An Angular Multigrid Acceleration Method for S n Equations with Highly Forward-Peaked Scattering

Transcription

1 An Angular Multigrid Acceleration Method for S n Equations with Highly Forward-Peaked Scattering Bruno Turcksin & Jean C. Ragusa & Jim E. Morel Texas A&M University, Dept. of Nuclear Engineering Bruno Turcksin & Jean C. Ragusa & Jim E. Morel Angular Multigrid Acceleration Method for S n Equations 1 / 20

2 Outline 1 Introduction Motivation Equation Previous attempts 2 Angular multigrid with Krylov solver Angular multigrid with Krylov solver Results Algebraic multigrid for DSA 3 Conclusions Conclusions References Bruno Turcksin & Jean C. Ragusa & Jim E. Morel Angular Multigrid Acceleration Method for S n Equations 2 / 20

3 Motivation Coupled photon-electron transport has many applications such as radiotherapy, electronics hardening, etc. Electrons interact through Coulomb interactions highly anisotropic scattering. Standard acceleration schemes (Diffusion Synthetic Acceleration) focus on the isotropic part of the flux inefficient for electron transport. Angular multigrid technique proven to be efficient in one dimension [Morel and Manteuffel, 1991] but unstable in the multidimensional case need an efficient way to stabilize the scheme [Pautz et al., 1999]. Bruno Turcksin & Jean C. Ragusa & Jim E. Morel Angular Multigrid Acceleration Method for S n Equations 3 / 20

4 Continuous equation Ω Ψ(r,Ω,E) + Σ t (r,e)ψ(r,ω,e) = Σ s (r,ω Ω,E E)Ψ(r,Ω,E )de dω + Q(r,Ω,E) 4π 0 (1) where : Ψ is the angular flux. Σ t is the macroscopic total cross section. Σ s is the macroscopic differential scattering cross section. Q is a volumetric source. Ω = (µ,φ) with µ the cosine of the directional polar angle and φ is the directional azimuthal angle. Bruno Turcksin & Jean C. Ragusa & Jim E. Morel Angular Multigrid Acceleration Method for S n Equations 4 / 20

5 Discretized equation The angular discretization (S n ) of the multigroup equation is : where : Y m l Ω d Ψ g d (r) + Σg t (r)ψg d (r) = G L l g =0 l=0 m= l (2l + 1) Σ g g s,l (r)φ g l,m 4π (r)y m l (Ω d ) + Q g d (r) (2) are the spherical harmonics. Σ s,l is the Legendre moment of degree l of the scattering cross section. Ψ d is the angular flux on direction d. Bruno Turcksin & Jean C. Ragusa & Jim E. Morel Angular Multigrid Acceleration Method for S n Equations 5 / 20

6 Source Iteration The previous equation can be written as : LΨ = MΣDΨ + Q (3) The Source Iteration (SI) method works as follows : LΨ k = MΣDΨ k 1 + Q (4) Source Iteration with DSA/P1SA can be written as : Φ k+1/2 = DL 1 MΣΦ k + DL 1 Q (5) ) δφ k = A (Φ k+1/2 Φ k (6) Φ k+1 = Φ k+1/2 + δφ k (7) where : Φ k = DΨ k is the vector of all moments of the flux. A is the DSA/P1SA operator. Bruno Turcksin & Jean C. Ragusa & Jim E. Morel Angular Multigrid Acceleration Method for S n Equations 6 / 20

7 Angular multigrid for one dimension problem In one dimension, angular multigrid works in a similar way as DSA/P1SA but : A is a sequence of transport sweep operators with half the number of angles. there is a correction for each S n transport equation: δφ k 1 = T n/2δφ k 0, δφk 2 = T n/4δφ k 1, etc. at the coarsest level (δφ k I ), we use a P1SA equation with δφk I 1 is given by a T 4. the optimal transport correction has to be used for each correction. Bruno Turcksin & Jean C. Ragusa & Jim E. Morel Angular Multigrid Acceleration Method for S n Equations 7 / 20

