Cumulant-based LBM. Repetition: LBM and notation. Content. Computational Fluid Dynamics. 1. Repetition: LBM and notation Discretization SRT-equation

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1 Content Cumulant-based LBM Prof B Wohlmuth Markus Muhr 25 July epetition: LBM and notation Discretiation ST-equation 2 MT-model Basic idea Moments and moment-transformation 3 Cumulant-based LBM Basic idea and technical tools Derivation of the cumulants Cumulant-transformation Cumulant collision operator 4 Validation aka colorful pictures Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 1/60 Computational Fluid Dynamics Numerical simulation of flows epetition: LBM and notation Calculation of the velocity field Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 2/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 3/60

2 Computational Fluid Dynamics Discretiation Numerical simulation of flows In space Calculation of the velocity field In velocity (D2Q9-model) c 6 c 2 c 5 c 3 c 1 c 7 c 4 c 8 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 4/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 5/60 (Discrete) Boltmann-equation (Discrete) Boltmann-equation Considering probabilities of an encounter fi Considering probabilities of an encounter fi f 6 f 2 f 5 f 6 f 2 f 5 f 3 f 1 f 3 f 1 f 7 f 4 f 8 Derivation of a (discrete) transport-equation for the distributions f i : f i (t, x) t + c i x f i (t, x) = Q(f i ) 1 τ C (f i (t, x) f eq i (t, x) ) f 7 f 4 f 8 Derivation of a (discrete) transport-equation for the distributions f i : f i (t, x) t + c i x f i (t, x) = Q(f i ) 1 τ C (f i (t, x) f eq i (t, x) ) Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 6/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 6/60

3 Equilibrium distributions f eq i ST-equation Maxwell-distribution as natural velocity-distribution of an ideal gas: ( ) ) m 3/2 m ξ u 2 M(ξ, ρ, u, T ) = ρ exp ( 2π k B T 2k B T Discretiation of the derivative by finite differences f i (t + t, x + c i t) = f i (t, x) 1 τ (f i (t, x) f eq i (t, x) ) Interpretation as collision and streaming: Taylor expansion up to second order ξ = ci : ( f eq i = ρ ω i c i u c 2 3 u 2 2c ( ci c 4 u ) ) 2 f 6 f 3 f 2 f 5 f 1 f 7 f 4 f 8 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 7/60 ST-equation ST-equation Discretiation of the derivative by finite differences f i (t + t, x + c i t) = f i (t, x) 1 τ (f i (t, x) f eq i (t, x) ) Discretiation of the derivative by finite differences f i (t + t, x + c i t) = f i (t, x) 1 τ (f i (t, x) f eq i (t, x) ) Interpretation as collision and streaming: Interpretation as collision and streaming: f 6 f 2 f f 6 f 2 f f 3 f f 3 f f 7 f 4 f 8 f7 f4 f8 f 7 f 4 f 8 f7 f4 f8 f7 f4 f8 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 8/60

4 ST-equation In three dimensions Discretiation of the derivative by finite differences f i (t + t, x + c i t) = f i (t, x) 1 τ (f i (t, x) f eq i (t, x) ) Interpretation as collision and streaming: Analogously in three dimensions (D3Q27-model) f 6 f 3 f 2 f 5 f f 7 f 4 f 8 f7 f4 f8 f7 f4 f y x Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 8/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 9/60 1 Notation Convenient notation by index-tupel: f 6 f 2 f 5 f 11 f 01 f 11 f 3 f 1 f 7 f 4 f 8 f 10 f 1 1 f 0 1 f 10 f 1 1 MT-model f i f ij with i, j { 1, 0, 1} Analogously in three dimensions: fi f ijk with i, j, k { 1, 0, 1} Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 10/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 11/60

5 Basic idea of MT (1) Basic idea of MT (1) The ST-equation reads: The ST-equation reads: f i (t + t, x + c i t) = f i (t, x) 1 τ (f i (t, x) f eq i (t, x) ) f i (t + t, x + c i t) = f i (t, x) 1 τ (f i (t, x) f eq i (t, x) ) Idea: Mix the relaxation terms Idea: Mix the relaxation terms f i (t + t, x + c i t) = f i (t, x) n V j=0 1 ( ) f j (t, x) f eq j (t, x) τ ij f i (t + t, x + c i t) = f i (t, x) n V j=0 1 ( ) f j (t, x) f eq j (t, x) τ ij In matrix-vector-form f i (t + t, x + c i t) = f i (t, x) S ( fi (t, x) f eq i (t, x) ) Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 12/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 12/60 Basic idea of MT (1) Basic idea of MT (2) The ST-equation reads: f i (t + t, x + c i t) = f i (t, x) 1 τ (f i (t, x) f eq i (t, x) ) In matrix-vector-form f (t + t, x + c t) = f (t, x) S (f (t, x) f eq (t, x)) Idea: Mix the relaxation terms f i (t + t, x + c i t) = f i (t, x) n V j=0 1 ( ) f j (t, x) f eq j (t, x) τ ij Diagonalie the matrix S: f (t+ t, x +c t) = f (t, x) M 1Ŝ (Mf (t, x) Mf eq (t, x)) In matrix-vector-form f i (t + t, x + c i t) = f i (t, x) S ( fi (t, x) f eq i (t, x) ) Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 12/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 13/60

