Revisiting the Least-Squares Procedure for Gradient Reconstruction on Unstructured Meshes

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 Fik = F(uL) + F(uR) + T |L| T-1 (uL –uR) - 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 1st and 2nd order schemes for strong shock capturing

7. Entropy Fix L 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|, d(|u|+c)) • u+c = sign(u+c) * max(|u+c|, d(|u|+c)) • u – c = sign(u -c) * max(|u-c|, d(|u|+c))

8. Entropy Fix • u = sign(u) * max(|u|, d(|u|+c)) • u+c = sign(u+c) * max(|u+c|, d(|u|+c)) • u – c = sign(u -c) * max(|u-c|, d(|u|+c)) d = 0.1 : typical value for enhanced robustness d = 1.0 : Scalar dissipation - L becomes scaled identity matrix • T |L| 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 • a = 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 • 1st order normal dissip. < 2nd order streamwise dissp.

30. Flow Alignment • Entropy fix: ufix = sign(u) . min (|u|, d(|u|+c)) • For aligned flow • Large increase in ufix for small values of d • 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 1st 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.