The Adaptive Collision Source Method for Discrete Ordinates Radiation Transport William J. Walters1,∗, Alireza Haghighat Virginia Tech Nuclear Science and Engineering Lab, Nuclear Engineering Program, Mechanical Engineering Dept., Arlington, VA, USA
Abstract A novel collision source method has been developed to solve the Linear Boltzmann Equation (LBE) more efficiently by adaptation of the angular quadrature order. The angular adaptation method is unique in that the flux from each scattering source iteration is obtained, with potentially a different quadrature order used for each. Traditionally, the flux from every iteration is combined, with the same quadrature applied to the combined flux. Since the scattering process tends to distribute the radiation more evenly over angles (i.e., make it more isotropic), the quadrature requirements generally decrease with each iteration. This method allows for an optimal use of processing power, by using a high order quadrature for the first iterations that need it, before shifting to lower order quadratures for the remaining iterations. This is essentially an extension of the first collision source method, and is referred to as the adaptive collision source (ACS) method. The ACS methodology has been implemented in the 3-D, parallel, multigroup discrete ordinates code TITAN. This code was tested on a several simple and complex fixed-source problems. The ACS implementation in TITAN has shown a reduction in computation time by a factor of 1.5-4 on the fixed-source test problems, for the same desired level of accuracy, as compared to the standard TITAN code. Keywords: Discrete ordinates, angular quadrature, radiation transport, collision source, TITAN, adaptive quadrature
1. Introduction The discrete ordinates (SN ) method is one of the standard methods to discretize the angular variable in the Linear Boltzman Equation (LBE) that governs radiation transport, and is used in many production radiation transport codes[1, 2, 3, 4]. In the discrete ordinates method, the LBE is solved over a set of angular directions Ωi (also called ordinates), with corresponding weights wi [5, 6]. This combination of (Ωi , wi ) is called an angular quadrature set. A quadrature set allows the angular integrals to be converted into sums over the angles of the set. Selection of an appropriate angular quadrature set is one of the difficulties with the discrete ordinates method[7, 8]. Depending on the problem, the flux may vary greatly in direction (that is, very anisotropic). An anisotropic function would require many quadrature points in order to properly resolve the integrals. Using a quadrature order that is too low will result in large errors from so-called ”ray effects”, while using an order too high greatly increases computation time[9]. There have been several methods developed to optimize the efficiency of the angular quadrature in discrete ordinates calculations. The first is simply in the selection of better general quadrature sets that can more ∗ Corresponding
author Email address:
[email protected] (William J. Walters) 1 Present address: Department of Mechanical and Nuclear Engineering, The Pennsylvania State University, University Park, PA, USA Preprint submitted to Annals of Nuclear Energy
accurately integrate the angular flux for the same total number of directions[10, 11, 12]. Another method is local angular refinement[13, 14, 15, 16], which involves adding quadrature points in angular directions of high anisotropy, while leaving a coarse distribution of points in the more isotropic directions. This has been implemented in the static, user defined case in production codes[1, 2] and some work has been done to perform this refinement adaptively[17, 18, 19, 10]. There is also the so-called firstcollision source method[20, 21], which is more closely related to the subject of this paper. This involves calculating the un-collided flux using a high-order transport method and then using this to generate a first-collision source, which is used to start a low order transport calculation to solve for the collided flux and thus save on computation time. This method has been implemented with success into several transport codes[4, 7, 22, 23]. The new adaptive collision source (ACS) method described in this paper builds off the first-collision source method by separating not just the uncollided flux, but also the once-collided flux, twice-collided flux, etc. At each transport iteration, only the i’th collided flux is solved for, and the possibility of using different angular quadrature orders for each is allowed. An intelligent scheme is also developed with which to choose, on-the-fly, an appropriate angular quadrature for each iteration. This can achieve the good speedups of the first-collision source method, but is more robust to a wide variety of problems for which the first collision source is not as effective. March 15, 2017
This paper is organized as follows: first, some background theory is given. Next the first-collision source method is discussed, followed by its extension to the new adaptive collision source (ACS) method[24, 25, 26]. For the ACS method, the general theory will be discussed followed by the the implementation of the ACS algorithm into the TITAN transport code. Finally, the results of the ACS method will be compared to the standard TITAN source iteration on several test problems.
2.3. Source iteration The LBE can be written in operator form as: HΨ = SΨ + Q0
(4)
Where, the streaming-collision operator H is defined as in Eq. 5. ˆ · ∇ + σ(~r, E) H=Ω
(5)
The scattering operator S is defined in Eq. 6.
2. Theory
Z∞ S=
2.1. The linear Boltzmann Equation
0
The steady-state LBE[27] for a fixed source problem describes the balance of the angular flux Ψ(~r, E, Ω) in a phase space (d3~r dE dΩ).
dE 0
Z
dΩσs (~r, E 0 → E, Ω0 · Ω)
(6)
4π
In the standard source iteration method, an initial flux Ψ(0) (usually 0) is assumed, then the flux is calculated assuming a constant scattering source SΨ, then the source is updated and the flux recalculated.
ˆ · ∇Ψ(~r, E, Ω) + σ(~r, E)Ψ(~r, E, Ω) = Ω Q0 (~r, E, Ω) Z∞ Z ˆ 0 · Ω)Ψ(~ ˆ + dE 0 dΩ0 σs (~r, E 0 → E, Ω r, E, Ω) 0
HΨ(i) = SΨ(i−1) + Q0
(1)
Where, Ψ(i) is the flux after iteration i. This operation, the solving of Eq 7, is commonly referred to as a transport sweep. After a sufficient iteration, both sides of the equation will converge to within some given tolerance.
4π
An important quantity derived from the angular flux is the scalar flux φ, defined in Eq. 2.
2.4. First Collision Source Method
Z φ(~r, E) =
dΩΨ(~r, E, Ω)
(7)
In the first-collision source (FCS) method, the flux is split into the uncollided (Ψu ) and collided (Ψc ) fluxes, as in Eq. 8. Ψ = Ψc + Ψu (8)
(2)
4π
2.2. Angular quadrature If we insert the above Eq. 8 into Eq. 4, we can arrive at a set of coupled transport equations in Eq. 9 and Eq. 10
For the discrete ordinates method, consider that the LBE holds over a set of discrete angles (or ordinates). This allows an integral over angle to be replaced by a weighted sum over the discrete ordinates: Z dΩf (Ω) = 4π
M X
wm f (Ωm )
(3)
HΨu = Q0
(9)
HΨc = Qc + SΨc
(10)
Where, the first-collision source Qc is defined in Eq. 11
m
Qc = SΨu Where, m is the ordinate index, M is the total number of ordinates, Ωm is the angle of the m’th ordinate and wm is the weight of the m’th ordinate. The set of weights and angles (called a quadrature set) must be chosen carefully to ensure conservation of various integral properties such as current and scalar flux moments. In this work, LegendreChebyschev quadrature sets are used[16, 15], denoted by SN where N is called the quadrature order. The total number of ordinates M in the set in a 3-D geometry given by M = N (N + 2). For example, the S20 quadrature set has M = 20(20 + 2) = 440 ordinates, while S6 has only 48. This can mean a huge difference in computation time between these sets.
