PHYSOR 2018: Reactor Physics paving the way towards more efficient systems Cancun, Mexico, April 22-26, 2018
GROUP-WISE TALLY SCHEME OF INCREMENTAL MIGRATION AREA FOR CUMULATIVE MIGRATION METHOD Zhaoyuan Liu, Kord Smith and Benoit Forget Department of Nuclear Science and Engineering Massachusetts Institute of Technology, Cambridge, Massachusetts, USA
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ABSTRACT A new group-wise tally method to compute multi-group diffusion coefficients and transport cross sections using Cumulative Migration Method (CMM) is proposed in this paper. By tallying the incremental migration area, CMM can be implemented in a more general and efficient manner. More importantly, the new tally scheme enables CMM to be used for computing heterogeneous multi-group diffusion coefficients and transport cross sections. Proper treatment of the tallies for region interface crossing neutrons is a key problem for further generalization of CMM. A straight-forward approach of separating the regional tally of incremental migration area at the intersection point is tested on the two-region “sanity check” problem. KEYWORDS: cumulative migration methods, diffusion coefficients, transport cross sections, group-wise tally
1. INTRODUCTION In deterministic neutronics simulation of nuclear reactor physics, multi-group diffusion coefficients and transport cross sections are key parameters for diffusion and transport calculations, respectively. There are many existing methods on computing these parameters, such as the well-known outscatter approximation method, Bell-Hansen-Sandmeier approximation method [1], in-flow transport approximation method [2], etc. Comprehensive work of testing and comparing among these methods for computing both diffusion coefficients in diffusion calculation and transport cross sections in transport calculation have been investigated over the past years [3–7]. However, unlike traditional multi-group cross sections, such as total cross sections and scattering cross sections, there is no direct way to generate accurate diffusion coefficients and transport cross section using tallies from Monte Carlo codes. Most methods require some amount of post-processing for Monte Carlo tally results. The cumulative migration method (CMM) is a new method originally proposed for computing homogenized multi-group diffusion coefficients and transport cross sections in light water reactors that is both rigorous and computationally efficient [8,9]. It allows for direct tallying to the desired group structure and eliminates the sources of inaccuracy in commonly applied approximation methods. The newly developed method is directly applicable to lattice calculations performed by
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Liu, et al, Group-wise Tally Scheme for CMM Monte Carlo and is capable of computing rigorous homogenized diffusion coefficients for arbitrarily heterogeneous lattices. Directional diffusion coefficients can also be computed in a natural approach using CMM. The improvement of accuracy for eigenvalue and power distribution using CMM has been demonstrated on various scales of test problems [9,10], including the BEAVRS PWR benchmark problem [11]. In this paper a new group-wise tally method for CMM will be introduced which is equivalent to the cumulative-group tally and enables its generalization to heterogeneous tallies. It makes the tally of migration area in Monte Carlo more consistent and general compared with other conventional tallies, allowing for simple integration into existing Monte Carlo codes. The Section 2 in this paper will introduce the methodology of this new group-wise tally scheme using incremental migration area. Implementation algorithm and corresponding pseudo-code can be found in Section 3 with two examples. The generalization of CMM towards heterogeneous problems and current challenges are discussed in Section 4. Conclusions are summarized in Section 5. 2. METHODOLOGY 2.1. Cumulative Migration Method Based on the theory of migration area [12], CMM introduced a new concept of cumulative group quantities. Traditional multi-group energy structures usually treat each energy range individually, while each cumulative group covers a broader energy range, always starting from the highest energy to the energy of the lower bound of the group of interest. For instance, in the illustration of energy group structure in Figure 1, the energy range of group g can be represented as [Eg , Eg−1 ], with Eg denoting for the energy value of lower boundary for group g. With Emax representing the highest energy, then in the concept of cumulative energy groups, the energy band covered by cumulative group g is [Eg , Emax ].
