MCRT: L4 A Monte Carlo Scattering Code

Transcription

1 MCRT: L4 A Monte Carlo Scattering Code Plane parallel scattering slab Optical depths & physical distances Emergent flux & intensity Internal intensity moments

2 Constant density slab, vertical optical depth τ max = n σ z max Normalized length units z = z / z max, so 0 < z < 1 Emit packets Packets scatters in slab until: absorbed: terminate, start new packet z < 0: re-emit packet z > 1: escapes, bin packet Loop over packets Pick optical depths, test for absorption, test if still in slab

3 θ Bin this packet in angle z = z max τ max = n σ z max Packet absorbed Start next packet z = 0 Re-start this packet

4 Emitting Packets: Packets need an initial starting location and direction. Uniform specific intensity from a surface. Start packet at (x, y, z) = (0, 0, 0) I ν ( µ de de dn ) = µ I ( µ ) µ dadt dν dω dadt dν dω dω ν Emission direction: sample φ = πξ Sample µ from P(µ) = µ I (µ) using cumulative distribution. Normalization: emitting outward from lower boundary, so 0 < µ < 1 µ ξ = P( µ ) dµ P( µ ) dµ = µ µ = ξ n x = sinθ cosφ n y = sinθ sinφ n z = cosθ

5 Distance Traveled: Random optical depth τ = -log ξ, and τ = n σ L, so distance traveled is: L = τ τ max z max x = x + L n x y = y + L n y z = z + L n z Scattering: Assume isotropic scattering, so new packet direction is: θ = φ = cos 1 (ξ 1) π ξ n x = sinθ cosφ n y = sinθ sinφ n z = cosθ Absorb or Scatter: Scatter if ξ < a, otherwise packet absorbed, exit do while in slab loop and start a new packet

6 Structure of FORTRAN program: do i = 1, npackets 1 call emit_packet do while ( (z.ge. 0.).and. (z.le. 1.) )! packet is in slab L = -log(ran) * zmax / taumax z = z + L * nz! update packet position, x,y,z if ((z.lt.0.).or.(z.gt.zmax)) goto! packet exits if (ran.lt. albedo) then call scatter else goto 3! terminate packet end if end do if (z.le. 0.) goto 1! re-start packet bin packet according to direction 3 continue! exit for absorbed packets, start a new packet end do

7 Fortran functions and subroutines: Pass variables and a FUNCTION performs maths, algebra, geometry, to return a single result, e.g., random number generators. SUBROUTINES are similar, but can return many different results. The order that variables are passed must be the same in the call statement and the subroutine. In main program have the call statement: call emit_packet(iseed,x,y,z,nx,ny,nz) subroutine emit_packet(iseed,x,y,z,nx,ny,nz) implicit none integer iseed real x,y,z,nx,ny,nz,ran Set packet position and direction using equations return end

8 Binning Packets Bin escaping packets according to direction of travel in φ (azimuth) and θ (polar). Bins of equal φ and equal [cos(θ)], gives bins of equal solid angle. Can also have internal spatial bins to sum quantities for computing number of interactions, fluxes, pressures, etc. See lectures on 3D grid codes. Can also bin in time when doing time-dependent simulations. In general the time for a particle to traverse a distance is simply t = d / v, where v is photon or particle speed in the medium.

9 Error Estimation Recall expected value (mean), variance, standard deviation: n E(x) = x = 1 x n i var(x) = σ = 1 i=1 n Where x can be number of photons in a bin (spatial or angular); number of scatterings per photon; counters for mean intensity, fluence, number of absorptions, radiation pressure; etc, etc Compute sum of x and x in each bin (angular or spatial) and estimate σ using: σ = var(x) = E(x ) [ E(x) ] n ( x i x) i=1 Variance of the mean is σ / n Standard deviation of the estimate of the mean is σ / n 1/

10 Random walks Net displacement of a single photon from starting position after N mean free paths between scatterings is: Square and average to get distance R travelled : The cross terms are all of the form: R = r 1 + r r N l * R = r 1 + r r N r 1 r = r 1 r cosδ + r 1 r +... where δ is the angle of deflection during the scattering. For isotropic scattering, <cos δ> = 0, cross-terms vanish.

11 Thus, for a random walk we have l * R = r 1 + r r N α l * = τ max = Nα r = N τ N = τ max / τ = τ max / Using: τ = p(τ )τ dτ = e τ τ dτ = 0 0 If the medium is optically thin, then the probability of scattering is Using 1 e τ τ then N τ, τ << 1 τ 1 e Therefore N τ +τ / will be roughly correct for any optical depth