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Technische Universitat Munchen
LB@TUM - OuttakesEindhoven
Philipp Neumann
28.02.2011
P. Neumann: LB@TUM - Outtakes
Eindhoven, 28.02.2011 1
Technische Universitat Munchen
Outline
Block-Structured Adaptive LB Simulations in Peano
Coupling of a Finite Element (FE) Based Navier-Stokes (NS)Solver and a Lattice Boltzmann (LB) Automaton
P. Neumann: LB@TUM - Outtakes
Eindhoven, 28.02.2011 2
Technische Universitat Munchen
Where I’m from - SCCS
• Chair of Scientific Computing, Department for Computer Science
• 34 people, Head: Prof. Bungartz
P. Neumann: LB@TUM - Outtakes
Eindhoven, 28.02.2011 3
Technische Universitat Munchen
Where I’m from - SCCS
What do we actually do?• CFD (Navier-Stokes, Lattice Boltzmann)
• Interpolation techniques for high dimensional problems (sparse grids)
• Hardware aware programming (GPUs etc.)
• Efficient linear solvers for special applications
• Molecular Dynamics
And what do I do?Coupling between scales in fluid dynamics→ Macro-Meso coupling (Navier-Stokes⇔ Lattice Boltzmann)→ Meso-Micro coupling (Lattice Boltzmann⇔ Molecular Dynamics)
P. Neumann: LB@TUM - Outtakes
Eindhoven, 28.02.2011 4
Technische Universitat Munchen
Peano: Grid syntax
Fig.1: Peano grid.P. Neumann: LB@TUM - Outtakes
Eindhoven, 28.02.2011 5
Technische Universitat Munchen
I’m still LB from the block... similarities cannot be excluded;-)
Fig.2: Block-structure.Black: Peano cell.Blue: Peano vertex and associatedLB block.
Motivation:
• Reduce costs per grid traversal,increase computational load pervertex
• Reduce number of grid traversals(interpolations etc.)
Current state of development:
• 2D/ 3D support, different velocitydiscretisation schemes (D2Q9,D3Q15, D3Q19, D3Q27)
• Different blocksizes (multiples of 3,blocksize ≥ 3)
• P2: Parallel simulations (MPI based)(2D XX, 3D X)
• P1: Adaptive simulations (volumetricconcept)
P. Neumann: LB@TUM - Outtakes
Eindhoven, 28.02.2011 6
Technische Universitat Munchen
Dynamic block adaptivity
• Integration of FSI coupling tool preCICE
• Remove/ add blocks of LB cells where necessary
• Example: Moving sphere (2D, 3 grid levels)
P. Neumann: LB@TUM - Outtakes
Eindhoven, 28.02.2011 7
Technische Universitat Munchen
Coupling Finite-Element Navier-Stokes (NS-FE)and Lattice Boltzmann
• Incompressible Navier-Stokes equations:
∇ · u = 0
∂tu + u · ∇u = −∇p + 1Re ∆u
• Lattice Boltzmann method:
fi (x + cidt , t + dt) = fi + ∆i (f − f eq)ρ =
∑i fi
ρu =∑
i fici
P. Neumann: LB@TUM - Outtakes
Eindhoven, 28.02.2011 8
Technische Universitat Munchen
Grid structure
Fig.3: Cartesian Grid for hybrid LB-NS simulations.
Cells:Dark blue - NS regionRed - LB overlapYellow - NS overlapLight blue - LB regionVertices:Dark blue - NS or LB regionRed - LB “boundary”Green - NS “boundary”
Location of degrees of freedom:• Cell centered: f1, ..., fQ , ρLB, uLB, pNS
• Vertices: uNS
P. Neumann: LB@TUM - Outtakes
Eindhoven, 28.02.2011 9
Technische Universitat Munchen
Transfer: LB→ NS
• LB→ NS: f1, ..., fQ → uNS
• Note: No need for pressure coupling
Thus:1 Compute ρLB, ρLB · uLB ⇒ uLB
2 Scale uLB to NS scale (→ dx ,dt)⇒ uLBscaled
3 Interpolate uNS from uLBscaled
P. Neumann: LB@TUM - Outtakes
Eindhoven, 28.02.2011 10
Technische Universitat Munchen
Transfer: NS→ LB
• NS→ LB: uNS,pNS → f1, ..., fQpNS ∈ R,uNS ∈ RD, D space dimension
• Problem: Q > D + 1• Basic idea: fi (x , t) = f eq
i (x , t) + f neqi (x , t)
• f eqi = f eq
i (ρLB,uLB)
1 Interpolate and scale uNS at cell centers⇒ uLB
2 Scale pNS ⇒ pNSscaled
3 Problem: pNSscaled = pLB + pofs
→ Choose pofs := 1N
N∑n=1
pNSscaledn ,
pNSscaledn pressure on interface layer between LB and NS domain (cf. [1])
4 Compute pLB and ρLB := pLB
c2s⇒ f eq
i
• Problem: Where do we get f neqi from?
P. Neumann: LB@TUM - Outtakes
Eindhoven, 28.02.2011 11
Technische Universitat Munchen
Transfer: Construction of f neq (1)
• Our approach: Choose f neqi as small as possible, such that
mass, momentum and stresses are conserved at theinterface
Minimise g(f neq) : RQ → R, s.t.∑i
f neqi = 0 mass, 1 equation
∑i
f neqi ciα = 0 momentum, D equations
∑i
f neqi ciαciβ = − 2
2− 1τ
ν(
∂uβ
∂xα+ ∂uα
∂xβ
)stresses,
D(D+1)/2 equations
with (lattice) viscosity ν
P. Neumann: LB@TUM - Outtakes
Eindhoven, 28.02.2011 12
Technische Universitat Munchen
Transfer: Construction of f neq (2)
Possible approaches:
1 g(f neq) := ‖f neq‖22 =
Q∑i=1
f neq2
i
2 From [2]: (Local) Knudsen number Kn =∣∣∣ f neq
f eq
∣∣∣ << 1
→ g(f neq) :=∥∥∥ f neq
f eq
∥∥∥2
2=
Q∑i=1
(f neqif eqi
)2
3 Approximate approach no. 2 in the const-density-zero-velocitylimit, that is f eq
i ≈ wi , wi lattice weights
P. Neumann: LB@TUM - Outtakes
Eindhoven, 28.02.2011 13
Technische Universitat Munchen
Poiseuille flow Re= 1 (1)
Fig.4: Velocity profile in the middle of the channel (40 × 40 cells, LB region:
16 × 16 cells), adapted velocity.
Fig.5: Velocity profile in the middle of the channel (40 × 40 cells, LB region:
16 × 16 cells), adapted viscosity.
P. Neumann: LB@TUM - Outtakes
Eindhoven, 28.02.2011 14
Technische Universitat Munchen
Poiseuille flow Re=1 (2)
Fig.6: Error ‖u(t + dt) − u(t)‖2 vs. timesteps
P. Neumann: LB@TUM - Outtakes
Eindhoven, 28.02.2011 15
Technische Universitat Munchen
Thank you!
P. Neumann: LB@TUM - Outtakes
Eindhoven, 28.02.2011 16
Technische Universitat Munchen
References
[1] J. Latt, B. Chopard, and P. Albuquerque.Spatial coupling of a lattice boltzmann fluid model with a finite difference navier-stokes solver.2005.http://www.citebase.org/abstract?id=oai:arXiv.org:physics/0511243.
[2] S. Succi.Modern particle methods for complex flow simulation.Presentation, RealityGrid Annual Workshop 2003.
P. Neumann: LB@TUM - Outtakes
Eindhoven, 28.02.2011 17