C1.3 Flow over a NACA0012 Airfoil

University of Michigan XFlow GroupCorrespondence: Marco Ceze, mceze@umich.edu PI: Krzysztof Fidkowski

1. Code description

XFlow is a high-order discontinuous Galerkin (DG) finite element solver written in ANSI C, intended tobe run on Linux-type platforms. Relevant supported equation sets include compressible Euler, Navier-Stokes, and RANS with the Spalart-Allmaras model. High-order is achieved compactly within elementsusing various high-order bases on triangles, tetrahedra, quadrilaterals, and hexahedra. Parallel runs aresupported using domain partitioning and MPI communication. Visual post-processing is performedwith an in-house plotter and with TecPlot. Output-based hp-adaptivity is available using discreteadjoints.

2. Case summary

A physicality-constrained pseudo-transient continuation method [2] was used to converge the initial,p = 1, solution initialized by uniform flow at M∞ = 0.2, Re = 9 × 106, and α = 16◦. Line-preconditioned GMRES was used as the linear solver in pseudo-transient backward Euler nonlinearsteps. Output convergence is sought via hp adaptation [1].

Runs were performed on the Flux supercomputing clusters at the University of Michigan. Theruns used 12 cores and the mesh was partitioned using a node-edge weighted partitioning method [2].On one core of the Flux machine, one TauBench unit is equivalent to 9.08 seconds of compute time.

The CPU time expressed in terms of work units in the figures below is cumulative and, hence, itaccounts for the total time taken from the beginning of the calculation. These time stamps includethe time taken for the primal and dual solves, for error estimation, and for mesh adaptation. On onecore of the Nyx machine, one TauBench unit is equivalent to 16.5 seconds of compute time.

3. Meshes

The initial mesh used in the adaptive runs has 720 quartic quadrilaterals. This mesh was generatedagglomerating 4x4 patches of linear elements from a multiblock mesh. The outer boundary is located1000 chord-lengths away from the body as requested in the case specification. Figure 1 shows a zoomand a global view of the mesh.

4. Results

The figures below present the results requested for all three conditions in this test case.For the subsonic inviscid case, output errors were obtained relative to truth solutions computed

from adjoint-based h-adaptive runs using p = 2 on the ref3 mesh as a starting point. Note in Figure2 the reference lift value may not be adequate for the runs presented here because both lift-based runsseem to reach an error plateau of ∼ 4×10−4. For the other runs, we did not compute reference values,hence, output values are report rather than their errors.

References

[1] Marco Ceze and Krzysztof J. Fidkowski. Anisotropic hp-adaptation framework for functionalprediction. AIAA Journal, 51(2):492–509, February 2013.

2013 High-Order CFD Workshop 1 University of Michigan XFlow Group

Figure 1: Initial quartic mesh (720 elements).

[2] Marco Antonio de Barros Ceze. A Robust hp-Adaptation Method for Discontinuous Galerkin Dis-cretizations Applied to Aerodynamic Flows. PhD thesis, The University of Michigan, 2013.

2013 High-Order CFD Workshop 2 University of Michigan XFlow Group

10−2

10−4

10−3

10−2

10−1

1/nDOF1/2

|Lift

Co

eff

icie

nt

err

or|

truth Cl = 2.864792e−01

Lift−based Anisotropic hp (p = 1−7)Lift−based Isotropic h (p = 1)

Drag−based Anisotropic hp (p = 1−7)Drag−based Isotropic h (p = 1)

(a) Lift convergence with respect to DOF’s

100

101

102

103

104

10−4

10−3

10−2

10−1

Work Units

|Lift

Co

eff

icie

nt

err

or|

truth Cl = 2.864792e−01

(b) Lift convergence with respect to workunits

10−2

10−6

10−5

10−4

10−3

10−2

1/nDOF1/2

|Dra

g C

oe

ffic

ien

t e

rro

r|

truth Cd = 2.140579e−06

Lift−based Anisotropic hp (p = 1−7)Lift−based Isotropic h (p = 1)

