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Mesh G eneration and Delaunay-Based Meshes

Mesh G eneration and Delaunay-Based Meshes. Jernej Barbic Computer Science Department Carnegie Mellon University. Outline Introduction Delaunay triangulation Ruppert’s algorithm Result on runtime analysis of Ruppert’s algorithm Conclusion.

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Mesh G eneration and Delaunay-Based Meshes

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  1. Mesh Generation and Delaunay-Based Meshes Jernej Barbic Computer Science Department Carnegie Mellon University

  2. Outline • Introduction • Delaunay triangulation • Ruppert’s algorithm • Result on runtime analysis of Ruppert’s algorithm • Conclusion

  3. Motivation: Temperature on the Surface of a Lake No analytical solution Need to approximate…

  4. How to approximate?

  5. How to approximate? First guess: uniformly. Runs too slow!

  6. No need to consider only uniform meshes… Too much computation Computation fast, but inaccurate Optimal solution BUT MUST AVOID SMALL ANGLES !!

  7. Outline • Introduction • Delaunay triangulation • Ruppert’s algorithm • Result on runtime analysis of Ruppert’s algorithm • Conclusion

  8. How to connect the dots into triangles?

  9. How to connect the dots into triangles? One possibility:

  10. How to connect the dots into triangles? Another possibility: Which triangulation makes minimum angle as large as possible?

  11. Boris Nikolaevich Delaunay (1890-1980) : Include a triangle iff there are no vertices inside its circumcircle.

  12. Delaunay Triangulation Further example:

  13. Delaunay Triangulation Negative example:

  14. Let’s try it for Lake Superior… Delaunay triangulationof Lake Superior … and we get Lake Inferior …

  15. Outline • Introduction • Delaunay triangulation • Ruppert’s algorithm • Result on runtime analysis of Ruppert’s algorithm • Conclusion

  16. How to avoid skinny triangles? Ruppert’s idea (1993) : Add triangle’s circumcenter into the mesh. Why does this make sense?

  17. How to avoid skinny triangles? Ruppert’s idea (1993) : Add triangle’s circumcenter into the mesh. Why does this make sense?

  18. Ruppert’s Algorithm Input: a set of points in the plane, minimum angle 0 Output: a triangulation, with all angles 0

  19. Ruppert’s Algorithm Input: a set of points in the plane, minimum angle 0 Output: a triangulation, with all angles 0 • Algorithm always terminates for 0< 20.7º. • Bigger minimum angles 0are harder. • Size of mesh is optimal up to a constant factor.

  20. Let’s demonstrate Ruppert on an example…

  21. Progress of Ruppert’s algorithm minimum allowed angle = 13 º Algorithm completed its task.

  22. Outline • Introduction • Delaunay triangulation • Ruppert’s algorithm • Result on runtime analysis of Ruppert’s algorithm • Conclusion

  23. Running Time Analysis Delaunay triangulation O(n log n)

  24. Running Time Analysis Delaunay triangulation O(n log n) Ruppert’s algorithm My result: (m2)

  25. We proved: worst-case bound of O(m2) is tight. Input point set (no edges in the input): Number of input points: n Number of output points: m=O(n)

  26. Delaunay triangulation of the input: Lots of skinny triangles. We choose to split theone shaded in grey.

  27. Big “fan” of triangles appears: high cost

  28. We have established a self-repeating pattern …

  29. Lots of work continues…

  30. Total work proven to be: (m2) Number of input points: O(n) Number of output points: m=O(n)

  31. However, in practice algorithm works well. Let’s try Ruppert on Lake Superior…

  32. Ruppert’s Algorithm on Lake Superior 0 = 25º

  33. Solve for temperature…

  34. Outline • Introduction • Delaunay triangulation • Ruppert’s algorithm • Result on runtime analysis of Ruppert’s algorithm • Conclusion

  35. Conclusion • Worst-case behavior of Ruppert’s algorithm is quadratic. • In practice, Ruppert’s algorithm performs well. • Main Delaunay idea: maximize minimum angle • Generating good meshes is an important problem.

  36. Applications and Future Work • Triangle software http://www-2.cs.cmu.edu/~quake/triangle.html • Find good strategies for selecting skinny triangles • Characterize input meshes that exhibit slow runtime

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