6 Basic idea of MT (2) Basic idea of MT (2) In matrix-vector-form In matrix-vector-form f (t + t, x + c t) = f (t, x) S (f (t, x) f eq (t, x)) f (t + t, x + c t) = f (t, x) S (f (t, x) f eq (t, x)) Diagonalie the matrix S: f (t+ t, x +c t) = f (t, x) M 1Ŝ (Mf (t, x) Mf eq (t, x)) Diagonalie the matrix S: f (t+ t, x +c t) = f (t, x) M 1Ŝ (Mf (t, x) Mf eq (t, x)) Interpretation as transformation M : f m Interpretation as transformation M : f m f (t + t, x + c t) = f (t, x) M 1Ŝ (m(t, x) meq (t, x)) f (t + t, x + c t) = f (t, x) M 1Ŝ (m(t, x) meq (t, x)) Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 13/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 13/60 Moments Moments Definition: (aw)-moments For α, β, γ = 0, 1, 2, one defined the (raw)-moments of order α + β + γ as follows: m αβγ = In matrix-vector-form (in 2D) m 00 m 10 m 01 m 20 m 02 m 11 m 21 m 12 m 22 = ( ) α ( ) β ( ) γ c ijk c ijk c ijk f ijk = x y i α j β k γ f ijk Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 14/60 f 0 f 1 f 2 f 3 f 4 f 5 f 6 f 7 f 8 Definition: (aw)-moments For α, β, γ = 0, 1, 2, one defined the (raw)-moments of order α + β + γ as follows: m αβγ = In matrix-vector-form (in 2D) m 00 m 10 m 01 m 20 m 02 m 11 m 21 m 12 m 22 = ( ) α ( ) β ( ) γ c ijk c ijk c ijk f ijk = x y i α j β k γ f ijk Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 14/60 f 0 f 1 f 2 f 3 f 4 f 5 f 6 f 7 f 8

7 Interpretation of the moments Interpretation of the moments The moments correspond to different physical quantities like: The moments correspond to different physical quantities like: m 00 = 1 i,j= 1 f ij = ρ density c 6 c 3 c 2 c 5 c 1 m 00 = 1 i,j= 1 f ij = ρ density c 6 c 3 c 2 c 5 c 1 m 10 = 1 ) (c ij i,j= 1 m01 = 1 i,j= 1 (c ij ) x y c 7 c 4 c 8 f ij = ρ u x x-momentum f ij = ρ u y y-momentum m 10 = 1 i,j= 1 m01 = 1 i,j= 1 (c ij ) (c ij ) x y c 7 c 4 c 8 f ij = ρ u x x-momentum f ij = ρ u y y-momentum Some moments have to be combined for a reasonable interpretation ( m20 + m 02 = 1 ( ) 2 ( ) ) 2 c ij + c ij f ij E kin x y i,j= 1 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 15/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 15/60 Interpretation of the moments Moment-transformation The moments correspond to different physical quantities like: m 00 = 1 i,j= 1 m 10 = 1 i,j= 1 m01 = 1 i,j= 1 f ij = ρ density (c ij ) (c ij ) x y c 6 c 3 c 2 c 5 c 1 c 7 c 4 c 8 f ij = ρ u x x-momentum f ij = ρ u y y-momentum Some moments have to be combined for a reasonable interpretation ( m20 + m 02 = 1 ( ) 2 ( ) ) 2 c ij + c ij f ij E kin x y i,j= 1 The respective linear combination of matrix-lines yields the moment-transformationmatrix M 0 M 1 M 2 M 3 M 4 M 5 M 6 M 7 M 8 = m 00 m 10 m 01 m 20 + m 02 m 20 m 02 m 11 m 21 m 12 m 22 = Further linear combinations of (raw)-moments are possible as for example row-wise orthogonaliation f 0 f 1 f 2 f 3 f 4 f 5 f 6 f 7 f 8 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 15/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 16/60

8 MT-algorithm MT-algorithm Transform the given distributions f into moments M = M f und M eq = M f eq Transform the given distributions f into moments M = M f und M eq = M f eq elax the moment Mi with frequencies ω i M 0 M 8 = ω 0 ω 8 M 0 M eq 0 M 8 M eq 8 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 17/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 17/60 MT-algorithm MT-algorithm Transform the given distributions f into moments M = M f und M eq = M f eq Transform the given distributions f into moments M = M f und M eq = M f eq elax the moment Mi with frequencies ω i elax the moment Mi with frequencies ω i M 0 M 8 = ω 0 ω 8 M 0 M eq 0 M 8 M eq 8 M 0 M 8 = ω 0 ω 8 M 0 M eq 0 M 8 M eq 8 Transform the relaxed moments back into distribution space = M 1 M Transform the relaxed moments back into distribution space = M 1 M Collision update and streaming step f (t+ t, x +c t) = f (t, x) M 1Ŝ (Mf (t, x) Mf eq (t, x)) Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 17/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 17/60