Drag−based Anisotropic hp (p = 1−7)Drag−based Isotropic h (p = 1)

(c) Drag convergence with respect to DOF’s

100

101

102

103

104

10−6

10−5

10−4

10−3

10−2

Work Units

|Dra

g C

oe

ffic

ien

t e

rro

r|

truth Cd = 2.140579e−06

(d) Drag convergence with respect to workunits

Figure 2: M = 0.5, α = 2o, inviscid: lift and drag coefficient histories for lift and drag basedadaptations. Dashed lines correspond to output correction by its error estimate.

2013 High-Order CFD Workshop 3 University of Michigan XFlow Group

10−3

10−2

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

1/nDOF1/2

Lift

Co

eff

icie

nt

Lift−based Anisotropic hp (p = 1−7)Lift−based Isotropic h (p = 1)

Drag−based Anisotropic hp (p = 1−7)Drag−based Isotropic h (p = 1)

(a) Lift convergence with respect to DOF’s

101

102

103

104

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Work Units

Lift

Co

eff

icie

nt

(b) Lift convergence with respect to workunits

10−3

10−2

0.05

0.055

0.06

0.065

0.07

0.075

0.08

1/nDOF1/2

Dra

g C

oe

ffic

ien

t

Lift−based Anisotropic hp (p = 1−7)Lift−based Isotropic h (p = 1)

Drag−based Anisotropic hp (p = 1−7)Drag−based Isotropic h (p = 1)

(c) Drag convergence with respect to DOF’s

101

102

103

104

0.05

0.055

0.06

0.065

0.07

0.075

0.08

Work Units

Dra

g C

oe

ffic

ien

t

(d) Drag convergence with respect to workunits

Figure 3: M = 0.5, α = 1o, Re = 5, 000: lift and drag coefficient histories for lift and drag basedadaptations. Dashed lines correspond to output correction by its error estimate.

2013 High-Order CFD Workshop 4 University of Michigan XFlow Group

10−2

0.29

0.3

0.31

0.32

0.33

0.34

0.35

0.36

1/nDOF1/2

Lift C

oeffic

ient

(a) Lift convergence with respect to DOF’s

101

102

103

104

0.29

0.3

0.31

0.32

0.33

0.34

0.35

0.36

Work Units

Lift

Co

eff

icie

nt

Lift−based Anisotropic hp (p = 1−3)Lift−based Isotropic h (p = 1)

Drag−based Anisotropic hp (p = 1−3)Drag−based Isotropic h (p = 1)

(b) Lift convergence with respect to workunits

10−2

0.022

0.023

0.024

0.025

0.026

0.027

0.028

1/nDOF1/2

Dra

g C

oe

ffic

ien

t

(c) Drag convergence with respect to DOF’s

101

102

103

104

0.022

0.023

0.024

0.025

0.026

0.027

0.028

Work Units

Dra

g C

oe

ffic

ien

t

Lift−based Anisotropic hp (p = 1−3)Lift−based Isotropic h (p = 1)

Drag−based Anisotropic hp (p = 1−3)Drag−based Isotropic h (p = 1)

(d) Drag convergence with respect to workunits

Figure 4: M = 0.8, α = 1.25o, inviscid: lift and drag coefficient histories for lift and drag basedadaptations.

2013 High-Order CFD Workshop 5 University of Michigan XFlow Group

Case 1.3: Flow Over a NACA 0012 Airfoil

University of Michigan - XFlow

Department of Aerospace EngineeringUniversity of Michigan

UofM 2nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C1.3 1/32

XFlow Code

Discontinuous Galerkin spatial discretization.Physicality-constrained pseudo-transient continuation (CPTC) fortime integration.Exact Jacobian with element-line-Jacobi preconditioner andGMRES linear solver.Roe solver for inviscid flux and BR2 for viscous discretization.MPI parallelization.Node-edge weighted mesh partitioning.Support for curved meshes.Oliver’s version of Spalart-Allmaras turbulence model.Shock-capturing via element-wise constant artificial viscosity.Output-based anisotropic hp-adaptation.