(11)
Eq. 9 requires no iterations on the scattering source, and could be solved relatively quickly using a high-order method (e.g., high order SN or ray-tracing). Eq. 10 looks exactly like the standard LBE, except the independent source Q0 is replaced with the first-collision source derived from the un-collided flux. This collided flux can then be solved using a lower quadrature order SN in a standard source iteration method relatively quickly. Again, the motivation behind this is that the un-collided flux has a much higher angular variation than the collided flux, so generally requires a more robust treatment of the angular variable than does the collided flux. 2
3. Adaptive Collision Source (ACS) Methodology
The equations solved at each iteration for the Source Iteration (SI) procedure are shown in Table 1. At each iteration (or equivalently, number of scatters), the streamingcollision operator H is acting on the entire flux, and always with the same quadrature level. The equations solved for in the First Collision Source (FCS) method are shown in Table 2. Here, the uncollided flux (i.e., Ψ0 ) is solved for using one quadrature level, while the remaining collided fluxes (Ψi ) are combined and solved using a second quadrature level (of lower order). Since Ψ0 has both the highest magnitude and the most anisotropy, the high level quadrature can be used for it, while saving time using a lower quadrature for the rest. Finally, the new ACS method is shown in Table 3. Every scattering order is solved for separately, potentially each with a different quadrature order (although for this example it is limited to three sets). In many problems, the first-collision source (SΨ0 ) can itself be localized and/or anisotropic, and thus still have high quadrature requirements. The ACS method gets around this problem by separating all collided source terms so that the lowest required quadrature can be used at every step. If this would introduce a high amount of error, then ACS will still operate at a high quadrature until a low amount of error is estimated.
The new ACS methodology builds on the first-collision source method to a type of n’th-collision source method using discrete ordinates. Instead of splitting the flux up into uncollided flux and collided flux, we expand the total flux (denoted as Ψt,n ) into the fluxes of different collision source order (i.e., 0 to n). Ψt,n = Ψ0 + Ψ1 + Ψ2 + ...Ψn
(12)
We truncate the series at the n’th collision. The uncollided flux and collided flux can be defined in terms of the i’th collided fluxes as follows: Ψu = Ψ0
(13)
Ψc = Ψ1 + Ψ2 + ...Ψn
(14)
Now we arrive at a similar formulation as the first-collision source method: HΨi = QΨi (15) For i = 0, Q0 is the independent source for the problem. For i > 0, the i’th-collision scattering source Qi is defined as: Qi = SΨi−1 (16)
3.2. Angular Quadrature Order Adaptation Criteria In order to change quadrature for each scattering iteration, we need some criteria to decide when and how to do so. If we want to estimate the error produced by changing the quadrature from SN to SN 0 at a certain iteration, there are two sources of error that we must consider. The first is the error in the flux of each iteration due to the quadrature change itself, which we will denote as quadrature error or quad . The second is the iterative error, denoted as iter , which represents the cumulative amount of flux that has yet to be calculated through the scattering iterations. We can define the relative angular quadrature error from changing quadrature from order N to N 0 in Eq 17.
The idea for the ACS method is to use the discrete ordinates method to solve for each scattering collision order Ψi separately, then sum them to get the total flux Ψt,n in Eq. 12. For the calculation of each Ψi , i.e., transport sweep, a different angular quadrature set can be used. Since we have separated each i’th collision flux from each other, Ψi depends only on the scattering source Qi , which is dependent only on the flux moments φkl,i−1 , and not explicitly on the angular flux Ψi−1 . This means that the transport sweep for Ψi can be performed using a different quadrature set, since Qi as computed is a continuous function in angle (since it is based on spherical harmonics, unlike Ψi−1 ) and can easily be evaluated at the new quadrature angles. The flux moments φkl,i−1 are all that need to be stored between iterations (at least for vacuum boundary conditions). Care must be taken to ensure that the lowest quadrature order can still adequately integrate the highest order of spherical harmonic expansion. It is important to note that Eq. 15 is similar to the standard LBE except that the independent source is only included when i = 0. As such, this method can be implemented into an existing transport codes by including the independent source in only the first iteration and keeping track of the total flux sum.
0
quad =
N φN i − φi N φi
(17)
We define the relative iterative error at iteration i (as compared to an infinite number of iterations) in Eq. 18. iter =
φt,i − φt,∞ φt,∞
(18)
The total relative error is defined in Eq. 19. 0
0
N N N N N (φN 0 + ...φi + φi+1 + ...φ∞ ) − (φ0 + ...φ∞ ) N φN 0 + ...φ∞ (19) This can also be written as in Eq. 20.
