On the existence of strongly regular families of triangulations for domains with a piecewise smooth boundary

In this note we present an algorithm for a construction of strongly regular families of triangulations for planar domains with a piecewise curved boundary. Some additional properties of the resulting triangulations are considered.     

Analysis of an algorithm for generating locally optimal meshes for approximation by discontinuous piecewise polynomials

This paper discusses the problem of constructing a locally optimal mesh for the best approximation of a given function by discontinuous piecewise polynomials. In the one-dimensional case, it is shown that, under certain assumptions on the approximated function, Baines' algorithm [M.J. Baines, Math. Comp., 62 (1994), pp. 645-669] for piecewise linear or piecewise constant polynomials produces a mesh sequence which converges to an optimal mesh. The rate of convergence is investigated. A two-dimensional modification of this algorithm is proposed in which both the nodes and the connection between the nodes are self-adjusting. Numerical results in one and two dimensions are presented.

     

Distributional properties of the three-dimensional Poisson Delaunay cell

This paper gives distributional properties of geometrical characteristics of the Delaunay tessellation generated by a stationary Poisson point process in ³. The considerations are based on a well-known formula given by Miles which describes the size and shape of the typical three-dimensional Poisson Delaunay cell. The results are the probability density functions for its volume, the area, and the perimeter of one of its faces, the angle spanned in a face by two of its edges, and the length of an edge. These probability density functions are given in integral form. Formulas for higher moments of these characteristics are given explicitly.     

用于高速可压缩流体分析的带多维耗散格式的自适应Delaunay三角剖分

利用自适应Delaunay三角剖分并结合胞格中心迎风算法,分析非粘滞高速可压缩流体问题.推导了多维耗散格式,并采用非结构化三角网格的迎风算法,改善了激波的计算结果.解精度评价中引入误差估计,在网格重划分算法中,解梯度变化大的区域生成小单元格,解梯度变化小的区域使用大单元格.该格式能进一步推广到高阶时空的解精度分析中.通过稳态和不稳态的高速可压缩流体超音速激波和激波传播特性的分析,可以评估该算法的效率.     

A Monotonicity Property for Weighted Delaunay Triangulations

Given a triangulation of points in the plane and a function on the points, one may consider the Dirichlet energy, which is related to the Dirichlet energy of a smooth function. In fact, the Dirichlet energy can be derived from a finite element approximation. S. Rippa showed that the Dirichlet energy (which he refers to as the “roughness”) is minimized by the Delaunay triangulation by showing that each edge flip which makes an edge Delaunay decreases the energy. In this paper, we introduce a Dirichlet energy on a weighted triangulation which is a generalization of the energy on unweighted triangulations and an analogue of the smooth Dirichlet energy on a domain. We show that this Dirichlet energy has the property that each edge flip which makes an edge weighted Delaunay decreases the energy. The proof is done by a direct calculation, and so gives an alternate proof of Rippa’s result.     

A note on tropical triangles in the plane

We define transversal tropical triangles （affine and projective） and characterize them via six inequalities to be satisfied by the coordinates of the vertices. We prove that the vertices of a transversal tropical triangle are tropically independent and they tropically span a classical hexagon whose sides have slopes ∞, 0, 1. Using this classical hexagon, we determine a parameter space for transversal tropical triangles. The coordinates of the vertices of a transversal tropical triangle determine a tropically regular matrix. Triangulations of the tropical plane are obtained.     

A calculus for ideal triangulations of three‐manifolds with embedded arcs

Refining the notion of an ideal triangulation of a compact three‐manifold, we provide in this paper a combinatorial presentation of the set of pairs (M,α), where M is a three‐manifold and α is a collection of properly embedded arcs. We also show that certain well‐understood combinatorial moves are sufficient to relate to each other any two refined triangulations representing the same (M,α). Our proof does not assume the Matveev–Piergallini calculus for ideal triangulations, and actually easily implies this calculus. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

一种光滑粒子流体动力学数据的后处理方法

提出一种光滑流体动力学数据后处理的三角网格化方法.首先确定流场中每个粒子作用域内的配对粒子集;再以当前粒子的配对粒子集为限定条件进行Delaunay三角剖分,可得到由粒子作为节点的三角形单元集;最后对"虚假"单元做过滤处理.算例表明复杂边界条件和严重变形的流体区域上的SPH计算结果都可用该方法处理.     

基于Kinect传感器多深度图像融合的物体三维重建

物体的三维重建技术一直是计算机视觉领域研究的热点问题,提出一种利用Kinect传感器获取的深度图像实现多幅深度图像融合完成物体三维重建的方法。在图像空间中对深度图像进行三角化,然后在尺度空间中融合所有三角化的深度图像构建分层有向距离场（hierarchical signed distance field）,对距离场中所有的体素应用整体Delaunay三角剖分算法产生一个涵盖所有体素的凸包,并利用Marching Tetrahedra算法构造等值面,完成物体表面重建。实验结果表明,该方法利用Kinect传感器采集的不同方向37幅分辨率为640480的深度图像完成目标物体的三维重建,仅需要48 s,并且得到非常精细的重建效果。     

On the Approximation of the Derivatives of Spline Quasi-Interpolation in Cubic Spline $S_3^{1,2}(∆_{mn}^{(2)})$

In this paper, based on the basis composed of two sets of splines with distinct local supports, cubic spline quasi-interpolating operators are reviewed on nonuniform type-2 triangulation. The variation diminishing operator is defined by discrete linear functionals based on a fixed number of triangular mesh-points, which can reproduce any polynomial of nearly best degrees. And by means of the modulus of continuity, the estimation of the operator approximating a real sufficiently smooth function is reviewed as well. Moreover, the derivatives of the nearly optimal variation diminishing operator can approximate that of the real sufficiently smooth function uniformly over quasi-uniform type-2 triangulation. And then the convergence results are worked out.