UofM 2nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C1.3 2/32

Initial Mesh720 quartic elements generated via agglomeration.Outer boundary located 1000 chord-lengths away from airfoil.

UofM 2nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C1.3 3/32

Mesh Improvement MethodThe output error is approximated as:

δJ ≈ −∑

κH∈T H

RκH (uH,p,ψH,p+1 − ψH,p).

The adaptive indicator is defined as:

ηκH =∣∣∣RκH (uH,p,ψH,p+1 − ψH,p)

∣∣∣.ψH,p+1 is approximated by 15 lean-Jacobi smoothing iterations.f adapt = 10% of the elements with the highest ηκH are refined ateach step.A set of discrete refinement options is considered, e.g.:

pp

(a) x

p

p

(b) y

p

p p

p

(c) xy

p+1

(d) p

UofM 2nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C1.3 4/32

hp-AdaptationQuestion:Which direction should we refine?

Our proposition:Rank the refinement options based on a function:

m(i) =b(i)c(i)

c(i) is a measure of the computational cost of refinement option i(e.g. non-zeros in the Jacobian).b(i) measures the gain in accuracy due to the refinement option i(e.g. output sensitivity to residual perturbations).Balance between high-cost-low-error and low-cost-high-erroroptionsChoose the option with the highest m(i).

UofM 2nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C1.3 5/32

Residual ConvergenceCase (a): Subsonic inviscid flow:

M = 0.5, α = 2o, inviscid.

0 50 100 150 200 250 30010

−15

10−10

10−5

100

105

Nonlinear iteration

|R|

Drag−based Anisotropic hp

Drag−based Isotropic h (p=1)

Lift−based Anisotropic hp

Lift−based Isotropic h (p=1)

UofM 2nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C1.3 6/32

Drag Convergence versus DOF

Case (a): Subsonic inviscid flow:M = 0.5, α = 2o, inviscid.Dashed lines correspond to output correction by its error estimate.

10−2

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

1/nDOF1/2

Dra

g C

oe

ffic

ien

t

Lift−based Anisotropic hp (p = 1−7)

Lift−based Isotropic h (p = 1)

Drag−based Anisotropic hp (p = 1−7)

Drag−based Isotropic h (p = 1)

Global

10−2

0

1

2x 10

−4

1/nDOF1/2

Dra

g C

oe

ffic

ien

t

Zoom

UofM 2nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C1.3 7/32

Drag Convergence versus WorkunitsCase (a): Subsonic inviscid flow:

M = 0.5, α = 2o, inviscid.Dashed lines correspond to output correction by its error estimate.Runs used 12 cores (1 node) of UM’s Flux machine (wu = 9.08s).

100

101

102

103

104

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

Work Units

Dra

g C

oeffic

ient

Lift−based Anisotropic hp (p = 1−7)

Lift−based Isotropic h (p = 1)

Drag−based Anisotropic hp (p = 1−7)

Drag−based Isotropic h (p = 1)

Global

101

102

103

104

0

1

2x 10

−4

Work Units

Dra

g C

oeffic

ient

ZoomUofM 2nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C1.3 8/32

Lift Convergence versus DOF

Case (a): Subsonic inviscid flow:M = 0.5, α = 2o, inviscid.Dashed lines correspond to output correction by its error estimate.

10−2

0.26

0.265

0.27

0.275

0.28

0.285

0.29

0.295

0.3

1/nDOF1/2

Lift

Co

eff

icie

nt

Lift−based Anisotropic hp (p = 1−7)

Lift−based Isotropic h (p = 1)

Drag−based Anisotropic hp (p = 1−7)

Drag−based Isotropic h (p = 1)

Global

10−2

0.283

0.2835

0.284

0.2845

0.285

0.2855

0.286

0.2865

1/nDOF1/2

Lift

Co

eff

icie

nt

Zoom

UofM 2nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C1.3 9/32

Lift Convergence versus WorkunitsCase (a): Subsonic inviscid flow:

M = 0.5, α = 2o, inviscid.Dashed lines correspond to output correction by its error estimate.Runs used 12 cores (1 node) of UM’s Flux machine (wu = 9.08s).

