An Evaluation of the GCI for Unstructured Grids

Francisco Elizalde-BlancasDepartment of Mechanical Engineering

University of GuanajuatoSalamanca, Guanajuato, Mexico

Patrick J. RoacheConsultant

Socorro, New Mexico, U.S.A.

___We gratefully acknowledge suggestions from I. Celik, L. Eça, and M. Hoekstra

Short Answer (1 slide talk)

● V1 - GCI < VREF < V1 + GCI for ~ 95% of cases

● 3 physical parameters CD and 6 grids

● Iteration convergence criterion: all residuals < 10-8

● Observed p's were poor, ~ 1/2

● Used simple 1-parameter GCI with Fs = 1.25 (not 1.5 as recommended [VVCSE 1998])

● Structured grid cases: passed 57/57

● Unstructured grid cases: passed 38/39

Abstract

Application of the Grid Convergence Index (GCI) uncertainty estimator to non-similar sets of unstructured grids using a simple effective grid refinement ratio is evaluated for 3D simulations of fuel cells. The GCI attained the target of roughly 95% coverage, even though the observed p's were < 1, using only the Factor of Safety Fs = 1.25, standard for structured grids.

Simple 1-parameter GCI is based on generalized Richardson Extrapolation + an empirically determined Factor of Safety Fs.

Goal - an uncertainty estimate (or error bar) with roughly 95% coverage of solution numerical error.

Recommended Fs = 1.25 with grid triplet(s) giving good observed p.

Theory depends on similarity of the grid triplet(s).

Strict geometric similarity is difficult for 2D unstructured grids and not possible for 3D.

Strict geometric similarity means a grid refinement factor r12 exists, between a (finer) grid 1 and 2, and is constant in space.

There is no necessity that r be constant over a grid triplet {123}, so r12 ≠ r23 is OK.

For small departures from strict similarity we expect [VVCSE 1998] that an effective r could be defined and used to calculate a GCI for unstructured grid sets.

Some grid generation software accepts user input of spatially varying nominal element density which is targeted in the grid genertion process.

If the mesh density distribution function of one mesh is uniformly reduced everywhere by the same factor to produce a second mesh, that factor will be an approximate or effective r for the two grid set.

Can we get away with this ?

Can we get away with this ?Answer: it depends …

Some previous relevant work ...

The following evaluated GCI.

[13] Pelletier, D. and Ignat, L. (1995), “On the Accuracy of the Grid Convergence Index and the Zhu-Zienkiewicz Error Estimator”

[6] K. Dowding in Section 7.2 of ASME V&V 20-2009.

[16] Freitas, C. J., "Application of V&V 20 to the Simulation of Propagating Blast Loads in Multi-compartment Structures", in ASME Verification and Validation Symposium 2012.

Some previous relevant work …The following evaluated consistency of observed p for either

unstructured or structured grids with defects in similarity.

[17] João Filipe Pedro, et al. (2012),Boundary Elements Method for Three-dimensional Potential Flow based on Unstructured Meshes, ...

[18] Luís Eça, et a. (2012),"On the Characterization of Grid Density in Grid Refinement Studies for Discretization Error Estimation"

[19] Luís Eça, et al. (2012), "A Code Verification Exercise for a Boundary Element Method Based on (Un)structured Grids," in ASME Verification and Validation Symposium 2012.

[20] Hay, A. and Pelletier, D. (2007), “Code and Solution Verification of an Adaptive Finite Element Turbulent Flow Solver.”

[21] Celik, I. and Karatekin, O. (1997), “Numerical Experiments on Application of Richardson Extrapolation to a Turbulent Flow with Separation”

Solid Oxide Fuel Cell simulation

● SOFC fuel cell converts chemical energy of hydrogen directly to voltage

● steady state conservation equations: continuity, momentum, energy, species transport, charge

● fuel stream: humidified hydrogen (90% H2 and 10% H2O by volume)

● oxidant stream to cathode: air.● Both streams were fed at 1073 K

Solid Oxide Fuel Cell simulation

● fuel cell dimensions 1 mm x 1mm x 40 mm● height and width of fuel and air channels are

0.3 mm and 0.6 mm respectively● thickness of anode and cathode = 0.1 mm● tortuosity = 3 for both electrodes● porosities of anode = 0.38, cathode = 0.3● 150 μm thick electrolyte

Solid Oxide Fuel Cell simulation

● CFD solution gives output voltage V for input operating current density CD (or i)

● 3 values of physical parameter CD [A/cm2] = 0.0625, 0.1875, 0.3125

● Activation, ohmic, or concentration loss dominate

●

Unstructured X-Ygrid, 20x20

●

 Table 1 Voltage (V) for fuel cell simulation, structured grids.

