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Unformatted text preview: From the SIGGRAPH 2007 conference proceedings Isosurface Stuffing: Fast Tetrahedral Meshes with Good Dihedral Angles Franc ¸ois Labelle Jonathan Richard Shewchuk University of California at Berkeley 160 180 140 20 40 60 80 100 120 158.2 15.2 Figure 1: A 134,400-tetrahedron mesh produced by isosurface stuffing, with cutaway views. At the lower right is a histogram of tetrahedron dihedral angles in 2 ◦ intervals; multiply the heights of the red bars by 20. (Angles of 45 ◦ , 60 ◦ , and 90 ◦ occur with high frequency.) The extreme dihedral angles are 15 . 2 ◦ and 158 . 2 ◦ . This mesh took 55 seconds to generate on a Mac Pro with a 2.66 GHz Intel Xeon processor, but the mesh generation time was only 642 milliseconds; nearly all the time was spent in the isosurface evaluation code. Abstract The isosurface stuffing algorithm fills an isosurface with a uni-formly sized tetrahedral mesh whose dihedral angles are bounded between 10 . 7 ◦ and 164 . 8 ◦ , or (with a change in parameters) be-tween 8 . 9 ◦ and 158 . 8 ◦ . The algorithm is whip fast, numerically ro-bust, and easy to implement because, like Marching Cubes, it gener-ates tetrahedra from a small set of precomputed stencils. A variant of the algorithm creates a mesh with internal grading: on the bound-ary, where high resolution is generally desired, the elements are fine and uniformly sized, and in the interior they may be coarser and vary in size. This combination of features makes isosurface stuff-ing a powerful tool for dynamic fluid simulation, large-deformation mechanics, and applications that require interactive remeshing or use objects defined by smooth implicit surfaces. It is the first al-gorithm that rigorously guarantees the suitability of tetrahedra for finite element methods in domains whose shapes are substantially more challenging than boxes. Our angle bounds are guaranteed by a computer-assisted proof. If the isosurface is a smooth 2-manifold with bounded curvature, and the tetrahedra are sufficiently small, then the boundary of the mesh is guaranteed to be a geometrically and topologically accurate approximation of the isosurface. CR Categories: I.3.5 [Computer Graphics]: Computational Ge-ometry and Object Modeling—Geometric algorithms Keywords: isosurface, tetrahedral mesh generation, dihedral angle 1 Introduction Finite element methods are the most popular techniques for numer-ical simulation of the partial differential equations governing phys-ical phenomena such as fluid flow, mechanical deformation, and diffusion. Most objects worth simulating have complicated shapes. To make them amenable to analysis, modelers decompose them into simple shapes called elements , which commonly are tetrahedra....
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This note was uploaded on 05/20/2011 for the course CAP 6701 taught by Professor Staff during the Spring '08 term at University of Florida.
- Spring '08