Multigrid discontinuous Galerkin method for multigroup particle - PowerPoint PPT Presentation

Multigrid discontinuous Galerkin method for multigroup particle transport Pablo Lucero Interdisciplinary Research Center for Scientific Computing Universit¨ at Heidelberg DEAL.II Workshop 2015 Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

Radiative transfer in astrophysics Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

Radiative transfer in climatology Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

Radiative transfer in neutron and gamma transport Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

Particle density description Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

Transport equation Ψ ′ = Ψ( Ω ′ , E ′ , x ) Ω = ( ψ, φ ) Ψ = Ψ( Ω , E , x ) σ s ( Ω ′ , E ′ ) = σ s ( Ω ′ → Ω , E ′ → E , x ) σ T = σ T ( E , x ) q = q ( Ω , E , x ) Transport equation � E max � σ s ( Ω ′ , E ′ ) Ψ ′ d Ω ′ dE ′ = q , Ω · ∇ Ψ + σ T Ψ − 0 S ∀ ( Ω , E , x ) ∈ S × ( 0 , E max ] × D Boundary condition Ψ( Ω , E ) = 0 ∀ ( Ω , E ) ∈ S × ( 0 , E max ] × ∂ D , Ω · n < 0 Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

Transport equation with fission νσ ′ f = νσ f ( E ′ ) χ = χ ( E ) Transport equation � E max � E max � σ s ( Ω ′ , E ′ ) Ψ ′ d Ω ′ dE ′ − χ f Ψ ′ dE ′ = q νσ ′ Ω · ∇ Ψ + σ T Ψ − 0 S 0 Eigenvalue problem � E max � E max � σ s ( Ω ′ , E ′ ) Ψ ′ d Ω ′ dE ′ = 1 νσ ′ f Ψ ′ dE ′ Ω · ∇ Ψ + σ T Ψ − χ k eff 0 S 0 Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

Diffusion approximation � � J = J ( E , x ) = Ω Ψ d Ω Φ = Φ( E , x ) = Ψ d Ω S S Ω · ∇ Ψ = ∇ · ( Ω Ψ) = ∇ · J J ≈ − D ( E , x ) ∇ Φ Diffusion equation � E max σ s ( E ′ ) Φ ′ dE ′ = q −∇ · ( D ∇ Φ) + σ T Φ − 0 � E max � E max σ s ( E ′ ) Φ ′ dE ′ − χ f Φ ′ dE ′ = q νσ ′ −∇ · ( D ∇ Φ) + σ T Φ − 0 0 � E max � E max σ s ( E ′ ) Φ ′ dE ′ = 1 νσ ′ f Φ ′ dE ′ −∇ · ( D ∇ Φ) + σ T Φ − χ k eff 0 0 Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

Angle description Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

Angle collocation S n � E max � σ s ( Ω ′ , E ′ ) Ψ ′ d Ω ′ dE ′ = q , Ω · ∇ Ψ + σ T Ψ − 0 S Ψ i ′ = Ψ( Ω i ′ , E ′ , x ) Ψ i = Ψ( Ω i , E , x ) s ( E ′ ) = σ s ( Ω i ′ → Ω i , E ′ → E , x ) σ i ′ i q i = q ( Ω i , E , x ) � E max � E max n � σ s ( Ω ′ , E ′ ) Ψ ′ d Ω ′ dE ′ ≈ � ω i ′ σ i ′ i s ( E ′ ) Ψ i ′ dE ′ 0 S 0 i ′ = 1 Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

Angle collocation S n � E max � σ s ( Ω ′ , E ′ ) Ψ ′ d Ω ′ dE ′ = q , Ω · ∇ Ψ + σ T Ψ − 0 S  � E max n s ( E ′ ) Ψ i ′ dE ′ = q 1 � ω i ′ σ i ′ 1  Ω 1 · ∇ Ψ 1 + σ T Ψ 1 −     0  i ′ = 1     ...     � E max n    s ( E ′ ) Ψ i ′ dE ′ = q i � ω i ′ σ i ′ i Ω i · ∇ Ψ i + σ T Ψ i − 0  i ′ = 1     ...     � E max n   s ( E ′ ) Ψ i ′ dE ′ = q n  � ω i ′ σ i ′ n Ω n · ∇ Ψ n + σ T Ψ n −     0  i ′ = 1 Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

Multigroup Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

Multigroup � E max n s ( E ′ ) Ψ i ′ dE ′ = q i , � ω i ′ σ i ′ i Ω i · ∇ Ψ i + σ T Ψ i − 0 i ′ = 1 ( 0 , E max ] = ( 0 , E 1 ] ∪ ... ∪ ( E g − 1 , E g ] ∪ ... ∪ ( E G − 1 , E max ] � E g σ T ( E , x ) dE � E g E g − 1 σ i , g Ψ i , g = Ψ( Ω i , E , x ) dE T = Ψ i , g E g − 1 � E g � E max n � ω i ′ σ i ′ i s ( E ′ ) Ψ i ′ dE ′ dE 0 E g − 1 i ′ = 1 σ i ′ i , g ′ g = s Ψ i , g ′ Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

