SUPG

1. SUPG STABILIZATION PARAMETERS CALCULATED FROM THE QUADRATURE-POINT COMPONENTS OF THE ELEMENT-LEVEL MATRICES ECCOMAS 2004 J. ED AKIN TAYFUN TEZDUYAR Mechanical Engineering, Rice University Houston, Texas akin@rice.edu http://www.mems.rice.edu/TAFSM/

9. Quadrature-point baseddefined: c = ∑qcq ,k~ = ∑qk~q (S1)q = || cq || / || k~q || I = ∑q (S1)q f (cq , k~q , …) Element work arrays:cqandk~q

11. Solution surface from Q4 elements and element-basedS1

12. Contours for mesh Q4:SUGN1 (top), element based S1 (lower left), quadrature-point based S1 (lower right)

13. Contours for T3 mesh:SUGN1 (top), element basedS1 (lower left), quadrature-point basedS1 (lower right)

14. Quadrature-point-based S1 contours for: Q4 mesh (top), Q9 mesh (lower left), Q16 mesh (lower right)

15. Element-based S1 contours for: Q4 mesh (top), Q9 mesh (lower left), Q16 mesh (lower right)

16. Quadrature-point-based S1 contours for: T3 mesh (top), T6 mesh (lower left), T10 mesh (lower right)

17. Element-based S1 contours for: T3 mesh (top), T6 mesh (lower left), T10 mesh (lower right)

18. Constant y-plane solution for S1 from:Q4 mesh (top), T3 mesh (bottom)

19. Constant y-plane solution for Q4, Q9, Q16 for  S1:quadrature-point-based (top), element-based (bottom)

20. Constant y-plane solution for T3, T6, T10 for S1:quadrature-point-based (top), element-based (bottom)

21. Constant x-plane solution for quadrature-point-based S1:Q4, Q9, Q16 meshes (top), T3, T6, T10 meshes (bottom)

22. Discrete Point Values of • The previous contours hide information because they omit the mesh detail and are “smoothed” through different point locations: • Element-based at element centroid • Quadrature-point-based positions • Nodal-point-based positions

23. Finest T3 mesh followed by zoom-in by center flow rotation point for  contour of: T3, T6, T10 elements, Q4, Q8, Q16 elements. Local Tau values are generally proportional to element length through the point, in the direction of the local velocity.

24. Full T3 Mesh

25. Zoom on T3 Mesh

26. Zoom on T6 Mesh

27. Full zoom T6 point values

28. Zoom on T10 Mesh

29. Zoom on Q8 Mesh

30. Full zoom on Q8 point values

31. Zoom on Q16 mesh

32. Conclusions Quadrature-point norm based  values are efficient to compute. They yield higher  values in local regions where the unit vector s or r rapidly changes its direction. The element-based norm  is a general framework that automatically accounts for local length scales. Both norm-based methods decrease  as the element polynomial order increases. Our favorite: element-based norm 

33. APPENDIX 1Regular T6 mesh uniform flow field test angles: 0 (horizontal edges), 23 (centroid), 45 (long edge), 90 (vertical edges)

34. 0 degrees

35. 23 degrees

36. 45 degrees

37. 90 degrees

38. APPENDIX 2 Results from previous ways to evaluate: