TORT solutions for the 3D radiation transport benchmarks for simple geometries with void region

Progress in Nuclear Energy, Vol.39, No. 2, pp. 155-166,2001

Pergamon www.elsevier.com/locate/pnucene

PII:

S0149-1970(01)00009-9

©2001 ElsevierScienceLtd.All nghtsreserved Printedin GreatBritain 0149-1970/01/$ - see frontmatter

T O R T S O L U T I O N S F O R T H E 3D R A D I A T I O N T R A N S P O R T BENCHMARKS

FOR SIMPLE GEOMETRIES

WITH VOID REGION

YOUSRY Y. AZMY, FRANZ X. GALLMEIER, DICK A. LILLIE

Oak Ridge National Laboratory Oak Ridge, Tennessee 37831-6363, USA [email protected]; [email protected]; [email protected]

Abstract - - We present the solutions for the set of three-dimensional radiation transport Benchmark problems obtained with the TORT transport code using its three optional methods: Theta Weighted (0W), Linear Nodal (LN), and Linear Characteristic (LC). Only the cases with 50% scattering are presented in this paper since the nonscattering cases are bound to suffer severe ray effects. By solving the problems on a sequence of refined meshes we illustrate that for some points defined in the benchmark the solution converges with mesh refinement. However, the solution at most points does not converge with mesh refinement, and we illustrate that this is a consequence of ray effects in the void region. Also, we compare TORT's solutions to the Monte Carlo reference solution and observe that even when TORT's solution converges with mesh refinement, it usually does not converge to the Monte Carlo reference. This behavior also results from ray effects, and therefore we conjecture it will appear in varying degrees in all discrete ordinates accurate solutions because ray effects exist in the exact solution of the discrete ordinates equations. While this result is disappointing from the benchmarking point of view, it bodes well for TORT's ability to produce highly accurate solutions to the discrete ordinates approximation. Eliminating ray effects requires extensions of the solution algorithm, e.g. via a first collision source, while preserving the desirable features of the discrete ordinates methodology.© 2001 ElsevierScience Ltd. All rights reserved. Keywords: TORT; Discrete Ordinates; Radiation Transport; Benchmarks; Ray Effects 1

INTRODUCTION

The discrete ordinates, three-dimensional transport code TORT (Rhoades, 1997) was used to solve the three benchmark problems proposed by Keisuke Kobayashi (Kobayashi, 2001). Due to anticipated difficulties from ray effects for the pure absorber cases with standard angular quadratures a corresponding set of Benchmark problems with 50% scattering was proposed with the expectation that ray effects should be weak at such level of scattering. Hence, we focused only on the 50% scatterer cases at this stage. While the results presented in (Kobayashi, 2001) indicate the high accuracy of the TORT solu155

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tions, here we illustrate that in spite of the 50% scattering, ray effects are still manifested in the void regions causing poor convergence of the solution with mesh refinement. We solved the three benchmark problems using an $16 standard quadrature with TORT's Theta-Weighted (0W), Linear Nodal (LN), and Linear Characteristic (LC) methods. For historic reasons TORT restricts the use of the LN method to cases with anisotropic scattering, hence we introduced a zero P1 component for the scattering cross sections for such runs. [Note the effect of this modification on the CPU time reported for such runs: it is longer than necessitated for a given problem because the code computes the P1 moments which are all zeros]. All cases were converged to a tight criterion of 10 -4 with the inner iterations accelerated with TORT's Partial Current Rebalance (PCR) scheme. Each one of the Benchmark problems was solved on a set of five uniform grids whose cells are cubes of size 1/h m, h -- 10, 30, 50, 70, and 90. The reason for this choice of the computational mesh is to position one cell in every mesh with its center coincident with each point defined in the Benchmark Problem specification (Kobayashi, 2001). Clearly this produces values that only approximate the point value of the scalar flux at the desired points since TORT only computes cell-averaged values of the scalar flux. The purpose of this paper is twofold: First, we report comprehensively on the TORT solutions on all meshes attempted for all benchmark points in the 50% scatterer cases. Second, we illustrate the adverse influence of ray effects on the solution accuracy at many benchmark points even in the presence of scattering, an influence which we believe must contaminate all accurate solutions based on the discrete ordinates approximation. The remainder of this paper is organized as follows. Section 2 contains plots of TORT's solution as a function of cell size for each benchmark point in each of the 50% scatterer benchmark problems. The computational requirements for obtaining these solutions are summarized in Sec. 3. Finally, we present a discussion of the solutions in Sec. 4, and close with our conclusions in Sec. 5. 2