9 MT-algorithm Pro/Con MT-model Transform the given distributions f into moments M = M f und M eq = M f eq elax the moment Mi with frequencies ω i M 0 M 8 = ω 0 ω 8 M 0 M eq 0 M 8 M eq 8 Transform the relaxed moments back into distribution space = M 1 M Collision update and streaming step f (t+ t, x +c t) = f (t, x) M 1Ŝ (Mf (t, x) Mf eq (t, x)) More tuning-parameter ωi as for ST Better insight into the model by identification with physical quantities Linear transformations statistical dependencies among the moments ( dependencies between ω i ) Physics gets lost by (wrong) variation of parameters How to choose them? Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 17/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 18/60 Pro/Con MT-model More tuning-parameter ωi as for ST Better insight into the model by identification with physical quantities Linear transformations statistical dependencies among the moments ( dependencies between ω i ) Physics gets lost by (wrong) variation of parameters How to choose them? Cumulant-based LBM Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 18/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 19/60

10 Basic idea of cumulant-based LBM Basic idea of cumulant-based LBM Goal: Find a transformation to statistically independent quantities C m Mathematically this is described by a factoriation of the probability-density f ijk = n V m=0 F m (C m ) Goal: Find a transformation to statistically independent quantities C m Mathematically this is described by a factoriation of the probability-density f ijk = n V m=0 F m (C m ) Efficient implementation of that (nonlinear) transformation Efficient implementation of that (nonlinear) transformation Main tools: Multivariate Taylor-expansion, Laplace-transformation and distribution-theory Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 20/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 20/60 Basic idea of cumulant-based LBM Goal: Find a transformation to statistically independent quantities C m Mathematically this is described by a factoriation of the probability-density f ijk = n V m=0 F m (C m ) Distributions ecall from measure theory: Lebesgue-space L 2 : L 2 () := { f : f (x) 2 dx < } Consider the following functional G : L 2 (), G(f ) := 1 1 f (x) dx Efficient implementation of that (nonlinear) transformation Main tools: Multivariate Taylor-expansion, Laplace-transformation and distribution-theory Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 20/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 21/60

11 Distributions ecall from measure theory: Lebesgue-space L 2 : L 2 () := { f : f (x) 2 dx < } Distributions ecall from measure theory: Lebesgue-space L 2 : L 2 () := { f : f (x) 2 dx < } Consider the following functional Consider the following functional G : L 2 (), G(f ) := 1 1 f (x) dx G : L 2 (), G(f ) := 1 1 f (x) dx One searches for a g L 2 () satisfying: G(f ) = g(x) f (x) dx f L 2 () Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 21/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 22/60 Distributions ecall from measure theory: Lebesgue-space L 2 : L 2 () := { f : f (x) 2 dx < } Distributions ecall from measure theory: Lebesgue-space L 2 : L 2 () := { f : f (x) 2 dx < } Consider the following functional Consider the following functional G : L 2 (), G(f ) := 1 1 f (x) dx G : L 2 (), G(f ) := 1 1 f (x) dx One searches for a g L 2 () satisfying: One searches for a g L 2 () satisfying: G(f ) = g(x) f (x) dx f L 2 () G(f ) = g(x) f (x) dx f L 2 () Solution: g(x) = 1[ 1,1] (x) Solution: g(x) = 1[ 1,1] (x) Interpretation of g as functional (ies representation theorem) Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 22/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 22/60

12 Distributions ecall from measure theory: Lebesgue-space L 2 : L 2 () := { f : f (x) 2 dx < } Consider the following functional Dirac-sequence Interprete the functions gn (x) = n 2 1 [ 1 n, n] 1 (x), n N as functionals G : L 2 (), G(f ) := 1 1 f (x) dx One searches for a g L 2 () satisfying: G(f ) = g(x) f (x) dx f L 2 () Solution: g(x) = 1[ 1,1] (x) Interpretation of g as functional (ies representation theorem) Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 22/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 23/60 Delta-distribution δ Delta-distribution δ For continuous functions f there holds: For continuous functions f there holds: g n (x) f (x) dx = n 2 1 n 1 n f (x) dx g n (x) f (x) dx = n 2 1 n 1 n f (x) dx = n 2 ( ( ) ( 1 F F 1 )) n n = n 2 ( ( ) ( 1 F F 1 )) n n ( ) = F 1 n F 2 1 n ( ) 1 n n F (0) = f (0) Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 24/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 24/60

13 Delta-distribution δ Delta-distribution δ For continuous functions f there holds: For continuous functions f there holds: g n (x) f (x) dx = n 2 1 n 1 n f (x) dx g n (x) f (x) dx = n 2 1 n 1 n f (x) dx = n 2 ( ( ) ( 1 F F 1 )) n n = n 2 ( ( ) ( 1 F F 1 )) n n ( ) = F 1 n F 2 1 n ( ) 1 n n F (0) = f (0) ( ) = F 1 n F 2 1 n ( ) 1 n Define the delta-distribution δ(x) := lim g n(x) = g (x) n by: δ(x) f (x) dx = f (0) n F (0) = f (0) Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 24/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 24/60 Delta-distribution δ Delta-function For continuous functions f there holds: g n (x) f (x) dx = n 2 = n 2 1 n 1 n f (x) dx ( ( ) ( 1 F F 1 )) n n ( ) = F 1 n F 2 1 n ( ) 1 n Define the delta-distribution δ(x) := lim g n(x) = g (x) n by: δ(x) f (x) dx = f (0) n F (0) = f (0) Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 24/60 Considered as a function there holds: δ(x) = { 0 if x 0 if x = 0 Which often gets normed to: δ(x) = { 0 if x 0 1 if x = 0 What you should remember for sure: δ(x) f (x) dx = f (0) Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 25/60

