lb@tum - outtakes - eindhoven · technische universit¨at munc¨ hen lb@tum - outtakes eindhoven...

Technische Universitat Munchen

LB@TUM - OuttakesEindhoven

Philipp Neumann

28.02.2011

P. Neumann: LB@TUM - Outtakes

Eindhoven, 28.02.2011 1


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


Eindhoven, 28.02.2011 2


Where I’m from - SCCS

• Chair of Scientific Computing, Department for Computer Science

• 34 people, Head: Prof. Bungartz


Eindhoven, 28.02.2011 3


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)


Eindhoven, 28.02.2011 4


Peano: Grid syntax

Fig.1: Peano grid.P. Neumann: LB@TUM - Outtakes

Eindhoven, 28.02.2011 5


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)


Eindhoven, 28.02.2011 6


Dynamic block adaptivity

• Integration of FSI coupling tool preCICE

• Remove/ add blocks of LB cells where necessary

• Example: Moving sphere (2D, 3 grid levels)


Eindhoven, 28.02.2011 7


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


Eindhoven, 28.02.2011 8


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


Eindhoven, 28.02.2011 9


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


Eindhoven, 28.02.2011 10


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?


Eindhoven, 28.02.2011 11


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


Eindhoven, 28.02.2011 12


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


Eindhoven, 28.02.2011 13


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.


Eindhoven, 28.02.2011 14


Poiseuille flow Re=1 (2)

Fig.6: Error ‖u(t + dt) − u(t)‖2 vs. timesteps


Eindhoven, 28.02.2011 15


Thank you!


Eindhoven, 28.02.2011 16


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.


Eindhoven, 28.02.2011 17

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