AC274:Particles in Cell
(phase-space fluids)
Systems with soft (long-range) interations:
Grazing collisions (small deflections)
Each particle feels a collective soft interaction
Plasmas, AstrophysicsBiological fluids,Electrophoresis,
Phase-space fluids
Particle-Particle: hard-core
Field-Particle: smooth
Particle: mesh-free
Field: mesh bound
Electro-biofluids: Ion Channels
Vlasov-Poisson-Boltzmann
Smooth force:
Length scales and resolution
1. Charge assign to the grid (P2G)
2. Solve field eq.s on the grid (GG)
3. Compute forces on the particles (G2P)
4. Move particles (PP)
PIC: solution procedure
G-P duality
Transfer operators: Weight and Green
P2G:
G2P:
Charge
PotentialForce
PIC algorithm
2.GG: Solve Poisson:
1. GG: Charge assign:
Vlasov-Poisson
3. GP: Force interpolation
4. PP: Move Particles
Charge assign: scatter, P2G
Solve Poisson, GG
Potential to Force (GG)
Force transfer G2P
Charge Assign: Locate
Grid NE
SE
NW
SW
P=(x,y)
Locating particles in arbitrary grids
Charge Assign: NGP
2
1
3
0
P
Charge Assign: PWC
NE
SE
NW
SW
P
Charge Assign: PWlin
NE=2NW=3
p=(x,y)
SW=0 SE=1a
b
Force interp: NGP
NE
SE
NW
SW
Force interp: PWC
NE
SE
NW
SW
Force interp: PWL
2
1
3
0 a
bp
Charge collect: receive
Charge assign: NGP
8
76
5
4
3
2
1
Charge assign: PWC
8
76
5
4
3
2
1
Shape functions: 1d
Spurious Forces and Grid Independence
The self-force problem
The self-force problem
ng=2
The self-force problem
PwL: -a*(1-a)+(1-a)*a; = 0 for any a
PwC: -a*1/2+(1-a)*1/2=1/2-a; = 0 only for a=1/2
NGP: -a*1, or (1-a)*1; = 0 never
BUT: if <a>=1/2 both pwc and ngp give zero on average!
Spurious forces
Choosing the kernel
Use the same kernel
Or:Force kernel G equal or lower order than charge kernel W
Poisson Solver
Spectral (if you can…)
Poisson Solver: relax
Poisson Solver: RES
Special case: Laplace
Poisson Solver: bc TBD
Thomas algorithm
Bwd sweep
Fwd sweep
Position Verlet
Staggered: (4th, reversible)
Q: How do we start-off?
PIC methods scale like NP*NG instead of NP*NP
Very useful for phase-space fluids with soft interactions: plasmas, astrophysics, electro-bioflows
Modern developments:Unstructured, Moving Grids, parallel solvers, totrack STRONG inhomogeneities
P3M: Particle-Particle (dense) + Particle-Mesh
Summary
Nearest grid point
i=nint(x/dx)
Phi(i)=phi(x)
CIC=PWC
Phi(i)=phi(x)/2Phi(i+1)=phi(x)/2
p=x/dx-i p>0: p<0:
CIC Force transfer
CIC Force transfer