total =
3.1. Iteration Procedure Recall the formulation for the general source iteration scheme: HΨ(i) = SΨ(i−1) + Q0
for i = 0 to n
0
total =
(7) 3
0
N N N (φN t,i − φt,∞ ) − (φt,i − φt,∞ ) φN t,∞
(20)
Table 1: Iteration Progression for the Standard Source Iteration
# Scatters
Equation Solved
Quadrature Level
0 1 2 ... n
HΨ0 = Q0 H(Ψ0 + Ψ1 ) = Q0 + SΨ0 H(Ψ0 + Ψ1 + Ψ2 ) = Q0 + S(Ψ0 + Ψ1 ) ... H(Ψ0 + Ψ1 + ...Ψn ) = Q0 + S(Ψ0 + ...Ψn−1 )
1 1 1 ... 1
Table 2: Iteration Progression for the First Collision Source Method
# Scatters
Equation Solved
Quadrature Level
0 1 2 ... n
HΨ0 = Q0 HΨ1 = SΨ0 H(Ψ1 + Ψ2 ) = S(Ψ0 + Ψ1 ) ... H(Ψ1 + ...Ψn ) = S(Ψ0 + ...Ψn−1 )
1 2 2 ... 2
Table 3: Iteration Progression for the Adaptive Collision Source Method
# Scatters
Equation Solved
Quadrature Level
0 1 2 ... n
HΨ0 = Q0 HΨ1 = SΨ0 HΨ2 = SΨ1 ... HΨn = SΨn−1
1 1 or 2 1 or 2 or 3 ... 1 or 2 or 3
calculated at the new quadrature (iter ). Once we have this error estimate, then we have the criteria for changing quadrature (i.e., lower the quadrature in iteration (i + 1) if (total < , for some user-set tolerance ). Now we will discuss how we calculate the error estimates quad and iter . 3.3. Angular Quadrature Error
Since radiation from a source necessarily spreads out in space during free-flight, and then becomes spread out in angle during collisions, we can assume that the angular anisotropy (and hence angular quadrature requirement) is a decreasing function of iterations. Using this principle, we Or, can develop a simple technique to decide on what quadrature should be used. If we define the scalar flux (i.e., 0’th N N N (φN t,i − φt,∞ ) − (1 + quad,i )φt,i − (1 + quad,∞ )φt,∞ order flux moment) for Ψi using quadrature SN : total = φN t,∞ M (21) X φN w m ΨN (24) Since the source is spreading out and becoming more i = i,m m=1 isotropic with each iteration, we can assume that the relative quadrature error only goes down with the number of To estimate the error of using a new quadrature set, iterations. the current iteration angular flux ΨN i,m is projected onto N0 the new quadrature set to yield Ψi,m0 . The projection quad,i+1 ≤ quad,i (22) technique used here is the same as is used to project beThis allows Eq. 21 to be simplified to: tween regions with different quadrature in TITAN. In this technique the flux in each ordinate of the new quadrature N equal a weighted sum of the three nearest neighbors of the quad (φN t,i − φt,∞ ) total ≤ previous quadrature, as in Eq. 25[28, 29, 2]. N φ t,∞
total ≤ quad · iter
(23) 1
0
The total relative error is equal to the quadrature error per iteration (quad ) times the amount of flux yet to be
ΨN i,m0 = 4
1 θ12
+
1 θ22
+
1 θ32
ΨN ΨN ΨN i,n1 i,n i,n + 22 + 23 2 θ1 θ2 θ3
! (25)
The scalar flux using this new angular flux is then calculated: M0 X 0 N0 φi = wm0 ΨN (26) i,m0
Ψ0 (i.e., the uncollided flux) would be calculated using the first quadrature set (e.g., S20 in this case). The error estimate described above would then be used to estimate the effect of lowering the quadrature to the next level (S14 ). If this is below the user-prescribed tolerance , then Ψ1 would be calculated using this new lower quadrature set. This angular quadrature adaptation would continue until the lowest quadrature order is reached (S6 in this example), which would then be used for the remainder of the calculation. It makes intuitive sense to set the adaptive tolerance to be somewhere on the order of the iterative tolerance for a given problem. We will examine the effect of changing this parameter later in the paper.
m0 =1 0
Now we have φN and φN i i to use in Eq. 17. 3.4. Iterative Error A source iteration of the LBE should exhibit linear monotonic convergence, since each iteration simply adds the next-collided flux, which is always less than the last (unless the system is super-critical). After a sufficient number of iterations, the flux in each successive iteration should be decaying by a constant factor Λ (i.e., the spectral radius)[30] as in Eq. 27. φi+1 = Λφi
3.6. ACS Implementation
(27)
In order to implement this algorithm, the TITAN 3D parallel hybrid particle transport code[29, 2, 28] was adapted. TITAN numerically solves the time-independent LBE using the discrete ordinates method, or the characteristic method, or a combination thereof. Only the discrete ordinates portion of the code is used here. TITAN has the built-in ability to use different quadrature orders in different regions and perform quadrature projection, so was an ideal choice for this work. The formulation for the ACS method (Eq. 15 is similar to the standard Source Iteration, Eq. 1. Because of this, the core structure of the transport code, including transport sweeping, calculation of scattering sources, etc, can remain unchanged. The changes required to the code can be summarized as follows:
That Λ is the spectral radius can be more easily seen if we rewrite Eq. 27 in terms of the cumulative flux φt,i . If these values are summed to infinity, an estimate of the remaining flux can be calculated (i.e., the iterative error). φt,∞ − φt,i =
∞ X
Λj φi
(28)
j=1
This sum can be expressed as follows. φt,∞ − φt,i =
Λi φi 1 − Λi
(29)
Where, Λi , the estimate of the spectral radius at iteration i, is defined as in Eq. 30. Λi =
φi φi−1
• Angular fluxes Ψ and flux moments φ become the current iteration fluxes Ψi and φi
(30) • Added new variables including the cumulative fluxes Ψt,i and φt,i
So we can now calculate the relative iterative error iter by combining Eq 30 and Eq. 29 and inserting it into Eq. 18. With the additional assumption that the denominator of Eq. 18 (φt,∞ ) is approximately equal to φt,i , we arrive at Eq. 31 for the relative iterative error. φi /φi−1 φi iter = (31) 1 − φi /φi−1 φt,i
• The flux iterative difference is redefined as f lux = φi /φt,i • Flux is always initialized to 0 • Independent source Qg , upscatter, and down-scatter Sg0 →g are only used in the first iteration (i.e., for Ψ0 ) • Added error calculations for the adaptation criteria
In this equation, the second factor is recognized as the standard flux change in one iteration, while the first factor accounts for the rest of the iterations. Of course, these equations are only valid if φi /φi−1 < 1. If this condition is not met, then the solution has not converged sufficiently, and the error is set to 100%.
• If the estimated error is less than the tolerance, the quadrature order is lowered • When a new energy group is started, the quadrature is reset to the highest level
3.5. Angular Quadrature Order Adaptation
3.6.1. Parallelization Parallelization in standard TITAN is done by angular decomposition of individual ordinates. This means that each processor is assigned a set of ordinates. With each iteration, transport sweeps are performed for each ordinate
To perform the angular quadrature adaptation, we describe a list of quadrature orders for the problem (e.g., [S20 , S14 , S10 , S6 ]). This set is arbitrary, user-defined, and the only restriction is that the order be decreasing. 5
subset and the flux moments are calculated for the subset. At the end of each iteration, the flux moments from all subsets are combined and communicated between all processors using MPI[31] to get the total flux moments. With vacuum boundaries, the angular decomposition is done the same as TITAN, requiring communication of the flux moments (φkl ). When the quadrature order is changed, the new set of ordinates is re-allocated among the processors. This requires little extra computation and no parallel communication. However, with every iteration, the scalar flux error estimate (quad ) must also be communicated among processors. This increases communication time by a relatively small amount. Results for the parallel performance of the ACS method for relatively small problems have been shown to be excellent[25]. On a test problem with 16 processors, parallel fractions of 98% and 97% were observed for vacuum and reflective boundaries, respectively.