Multigroup � E max n s ( E ′ ) Ψ i ′ , g ′ dE ′ = q i , � ω i ′ σ i ′ i Ω i · ∇ Ψ i + σ T Ψ i − 0 i ′ = 1  G n  � � ω i ′ σ i ′ i , g ′ 1  Ψ i ′ , g ′ = q i , 1 Ω i · ∇ Ψ i , 1 + σ T , 1 Ψ i , 1 −   s    g ′ = 1 i ′ = 1     ...       G n   � � ω i ′ σ i ′ i , g ′ g Ψ i ′ , g ′ = q i , g Ω i · ∇ Ψ i , g + σ T , g Ψ i , g − s  g ′ = 1 i ′ = 1     ...      G n    � � ω i ′ σ i ′ i , g ′ G Ω i · ∇ Ψ i , G + σ T , G Ψ i , G − Ψ i ′ , g ′ = q i , G   s    g ′ = 1 i ′ = 1  Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

Discrete angle and energy system  i ′ , 1 i ′ , 1 i ′ , 1   1 , 1 ... ... n , 1   1 , 1 ... ... n , 1   1 , 1 ... ... n , 1  i ′ , 1 i ′ , 1 i ′ , 1 ... ... ... ... ... ... ... ... ... ... ... ...           i ′ , i   i ′ , i   i ′ , i   1 , i ... ... n , i ... 1 , i ... ... n , i ... 1 , i ... ... n , i           i ′ , 1   i ′ , 1   i ′ , 1   ... ... ... ... ... ... ... ... ... ... ... ...           i ′ , 1 i ′ , 1 i ′ , 1 1 , n ... ... n , n 1 , n ... ... n , n 1 , n ... ... n , n   1 , 1 g ′ , 1 G , 1    ... ... ... ... ...    i ′ , 1 i ′ , 1 i ′ , 1        1 , 1 ... ... n , 1 1 , 1 ... ... n , 1 1 , 1 ... ... n , 1    i ′ , 1 i ′ , 1 i ′ , 1  ... ... ... ... ... ... ... ... ... ... ... ...           i ′ , i   i ′ , i   i ′ , i   1 , i ... ... n , i ... 1 , i ... ... n , i ... 1 , i ... ... n , i            i ′ , 1   i ′ , 1   i ′ , 1   ... ... ... ... ... ... ... ... ... ... ... ...          i ′ , 1 i ′ , 1 i ′ , 1  1 , n ... ... n , n 1 , n ... ... n , n 1 , n ... ... n , n   g ′ , g 1 , g G , g   ... ... ... ... ...     i ′ , 1 i ′ , 1 i ′ , 1  1 , 1 ... ... n , 1   1 , 1 ... ... n , 1   1 , 1 ... ... n , 1      i ′ , 1 i ′ , 1 i ′ , 1 ... ... ... ... ... ... ... ... ... ... ... ...            i ′ , i   i ′ , i   i ′ , i   1 , i ... ... n , i ... 1 , i ... ... n , i ... 1 , i ... ... n , i           i ′ , 1   i ′ , 1   i ′ , 1   ... ... ... ... ... ... ... ... ... ... ... ...          i ′ , 1 i ′ , 1 i ′ , 1 1 , n ... ... n , n 1 , n ... ... n , n 1 , n ... ... n , n 1 , G g ′ , G G , G Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

Discontinuous Galerkin finite elements G n � � ω i ′ σ i ′ i , g ′ g Ω i · ∇ Ψ i , g ( x ) + σ T , g ( x )Ψ i , g ( x ) − ( x )Ψ i ′ , g ′ ( x ) = q i , g ( x ) s g ′ = 1 i ′ = 1 v ∈ L 2 ( D ) � � � V h = � v | K ∈ P K } := v 1 + v 2 } := v 1 n 1 + v 2 n 2 { { v } { { v n } 2 2 Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

Discontinuous Galerkin finite elements   G n G n � � � � � � ω i ′ σ i ′ i , g ′ g  v i , g d x a h ( ψ, v ) = ω i  Ω i · ∇ ψ i , g + σ T , g ψ i , g − ψ i ′ , g ′ s K K ∈ T h g = 1 i = 1 g ′ = 1 i ′ = 1 G � � � � + ω i | Ω i · n | ψ i v i d x + b h ( ψ, v ) F g = 1 Ω i · n ≤ 0 F ∈ F b h G n � � � 4 � � � b h ( ψ, v ) = ω i max { 4 , σ s h } | Ω i · n | { { ψ i , g n } }{ { v i , g n } } − 2 Ω i · { { ψ i , g n } }{ { v i , g } } d x F g = 1 i = 1 F ∈ F j h Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

Multigrid preconditioner with Schwarz smoothers Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015

Source iteration? Coupled D=1e-05 Source Iteration D=1e-05 1e+06 50 1e+06 50 10000 10000 40 40 Frequency redistribution Frequency redistribution 100 100 30 30 1 1 20 20 0.01 0.01 10 10 0.0001 0.0001 1e-06 0 1e-06 0 1e-06 0.0001 0.01 1 100 10000 1e+06 1e-06 0.0001 0.01 1 100 10000 1e+06 Absorption Absorption Pablo Lucero (IWR) MGDG for multigroup particle transport DEAL.II Workshop 2015