American Institute of Aeronautics and Astronautics1

On the a Posteriori Error Estimation in Mesh Adaptation toImprove CFD Solutions

Lakhdar Remaki*, S. Nadarajah†, Wagdi G. Habashi‡

NSERC-J. Armand Bombardier Industrial Research Chair, Computational Fluid Dynamics Laboratory, McGillUniversity 688 Sherbrooke Street West, 7th floor, Montreal, QC, Canada H3A 2S6

M.C. Bogstad§, C. Kho¶, F. Mokhtarian#

Advanced Aerodynamics, Bombardier Aerospace, 400 Côte-Vertu West, Dorval, QC, Canada, H4S 1Y9

The objective of this work is to study the impact of mesh adaptation on computationalestimates of lift and drag coefficients. The convergence of these quantities with successiveadaptation is demonstrated by comparing to experimental results. Optimization ofadaptation parameters to accelerate the convergence is also examined.

Nomenclature H = the Hessian matrix P = pressure M = Mach number(.)ρ = Spectral radius

I. Introductionuring a typical CFD analysis, a large effort is devoted to the mesh generation process. Not only is meshgeneration time consuming, but also the resulting mesh may not be completely appropriate to sufficiently

resolve all the details of the flow. Mesh adaptation methodology is a powerful tool, not only to improve solveraccuracy but also to allow simulations starting from an arbitrarily coarse initial grid. In Refs.1,2, a robust andefficient 3D automatic mesh adaptation methodology, with CAD integrity, OptiGrid, has been developed. Theadaptation procedure uses an a posteriori interpolation error estimate, whose magnitude and direction are controlledby the matrix of local second derivatives of a selected flow variable. This error is projected over the mesh edges, anddrives the nodal movement algorithm, as well as the edge refinement, coarsening and face swapping strategies. Verygood results have been obtained using this mesh approach1,2,3,4,5.

The adaptation process, however, can take several solution-adaptation cycles depending on the quality of the initialgrid and the complexity of the test case. In this paper we will first show the efficiency of the mesh adaptation inensuring accuracy of the lift and drag coefficients, the bottom line of most aerodynamic calculations. To improvethe performance of the mesh adaptation procedure and reduce the number of adaptation cycles, we propose a newvariable for adaptation that better represents global physical features, as well as a modification of the metric takingaccount of the first derivatives which have so far been neglected in the error indicator computation. The efficiencyof the proposed improvements is demonstrated on two test cases: a low subsonic test case, being the NASA Semi-Span flap test case6,7,8 and a transonic test case, being the RAE2822 airfoil9. * Research Associate, NSERC-J.-Armand Bombardier Industrial Research Chair, Computational Fluid DynamicsLaboratory, McGill, lakhdar@cfdlab.mcgill.ca.† Professor, NSERC-J.-Armand Bombardier Industrial Research Chair, Computational Fluid Dynamics Laboratory,McGill, siva.nadarajah@mcgill.ca.‡ Professor and Director, NSERC-J.-Armand Bombardier Industrial Research Chair, Computational Fluid DynamicsLaboratory, McGill, wagdi.habashi@mcgill.ca, Associate Fellow AIAA.§ Engineering Specialist, Advanced Aerodynamics Bombardier Aerospace, Mark.Bogstad@canadair.ca.¶ Engineering Professional, Advanced Aerodynamics Bombardier Aerospace, cedric.kho@canadair.ca.# Section Chief, Advanced Aerodynamics Bombardier Aerospace, farzad.mokhtarian@canadair.ca.

D

44th AIAA Aerospace Sciences Meeting and Exhibit9 - 12 January 2006, Reno, Nevada

AIAA 2006-890

Copyright © 2006 by the Computational Fluid Dynamics Laboratory, McGill University. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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In Section 2, a general overview of the mesh optimization methodology is given. The impact of mesh adaptationon lift coefficient convergence is demonstrated in Section 3. In Section 4 the proposed improvements are developedand validated for the same test case finally, conclusions are drawn in Section 5.