TORT'S SOLUTION TO THE 50% SCATTERER BENCHMARK PROBLEMS

In this section we present plots of the scalar flux obtained by TORT with 0W, LN, and LC as a function of cell size for each benchmark point. Each plot includes also the reference Monte Carlo value (Nagaya, 2001) to within two fractional standard deviations around the mean value. In the following plots we adopt a sequential indexing scheme of the benchmark points following the same order in which they were defined in (Kobayashi, 2001). Figures 1-3 depict the solutions to problem 1.ii; Figures 4 and 5 depict the solutions for problem 2.ii; and Figures 6-8 depict the solutions for problem 3.ii. It is worth noting that TORT computes the scalar flux averaged over computational cells, not at points as specified in the benchmark. Hence, the values plotted in the figures of this section are approximations of the Monte Carlo reference solution. A version of the MCNPX code (Hughes, 1998) upgraded with a mesh tally feature (Snow, 1998) was used to obtain Monte Carlo average fluxes over cells of 0.02 × 0.02 × 0.02 m 3 throughout the geometry of Benchmark problems 1.ii and 2.ii. These results compare well with the Monte Carlo reference solution and justify performing a comparison of the TORT cell-averaged fluxes in the refined meshes against point fluxes in the reference solution. 3

COMPUTATIONAL REQUIREMENTS

All results presented in the previous section were obtained on an IBM RISC 6000 Model 43P-260 workstation running the AIX 4.3.3 operating system, using 32 bit arithmetic. In Table 1 we summarize the CPU time in minutes, and the required memory in MWords for each of the runs performed. In all cases TORT was able to fit the entire problem within the size of core

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The choice of high angular quadrature order, fine mesh, and tight convergence criterion was motivated by our desire to produce a high quality solution to each problem in order to qualify as a b e n c h m a r k solution. As expected, these consume large computational resources, i.e. long CPU time and large memory requirement. This should not be viewed as an indication of inefficiency of TORT since in real applications one would select input parameters that are more in tune with the accuracy and precision expectations of the application. For example, fine meshes and high quadrature orders as high as employed here are normally reserved for more complex geometric configurations and are often motivated by the desire to resolve geometric detail rather than achieve higher accuracy. Also, given the accuracy expectations in most applications, the convergence criterion is usually set to 10 -3 or larger, thereby potentially requiring fewer iterations to achieve convergence than reported here. Hence, the reader should view the performance figures presented here in light of these parameter settings that while suitable for a benchmark exercise are unnecessary in most applications. 4

DISCUSSION OF RESULTS

It is evident from the results presented in Sec. 2 that the TORT computed scalar flux for most of the benchmark points does not converge with mesh refinement. Even when it does converge, it converges to a value that is often different from the Monte Carlo computed reference value. On the other hand, in the cases where all 0W, LN, and LC solutions converge with mesh refinement, they converge to the same solution thus lending some confidence in their accuracy. Initially we conjectured that this behavior results from the nature of the TORT solution, namely that it comprises cell-averaged scalar fluxes rather than the point values representing the Monte Carlo reference solution. In contrast, if we fix the volume over which the average flux is computed, e.g. a 0.1-m cube centered at each point specified in the Benchmark, the scalar flux converges with mesh refinement for almost all cases. Since most deterministic transport codes compute cell-averaged, rather than point values, of the scalar flux, perhaps it would be generally more consistent to define the benchmark values as volume averages over specified regions.