14 Delta-function epetition: Fourier-transform Considered as a function there holds: δ(x) = { 0 if x 0 if x = 0 Which often gets normed to: δ(x) = { 0 if x 0 1 if x = 0 The Fourier-transform of f : C is (modulo scaling) defined by: f (ω) = F[f (x)](ω) := f (x) e iωx dx f (ω) tells, how strong e iω is present in the frequency-spectrum of f What you should remember for sure: δ(x) f (x) dx = f (0) Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 25/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 26/60 epetition: Fourier-transform Laplace-transformation The Fourier-transform of f : C is (modulo scaling) defined by: f (ω) = F[f (x)](ω) := f (x) e iωx dx f (ω) tells, how strong e iω is present in the frequency-spectrum of f δ(ω 1) δ(ω + 1) Example: F[sin(x)](ω) 2i Instead of projection onto oscillations {e iω } ω one projects onto damping-functions {e s } s + F (s) := L[f (x)](s) := 0 f (x) e sx dx Laplace-transformations of a few common functions: L[x n ](s) = n! s n+1 L[sin(ax)](s) = a s 2 +a 2 L[ln(ax)](s) = 1 s (ln(s/a) + γ) Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 27/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 28/60

15 Laplace-transformation Laplace-transformation Instead of projection onto oscillations {e iω } ω one projects onto damping-functions {e s } s + F (s) := L[f (x)](s) := 0 f (x) e sx dx Instead of projection onto oscillations {e iω } ω one projects onto damping-functions {e s } s + F (s) := L[f (x)](s) := 0 f (x) e sx dx Laplace-transformations of a few common functions: L[x n ](s) = n! s n+1 L[sin(ax)](s) = a s 2 +a 2 L[ln(ax)](s) = 1 s (ln(s/a) + γ) Laplace-transformations of a few common functions: L[x n ](s) = n! s n+1 L[sin(ax)](s) = a s 2 +a 2 L[ln(ax)](s) = 1 s (ln(s/a) + γ) Certain features as for example the shift-property hold true: Certain features as for example the shift-property hold true: L[f (x a)](s) = e as F (s) L[f (x a)](s) = e as F (s) Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 28/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 28/60 Laplace-transformation of the delta-distribution Laplace-transformation of the delta-distribution Apply the definition of the (two-sided) Laplace-transform: Apply the definition of the (two-sided) Laplace-transform: L[δ(x)](s) = δ(x) e sx dx L[δ(x)](s) = δ(x) e sx dx Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 29/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 29/60

16 Laplace-transformation of the delta-distribution Laplace-transformation of the delta-distribution Apply the definition of the (two-sided) Laplace-transform: Apply the definition of the (two-sided) Laplace-transform: L[δ(x)](s) = δ(x) e sx dx = e s 0 = 1 L[δ(x)](s) = δ(x) e sx dx = e s 0 = 1 Transformation of the shifted delta-distribution by the shift-property: L[δ(x a)](s) = e as L[δ(x)](s) Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 30/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 30/60 Laplace-transformation of the delta-distribution Laplace-transformation of the delta-distribution Apply the definition of the (two-sided) Laplace-transform: L[δ(x)](s) = δ(x) e sx dx = e s 0 = 1 Apply the definition of the (two-sided) Laplace-transform: L[δ(x)](s) = δ(x) e sx dx = e s 0 = 1 Transformation of the shifted delta-distribution by the shift-property: L[δ(x a)](s) = e as L[δ(x)](s) Transformation of the shifted delta-distribution by the shift-property: L[δ(x a)](s) = e as L[δ(x)](s) = e as Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 30/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 31/60

17 Derivation of the cumulants (1) - Analytiation Derivation of the cumulants (1) - Analytiation Want to capture the discrete velocity-distributions analytically Want to capture the discrete velocity-distributions analytically f 11 f 01 f 11 f 11 f 01 f 11 f 10 f 10 f 10 f 10 f 1 1 f 0 1 f 1 1 f 1 1 f 0 1 f 1 1 This can be done by the peak-function f (ξ): This can be done by the peak-function f (ξ): f (ξ) = f (ξ x, ξ y, ξ ) = f ijk δ(ic ξ x ) δ(jc ξ y ) δ(kc ξ ) f (ξ) = f (ξ x, ξ y, ξ ) = f ijk δ(ic ξ x ) δ(jc ξ y ) δ(kc ξ ) Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 32/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 32/60 Derivation of the cumulants (1) - Analytiation Derivation (2) - Laplace-transform Want to capture the discrete velocity-distributions analytically f 11 f 01 f 11 To increase regularity one performs the two-sided Laplace-transform of f (ξ): f 10 f 1 1 f 0 1 f 10 f 1 1 L[f (ξ)](ξ) = f ijk L[δ(ic ξ x )]L[δ(jc ξ y )]L[δ(kc ξ )] This can be done by the peak-function f (ξ): f (ξ) = f (ξ x, ξ y, ξ ) = f ijk δ(ic ξ x ) δ(jc ξ y ) δ(kc ξ ) = f ijk e Ξ x ic e Ξ y jc e Ξ kc =: F (Ξ) Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 33/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 34/60