II. Mesh Optimization MethodologyThe mesh adaptation methodology described herein is embedded in OptiGrid, a software that offers automatic

solution-based anisotropic mesh adaptation, as well as geometrical mesh smoothing of the initial mesh, even beforea solution is attempted. The latter feature is quite important, as it is not unusual for many CFD codes to simply failbecause of a poor initial grid: volumes that are inappropriate, too skewed, have jerky gradients or are degenerate.Once a solution is launched, the basic adaptation operations then include node movement, edge refinement,coarsening and swapping, for hybrid grids consisting of any combination of tetrahedra, prisms, hexahedra andpyramids.The solution-based adaptation is driven by an a posteriori error estimate based on the Hessian H of a selected scalarflow variable, since, for FEM-FVM solvers with linear basis functions, the truncation error is dominated by thesecond derivatives. When H is positive definite it defines a metric, therefore the error along a given vector could beinterpreted as the length of the vector measured by the Hessian metric. Consequently, a metricT is derived from the

Hessian by considering its absolute value RRHT t Λ== , with R being the eigenvectors matrix of H and

€

Λ = diag λ1 , λ2 , λ3{ } the matrix composed by the absolute values of the eigenvalues ofH . The eigenvectors

of M give the local direction of the stretching, while the eigenvalues give its local magnitude, thus graduallycreating anisotropy. The goal of the adaptation is to equi-distribute the error on the adapted grid, where the erroralong an edge in the Riemannian metric is computed as:

∫=1

0

)( dsxsTxTvv

ε

where

€

r x is the vector that defines the edge.

The sequence of operations begins with node movement, edge refinement and edge swapping on solid boundaries, tosatisfy a minimum and maximum edge length constraint, as well as a curvature constraint, yielding substantialsurface CAD improvements. The process then continues with node movement in the entire domain, followed byrefinement and coarsening, then swapping, before concluding with additional node movement. OptiGrid preservesCAD integrity by reprojecting boundary points onto the original surfaces during the adaptation process.

The next sections discuss the impact of adaptation on the convergence of the solution as well as strategies forcost-efficient mesh optimization. The intent is to minimize the number of solver-adaptation cycles by improvingsome parameters of the adaptation procedure.

III. Mesh Adaptation Impact on the Lift CoefficientThe NASA semi-span flap6,7,8 test case was chosen for evaluating the impact of the mesh adaptation on theprediction of lift and drag coefficients. This test case is a NACA 632-215 Mod B airfoil with a 30%, half-spanslotted flap at 30 degrees. The flow conditions correspond to the experiments carried out in the NASA 7x10-footwind tunnel1 at Mach = 0.22, AoA = 10o, Rec = 3.7x106, based on the undeflected chord. The Spalart-Allmarasturbulence model is used. The metric as described in section II is used and the Mach number is chosen as theadaptation variable, having been demonstrated in the literature1,2,3 to be appropriate for various applications. Thechordwise pressure distributions were compared to the experimental results8. Of particular interest are four span-wise positions located on both sides of the flap tip (i.e. h = 0.39, 0.47, 0.53, 0.61 in Figure 1). Figure2a and Figure2bshow the original grid and the adapted one (after 8 cycles of adaptation) and the corresponding pressure distributionat the stations of interest. Figure 3 shows the evolution of the lift coefficient with the number of adaptation cycles.One can see first that the obtained results match well the experimental results, and as the adaptation proceeds, thelift coefficient converges to the experimental value. As a conclusion, this test case clearly demonstrates that meshadaptation strategy is an efficient and robust method to ensure the convergence of the solution (to the experimentallyobtained values) and to correctly capture physical features that are of a high importance in industrial applications.However, as we can see, 8 cycles were necessary to achieve convergence, which could be considered somewhatcostly in terms of time, especially when we deal with large test cases.

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IV. Mesh Parameters ImprovementAs mentioned above, it is of great interest if we could improve the performance of the mesh adaptation

procedure in terms of processing time. In this study, we focus on two important and sensitive parameters; theadaptation variable and the generated metric.

A. The Adaptation VariableIt is commonly accepted that the Mach number (M ) is an appropriate adaptation variable in many situations.