Figure 1: Illustration of cumulative energy groups compared with conventional individual energy groups. (The total number of individual groups is G in this example, and EG is usually 0 as the minimum energy value.) Based on this concept, the relationship of cumulative migration area (Mgc )2 , cumulative diffusion coefficient Dgc , and cumulative removal cross section Σcr,g can be expressed as Proceedings of the PHYSOR 2018, Cancun, Mexico 2513
Reactor Physics paving the way towards more efficient systems
(Mgc )2
Dgc 1
= c = (rg )2 Σr,g 6
(1)
where the superscript c indicates that all the quantities in this equation are for the cumulative group g. In this equation, rg is the crow flight length from the neutron’s birth position to the position where it is removed from the cumulative group g. The second relationship of equality can be derived using the analytical diffusion solution of spatial flux distribution from a point source in infinite homogeneous medium [12]. Equation (1) provides a scheme for computing cumulative-group diffusion coefficients because rg is a quantity that can be tallied directly in Monte Carlo codes. And using CMM, then group-wise diffusion coefficients Dg can be calculated by “unfolding” cumulative-group diffusion coefficients Dgc using flux-weighting as shown in Equation (2). g P
Dgc
=
D g 0 φg 0
g 0 =1 g P
(2) φg 0
g 0 =1
2.2. Incremental Migration Area As shown in Section 2.1, the tally of cumulative migration area is different to general tallies in Monte Carlo codes. It is natural to interpret and implement the tally for cumulative migration area in a “cumulative” way, which could involve an iteration over multiple groups for each reaction. However, it is unnatural to execute inner loops in Monte Carlo tallies and difficult to generalize these cumulative results to heterogeneous regions. In a previous paper [13], an alternative approach of vector-connected group-wise tally scheme was introduced to solve this issue, but the realization was still indirect and required lots of post-processing. Instead, an purely group-wise approach was found to be equivalent for computing multigroup diffusion coefficients. This new approach makes use of the incremental migration area ∆Mg2 , which is the mean increment of migration area in each group and on average it is equal to h(rg )2 i − h(rg−1 )2 i. Using the incremental migration area, the multi-group diffusion coefficients can be computed simply as Dg = ∆Mg2 Σr,g
(3)
In Equation (3), Σr,g is the group-wise removal cross section. The removal cross section accounts for all the removal reactions in each group, which includes the absorption, down-scatter and up-scatter reactions. By definition, the group-wise removal cross section can be computed as in Equation 4. Proceedings of the PHYSOR 2018, Cancun, Mexico 2514
Liu, et al, Group-wise Tally Scheme for CMM This is in the same form as other conventional group-wise cross sections, with Rr,g representing the removal reaction rates in group g and φg representing the flux in group g.
Σr,g =
Rr,g φg
(4)
2.3. Equivalence between Cumulative and Incremental Tallies Compared with the previous cumulative or vector-connected group-wise tallies, the new group-wise tallies of incremental migration area is simpler and more straight-forward. Moreover, it can be shown by examples that they are essentially equivalent and the results of cumulative migration area and resulting diffusion coefficients are exactly the same. With the group-wise incremental migration area ∆Mg2 tallied using this new method, the cumulativegroup migration area (Mgc )2 can be computed by the following equation:
(Mgc )2 Ngc
=
g X
∆Mg20 Ng0
(5)
g 0 =1
In Equation (5), Ngc is the number of removal reactions in cumulative group g, and Ng is the number of removal reactions in each individual group g. One warning should be mentioned here is that the relationship between Ngc and Ng is not in the form of a simple summation. Because down-scatter and up-scatter reactions from group g may contribute to Ng , but may not contribute to Ngc , and vice versa. For example, an up-scatter reaction from group 3 to group 2 will be counted in N3 , since the neutron left group 3 after this reaction. But it will not be counted in N3c , because it still stays in cumulative group 3. The equivalence between cumulative group tallies and the new purely group-wise incremental tallies can be better explained with the examples in the next section. 3. IMPLEMENTATION 3.1. Tally of Incremental Migration Area Based on the method introduced in Section 2.2, CMM can be implemented in a purely group-wise approach in the same manner of other conventional tallies in Monte Carlo codes. One can tally and compute multi-group diffusion coefficients using CMM according to Equation (3), but actually the algorithm shown in Algorithm 1 provides a more straight-forward tally process, without the necessity of computing removal cross sections Σr,g . The pseudo-code for tallying incremental migration area in a purely group-wise manner is shown in Algorithm 1. This pseudo-code is intended to illustrate the tally process in an analog way for neutrons with unity weight, and it should be mentioned that it is not optimal for efficiency. Proceedings of the PHYSOR 2018, Cancun, Mexico 2515