All iterative solution residuals < 10-8

grid density Voltage at CD = 0.3125

20x20x40 0.15890732

30x30x60 0.16420840

40x40x80 0.16775820

60x60x120 0.17251435

80x80x160 0.17563072

120x120x240 0.17947632

VREF 0.202243....

 Table 2 Voltage (V) for fuel cell simulation, unstructured grids.

All iterative solution residuals < 10-8

grid density Voltage at CD = 0.3125

20x20x40 0.15869132

30x30x60 0.16414417

40x40x80 0.16770338

60x60x120 0.17249109

80x80x160 0.17560072

120x120x240 non-convergent

VREF 0.202243....

 Grid triplet notation

20 grid triplets evaluated from set of 6 grids.

“6812” => (60x60x120, 80x80x160,120x120x240)

grid density

20x20x40

30x30x60

40x40x80

60x60x120

80x80x160

120x120x240

Defining effective r's

Unstructured grid generation algorithm used cannot maintain the precise number of cells.

Effective r's were defined based on nominal grid densities on the boundaries.

This gives same effective r's for unstructured as nominal structured grid triplets.

This is possibly the most to be expected from an analyst working on a practical problem.

Defining effective r's

We also calculated effective r's based on the ratio of number of 2-D elements and compared* these results with those based on number of boundary elements. These two approaches produced agreement to <1% in 9/15 triplets and max = .52%, giving some indication that the commercial grid generation algorithm was doing a good job, at least in an average sense.

___*Suggested by L. Eça.

Table 3 (part 1/2)

Observed p for CD = 0.0625 A/cm2 (easiest physical) structured grids

Grid Triplet Observed p

234 0.5389 r12 = 1.333... r23 = 1.5

236 0.5137 r12 = 2 r23 = 1.5

238 0.5059 r12 = 2.666... r23 = 1.5

2312 0.5070 r12 = 4 r23 = 1.5

246 0.4946 r12 = 1.5 r23 = 2

248 0.4912 (0.491156) simple doubling: r12=r23=2

2412 0.4975 r12 = 3 r23 = 2

268 0.4860 r12 = 1.333... r23 = 3

2612 0.4994 r12 = 2 r23 = 3

2812 0.5097 r12 = 1.5 r23 = 4

Table 3 (part 2/2)

Observed p for CD = 0.0625 A/cm2 (easiest physical) structured grids

Grid Triplet Observed p

346 0.4662 r12 = 1.5 r23 = 1.333...

348 0.4692 r12 = 2 r23 = 1.333...

3412 0.4833 r12 = 3 r23 = 1.333...

368 0.4732 r12 = 1.333... r23 = 2

3612 0.4946 (0.494598) simple doubling: r12=r23=2

3812 0.5109 r12 = 1.5 r23 = 2.666...

468 0.4764 r12 = 1.333... r23 = 1.5

4612 0.5029 r12 = 2 r23 = 1.5

4812 0.5230 r12 = 1.5 r23 = 2

6812 0.5530 r12 = 1.5 r23 = 1.333...

At present we have no explanation for the low values of observed p.

B.C. 's ?

Singularity ?

Coupling ?

Probably not incomplete iteration. The answers did not change when tolerance was changed from 10-5 to 10-8.

6 grids provide for 20 possible grid triplets.

Extremes will result in grid refinement factors r larger than commonly used.

E.g., triplet 2312 gives r12 = 4 and r23 = 1.5

These provide a rather stringent test of the unstructured GCI concept, in our opinion.

Table 8 (part ½)Observed p for CD = 0.3125 A/cm2 (hardest physical)

unstructured grids (no convergence on grid 12)

Grid Triplet Observed p

234 0.2402 r12 = 1.333... r23 = 1.5

236 0.2021 r12 = 2 r23 = 1.5

238 0.2054 r12 = 2.666... r23 = 1.5

2312 non-convergent r12 = 4 r23 = 1.5

246 0.1746 r12 = 1.5 r23 = 2

248 0.1905 (0.190490) simple doubling: r12=r23=2

2412 non-convergent r12 = 3 r23 = 2

268 0.2136 r12 = 1.333... r23 = 3

2612 non-convergent r12 = 2 r23 = 3

2812 non-convergent r12 = 1.5 r23 = 4

Table 8 (part ½)Observed p for CD = 0.3125 A/cm2 (hardest physical)

unstructured grids (no convergence on grid 12)

Grid Triplet Observed p

346 0.1348 r12 = 1.5 r23 = 1.333...