However, an increasing number of mesh adaptation iterations may be required to obtain a converged solution forlow subsonic flows where Mach number gradients are less dominant (than in transonic flows). Furthermore, forviscous flows, the Mach number is theoretically zero at the surface and therefore an optimal surface grid cannot beachieved when using the Mach number to drive the mesh optimization process. On the other hand, the Mach numberdoes capture a strong gradient in the boundary layer, which needs to be preserved. It would also be undesirable touse a flow parameter such as pressure to drive the adaptation, as the lack of a gradient in the boundary layer willdestroy the fine grid spacing throughout the boundary layer.

Figure 1: NASA Semi-Span Flap; Pressure Measurement Stations

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Figure 2a: The Cp profile compared to experimental results and the adapted grid after 8cycle of adaptation

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Figure 2b: The Cp profile compared to experimental results and the adapted grid after 8cycle of adaptation

Figure 3: Evolution of the lift coefficient with the adaptation cycles

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In the new approach, it is therefore proposed to combine the pressure ( P ) and the Mach number (M ) via theircorresponding adaptation metrics, in the following way.

First both variables are normalized to the range

€

0,1[ ] , and the corresponding metrics PT , MT are created as described

in II. The new metric derived from the two variables is defined as:

PMPMTTT α+=

,

where)()(

)(

PMinPMax

PMinPP

−

−= and

)()(

)(

MMinMMax

MMinMM

−

−=

PMααα /=

∑=

=nodesN

jP

nodesP

jTN 1

))((1

ρα , ∑=

=nodesN

jM

nodesM

jTN 1

))((1

ρα , nodeN is the number of total nodes, ))(( jTP

ρ and

))(( jTM

ρ are the spectral radius of )( jTP and )( jT

M

By combining the pressure and the Mach number through the metric PMT

, , the nature of the boundary layer is

preserved through the Mach number, and the surface information is preserved via the pressure distribution. For lowspeed flows, there can be local pockets of transonic/supersonic flow at the leading edges, for example. In otherregions, the variation of Mach number diminishes to a point where it is no longer suitable for mesh adaptation. Insuch regions the pressure terms takes over as the driving parameter for the mesh optimization process.

A. Metric ModificationThe metric described in section II uses the second derivatives to compute the error indicator. It is obvious that tocapture high gradients, a certain level of node concentration is necessary. In regions of high gradient the solutioncould be linear or close to linear, causing the second derivatives to vanish. Conversely, the gradient vanishes inregions of extrema, which correspond to regions of maximum of curvature. To improve the quality of the adaptedmesh, therefore, we need to combine both the Hessian and the gradient to compute the error indictor. Since Hessianand gradient generally lie in different ranges, some normalization and balancing between the two associated metricsis necessary. Based on that, we propose the following formulation:

GxTT )(α+=

G is the normalized gradient metric given by:

€

G =(Maxx

ρ(H (x)) − Minxρ (H (x)))

(Maxx

ρ(J (x)) − Minxρ (J (x)))

J − (Minx

ρ(J(x))I

+ (Min

xρ (J (x))I

)( SSJ T ∇•∇=S is the variable of adaptation

I is the identity matrixα is a coefficient that prevent the gradient from dominating the Hessian and given by:

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=

))((

))((,1)(

xGMax

xTMinx

xρ

ρα

B. Test casesThe calculations on the NASA Semi-Span test case are repeated with the new adaptation variable and the correctedmetric. Figure 4 shows the variation of the lift coefficient as a function of the number of adaptation cycles, using thenew adaptation variable and metric, in comparison to the previous results. The number of cycles is reduced to almosta half, and, moreover, and moreover, the required number of nodes is also reduced to almost a half as shown in table1. This demonstrates clearly the improvements of the new variable of adaptation and the modified metric.

To show the effect of the mesh adaptation on the drag coefficient, a transonic turbulent flow case was selected. TheRAE28229 was analyzed at M = 0.73, AoA = 2.79, Rec =6.5 106, using the new adaptation variable and modifiedmetric. The adapted grid (Figure 5) shows that the nodes are more concentrated in the shock region, the leading edgeand on the boundary layers, as compared to the adaptation without the proposed modifications. In addition, thenumber of nodes is not increased. Figure 6 shows the Cp distributions on the original grid and after one cycle ofadaptation; both results match well the experimental data. However, the drag coefficient value, obtained on theadapted grid, correlates much better with the experimentally measured values, as shown in Table 2.

