53 research outputs found
Studies of several tetrahedralization problems
The main purpose of decomposing an object into simpler components is to simplify a
problem involving the complex object into a number of subproblems having simpler
components. In particular, a tetrahedralization is a partition of the input domain in
R3 into a number of tetrahedra that meet only at shared faces. Tetrahedralizations
have applications in the finite element method, mesh generation, computer graphics,
and robotics.
This thesis investigates four problems in tetrahedralizations and triangulations.
The first problem is on the computational complexity of tetrahedralization detections.
We present an O(nm log n) algorithm to determine whether a set of line segments .C
is the edge set of a tetrahedralization, where m is the number of segments and n is
the number of endpoints in .C. We show that it is NP-complete to decide whether .C
contains the edge set of a tetrahedralization. We also show that it is NP-complete to
decide whether .C is tetrahedralizable. The second problem is on minimal tetrahedralizations.
After deriving some properties of the graph of polyhedra, we identify a class of polyhedra and show that this class of polyhedra can be minimally tetrahedralized
in O(n²) time. The third problem is on the tetrahedralization of two nested convex
polyhedra. We give a method to tetrahedralize the region between two nested convex
polyhedra into a linear number of tetrahedra without introducing Steiner points.
This result answers an open problem raised by Bern [16]. The fourth problem is on
the lower bound for β-skeletons belonging to minimum weight triangulations. We
prove a lower bound on β (β = [one sixth times the square root of two times the square root of 3] + 45 such that if β is less than this value,
the β-skeleton of a point set may not always be a subgraph of the minimum weight
triangulation of this point set. This result settles Keil's conjecture [62]
The existence of triangulations of non-convex polyhedra without new vertices
It is well known that a simple three-dimensional non-convex polyhedron may not be triangulated without using new vertices (so-called {\it Steiner points}). In this paper, we prove a condition that guarantees the existence of a triangulation of a non-convex polyhedron (of any dimension) without Steiner points. We briefly discuss algorithms for efficiently triangulating three-dimensional polyhedra
TetGen: A quality tetrahedral mesh generator and a 3D Delaunay triangulator (Version 1.5 — User’s Manual)
TetGen is a software for tetrahedral mesh generation. Its goal is to generate good quality tetrahedral meshes suitable for numerical methods and scientific computing. It can be used as either a standalone program or a library component integrated in other software. The purpose of this document is to give a brief explanation of the kind of tetrahedralizations and meshing problems handled by TetGen and to give a fairly detailed documentation about the usage of the program. Readers will learn how to create tetrahedral meshes using input files from the command line. Furthermore, the programming interface for calling TetGen from other programs is explained
On Monotone Sequences of Directed Flips, Triangulations of Polyhedra, and Structural Properties of a Directed Flip Graph
This paper studied the geometric and combinatorial aspects of the classical
Lawson's flip algorithm in 1972. Let A be a finite set of points in R2, omega
be a height function which lifts the vertices of A into R3. Every flip in
triangulations of A can be associated with a direction. We first established a
relatively obvious relation between monotone sequences of directed flips
between triangulations of A and triangulations of the lifted point set of A in
R3. We then studied the structural properties of a directed flip graph (a
poset) on the set of all triangulations of A. We proved several general
properties of this poset which clearly explain when Lawson's algorithm works
and why it may fail in general. We further characterised the triangulations
which cause failure of Lawson's algorithm, and showed that they must contain
redundant interior vertices which are not removable by directed flips. A
special case if this result in 3d has been shown by B.Joe in 1989. As an
application, we described a simple algorithm to triangulate a special class of
3d non-convex polyhedra. We proved sufficient conditions for the termination of
this algorithm and show that it runs in O(n3) time.Comment: 40 pages, 35 figure
Lattice cleaving: a multimaterial tetrahedral meshing algorithm with guarantees
pre-printWe introduce a new algorithm for generating tetrahedral meshes that conform to physical boundaries in volumetric domains consisting of multiple materials. The proposed method allows for an arbitrary number of materials, produces high-quality tetrahedral meshes with upper and lower bounds on dihedral angles, and guarantees geometric fidelity. Moreover, the method is combinatoric so its implementation enables rapid mesh construction. These meshes are structured in a way that also allows grading, to reduce element counts in regions of homogeneity. Additionally, we provide proofs showing that both element quality and geometric fidelity are bounded using this approach
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