CONSTRUCTING CONSTRAINED DELAUNAY TETRAHEDRALIZATIONS OF VOLUMES BOUNDED BY PIECEWISE SMOOTH SURFACES

2011 ◽  
Vol 21 (05) ◽  
pp. 571-594 ◽  
Author(s):  
SERGE GOSSELIN ◽  
CARL OLLIVIER-GOOCH
Keyword(s):  
Poor Quality ◽  
Sampling Procedure ◽  
Piecewise Smooth ◽  
Surface Patch ◽  
Smooth Surfaces ◽  
Delaunay Refinement ◽  
Protection Scheme ◽  
Delaunay Tetrahedralization ◽  
Boundary Curves ◽  
Refinement Algorithm

This article presents an algorithm to construct constrained Delaunay tetrahedralizations of geometric domains bounded by piecewise smooth surfaces. Meshes are built from the bottom-up by first discretizing the boundary curves and then by sampling the smooth surfaces. The sampling procedure refines the Delaunay triangulation restricted to these surfaces, targeting topological violations and poor quality triangles. Unlike previously published algorithms adopting a similar approach, we propose to sample each smooth surface patch independently. This obviates the need for a boundary protection scheme around small dihedral angles in the input and can also lead to coarser constraining triangulations. Starting from a Delaunay tetrahedralization of the point samples, a combination of mesh reconfigurations and vertex insertions is then used to obtain a tetrahedralization constrained to the boundary surfaces. The algorithm is designed to produce tetrahedralizations that can be used in conjunction with a Delaunay refinement algorithm implemented on a Bowyer-Watson framework.

scholarly journals PARALLEL DELAUNAY REFINEMENT: ALGORITHMS AND ANALYSES

2007 ◽  
Vol 17 (01) ◽  
pp. 1-30 ◽  
Author(s):  
DANIEL A. SPIELMAN ◽  
SHANG-HUA TENG ◽  
ALPER ÜNGÖR
Keyword(s):  
Mesh Size ◽  
Constant Factor ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Delaunay Refinement ◽  
Steiner Points ◽  
Input Domain ◽  
Set Of Points ◽  
Refinement Algorithm

We present a parallel Delaunay refinement algorithm for generating well-shaped meshes in both two and three dimensions. Like its sequential counterparts, the parallel algorithm iteratively improves the quality of a mesh by inserting new points, the Steiner points, into the input domain while maintaining the Delaunay triangulation. The Steiner points are carefully chosen from a set of candidates that includes the circumcenters of poorly-shaped triangular elements. We introduce a notion of independence among possible Steiner points that can be inserted simultaneously during Delaunay refinements and show that such a set of independent points can be constructed efficiently and that the number of parallel iterations is O( log 2Δ), where Δ is the spread of the input — the ratio of the longest to the shortest pairwise distances among input features. In addition, we show that the parallel insertion of these set of points can be realized by sequential Delaunay refinement algorithms such as by Ruppert's algorithm in two dimensions and Shewchuk's algorithm in three dimensions. Therefore, our parallel Delaunay refinement algorithm provides the same shape quality and mesh-size guarantees as these sequential algorithms. For generating quasi-uniform meshes, such as those produced by Chew's algorithms, the number of parallel iterations is in fact O( log Δ). To the best of our knowledge, our algorithm is the first provably polylog(Δ) time parallel Delaunay-refinement algorithm that generates well-shaped meshes of size within a constant factor of the best possible.

A 2-D Delaunay Refinement Algorithm Using an Initial Prerefinement From the Boundary Mesh

2008 ◽  
Vol 44 (6) ◽  
pp. 1418-1421 ◽  
Author(s):  
M.M. Sakamoto ◽  
J.R. Cardoso ◽  
J.M. Machado ◽  
M. Salles
Keyword(s):  
Delaunay Refinement ◽  
Refinement Algorithm

The Somigliana identity on piecewise smooth surfaces

1981 ◽  
Vol 11 (4) ◽  
pp. 403-423 ◽  
Author(s):  
Friedel Hartmann
Keyword(s):  
Piecewise Smooth ◽  
Smooth Surfaces

scholarly journals The Development of Log Aesthetic Patch and Its Projection onto the Plane

Mathematics ◽  
2022 ◽  
Vol 10 (1) ◽  
pp. 160
Author(s):  
Yee Meng Teh ◽  
R. U. Gobithaasan ◽  
Kenjiro T. Miura ◽  
Diya’ J. Albayari ◽  
Wen Eng Ong
Keyword(s):  
Outer Part ◽  
Geodesic Curvature ◽  
Surface Patch ◽  
Hyperbolic Paraboloid ◽  
Shipbuilding Industry ◽  
First Derivative ◽  
Cold Bending ◽  
Lines Of Curvature ◽  
New Type ◽  
Boundary Curves

In this work, we introduce a new type of surface called the Log Aesthetic Patch (LAP). This surface is an extension of the Coons surface patch, in which the four boundary curves are either planar or spatial Log Aesthetic Curves (LACs). To identify its versatility, we approximated the hyperbolic paraboloid to LAP using the information of lines of curvature (LoC). The outer part of the LoCs, which play a role as the boundary of the hyperbolic paraboloid, is replaced with LACs before constructing the LAP. Since LoCs are essential in shipbuilding for hot and cold bending processes, we investigated the LAP in terms of the LoC’s curvature, derivative of curvature, torsion, and Logarithmic Curvature Graph (LCG). The numerical results indicate that the LoCs for both surfaces possess monotonic curvatures. An advantage of LAP approximation over its original hyperbolic paraboloid is that the LoCs of LAP can be approximated to LACs, and hence the first derivative of curvatures for LoCs are monotonic, whereas they are non-monotonic for the hyperbolic paraboloid. This confirms that the LAP produced is indeed of high quality. Lastly, we project the LAP onto a plane using geodesic curvature to create strips that can be pasted together, mimicking hot and cold bending processes in the shipbuilding industry.

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