Nonlinear Computational Methods for Simulating Interactions of Radiation with Matter in Physical Systems

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1 Nonlinear Computational Methods for Simulating Interactions of Radiation with Matter in Physical Systems Dmitriy Y. Anistratov Department of Nuclear Engineering North Carolina State University NE Seminar, Oregon State University October 14,

2 Introduction The simulation of interaction of particles with matter is a challenging problem. The transport equation is a basis for mathematical models of this physical phenomenon. Transport problems are difficult to solve high dimensionality, an integro-differential equation, coefficients depend on the state of matter the state of matter is affected by fluxes of particles. Applications: reactor physics, astrophysics (stars), plasma physics (laser fusion), atmospheric sciences. 2

3 Introduction There exists a family of efficient nonlinear methods for solving the transport equation: Quasidiffusion method (V. Gol din, 1964), Nonlinear Diffusion Acceleration (K. Smith, 2002), Flux methods (T. Germogenova, V. Gol din, 1969). These methods are defined by a system of nonlinearly coupled high-order and low-order problems that is equivalent to the original linear transport problem. The Nonlinear Projective Iteration (NPI) methods possess certain advantages for their use in multiphysics applications. The low-order equations of NPI methods can be used to formulate approximate particle transport models. These low-order equations can also be utilized as a basis for development of hybrid Monte Carlo computational methods. The NPI methods are distinct from each other by the definition of the low-order equations which gives rise to differences in features of these methods. 3

4 Introduction Outline A Tutorial on nonlinear methods for solving the transport equation The Quasidiffusion (QD) method for transport problems in 2D Cartesian geometry on grids composed of arbitrary quadrilaterals Nonlinear Weighted Flux (NWF) methods for particle transport problems in 2D Cartesian geometry on orthogonal grid This work was performed in collaboration with my Ph.D. students at NCSU William A. Wieselquist (to graduate in 2008 Paul Scherrer Institute, Switzerland) Loren Roberts (graduated in 2008 Baker Hughes Inc.) 4

5 Transport Problem Let us consider the following single-group slab geometry transport problem with isotropic scattering and source: μ x ψ(x, μ)+σ t(x)ψ(x, μ) = 1 1 Σs (x) 2 μ 1, 0 x L, ψ(x, μ )dμ + Q(x), φ(x) = 1 ψ(x, μ)dμ, J(x) = 1 μψ(x, μ)dμ, with reflective left boundary ψ(0,μ)=ψ(0, μ), 0 <μ 1, and vacuum boundary condition on the right ψ(l, μ) =0, μ<0. 5

6 Low-Order Equations The transport equation is integrated over μ 1 with weights 1 and μ dj dx +(Σ t Σ s )φ = Q, d 1 μ 2 ψdμ +Σt J =0, dx where φ(x) = 1 ψ(x, μ)dμ, J(x) = 1 μψ(x, μ)dμ. What should we do with the extra moment? 1 μ 2 ψdμ =? 6

7 Approximate Closure: Diffusion Theory Diffusion theory (approximation) ψ(x, μ) = 1 2 (φ(x)+3μj(x)), 1 μ 2 ψdμ = 1 3 φ(x). P 1 equations Diffusion equation dj dx +(Σ t Σ s )φ = Q, 1 dφ 3 dx +Σ tj =0, d 1 dφ dx 3Σ t dx +(Σ t Σ s )φ = Q, 7

8 Approximate Closure: Variable Eddington Factor Method 1 μ 2 ψdμ F(x)φ(x) Minerbo closure (approximation) ψ(x, μ) =α(x)e μβ(x) F(x) =1 2 J(x) Z(x) φ(x), Kershaw closure (approximation) ( F(x) = 1 ( ) ) 2 J(x) φ(x) Levermore-Pomraning closure (approximation) F(x) = J(x) φ(x) coth(z(x)), coth(z) 1 Z = J φ P 1 -like nonlinear low-order equations dj dx +(Σ t Σ s )φ = Q, d dx (Fφ)+Σ tj =0, F = F(φ, J) 8

9 Quasidiffusion (QD) Method: Exact Closure The Quasidiffusion method (exact) 1 1 μ 2 ψdμ μ 2 ψdμ = 1 ψdμ The quasidiffusion factor. E(x) = μ 2 ψdμ ψdμ ψdμ = E(x)φ(x), =< μ 2 > Quasidiffusion low-order equations dj dx +(Σ t Σ s )φ = Q, deφ dx +Σ tj =0, E = E[ψ]. 9

10 System of Equations of the Quasidiffusion (QD) Method The transport (high-order) equation μ x ψ(x, μ)+σ tψ(x, μ) = 1 ( Σs φ(x)+q ). 2 QD (variable Eddington) factors E(x) = 1 1 μ 2 ψ(x, μ)dμ ψ(x, μ)dμ, C L = the low-order QD equations d dx J(x)+(Σ t Σ s )φ(x) =Q, ( ) d E(x)φ(x) +Σ t J(x) =0. dx 0 0 μψ(l, μ)dμ ψ(l, μ)dμ. J(0) = 0, J(L) =C L φ(l). 10