18 Derivation (2) - Laplace-transform Derivation (3) - statistical independence To increase regularity one performs the two-sided Laplace-transform of f (ξ): L[f (ξ)](ξ) = = f ijk L[δ(ic ξ x )]L[δ(jc ξ y )]L[δ(kc ξ )] f ijk e Ξ x ic e Ξ y jc e Ξ kc =: F (Ξ) Applying the statistical independence (which one aims to have) in the frequency space: And using the logarithm F (Ξ x, Ξ y, Ξ ) = ln (F (Ξ x, Ξ y, Ξ )) = n V m=0 n V m=0 F m (C m ) ln (F m (C m )) Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 34/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 35/60 Derivation (4) - Taylor-expansion Derivation (4) - Taylor-expansion The multivariate Taylor-expansion of ln (F (Ξx, Ξ y, Ξ )) around Ξ = 0 is: α+β+γ ln(f (Ξ x, Ξ y, Ξ )) = α+β+γ 0 1 α!β!γ! Ξ α x Ξ β y Ξ γ ln(f (Ξ x, Ξ y, Ξ )) Definition: Cumulants For α, β, γ = 0, 1, 2, one defines the cumulants of order α + β + γ as follows: c αβγ := c (α+β+γ) α+β+γ Ξ α x Ξ β y Ξ γ ln(f (Ξ x, Ξ y, Ξ )) Ξ α x Ξβ y Ξ γ The multivariate Taylor-expansion of ln (F (Ξx, Ξ y, Ξ )) around Ξ = 0 is: α+β+γ ln(f (Ξ x, Ξ y, Ξ )) = α+β+γ 0 1 α!β!γ! Ξ α x Ξ β y Ξ γ ln(f (Ξ x, Ξ y, Ξ )) Definition: Cumulants For α, β, γ = 0, 1, 2, one defines the cumulants of order α + β + γ as follows: c αβγ := c (α+β+γ) α+β+γ Ξ α x Ξ β y Ξ γ ln(f (Ξ x, Ξ y, Ξ )) Ξ α x Ξβ y Ξ γ Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 36/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 36/60

19 Efficient calculation via moments Where the magic happens Conduct the Taylor-expansion without the logarithm: α+β+γ F (Ξ x, Ξ y, Ξ ) = α+β+γ 0 1 α!β!γ! Ξ α x Ξ β y Ξ γ F (Ξ x, Ξ y, Ξ ) Analogously consider the coefficients: α+β+γ Ξ α x Ξ β y Ξ γ F (Ξ x, Ξ y, Ξ ) Ξ α x Ξ β y Ξ γ Explicit calculation of these coefficients yields: α+β+γ Ξ α x Ξ β y Ξ γ F (Ξ x, Ξ y, Ξ ) = = α+β+γ Ξ α x Ξ β y Ξ γ f ijk e Ξ x ic e Ξ y jc e Ξ kc f ijk ( ic) α e Ξ x ic ( jc) β e Ξ y jc ( kc) γ e Ξ kc Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 37/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 38/60 Where the magic happens Where the magic happens Explicit calculation of these coefficients yields: α+β+γ Ξ α x Ξ β y Ξ γ F (Ξ x, Ξ y, Ξ ) = = α+β+γ Ξ α x Ξ β y Ξ γ f ijk e Ξ x ic e Ξ y jc e Ξ kc f ijk ( ic) α e Ξ x ic ( jc) β e Ξ y jc ( kc) γ e Ξ kc = ( c) α+β+γ i α j β k γ f ijk Explicit calculation of these coefficients yields: α+β+γ Ξ α x Ξ β y Ξ γ F (Ξ x, Ξ y, Ξ ) = = α+β+γ Ξ α x Ξ β y Ξ γ f ijk e Ξ x ic e Ξ y jc e Ξ kc f ijk ( ic) α e Ξ x ic ( jc) β e Ξ y jc ( kc) γ e Ξ kc = ( c) α+β+γ i α j β k γ f ijk = ( c) α+β+γ m αβγ Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 38/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 38/60

