Recent developments for the multigrid scheme of the DLR ...ossanworld.com/.../NIACFDS2013-04-09_Axel_TAUmultigrid.pdf2013/04/09 · Recent developments for the multigrid scheme of

Recent developments for the multigrid scheme of the DLR TAU-Code

www.DLR.de • Chart 1 > 21st NIA CFD Seminar > Axel Schwöppe • Recent developments for the multigrid scheme of the DLR TAU-Code > Apr 11, 2013

Axel Schwöppe Institute of Aerodynamics and Flow Technology Center of Computer Applications in Aerospace Science and Engineering - C2A2S2E

Content

• Introduction (repeat) • Investigated multigrid components • Coarse grid discretization • Semi-coarsening • Prolongation • Order of fine grid turbulence equation • Summary and open questions


Repeat: Difficulties using TAU multigrid • Stall of residual after several

iteration steps • Inaccurate coefficients • Default agglomeration does not

take account of semi-coarsening or line-coarsening






Stefan Langer




• Full multigrid does not provide a sufficient start solution

• Some cases need a smaller CFL number and/or more artificial dissipation for multigrid than for singlegrid to converge


Repeat: Questions

Where does the improvement come from? • Aggregation = Galerkin projection • Line-coarsening • Retain fine grid geometry

Are there other points for improvement? • Coarse grid discretization • Order of prolongation


Fine grid cells Agglomerated cell

Stefan Langer

1. Inject coarse grid values to finest grid level

2. Compute fluxes over all edges 3. Coarse grid residual = sum over

boundary edges of fused cells

Repeat: Questions

Where does the improvement come from? • Aggregation = Galerkin projection • Line-coarsening • Retain fine grid geometry

Are there other points for improvement? • Coarse grid discretization • Order of prolongation


Test case: 2D Zero Pressure Gradient Flat Plate

• Turbulence Modeling Resource: http://turbmodels.larc.nasa.gov

• 3 finest quadrilateral grids • 137x97 • 273x193 • 545x385

• TAU singlegrid converges on each

grid • TAU multigrid converges

sometimes using adjusted parameter setting dependent on grid level


http://turbmodels.larc.nasa.gov/

Investigated multigrid components

Implemented/tested • Semi-coarsening • Fixed first cell layer at wall on coarse meshes • Linear interpolation for prolongation • Face-Tangent and Edge-Normal discretization of coarse diffusive terms • Time step control (linear system of relaxation scheme) • Limiting of corrections of main and turbulence updates

Pitfalls • Inconsistent low Mach number preconditioning for LUSGS • SGS relaxation solver can fail in parallel mode


Check of coarse grid discretization again

Convective terms • Central scheme with 1st order dissipation • 1st order upwind schemes

Viscous terms • Averaged gradients without correction

Gradient construction • Green-Gauss

Turbulence equations • (Fully coupled)


( )jiij

ijij

ijijijijij

WWW

ex

UUeWWW

∇+∇=∇

∆

−−⋅∇−∇=∇

21

Check of coarse grid discretization again


Viscous terms • Averaged gradients without correction


Turbulence equations • (Fully coupled without sources)


( )jiij

ijij

ijijijijij

WWW

ex

UUeWWW

∇+∇=∇

∆

−−⋅∇−∇=∇

21

Coarse grid discretization components Test case: flat plate


Fine grid Coarse grid

GC viscous fluxes

GC turb. diffusion

Turbulent Sources

GC viscous fluxes

GC turb. diffusion

Solution process

1 1 1 0 0 0 stalls

2 1 1 1 1 1 converges


4 1 1 0 0 1 stalls

5 1 1 0 1 0 stalls

6 1 1 0 1 1 stalls

7 1 1 1 0 0 stalls


9 1 1 1 1 0 stalls

10 0 0 0 0 0 stalls


Coarse grid discretization components Test case: 2D Zero Pressure Gradient Flat Plate • Row 1

Current TAU discretization

• Row 2 Consistent fine and coarse grid discretization

• Row 3 Without face gradient correction for fine and coarse grid discretization


Corrected coarse grid discretization


Viscous terms • Averaged gradients + face-tangent or edge-normal argumentation


Turbulence equations • Fully coupled + sources


Test case: 2D Zero Pressure Gradient Flat Plate Corrected coarse grid discretization • Turbulence Modeling Resource:

http://turbmodels.larc.nasa.gov • 3 finest quadrilateral grids

• 137x97 • 273x193 • 545x385

• TAU multigrid converges using

same parameter setting on each grid level


http://turbmodels.larc.nasa.gov/

Intermediate results

Corrected coarse grid discretization • Seems to be much more stable

• Test case flat plate: same parameter setting on each grid level • 3D test cases: same CFL-number as for singlegrid can be used • Same artificial dissipation levels can be used for single and multigrid

