Evaluation of Ray Effects in Linear Transport Problems

1. Evaluation of Ray Effects in Linear Transport Problems A. Barbarinoa, S. Dullaa, A.K. Prinjab, P. Ravettoa aPolitecnico di Torino, Torino, Italy bUniversity of New Mexico, Albuquerque, NM

2. Motivation • Interest in a thorough analysis of ray effects, associated to the discrete treatment of the angular variable, appearing in different situations when dealing with linear transport • Identification of the different types of ray effect that can be encountered • Quantification of these effects depending on the system configuration • Objective of using, as much as possible, explicit analytical expressions (Fourier transforms, analytical solutions with simplifying hypotheses …)

3. Background • Paper presented at ANS Annual Meeting 2012: “On the evaluation of ray effects in multidimensional and time-dependent transport problems” • based mainly on the use of Fourier transform for the operator analysis • Aimed at comparing the ray effect appearing in multidimensional steady-state problems to the time ray effect appearing also in 1D configurations • The interest of the subject got more people, and more ideas, involved, leading to the current presentation

4. Contents - I • The transport equation in • x - y spatial ray effect • x – t  time ray effect • Solution in the Fourier transformed space • Definition of ray-effect indicator IRE • The transport equation in • x – t without scattering • Analytical solution in the direct space • Comparison of performance of different angular quadratures

5. Contents - II • Q: Is there a “ray effect” also associated to the spatial discretization ? • 1D slab transport spatially discretized • No scattering • Analytical explicit solution in the direct space • x – y transport spatially discretized • Angle dependence kept continuous • Operator observation in Fourier transformed space • x – y transport without scattering • Exact solution attempted in the direct space • Discretized solution in explicit form • Identification of “ray effect like” phenomena

6. Chapter 1: ray effect VS time ray efffect • Is it possible to compare the ray effect appearing in x - y configurations with the time ray effect of the x - t slab case ? • The mathematical nature of the equation is the same, but a different physics is described • The final objective is still to identify some sort of a quantitative indicator of the amount of distortion introduced • To this objective, both models have been studied performing a Fourier transform in all dimensions •  observe distortion of the transport operator

7. Ray effect in multi-D transport Discretization of the angle prevents neutrons to reach certain points in space (especially if scattering is low) Example of a 2D x - y case (disregarding the polar angle): Physical propagation of neutrons Propagation distortion associated to an S2 angular discretization

8. Time-dependent ray effect Discretization of the angle prevents neutrons to reach all possible points in space at a certain time Physics: at , neutrons can reach all points such that SN distortion: At , all neutrons reach the same point

9. Transport equation in x-y flatland Polar angle is neglected (but the ray effect characteristics are retained) The mathematical “complication” is tackled moving to the Fourier-transformed space:

10. Transport equation in x-y flatland The scalar flux can be made explicit easily by angular integration, allowing to identify the Green function of the problem (transfer function): Since the function depends only on the length of the ω vector, we see a circular symmetry in the transformed space

11. SN discretization in x-y The angular integration is substituted by a sum, and the corresponding transfer function is distorted: The function depends on the two components separately  no circular symmetry The comparison of this function with the exact version allows for quantitative measurements of the amount of ray effect appearing

12. SN discretization in x-y σ=1, c=0.99 σ=1, c=0.5 exact S4

13. The integral indicator of ray effect in x - y The work in the transformed space allows for a more manageable definition of an integral parameter to quantify the ray effect The idea is to measure the distortion of the SN isolines with respect to the exact ones The indicator selected (IRE) is the difference in the distance from the center of an isoline, thus depending on

14. The integral indicator of ray effect in x - y

15. Time-dependent transport The time-dependent one-dimensional transport equation in slab geometry is now considered: The mathematical structure of this equation is similar to the x - y case, but the implications of the discretization of the angular variable are different

16. Fourier transform for x - t problem Again, the use of the Fourier transform allows the determination of the Green function and the corresponding SN discretized version

17. Fourier transfer function in x - t transport exact S4

18. Fourier transfer function in x - t transport No circular symmetry as in x - y case We proceed to a change of coordinate to regain a circular symmetry for the exact case The same mapping is applied to the SN discretized transfer function

19. Fourier transfer function in x - t transport exact S2 Samelogicasbefore for the definition of the integralindicator IRE

20. The integral indicator of ray effect in x - t

21. Chapter 2: time ray efffect in the direct space • In order to have results of easier interpretation, the problem is approached in the direct space • x-t problem without scattering • Analytical solution with Laplace transform in time for a source problem • Comparison of different quadrature sets to the exact transport solution • Transport model considered • Solution obtained starting from the Green function of the problem