Reactor Physics paving the way towards more efficient systems Algorithm 1 Tally process of incremental migration area ∆Mg2 (in Python syntax) 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18:
procedure TALLY IMA for neutron in allNeutrons: (xbirth , ybirth , zbirth ) = neutron.getBirthPosition() while neutron.isAlive(): # record the information of this neutron’s state before each collision state pre = neutron.state (xpre , ypre , zpre ) = state pre.coordinates 2 rpre = (xpre − xbirth )2 + (ypre − ybirth )2 + (zpre − zbirth )2 gpre = state pre.group # execute one transport flight of this neutron neutron.transport() # get the information of the new state state new = neutron.state (xnew , ynew , znew ) = state new.coordinates 2 rnew = (xnew − xbirth )2 + (ynew − ybirth )2 + (znew − zbirth )2 gnew = state new.group # update the tally of incremental migration area 2 2 ) T[gpre ] += 61 (rnew − rpre
The array T in the last line of Algorithm 1 stores the raw summation of the tally results of incremental migration area in each group. Like common Monte Carlo tallies, once all neutron histories have been finished, the multi-group incremental migration area can be computed as
∆Mg2 = RR
T [g] Rr,g dt dv
(6)
But combined with Equation (3) and (4), it becomes obvious that
Dg = ∆Mg2 Σr,g = RR
T [g] T [g] Σr,g = RR Rr,g dt dv φg dt dv
(7)
As a result, multi-group diffusion coefficients no longer depend on the removal cross sections. The removal cross sections was left in previously to illustrate the equivalence of the new method. Another important notice about Algorithm 1 is that in the last line (the tally step), the group index of the tally is gpre but not gnew , since during the neutron’s last flight its energy is in the previous group (gpre ). Proceedings of the PHYSOR 2018, Cancun, Mexico 2516
Liu, et al, Group-wise Tally Scheme for CMM 3.2. Examples of Tallies As mentioned in Section 2.3, the new group-wise tally method of CMM is essentially equivalent to previous cumulative tally method, and the results of cumulative migration area and resulting diffusion coefficients are exactly the same. In this section, examples of two individual neutrons are provided to show the equivalence. 3.2.1. Example 1: a pure down-scatter case The first example neutron is assumed to undergo a series of pure down-scatter reactions and its traveling history is depicted on the left of in Figure 2. This figure displays the neutron’s trajectory in 2D for simplicity, which can be seen as projecting the real 3D flight trajectory to a 2D plane. In addition, to better illustrate a neutron’s spatial movement and energetic variation simultaneously, the trajectory (broken lines in the figure) is assigned with different colors in accordance with the neutron’s energy group during that segment of flight. Similar to commonly-used color temperature schemes, warmer colors are assigned to groups of higher energy, and vice versa. This example neutron born at spatial point O (source origin) undergoes a series of scattering reactions and finally gets absorbed at point C. The original energy of this neutron at emission belongs to group 1 in a given multi-group energy structure. As the neutron travels, its energy varies as scattering reactions occur. At point A, a down-scatter reaction slows down the neutron so that its energy after the reaction belongs to group 3. Similarly, at point B, the neutron’s energy is changed from group 3 to group 4. Finally, after a few more scattering reactions (but the energy group is no longer changed), the neutron is absorbed at point C. For this example neutron in Figure 2, the tallies of migration areas for cumulative group 1, 3 # »2 # »2 # »2 and 4 are r#»1 2 (OA ), r#»3 2 (OB ) and r#»4 2 (OC ), respectively. These tallied data will be used for computing the mean migration area of all cumulative groups (Mgc )2 . Then cumulative group diffusion coefficients Dgc can be computed using Equation 1. The cumulative-group tally and group-wise tally processes for this example neutron are listed side-by-side in Table 1. As shown in the table, the new group-wise tally based on incremental migration area eliminates the redundancy in the cumulative-group tally. And the results of all (Mgc )2 on the left can be obtained using the group-wise tallies on the right by Equation (5). Taking an example of group 3, it is obvious from the table that (M3c )2 = T1 N1 + T2 N2 + T3 N3 . (N3c = 1, N1 = 1, N2 = 0, N3 = 1.) 