% 7% 6% Ω Φ( Ω) = % 0.08% 0.07%

18 Penning-Malmberg Trap Crystal Formation Bounce Time >>ExB Rotation dx r dt = V r EXB, r V EXB = QuickTime?and a decompressor are n r E B r r B 2 Experiment τ = 0 τ = 0.5 τ = 1 τ = 10 Simulation τ = 0 τ = 0.5 τ = 1 τ = 3

19 Future Work: Grid-Free EM H = ε t E + J, µ t H = E, εe = J = q i i q i i Maxwell f i dv, µh = 0, v f i dv wo Approaches Three Ideas Time Dependent Green s 1. Taylor Expand in Time Time Independent Green s Function} 2. Mesh (P3M) 3. Adaptive Particles

20 Time Clustering Consider Free Space Wave Equation: The free space Green s function: G(r,r' t,t') = 1 4π IDEA 1: Taylor Expand in Time and Space ( ) r r' δ t t' r r' u tt u = s(r,t) 1 ε 1 e 4π Solution treated as Free Space Plus Homogenous Cluster in Space Time Delay ~ r-r, Cluster in Time t t' (r r') 2 +ε 2 ( ) 1 2 ((r r') 2 + ε 2 ) 1 2 For Static Meshes (Plane Wave Expansion) B. Shanker, A. Ergin and E. Michielssen, J. Opt. Soc. Am (2002) N. Gres, A. Ergin, E. Michielssen and B. Shanker, R. Science (2001) 2 ε

21 Boundary Element Methods IDEA 2: Particle-Particle Particle-Mesh Solve on a Mesh, then correct for local interactions using BIM ω >> x Magnetrons: Looks Static (Existing Ideas) Use Mesh to Approximately Track Time History (Symmetry of G) Summer: AFRL

22 Adaptive Particle Method IDEA 3: Finite Difference in Time - Invert Spatial Operator Using G u tt = u Define: ζ = G Ω ( )dv ζ u n +1 2ζ u n + ζ u n 1 = dt 2 ζ u n u n +1 2u n + u n 1 dt 2 u n, where: G = δ x x o ( ) = dt 2 u n ( u n G G u n ) n ds Ω Boundary Element Methods ζ u n +1 = 2ζ u n ζ u n 1 + dt 2 u n + BEM

23 Adaptive Particles Lagrangian : u n (x) = α jn δ(x x j n ) N j=1 Then at the spatial collocation points x i n +1 N ζ u k n (x +1 i ) = α k n j G(x +1 i x k j ), k n 1, n,n +1 j=1 Unknowns: α i n +1 where: A k n +1 M α i n +1 M = bn +1 { } [ n +1 A ] k = G( n x +1 k i x ) j

24 Goal Multi-Scale Lagrangian Simulation Framework for Fluid-Kinetic Electormagnetic Plasma Simulations

25 Conclusions Electrostatic Results Have a GOOD Start christlieb@math.msu.edu A lot of work to be done, i.e., Need GOOD Grid Free Fluid Solver Increase Efficiency of Field Solver For EM, Looks Reasonable, but cleverness will lie in time history