• Full multigrid provides much better start solution

Additional questions • Quality of coarse grid gradients (Green Gauss, Least Squares, …) • Agglomeration: cell centers of coarse grid cells






Additional questions • Quality of coarse grid gradients (Green Gauss, Least Squares, …) • Agglomeration: centers of coarse grid cells


Semi-coarsening of TAU Schematic • A type of semi-coarsening for the

advancing front algorithm • Used in structured grid parts

(hexahedrons, prisms) • Controlled by parameter 𝑠

• From free neighbors select the neighbor whose facet fulfill:


b 1 0 2

𝑚𝑚𝑚 𝑚0,𝑏0 ∙ 𝑠 < 𝑚1,𝑏2

free neighbor seeding volume already fused

a

Semi-coarsening of TAU Test case: flat plate


Grid level 1

Semi-coarsening off Semi-coarsening 0.5



Grid level 2




Grid level 3


Semi-coarsening of TAU Test case: flat plate • Obvious improvement of

converges

• 𝑠 has influence on • thickness of ‘semi-coarsening’

in boundary layer • number of coarse grid cells


Structured grid coarsening of TAU Test case: flat plate • Obvious improvement of

converges by semi-coarsening

• 𝑠 has influence on • thickness of ‘semi-coarsening’

in boundary layer • number of coarse grid cells

and thus on runtime


Prolongation Test case: flat plate

Coarse grid corrections • Added to finer grid by injection

(constant) • Smoothed using an explicit

Laplacian type smoother

Tested • Linear interpolation using triangle

interpolation


Order of fine grid turbulence equation Test case: RAE2822

Convective terms of fine grid discretization • Main equations : 2nd order • Turbulence equation: 1st / 2nd order

Convective terms of coarse grid discretization • Main equations : 1st order • Turbulence equation: 1st order


3D test case: NASA trap wing configuration (High Lift Prediction Workshop) Grid • Structured coarse mesh (by JAXA) • 12 million points

Flow field • AoA = 13 • Ma = 0.2 • Re = 4.3e6




• Singlegrid converges




• 3w stalls




• Turbulence equation of 2v stalls


Summary

Coarse grid discretization • Sources of turbulence equation are required • Face gradient: Average of gradients requires correction (FT or EN) Agglomeration • Some type of semi-coarsening (line-coarsening) is very helpful Prolongation • Order of interpolation influences convergence Fine/coarse grid discretization • Order of fine grid turbulence equation influences multigrid convergence

There are still many open questions


Open questions

Influence of • Retaining fine grid geometry • Order of prolongation (3D) • Gradients of coarse grid discretization • Order of turbulence equations Benefit of ‘Galerkin-Projection’ • Retains fine grid geometry • Does not need new location of coarse cell centers • Does not support linear interpolation for prolongation Efficiency • Cycle strategy • Multigrid vs. singlegrid


Recent developments for the multigrid scheme of�the DLR TAU-CodeContentRepeat: Difficulties using TAU multigrid�Repeat: Difficulties using TAU multigrid�Repeat: Difficulties using TAU multigrid�Repeat: Difficulties using TAU multigrid�Repeat: QuestionsRepeat: QuestionsTest case: 2D Zero Pressure Gradient Flat PlateInvestigated multigrid componentsCheck of coarse grid discretization againCheck of coarse grid discretization againCoarse grid discretization components�Test case: flat plateCoarse grid discretization components �Test case: 2D Zero Pressure Gradient Flat PlateCorrected coarse grid discretizationTest case: 2D Zero Pressure Gradient Flat Plate�Corrected coarse grid discretizationIntermediate resultsIntermediate resultsIntermediate resultsIntermediate resultsIntermediate resultsIntermediate resultsSemi-coarsening of TAU�SchematicSemi-coarsening of TAU�Test case: flat plateSemi-coarsening of TAU�Test case: flat plateSemi-coarsening of TAU�Test case: flat plateSemi-coarsening of TAU�Test case: flat plateStructured grid coarsening of TAU�Test case: flat plateProlongation�Test case: flat plateOrder of fine grid turbulence equation�Test case: RAE28223D test case: NASA trap wing configuration�(High Lift Prediction Workshop)3D test case: NASA trap wing configuration�(High Lift Prediction Workshop)3D test case: NASA trap wing configuration�(High Lift Prediction Workshop)3D test case: NASA trap wing configuration�(High Lift Prediction Workshop)SummaryOpen questions

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Recent developments for the multigrid scheme of the DLR ...ossanworld.com/.../NIACFDS2013-04-09_Axel_TAUmultigrid.pdf2013/04/09 · Recent developments for the multigrid scheme of