(a) 3-D Geometry
4. Results The newly developed ACS algorithm was compared to the standard TITAN source iteration on several test problems, two of which are presented here. First, a very simple, homogeneous, single group box-in-a-box problem, and second, complex, heterogeneous multi-group problem based on the VENUS-2 reactor dosimetry benchmark. 4.1. Box Problem In order to perform detailed tests on the algorithm, a simple model is used, consisting of a cube of scattering material, with a small cube of source material in the middle. This simple model will allow the testing of various parameters without added difficulties of complex geometry. A 3-D depiction of the problem is shown in Figure 1, along with the mesh distribution (constant 0.063 cm size). This mesh size was chosen to keep the spatial discretization error small compared to the angular discretization error. The large box (blue) is of size 11cm· 11cm · 11cm, while the source box (red) is of size 1cm· 1cm · 1cm. For simplicity, only one energy group was used, with total cross sections always equal to 1 cm−1 . Three materials were examined, with scattering ratios (c = σs /σt ) of 0.1, 0.5 and 0.9. Table 4 lists the cross sections used for different materials in this problem. In order to test the ACS method, there are three parameters to examine. These are the quadrature order (including the first and second quadrature orders for ACS), and adaptive tolerance (). The quadrature order is varied between 4 and 54 (corresponding to 24 to 3024 ordinates), and the adaptive tolerance is varied from 10−4 to 1. The quadrature orders were selected so as to approximately double the number of ordinates in each step.
(b) Mesh distribution Figure 1: Box-in-a-box problem geometry
4.2. Reference TITAN Calculations TITAN calculations were performed for all quadrature orders. Plots of the scalar flux over all quadratures are shown in Figure 2 for material 2, with scattering ratio of 0.5. The flux magnitude drops by several orders of magnitude outside the source region. The angular discretization errors, seen as “ray effects” are easily visible at the lower quadrature levels. This is more pronounced in streamingdominated material (i.e., material 1, c = 0.1), and much less pronounced in highly scattering material (material 3, c = 0.9). These errors get smaller as the quadrature order increases. To compare between angular discretizations, the scalar flux is used as the metric. As a metric of error, the root mean square relative error (RMSE) of the scalar flux is calculated, as compared to the reference case (standard source iteration, 0.063cm mesh size, S54 quadrature). The RMSE is defined in Eq. 32. 6
Table 4: Cross Sections for Box Problem
Material ID
Total σt (cm−1 )
Absorption σa (cm−1 )
Scatter σs0 (cm−1 )
P1 Scattering Moment σs1 (cm−1 )
1 2 3
1.0 1.0 1.0
0.9 0.5 0.1
0.1 0.5 0.9
0.1 0.5 0.9
S4
S6
S10
S14
S20
S30
S40
S54 Flux 1e−01 1e−02 1e−03 1e−04 1e−05
Figure 2: TITAN scalar flux plots for the box problem at different quadrature levels, material 2 (c = 0.5)
v u 2 J u1 X φj − φj,ref t RM SE = J j=1 φj,ref
increases and the computation time decreases approximately in proportion to the number of ordinates. With the quadrature at S40 , an error of 0.29% is acheived in 1277s, compared to S6 with 30% error in 44s. Both the error and computation time show changes of several orders of magnitude, and show the importance of even relatively small changes in quadrature order.
(32)
Where, j is the spatial index, J is the total number of spatial elements, φj is the scalar flux of the case in question, and φj,ref is the scalar flux of the reference case. The RMSE and computation time the material 2 standard TITAN calculations are shown in Table 5. As expected, when the quadrature order is lowered, the error 7
Table 5: Reference TITAN Results for Box Problem with Material 2 (c = 0.5)
Quadrature Order
# of ordinates
RMSE (%)
Time (s)
54 40 30 20 14 10 6
3024 1680 960 440 224 120 48
0.29% 0.36% 1.34% 5.58% 13.25% 30.40%
2299.5 1276.7 732.7 354.6 183.9 99.9 43.6
4.3. Analysis of TITAN with the ACS Algorithm For any problem, there are trade-offs between accuracy and speed. When comparing methods, it will be important to compare computation time for a given level of error (or vice versa). The best method will be the one that minimizes computation time for a given level of error (or vice versa). This section will examine the computational efficiency of SI and ACS for several parameters, including the adaptive tolerance, material scattering ratio, and quadrature orders. In order to compare the methods, the computation time will be plotted against the RMSE of that calculation for all parameters. For a result to be “good”, the ACS results should appear to be below the standard SI results (that is, have a lower computation time for a given error). In the first test, the primary ACS quadrature is set at S30 , and the second quadrature is varied from S20 to S4 , for various adaptive tolerances. This is compared to SI with quadrature fixed at between S30 and S4 . The results comparing computation time and MRE are shown in Figures 3 to 5. In these figures, each color represents a different adaptive tolerance. Each point within these colors represents a different secondary quadrature (the first quadrature is fixed at S30 ). For the low scatter case (c = 0.1, Figure 3), most of the points are clustered around the error level of the standard S30 case. This is because problem is dominated by the uncollided flux, which is being calculated at S30 for all the ACS cases. Only when the adaptive tolerance is 1 (which makes it essentially the first collision source method), with a low second order quadrature does the error get much higher. For all ACS parameters, though, the computation time is lower for a given level of error (or vice-versa). In the medium scatter case (c = 0.5, Figure 4), the points start to impact of the ACS parameters shows a little more variation in the result. In the most extreme case of = 1 and a quadrature of S30 /S4 , the results are worse than the standard source iteration. However, all “reasonable” parameter combinations yield a good speedup. Finally, the high scatter case (c = 0.9, Figure 5), the importance of the later iterations become more important. It is seen that the extreme quadrature combinations (e.g., S30 /S4 ) are not very efficient, so it is recommended that
a quadrature order approximately half of the primary be chosen. Again, however, all reasonable parameter choices give a good speedup. Effect of Second Quadrature on Efficiency 1000
Material 1 (c = 0.1) ●
S30
500
●
200
S20
S14
100
●
●
S10
50
CPU Time (s)
●
Standard ACS ε = 100 ACS ε = 10−1 ACS ε = 10−2 ACS ε = 10−3 ACS ε = 10−4
S6
ACS Quadratures S30 S20
S30 S14
S30 S10
●
S30 S6
S30 S4
S4
●
0.02
0.05
0.10
0.20
0.50
1.00
2.00
RMS Error
Figure 3: Effect of second quadrature on ACS efficiency for box material 1 (c=0.1)
For the next test, the first/second quadratures are set to fixed, reasonable levels (the second quadrature order is approximately half of the first, as given in Table 6). This gives a ratio of ordinates in the high/low quadrature of about 4, which was shown to give good results in the previous section. The adaptive tolerance is then varied for each of these combinations. These results are shown in Figures 6 to 8. This time the color in the plot indicates the quadrature orders, while the points within each color indicate the adaptive tolerance. It is noted that for each case, the lower bound of the error on the ACS method is dictated by the highest quadrature. The low tolerance ACS error lines up very well with the standard source iteration error at the same quadrature. For the low scatter case (c = 0.1, Figure 3), it is again noticed that the tolerance does not have a large effect on the error. It does, however, slightly reduce the com8
2000
Effect of Second Quadrature on Efficiency Material 2 (c = 0.5) ●
Table 6: Primary and Secondary Quadratures for the ACS Method in the Box Problem
S30 Standard ACS ε = 100 ACS ε = 10−1 ACS ε = 10−2 ACS ε = 10−3 ACS ε = 10−4
1000
●
S20
●
Quad. Order (# of ordinates) Primary
S14
200
●
30 20 14 10 8 6
S10
100
CPU Time (s)
500
●
Secondary
(960) (440) (224) (120) (80) (48)
14 10 6 4 4 4
(224) (120) (48) (24) (24) (24)
S6
●
ACS Quadratures S30 S20
S30 S14
S30 S10
S30 S6
S30 S4
50
S4
●
0.010
0.020
0.050
0.100
0.200
tolerance to use.