20 Where the magic happens Cumulants vs Moments Explicit calculation of these coefficients yields: α+β+γ Ξ α x Ξ β y Ξ γ F (Ξ x, Ξ y, Ξ ) = α+β+γ Ξ α x Ξ β y Ξ γ f ijk e Ξ x ic e Ξ y jc e Ξ kc The (raw)-moments were shown to be: m αβγ = ( c) (α+β+γ) α+β+γ Ξ α x Ξ β y Ξ γ F (Ξ x, Ξ y, Ξ ) = f ijk ( ic) α e Ξ x ic ( jc) β e Ξ y jc ( kc) γ e Ξ kc = ( c) α+β+γ = ( c) α+β+γ m αβγ i α j β k γ f ijk The cumulants: c αβγ = c (α+β+γ) α+β+γ Ξ α x Ξ β y Ξ γ ln(f (Ξ x, Ξ y, Ξ )) Idea: By applying the chain rule one should be able to trace back the cumulants to the moments Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 38/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 39/60 Cumulants vs Moments Cumulant-transformation (1) The (raw)-moments were shown to be: m αβγ = ( c) (α+β+γ) α+β+γ Ξ α x Ξ β y Ξ γ F (Ξ x, Ξ y, Ξ ) The cumulants: c αβγ = c (α+β+γ) α+β+γ Ξ α x Ξ β y Ξ γ ln(f (Ξ x, Ξ y, Ξ )) We conduct the idea on the example c100 : c 100 = c (1+0+0) 1 Ξ 1 x Ξ 0 y Ξ 0 ln(f (Ξ x, Ξ y, Ξ )) = c 1 1 F (0, 0, 0) 1 F (Ξ x, Ξ y, Ξ ) Ξ 1 x Idea: By applying the chain rule one should be able to trace back the cumulants to the moments Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 39/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 40/60

21 Cumulant-transformation (1) Cumulant-transformation (1) We conduct the idea on the example c100 : c 100 = c (1+0+0) 1 Ξ 1 x Ξ 0 y Ξ 0 ln(f (Ξ x, Ξ y, Ξ )) = c 1 1 F (0, 0, 0) 1 F (Ξ x, Ξ y, Ξ ) Ξ 1 x = c 1 1 F (0, 0, 0) m 100 ( c) 1 = 1 F (0, 0, 0) m 100 We conduct the idea on the example c100 : c 100 = c (1+0+0) 1 Ξ 1 x Ξ 0 y Ξ 0 ln(f (Ξ x, Ξ y, Ξ )) = c 1 1 F (0, 0, 0) 1 F (Ξ x, Ξ y, Ξ ) Ξ 1 x = c 1 1 F (0, 0, 0) m 100 ( c) 1 = 1 F (0, 0, 0) m 100 With F (0, 0, 0) = f ijk e 0 e 0 e 0 = ρ also: c 100 = 1 ρ m 100 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 40/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 40/60 Cumulant-transformation (1) Cumulant-transformation (1) We conduct the idea on the example c100 : c 100 = c (1+0+0) 1 Ξ 1 x Ξ 0 y Ξ 0 ln(f (Ξ x, Ξ y, Ξ )) = c 1 1 F (0, 0, 0) 1 F (Ξ x, Ξ y, Ξ ) Ξ 1 x = c 1 1 F (0, 0, 0) m 100 ( c) 1 = 1 F (0, 0, 0) m 100 With F (0, 0, 0) = f ijk e 0 e 0 e 0 = ρ also: c 100 = 1 ρ m 100 Define: Cαβγ = ρc αβγ, then = C 100 = m 100 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 40/60 We conduct the idea on the example c100 : c 100 = c (1+0+0) 1 Ξ 1 x Ξ 0 y Ξ 0 ln(f (Ξ x, Ξ y, Ξ )) = c 1 1 F (0, 0, 0) 1 F (Ξ x, Ξ y, Ξ ) Ξ 1 x = c 1 1 F (0, 0, 0) m 100 ( c) 1 = 1 F (0, 0, 0) m 100 With F (0, 0, 0) = f ijk e 0 e 0 e 0 = ρ also: c 100 = 1 ρ m 100 Define: Cαβγ = ρc αβγ, then = C 100 = m 100 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 40/60

22 Cumulant-transformation (2) Half-time score Analogous procedure for the remaining cumulants: C 100 = m 100 C 010 = m 010 C 001 = m 001 C 200 = m ρ m2 100 C 020 = m ρ m2 010 C 002 = m ρ m2 001 What do we have so far? How far do we come with that, starting with the current distributions? Transform distributions f to (raw)-moments m = M f Transform (raw)-moments m to cumulants C = K(m) = K(M f ) = C(f ) C 110 = m ρ m 100m 010 C 101 = m ρ m 100m 001 [Ŝ ] f (t+ t, x +c t) = f (t, x) C 1 (C(f (t, x)) C(f eq (t, x))) C 210 = m ρ 2 m2 100m ρ m 110m ρ m 200m 010 Efficient transformation of equilibria Choice of relaxation parameters ωi Inverse mapping C 1 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 41/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 42/60 Cumulant equilibria - Derivation Cumulant equilibria - Derivation The equilibria can be calculated in advance The equilibria can be calculated in advance Laplace-transformation of Maxwell-distribution: F eq (Ξ) = L [ M(ξ) ] (Ξ) = M(ξ) e Ξ ξ dξ x dξ y dξ = ρ ( ) 1 exp C 4C (Ξ2 x + Ξ 2 y + Ξ 2 ) Ξ x u x Ξ y u y Ξ u Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 43/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 43/60