11 QD Method: Iteration Process High-order problem (transport sweep) μ x ψ(s+1/2) +Σ t ψ (s+1/2) = 1 2 (Σ sφ (s) + Q), Calculation of QD factors E (s+1/2) (x) = 1 μ2 ψ (s+1/2) (x, μ)dμ 1 ψ(s+1/2) (x, μ)dμ, C L = Low-order QD problem d dx J (s+1) +(Σ t Σ s )φ (s+1) = Q, d ) (E (s+1/2) φ (s+1) +Σ t J (s+1) =0, dx J (s+1) (0) = 0, 0 0 μψ (s+1/2) (L, μ)dμ ψ (s+1/2) (L, μ)dμ. J (s+1) (L) =C (s+1/2) L φ (s+1) (L). 11

12 Nonlinear Diffusion Acceleration (NDA) High-order problem (transport sweep) Calculation of D (s+1/2) J (s+1/2) (x) = μ x ψ(s+1/2) +Σ t ψ (s+1/2) = 1 2 (Σ sφ (s) + Q), 1 μψ (s+1/2) (x, μ)dμ, φ (s+1/2) (x) = D (s+1/2) = 1 φ (s+1/2) ( 1 J (s+1/2) + 1 ) dφ (s+1/2) 3Σ t dx Low-order NDA problem d dx J (s+1) +(Σ t Σ s )φ (s+1) = Q, 1 dφ (s+1) 3Σ t dx + D (s+1/2) φ (s+1) + J (s+1) =0, (Kord Smith) ψ (s+1/2) (x, μ)dμ,. 12

13 QD Method for the Multidimensional Transport Equation The transport (high-order) equation Ω ψ (k+1/2) ( r, Ω) + σ t ( r)ψ (k+1/2) ( r, Ω) = 1 4π σ s( r)φ (k) ( r)+ 1 4π q( r). The QD (Eddington) tensor E (k+1/2) αβ ( r) = 4π Ē = ( ) Exx E xy E xy E yy Ω α Ω β ψ (k+1/2) ( r, Ω)d Ω / 4π E αβ ( r) = average value of Ω α Ω β. The low-order QD (LOQD) equations J (k+1) ( r)+σ a ( r)φ (k+1) ( r) =q( r), J (k+1) ( r) = 1 ) (Ē (k+1/2) ( r)φ k+1 ( r). σ t ( r), ψ (k+1/2) ( r, Ω)d Ω, α,β = x, y, 13

14 Discretization of the LOQD Equations The unknowns: φ i, φ iω,andj iω = J iω n iω, The balance equation in ith cell J iω A iω + σ a,i φ i V i = q i V i, where ω cell face. ω Jim Morel (2002) proposed a cell-centered discretization for the diffusion equation on meshes of arbitrary polyhedrons. { φ} iω =(φ iω φ i ) α iω + 1 (φ iω ) β iωω, V i ω where α iω and β iωω are based on geometry. We apply Morel s method J iω = 1 { } (Ēφ). σ t,i iω { } ({ } { } ) ({ } Exx φ Exy φ Exy φ (Ēφ) = e x + + e y iω x iω y iω x iω + { } ) Eyy φ y iω. 14

15 Discretization of the LOQD Equations Conditions at interfaces between cells Strong current and weak scalar flux continuity conditions J R = J 2L J R = J 3L φ R A R = φ 2L A 2L + φ 3L A 3L 15

16 Analytic Test Problem for the LOQD Equations 0 x 1, 0 y 1. σ t =1, σ a =0.5. Analytic solution [ ( φ(x, y) =5 tanh 100 x 2) 1 2 ( y 1 ) ] 2 2 The QD tensor (E αβ (x, y)) is given in analytic form φ(x, y) E xx (x, y) E xy (x, y) 16

17 Results on 3-Level Grids Relative error in the cell-average scalar flux Grid 1: 40 cells orthogonal grid randomized (20%) grid 17

18 Results on 3-Level Grids Relative error in the cell-average scalar flux Grid 2: 160 cells orthogonal grid randomized (20%) grid 18

19 Results on 3-Level Grids Relative error in the cell-average scalar flux Grid 3: 640 cells orthogonal grid randomized (20%) grid 19

20 Results on 3-Level Grids Relative error in the cell-average scalar flux Grid 4: 2560 cells orthogonal grid randomized (20%) grid 20

21 Transport Test Problem 21

22 Numerical Results Results of Single Level Grids Orthogonal Grid Randomized Grid Relative Difference F φ L φ R F φ L φ R F φ L φ R e e e-2-4.0e-3 2.2e e e e-3-2.2e-3 1.4e e e e-3-2.4e-4 5.0e e e e-3 1.2e-4 8.4e-4 Numerically Estimated Spatial Order of Convergence Orthogonal Grid Randomized Grid F φ L φ R F φ L φ R single-level two-level (with refinement on the left) two-level (with refinement on the right)