0.500 1000
0.005
RMS Error
●
Effect of Tolerance on Efficiency Material 1 (c = 0.1)
S30 ●
500
Figure 4: Effect of second quadrature on ACS efficiency for box material 2 (c=0.5)
S20
1000
●
S14
50
2000
●
Standard ACS ε = 100 ACS ε = 10−1 ACS ε = 10−2 ACS ε = 10−3 ACS ε = 10−4
200
S30 ●
●
●
●
S10
S6
ACS Tolerances (ε)
10−4
S10
S14
10−3
10−2
10−1
●
100
S4
●
500
CPU Time (s)
S20
100
●
●
CPU Time (s)
5000
Effect of Second Quadrature on Efficiency Material 3 (c = 0.9)
Standard S30 − S4 ACS S30 S14 ACS S20 S10 ACS S14 S6 ACS S10 S4
0.02
0.05
0.10
S6
●
200
ACS Quadratures S30 S20
S30 S14
S30 S10
S30 S6
2e−03
1.00
2.00
Figure 6: Effect of adaptive tolerance on ACS efficiency for box material 1 (c=0.1)
S4
1e−03
0.50
S30 S4
●
5e−04
0.20 RMS Error
5e−03
1e−02
2e−02
5e−02
1e−01
From all these results, ACS is more efficient, but difficult how to tell by how much, since the calculations have different errors as well as computation times. In order to do a fair comparison, a new metric is introduced, which is termed the equal-error speedup (EES), and is defined for some computation a in Eq. 33.
RMS Error
Figure 5: Effect of second quadrature on ACS efficiency for box material 3 (c=0.9)
putation time by spending fewer iterations at the higher quadrature order. In the medium scatter case (c = 0.5, Figure 4), the failings of the first collision source (i.e., = 1) become clear. These points all have significantly higher error for little gain in computation time compared to the lower tolerance levels. Finally, the high scatter case (c = 0.9, Figure 5), the cases get quite spread out, with all ACS cases being more efficient than the standard code. The tolerance has a large effect on the result, though it is unclear what is the optimal
EESa =
TSI (RM SEa ) Ta
(33)
In this equation, Ta and RM SEa are the computation time and RMSE for the calculation a in question. TSI (x) is the computation time for an SI calculation interpolated to the error value of x. It can be visualized as the lines connecting the SI points in the Figures 3 to 8. By definition, all SI calculations will have EES = 1, and ACS calculations will have EES > 1 if they are more efficient, and EES < 1 if they are less efficient. The higher the 9
2000
Effect of Tolerance on Efficiency Material 2 (c = 0.5) ●
1000
●
S20
500
●
●
Standard S30 − S4 ACS S30 S14 ACS S20 S10 ACS S14 S6 ACS S10 S4
S14
●
200
CPU Time (s)
varies from case to case, but should certainly be less than 1. If high accuracy is required in highly scattering problems, then a low tolerance should be used. Another thing to note is that for the S6 cases, there is not much speedup possible since the only option below S6 is S4 , which is not very much faster. Regardless of parameters chosen, ACS can get an answer with the same level of error in a much shorter amount of time.
S30
S10
●
ACS Tolerances (ε)
10−4
10−3
10−2
10−1
Equal−Error Speedup
100
3.0 S6
100
2.5
50
S4
●
0.005
0.010
Tol ● 1e−04 0.001 0.01 0.1 1
0.020
0.050
0.100
0.200
0.500
Quadrature ● 6 ● 10 ● 14 ● 20 ● 30
2.0
RMS Error
Figure 7: Effect of adaptive tolerance on ACS efficiency for box material 2 (c=0.5)
●
1.5
●
● ●
5000
Effect of Tolerance on Efficiency Material 3 (c = 0.9) ●
0.1 ●
2000
●
S20
1000
●
1.0
RMS Error
Standard S30 − S4 ACS S30 S14 ACS S20 S10 ACS S14 S6 ACS S10 S4
Figure 9: Equal-Error Speedup for all ACS parameters on box material 1 (c=0.1)
S14
●
S10
500
CPU Time (s)
●
1.0
S30
3.0
200
ACS Tolerances (ε)
10−4
10−3
10−2
10−1
Equal−Error Speedup
S6
●
100
2.5
S4
●
5e−04
1e−03
2e−03
5e−03
1e−02
2e−02
5e−02
Tol ● 1e−04 0.001 0.01 0.1 1
1e−01
2.0
RMS Error
Figure 8: Effect of adaptive tolerance on ACS efficiency for box material 3 (c=0.9)
Quadrature ● 6 ● 10 ● 14 ● 20 ● 30
● ●
● ●
1.5
●
0.01
EES of the method, the better. For example, an EES of 2 would indicate that for the same error, ACS performed 2 times faster than the standard method. The calculated EES values for the fixed-quadratureratio cases (i.e., all the cases from Figures 6 to 8) are shown in Figures 9 to 11 for materials 1 to 3, respectively. For these figures, the shape of the point represents adaptive tolerance value, while the color indicates the primary quadrature order. From these plots, it is fairly clear that almost all the ACS calculations show an equalerror speedup of between 1.5 and 4. The optimal tolerance
0.10
RMS Error
Figure 10: Equal-Error Speedup for all ACS parameters on box material 2 (c=0.5)
4.4. VENUS-2 Dosimetry Benchmark In the previous section, the ACS method was tested on a very simple box problem. Now the intention is to show 10
Equal−Error Speedup
for spatial differencing. For angular quadrature, LegendreChebyschev (Pn-Tn) quadrature sets of order 4 to 16 (corresponding to 24 - 288 discrete ordinates) were used. The flux convergence tolerance was set at 5×10−4 for all cases. Reference 26-group flux results were obtained using the standard TITAN code with a S16 quadrature set. Fluxes for energy groups 7 (4.97-6.03 MeV) and 18 (1.0 - 1.35 MeV) are shown in Figures 14 and 15, respectively. Other energy groups look similar, and plots of these are omitted for brevity. As expected, the flux is highest in the core, dips in the center and in the Pyrex pins, and drops quickly outside of the core. Using these fluxes, coupled with the dosimeter cross sections, reaction rates were obtained for every dosimeter location. Results for the 27 Al(n, α) reaction are shown in Figure 16 for all locations. These other dosimeters show similar reaction rates, but are