1. Revisiting the Least-Squares Procedure for Gradient Reconstruction on Unstructured Meshes Dimitri J. Mavriplis National Institute of Aerospace Hampton, VA 23666

2. Motivation Originated from study of matrix dissipation versus upwind schemes for unstructured mesh RANS solver Least Squares Gradient now standard technique for higher order accuracy with upwind schemes Unexpected behavior observed (with entropy fix) 1 week project  3 month investigation

3. Summary of Findings Least squares gradient construction may under-predict gradients by orders of magnitude (~100% error) –Vertex, cell centered, simplicial, mixed elements Subtle mechanism –Apparently has gone unnoticed in literature –May not show up in standard test cases Similar results: N.B. Petrovskaya: ``The impact of grid cell geometry on the least squares gradient reconstruction’’, Keldysh Institute of Applied Math., Russian Academy of Sciences, April 2003

4. Spatial Discretization Mixed Element Meshes –Tetrahedra, Prisms, Pyramids, Hexahedra Control Volume Based on Median Duals –Fluxes based on edges –Single edge-based data-structure represents all element types F ik = F(u L ) + F(u R ) + T  T -1 (u L –u R ) - Upwind discretization - Matrix artificial dissipation

5. Upwind Discretization First order scheme Second order scheme Gradients evaluated at vertices by Least-Squares Limit Gradients for Strong Shock Capturing

6. Matrix Artificial Dissipation First order scheme Second order scheme By analogy with upwind scheme: Blending of 1 st and 2 nd order schemes for strong shock capturing

7. Entropy Fix  matrix: diagonal with eigenvalues: u, u, u, u+c, u-c Robustness issues related to vanishing eigenvalues Limit smallest eigenvalues as fraction of largest eigenvalue: |u| + c –u = sign(u) * max(|u|,  (|u|+c)) –u+c = sign(u+c) * max(|u+c|,  (|u|+c)) –u – c = sign(u -c) * max(|u-c|,  (|u|+c))

8. Entropy Fix –u = sign(u) * max(|u|,  (|u|+c)) –u+c = sign(u+c) * max(|u+c|,  (|u|+c)) –u – c = sign(u -c) * max(|u-c|,  (|u|+c))  = 0.1 : typical value for enhanced robustness  = 1.0 : Scalar dissipation  becomes scaled identity matrix –T |  | T -1 becomes scalar quantity –Simplified (lower cost) dissipation operator Applicable to upwind and art. dissipation schemes

9. Green-Gauss Gradient Construction –Contour integral around control volume –Generally NOT Exact for linear functions Only for vertex discretizations on triangles/tetrahedra –Accuracy dependant on cell shapes –Poor solver robustness reported for RANS cases

10. Least Squares Gradient Construction –Formally unrelated to grid topology –Natural to base point sample on grid stencil –Exact for linear functions on all grid/discretization types –More accurate gradients on distorted meshes –Reported to be more robust for viscous flows

11. Drag Prediction Workshop I DLR-F4: Mach=0.75, CL=0.6, Re=3M Baseline grid: 1.65 million vertices, mixed elements

12. Comparison of Discretization Formulation (Art. Dissip vs. Grad. Rec.) Least squares approach slightly more diffusive Extremely sensitive to entropy fix value

13. Reduce to Simpler 2D Case RAE 2822 Airfoil, Mach=0.73, alpha=2.31, Re= 6.5M Least-square gradient upwind scheme with entropy fix overly diffusive

14. Gradient Accuracy Study Least-Squares, Green-Gauss, Finite Difference Discretization type (cell-vertex), element type Exact analytic function (non-linear) –Compute exact error –Function similar to flow gradients –Boundary layer regions

15. Distance Function: D(x,y) Similar to boundary layer velocity gradients Available (required by turbulence model) Approximately linear: Good accuracy of estimated gradient with all methods

16. Non-Linear Function Non-linear function required for adequate test –  = 200 (reduces roundoff error for small D) Exact Gradient : Since

17. Gradient Error Study Compare calculated and exact Gradient of function F at vertices of mixed element unstructured mesh (quadrilateral elements near airfoil surfaces)

18. Vertex Discretization on Quadrilaterals Unweighted Least Squares Gradients under- predicted by order of magnitude in inner BL

19. Simpler Flat Plate Geometry Rounded/Tapered Leading Edge

20. Flat Plate Geometry Unweighted Least Squares gradients underpredicted up to point of vanishing curvature

21. Accuracy Failure Mechanism All stencil points contribute equally (unweighted) Upstream/Downstream Points contribute to –H > h (due to surface curvature)

22. Grid Requirements for Unweighted LS h > H for accurate grads eg: Unit circle, 100 surface points: h > 10 -4 Inv.Distance weighting OK –S >> h

23. Vertex Discretization on Quadrilaterals Unweighted Least Squares Gradients under- predicted by order of magnitude in inner BL

24. Vertex Discretization on Triangles Similar behavior to vertex discretization on quadrilaterals

25. Cell Centered Discretizations Cell-centered on quads: similar to vertex-based stencil Cell-centered on triangles: No close neighbors

26. Cell-Centered on Triangles Unweighted and Weighted Least Squares Inadequate Green-Gauss varies by 10% depending on diagonal edge orientation

27. Effect on Solution Accuracy How can good solution accuracy be obtained in the presence of poor gradient estimates ? Why is accuracy so sensitive to small values of entropy fix ? Flow alignment phenomena Occurs in exact same regions as inaccurate gradients Inner BL region

28. Flow Alignment Flow solution on RAE Airfoil Grid at x=0.3 Normal velocity << Streamwise velocity Normal convective eigenvalues (u.ds) can be largest (stiff)

29. Flow Alignment Normal dissipation << streamwise dissipation 1 st order normal dissip. < 2nd order streamwise dissp.

30. Flow Alignment Entropy fix: u fix = sign(u). min (|u|,  (|u|+c)) For aligned flow –Large increase in u fix for small values of  –Explains solution sensitivity to entropy fix Flow alignment irrelevant for acoustic modes –Good overall accuracy retained in spite of poor resolution of acoustic modes in BL (?)

31. Implications Weighted LS gradients for vertex discretizations –Accurate gradients –Reduced sensitivity to entropy fix

32. Implications Unweighted LS more accurate on isotropic grids Unweighted LS inaccurate on stretched meshes –Effect mitigated by flow alignment Inaccurate grads only in presence of curvature –Problem not seen for flat plate BL test case

33. Implications Weighted LS or Green-Gauss gradients more accurate overall –Robustness issues reported Unweighted LS Grads more robust –Not because of superior gradient estimates –Because solution is 1 st order (limited) in BL Viscous (NS) terms based on LS grads could pass flat plate test, but be disastrous

34. Conclusions Unweighted LS grads acceptable –Must be used only for reconstruction in convective terms –No entropy fix Weighted LS grads offer superior accuracy –Result in well conditioned system of equations for gradient calculation Stencils require close normal neighbor points –Semi-structured BL meshes Robustness issues remain (further investigation) Alternate construction techniques (further investigation) –Dimensional splitting –Gradient projection (Desideri), SLIP (Jameson) –Other approaches (Frink, Rausch, Batina and Yang) etc.