23 Nonlinear Weighted Flux (NWF) Methods The transport (high-order) equation The factors G m ( r) =Γ m F α m ( r) =Γ m Ω ψ( r, Ω) + σ t ( r)ψ( r, Ω) = 1 4π σ s( r)φ( r)+ 1 4π q( r). ω m ω m w(ω x, Ω y )ψ( r, Ω)d Ω / Ω α w(ω x, Ω y )ψ( r, Ω)d Ω ψ( r, Ω)d Ω, Γ m = / ω m dω ω m w(ω x, Ω y )dω. ω m ψ( r, Ω)d Ω, α = x, y, The low-order NWF equations φ m ( r) = ω m ψ( r, Ω)d Ω, m =1,...,4 ν x m ( ) F x m φ m + ν y m x y ( F y m φ m ) + σt G m φ m = 1 4 (σ sφ + q), φ = 4 m=1 φ m., ν x 1 = ν y 1 =1, νx 2 =, ν y 2 =1,νx 3 = ν y 3 =, νx 4 =1, ν y 4 =, 23

24 Asymptotic Diffusion Limit of the Transport Eqation We define a small parameter ε, scaled cross sections and source: Consider that ε 0. σ t = ˆσ t ε, σ a = εˆσ a, q = εˆq, We now assume that the solution can be expanded in power series of ε ψ = ε n ψ [n], n=0 The leading-order solutio of the transport the equation meets to the diffusion equation 1 ψ [0] x3σ t x 1 ψ [0] y 3σ t y + σ aψ [0] = q. 24

25 Asymptotic Diffusion Limit of the Low-Order NWF Equations We consider various weights w(ω x, Ω y )=1 w(ω x, Ω y )= Ω x + Ω y w(ω x, Ω y )=1+ Ω x + Ω y w(ω x, Ω y )=1+β( Ω x + Ω y ) Values of the diffusion coefficients (D) for specific NWF methods Weight 1 Ω x + Ω y 1+ Ω x + Ω y 1+β( Ω x + Ω y ) ( 1 D π+2 ) 2 1 ( 4σ t 3π σ t 1 4+5π ) σ t 12π σ t σ t 3σ t 25

26 Numerical Results Test problem A unit square domain σ t = 1 ɛ, σ a = ɛ, andq = ɛ ɛ =10 2, 10 3, 10 4, and 10 5 Boundary conditions are vacuum. A uniform spatial mesh consists of equal cells. The compatible product quadruple-range quadrature set uses 3 polar and 3 azimuthal angles per octant. Relative errors of the cell-average scalar flux in the cell located at the center Weight ɛ 1 Ω x + Ω y 1+ Ω x + Ω y 1+β( Ω x + Ω y ) E E E E E E E E E E E E E E E E-3 26

27 Asymptotic Boundary Condition There exist a boundary layer with the width of order of ε. The scalar flux at the boundary of the diffusive domain φ [0] (X, y) =2 W ( n Ω )ψ in (X, y, Ω)d Ω, n Ω<0 It is possible to improve the performance of the NWF method by using w(ω x, Ω y )=1+λ( Ω x + Ω y )+κ Ω x Ω y Relative errors of the asymptotic boundary conditions for the NWF methods BLD Transport Weight τ (mfp) Method 1 Ω x + Ω y 1+ Ω x + Ω y w β w λ,κ % 0.033% 0.033% 0.033% 0.033% % % % % % % % % % % % % 0.697% % % % % % 1.36% % % % % % 1.25% % % % % % 0.755% 27

28 Problem with an Unresolved Boundary Layer A domain 0 x, y 11 with two subregions: 1. 0 x 1, σ t =2,σ s =0,q =0,Δx =0.1, and Δy =1, 2. 1 x 11, σ t = σ s = 100, q =0,andΔx =Δy =1. There is an isotropic incoming angular flux with magnitude 1 boundary 2π on the left Other boundary condition are vacuum. This problem enables one to test the ability of a method to reproduce an accurate diffusion solution in the interior of the diffusive region with the spatially unresolved boundary layer. 28

29 Problem with an Unresolved Boundary Layer 29

30 Conclusions We have developed a new method for solving the LOQD equations on arbitrary quadrilateral and AMR-like grids The resulting QD method for solving transport problems demonstrated good performance on both randomized and AMR-like grids. Observe second order spatial convergence in numerical tests We defined a new parameterized family of nonlinear flux methods for solving the 2D transport equation. The asymptotic diffusion analysis enabled us to find a particular method of this family the solution of which satisfies a good approximation of the diffusion equation in diffusive regions. As a result, we developed a 2D flux method with an important feature for solving transport problems with optically thick regions. Note that it is possible to formulate a linear version of the proposed methods. 30