CAE: FEM - Paving

1
CAE FEM - Paving

• Several examples for demonstration of paving
  algorithms available. Mesh quality, robustness,
  speed are of importance
• Example A has a single square exterior boundary
  and a single interior skewed boundary
• Alternate paving rows overlap (Figure 3.1)

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Paving Examples

• Two exterior paving boundaries are formed
• One completely paved before the other boundary
• Continue to close all exterior boundaries
• Cleanup does little to modify nodes. Boundary
  sensitivity is seen. Irregular nodes moved to
  interior

3
Paving Examples

• Example B shows an exterior boundary and two
  interior boundaries (Figure 3.2)
• Mesh densities very different on concave
  boundary. Upper slot matches concave boundary in
  density
• Lower slot matches rest of exterior boundary

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Paving Examples - transition

• Exterior boundary has non-uniform node spacing.
  Concave arc twice as dense as other portions of
  boundary
• Dissimilar element sizes overlap during paving.
  Seaming connects nodes nicely
• minimum of irregular nodes are formed
• large elements are reduced in size with wedges

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Paving Examples - transition

• Boundary sensitivity is demonstrated. Meshes near
  boundaries are well-formed
• Example C demonstrates an odd shaped exterior
  boundary (Figure 3.3)
• Two sharp corners found in exterior
• Interior boundary very close to exterior boundary

6
Paving Examples cleanup

• Sharp angles found and cleanup will be required
  to create good quality elements
• The seam created in first row blunts sharp
  corners and regular mesh results in later steps
• Distorted elements formed due to mesh overlap.
  Remains with mesh for most of the steps

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Paving Examples gradation

• Cleanup eliminates most bad elements in example
  C.
• Example D has simple exterior boundary. No
  interior boundaries.
• Extreme gradation in density of nodes is applied
  along the boundary. (figure 3.4)
• Nodes along notch at upper right are 20 times
  denser than the rest of the boundary.

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Paving Examples Large gradations

• Paving handles large gradations
• disproportionate mesh rows interact and the
  resulting meshes are kept valid
• cleanup dramatically improves quality
• dense mesh near smaller meshes. Large element row
  moves farther from boundary than a small element
  row

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Paving Performance quality

• Goal is to generate all quad (4 nodes) elements
  in any arbitrary geometry/topology
• quality of generated mesh is critical (aspect
  ratio, distortion, interior angle etc.,)
• robustness of algorithm is critical
• speed of mesh generation is critical

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Paving Performance quality

• Meshes are boundary sensitive, orientation
  insensitive, and minimize placement of irregular
  nodes (less than or greater than 4 elements at a
  node)
• distortion measure if used to check quality
• other factors such as type of analysis, element
  size, loading affect simulation results.

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Paving Robustness

• Example A has distortion D0.02 (average) and
  D0.26(max)
• Example B has distortion D0.09 (average) and
  D1.09(max)
• Example C has distortion D0.11 (average) and
  D1.13(max)
• Example D has distortion D0.30 (average) and
  D1.96(max)

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Paving Robustness, Speed

• Robustness is quite good upto 301 transition
  ratio (need adjustment of nodes on permanent
  boundary)
• Long and thin sections proceed quicker than large
  open boundaries (results in many small outer
  boundaries for former case)
• Linear relation with number of nodes and cpu
  seconds. Geometric complexity counts

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3D meshing tet-meshing

• Tetrahedral meshes are generated from a closed
  set of meshes
• 3D meshing techniques for tet-meshes are fully
  automated
• hex meshers are relatively harder to automate and
  a very few are available in the market

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3D meshes - hex/tet meshes

• Paper on hex/tet meshing benefits
• tet 10 element formulations are shown to be
  comparable in performance to hex 20 elements on
  simple studies
• Adaptive meshing with h-method meshing as well as
  p-method meshing show good potential for
  automation

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Fergusons vs. Bezier

• Form of the curve/surface equation
• physical meaning of vector coefficients
• bezier more suitable for curve design rather than
  curve fit
• bezier contained within characteristic polygon
  (hull convexity property)
• criteria for loop formation varies

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Fegusons vs. Bezier

• Corner vertices lie on bezier surface.
• Bezier suited well to study conic curves and
  surfaces
• Matrix notational differences