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Delaunay triangulation

About: Delaunay triangulation is a research topic. Over the lifetime, 5816 publications have been published within this topic receiving 126615 citations. The topic is also known as: Delone triangulation.


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TL;DR: In this paper, several functionals on the set of all triangulations of the finite system of points in d-space achieving global minimum on the Delaunay triangulation were defined.
Abstract: We defined several functionals on the set of all triangulations of the finite system of points in d-space achieving global minimum on the Delaunay triangulation (DT). We consider a so called "parabolic" functional and prove it attains its minimum on DT in all dimensions. As the second example we treat "mean radius" functional (mean of circumcircle radii of triangles) for planar triangulations. As the third example we treat a so called "harmonic" functional. For a triangle this functional equals the sum of squares of sides over area. Finally, we consider a discrete analog of the Dirichlet functional. DT is optimal for these functionals only in dimension two.

44 citations

Journal ArticleDOI
TL;DR: In this paper, a numerical approach for the topological optimization of 2D linear elastic problems using boundary elements is presented, where models are discretized using linear elements and a periodic distribution of internal points over the domain.
Abstract: Topological optimization provides a powerful framework to obtain the optimal domain topology for several engineering problems. The topological derivative is a function which characterizes the sensitivity of a given problem to the change of its topology, like opening a small hole in a continuum or changing the connectivity of rods in a truss. A numerical approach for the topological optimization of 2D linear elastic problems using boundary elements is presented in this work. The topological derivative is computed from strain and stress results which are solved by means of a standard boundary element analysis. Models are discretized using linear elements and a periodic distribution of internal points over the domain. The total potential energy is selected as cost function. The evaluation of the topological derivative is performed as a post-processing procedure. Afterwards, material is removed from the model by deleting the internal points and boundary nodes with the lowest values of the topological derivate. The new geometry is then remeshed using a weighted Delaunay triangularization algorithm capable of detecting “holes” at those positions where internal points and boundary points have been removed. The procedure is repeated until a given stopping criterion is satisfied. The proposed strategy proved to be flexible and robust. A number of examples are solved and results are compared to those available in the literature.

44 citations

Journal ArticleDOI
TL;DR: The vertex set of the Capacity-Constrained Delaunay Triangulation (CCDT) is shown to have good blue noise characteristics, comparable in quality to those of state-of-the-art methods, achieved at a fraction of the runtime.

44 citations

Journal ArticleDOI
TL;DR: In this paper, a constrained boundary recovery method for three dimensional Delaunay triangulations is presented, which successfully resolves the difficulties related to the minimal addition of Steiner points and their good placement.
Abstract: SUMMARY A new constrained boundary recovery method for three dimensional Delaunay triangulations is presented. It successfully resolves the difficulties related to the minimal addition of Steiner points and their good placement. Applications to full mesh generation are discussed and numerical examples are provided to illustrate the effectiveness of guaranteed recovery procedure. Copyright 2004 John Wiley & Sons, Ltd.

44 citations

Posted Content
TL;DR: The numerical behavior of the method is studied, showing empirically that it converges under refinement, and the construction of intrinsic Delaunay triangulations are augmented so that they can be used in the context of tangent vector field processing.
Abstract: This paper describes a method for efficiently computing parallel transport of tangent vectors on curved surfaces, or more generally, any vector-valued data on a curved manifold. More precisely, it extends a vector field defined over any region to the rest of the domain via parallel transport along shortest geodesics. This basic operation enables fast, robust algorithms for extrapolating level set velocities, inverting the exponential map, computing geometric medians and Karcher/Frechet means of arbitrary distributions, constructing centroidal Voronoi diagrams, and finding consistently ordered landmarks. Rather than evaluate parallel transport by explicitly tracing geodesics, we show that it can be computed via a short-time heat flow involving the connection Laplacian. As a result, transport can be achieved by solving three prefactored linear systems, each akin to a standard Poisson problem. To implement the method we need only a discrete connection Laplacian, which we describe for a variety of geometric data structures (point clouds, polygon meshes, etc). We also study the numerical behavior of our method, showing empirically that it converges under refinement, and augment the construction of intrinsic Delaunay triangulations (iDT) so that they can be used in the context of tangent vector field processing.

44 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202393
2022203
2021130
2020185
2019204
2018223