sensitive to slightly different energy groups. Calculations were also performed using the standard TITAN code for lower quadrature orders (S4-S10). Relative errors compared to the S16 case for S4 and S10 are given in Figure 17 for 27 Al(n, α). As expected, the error decreases with quadrature order, and is highest in areas far from the source and dominated by streaming and high angular dependency. To compare between cases, we consider the RMS relative error of all dosimeter locations and types. The maximum error is also considered. These errors, along with computation times for the TITAN calculations are given in Table 7. As expected, the errors consistently decrease while the computation time increases with the quadrature order. The “acceptable” quadrature level would depend on the required accuracy of the computation. These calculations were repeated using the ACS algorithm using several quadrature combinations from S10 to S4 and and tolerances from 10−3 to 10−1 . Results from the ACS calculations including computation times, RMS errors, and equal-error speedups are given in Table 8. With increasing , more iterations are performed at the lower quadrature, and consequently the computation time decreases while the error increases. With a reasonable level of the adaptive tolerance, the error goes up very little, while the computation time decreases by a significant amount. An equal-error speedup is obtained for all ACS cases, up to a factor of 1.6. All cases offered a more efficient solution than the alternative of simply uniformly changing the quadrature. It is also noted that regardless of chosen parameters (quadrature and tolerance), ACS performs better, so a priori knowledge of some optimal parameters is not required to obtain a speedup. It is noted that the speedups seen here are more modest (around 1.5) compared to the box problem (around 3). This is most likely due to the relatively large source region that makes the quadrature requirements generally quite low. The ACS algorithm is generally better when the uncollided flux has fairly high levels of anisotropy.
Tol ● 1e−04 0.001 0.01 0.1 1
2.5
●
2.0
Quadrature ● 6 ● 10 ● 14 ● 20 ● 30
● ●
●
1.5
●
0.001
0.010
RMS Error
Figure 11: Equal-Error Speedup for all ACS parameters on box material 3 (c=0.9)
how it performs on a more realistic, complex problem. The VENUS-2 MOX-fueled reactor dosimetry benchmark[32] was issued by the OECD/NEA based on the VENUS reactor owned and operated by SCK•CEN in Belgium. The objective was to validate and compare nuclear data and transport codes for MOX-fueled dosimetry calculations. Our goal is to compare the ACS method to the standard source iteration in TITAN. As such, the actual benchmark performance and accuracy is not the issue - just the difference between the two methods. Using the benchmark information, a model was generated for TITAN. The mesh and materials for an x-y slice at the core region are shown in Figure 12. A full description of the model can be found in[26, 32]. The 34 dosimeter locations are overlaid over the mesh distribution. Locations 1-19 indicate regions surrounded by water, while locations 20-34 indicate regions surrounded by steel. The fission source distribution used for the model is shown in Figure 13. Again, vertical diagrams are omitted for brevity. For benchmarking, six threshold interactions including 58 Ni(n, p), 115 In(n, n0 ), 103 Rh(n, n0 ), 64 Zn(n, p), 237 Np(n, f ), and 27 Al(n, α) are used to determine dose at 34 locations shown in Figure 12. These dosimeters have various threshold energies that are all above 0.1M eV . Multi-group cross sections were derived from the 47group BUGLE-96[33] cross-section library that is based on the ENDF/B-VI data set[34]. The fuel region was homogenized by volume fraction of fuel, clad and water. The dosimeter cross sections were also obtained from BUGLE. Only the first 26 energy groups from BUGLE were used, which corresponds to 0.11 to 20 MeV. The mesh size used was 0.315 cm in the X/Y directions and 0.63 cm in the Z direction, for a total of 557,583 spatial meshes. The diamond difference (DD) scheme is used 11
70.00
Material UO2 Fuel 3.3 w/o UO2 Fuel 4.0 w/o MOX Fuel Pyrex Steel Baffle Steel Barrel Steel Pad Steel Vessel Water Air
Y (cm)
19
●
40.66
29
●
17 28 ● ● 34 ●16 ●27 ● 33 ● 26 32 15 ● ● ● 25 31 ● 14 ● ● 24 30 ● ● 13 ● 9 8 7 ● ● ● 23 6 12 ● ● ● 22 ●
18.90
●
5
●
4
●
3
18
●
11 ●21
●
20 ●
0.00 0.00
●
2
●
1
18.90
10
●
40.66
70.00
X (cm) Figure 12: VENUS-2 mesh and dosimeter Layout Table 7: SI Algorithm Performance for VENUS problem
Quad. Order
CPU Time (min)
Inner Iterations
RMS Error
Max Error
S16 S10 S8 S6 S4
1100.2 452.5 303.9 187.3 96.3
381 381 382 389 393
0.37% 0.56% 0.93% 3.53%
1.91% 2.88% 4.22% 13.43%
Table 8: ACS Algorithm Performance for VENUS problem
Quad. Order
ACS Tol.
Time (min)
# Iterations at: Quad. 1 Quad. 2 Total
RMS Error %
MAX Error %
Equal-Error Speedup
S10 /S4 S10 /S4 S10 /S4
10−3 10−2 10−1
296.3 188.1 115.9
211 80 26
182 310 366
393 390 392
0.43% 0.59% 2.03%
2.35% 2.38% 6.63%
1.36 1.58 1.28
S8 /S4 S8 /S4 S8 /S4
10−3 10−2 10−1
200.0 130.6 103.2
243 110 30
154 285 361
397 395 391
0.60% 1.06% 2.09%
3.57% 3.58% 6.68%
1.46 1.40 1.42
S6 /S4 S6 /S4 S6 /S4
10−3 10−2 10−1
144.2 113.9 95.1
234 111 28
158 281 365
392 392 393
0.98% 1.04% 1.89%
4.79% 4.78% 5.49%
1.29 1.61 1.61
5. Discussion
stant quadrature using a source iteration procedure. There are a few factors that make the method more or less effective than other potential methods.
