Mesh Improvement by Subdivision of Tetras into Hexes

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Mesh Improvement by Subdivision of Tetras into
Hexes

• Pedro V Marcal
• MPACT Corp.
• San Diego, CA.

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Objective Improvement of meshes for adaptive
analysis

• Subdivide tetras into hexes. Take advantage of
  automatic CAD based tetra mesh generation.
• Compare error norms in an environment without
  curvature.
• Support results with consideration of truncation
  errors in element
• Evaluate strategy for adaptive analysis

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Subdivision of beam

• Built in beam with shear loading at end
• Error norms, same interpolation functions for
  stress as for displacement. Specific energy
• Tetra mesh ,101 nodes
• error 0.64
• Hex mesh, 1274 nodes
• error 0.11

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Beam results

• Tetra p2 538 496 0.00134 0.44

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Note on Beam Results

• Results for hexes is surprising, since elements
  with bad aspect ratios arte expected to produce
  bad results.
• In the beam, having isolated most of the other
  factors such as surface curvature, uneven meshes,
  so we can compare results of tetras and hexes.
• Next examine truncation errors.

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Truncation errors in elements

• The effects of the truncation can be seen more
  clearly by considering the two dimensional case.
  Many writers have termed the truncation in the
  simplex triangular elements as second order
  because the terms in the first order is complete.
  But this is true not only along the edges of the
  triangle.
• We assume u a0 a1 x a2 y

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Truncation errors continued

• Along any other line emanating from the corner,
  we can substitute ymx and obtain a truncation in
  the quadratic term in x.
• In the case of the rectangular element, the
  truncation is quadratic along the edges
• We assume u a0 a1 x a2 y a3 x y
• Along any other line emanating from the corner,
  we again assume that ymx and obtain a truncation
  in the cubic term of x

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Truncation errors continued

• We observe that the truncation order is enhanced
  as we move away from the edges. This explains the
  superiority of the quadrilateral elements (when
  we use isoparametric elements).
• Is the xy term a linear or quadratic term?

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Local and global errors

• Finally, we consider the application of the error
  norm for these meshes in adaptive analysis. The
  error norm is a strain energy based local error.
  In order to give it a globally relative measure
  we weigh the error norm by its equivalent stress.
  
• This allows us to select the node with the
  maximum weighted error and subdivide the elements
  connected to that node.

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Local weighted error subdivision

• Error for beam, initial mesh on left
• Error for local division below

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Local and global errors (cont.)

• This improves the performance and reduces the
  number of nodes required for convergence. The
  maximum stress changed from 240 to 335. The
  maximum weighted error changed from 23 to 6.4 and
  the peak location moved to the end of the beam,
  the weighted peak improvement given by a full
  mesh subdivision. The number of nodes used for
  this refinement was 110 versus the 589 for the
  full subdivision.

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Conclusions

• Tetra to Hex Subdivision results in significant
  mesh improvement.
• Truncation errors in a hex element improves in
  its interior.
• Local improvement of peak weighted error norms
  provide the equivalent of a full mesh
  subdivision. It should precede a tet to hex
  subdivision.