Number ofnodes for each

cycle ofadaptationbefore the

modifications

599835nodes

856661nodes

733385nodes

787886nodes

791878nodes

870113nodes

816361nodes

1051845nodes

1040535nodes

Number ofnodes for each

cycle ofadaptationafter the

modifications

599835nodes

486780nodes

328517nodes

578139nodes

570578nodes

673806nodes

Cycles Cycle 0(InitialGrid)

Cycle 1 Cycle 2 Cycle 3 Cycle 4 Cycle 5 Cycle 6 Cycle 7 Cycle 8

Figure 4: Evolution of the lift coefficient with the adaptation cycles (newversus old adaptation variable and metric).

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Table1: Evolution of the number of nodes with and without the proposed modifications

Figure 6: The cp profile compared to experimental results

Solution on the original grid Solution on the first adapted grid

Figure 5: Grid comparison with and without the proposed modifications

Adapted Grid withoutmodifications

Adapted Grid withmodifications

Original Grid

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Experimental results FENSAP Results onOriginal Grid

FENSAP Results afterone Cycle of Adaptation

Cl 0.803 0.802 0.801Cd 0.0168 0.020 0.0174

Table 2: Lift and drag coefficients on the original and the adapted grid, compared to experimental results

V. ConclusionThis paper investigated the impact of mesh adaptation on the calculation of lift and drag coefficients that are ofimportance for industry. The results obtained through the mesh adaptation strategies were compared to experimentalresults.It has been demonstrated that, by generating a metric that combines pressure and Mach number in a unique way andintroducing a new metric that takes of account the information from the gradient, the convergence of the meshoptimization process can significantly be accelerated. The impact of the new parameters on grid qualityimprovement and convergence acceleration has also been demonstrated, for both low subsonic and transonic testcases.

References1Habashi, W.G., Dompierre, J., Bourgault, Y., Fortin, M. and Vallet, M-G., “Certifiable Computational Fluid Dynamics

Through Mesh Optimization”, Special Issue on Credible Computational Fluid Dynamics Simulation, AIAA Journal, Vol. 36, No.5, pp. 703-711, 1998.

2Wagdi G. Habashi et al. Anisotropic Mesh Optimization: Towards a Solver-Independent and Mesh-Independent CFD, inthe von Karman Institute for Fluid Dynamics Lecture Series 1996-06, Computational Fluid Dynamics, Montreal, August 1996.

3Remaki, L., Lepage, C.Y. and Habashi, W.G., “Efficient Anisotropic Mesh Adaptation on Weak and Multiple Shocks,”AIAA Paper 2004-0084.

4Remaki and Habashi, W.G., “Efficient Anisotropic Mesh Adaptation on Weak and Multiple Shocks,” AIAA Paper 2004-0084.

5Lepage, C.Y., Suerich-Gulick, F. and Habashi, W.G., “Anisotropic 3-D Mesh Adaptation on Unstructured Hybrid Meshes”,AIAA Paper 2002-0859.

6M.C. Bogstad, C. Kho, F. Mokhtarian, L. Remaki, C. Y. Lepage, W. G. Habashi, “Geometrical- and Solution-based MeshAdaptation on a NASA Semi-Span Flap”, CASI 2003.

7Bruce L. Storms, Timothy Takahashi, James C. Ross, “Aerodynamic Influence of a Finite-Span Flap on a Simple Wing”,SAE paper 951977, September 1998.

8Bruce L. Storms, James C. Ross et al. - An Aeroacoustic Study of an Unswept Wing with a Three-Dimensional High-liftSystem. NASA/TM-1998-112222.

9S. S. Davis, “NACA 64A010 (NASA Ames model) oscillatory pitching”, AGARD Report 702, August 1982. InCompendium of Unsteady Aerodynamic Measurements.