From the tested problems, it is clear that the ACS method offers a good speedup compared to using a con12
70.00
Source
40.66
0.006
Flux 1e−01 1e−02 1e−03 1e−04 1e−05
40.66
0.004
y
Y (cm)
70.00
0.002 0.000 18.90
18.90
0.00 0.00
18.90
40.66
70.00
0.00 0.00
X (cm)
18.90
40.66
70.00
x
Figure 13: VENUS-2 source distribution Figure 15: VENUS-2 reference flux distribution, group 18 (1.00-1.35 MeV) 70.00 ● ● ● ●
Dosimeter Response (Arbitrary Units)
10−4
Flux 1e−02
40.66
y
1e−03 1e−04 1e−05
18.90
0.00 0.00
18.90
40.66
● ● ●
Figure 14: VENUS-2 reference flux distribution, group 7 (4.97-6.03 MeV)
● ● ●
●
●
10−5
●
● ●
● ● ● ●
● ●
●
● ● ● ● ● ●
●
Steel Steel Locations Locations (1−19) (1−19) II
VI
Inner Baffle
Outer Baffle
Figure 16:
A localized source means high anisotropy for the first few collided fluxes. This is advantageous to the ACS method as the optimal quadrature changes significantly and time can be saved by changing quadratures. With a more disperse source (as in the VENUS case), there is little advantage since the quadrature requirements do not change significantly with the number of collisions. For low-scattering problems, the flux is dominated by the un-collided component. This is good for ACS, but it also means that the first-collision source method will do almost as well. ACS has the advantage over FCS in that you don’t have to know this about the problem - the adaptive algorithm will automatically pick a good option. For high-scattering problems, the flux is dominated the scattered flux, for which the quadrature requirements do not change significantly. This brings up one of the downsides of the ACS method. Since the flux is being separated and each collision solved for individually, this precludes
●
●
10−6
70.00
x
● ●
VIII
●
X I
Water Water Locations Locations (20−34) (20−34) IX
Core Neutron Central Water Barrel Pad Hole Gap
27 Al(n, α)
VII Reflector
reaction rates at all locations
the option of using possible acceleration techniques such as Diffusion Synthetic Acceleration (DSA)[35] or Krylov methods[36]. A constant quadrature using these methods can probably outperform the ACS method on highly diffusive problems due to the slow convergence of the scattering source. For medium-scattering problems, the flux is not dominated by any component. This means that the firstcollision method can be inefficient since the first few collisions can still be significantly anisotropic. Traditional acceleration methods such as DSA are not as impactful since the source iteration solution converges relatively quickly. Given these observations, the ACS method performs best compared to the alternatives on localized-source problems with low-to-medium levels of scattering. On other problems, ACS still performs very well compared to the 13
15 5 0
● ● ● ● ● ●
● ● ● ●
●
●
● ●
● ● ●
●
●
●
●
●
● ●
● ● ●
● ● ● ● ● ●
−5
●
Acknowledgments
−10
Dosimeter Response Error (%)
10
●
very stable. There were no convergence problems noted on any problem that was given to it. The ACS method does not require any special knowledge beyond that of being able to run a standard discrete ordinates code. The two arbitrary parameters, additional quadrature order(s) and adaptive tolerance, do not need to be carefully chosen. If the user picks reasonable values (a second quadrature approximately half the order of the first, and adaptive tolerance in the range of 0.001-0.1), then a good speedup should be obtainable.
S4 Error S10 Error
−15
Steel Steel Locations Locations (1−19) (1−19) II
VI
Inner Baffle
Outer Baffle
Figure 17:
27 Al(n, α)
VIII
Water Water Locations Locations (20−34) (20−34) X I
IX
Core Neutron Central Water Barrel Pad Hole Gap
The authors would like to thank Dr. Ce Yi, author of the TITAN code, for his insights into the structure and function of the code. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
VII Reflector
reaction rates errors by quadrature order
References
standard source iteration.
[1] G. Sjoden, A. Haghighat, PENTRAN - a 3-D cartesian parallel SN code with angular, energy, and spatial decomposition, in: Proceedings of the Joint International Conference on Mathematical Methods and Supercomputing in Nuclear Applications, Saratoga Springs, NY, Oct. 6-10,, Vol. 2, 1997, pp. 1267–1276. [2] C. Yi, A. Haghighat, A three-dimensional block-oriented hybrid discrete ordinates and characteristics method, Nuclear Science and Engineering 164 (3) (2010) 221–247. [3] W. A. Rhoades, R. L. Childs, The TORT three-dimensional discrete ordinates neutron/photon transport code, Tech. rep., Oak Ridge National Lab., TN (USA) (1987). [4] T. A. Wareing, J. M. McGhee, J. E. Morel, ATTILA: A 3-D unstructured tetrahedral-mesh Sn code, in: 3-D Deterministic Radiation Transport Computer Programs: Features, Applications and Perspectives: 2-3 December 1996, OECD Chˆ ateau de la Muette, Paris, France, Organisation for Economic Cooperation and Development, 1997, p. 49. [5] S. Chandrasekhar, Radiative transfer, New York: Dover, 1950. [6] B. Carlson, Transport theory: Discrete ordinates quadrature over the unit sphere., Tech. rep., Los Alamos Scientific Lab., N. Mex. (1970). [7] K. D. Lathrop, Remedies for ray effects, Nuclear Science and Engineering 45 (1971) 255–268. [8] E. Lewis, W. Miller, Computational methods of neutron transport, John Wiley and Sons, Inc., New York, NY, 1984. [9] K. Lathrop, Ray effects in discrete ordinates equations, Nuclear Science and Engineering 32 (1968) 357–269. [10] J. J. Jarrell, M. L. Adams, Discrete-ordinates quadrature sets based on linear discontinuous finite elements, in: International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering (M&C 2011), Rio de Janeiro, RJ, Brazil, 2011. [11] C. D. Ahrens, Highly efficient, exact quadratures for threedimensional discrete ordinates transport calculations, Nuclear Science and Engineering 170 (1) (2012) 98–101. [12] K. Manalo, C. D. Ahrens, G. E. Sjoden, Advanced quadratures for three-dimensional discrete ordinate transport simulations: A comparative study, Annals of Nuclear Energy 81 (2015) 196– 206. [13] G. Longoni, A. Haghighat, Development of new quadrature sets with the ‘ordinate splitting’ technique, in: Proceedings of the 2001 ANS International Meeting on Mathematical Methods for Nuclear Applications (M&C),Salt Lake City, Utah, USA, Sept. 2001, 2001.