100

101

102

103

104

0.26

0.265

0.27

0.275

0.28

0.285

0.29

0.295

0.3

Work Units

Lift C

oeffic

ient

Lift−based Anisotropic hp (p = 1−7)

Lift−based Isotropic h (p = 1)

Drag−based Anisotropic hp (p = 1−7)

Drag−based Isotropic h (p = 1)

Global

101

102

103

104

0.283

0.2835

0.284

0.2845

0.285

0.2855

0.286

0.2865

Work Units

Lift

Co

eff

icie

nt

Zoom

UofM 2nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C1.3 10/32

Drag-adapted MeshesCase (a): Subsonic inviscid flow:

M = 0.5, α = 2o, inviscid.Mach contour lines.

Drag-based aniso-hp Drag-based iso-h (p = 1)

UofM 2nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C1.3 11/32

Lift-adapted MeshesCase (a): Subsonic inviscid flow:

M = 0.5, α = 2o, inviscid.Mach contour lines.

Lift-based aniso-hp Lift-based iso-h (p = 1)

UofM 2nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C1.3 12/32

Drag-based hp adaptation statisticsCase (a): Subsonic inviscid flow:

Refinement optionsAdaptive step iso-h sc-h iso-p

1 0.000 6.429 × 10−1 3.571 × 10−1

2 0.000 5.714 × 10−1 4.286 × 10−1

3 0.000 6.000 × 10−1 4.000 × 10−1

4 0.000 5.882 × 10−1 4.118 × 10−1

5 0.000 8.333 × 10−1 1.667 × 10−1

6 0.000 7.000 × 10−1 3.000 × 10−1

7 0.000 8.571 × 10−1 1.429 × 10−1

8 0.000 7.826 × 10−1 2.174 × 10−1

9 0.000 8.800 × 10−1 1.200 × 10−1

10 3.571 × 10−2 7.857 × 10−1 1.786 × 10−1

11 0.000 9.355 × 10−1 6.452 × 10−2

12 2.857 × 10−2 8.286 × 10−1 1.429 × 10−1

13 0.000 8.462 × 10−1 1.538 × 10−1

UofM 2nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C1.3 13/32

Lift-based hp adaptation statisticsCase (a): Subsonic inviscid flow:

Refinement optionsAdaptive step iso-h sc-h iso-p

1 0.000 2.857 × 10−1 7.143 × 10−1

2 0.000 5.000 × 10−1 5.000 × 10−1

3 0.000 6.667 × 10−1 3.333 × 10−1

4 0.000 4.375 × 10−1 5.625 × 10−1

5 0.000 8.235 × 10−1 1.765 × 10−1

6 1.111 × 10−1 5.556 × 10−1 3.333 × 10−1

7 1.000 × 10−1 8.500 × 10−1 5.000 × 10−2

8 0.000 8.696 × 10−1 1.304 × 10−1

9 0.000 7.200 × 10−1 2.800 × 10−1

10 3.704 × 10−2 8.148 × 10−1 1.481 × 10−1

11 3.226 × 10−2 9.677 × 10−1 0.00012 0.000 8.857 × 10−1 1.143 × 10−1

13 0.000 8.205 × 10−1 1.795 × 10−1

UofM 2nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C1.3 14/32

Residual ConvergenceCase (b): Transonic inviscid flow:

M = 0.8, α = 1.25o, inviscid.