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Table 1. Execution time (min) and memory requirement (MW) on an IBM RISC 6000 Model 43P-260 workstation running the AIX 4.3.3 operating system for 0W, LN, and LC

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However, as explained in Sec. 2, MCNPX calculations for Benchmark 2.ii where the average flux was computed over cell volumes in the 30 x 50 × 30 mesh provided evidence that contradicts our conjecture. In order to understand these two aspects of the solution behavior with mesh refinement, i.e. the lack of convergence and/or convergence to values different from the reference solution, we examined plots of the scalar flux computed by LN for Benchmark 2.ii on the 54 x 90 x 54 mesh. Color coded plots of the scalar flux at the x=0.05 m and the y--0.95 m planes are presented in Figs. 9 and 10, respectively. These figures clearly exhibit ray effects that are most pronounced in the void tube. For example, the low flux region along the end of the y-axis in Fig. 9, and the offset high flux region in the square centered at x=0.05 m, z=0.05 m in Fig. 10 are clear manifestations of ray effects which persist in spite of the 50% scattering ratio due to the small total cross section in the void. In essence, in the void region the distributed, isotropic scattering source is dominated by the streaming into individual cells along discrete ordinates, or rays. TORT effectively constructs the streaming from the source region as broad parallel beams emanating from cells in the cross sectional area of the source cube along the directions of the discrete ordinates in the angular quadrature. These beams dominate the flux distribution in the voided regions as can be seen from Figs. 9 and 10. The two beams with directions nearest to the y-axis (recall the reflective boundary condition at z=O) overlap partially and form the centerline peak at y=0.6 m. The overlap of the two neighboring beams near the y-axis pointing into the positive z - y plane indicates another peak at z=O.1 m and y=0.5 m. The flux valley extending in the centerline at y values higher than 0.6 m is an area that no beam can reach directly from the source and therefore only sees scattered contributions. This pattern explains the convergence of the LN-computed ¢(6) in Fig. 4 to a value lower than the reference solution as the cell containing the Benchmark point lies in the flux valley described above. The lack of convergence in some cases also can be explained by ray effects as follows. Consider for example the square centered at x=0.05 m, z=0.05 m in Fig. 10. The shift in the high flux off the

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axis of the void tube, i.e. the y-axis, results from the incomplete illumination of the entire area of this square directly by the source. Indeed the offset of the lower left corner of the high flux region in this square from the y-axis forms an angle that agrees very well with the closest discrete ordinate to the y-axis. This feature of the ray effects tends to cluster the particles near the discrete ordinate involved, essentially depriving cells that are not directly illuminated from receiving their share of particles they would have received had they been connected to the source region with one or more angles. Since the transport method conserves particles on a per cell basis, it follows that the flux is overestimated in cells that are directly illuminated by the source. As the mesh is refined, given the same angular quadrature, the fractional area that is directly illuminated decreases, i.e. sharper resolution, thereby increasing the overestimation of the flux beyond the reference value. Even though in some cases it appears that the flux is not converging with mesh refinement, e.g. ¢(10) in Benchmark 2.ii, we conjecture that it eventually will converge to a value that is approximately four times that of the reference value, the reciprocal of the ratio of the directly illuminated area to the total square area in Fig. 10. The values plotted in Fig. 4 are too far from this limit, hence the appearance of lack of convergence. Based on the above discussion we believe that increasing the quadrature order, as typically done in such cases, will not satisfactorily resolve ray effects for all benchmark points. The only way to accomplish this here is to tailor a special quadrature for each benchmark problem so that each void cell containing a benchmark point is directly illuminated by the source. Clearly this would require an unrealistically large number of angles, and does not provide a procedure that is easily generalizable to other applications. It is worth noting that in many practical applications raising the order of the angular quadrature satisfactorily resolves ray effects. In our judgement, it is the peculiarity of this Benchmark exercise, namely the large number of pointwise fluxes in void cells to be determined, that precipitates the behavior noted above. Another practical, and well-accepted procedure to deal with ray effects that is available in most production discrete ordinates code packages is to break up the flux into an uncollided and fully collided components. The uncollided flux is typically computed by a standard ray-tracing technique in all computational cells then used to construct a first collision source that is naturally distributed throughout the problem domain. It follows that the fully collided flux calculated using standard Sn methods from this first collision source does not exhibit strong ray effects (even though secondary ray effects are inevitable). The complete solution is composed from the sum of the uncollided and fully collided fluxes for each energy group and computational cell. Such capability is provided to TORT users via the recently developed code GI~TUNCL3D (Lillie, 1998) that is distributed in the DOORS package, as well as other independently developed software, FNSUNCL3 (Konno, 2001). 5

