Visualization of Implicit Surfaces Using Adaptive Tetrahedrizations

Müller, Heinrich; Wehle, Marko

doi:10.1109/dagstuhl.1997.1423119

Cited by 12 publications

(6 citation statements)

References 13 publications

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“…Remark 1. The following fact is taken from the marching tetrahedra algorithm [43] (which is a variant of the marching cubes algorithm [38]).…”

Section: Intersections With γmentioning

confidence: 99%

“…The two classic examples of marching tetrahedra [43]. The signs of the level set function are given at each vertex.…”

Section: Intersections With γmentioning

confidence: 99%

“…Other methods take an optimization viewpoint [53,24,32,41,40,23], while others [12,13,2] use a variational form to minimize the interpolation error by local remeshing. Some methods use specific tilings of 3-D space [62,22] or marching cubes [38] or marching tetrahedra [43]. Another option for mesh generation uses implicit (level set) functions to create conforming meshes [46,47,45,10,50,51,63,35,36] as well as adaptive methods to create meshes adapted to the local feature size [7,28,33].…”

Section: Introductionmentioning

confidence: 99%

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Tetrahedralization of Isosurfaces with Guaranteed-Quality by Edge Rearrangement (TIGER)

Walker¹

2013

SIAM J. Sci. Comput.

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We present a method for generating three-dimensional (3-D) unstructured tetrahedral meshes of solids whose boundary is a smooth surface. The method uses a background grid (bodycentered-cubic (BCC) lattice) from which to build the final conforming 3-D mesh. The algorithm is fast and robust and provides useful guaranteed dihedral angle bounds for the output tetrahedra. The dihedral angles are bounded between 8.5 • and 164.2 •. If the lattice spacing is smaller than the "local feature size," then the dihedral angles are between 11.4 • and 157.6 • (cf. Labelle and Shewchuk [SIGGRAPH '07, ACM, New York, 2007]). The method is simple to implement and performs no extra refinement of the background grid. The most complicated mesh transformations are 4-4 edge flips. Moreover, the only parameter in the method is the BCC lattice spacing. If the surface has bounded curvature and if the background grid is sufficiently fine, then the boundary of the output mesh is guaranteed to be a geometrically and topologically accurate approximation of the solid surface. Applications of the method are in free boundary flows, modeling deformations, shape optimization, and anything that requires dynamic meshing, such as virtual surgery.

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“…Remark 1. The following fact is taken from the marching tetrahedra algorithm [43] (which is a variant of the marching cubes algorithm [38]).…”

Section: Intersections With γmentioning

confidence: 99%

“…The two classic examples of marching tetrahedra [43]. The signs of the level set function are given at each vertex.…”

Section: Intersections With γmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Tetrahedralization of Isosurfaces with Guaranteed-Quality by Edge Rearrangement (TIGER)

Walker¹

2013

SIAM J. Sci. Comput.

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“…The first step of creating an adaptive partition of space was taken in early works [Blo88,MS91]. With a few exceptions, such as [MW97,GK04], researchers use octrees to partition space. Some methods extend MC to octrees through crack-patching between multi-resolution cells [WKE99,VT01] or bending high-resolution vertices to adjacent, low-resolution contours [SFYC96].…”

Section: Figurementioning

confidence: 99%

Isosurfaces Over Simplicial Partitions of Multiresolution Grids

Manson

Schaefer

2010

Computer Graphics Forum

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“…The domain-approximating refinement for tetrahedron-based meshes is extensively discussed in other works. [27][28][29][30][31][32] The higher-order approximation of curved surfaces on the basis of such tetrahedron meshes is also pursued in other works. [33][34][35] The refinement of hexahedrons to approximate a given domain and its surface was depicted in the work of Engwer 32 and more thoroughly discussed in the work of Wang.…”

Section: Introductionmentioning

confidence: 99%

Marching volume polytopes algorithm

Byfut

Hellwig

Schröder

2018

Numerical Meth Engineering

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This paper presents an algorithm for the refinement of two-or threedimensional meshes with respect to an implicitly given domain, so that its surface is approximated by facets of the resulting polytopes. Using a Cartesian grid, the proposed algorithm may be used as a mesh generator. Initial meshes may consist of polytopes such as quadrilaterals and triangles, as well as hexahedrons, pyramids, and tetrahedrons. Given the ability to compute edge intersections with the surface of an implicitly given domain, the proposed marching volume polytopes algorithm uses predefined refinement patterns applied to individual polytopes depending on the intersection pattern of their edges. The refinement patterns take advantage of rotational symmetry. Since these patterns are applied independently to individual polytopes, the resulting mesh may encompass the so-called orientation problem, where two adjacent polytopes are rotated against one another. To allow for a repeated application of the marching volume polytopes algorithm, the proposed data structures and algorithms account for this ambiguity. A simple example illustrates the advantage of the repeated application of the proposed algorithm to approximate domains with sharp corners. Furthermore, finite element simulations for two challenging real-world problems, which require highly accurate approximations of the considered domains, demonstrate its applicability. For these simulations, a variant of the fictitious domain method is used. KEYWORDSfictitious domain method, finite element method, marching cubes, marching tetrahedrons, mesh generation, volume mesh Delaunay, quadtree/octree, and advancing-front methods for mesh generation. Using quadtree or octree techniques, 2-6 Cartesian grids covering a given domain are repeatedly refined via isotropic bisections of quadrilaterals into four similar sub-quadrilaterals or of hexahedrons into eight similar sub-hexahedrons for

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Visualization of Implicit Surfaces Using Adaptive Tetrahedrizations

Cited by 12 publications

References 13 publications

Tetrahedralization of Isosurfaces with Guaranteed-Quality by Edge Rearrangement (TIGER)

Tetrahedralization of Isosurfaces with Guaranteed-Quality by Edge Rearrangement (TIGER)

Isosurfaces Over Simplicial Partitions of Multiresolution Grids

Marching volume polytopes algorithm

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