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SUPGSTABILIZATION PARAMETERS

CALCULATED FROM THE QUADRATURE-POINT COMPONENTS OF THE ELEMENT-LEVEL MATRICES

ECCOMAS 2004

J. ED AKINTAYFUN TEZDUYAR

Mechanical Engineering, Rice UniversityHouston, Texas

akin@rice.edu

http://www.mems.rice.edu/TAFSM/

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Quadrature-point based defined:

c = ∑q c q , k~ = ∑q k~ q

(S1)q = || cq || / || k~q ||

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 from:Q4 mesh (top), T3 mesh (bottom)

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

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

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Constant x-plane solution for quadrature-point-based S1:Q4, 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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