8 Angular multigrid for one dimension problem Let s define the optimal transport correction for a P N 1 expansion of the cross sections is given by : Σ MG, s,l = Σ s,l Σ s,n/2 + Σ s,n 1 2 for k = 0,...,N 1 (8) This correction is optimal because it minimizes high-frequency angular errors. These errors are minimized according to : ρ s = max ( Σ s,n/2 /Σ s,0, Σ s,n/2+1 /Σ s,0,..., Σ s,n 1 /Σ s,0 ) (9) Bruno Turcksin & Jean C. Ragusa & Jim E. Morel Angular Multigrid Acceleration Method for S n Equations 8 / 20

9 Angular multigrid for one dimension problem The scheme works as follows : 1 Perform a transport sweep for the S n equations. 2 Perform a transport sweep for the S n/2 equations with a P n/2 1 expansion of the S n residual as the inhomogeneous source. 3 Continue coarsening the angular grid by a factor 2 until a sweep has been performed for the S 4 equations. 4 Solve the P1SA equation with a P 1 expansion of for the S 4 residual as the inhomogeneous source. 5 Add the Legendre moments of the diffusion solution to the Legendre moments of the S 4 iterate to obtain the accelerated S 4 iterate. 6 Continue adding the corrections from each coarse grid to the finer grid above until the accelerated S n iterate has been obtained. Bruno Turcksin & Jean C. Ragusa & Jim E. Morel Angular Multigrid Acceleration Method for S n Equations 9 / 20

10 Angular multigrid for one dimension problem Previous results in one dimension were very encouraging : if Fokker-Planck cross sections are used to approximated electron cross section, the spectral radius (ρ) of the angular multigrid method is bounded by 0.6 while the spectral radius of the DSA can be arbitrary close to 1. Scheme Quadrature order Computational ρ Theoretical ρ DSA MG Table: Performance of angular multigrid and DSA[Morel and Manteuffel, 1991] Bruno Turcksin & Jean C. Ragusa & Jim E. Morel Angular Multigrid Acceleration Method for S n Equations 10 / 20

11 Angular multigrid for multidimensional problem P1SA is unstable if Σ s,1 Σ t 0.5 P1SA cannot be used for electron transport. Pautz et al[pautz et al., 1999] adapted the angular multigrid method to multidimensions by replacing the P1SA by DSA and using the transport operator until S 2 instead of S 4. However the scheme is unstable need a filter to stabilize the method deteriorate the convergence speed. The spectral radius can become arbitrary close to one. Bruno Turcksin & Jean C. Ragusa & Jim E. Morel Angular Multigrid Acceleration Method for S n Equations 11 / 20

12 Angular multigrid with Krylov solver To stabilize the angular multigrid method, Krylov solver (e.g. GMRES) can be used instead of a filter + SI. When using a Krylov solver equation (3) : LΨ = MΣDΨ + Q has to be rewritten as : (I T )Φ = DL 1 Q (10) where I is the identity matrix, T is DL 1 MΣ and the previous equation is refered to as SI-preconditioned. If A is a preconditioner, the preconditioned equation is given by : A(I T )Φ = ADL 1 Q (11) Bruno Turcksin & Jean C. Ragusa & Jim E. Morel Angular Multigrid Acceleration Method for S n Equations 12 / 20

13 Angular multigrid with Krylov solver When using the angular multigrid method with a Krylov solver, the preconditioner is given by : A = I +P n/2 n T n/2 (I +P n/4 n/2 T n/4 (...(I +P 0 2 T 0 R 2 0 )...))R n n/2 (12) where : P n/2 n is the projection matrix of Φ n/2 to Φ n. R n n/2 is the restriction matrix of Φ n to Φ n. Bruno Turcksin & Jean C. Ragusa & Jim E. Morel Angular Multigrid Acceleration Method for S n Equations 13 / 20

14 Homogeneous medium Set-up : the domain is 5cm by 5cm and there are 50 by 50 cells. We use Fokker-Planck cross-section ( Σ s,l = α 2 (L(L + 1) l(l + 1)) with α = 1. The quadrature used is a Galerkin Gauss-Legendre-Chebyshev quadrature. The spatial discretization uses linear discontinuous finite elements. There is a uniform source of intensity 10. The relative tolerance on the solver is 10 4.We compare the number of GMRES iterations and the elapsed time for : SI preconditioning (SI). DSA preconditioning (DSA). angular multigrid preconditioning (MG). Bruno Turcksin & Jean C. Ragusa & Jim E. Morel Angular Multigrid Acceleration Method for S n Equations 14 / 20