CONCLUSION

We presented a comprehensive report on the TORT solutions for the Kobayahsi 3D Benchmark problems with 50% scattering. Our study included an investigation of the behavior of the solution at the defined benchmark points with mesh refinement, a necessary step to qualify confidence in the numerical solution. This revealed the persistence of ray effects, in spite of the reasonably large scattering ratio, in two ways: lack of convergence, and convergence to a value different from the reference Monte Carlo value, at most benchmark points. Perhaps this suggests that the scattering ratio is a better indicator of the influence of ray effects on quantities that are spread over a reasonable range in space. In contrast, local (or pointwise) quantities, like the benchmark fluxes examined here, seem to suffer from ray effects when the streaming source overwhelms the scattering source in their immediate neighborhood regardless of the magnitude of the scattering ratio. In a separate study, Konno independently recognized the adverse effect of ray effects on the TORT solutions, and used the first collision source code FNSUNCL3 in combination with TORT to obtain more accurate results. (Konno, 2001) The reader is referred to (Konno, 2001) for TORT results with a first collision source, and is advised that these

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results must be close to their counterparts computed with GRTUNCL3D (Lillie, 1998). Since Konno computed and reported such results first, we deemed it unnecessarily repetitious to report the same for the GRTUNCL3D-TORT combination. Since the exact solution of the discrete ordinates equations in a void exhibits ray effects we conclude that TORT's solutions, while defective from the benchmarking point of view, are faithful to the discrete angle model it is based on. In fact the presence of ray effects in the solutions implies that numerical diffusion, typically the culprit in many shielding computations, is under control, thus providing evidence to the high accuracy of TORT. Indeed a solution to the straight discrete ordinates equations that does not exhibit ray effects in void regions, while perhaps close to the benchmark value, must be viewed with suspicion. It is either an inaccurate solution, i.e. strong numerical diffusion smears out the ray effects, or it is solving equations that are not the discrete ordinates equations. The proper way to deal with ray effects is via a first collision source that hopefully spreads out the local uncollided source thereby directly illuminating all points in space, and more accurately computing the streaming component at points where it dominates the flux.(Konno, 2001) We have presented arguments to the effect that increasing the quadrature order farther in the present set of Benchmark problems will not sufficiently suppress ray effects at all benchmark points unless each such point is directly illuminated by the source. References Hughes, Grady, et al (1998) Recent Developments in MCNPX, Proc. ANS Topl. Mtg. Nucl. Appl. Accelerator Tech., Gatlinburg, TN, Sept. 20-23, 1998, American Nuclear Society (1998). Kobayashi, K., Sugimura, N., Nagaya, Y. (2001) 3D Radiation Transport Benchmarks for Simple Geometry's with Void Region, this volume of Progress in Nuclear Energy (2001). Konno, C. (2001) TORT Solutions with FNSUNCL3 on 3D Radiation Transport Benchmarks for Simple Geometries with Void Region, this "volume of Progress in Nuclear Energy (2001). Lillie, R. A. (1998) GRTUNCL3D: A Discontinuous Mesh Three-Dimensional First Collision Source Code, Proc. ANS Topl. Mtg. Radiation Protection and Shielding, Nashville, TN, April 19-23, 1998, Vol. I, p. 368, American Nuclear Society (1998). Nagaya, Y. (2001) Reference solutions for 3-D Radiation Transport Benchmarks by a Monte Carlo Code GMVP, this volume of Progress in Nuclear Energy (2001). Rhoades, W. A., Simpson, D. B. (1997) The TORT Three-Dimensional Discrete Ordinates Neutron/Photon Transport Code, ORNL/TM-13221 (1997). Snow, E. C. (1998) Mesh Tallies and Radiography Images for MCNPX, 4th Workshop on Simulating Accelerator Radiation Environments, Knoxville, TN, Sept. 14-16, 1998, p. 113 (1998).