23 Cumulant equilibria - Derivation Cumulant equilibria - Derivation The equilibria can be calculated in advance Laplace-transformation of Maxwell-distribution: F eq (Ξ) = L [ M(ξ) ] (Ξ) = M(ξ) e Ξ ξ dξ x dξ y dξ = ρ ( ) 1 exp C 4C (Ξ2 x + Ξ 2 y + Ξ 2 ) Ξ x u x Ξ y u y Ξ u The equilibria can be calculated in advance Laplace-transformation of Maxwell-distribution: F eq (Ξ) = L [ M(ξ) ] (Ξ) = M(ξ) e Ξ ξ dξ x dξ y dξ = ρ ( ) 1 exp C 4C (Ξ2 x + Ξ 2 y + Ξ 2 ) Ξ x u x Ξ y u y Ξ u Apply the logarithm: ( ) ρ ln(f eq (Ξ)) = ln Ξ x u x Ξ y u y Ξ u + c2 ( s Ξ 2 2 x + Ξ 2 y + Ξ 2 ) ρ 0 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 43/60 Apply the logarithm: ( ) ρ ln(f eq (Ξ)) = ln Ξ x u x Ξ y u y Ξ u + c2 ( s Ξ 2 2 x + Ξ 2 y + Ξ 2 ) ρ 0 Taylor-expansion can be skipped, since this is a polynomial anyway Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 43/60 Cumulant equilibria - Derivation Cumulant equilibria The equilibria can be calculated in advance Laplace-transformation of Maxwell-distribution: F eq (Ξ) = L [ M(ξ) ] (Ξ) = M(ξ) e Ξ ξ dξ x dξ y dξ = ρ ( ) 1 exp C 4C (Ξ2 x + Ξ 2 y + Ξ 2 ) Ξ x u x Ξ y u y Ξ u Apply the logarithm: ( ) ρ ln(f eq (Ξ)) = ln Ξ x u x Ξ y u y Ξ u + c2 ( s Ξ 2 2 x + Ξ 2 y + Ξ 2 ) ρ 0 Taylor-expansion can be skipped, since this is a polynomial anyway Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 43/60 Only a certain few cumulants have an equilibrium not equal 0: c eq 000 = ln(ρ/ρ 0) c eq 100 = c 1 u x c eq 010 = c 1 u y c eq 001 = c 1 u c eq 200 = c 2 c 2 s c eq 020 = c 2 c 2 s c eq 002 = c 2 c 2 s For all others, there holds: c eq αβγ = 0 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 44/60

24 Half-time score Choice of relaxation parameters What do we have so far? How far do we come with that, starting with the current distributions? Transform distributions f to (raw)-moments m = M f Transform (raw)-moments m to cumulants C = K(m) = K(M f ) = C(f ) [Ŝ ] f (t+ t, x +c t) = f (t, x) C 1 (C(f (t, x)) C(f eq (t, x))) For the most cumulants the equilibrium is 0, hence one gets for example: C 110 = ω 1 C 110 C 101 = ω 1 C 101 C 011 = ω 1 C 011 C 222 = ω 10 (C 222 0) Efficient transformation of equilibria Choice of relaxation parameters ωi Inverse mapping C 1 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 45/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 46/60 Choice of relaxation parameters For the most cumulants the equilibrium is 0, hence one gets for example: C 222 = ω 10 (C 222 0) Choice of relaxation parameters For the most cumulants the equilibrium is 0, hence one gets for example: C 222 = ω 10 (C 222 0) C 110 = ω 1 C 110 C 211 = ω 8 C 211 C 110 = ω 1 C 110 C 211 = ω 8 C 211 C 101 = ω 1 C 101 C 121 = ω 8 C 121 C 101 = ω 1 C 101 C 121 = ω 8 C 121 C 011 = ω 1 C 011 C 112 = ω 8 C 112 C 011 = ω 1 C 011 C 112 = ω 8 C 112 Those with non vanishing equilibrium one can combine: C 200 C 020 = ω 1 (C 200 C 020 ) C 200 C 002 = ω 1 (C 200 C 002 ) Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 46/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 46/60

25 Choice of relaxation parameters Half-time score For the most cumulants the equilibrium is 0, hence one gets for example: C 110 = ω 1 C 110 C 222 = ω 10 (C 222 0) C 211 = ω 8 C 211 What do we have so far? How far do we come with that, starting with the current distributions? Transform distributions f to (raw)-moments m = M f Transform (raw)-moments m to cumulants C = K(m) = K(M f ) = C(f ) C 101 = ω 1 C 101 C 011 = ω 1 C 011 C 121 = ω 8 C 121 C 112 = ω 8 C 112 [Ŝ ] f (t+ t, x +c t) = f (t, x) C 1 (C(f (t, x)) C(f eq (t, x))) Those with non vanishing equilibrium one can combine: C 200 C 020 = ω 1 (C 200 C 020 ) C 200 C 002 = ω 1 (C 200 C 002 ) Efficient transformation of equilibria Choice of relaxation parameters ωi Inverse mapping C 1 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 46/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 47/60 Half-time score Advantages of Cumulant-based LBM What do we have so far? How far do we come with that, starting with the current distributions? Transform distributions f to (raw)-moments m = M f Transform (raw)-moments m to cumulants C = K(m) = K(M f ) = C(f ) [Ŝ ] f (t+ t, x +c t) = f (t, x) C 1 (C(f (t, x)) C(f eq (t, x))) Efficient transformation of equilibria Choice of relaxation parameters ωi Inverse mapping C 1 Better representation of physical processes due to statistical independence Equilibria of cumulants are almost all 0 More (real) degrees of freedom than in classical MT Gradual improvement of the method by asymptotic analysis and elimination of error-terms Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 48/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 49/60