0 500 1000 1500 2000 250010

−15

10−10

10−5

100

105

Nonlinear iteration

|R|

Drag−based Anisotropic hp

Drag−based Isotropic h (p=1)

Lift−based Anisotropic hp

Lift−based Isotropic h (p=1)

UofM 2nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C1.3 15/32

Drag Convergence versus DOF

Case (b): Transonic inviscid flow:M = 0.8, α = 1.25o, inviscid.Dashed lines correspond to output correction by its error estimate.

10−2

0.022

0.023

0.024

0.025

0.026

0.027

0.028

1/nDOF1/2

Dra

g C

oe

ffic

ien

t

Lift−based Anisotropic hp (p = 1−3)

Lift−based Isotropic h (p = 1)

Drag−based Anisotropic hp (p = 1−3)

Drag−based Isotropic h (p = 1)

Global

10−2

0.022

0.0222

0.0224

0.0226

0.0228

0.023

1/nDOF1/2

Dra

g C

oe

ffic

ien

t

Zoom

UofM 2nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C1.3 16/32

Drag Convergence versus WorkunitsCase (b): Transonic inviscid flow:

M = 0.8, α = 1.25o, inviscid.Dashed lines correspond to output correction by its error estimate.Runs used 12 cores (1 node) of UM’s Flux machine (wu = 9.08s).

101

102

103

104

0.022

0.023

0.024

0.025

0.026

0.027

0.028

Work Units

Dra

g C

oeffic

ient

Lift−based Anisotropic hp (p = 1−3)

Lift−based Isotropic h (p = 1)

Drag−based Anisotropic hp (p = 1−3)

Drag−based Isotropic h (p = 1)

Global

101

102

103

104

0.022

0.0222

0.0224

0.0226

0.0228

0.023

Work Units

Dra

g C

oe

ffic

ien

t

Zoom

UofM 2nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C1.3 17/32

Lift Convergence versus DOF

Case (b): Transonic inviscid flow:M = 0.8, α = 1.25o, inviscid.Dashed lines correspond to output correction by its error estimate.

10−2

0.29

0.3

0.31

0.32

0.33

0.34

0.35

0.36

1/nDOF1/2

Lift

Co

eff

icie

nt

Lift−based Anisotropic hp (p = 1−3)

Lift−based Isotropic h (p = 1)

Drag−based Anisotropic hp (p = 1−3)

Drag−based Isotropic h (p = 1)

Global

10−2

0.33

0.335

0.34

0.345

0.35

1/nDOF1/2

Lift

Co

eff

icie

nt

Zoom

UofM 2nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C1.3 18/32

Lift Convergence versus WorkunitsCase (b): Transonic inviscid flow:

M = 0.8, α = 1.25o, inviscid.Dashed lines correspond to output correction by its error estimate.Runs used 12 cores (1 node) of UM’s Flux machine (wu = 9.08s).

101

102

103

104

0.29

0.3

0.31

0.32

0.33

0.34

0.35

0.36

Work Units

Lift

Co

eff

icie

nt

Lift−based Anisotropic hp (p = 1−3)

Lift−based Isotropic h (p = 1)

Drag−based Anisotropic hp (p = 1−3)

Drag−based Isotropic h (p = 1)

Global

101

102

103

104

0.33

0.335

0.34

0.345

0.35

Work Units

Lift C

oeffic

ient

Zoom

UofM 2nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C1.3 19/32

Drag-adapted MeshesCase (b): Transonic inviscid flow:

M = 0.8, α = 1.25o, inviscid.Mach contour lines.

Drag-based aniso-hp Drag-based iso-h (p = 1)

UofM 2nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C1.3 20/32

Lift-adapted MeshesCase (b): Transonic inviscid flow:

M = 0.8, α = 1.25o, inviscid.Mach contour lines.