22. Laplace Effect of quadrature Time ray efffect in the direct space • Point source in the phase space • Green function in transformed space • Response to a generic source – angular flux • Response to a generic source – scalar flux

23. exact SN Time ray efffect in the direct space • Different source configurations considered • Gauss-Legendre quadrature compared to Gauss-Lobatto Legendre • Case 1: Normal spatial distribution, angular isotropic, pulsed in time

24. Time ray efffect in the direct space

25. Time ray efffect in the direct space Case 2: Normal spatial distribution, pulsed in time, defined only for • The discretization of the angle may disturb the angular characteristics of the solution • If the discrete ordinates are outside the interval [a; b] the solution is null (unphysical) • The preservation of the number of particles is no longer ensured

26. Time ray efffect in the direct space S8 solution 0.8 < μ < 1.0 Only one ordinate in the interval

27. Chapter 3: effects of spatial discretization • Q: can the distortions associated to the spatial discretization be characterized as ray effects ? • Analysis of transport problems when the spatial discretization is introduced • x-y transport in the transformed space • 1D slab transport w/o scattering in the direct space • x-y transport w/o scattering in the direct space

28. x-y spatially discretized transportin the transformed space • Spatial derivatives approximated with finite differences • Then the Fourier transform is applied, knowing that • And the operator form in the transformed space is (w/o scattering, but easy to introduce)

29. x-y spatially discretized transportin the transformed space Spatial Discretization only Pattern depending on the discretization step Circular symmetry not preserved … may this lead to a ray-effect phenomenon ?

30. x-y spatially discretized transportin the transformed space S2 angular treatment introduced in the spatially discretized operator Complex patterns appearing The characteristic rays (as seen previously) associated to SN are anyway visible The interpretation of these results is still complicated let’s go back to the direct space

31. 1D slab transport w/o scattering in the direct space • Analytical solution (isotropic source and BC) • Spatially discretized form (standard DD scheme) S x x0 0

32. The Yanghui triangle 1D slab transport w/o scattering in the direct space • The transport sweeping implies the contribution of all fluxes in points 1..i-1 at the point i • The back- substitution of the fluxes provides a general expression depending only on the source and BC (let’s drop the source to simplify the formulae …) A recurrence relation appears … Ex:

33. 1D slab transport w/o scattering in the direct space • This explicit form of the numerical solution of the transport sweep allows to • Proof that the numerical scheme converges to • Have a continuous dependence on μ, therefore the distortions appearing in the solution are due to the spatial discretization only • Evaluate the scalar flux by analytical integration on μ … and then also perform a numerical quadrature, recovering the SN formulation of the problem • Get some nice insight into the behavior of the solution …

34. S x x0 0 1D slab transport w/o scattering in the direct space • Angular fluxes (continuously dependent on μ) • As expected: • Discrepancies in the x-discretized solution appear for small values of μ • The amount of the error decreases at larger distances

35. 1D slab transport w/o scattering in the direct space Scalar flux

36. y y0 x 2D transport (flatland) w/o scattering in the direct space • The 2D problem tackled in the transformed space is here resolved in the direct form, to check how the propagation may be affected also by the spatial discretization • No scattering, no source • localized BC • Solution discretized in space and continuous in angle, allowing to distinguish spatial and angular discretization effects

37. (x,y) y (x,y) α1 α1 y0 α0 x α0 y y0 x 2D transport (flatland) w/o scattering in the direct space • Analytical solution

38. 2D transport (flatland) w/o scattering in the direct space • Space-discretized solution • Standard DD in x and y • The resulting function has been integrated numerically on the angle using a fine mesh • Parametric study on the mesh dimension

39. 2D transport (flatland) w/o scattering in the direct space Oscillations are mainly due to the linear approx within the mesh + balance preservation on the mesh volume

40. 2D transport (flatland) w/o scattering in the direct space • Viceversa • Continuous space • Discretized angle  quadrature formulae • Uniform spacing on the interval [-π/2 : π/2] • GL weight and ascissae in the same interval S6 S2

41. 2D transport (flatland) w/o scattering in the direct space S4 Gauss Legendre scheme Uniform angular spacing

42. 2D transport (flatland) w/o scattering in the direct space S8 Gauss Legendre scheme Uniform angular spacing

43. Comments and conclusions • The objective of this work was to analyze the different aspects of ray effects in linear transport • The work has been proven useful for • Obtaining a physical insight in different model configurations • Trying to predict the kind of results that can be obtained with numerical solutions • quantifying the ray effects with suitable indicator definitions • In perspective, trying to devise mitigation techniques for such effects • Last but not least, the results can be fruitfully used for their educational value