26 Synthesis of Navier-Stokes-equation Synthesis of Navier-Stokes-equation By a Chapman-Enskog expansion one can derive the Navier-Stokes equations: u = 0 u t + (u ) u = 1 ρ p + 1 ( 1 1 ) u + f 3 ω 1 2 ρ }{{} =ν By a Chapman-Enskog expansion one can derive the Navier-Stokes equations: u = 0 u t + (u ) u = 1 ρ p + 1 ( 1 1 ) u + f 3 ω 1 2 ρ }{{} =ν With ω1 one can simulate fluids of different viscosity Only ω1 directly influences the solution With ω1 one can simulate fluids of different viscosity Only ω1 directly influences the solution Further parameters influence higher order error terms (set them to 1 in the following) Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 50/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 50/60 Synthesis of Navier-Stokes-equation By a Chapman-Enskog expansion one can derive the Navier-Stokes equations: u = 0 u t + (u ) u = 1 ρ p + 1 ( 1 1 ) u + f 3 ω 1 2 ρ }{{} =ν Validation aka colorful pictures With ω1 one can simulate fluids of different viscosity Only ω1 directly influences the solution Further parameters influence higher order error terms (set them to 1 in the following) Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 50/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 51/60

27 Turbulence measure The eynolds-number is dimensionless parameter of a flow e = Comparison: BGK Kumulanten U L ν At some specific threshold e the laminar-turbulent transition starts a) BGK-ST-LBM Traditional LBM methods have problems with high eynolds-numbers: Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 52/60 (Picture source [5]) Kumulanten-basierte LBM b) Cumulant-based LBM Prof B Wohlmuth Markus Muhr (Picture source [5]) 53/60 Turbulence resolution in 3D Comparison ST- and cumulant-based LBM for a cylinder flow at e = 8000 Turbulence resolution in 3D epresentation of (turbulence) vortices around a sphere flow: (Picture source [5]) (Picture source [5]) Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 54/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 55/60

28 Turbulence resolution in 3D The highlights MT-like-method with nonlinear transformation to statistical independent quantities Mathematical methods: I I I Distributions as generalied functions (Analytiation) Laplace-transform Comparison of Taylor-coefficients (Picture source [5]) Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 56/60 Kumulanten-basierte LBM I Mathematical methods: I Distributions as generalied functions (Analytiation) Laplace-transform Comparison of Taylor-coefficients Prof B Wohlmuth Markus Muhr MT-like-method with nonlinear transformation to statistical independent quantities Mathematical methods: I I I Advantages especially at high eynolds-numbers (stability and adaptivity) Kumulanten-basierte LBM 57/60 The highlights MT-like-method with nonlinear transformation to statistical independent quantities I Markus Muhr The highlights Prof B Wohlmuth 57/60 Distributions as generalied functions (Analytiation) Laplace-transform Comparison of Taylor-coefficients Advantages especially at high eynolds-numbers (stability and adaptivity) Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 57/60

29 Literature I Literature II d Humières, Dominique: Multiple relaxation time lattice Boltmann models in three dimensions Philosophical Transactions of the oyal Society of London A: Mathematical, Physical and Engineering Sciences, 360(1792): , 2002 Dubois, François: Equivalent partial differential equations of a lattice Boltmann scheme Computers & Mathematics with Applications, 55(7): , 2008 Geier, M: Ab initio derivation of the cascaded Lattice Boltmann Automaton University of Freiburg IMTEK, 2006 Geier, Martin, Andreas Greiner und Jan G Korvink: Cascaded digital lattice Boltmann automata for high eynolds number flow Physical eview E, 73(6):066705, 2006 Geier, Martin, Martin Schönherr, Andrea Pasquali und Manfred Krafcyk: The cumulant lattice Boltmann equation in three dimensions: Theory and validation Computers & Mathematics with Applications, 70(4): , 2015 Ning, Yang und Kannan N Premnath: Numerical Study of the Properties of the Central Moment Lattice Boltmann Method arxiv preprint arxiv: , 2012 ohm, Florian: A commented Python Script for LBM-Flow-Simulations Lecture on, 2015 Wolf-Gladrow, Dieter A: Lattice-gas cellular automata and lattice Boltmann models: An Introduction Nummer 1725 Springer Science & Business Media, 2000 Hänel, Dieter: Molekulare Gasdynamik: Einführung in die kinetische Theorie der Gase und Lattice-Boltmann-Methoden Springer-Verlag, 2006 Lukacs, Eugene: Characteristics functions Griffin, London, 1970 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 58/60 Kumulanten-basierte LBM Prof B Wohlmuth Markus Muhr 59/60