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  [21, 22]. Let A be a finite point set in R^2 and ω : A → R be a height function which lifts the vertices of A into R^3. Every flip in triangulations of A can be assigned a direction [6, Definition 6.1.1]. A sequence of directed flips is monotone if all its flips follow the same direction. We first established a relatively obvious relation between monotone sequences of directed flips on triangulations of A and triangulations of the lifted point set A^ω in R^3. 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 of this result in 3d has been shown in [19]. As an application, we described a simple algorithm to triangulate a special class of 3d non-convex polyhedra without using additional vertices. We prove sufficient conditions for the termination of this algorithm, and show it runs in O(n^3) time, where $n$ is the number of input vertices

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

Adaptive tetrahedral mesh generation by constrained delaunay refinement

This paper discusses the problem of refining a constrained Delaunay tetrahedralization (CDT) for adaptive numerical simulation. A simple and efficient algorithm which makes use of the classical Delaunay refinement scheme is proposed. It generates an isotropic tetrahedral mesh corresponding to a sizing function which can be either user-specified or automatically derived from the input CDT. The quality of the produced meshes is guaranteed, i.e., most output tetrahedra have their circumradius-to-shortest-edge ratios bounded except those in the neighborhood of small input angles. Good mesh conformity can be obtained for smoothly changing sizing information. The algorithm has been implemented. Various examples are provided to illustrate its theoretical aspects as well as practical performance

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

TetGen, towards a quality tetrahedral mesh generator

TetGen is a C++ program for generating quality tetrahedral meshes aimed to support numerical methods and scientific computing. It is also a research project for studying the underlying mathematical problems and evaluating algorithms. This paper presents the essential meshing components developed in TetGen for robust and efficient software implementation. And it highlights the state-of-the-art algorithms and technologies currently implemented and developed in TetGen for automatic quality tetrahedral mesh generation

Adaptive tetrahedral mesh generation by constrained Delaunay refinement

This paper discusses the problem of refining a constrained Delaunay tetrahedralization (CDT) for adaptive numerical simulation. A simple and efficient algorithm which makes use of the classical Delaunay refinement scheme is proposed. It generates an isotropic tetrahedral mesh corresponding to a sizing function which can be either user-specified or automatically derived from the input CDT. The quality of the produced meshes is guaranteed, i.e., most output tetrahedra have their circumradius-to-shortest-edge ratios bounded except those in the neighborhood of small input angles. Good mesh conformity can be obtained for smoothly changing sizing information. The algorithm has been implemented. Various examples are provided to illustrate its theoretical aspects as well as practical performance

Tetrahedral mesh improvement using moving mesh smoothing, lazy searching flips, and RBF surface reconstruction

Given a tetrahedral mesh and objective functionals measuring the mesh quality which take into account the shape, size, and orientation of the mesh elements, our aim is to improve the mesh quality as much as possible. In this paper, we combine the moving mesh smoothing, based on the integration of an ordinary differential equation coming from a given functional, with the lazy flip technique, a reversible edge removal algorithm to modify the mesh connectivity. Moreover, we utilize radial basis function (RBF) surface reconstruction to improve tetrahedral meshes with curved boundary surfaces. Numerical tests show that the combination of these techniques into a mesh improvement framework achieves results which are comparable and even better than the previously reported ones.Comment: Revised and improved versio

Brain regions and cell type specific Wnt signalling changes in Parkinson’s disease mouse models

Parkinson’s disease (PD) is a late onset neurodegenerative disease characterised by the loss of dopaminergic neurons with motor and cognitive symptoms. Different mutations have been identified as a risk factor or direct cause of the disease. LRRK2 gene mutation is a major cause of sporadic and inherited Parkinson’s disease (PD), but the exact mechanism of how LRRK2 mutation causes PD remains to be revealed. LRRK2 is a huge complex protein with both GTPase and Kinase domains. G2019S is the most common LRRK2 mutation on the kinase domain. There is accumulate evidence showing LRRK2 as a scaffolding protein interacts with canonical and noncanonical Wnt signalling pathways. These pathways play an important role on immune responses, nerves system development as well as neuronal maintenance./ This project aims to study how LRRK2 influence Wnt signalling pathways activities, we used LRRK2 wild type (WT), LRRK2 knock-out (KO) and G2019S knock-in (KI) mouse models in the project. We identified the brain regions with Wnt and NFAT signalling activities by applying biosensor system via lentiviral construct transduction into the brain at P0 and investigated the signalling activation by immunohistochemistry at 6 months old. We discovered LRRK2 KO and G2019S KI alter Wnt signalling activity in several brain regions including the PD important striatum. mRNA and protein expression level analysis in selected brain regions showed a region specific dysregulation of Wnt signalling cascade components, the dysregulation was differed between male and female mice./ We discovered Wnt and NFAT signalling activity might be higher in glial cells than neurons in primary culture experiment, which lead us to put our focus on astrocytes. LRRK2 KO and G2019S mutation caused changes in Wnt and NFAT signalling activities in astrocytes under basal and stimulated conditions. These differences were reflected in mRNA expression levels of signalling mediators./ Taken together, these data suggest astrocytes might hold a key insight towards a better understanding of the correlation between Wnt signalling dysregulation and PD progression

3D boundary recovery by constrained Delaunay tetrahedralization

Three-dimensional boundary recovery is a fundamental problem in mesh generation. In this paper, we propose a practical algorithm for solving this problem. Our algorithm is based on the construction of a {\it constrained Delaunay tetrahedralization} (CDT) for a set of constraints (segments and facets). The algorithm adds additional points (so-called Steiner points) on segments only. The Steiner points are chosen in such a way that the resulting subsegments are Delaunay and their lengths are not unnecessarily short. It is theoretically guaranteed that the facets can be recovered without using Steiner points. The complexity of this algorithm is analyzed. The proposed algorithm has been implemented. Its performance is reported through various application examples