Lift-based aniso-hp Lift-based iso-h (p = 1)

UofM 2nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C1.3 21/32

Drag-based hp adaptation statisticsCase (b): Transonic inviscid flow:

Refinement optionsAdaptive step iso-h sc-h iso-p

1 0.000 7.857 × 10−1 2.143 × 10−1

2 0.000 8.667 × 10−1 1.333 × 10−1

3 0.000 6.250 × 10−1 3.750 × 10−1

4 0.000 6.667 × 10−1 3.333 × 10−1

5 0.000 5.500 × 10−1 4.500 × 10−1

6 0.000 7.727 × 10−1 2.273 × 10−1

7 0.000 6.154 × 10−1 3.846 × 10−1

8 0.000 6.897 × 10−1 3.103 × 10−1

9 0.000 9.412 × 10−1 5.882 × 10−2

10 2.439 × 10−2 9.268 × 10−1 4.878 × 10−2

11 0.000 9.423 × 10−1 5.769 × 10−2

UofM 2nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C1.3 22/32

Lift-based hp adaptation statisticsCase (b): Transonic inviscid flow:

Refinement optionsAdaptive step iso-h sc-h iso-p

1 0.000 6.429 × 10−1 3.571 × 10−1

2 0.000 4.286 × 10−1 5.714 × 10−1

3 0.000 4.667 × 10−1 5.333 × 10−1

4 0.000 5.625 × 10−1 4.375 × 10−1

5 0.000 4.706 × 10−1 5.294 × 10−1

6 0.000 6.111 × 10−1 3.889 × 10−1

7 0.000 6.316 × 10−1 3.684 × 10−1

8 4.762 × 10−2 5.714 × 10−1 3.810 × 10−1

9 8.696 × 10−2 8.261 × 10−1 8.696 × 10−2

10 0.000 8.462 × 10−1 1.538 × 10−1

11 0.000 8.214 × 10−1 1.786 × 10−1

12 3.226 × 10−2 6.452 × 10−1 3.226 × 10−1

13 5.714 × 10−2 7.143 × 10−1 2.286 × 10−1

UofM 2nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C1.3 23/32

Residual ConvergenceCase (c): Subsonic viscous flow:

M = 0.5, α = 1o, Re = 5,000.

0 50 100 150 200 250 300 35010

−15

10−10

10−5

100

105

Nonlinear iteration

|R|

Drag−based Anisotropic hp

Drag−based Isotropic h (p=1)

Lift−based Anisotropic hp

Lift−based Isotropic h (p=1)

UofM 2nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C1.3 24/32

Drag Convergence versus DOF

Case (c): Subsonic viscous flow:M = 0.5, α = 1o, Re = 5,000.Dashed lines correspond to output correction by its error estimate.

10−3

10−2

0.05

0.055

0.06

0.065

0.07

0.075

0.08

1/nDOF1/2

Dra

g C

oe

ffic

ien

t

Lift−based Anisotropic hp (p = 1−7)

Lift−based Isotropic h (p = 1)

Drag−based Anisotropic hp (p = 1−7)

Drag−based Isotropic h (p = 1)

Global

10−3

10−2

0.055

0.0555

0.056

0.0565

1/nDOF1/2

Dra

g C

oe

ffic

ien

t

Zoom

UofM 2nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C1.3 25/32

Drag Convergence versus WorkunitsCase (c): Subsonic viscous flow:

M = 0.5, α = 1o, Re = 5,000.Dashed lines correspond to output correction by its error estimate.Runs used 12 cores (1 node) of UM’s Flux machine (wu = 9.08s).

101

102

103

104

0.05

0.055

0.06

0.065

0.07

0.075

0.08

Work Units

Dra

g C

oeffic

ient

Lift−based Anisotropic hp (p = 1−7)

Lift−based Isotropic h (p = 1)

Drag−based Anisotropic hp (p = 1−7)

Drag−based Isotropic h (p = 1)

Global

101

102

103

104

0.055

0.0555

0.056

0.0565

Work Units

Dra

g C

oe

ffic

ien

t

Zoom

UofM 2nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C1.3 26/32

Lift Convergence versus DOF

Case (c): Subsonic viscous flow:M = 0.5, α = 1o, Re = 5,000.Dashed lines correspond to output correction by its error estimate.