6. Conclusions In all types of calculations, there is a trade-off between speed and accuracy. More accurate solutions are always obtainable by taking more computational effort. In the development of new methods, the important thing is the efficiency - that is, what is the speed for a given level of accuracy, or vice versa. In this paper, the ACS method was developed to improve the efficiency of discrete ordinates calculations by optimizing the angular quadrature. By separating the flux components by collision order, it is possible to use high-order quadrature initially when it is most needed, and shift to lower order quadrature for the remaining calculations. This quadrature adaptation is performed based on an intelligent automated estimate of the error and a user-adjusted tolerance. Although the method requires user-defined quadrature orders and adaptive tolerance, the choices of these do not greatly affect the efficiency of the adaptive collision source method as opposed to the alternative of using a uniform quadrature set. The ACS method was developed and incorporated into the 3-D, multi-group, parallel, hybrid transport code TITAN and tested on a variety of problems. The 3-D, multigroup implementation means that ACS is practically applicable to real-world problems and is not just a theoretical method. For shielding problems that were tested (a simple box-in-a-box problem and the VENUS-2 dosimetry benchmark), the ACS method proved to be very efficient. Depending on the problem specifics and the parameters chosen, ACS can perform between 1.5-4 times faster than the standard source iteration. Equivalently, an accuracy up to 4 times better is obtainable in the same amount of computation time. Further, the method appears to be 14
[14] G. Longoni, A. Haghighat, Development and application of the regional angular refinement technique and its application to non-conventional problems, in: Proceedings of PHYSOR 2002, Seoul, Korea, October 2002, 2002. [15] G. Longoni, A. Haghighat, Simulation of a CT-scan device with PENTRAN using the new Pn-Tn quadrature set and an angular regional refinement technique, in: Proceedings of the 12th Biennial RPSD Topical Meeting, Santa Fe, NM, April, 2002. [16] G. Longoni, Advanced quadrature sets, acceleration, and preconditioning techniques for the discrete ordinates method in parallel computing environments, Ph.D. thesis, University of Florida (2004). [17] J. C. Stone, M. L. Adams, Progress on adaptive discreteordinates algorithms and strategies, in: Nuclear Mathematical and Computational Sciences Meeting, Gatlinburg, TN, 2003. [18] J. Stone, Adaptive discrete-ordinates algorithms and strategies, Ph.D. thesis, Texas A&M University (2007). [19] J. J. Jarrell, An adaptive angular discretization method for neutral-particle transport in three-dimensional geometries, Ph.D. thesis, Texas A&M University (2010). [20] R. Alcouffe, A first collision source method for coupling monte carlo and discrete ordinates for localized source problems, in: Monte-Carlo Methods and Applications in Neutronics, Photonics and Statistical Physics, Vol. 240, Springer Berlin / Heidelberg, 1985, pp. 352–366. [21] R. Alcouffe, R. O’Dell, F. Brinkley Jr, A first-collision source method that satisfies discrete S n transport balance, Nuclear Science and Engineering 105 (2) (1990) 198–203. [22] T. Wareing, J. Morel, D. Parsons, A first collision source method for ATTILA, an unstructured tetrahedral mesh discrete ordinates code, Proceedings of the 1998 American Nuclear Society Radiation Protection and Shielding Division Topical Conference, Nashville TN 1 (1998) 376–382. [23] R. Lillie, GRTUNCL3D: A discontinuous mesh threedimensional first collision source code, in: Proceedings of the 1998 American Nuclear Society Radiation Protection and Shielding Topical Conference, Nashville, TN, April 19-23, (1998), pp. 1–5. [24] W. J. Walters, A. Haghighat, Development of the adaptive collision source (ACS) method for discrete ordinates, in: International Conference on Mathematics and Computational Methods Applied to Nuclear Science & Engineering (M&C 2013), Sun Valley, Idaho, USA, May 5-9, 2013. [25] W. J. Walters, A. Haghighat, Performance of the adaptive collision source (ACS) method for discrete ordinates in parallel environments, in: SNA+ MC 2013-Joint International Conference on Supercomputing in Nuclear Applications+ Monte Carlo, Paris, France, Oct 27-31, EDP Sciences, 2013, p. 04106. [26] W. J. Walters, A. Haghighat, ACS algorithm in discrete ordinates for pressure vessel dosimetry, in: Proc. 15th International Symposium on Reactor Dosimetry, Aix en Provence, France, May 18-23, 2014. [27] G. I. Bell, S. Glasstone, Nuclear Reactor Theory, Van Nostrand Reinhold Company, 1970. [28] C. Yi, A. Haghighat, A hybrid block-oriented discrete ordinates and characteristics method algorithm for solving linear boltzmann equation, in: Proc. Int. Conf. M&C, Vol. 77, 2007. [29] C. Yi, Hybrid discrete ordinates and characteristics method for solving the linear boltzmann equation, Ph.D. thesis, University of Florida (2007). [30] W. Oberkampf, C. Roy, Verification and validation in scientific computing, Cambridge University Press, Cambridge, 2010. [31] W. Gropp, E. Lusk, N. Doss, A. Skjellum, A high-performance, portable implementation of the mpi message passing interface standard, Parallel computing 22 (6) (1996) 789–828. [32] C. Y. Han, C. ho Shin, H.-C. Kim, J. K. Kim, N. Messaoudi, B.-C. Na, VENUS-2 MOX-fuelled reactor dosimetry calculations benchmark specification, Tech. rep., OECD/NEA, NEA/NSC/DOC(2004)6 (2004). [33] J. E. White, D. T. Ingersoll, C. O. Slater, R. W. Roussin, BUGLE-96: a revised multigroup cross section library for LWR
applications based on endf/b-vi release 3, Tech. rep., Oak Ridge National Lab. 37831-6363 (1996). [34] M. Chadwick, P. Obloˇ zinsk` y, M. Herman, N. Greene, R. McKnight, D. Smith, P. Young, R. MacFarlane, G. Hale, S. Frankle, et al., ENDF/B-VII: Next generation evaluated nuclear data library for nuclear science and technology, Nuclear data sheets 107 (12) (2006) 2931–3060. [35] R. E. Alcouffe, Diffusion synthetic acceleration methods for the diamond-differenced discrete-ordinates equations, Nuclear Science and Engineering 64 (2) (1977) 344–355. [36] M. L. Adams, E. W. Larsen, Fast iterative methods for discreteordinates particle transport calculations, Progress in nuclear energy 40 (1) (2002) 3–159.
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