PowerPointPrsentation

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Delaunay Triangulation (DT)
Lehrstuhl für Informatik 10 (Systemsimulation) Rus
lana Mys

• Introduction
• Delaunay-Voronoi based method
• Algorithms to compute the convex hull
• Algorithms for generating the DT
• Conclusion

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Two basic types of the mesh
Unstructured mesh
Structured mesh
Introduction
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Hybrid Approache
Outline of mesh generation approches
Introduction
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The Delaunay triangulation is closely related
geometrically to the Voronoi tessellation (also
known as the Direchlet or Theissen
tessellations).
Voronoi diagram
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The Delaunay triangulation is created by
connecting all generating points which share a
common tile edge. Thus formed, the triangle edges
are perpendicular bisectors of the tile edges.
This method has the O(log(n)) complexity.
Voronoi diagram
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The E satisfies min-max angle property iff either
the quadrilateral formed by the two triangles
sharing edge E is not strictly convex, or E is
the diagonal of quadrilateral which maximizes the
minimum of the six internal angles associated
with each of the two possible triangulations of
quadrilateral.
E
E
Non-convex quadrilateral
Maximization
Maximum-Minimum Angle Property
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Let E be an internal edge of triangulation. Then
E satisfies local empty-circle property iff the
circumcircle of any of two triangles sharing edge
E does not contain the vertex of the other
triangle in ist interior.
E
E
Not Delaunay triangulation
Delaunay triangulation
Local Empty-Circle Property
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Algorithms to computing the convex hull of set of
points

• Incremental and Gift wrapping algorithms
• the idea is to add the points one at a time
  updating the hull
• first step is to compute the area A of starting
  triangle as
• where xi and yi are the coordinates of triangle
  vertecies (in 2D)
• Divide and conquer algorithm O(nlog(n)))
• can be used if the domain is very large

If A 0 ? incremental algorithm (the points
occure in clockwise order) O(n2) If A 0 ? gift
wrapping algorithm (the points occure in
anti-clockwise order)
(if the of sides of the hull is h, then O(nh))
Algorithms to computing the convex hull
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Incremental and Gift wrapping algorithms

• Incremental

• Delete all edges from the hull that the new
  point can see
• Add two edges to connect the new point to the
  remainder of the old hull
• Repeat for the next point outside the hull

• Gift wrapping

• Find least point A (with minimum y-coordinate)
  as the starting point
• We can find B where all points lie to the left
  of AB by scanning through all the points
• Similarly, we can find C where all points lie to
  the left of BC and so on

Algorithms to computing the convex hull
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Divide and conquer algorithm
O(log(n))

• 2D-case

• First sort the points by x coordinate
• Divide the points into two equal sized sets L
  and R s.t. all points of L are to the left of
  the most leftmost points in R. Recursively find
  the convex hull of L and R

Algorithms to computing the convex hull
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Divide and conquer algorithm

• To merge the left hull and the right hull it is
  necessary to find the two red edges
• The upper common tangent can be found in linear
  time by scanning around the left hull in a
  clockwise direction and around the right hull in
  an anti-clockwise direction

• The two tangents divide each hull into two
  pieces. The edges belonging to one of these
  pieces must be deleted

Algorithms to computing the convex hull
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Divide and conquer algorithm

• 3D-case

• To merge in 3D need to construct a cylinder of
  triangles connecting the left hull and the right
  hull

• One edge of the cylinder AB is just the lower
  common tangent which can be computed just as in
  the 2D case

Algorithms to computing the convex hull
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Divide and conquer algorithm

• Next need to find a triangle ABC belonging to the
  cylinder which has AB as one of its edges. The
  third vertex of the triangle (C) must belong to
  either the left hull or the right hull. (In this
  case it belongs to the right hull)

• After finding triangle ABC we now have a new edge
  AC that joins the left hull and the right hull.
  We can find triangle ACD just as we did ABC.
  Continuing in this way, we can find all the
  triangles belonging to the cylinder, ending when
  we get back to AB.

Algorithms to computing the convex hull
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Algorithms to generate the DT

• Two-steps algorithm
• Computation of an arbitrary triangulation
• Optimization of triangulation to produce a
  Delaunay triangulation

O(n2)

• Incremental (Watsons) algorithm
• Modification of an existing Delaunay
  triangulation while adding a new vertex at a time

O(n2)

• Sloans algorithm
• Computation of Delaunay triangulation of
  arbitrary domain

Algorithms for genereting the (DT)
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Two-steps algorithm
Used to produce the coarse triangulation in
refinement simplification process

• Computation of an any arbitrary triangulation of
  domain
• Optimaze this triangulation to produce the DT

• Optimization step
• for every internal edge E do
• If E is not locally optimal, then swap it with
  the other diagonal
• of the quadrilateral formed by the two
  triangles sharing edge E
• Repeat the previose loop until one pass through
  the whole loop
• does not cause any edge swap

Algorithms for generating the DT
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Optimization step

• An edge pq is locally optimal iff
• Point s in the plane is outside circle prq
• Point l(s) in 3D space is above the oriented
  plane defined by the triple l(p), l(r) , l(q)

Algorithms for generating the DT
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Optimization step

• An edge pq is not locally optimal (it must be
  swapped) iff
• Point s in the plane is inside circle prq
• Point l(s) in 3D space is below the oriented
  plane defined by the triple l(p), l(r), l(q)

Algorithms for generating the DT
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Incremental (Watsons) algorithm

• Detection of the influence region
• Deletion of the triangles of this region
• Re-triangulate the region by join the point to
  the verticies of the influence polygon

Algorithms for generating the DT
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Incremental (Watsons) algorithm
After 100 insertion
Initial triangulation
After 200 insertion
After 314 insertion
Algorithms for generating the DT
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Sloans algorithm

• Nodalization of the domain
• Form a supertriangle by tree points, which is
  completely encompasses all points to be
  triangulated
• Create a triangle list array T, where the
  supertriangle is listed as the first triangle

• Introduce the first point into the
  supertriangle and generate three triangles by
  connecting the three vertices of the
  supertriangle
• Delete the supertiangle from the list and put
  there three newly formed triangles

Algorithms for generating the DT
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Sloans algorithm

• Introduce the next point for triangulation
• Optimize the grid

• Repeat the previous two steps until all N
  points are consumed
• Finally, all triangles, which contain one or
  more of the vertices of supertriangle, are
  removed

Algorithms for generating the DT
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Advantages of DT

• Wide range of utilization
• Irregular connectivity
• Number of neighboring nodes variable
• Equally easy (or difficult!) to generate for
  all domains
• Easy to vary point density
• Automatic mesh generation
• Elegant theoretical foundations
• Inherent grid quality

(Dis)advantages of DT

• Maximizes minimum angle in 2D, not in 3D
• Does not preserve original surface mesh
• Can produce sliver elements at the boundary

Conclusion
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Thank you for your attention !
Thank you for your attention !