SUPG

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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_at_rice.edu http//w
ww.mems.rice.edu/TAFSM/
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Quadrature-point based ? defined

• c ?q c q , k ?q k q
• (?S1)q cq / kq
• I ?q (?S1)q f (c q , k q , )
• Element work arrays c q and k q

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Solution surface from Q4 elements and
element-based ?S1
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Contours for mesh Q4?SUGN1 (top), element based
?S1 (lower left), quadrature-point based ?S1
(lower right)
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Contours for T3 mesh?SUGN1 (top), element based
?S1 (lower left), quadrature-point based ?S1
(lower right)
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Quadrature-point-based ?S1 contours for Q4 mesh
(top), Q9 mesh (lower left), Q16 mesh (lower
right)
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Element-based ?S1 contours for Q4 mesh (top),
Q9 mesh (lower left), Q16 mesh (lower right)
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Quadrature-point-based ?S1 contours for T3 mesh
(top), T6 mesh (lower left), T10 mesh (lower
right)
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Element-based ?S1 contours for T3 mesh (top),
T6 mesh (lower left), T10 mesh (lower right)
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Constant y-plane solution for ?S1 fromQ4 mesh
(top), T3 mesh (bottom)
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Constant y-plane solution for Q4, Q9, Q16 for ?
S1quadrature-point-based (top), element-based
(bottom)
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Constant y-plane solution for T3, T6, T10 for ?
S1quadrature-point-based (top), element-based
(bottom)
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Constant x-plane solution for quadrature-point-bas
ed ?S1Q4, Q9, Q16 meshes (top), T3, T6, T10
meshes (bottom)
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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

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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.

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Full T3 Mesh
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Zoom on T3 Mesh
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Zoom on T6 Mesh
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Full zoom T6 point values
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Zoom on T10 Mesh
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Zoom on Q8 Mesh
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Full zoom on Q8 point values
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Zoom on Q16 mesh
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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 ?

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APPENDIX 1Regular T6 mesh uniform flow field
test angles

• 0 (horizontal edges),
• 23 (centroid),
• 45 (long edge),
• 90 (vertical edges)

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0 degrees
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23 degrees
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45 degrees
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90 degrees
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APPENDIX 2

• Results from previous ways to evaluate ?

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