348 0.1690 r12 = 2 r23 = 1.333...

3412 non-convergent r12 = 3 r23 = 1.333...

368 0.2186 r12 = 1.333... r23 = 2

3612 non-convergent simple doubling: r12=r23=2

3812 non-convergent r12 = 1.5 r23 = 2.666...

468 0.2544 r12 = 1.333... r23 = 1.5

4612 non-convergent r12 = 2 r23 = 1.5

4812 non-convergent r12 = 1.5 r23 = 2

6812 non-convergent r12 = 1.5 r23 = 1.333...

Convergence results

3 physical parameters CD and 6 grids

Iteration convergence criterion: all residuals < 10-8

Structured grids: all grid cases converged.

Unstructured grids: all 6 grid cases converged for CD = 0.0625, 5/6 converged for CD = 0.1875 and 0.3125

The finest unstructured grid 120x120x240 had severe element aspect ratios.

VREF calculated by Richardson Extrapolation from the (structured) finest grid triplet 6812 using the observed p.

GCI results

● Target GCI performance: for any triplet,

V1 - GCI < VREF < V1 + GCI

in roughly 95% of the cases

● Available for GCI evaluations:

96 grid triplets*

57 on structured grids

39 on unstructured grids

____

* Note the “12” grid converged solution on 4 cases (3 structured for all CD's and 1 unstructured for smallest CD) are not incuded because the definition of VREF results in automatic “pass”. If “12” had converged, we would have 120 – 4 =116.

GCI results

● V1 - GCI < VREF < V1 + GCI

● Used simple 1-parameter GI with Fs = 1.25 (not 1.5 as recommended [VVCSE 1998])

● 3 values of physical parameter

● For all cases: passed 95/96

● Structured grid cases: passed 57/57

● Unstructured grid cases: passed 38/39

GCI results

● The one case that "failed" was the

– coarsest unstructured grid triplet (234)

– for the easiest physical case (CD = 0.0625)

– with anomalously highest observed p = 0.7292.● Repeatedly checked more stringent iterative convergence

● Increasing Fs from 1.25 to 1.46 => pass

– (1.46 < recommended Fs = 1.5 for unstructured grids)● Using p = 0.63 > all other oberved p with Fs = 1.25 => pass

● Any least squares evaluation would give p < 0.63 => pass

GCI results

We also evaluated the proposed method* of limiting p ≥ 0.5 and increasing Fs = 1.25 → 3.

Present observed p's are slowly varying, uniformly low. Experienced CFD analysts would perhaps trust the indicated convergence and not impose the limit.

This modified method gives a more conservative uncertainty estimate and of course results in no “failures”.

The number of activations of this more conservative method was 64/96, i.e. over-conservative.* Oberkampf and Roy [[7, p. 326]; see also Eça and Hoekstra [10]

LSQ-GCI results(by Luis Eça, IST and MARIN)

● “Standard” method, limit* on p >1/2, Fs = 3 *Present observed p's are slowly varying, uniformly low. Experienced CFD analysts would perhaps trust the indicated convergence and not impose the limit

● Structured 63/66 pass● Unstructured 34/34

● No limit on p, Fs = 1.25● Structured 66/66● Unstructured 34/34

Impediment To Fully (3/3)-UNSTRUCTURED Grids

Generation of grids with fully 3D "unstructuredness" or (3/3)-Unstructured grids caused difficulties.

Grids were more ifficult to generate.

Nominal grid densities were farther from the structured grid values.

Nominal refinement factors were less meaningful.

Fuel cell simulations did not converge iteratively, probably due to well-recognized difficulties with very high aspect ratio elements.

(Not a shortcoming of the GCI per se.)

Summary and Recommendations (1/2)

● Simple one-parameter GCI was applied to non-similar sets of unstructured grids using a simple effective grid refinement ratio.

● Results consistent* with target of ~ 95% coverage ● Achieved using only the Fs = 1.25 previously

determined for structured grid cases and fixed (i. e., not tweaked) since 1998.

● Earlier studies also encouraging, but still small samples.

● Efficacy will depend on grid generation to approximate a uniform distribution of r.

* but small sample

Summary and Recommendations (2/2)

● We still recommend Fs = 1.5 with unstructured refinement unless sampled high resolution tests on problem sets of interest continue to indicate that Fs = 1.25 is adequate.

● Our confidence is limited to solution functionals● Unstructured grids will dominate in 3D● LSQ-GCI, used on structured grids with defects in

similarity, are now highly developed.● Present results are encouraging for future

application of LSQ-GCI to unstructured grids and meshless methods.