CAE: Engineering Visualization

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CAE Engineering Visualization

• CAE results visualization includes display of a
  single variable within a 3-dimensional field
• Equivalent stress, temperature, estimated error
  of a solution are typically represented as a 2 or
  3-dimensional fields of a single variable

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CAE Results Visualization

• A scalar variable can be represented as a
  function of its position in space and time sigma
  F(x,y,z,t). Result values at the vertices of
  elements connected to vertices
• simple color coding or contour plot,
  3-dimensional variation of a single variable
  within a volume helps with evaluating true 3-d
  state of behaviour within a model

3
CAE Results Visualization

• 1-d scalar fields use RGB variations to show
  colors corresponding to scalar values
• 2-dimensional scalar fields represent a variable
  across one or more surfaces
• Element face color coding color of the visible
  element face is chosen from the average scalar
  value across this face.

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CAE Element face color coding

• Single centroidal value per face using values at
  the vertices of the face
• Normalized value is used to interpolate a color
  value from a color table (min to max)
• Only conservative and very approximate since
  actual behaviour is glossed over (stress values
  at gauss points and vertex locations are averaged
  out)

5
CAE Contour Display

• Contour display Based on contour lines or
  isovalue lines represent constant value across a
  surface field.
• Basic linear interpolation of contour levels It
  is assumed overall variation of the scalar value
  is more critical than variation within a single
  element.

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CAE Contour Display

• Piecewise contour interpolation on isoparametric
  elements More useful for P-version of finite
  element method (as compared to H-version).
• P-version works with higher order elements. Uses
  a small number of large elements.
• Variation within one element could be significant

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CAE Contour Visualization

• Direct contour generation is useful working on
  parametric surfaces
• Linear elements may do well with simple linear
  interpolation
• Contour Fringe Plot Discreet color filled
  regions correspond to ranges of result value
• Discritization into smaller triangles/polygon for
  color filling.

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CAE Optimize Visualization

• Optimize to process only visible exterior faces
  (free faces)
• Removal of duplicate Interior faces For solid
  elements, all faces sharing all common vertices
  will be interior. For surface elements either
  polygon may be visible
• Display time culling of rearward faces Based on
  polygon normal.

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CAE Results Visualization

• View specific and so must be recomputed for each
  view
• Direct color interpolation of scalar results
  Starts with polygon vertices and values at
  vertices and uses methods similar to Gouraud
  polygon shading.
• Bresenhams algorithm Used to find pixels making
  up each scan line

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CAE Delaunay Triangulation

• Fast algorithm to create triangles from a set of
  points. Most useful in graphics display of
  contours, tessellated surfaces, rendering
  algorithms
• Dirichlet tessellation is used with perpendicular
  bisectors of lines joining the neighbouring data
  points

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CAE Delaunay Triangulation

• Fig 1 shows the delaunay triangulation
• Dirichlet tesselation vertex is generally formed
  at a location equi-distant from each of the three
  data points forming the Delaunay triangle
• No Data point lies within the circle formed with
  vertices of any Delaunay triangle.

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CAE Delaunay Triangulation

• Delaunay triangulation is generally unique for
  planar arbitrary set of points
• For non-unique cases, it is considered degenrate
• Example a simple square with 4 vertex points as
  data points. Dirichlet vertex is at the square
  centroid. (Fig 2) 2 valid solutions

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CAE Delaunay Triangulation

• Avoids forming triangles with small included
  angles. Locally equi-angular
• For every convex quadrilateral formed by two
  adjacent triangles, the minimum of the 6 angles
  in the 2 triangles is greater than it would be if
  the alternate diagonal had been drawn and those
  triangles chosen.

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CAE Delaunay Algorithm

• Average case for planar delaunay triangulation is
  O(N power 1.2). Worst case is shown as O(N power
  3)
• Some codes available in public domain. Efficient
  and fast algorithms are key
• Introduce one data point at a time with an
  existing Delaunay triangle and an update is done.

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CAE Delaunay Algorithm

• 3 points chosen as a super triangle that
  encompasses ALL points to be triangulated
• Add a new point P and find a triangle that
  includes P. Create 3 new triangles by connecting
  P to 3 vertices. Original triangle is deleted.
• Update triangles by swapping (Lawsons method) (
  Figure 3)

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CAE Delaunay Swapping

• All triangles which are adjacent to edges,
  opposite to point P are stacked. Then, for each
  triangle, check is made to see if P lies inside
  the circle formed by the vertices of the
  triangle. (figure 3)
• If true, the triangle and adjacent triangle form
  a convex quadrilateral, with a need to swap
  diagonal to preserve the structure.

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CAE Delaunay Triangulation

• Replace 2 old triangles with 2 new ones
• Only finite number of swaps are needed
• Finally remove super triangle.
• Number of triangles is 2N1
• Algorithm does better than O(NN)
• Bin sort may be ignored -- longer times
• Table 2 for further timings