15 Homogeneous medium S 4 S 8 S 16 SI DSA MG SI DSA MG SI DSA MG Table: GMRES iterations S 4 S 8 S 16 SI DSA MG SI DSA MG SI DSA MG Table: Elapsed time (s) Bruno Turcksin & Jean C. Ragusa & Jim E. Morel Angular Multigrid Acceleration Method for S n Equations 15 / 20

16 Heterogeneous medium Set-up : the domain is 10cm by 10cm and there are 50 by 50 cells. We use a S 8 Galerkin Gauss-Legendre-Chebyshev quadrature. The spatial discretization uses linear discontinuous finite of elements. There is a uniform source of intensity 10. The relative tolerance on the solver is We use two repeated layers on 10 zones, both of them are Fokker-Planck cross sections : Medium 1 : Σ t = Σ s,0 = 720 Medium 2 : Σ t = Σ s,0 = 1/720 Figure: Setup of the problem Bruno Turcksin & Jean C. Ragusa & Jim E. Morel Angular Multigrid Acceleration Method for S n Equations 16 / 20

17 Heterogeneous medium SI MG Table: GMRES iterations SI MG Table: Elapsed time (s) Even in heterogeneous medium the scheme is efficient. Bruno Turcksin & Jean C. Ragusa & Jim E. Morel Angular Multigrid Acceleration Method for S n Equations 17 / 20

18 Algebraic multigrid for DSA At the coarsest level, we need to solve a diffusion equation, i.e., to solve a Symmetric Positive Definite (SPD) system. DSA used is discretized using discontinuous finite elements[wang and Ragusa, 2010]. Standard method to solve a SPD system is to use preconditioned conjugate gradient (PCG). Algebraic spatial multigrid methods can provide an excellent preconditioner. Bruno Turcksin & Jean C. Ragusa & Jim E. Morel Angular Multigrid Acceleration Method for S n Equations 18 / 20

19 Conclusions Angular multigrid is very efficient for one dimension problems but needs to be stabilized for multidimensional prolems. Krylov methods stabilize efficiently angular multigrid for multidimensional problems. Algebraic multigrid can advantageously preconditioned conjugate gradient to solve DSA if the storage of the matrix can be afforded. Bruno Turcksin & Jean C. Ragusa & Jim E. Morel Angular Multigrid Acceleration Method for S n Equations 19 / 20

20 References Adams, M. L. and Wareing, T. A. (1993). Diffusion-synthetic acceleration given anisotropic scattering, general quadratures, and multidimensions. Conference: American Nuclear Society(ANS) annual meeting, San Diego, 68. Bell, W. N., Olson, L. N., and Schroder, J. B. (2011). PyAMG: Algebraic Multigrid Solvers in Python v2.0. Release 2.0. Briggs, W. L., Henseon, V. E., and McCormik, S. F. (2000). A Multigrid Tutorial. Second edition. Morel, J. and Manteuffel, T. (1991). An Angular Multigrid Acceleration Technique S n Equations with Highly Forward-Peaked Scattering. Nuclear Science and Engineering, 107: Notay, Y. (2010). An aggregation-based algebraic multigrid method. Electronic Transactions on Numerical Analysis, 37: Pautz, S., Morel, J., and Adams, M. (1999). An Angular Multigrid Acceleration Method for S n Equations with Highly Forward-Peaked Scattering. Proceedings of the International Conference on Mathematics and Computation, Reactor Physics and Environmental Analyses in Nuclear Applications, Madrid, Spain, 1: Wang, Y. and Ragusa, J. (2010). Diffusion Synthetic Acceleration for High-Order Discontinuous Finite Element S n Transport Schemes and Application to Locally Refined Unstructured Meshes. Nuclear Science and Engineering, 166: Bruno Turcksin & Jean C. Ragusa & Jim E. Morel Angular Multigrid Acceleration Method for S n Equations 20 / 20