3.2.2. Example 2: a case with down-scatter and up-scatter reactions For the second example neutron as shown on the right of Figure 2, an up-scatter reaction takes place at point B, where the neutron’s energy group is changed from group 5 to group 3. The cumulative-group and group-wise tally results are listed side-by-side in Table 2. With the existence of up-scatter reactions, it becomes more complicated for cumulative tallies of migration area. Especially for the tally at point B, the tally of migration area for cumulative group Proceedings of the PHYSOR 2018, Cancun, Mexico 2517
Reactor Physics paving the way towards more efficient systems
Figure 2: 2D illustration of the spatial and energetic variation of two example neutrons undergoing a series of scatter reactions and finally absorbed. ( On the left: group 1 → group 3 → group 4; on the right: group 1 → group 5 → group 3. As in the legend, a 5-group energy structure is used for this example, with “warmer” colors representing higher energy groups. )
Table 1: The cumulative-group and group-wise tallies for the first example neutron, which undergoes a series of pure down-scatter reactions with the energy group transition history of (group 1) → (group 3) → (group 4). Group 1 2 3 4 5
(M1C )2 (M2C )2 (M3C )2 (M4C )2 (M5C )2
Cumulative-group Tally at A at B at C # »2 OA # »2 OA # »2 OB # »2 OC # »2 OC
T1 T2 T3 T4 T5
at A # »2 OA
Group-wise Tally at B at C
# »2 # »2 OB − OA
# » 2 # »2 OC − OB
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Table 2: The cumulative-group and group-wise tallies for the second example neutron, which undergoes a series of both down-scatter and up-scatter reactions with the energy group transition history of (group 1) → (group 5) → (group 3). Group 1 2 3 4 5
(M1C )2 (M2C )2 (M3C )2 (M4C )2 (M5C )2
Cumulative-group Tally at A at B at C # »2 OA # »2 OA # »2 # »2 # »2 OA −OB OC # »2 # »2 # »2 OA −OB OC # »2 OC
T1 T2 T3 T4 T5
at A # »2 OA
Group-wise Tally at B at C
# » 2 # »2 OC − OB # »2 # »2 OB − OA
# »2 3 and cumulative group 4 need to be subtracted by OB , accounting for the neutron’s return to cumulative group 3 and 4. On the other hand, the new group-wise tally method can be executed simply following the same principle in Algorithm 1, resulting simple and consistent tallies as the first example neutron. From Table 2, again taking the example of group 3 following Equation (5), one can also get that (M3c )2 = T1 N1 + T2 N2 + T3 N3 . (Now N3c = 1 − 1 + 1 = 1, N1 = 1, N2 = 0, N3 = 1. For N3c , the first count of 1 is caused by the down-scatter from group 1 to group 5; the second count of −1 is caused by the up-scatter “come-back” to cumulative group 3; and the third count of 1 is due to absorption at point C.) Similarly, for group 5 one can get (M5c )2 = T1 N1 + T2 N2 + T3 N3 + T4 N4 + T5 N5 . (N5c = 1, N1 = 1, N2 = 0, N3 = 1, N4 = 0, N5 = 1.) 4. TOWARDS HETEROGENEOUS TALLY With this new group-wise tally scheme using incremental migration area applied to CMM, the implementation of this method in Monte Carlo codes can be more general and efficient. But more importantly, it becomes achievable to separate the tallies of incremental migration area into multiple regions. This ability of separation enables the capability for computing heterogeneous multi-group diffusion coefficients and transport cross sections. Different from homogenized tallies, neutrons can travel from one tally region to another in heterogeneous tally problems. How to deal with the case when neutrons cross the interface of two adjacent tally regions is the key problem for further generalization of CMM. As the definition of migration area is referred to the neutron’s birth position, it poses challenge on how to calculate the incremental migration area when tallies are made in a region not containing the neutron’s birth position. Research about this problem is still on-going and the following treatment is just provided Proceedings of the PHYSOR 2018, Cancun, Mexico 2519
Reactor Physics paving the way towards more efficient systems
Figure 3: Model of the “sanity check” problem with an example of a neutron’s trajectory crossing the region interface at point X.
