1. T  AFSM 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/

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9. T  AFSM Quadrature-point based  defined: c = ∑ q c q, k ~ = ∑ q k ~ q (  S1 ) q = || c q || / || k ~ q || I = ∑ q (  S1 ) q f (c q, k ~ q, …) Element work arrays: c q and k ~ q

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

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

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

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

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

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

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

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

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

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

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

22. T  AFSM 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. T  AFSM 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. T  AFSM Full T3 Mesh

25. T  AFSM Zoom on T3 Mesh

26. T  AFSM Zoom on T6 Mesh

27. T  AFSM Full zoom T6 point values

28. T  AFSM Zoom on T10 Mesh

29. T  AFSM Zoom on Q8 Mesh

30. T  AFSM Full zoom on Q8 point values

31. T  AFSM Zoom on Q16 mesh

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

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35. T  AFSM 23 degrees

36. T  AFSM 45 degrees

37. T  AFSM 90 degrees

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

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