Correlated Sampling Fission Matrix Method

We have shown in previous sections of this chapter that either (i) subtracting two independent Monte Carlo simulations or (ii) combining source iteration and correlated sampling fails to estimate small perturbation effects with reasonable accuracy. Several references [Rie88, Sei91, Gal95] point out that for the calculation of Monte Carlo perturbation effects it is necessary to utilize the fission matrix method to perform the eigenvalue calculation.

To that end, two fission matrix equations,

K^up S^up = A^up S^up,

and

K^p S^p = A^p S^p,

for the unperturbed and perturbed systems respectively are needed to formulate and solve for the eigenvalue perturbations:

$${\Delta}K = K^p - K^{up}.$$ (75)

In the above equations, K is the eigenvalue, S is the fission source vector, and A is the fission matrix. One approach [Sei91, Gal95] utilizes the linearity of the transport equation and splits the fission matrix into two parts:

A^up = A₀ + A₁.

The fission matrix A₀ is formed by the particles that do not go through a perturbed cell during a random walk simulation, whereas the fission matrix A₁ is formed by the particles that arrive from any perturbed cell. Since the perturbation is confined to the perturbed cells, the matrix A₀ is equal in both the unperturbed and perturbed problems. The matrix A₁ is evaluated only for the perturbed problem. During particle tracking in the unperturbed system, all necessary information for particles that enter any perturbed cell are saved. At the end of the simulation in the unperturbed system, this information is then used to calculate to the matrix A₁ for the perturbed system.

We have taken a different approach in which the correlated sampling technique is applied to the fission matrix method [Rie88]. The actual Monte Carlo simulation is done in an artificial reference system with cross sections that are linear combination of the unperturbed and perturbed systems. The $\delta$-scatter cross sections in the forward direction are used for the reference, unperturbed, and perturbed systems as explained in section 3.3.1 of this chapter. For the determination of the fission matrix of the reference system, the first generation is started with an assumed source distribution and W^ref = 1, where W^ref is the weight of a reference system particle. Likewise, the initial weights of the unperturbed (W^up) and perturbed (W^p) particles are also set to unity. Particle tracking is carried out only in the reference system and the weights of the unperturbed and perturbed systems are modified by multiplying them with appropriate adjusting weight factors, WF^up and WF^p, respectively, during the simulation. At the end of the first generation, source normalization is done for the reference system particles to stabilize the neutron population. The source normalization has no direct effect on other systems, except that the next generation particles in the unperturbed and perturbed systems start out with the same weights as that of the reference system particles. For the second generation, the fission neutron production distribution obtained from the first generation is used for the reference system particles. The particles in the unperturbed and perturbed systems also use the same source distribution as that of the reference system for the second generation. This process is continued for specified number of generations. A few of the initial generations are discarded to avoid bias due to the initial source guess.

Now the fission rate is determined. Recall from our discussion on the fission matrix algorithm of chapter 2 that the fission rate is determined by the probability that a particle starting in volume element l generates a_l,m particles in element m, where a_l,m is an element of the fission matrix A. For the reference system, the matrix element a_l,m^ref is scored as,

$${a_{l,m}^{ref}}={a_{l,m}^{ref}} + W^{ref}{[{\nu^{ref}\Sigma_f^{ref}}] \over{[\Sigma_t^{ref} + \delta^{ref}]}}\;\;\;,$$ (77)

where $\nu^{ref}$ is the number of particles emerging from a fission reaction in the reference system and $\Sigma_f^{ref}$ is the macroscopic fission cross-section for the reference system. At the same time, the fission matrix elements for the unperturbed and perturbed fission matrices, A^up and A^p, are estimated as,

$${a_{l,m}^{up}}={a_{l,m}^{up}} + W^{up}{[{\nu^{up}\Sigma_f^{up}}] \over{[\Sigma_t^{ref} + \delta^{ref}]}}\;\;\;,$$ (78)

and

$${a_{l,m}^{p}}={a_{l,m}^{p}} + W^{p}{[{\nu^{p}\Sigma_f^{p}}] \over{[\Sigma_t^{ref} + \delta^{ref}]}}\;\;\;.$$ (79)

The dominant eigenvalues K^ref, K^up, and K^p of matrices A^ref, A^up and A^p, respectively, are determined numerically, after which ${\Delta}K$ can be calculated as,

$$\Delta{K} = K^{p} - K^{up}\;\;\;.$$ (80)

As before, the standard deviation of the single generation $\Delta$K is given by,  

$$\sigma_s = \sqrt {{{\sum\limits _{n=1}^{I_a}} {\Delta}{K_n^... ...s _{n=1}^{I_a}}{\Delta}{K_n})}^2 \over {I_a(I_a-1)}}} \;\;\;$$ (81)

and the standard deviation of the mean is given by  

$$\sigma = {\sigma_s \over {I_a}^{1 \over 2}}\;\;\;.$$ (82)

This $\sigma$ is provided with the numerical Monte Carlo results. Variance reduction schemes are applied as explained in section 3.3.2 of this chapter.

The perturbation problems of table 3.1 and 3.2 are now solved with the combined CSFM technique as described above. These results are shown in table 3.3. Similar to table 3.1 and 3.2, the Monte Carlo runs for table 3.3 utilize 30 inactive batches, 70 active batches and 2000 neutrons per batch.

 

1|cUnperturbed 1ccross sections: 1c 1c|

$$\Sigma_t \nu\Sigma_f \Sigma_s$$ 1c|

$$\Delta \Delta$$ 1|cProblem # 1|c|Perturbed cross sections

$$\Delta\Sigma_t \Delta\Sigma_a \mp$$ 1 -.008807

$$\delta^{ref}$$

$$\Delta\Sigma_t \Delta\Sigma_a \mp$$ 2 -.000888

$$\delta^{ref}$$

$$\Delta\Sigma_f {\pm}$$ 3 .000903

$$\delta^{ref}$$

Comparing the results of table 3.1 and table 3.2 with that of table 3.3 we observe that the CSFM technique provides significantly improved results compared with direct subtraction or the source iteration method with correlated sampling.