as a straight-forward approach which is under more testing. 4.1. The “Sanity Check” Problem Despite how to solve the interface crossing tally separation challenge, there is a simple “sanity check” problem to be tested first. The problem is essentially the same as an infinite-medium homogeneous-material problem but just divided into two regions. As shown in Figure 3, the geometry of this problem can be simply a box consisting of pure hydrogen atoms with reflective boundaries on all sides. It is divided into two symmetric parts by an “imaginary” plane in the middle. The left and right part of this model are exactly the same in material composition. For this problem, there is no doubt that the results of diffusion coefficients and transport cross sections tallied for the left and right semi-box should be exactly the same as the infinite-medium results. Since it has been demonstrated that CMM can generate accurate diffusion coefficients and transport cross sections for homogenized regions, the same results are expected for the two semi-boxes using heterogeneous tallies. 4.2. Straight-forward Interface Crossing Treatment and Results As shown in Figure 3, an example neutron cross the left and right region interface at point X during its flight from B to C. A straight-forward approach for this is to separate the tally of incremental # » 2 # »2 migration area right at the intersection point X by attributing (OX − OB ) to the left region and # »2 # » 2 (OC − OX ) to the right region. That is to say, in addition to the tally process triggered by each reaction, the neutrons’ states in Algorithm 1 will also be updated to make tallies in case of region interface crossing events. This method of interface crossing tally separation is straight-forward and intuitive, and as shown in Figure 4 it can get correct results for the “sanity check” problem. Figure 4 shows the 70Proceedings of the PHYSOR 2018, Cancun, Mexico 2520
Liu, et al, Group-wise Tally Scheme for CMM
1.2
combined left right
1
TCR
0.8
0.6
0.4
0.2
0 10 -10
10 -5 Energy (MeV)
10 0
Figure 4: Results of the 70-group transport corrections ratios (TCR = Σtr,g /Σt,g ) computed for the left and right part of the “sanity check” problem.
