FRP2 - Montecarlo Flashcards
Cosa è il collision kernel?
K(r,v | r’,v’)drdv è il prodotto tra i due kernel collision C(v | r’,v’)dv e transition T(r -> r’,v)dr
They both are less than one due to absorption and escaping
The kernel gets r and v as input and compute the exiting r’,v’ unless the particle escaped or got absorbed. In this case we proceed with the next one
Talk about montecarlo
E un metodo statistico per risolvere l’equazione di boltzmann in forma integrale.
Inzialmente sviluppato per steady state ora usato anche con cambiamenti di burnup.
la base è un sistema markoviano che sta in uno spazio delle fasia 6 variabili, spazio e velocità (vettoriali)
Quale è la procedura per il calcolo montecarlo?
Parto dal source term
Applico i kernel per ciascuna collisione fino a che non viene assorbito o esce dal sistema
Ripeto per il prossimo neutrone (o fotone)
Spiega la consistenza del metodo montecarlo
psi_n(r,v) è la collision density parziale, cioe una pdf che la particella abbia la n+1 collisione in (r,v).
La espandiamo in serie di Neumann
psi_n(r,v) = int( psi_(n-1)(r’,v’)K(r,v | r’,v’) dr’dv’)
Quindi sommando tutti i contributi
psi (r,v) = sum_n psi_n = psi_0 + sum_n psi_n
che è la stessa faccia di BE nella forma integrale [phi = Q* + K phi]
Come stimare gli integrali in MC?
Con il vaolre di aspettazione.
G = int_a^b g(x) f(x) dx
f(x) is the pdf of “hitting the target”
g(x) is the actual value that you get by “hitting it”
Iterated source method with montecarlo
Starting from a stationary system with uniform and isotropic neutrons coming from fission (this is an approx)
k1 is the first tentative multiplication factor
l1 is the number of location of fission
nu1 = nu/k1
simulation is performed with nu1
nu1 l1 is the neutrons produced by fission
if we are critical nu1 l1 = nu2 l2 = … so that
nu_i = nu/k_i and
nu_i = nu1 l1/l_i
quindi k_i = k1 l_i/l1
This works also for fastly diverging systems
What is the rejection method?
- define H so that H>= max(f(x))
- define g(x) = H in [a,b] and zero elsewhere
- Sample a uniform distribution in [a,b] so that x’ = Rand*(b-a) + a
- define surviving probability as P = f(x’) / H
- if P is smaller than a second random number than I keep the count else I discard it
What is the inverse transform method?
- define the cdf F(x)
- find x as F^-1(rand)
Example: if f(x) is the pdf f(x) = H(x) * sigma * exp(-Sigma x)
where H is heavyside
then rand = F(x) = 1-exp(-Sigma x)
x = …
What is letargy?
u = ln(E0/E)
High E means low letargy
Neutrons goes from low letargy to high letargy, when they interact.
It’s better to reduce the scale that I am sampling
How do we compute radiation crossing different materials?
free flight is:
x = -1/Sigma * ln(1-rand)
- Exhaust optical length:
chi = x Sigma1 = -ln(1-R)
chi12 = x12 Sigma1
if chi > chi12
then x = x12 + (chi-chi12)/Sigma2 - Re-extraction of free flight
x1 = -1/Sigma1 * ln(1-rand)
if x1 < x12
then x = x1 + (-1/Sigma2 * ln(1-rand)) = x1 + x2
What is the Figure Of Merit?
FOM = 1/(R^2 T)
T: computational time (propto N)
R: relative error = 1/(sqrt(N) mu)
mu: mean value
FOM measure the efficency we can achieve with an algorithm. It’s independent of N
What is the splitting procedure?
If we consider a heavy absorber very few neutrons will pass through, resulting in low FOM.
We divide the slab in many subslabs with different importance.
The particles that passes the first subslab get multiplied by three to improve statistics but their importance is divided by 3 to keep physical meaning. so on on the other slabs
What is the forced collision method?
if the slab is very thin (delta) the probebility of interaction is very low: P = 1-exp(-Sigma delta)
We set the probability to 1 and reduce accordingly the weight so that P * weight = constant
so the weight will become 1-exp(-Sigma delta)
What are the possible error sources in MC?
- Geometry approx
- Epistemic errors (things that I do not know)
- Sistematic errors
- Statistical errors (relative error R = 1/sqrt(N))