10−3

10−2

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

1/nDOF1/2

Lift C

oeffic

ient

Lift−based Anisotropic hp (p = 1−7)

Lift−based Isotropic h (p = 1)

Drag−based Anisotropic hp (p = 1−7)

Drag−based Isotropic h (p = 1)

Global

10−3

10−2

0.018

0.0185

0.019

0.0195

0.02

0.0205

0.021

0.0215

0.022

1/nDOF1/2

Lift

Co

eff

icie

nt

Zoom

UofM 2nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C1.3 27/32

Lift Convergence versus WorkunitsCase (c): Subsonic viscous flow:

M = 0.5, α = 1o, Re = 5,000.Dashed lines correspond to output correction by its error estimate.Runs used 12 cores (1 node) of UM’s Flux machine (wu = 9.08s).

101

102

103

104

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Work Units

Lift C

oeffic

ient

Lift−based Anisotropic hp (p = 1−7)

Lift−based Isotropic h (p = 1)

Drag−based Anisotropic hp (p = 1−7)

Drag−based Isotropic h (p = 1)

Global

101

102

103

104

0.018

0.0185

0.019

0.0195

0.02

0.0205

0.021

0.0215

0.022

Work Units

Lift

Co

eff

icie

nt

Zoom

UofM 2nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C1.3 28/32

Drag-adapted MeshesCase (c): Subsonic viscous flow:

M = 0.5, α = 1o, Re = 5,000.Mach contour lines.

Drag-based aniso-hp Drag-based iso-h (p = 1)

UofM 2nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C1.3 29/32

Lift-adapted MeshesCase (c): Subsonic viscous flow:

M = 0.5, α = 1o, Re = 5,000.Mach contour lines.

Lift-based aniso-hp Lift-based iso-h (p = 1)

UofM 2nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C1.3 30/32

Drag-based hp adaptation statisticsCase (c): Subsonic viscous flow:

Refinement optionsAdaptive step iso-h sc-h iso-p

1 0.000 8.214 × 10−1 1.786 × 10−1

2 0.000 6.500 × 10−1 3.500 × 10−1

3 0.000 8.462 × 10−1 1.538 × 10−1

4 0.000 7.887 × 10−1 2.113 × 10−1

5 0.000 8.354 × 10−1 1.646 × 10−1

6 0.000 8.046 × 10−1 1.954 × 10−1

7 0.000 8.438 × 10−1 1.562 × 10−1

8 0.000 7.757 × 10−1 2.243 × 10−1

9 0.000 8.000 × 10−1 2.000 × 10−1

10 0.000 8.120 × 10−1 1.880 × 10−1

11 0.000 7.432 × 10−1 2.568 × 10−1

12 0.000 7.636 × 10−1 2.364 × 10−1

13 0.000 8.197 × 10−1 1.803 × 10−1

UofM 2nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C1.3 31/32

Lift-based hp adaptation statisticsCase (c): Subsonic viscous flow:

Refinement optionsAdaptive step iso-h sc-h iso-p

1 0.000 6.786 × 10−1 3.214 × 10−1

2 0.000 7.966 × 10−1 2.034 × 10−1

3 0.000 8.281 × 10−1 1.719 × 10−1

4 0.000 7.714 × 10−1 2.286 × 10−1

5 0.000 7.922 × 10−1 2.078 × 10−1

6 0.000 8.235 × 10−1 1.765 × 10−1

7 1.064 × 10−2 7.766 × 10−1 2.128 × 10−1

8 0.000 8.173 × 10−1 1.827 × 10−1

9 0.000 8.000 × 10−1 2.000 × 10−1

10 0.000 7.578 × 10−1 2.422 × 10−1

11 0.000 8.369 × 10−1 1.631 × 10−1

12 6.369 × 10−3 7.452 × 10−1 2.484 × 10−1

13 5.747 × 10−3 7.529 × 10−1 2.414 × 10−1

UofM 2nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C1.3 32/32