group transport corrections ratios (TCR = Σtr,g /Σt,g ) computed for the left and right part of the “sanity check” problem using the heterogeneous tally capability for incremental migration area. As expected, TCR of the left and right part match each other and they are also consistent with the infinite-medium results (annotated as “combined” in the figure). But there are also some problems with this straight-forward interface crossing treatment. One big issue is that using each neutron’s birth position for computing incremental migration area, the tally in one region could be dependent on the neutron’s history in another region, just as the example shown in Figure 3. For the flight segment of XC in the right part, the corresponding tally for # »2 # » 2 incremental migration area (OC − OX ) is dependent on the location of point O in the left part. More research focused on these problems is on-going. 5. CONCLUSIONS To generalize the application of Cumulative Migration Method (CMM) to heterogeneous problems, a purely group-wise tally method to compute multi-group diffusion coefficients and transport cross sections is proposed in this paper. By talling a new quantity, the incremental migration area ∆Mg2 , CMM can be implemented in a more general and efficient manner. The new group-wise form of tallies for incremental migration area is quite simple and straight-forward, compared with the previous cumulative tallies or vector-connected group-wise tallies, but maintains complete equivalence and can generate the same results. More importantly, the tally of incremental migration area can be separated into multiple regions, which enables CMM to be used for computing heterogeneous multi-group diffusion coefficients Proceedings of the PHYSOR 2018, Cancun, Mexico 2521
Reactor Physics paving the way towards more efficient systems and transport cross sections. Proper treatment of the tallies for region interface crossing neutrons is a key problem for further generalization of CMM. A straight-forward approach of separating the tally of incremental migration area right at the intersection point can generate correct results for the two-region “sanity check” problem, and more verification and testing problems will be done in following steps. ACKNOWLEDGEMENTS The research work presented in this paper was supported by the US Department of Energy Nuclear Energy University Program contract DE-NE0008578 and the China Scholarship Council. REFERENCES [1] G. Bell, G. Hansen, and H. Sandmeier. “Multitable treatments of anisotropic scattering in SN multigroup transport calculations.” Nuclear Science and Engineering, volume 28(3), pp. 376–383 (1967). [2] R. Macfarlane, D. W. Muir, R. Boicourt, A. C. Kahler III, and J. L. Conlin. “The NJOY Nuclear Data Processing System, Version 2016.” Technical report, Los Alamos National Laboratory (LANL) (2017). [3] J. M. Pounders. Stochastically generated multigroup diffusion coefficients. Master’s thesis, Georgia Institute of Technology (2006). [4] A. Yamamoto, Y. Kitamura, and Y. Yamane. “Simplified treatments of anisotropic scattering in LWR core calculations.” Journal of nuclear science and technology, volume 45(3), pp. 217–229 (2008). [5] E. Fridman and J. Lepp¨anen. “On the use of the Serpent Monte Carlo code for few-group cross section generation.” Annals of Nuclear Energy, volume 38(6), pp. 1399–1405 (2011). [6] S. Choi, K. Smith, H. C. Lee, and D. Lee. “Impact of inflow transport approximation on light water reactor analysis.” Journal of Computational Physics, volume 299, pp. 352–373 (2015). [7] S. Choi, K. S. Smith, H. Kim, T. Tak, and D. Lee. “On the diffusion coefficient calculation in two-step light water reactor core analysis.” Journal of Nuclear Science and Technology, volume 54(6), pp. 705–715 (2017). [8] Z. Liu, K. Smith, and B. Forget. “A Cumulative Migration Method for Computing Rigorous Transport Cross Sections and Diffusion Coefficients for LWR Lattices with Monte Carlo.” pp. 2915–2930. PHYSOR (2016). [9] Z. Liu, K. Smith, B. Forget, and J. Ortensi. “Cumulative migration method for computing rigorous diffusion coefficients and transport cross sections from Monte Carlo.” Annals of Nuclear Energy, volume 112, pp. 507–516 (2018). [10] K. S. Smith. “Nodal diffusion methods and lattice physics data in LWR analyses: Understanding numerous subtle details.” Progress in Nuclear Energy (2017). [11] N. Horelik, B. Herman, B. Forget, and K. Smith. “Benchmark for evaluation and validation of reactor simulations (BEAVRS).” Technical report, American Nuclear Society, 555 North Kensington Avenue, La Grange Park, IL 60526 (United States) (2013). [12] J. R. Lamarsh. Introduction to Nuclear Reactor Theory. Addison-Wesley (1966). Proceedings of the PHYSOR 2018, Cancun, Mexico 2522
Liu, et al, Group-wise Tally Scheme for CMM [13] Z. Liu, K. Smith, and B. Forget. “Progress of Cumulative Migration Method for Computing Diffusion Coefficients with OpenMC.” M&C2017 - International Conference on Mathematics & Computational Methods Applied to Nuclear Science & Engineering (2017).
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